Solving Direct and Inverse Heat Conduction Problems

Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface......Page 4
Table of contents\r......Page 6
Nomenclature......Page 22
Part I. Heat Conduction Fundamentals......Page 26
1. Fourier Law......Page 27
Literature......Page 30
2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity\r......Page 31
2.2 Inner Energy Balance Equation......Page 33
2.2.1 Energy Balance Equations in Three Basic Coordinate Systems\r......Page 36
2.3 Hyperbolic Heat Conduction Equation......Page 40
2.4 Initial and Boundary Conditions......Page 41
2.4.2 Second Kind Boundary Conditions von Neumann Conditions)\r......Page 42
2.4.3 Third Kind Boundary Conditions......Page 43
2.4.4 Fourth Kind Boundary Conditions......Page 45
2.4.5 Non-Linear Boundary Conditions......Page 46
2.4.6 Boundary Conditions on the Phase Boundaries......Page 48
Literature......Page 50
3.1 Linearization of a Heat Conduction Equation......Page 52
3.2.1 A Body Model with a Lumped Thermal Capacity......Page 54
3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness\r......Page 56
3.2.3 Heat Conduction Equation for a Circular Fin with Uniform Thickness\r......Page 58
3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity......Page 60
Literature......Page 62
4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient\r......Page 63
4.2 One-Dimensional Inverse Transient Heat Conduction Problem\r......Page 65
Literature......Page 68
5. Variable Change......Page 69
Literature......Page 72
Part II. Exercises. Solving Heat Conduction Problems......Page 73
Exercise 6.1 Fourier Law in a Cylindrical Coordinate System\r......Page 74
Solution......Page 75
Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation\r......Page 76
Solution......Page 77
Solution......Page 78
Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall\r......Page 81
Solution......Page 82
Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe\r......Page 83
Solution......Page 84
Solution......Page 86
Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall\r......Page 89
Solution......Page 90
Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity\r......Page 91
Solution......Page 92
Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor\r......Page 95
Solution......Page 96
Exercise 6.10 Determining Heat FluxBy Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity\r......Page 98
Solution......Page 99
Solution......Page 101
Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources\r......Page 103
Solution......Page 104
Solution......Page 106
Solution......Page 108
Solution......Page 110
Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device\r......Page 113
Solution......Page 114
Solution......Page 116
Solution......Page 119
Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins\r......Page 120
Solution......Page 121
Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method\r......Page 123
Solution......Page 124
Solution......Page 129
Solution......Page 130
Solution......Page 135
Exercise 6.24 Overall HeatTransfer Coefficient for a Longitudinally Finned Pipewith a Scale Layer on an Inner Surface\r......Page 136
Solution......Page 138
Solution......Page 140
Solution......Page 143
Solution......Page 148
Solution......Page 152
Solution......Page 155
Literature......Page 159
Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness\r......Page 161
Solution......Page 162
Solution......Page 165
Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip\r......Page 168
Solution......Page 169
Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler\r......Page 176
Solution......Page 177
Literature......Page 180
Solution......Page 181
Solution......Page 183
Solution......Page 185
Literature......Page 189
Solution......Page 190
Solution......Page 192
Solution......Page 195
Solution......Page 198
Solution......Page 200
Literature......Page 201
Solution......Page 202
Solution......Page 204
Solution......Page 205
Literature......Page 206
Exercise 11.1 Description of the Control Volume Method......Page 207
Solution......Page 208
a) Heat balance equation- Cartesian coordinates......Page 209
b) Heat balance equation-cylindrical coordinates......Page 210
Solution......Page 212
Solution......Page 217
Solution......Page 222
Solution......Page 226
Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney\r......Page 233
Solution......Page 234
Solution......Page 239
Historical Development of FEM......Page 248
Solution......Page 249
Solution......Page 252
Solution......Page 256
a) Conductivity matrix [Kec] for a finite rectangular element \r......Page 263
b) Conductivity matrix [Kec] for a finite triangular element \r......Page 265
a) Rectangular finite element......Page 267
b) Triangular finite element......Page 269
a) Rectangular element......Page 271
b) Triangular element......Page 272
a) Finite rectangular element......Page 274
b) Finite triangular element......Page 276
Exercise 11.15 Methods for Building Global Equation System in FEM\r......Page 277
Solution......Page 278
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)\r......Page 282
Solution......Page 283
Solution......Page 289
Solution......Page 293
Solution......Page 303
Solution......Page 315
Solution......Page 318
Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS Program)\r......Page 321
Solution......Page 322
Literature......Page 325
Solution......Page 326
Solution......Page 331
Solution......Page 340
Exercise 12.4 Determining Temperature Distribution in a Square Region using Boundary Element Method\r......Page 344
Solution......Page 345
Literature......Page 348
Solution......Page 349
Solution......Page 352
Solution......Page 355
Solution......Page 356
Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple\r......Page 358
Solution......Page 359
Solution......Page 360
Solution......Page 361
Solution......Page 362
Solution......Page 363
Exercise 13.10 Measuring Heat Flux......Page 365
Solution......Page 366
Literature......Page 367
Solution......Page 368
Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature\r......Page 370
Solution......Page 371
Solution......Page 373
Solution......Page 375
Solution......Page 379
Solution......Page 381
Solution......Page 389
Exercise 14.8 Depth of Heat Penetration......Page 390
Solution......Page 391
Solution......Page 392
Solution......Page 393
Solution......Page 394
Solution......Page 395
Solution......Page 397
Literature......Page 398
Solution......Page 399
Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind\r......Page 408
Solution......Page 409
Solution......Page 412
Solution......Page 416
Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind\r......Page 426
Solution......Page 427
Solution......Page 430
Solution......Page 434
Solution......Page 442
Solution......Page 446
Solution......Page 450
Solution......Page 455
Solution......Page 458
Solution......Page 462
Solution......Page 466
Solution......Page 470
Solution......Page 474
Exercise 15.17 Calculating the Heating Rate of a Steel Shaft\r......Page 475
Solution......Page 476
Solution......Page 477
Solution......Page 478
Solution......Page 479
Literature......Page 481
Solution......Page 482
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\r......Page 485
Solution......Page 486
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\r......Page 489
Solution......Page 490
Solution......Page 492
Solution......Page 494
Solution......Page 497
Example 1......Page 498
Example 2......Page 499
Example 4......Page 500
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\r......Page 501
Solution......Page 502
Solution......Page 504
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 Calculationof This Temperature\r......Page 508
Solution......Page 509
Solution......Page 513
Solution......Page 516
Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time\r......Page 519
Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided\r......Page 520
Solution......Page 521
Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier\r......Page 522
Solution......Page 524
Literature......Page 526
Solution......Page 527
Exercise 17.2 Deriving a Formula for Heat Fluxon the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function\r......Page 530
Solution......Page 531
Solution......Page 533
Solution......Page 535
Solution......Page 539
Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method\r......Page 540
Solution......Page 541
Solution......Page 544
Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature\r......Page 547
Solution......Page 548
Literature......Page 551
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\r......Page 552
Solution......Page 553
Solution......Page 556
Solution......Page 558
Solution......Page 560
Solution......Page 566
Exercise 18.6 Determining Temperature and Heat Flux on the Surface of a Plate Front Faceon the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flowon the Plate Surface is in the Form of a Rectangular Pulse\r......Page 573
Solution......Page 574
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\r......Page 576
Solution......Page 577
Solution......Page 580
Literature......Page 582
Boundary Conditions of 1st and 3rd Kind\r......Page 584
Boundary Conditions of 2nd Kind\r......Page 585
Solution......Page 588
Solution......Page 591
Solution......Page 593
Solution......Page 595
Solution......Page 598
Solution......Page 601
Example 1......Page 605
Example 2......Page 606
Example 3......Page 609
Example 4......Page 610
Solution......Page 611
Literature......Page 613
21. Finite Difference Method......Page 616
Solution......Page 617
Exercise 21.2 Explicit Finite Difference Methodwith Boundary Conditions of 1st, 2nd and 3rd Kind......Page 621
Solution......Page 622
Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method\r......Page 627
Solution......Page 628
Solution......Page 633
Solution......Page 637
Solution......Page 641
Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program\r......Page 645
Solution......Page 646
Solution......Page 650
Solution......Page 655
Solution......Page 663
Literature......Page 667
Solution......Page 669
Solution......Page 672
Solution......Page 678
Solution......Page 681
a. One-dimensional elements......Page 684
b. Two-dimensional tetragonal elements......Page 685
c. Two-dimensional triangular elements......Page 686
Solution......Page 688
a. One-dimensional elements......Page 692
b. Tetragonal elements......Page 694
c. Triangular elements......Page 695
Exercise 22.7 Calculating Temperature in a Complex ShapeFin by Means of the ANSVS Program\r......Page 697
Solution......Page 699
Literature......Page 700
23. Numerical-Analytical Methods......Page 702
Crank-Nicolson Method......Page 703
Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method\r......Page 704
Solution......Page 705
Solution......Page 707
a. Approximating u(t) with a step function\r......Page 708
b. Approximating u(t) with a piecewise linear function\r......Page 710
Solution......Page 712
Solution......Page 718
Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines\r......Page 723
Solution......Page 724
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\r......Page 727
Solution......Page 729
Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines\r......Page 733
Solution......Page 734
Literature......Page 738
Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems\r......Page 741
Solution......Page 742
a. Division of an inverse region into two control volumes (Fig. 24.2a)\r......Page 744
b. Division of an inverse region into three control volumes (Fig. 24.2b)\r......Page 745
c. Division of an inverse region into four control volumes (Fig. 24.2c)\r......Page 746
Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems\r......Page 747
Solution......Page 748
Solution......Page 754
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 Shapeof a Rectangular Pulse\r......Page 759
Solution......Page 760
Solution......Page 767
Literature......Page 771
25. Heat Sources......Page 772
Solution......Page 774
Solution......Page 777
Solution......Page 779
Solution......Page 781
Solution......Page 784
Solution......Page 786
Solution......Page 788
Solution......Page 792
Solution......Page 796
Solution......Page 799
Literature......Page 803
26. Melting and Solidification (Freezing)......Page 805
Solution......Page 809
Solution......Page 814
Exercise 26.3 Derivation of a Formula that Describe Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer\r......Page 817
Solution......Page 818
Solution......Page 822
b. Cylinder......Page 824
Exercise 26.5 Ablation of a Semi-Infinite Body......Page 826
Solution......Page 827
Solution......Page 829
Solution......Page 831
Exercise 26.8 Calculating Accumulated Energy in a Melted Wax\r......Page 832
Solution......Page 833
Solution......Page 834
Literature......Page 835
A.1. Gauss Error Function......Page 836
A.2. Hyperbolic Functions......Page 838
A.3. Bessel Functions......Page 839
Literature......Page 840
B.1. Tables of Thermo-Physical Properties of Solids......Page 841
B.2. Diagrams......Page 860
Density p at temperature 20°C\r......Page 862
Specific heat capacity c in a temperature function......Page 863
Poisson ratio v in function of temperature......Page 864
Literature......Page 865
Appendix C. Fin Efficiency Diagrams (for Chap. 6, part II)......Page 866
Literature......Page 868
Appendix D. Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, Part II)......Page 869
Subprogram for solving linearalgebraic equations system using Gauss method\r......Page 881
Subprogram SOR section appendix f subprogram,for solving a linear algebraic equations system by means of over-relaxation method\r......Page 883
Subprogram for solving an ordinary differential equations system of 1st order using Runge-Kutta method of 4th order\r......Page 884
Appendix H. Determining inverse Laplace Transform (for Chap. 15, part II)......Page 885
Literature......Page 889