Solving Direct and Inverse Heat Conduction Problems

Solving Direct and Inverse Heat Conduction Problems

Jan Taler Piotr Duda Solving Direct and Inverse Heat Conduction Problems ~ Springer

Preface This book is devoted to the concept of simple and inverse heat conduction problems. The process of solving direct problems is based on the temperature determination when initial and boundary conditions are known, while the solving of inverse problems is based on the search for boundary conditions when temperature properties are known, provided that temperature is the function of time, at the selected inner points of a body. In the first part of the book (Chaps. 1-5), we have discussed theoretical basis for thermal conduction in solids, motionless liquids and liquids that move in time. In the second part of the book, (Chapters 6-26), we have discussed at great length different engineering problems, which we have presented together with the proposed solutions in the form of theoretical and mathematical examples. It was our intention to acquaint the reader in a step-by-step fashion with all the mathematical derivations and solutions to some of the more significant transient and steady-state heat conduction problems with respect to both, the movable and immovable heat sources and the phenomena of melting and freezing. Lots of attention was paid to non-linear problems. The methods for solving heat conduction problems, i.e. the exact and approximate analytical methods and numerical methods, such as the finite difference method, the finite volume method, the finite element method and the boundary element method are discussed in great detail. Aside from algorithms, applicable computational programs, written in a FORTRAN language, were given. The accuracy of the results obtained by means of various numerical methods was evaluated by way of comparison with accurate analytical solutions. The presented solutions not only allow to illustrate mathematical methods used in thermal conduction but also show the methods one can use to solve concrete practical problems, for example during the designing and life-time calculations of industrial machinery, combustion engines and in refrigerating and air conditioning engineering. Many examples refer to the topic of heating and thermo-renovation of apartment buildings. The methods for solving problems involved with welding and laser technology are also discussed in great detail. This book is addressed to undergraduate and PhD students of mechanical, power, process and environmental engineering. Due to the complexity of the heat conduction problems elaborated in this book, this edition can

vi Preface also serve as a reference book that can be used by nuclear, industrial and civil engineers. Jan Taler is the author of the theoretical part of this book, mathematical exercises (excluding 12.1 & 12.3), and C, D & H attachments (found at the back of this book). Piotr Duda wrote in the FORTRAN language all presented programs and solved with their help exercises 7.3, 11.2-11.7, 15.1, 15.2, 15.4, 15.5, 15.7, 15.8, 15.11, 15.13, 15.15, 16.5, 16.9, 16.10, 17.7, 18.5-18.8,21.5, 21.7-21.10, 22.7, 23.3-23.7, 24.4 and 24.5. He also carried out calculations using the following programs: ANSYS (in Exercises 11.18-11.22, 12.4, 21.9 and 25.10), BETIS (in Exercise 12.4) and MathCAD (in Exercises 14.10, 16.2,16.4,17.6 and 25.10). Furthermore, Piotr Duda is the author of Exercises 12.1 and 12.3, and attachments A, B, E, F and G. Krakow June, 2005. Jan Taler Piotr Duda

Contents Part I Heat Conduction Fundamentals 1 1 Fourier Law......3 Literature 6 2 Mass and Energy Balance Equations 7 2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity 7 2.2 Inner Energy Balance Equation u 9 2.2.1 Energy Balance Equations in Three Basic Coordinate Systems 12 2.3 Hyperbolic Heat Conduction Equation 16 2.4 Initial and Boundary Conditions 17 2.4.1 First Kind Boundary Conditions (Dirichlet Conditions) 18 2.4.2 Second Kind Boundary Conditions (von Neumann Conditions) 18 2.4.3 Third Kind Boundary Conditions 19 2.4.4 Fourth Kind Boundary Conditions 21 2.4.5 Non-Linear Boundary Conditions 22 2.4.6 Boundary Conditions on the Phase Boundaries 24 Literature 26 3 The Reduction of Transient Heat Conduction Equations and Boundary Conditions 29 3.1 Linearization of a Heat Conduction Equation 29 3.2 Spatial Averaging of Temperature 31 3.2.1 A Body Model with a Lumped Thermal Capacity 31 3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness 33 3.2.3 Heat Conduction Equation for a Round Fin with Uniform Thickness 35 3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity 37 Literature 39

viii Contents 4 Substituting Heat Conduction Equation by Two-Equations System 41 4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient 41 4.2 One-Dimensional Inverse Transient Heat Conduction Problem 43 Literature 46 5 Variable Change 47 Literature 50 Part II Exercises. Solving Heat Conduction Problems 51 6 Heat Transfer Fundamentals 53 Exercise 6.1 Fourier Law in a Cylindrical Coordinate System 53 Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation 55 Exercise 6.3 Heat Transfer Through a Flat Single-Layered and Double-Layered Wall 57 Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall 60 Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe 62 Exercise 6.6 Radiant Tube Temperature 65 Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall 68 Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity... 70 Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor 74 Exercise 6.10 Determining Heat Flux By Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity 77 Exercise 6.11 One-Dimensional Steady-State Plate Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources 80 Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources 82 Exercise 6.13 Inverse Steady-State Heat Conduction Problem in a Pipe 85 Exercise 6.14 General Equation of Heat Conduction in Fins 87 Exercise 6.15 Temperature Distribution and Efficiency of a Straight Fin with Constant Thickness 89

Contents ix Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device 92 Exercise 6.17 Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness 95 Exercise 6.18 Approximated Calculation of a Circular Fin Efficiency 98 Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins 99 Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method 102 Exercise 6.21 Calculating Rectangular Fin Efficiency 108 Exercise 6.22 Heat Transfer Coefficient in Exchangers with Extended Surfaces 109 Exercise 6.23 Calculating Overall Heat Transfer Coefficient in a Fin Plate Exchanger 114 Exercise 6.24 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe with a Scale Layer on an Inner Surface 115 Exercise 6.25 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe 119 Exercise 6.26 Determining One-Dimensional Temperature Distribution in a Flat Wall by Means of Finite Volume Method 122 Exercise 6.27 Determining One-Dimensional Temperature Distribution in a Cylindrical Wall by Means of Finite Volume Method 127 Exercise 6.28 Inverse Steady-State Heat Conduction Problem for a Pipe Solved by Space-Marching Method 131 Exercise 6.29 Temperature Distribution and Efficiency of a Circular Fin with Temperature-Dependent Thermal Conductivity 134 Literature 138 7 Two-Dimensional Steady-State Heat Conduction. Analytical Solutions 141 Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness 141 Exercise 7.2 Temperature Distribution in a Straight Fin with Constant Thickness and Insulated Tip 145 Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip 148

x Contents Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler 156 Literature 160 8 Analytical Approximation Methods. Integral Heat Balance Method 161 Exercise 8.1 Temperature Distribution within a Rectangular Cross- Section of a Bar 161 Exercise 8.2 Temperature Distribution in an Infinitely Long Fin of Constant Thickness 163 Exercise 8.3 Determining Temperature Distribution in a Boiler's Water-Wall Tube by Means of Functional Correction Method 165 Literature 169 9 Two-Dimensional Steady-State Heat Conduction. Graphical Method 171 Exercise 9.1 Temperature Gradient and Surface-Transmitted Heat Flow 171 Exercise 9.2 Orthogonality of Constant Temperature Line and Constant Heat Flux 173 Exercise 9.3 Determining Heat Flow between Isothermal Surfaces.. 176 Exercise 9.4 Determining Heat Loss Through a Chimney Wall; Combustion Channel (Chimney) with Square Cross-Section 179 Exercise 9.5 Determining Heat Loss Through Chimney Wall with a Circular Cross-Section 181 Literature 182 10 Two-Dimensional Steady-State Problems. The Shape Coefficient 183 Exercise 10.1 Buried Pipe-to-Ground Surface Heat Flow 183 Exercise 10.2 Floor Heating 185 Exercise 10.3 Temperature of a Radioactive Waste Container Buried Underground 186 Literature 187 11 Solving Steady-State Heat Conduction Problems by Means of Numerical Methods 189 Exercise 11.1 Description of the Control Volume Method 189 Exercise 11.2 Determining Temperature Distribution in a Square Cross-Section of a Long Rod by Means of the Finite Volume Method 194

Contents xi Exercise 11.3 A Two-Dimensional Inverse Steady-State Heat Conduction Problem 199 Exercise 11.4 Gauss-Seidel Method and Over-Relaxation Method 204 Exercise 11.5 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Uniform Thickness by Means of the Finite Volume Method 208 Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney 215 Exercise 11.7 Pseudo-Transient Determination of Steady-State Temperature Distribution in a Square Cross-Section of a Chimney; Heat Transfer by Convection and Radiation on an Outer Surface of a Chimney 221 Exercise 11.8 Finite Element Method 230 Exercise 11.9 Linear Functions That Interpolate Temperature Distribution (Shape Functions) Inside Triangular and Rectangular Elements 234 Exercise 11.10 Description of FEM Based on Galerkin Method 238 Exercise 11.11 Determining Conductivity Matrix for a Rectangular and Triangular Element 245 Exercise 11.12 Determining Matrix [:K/] in Terms of Convective Boundary Conditions for a Rectangular and Triangular Element 249 e Exercise 11.13 Determining Vector {f } Q with Respect to Volumetric and Point Heat Sources in a Rectangular and Triangular Element 253 e e Exercise 11.14 Determining Vectors {f } q and {f a } with Respect to Boundary Conditions of 2nd and 3rd Kind on the Boundary of a Rectangular or Triangular Element 256 Exercise 11.15 Methods for Building Global Equation System in FEM 259 Exercise 11.16 Determining Temperature Distribution in a Square Cross-Section of an Infinitely Long Rod by Means of FEM, in which the Global Equation System is Constructed using Method I (from Ex. 11.15) 264 Exercise 11.17 Determining Temperature Distribution in an Infinitely Long Rod with Square Cross-Section by Means of FEM, in which the Global Equation System is Constructed using MethodII (from Ex. 11.15) 271 Exercise 11.18 Determining Temperature Distribution by Means of FEM in an Infinitely Long Rod with Square Cross-Section, in which Volumetric Heat Sources Operate 275 Exercise 11.19 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Constant Thickness by Means of FEM 285

xii Contents Exercise 11.20 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Straight Fin with Constant Thickness (ANSYS Program) 297 Exercise 11.21 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Hexagonal Fin with Constant Thickness (ANSYS Program) 300 Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS program) 303 Literature 307 12 Finite Element Balance Method and Boundary Element Method 309 Exercise 12.1 Finite Element Balance Method 309 Exercise 12.2 Boundary Element Method 314 Exercise 12.3 Determining Temperature Distribution in Square Region by Means of FEM Balance Method 323 Exercise 12.4 Determining Temperature Distribution in a Square Region Using Boundary Element Method 327 Literature 331 13 Transient Heat Exchange Between a Body with Lumped Thermal Capacity and Its Surroundings 333 Exercise 13.1 Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings 333 Exercise 13.2 Heat Exchange between a Body with Lumped Thermal Capacity and Surroundings with Time-Dependent Temperature 336 Exercise 13.3 Determining Temperature Distribution of a Body with Lumped Thermal Capacity, when the Temperature of a Medium Changes Periodically 339 Exercise 13.4 Inverse Problem: Determining Temperature of a Medium on the Basis of Temporal Thermometer-Indicated Temperature History 340 Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple 342 Exercise 13.6 Determining the Time It Takes to Cool Body Down to a Given Temperature 344 Exercise 13.7 Temperature Measurement Error of a Medium whose Temperature Changes at Constant Rate 345 Exercise 13.8 Temperature Measurement Error of a Medium whose Temperature Changes Periodically 346

Contents xiii Exercise 13.9 Inverse Problem: Calculating Temperature of a Medium whose Temperature Changes Periodically, on the Basis of Temporal Temperature History Indicated by a Thermometer 347 Exercise 13.10 Measuring Heat Flux 349 Literature 351 14 Transient Heat Conduction in Half-Space 353 Exercise 14.1 Laplace Transform 353 Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature 355 Exercise 14.3 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Heat Flux 358 Exercise 14.4 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Temperature of a Medium 360 Exercise 14.5 Formula Derivation for Temperature Distribution in a Half-Space when Surface Temperature is Time-Dependent. 364 Exercise 14.6 Formula Derivation for a Quasi-Steady State Temperature Field in a Half-Space when Surface Temperature Changes Periodically 366 Exercise 14.7 Formula Derivation for Temperature of Two Contacting Semi-Infinite Bodies 374 Exercise 14.8 Depth of Heat Penetration 375 Exercise 14.9 Calculating Plate Surface Temperature under the Assumption that the Plate is a Semi-Infinite Body 377 Exercise 14.10 Calculating Ground Temperature at a Specific Depth 378 Exercise 14.11 Calculating the Depth of Heat Penetration in the Wall of a Combustion Engine 379 Exercise 14.12 Calculating Quasi-Steady-State Ground Temperature at a Specific Depth when Surface Temperature Changes Periodically 380 Exercise 14.13 Calculating Surface Temperature at the Contact Point of Two Objects 382 Literature 383 15 Transient Heat Conduction in Simple-Shape Elements 385 Exercise 15.1 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 3rd Kind 385 Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind 394

XIV Contents Exercise 15.3 Calculating Plate Surface Temperature and Average Temperature Across the Plate Thickness by Means of the Provided Graphs 398 Exercise 15.4 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind 402 Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind 412 Exercise 15.6 Calculating Temperature in an Infinitely Long Cylinder using the Annexed Diagrams 416 Exercise 15.7 Formula Derivation for a Temperature Distribution in a Sphere with Boundary Conditions of 3rd Kind 420 Exercise 15.8 A Program for Calculating Temperature Distribution and Its Change Rate in a Sphere with Boundary Conditions of 3rd Kind 428 Exercise 15.9 Calculating Temperature of a Sphere using the Diagrams Provided 432 Exercise 15.10 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind 436 Exercise 15.11 A Program and Calculation Results for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind 441 Exercise 15.12 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind 444 Exercise 15.13 Program and Calculation Results for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind 448 Exercise 15.14 Formula Derivation for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind 452 Exercise 15.15 Program and Calculation Results for Temperature Distribution in a Sphere with Boundary Conditions of 2nd kind 456 Exercise 15.16 Heating Rate Calculations for a Thick-Walled Plate 460 Exercise 15.17 Calculating the Heating Rate of a Steel Shaft 461 Exercise 15.18 Determining Transients of Thermal Stresses in a Cylinder and a Sphere 463 Exercise 15.19 Calculating Temperature and Temperature Change Rate in a Sphere 464

Contents xv Exercise 15.20 Calculating Sensor Thickness for Heat Flux Measuring 465 Literature 467 16 Superposition Method in One-Dimensional Transient Heat Conduction Problems 469 Exercise 16.1 Derivation of Duhamel Integral 469 Exercise 16.2 Derivation of an Analytical Formula for a Half-Space Surface Temperature when Medium's Temperature Undergoes a Linear Change in the Function of Time 472 Exercise 16.3 Derivation of an Approximate Formula for a Half-Space Surface Temperature with an Arbitrary Change in Medium's Temperature in the Function of Time 476 Exercise 16.4 Derivation of an Approximate Formula for a Half-Space Surface Temperature when Temperature of a Medium Undergoes a Linear Change in the Function of Time...479 Exercise 16.5 Application of the Superposition Method when Initial Body Temperature is Non-Uniform 481 Exercise 16.6 Description of the Superposition Method Applied to Heat Transfer Problems with Time-Dependent Boundary Conditions 484 Exercise 16.7 Formula Derivation for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse 488 Exercise 16.8 Formula Derivation for a Half-Space Surface Temperature with a Mixed Step-Variable Boundary Condition in Time 491 Exercise 16.9 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Triangular Pulse and the Calculation of this Temperature 495 Exercise 16.10 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Rectangular Pulse; Temperature Calculation 500 Exercise 16.11 A Program and Calculation Results for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse 503 Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time 506 Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided 507 Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier 509 Literature 513

xvi Contents 17 Transient Heat Conduction in a Semi-Infinite body. The Inverse Problem 515 Exercise 17.1 Measuring Heat Transfer Coefficient. The Transient Method 515 Exercise 17.2 Deriving a Formula for Heat Flux on the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function 518 Exercise 17.3 Deriving Heat Flux Formula on the Basis of a Measured and Polynomial-Approximated Half-Space Surface Temperature Transient 521 Exercise 17.4 Formula Derivation for a Heat Flux Periodically Changing in Time on the Basis of a Measured Temperature Transient at a Point Located under the Semi-Space Surface 523 Exercise 17.5 Deriving a Heat Flux Formula on the Basis of Measured Half-Space Surface Temperature Transient, Approximated by a Linear and Square Function 527 Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method 528 Graphical Method.~ 529 Numerical Method 529 Exercise 17.7 Determining Heat Flux on the Basis of a Measured Time Transient of the Half-Space Temperature, Approximated by a Piecewise Linear Function 532 Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature 535 Literature 539 18 Inverse Transient Heat Conduction Problems 541 Exercise 18.1 Derivation of Formulas for Temperature Distribution and Heat Flux in a Simple-Shape Bodies on the Basis of a Measured Temperature Transient in a Single Point 541 Plate 543 Cylinder 543 Sphere 544 Exercise 18.2 Formula Derivation for a Temperature of a Medium when Linear Time Change in Plate Surface Temperature is Assigned 545 Exercise 18.3 Determining Temperature Transient of a Medium for which Plate Temperature at a Point with a Given Coordinate Changes According to the Prescribed Function 547

Contents xvii Exercise 18.4 Formula Derivation for a Temperature of a Medium, which is Warming an Infinite Plate; Plate Temperature at a Point with a Given Coordinate Changes at Constant Rate 549 Exercise 18.5 Determining Temperature and Heat Flux on the Plate Front Face on the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flow on the Plate Surface is in the Form of a Triangular Pulse 555 Exercise 18.6 Determining Temperature and Heat Flux on the Surface of a Plate Front Face on the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flow on the Plate Surface is in the Form of a Rectangular Pulse 562 Exercise 18.7 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes in a Linear Way 565 Exercise 18.8 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes According to the Square Function Assigned 569 Literature 571 19 Multidimensional Problems. The Superposition Method 573 Exercise 19.1 The Application of the Superposition Method to Multidimensional Problems 573 Exercise 19.2 Formula Derivation for Temperature Distribution in a Rectangular Region with a Boundary Condition of 3rd Kind... 577 Exercise 19.3 Formula Derivation for Temperature Distribution in a Rectangular Region with Boundary Conditions of 2nd Kind 580 Exercise 19.4 Calculating Temperature in a Steel Cylinder of a Finite Height 582 Exercise 19.5 Calculating Steel Block Temperature 584 20 Approximate Analytical Methods for Solving Transient Heat Conduction Problems 587 Exercise 20.1 Description of an Integral Heat Balance Method by Means of a One-Dimensional Transient Heat Conduction Example 587 Exercise 20.2 Determining Transient Temperature Distribution in a Flat Wall with Assigned Conditions of 1st, 2nd and 3rd Kind 590 Exercise 20.3 Determining Thermal Stresses in a Flat Wall 600 Literature 600

xviii Contents 21 Finite Difference Method 605 Exercise 21.1 Methods of Heat Flux Approximation on the Plate surface 606 Exercise 21.2 Explicit Finite Difference Method with Boundary Conditions of 1st, 2nd and 3rd Kind 610 Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method 616 Exercise 21.4 Solving Two-Dimensional Problems by Means of the Implicit Difference Method 622 Exercise 21.5 Algorithm and a Program for Solving a Tridiagonal Equation System by Thomas Method 626 Exercise 21.6 Stability Analysis of the Explicit Finite Difference Method by Means of the von Neumann Method 630 Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program 634 Exercise 21.8 Calculating One-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program 639 Exercise 21.9 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program; Algebraic Equation System is Solved by Gaussian Elimination Method 644 Exercise 21.10 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program; Algebraic Equation System Solved by Over-Relaxation Method 652 Literature 656 22 Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM) 659 Exercise 22.1 Description of FEM Based on Galerkin Method Used for Solving Two-Dimensional Transient Heat Conduction Problems 659 Exercise 22.2 Concentrating (Lumped) Thermal Finite Element Capacity in FEM 662 Exercise 22.3 Methods for Integrating Ordinary Differential Equations with Respect to Time Used in FEM 668 Exercise 22.4 Comparison of FEM Based on Galerkin Method and Heat Balance Method with Finite Volume Method 671 Exercise 22.5 Natural Coordinate System for One-Dimensional, Two-Dimensional Triangular and Two-Dimensional Rectangular Elements 674

Contents xix Exercise 22.6 Coordinate System Transformations and Integral Calculations by Means of the Gauss-Legendre Quadratures 678 Exercise 22.7 Calculating Temperature in a Complex-Shape Fin by Means of the ANSYS Program 687 Literature 690 23 Numerical-Analytical Methods 693 Explicit Method 694 Implicit Method 694 Crank-Nicolson Method 694 Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge- Kutta Method 695 Exercise 23.2 Numerical-Analytical Method for Integrating a Linear Ordinary Differential Equation System 698 Exercise 23.3 Determining Steel Plate Temperature by Means of the Method of Lines, while the Plate is Cooled by air and Boiling Water 703 Exercise 23.4 Using the Exact Analytical Method and the Method of lines to Determine Temperature of a Cylindrical Chamber 709 Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines 714 Exercise 23.6 Determining Temperature Distribution in a cylindrical Chamber with Constant and Temperature Dependent Thermo- Physical Properties by Means of the Method of Lines 718 Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines 724 Literature 729 24 Solving Inverse Heat Conduction Problems by Means of Numerical Methods 733 Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems 733 Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems 739 Exercise 24.3 Weber Method Step-Marching Methods in Space 746 Exercise 24.4 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Measured Temperature on a Thermally Insulated Back Plate Surface; Heat Flux is in the Shape of a Rectangular Pulse 751

xx Contents Exercise 24.5 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Temperature Measurement on an Insulated Back Plate Surface; Heat Flux is in the Shape of a Triangular Pulse 759 Literature 763 25 Heat Sources 765 Exercise 25.1 Determining Formula for Transient Temperature Distribution Around an Instantaneous (Impulse) Point Heat Source Active in an Infinite Space 767 Exercise 25.2 Determining Formula for Transient Temperature Distribution in an Infinite Body Produced by an Impulse Surface Heat Source 770 Exercise 25.3 Determining Formula for Transient Temperature Distribution Around Instantaneous Linear Impulse Heat Source Active in an Infinite Space 772 Exercise 25.4 Determining Formula for Transient Temperature Distribution Around a Point Heat Source, which lies in an Infinite Space and is Continuously Active 774 Exercise 25.5 Determining Formula for a Transient Temperature Distribution Triggered by a Surface Heat Source Continuously Active in an Infinite Space 777 Exercise 25.6 Determining Formula for a Transient Temperature Distribution Around a Continuously Active Linear Heat Source with Assigned Power 'q per Unit of Length 779 Exercise 25.7 Determining Formula for Quasi-Steady-State Temperature Distribution Caused by a Point Heat Source with a Power 0 0 that Moves at Constant Velocity v in Infinite Space or on the Half Space Surface 781 Exercise 25.8 Determining Formula for Transient Temperature Distribution Produced by a Point Heat Source with Power Q o that Moves At Constant Velocity v in Infinite Space or on the Half Space Surface 785 Exercise 25.9 Calculating Temperature Distribution along a Straight Line Traversed by a Laser Beam 789 Exercise 25.10 Quasi-Steady State Temperature Distribution in a Plate During the Welding Process; a Comparison between the Analytical Solution and FEM 00.0000.00000 0 0 0 0 0 0 0 0 0 0 0 0.0 792 Literature 796

Contents xxi 26 Melting and Solidification (Freezing)................... 799 Exercise 26.1 Determination of a Formula which Describes the Solidification (Freezing) and Melting of a Semi-Infinite Body (the Stefan Problem) 803 Exercise 26.2 Derivation of a Formula that Describes the Solidification (Freezing) of a Semi-Infinite Body Under the Assumption that the Temperature of a Liquid Is Non-Uniform...808 Exercise 26.3 Derivation of a Formula that Describes Quasi-Steady- State Solidification (Freezing) of a Flat Liquid Layer 811 Exercise 26.4 Derivation of Formulas that Describe Solidification (Freezing) of Simple-Shape Bodies: Plate, Cylinder and Sphere 816 Exercise 26.5 Ablation of a Semi-Infinite Body 820 Exercise 26.6 Solidification of a Falling Droplet of Lead 823 Exercise 26.7 Calculating the Thickness of an Ice Layer After the Assigned Time 825 Exercise 26.8 Calculating Accumulated Energy in a Melted Wax 826 Exercise 26.9 Calculating Fish Freezing Time 828 Literature 829 Appendix A Basic Mathematical Functions 831 A.l. Gauss Error Function 831 A.2. Hyperbolic Functions 833 A.3. Bessel Functions 834 Literature 835 Appendix B Thermo-Physical Properties of Solids......837 B.1. Tables of Thermo-Physical Properties of Solids 837 B.2. Diagrams 856 B.3. Approximated Dependencies for Calculating Thermo-Physical Properties of a Steel [8] 858 Literature 861 Appendix C Fin Efficiency Diagrams (for Chap. 6, Part 11) 863 Literature 865 Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, part 11) 867 Appendix E Subprogram for Solving Linear Algebraic Equations System using Gauss Elimination Method (for Chap. 6, Part 11) 879 Appendix F Subprogram for Solving a Linear Algebraic Equations System by Means of Over-Relaxation Method 881

xxii Contents Appendix G Subprogram for Solving an Ordinary Differential Equations System of 1st order using Runge-Kutta Method of 4th Order (for Chap. 11, Part II) 883 Appendix H Determining Inverse Laplace Transform for Chap. 15, Part II) 885 Literature 889

Nomenclature a a,b A A b Bi C < c, c* C d,d d w d z d D e E E f F Fo g [g] h h h sl thermal diffusivity, m 2/s length of rectangle sides, m surface area, m' matrix of the algebraic equation coefficients vector of the coefficients from the right side of an equation Biot number alia specific heat, J/(kg K) specific heat at constant pressure, J/(kg K) specific heat at constant volume, J/(kg K) specific heat substitute, J/(kg K) an inverse matrix to A coefficients matrix diameter, m inner diameter, m outer diameter, m directional versor pulse duration, s coordinates versor longitudinal elasticity modulus (Young modulus), MPa distance of a temperature sensor from a solid's surface, m measured temperature, in C or K dimensionless measured temperature Fourier number (at/l 2 ) thickness, m column vector of temperature gradient enthalpy, J/kg height, m latent heat of melting (change in enthalpy due to a change from a solid to liquid phase), J/kg dimensionless heat flux enthalpy, J/kg current, A modified Bessel function of the first kind of order zero modified Bessel function of the first kind of order one Bessel function of the first kind of order zero Bessel function of the first kind of order one overall heat transfer coefficient, W/(m.K) dimensionless heat transfer coefficient modified Bessel function of the second kind of order zero modified Bessel function of the second kind of order one

xxiv Nomenclature L L Lc L i; Ly' t; m m n». ny' n, N j e p P P r;». q q r s s s S Ste t t t t t utr, T cz T m t; r, r, T p T sp length, m fin height, m equivalent fin height, m latent heat of melting, J/kg length in x, y, z direction, m mass, kg fin parameter (2a1At)0.5,11m outwardly directed unit normal vector to the control volume boundary directional cosinuses shape function for i- node of a finite element e pressure, MPa perimeter, m temperature-change duratioon, s perpendicular and linear pitch to the direction of air flow, m energy per unit of surface, J/m 2 a variable in Laplace transform q = ;;r; energy per unit of length, Jim heat flux, W/m 2 thermal load of heat furnace wall (absorbed heat flux), W/m 2 energy generation rate per unit volume (uniform within the body), W/m 3 heat flow, W radius, m inner surface radius, m outer surface radius, m heat resistance, KJW phase boundary position, m complex variable fin spacing, m shape coefficient, m Stefan number c (T - m To)lh sl time, s width, m fin width, m water-wall tube spacing, m time of heat flux step-change, s temperature, C or K fin base temperature, C or K fluid temperature, C or K melting temperature, C or K mean temperature over the wall thickness, C or K temperature in z-node, C or K external temperature during radiation heat transfer, K air temperature, C or K temperature of combustion gases, C or K

Nomenclature xxv To T T u u u(r, t) U U W Wx' wy' W z x,y,z X Y Yo Y 1 initial temperature, C or K mean temperature, C or K Laplace transform inner energy, J/kg transformed temperature, temperature difference (T - To)' K influence function (system response to unit step function) voltage, V Laplace transform specific volume, m 3/kg velocity, m/s temperature change rate, Kls or Klmin volume, m' width, m velocity components, m/s Cartesian coordinates dimensionless coordinate smoothed temperature, C or K Bessel function of the second kind of order zero Bessel function of the second kind of order one Greek symbols a f3 rs s ~t ~T ~,~r & e 1] 1] g e Jl nv ~ p p a heat transfer coefficient, W/(m 2 K) linear or volumetric thermal expansion coefficient, 11K unit step function (Heaviside function) depth of heat penetration, m Dirac function time step, s temperature difference, K spatial step in the x or r direction, m surface emissivity temperature measurement error, K fin efficiency similarity variable (dimensionless coordinate) angle coordinate dimensionless temperature excess temperature (temperature difference between true and initial temperature or between true and ambient temperature), K thermal diffusivity, m 2/s thermal conductivity, W/(m K) thermal conductivity in x, y, z direction, W/(m K) thermal conductivity tensor n-th root of a characteristic equation Poisson ratio a coordinate in moving coordinate system, m density, kg/m' dimensionless radius Stefan-Boltzmann constant, 5.67.10-8 W/(m 2 K4 )

xxvi Nomenclature radial, tangential and axial stresses respectively, MPa relaxation time, s time constant, s angle coordinate dissipation function, W/m 3 circular frequency, lis