Solving Direct and Inverse Heat Conduction Problems

✍ Scribed by Jan Taler, Piotr Duda

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 890
• Edition: 1
• Category: Library

✦ Synopsis

This book presents a solution for direct and inverse heat conduction problems, discussing the theoretical basis for the heat transfer process and presenting selected theoretical and numerical problems in the form of exercises with solutions. The book covers one-, two- and three dimensional problems which are solved by using exact and approximate analytical methods and numerical methods. An accompanying CD-Rom includes computational solutions of the examples and extensive FORTRAN code.

✦ Table of Contents

354033470X......Page 1
Solving Direct and
Inverse Heat Conduction
Problems......Page 3
Copyright Page
......Page 4
Preface......Page 5
Table of contents
......Page 7
Nomenclature......Page 23
Part I Heat Conduction Fundamentals......Page 27
1 Fourier Law......Page 28
Literature......Page 31
2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity
......Page 32
2.2 Inner Energy Balance Equation......Page 34
2.2.1 Energy Balance Equations in Three Basic Coordinate Systems
......Page 37
2.3 Hyperbolic Heat Conduction Equation......Page 41
2.4 Initial and Boundary Conditions......Page 42
2.4.2 Second Kind Boundary Conditions von Neumann Conditions)
......Page 43
2.4.3 Third Kind Boundary Conditions......Page 44
2.4.4 Fourth Kind Boundary Conditions......Page 46
2.4.5 Non-Linear Boundary Conditions......Page 47
2.4.6 Boundary Conditions on the Phase Boundaries......Page 49
Literature......Page 51
3.1 Linearization of a Heat Conduction Equation......Page 53
3.2.1 A Body Model with a Lumped Thermal Capacity......Page 55
3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness
......Page 57
3.2.3 Heat Conduction Equation for a Circular Fin with Uniform Thickness
......Page 59
3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity......Page 61
Literature......Page 63
4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient
......Page 64
4.2 One-Dimensional Inverse Transient Heat Conduction Problem
......Page 66
Literature......Page 69
5 Variable Change......Page 70
Literature......Page 73
Part II Exercises. Solving Heat Conduction Problems......Page 74
Exercise 6.1 Fourier Law in a Cylindrical Coordinate System
......Page 75
Solution......Page 76
Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation
......Page 77
Solution......Page 78
Solution......Page 79
Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall
......Page 82
Solution......Page 83
Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe
......Page 84
Solution......Page 85
Solution......Page 87
Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall
......Page 90
Solution......Page 91
Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity
......Page 92
Solution......Page 93
Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor
......Page 96
Solution......Page 97
Exercise 6.10 Determining Heat FluxBy Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity
......Page 99
Solution......Page 100
Solution......Page 102
Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources
......Page 104
Solution......Page 105
Solution......Page 107
Solution......Page 109
Solution......Page 111
Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device
......Page 114
Solution......Page 115
Solution......Page 117
Solution......Page 120
Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins
......Page 121
Solution......Page 122
Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method
......Page 124
Solution......Page 125
Solution......Page 130
Solution......Page 131
Solution......Page 136
Exercise 6.24 Overall HeatTransfer Coefficient for a Longitudinally Finned Pipewith a Scale Layer on an Inner Surface
......Page 137
Solution......Page 139
Solution......Page 141
Solution......Page 144
Solution......Page 149
Solution......Page 153
Solution......Page 156
Literature......Page 160
Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness
......Page 162
Solution......Page 163
Solution......Page 166
Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip
......Page 169
Solution......Page 170
Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler
......Page 177
Solution......Page 178
Literature......Page 181
Solution......Page 182
Solution......Page 184
Solution......Page 186
Literature......Page 190
Solution......Page 191
Solution......Page 193
Solution......Page 196
Solution......Page 199
Solution......Page 201
Literature......Page 202
Solution......Page 203
Solution......Page 205
Solution......Page 206
Literature......Page 207
Exercise 11.1 Description of the Control Volume Method......Page 208
Solution......Page 209
a) Heat balance equation- Cartesian coordinates......Page 210
b) Heat balance equation-cylindrical coordinates......Page 211
Solution......Page 213
Solution......Page 218
Solution......Page 223
Solution......Page 227
Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney
......Page 234
Solution......Page 235
Solution......Page 240
Historical Development of FEM......Page 249
Solution......Page 250
Solution......Page 253
Solution......Page 257
a) Conductivity matrix [Kec] for a finite rectangular element
......Page 264
b) Conductivity matrix [Kec] for a finite triangular element
......Page 266
a) Rectangular finite element......Page 268
b) Triangular finite element......Page 270
a) Rectangular element......Page 272
b) Triangular element......Page 273
a) Finite rectangular element......Page 275
b) Finite triangular element......Page 277
Exercise 11.15 Methods for Building Global Equation System in FEM
......Page 278
Solution......Page 279
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)
......Page 283
Solution......Page 284
Solution......Page 290
Solution......Page 294
Solution......Page 304
Solution......Page 316
Solution......Page 319
Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS Program)
......Page 322
Solution......Page 323
Literature......Page 326
Solution......Page 327
Solution......Page 332
Solution......Page 341
Exercise 12.4 Determining Temperature Distribution in a Square Region using Boundary Element Method
......Page 345
Solution......Page 346
Literature......Page 349
Solution......Page 350
Solution......Page 353
Solution......Page 356
Solution......Page 357
Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple
......Page 359
Solution......Page 360
Solution......Page 361
Solution......Page 362
Solution......Page 363
Solution......Page 364
Exercise 13.10 Measuring Heat Flux......Page 366
Solution......Page 367
Literature......Page 368
Solution......Page 369
Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature
......Page 371
Solution......Page 372
Solution......Page 374
Solution......Page 376
Solution......Page 380
Solution......Page 382
Solution......Page 390
Exercise 14.8 Depth of Heat Penetration......Page 391
Solution......Page 392
Solution......Page 393
Solution......Page 394
Solution......Page 395
Solution......Page 396
Solution......Page 398
Literature......Page 399
Solution......Page 400
Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind
......Page 409
Solution......Page 410
Solution......Page 413
Solution......Page 417
Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind
......Page 427
Solution......Page 428
Solution......Page 431
Solution......Page 435
Solution......Page 443
Solution......Page 447
Solution......Page 451
Solution......Page 456
Solution......Page 459
Solution......Page 463
Solution......Page 467
Solution......Page 471
Solution......Page 475
Exercise 15.17 Calculating the Heating Rate of a Steel Shaft
......Page 476
Solution......Page 477
Solution......Page 478
Solution......Page 479
Solution......Page 480
Literature......Page 482
Solution......Page 483
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
......Page 486
Solution......Page 487
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
......Page 490
Solution......Page 491
Solution......Page 493
Solution......Page 495
Solution......Page 498
Example 1......Page 499
Example 2......Page 500
Example 4......Page 501
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
......Page 502
Solution......Page 503
Solution......Page 505
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
......Page 509
Solution......Page 510
Solution......Page 514
Solution......Page 517
Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time
......Page 520
Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided
......Page 521
Solution......Page 522
Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier
......Page 523
Solution......Page 525
Literature......Page 527
Solution......Page 528
Exercise 17.2 Deriving a Formula for Heat Fluxon the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function
......Page 531
Solution......Page 532
Solution......Page 534
Solution......Page 536
Solution......Page 540
Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method
......Page 541
Solution......Page 542
Solution......Page 545
Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature
......Page 548
Solution......Page 549
Literature......Page 552
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
......Page 553
Solution......Page 554
Solution......Page 557
Solution......Page 559
Solution......Page 561
Solution......Page 567
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
......Page 574
Solution......Page 575
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
......Page 577
Solution......Page 578
Solution......Page 581
Literature......Page 583
Boundary Conditions of 1st and 3rd Kind
......Page 585
Boundary Conditions of 2nd Kind
......Page 586
Solution......Page 589
Solution......Page 592
Solution......Page 594
Solution......Page 596
Solution......Page 599
Solution......Page 602
Example 1......Page 606
Example 2......Page 607
Example 3......Page 610
Example 4......Page 611
Solution......Page 612
Literature......Page 614
21 Finite Difference Method......Page 617
Solution......Page 618
Exercise 21.2 Explicit Finite Difference Methodwith Boundary Conditions of 1st, 2nd and 3rd Kind......Page 622
Solution......Page 623
Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method
......Page 628
Solution......Page 629
Solution......Page 634
Solution......Page 638
Solution......Page 642
Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program
......Page 646
Solution......Page 647
Solution......Page 651
Solution......Page 656
Solution......Page 664
Literature......Page 668
Solution......Page 670
Solution......Page 673
Solution......Page 679
Solution......Page 682
a. One-dimensional elements......Page 685
b. Two-dimensional tetragonal elements......Page 686
c. Two-dimensional triangular elements......Page 687
Solution......Page 689
a. One-dimensional elements......Page 693
b. Tetragonal elements......Page 695
c. Triangular elements......Page 696
Exercise 22.7 Calculating Temperature in a Complex ShapeFin by Means of the ANSVS Program
......Page 698
Solution......Page 700
Literature......Page 701
23 Numerical-Analytical Methods......Page 703
Crank-Nicolson Method......Page 704
Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method
......Page 705
Solution......Page 706
Solution......Page 708
a. Approximating u(t) with a step function
......Page 709
b. Approximating u(t) with a piecewise linear function
......Page 711
Solution......Page 713
Solution......Page 719
Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines
......Page 724
Solution......Page 725
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
......Page 728
Solution......Page 730
Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines
......Page 734
Solution......Page 735
Literature......Page 739
Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems
......Page 742
Solution......Page 743
a. Division of an inverse region into two control volumes (Fig. 24.2a)
......Page 745
b. Division of an inverse region into three control volumes (Fig. 24.2b)
......Page 746
c. Division of an inverse region into four control volumes (Fig. 24.2c)
......Page 747
Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems
......Page 748
Solution......Page 749
Solution......Page 755
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
......Page 760
Solution......Page 761
Solution......Page 768
Literature......Page 772
25 Heat Sources......Page 773
Solution......Page 775
Solution......Page 778
Solution......Page 780
Solution......Page 782
Solution......Page 785
Solution......Page 787
Solution......Page 789
Solution......Page 793
Solution......Page 797
Solution......Page 800
Literature......Page 804
26 Melting and Solidification (Freezing)......Page 806
Solution......Page 810
Solution......Page 815
Exercise 26.3 Derivation of a Formula that Describe Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer
......Page 818
Solution......Page 819
Solution......Page 823
b. Cylinder......Page 825
Exercise 26.5 Ablation of a Semi-Infinite Body......Page 827
Solution......Page 828
Solution......Page 830
Solution......Page 832
Exercise 26.8 Calculating Accumulated Energy in a Melted Wax
......Page 833
Solution......Page 834
Solution......Page 835
Literature......Page 836
A.1. Gauss Error Function......Page 837
A.2. Hyperbolic Functions......Page 839
A.3. Bessel Functions......Page 840
Literature......Page 841
B.1. Tables of Thermo-Physical Properties of Solids......Page 842
B.2. Diagrams......Page 861
Density p at temperature 20°C
......Page 863
Specific heat capacity c in a temperature function......Page 864
Poisson ratio v in function of temperature......Page 865
Literature......Page 866
Appendix C Fin Efficiency Diagrams (for Chap. 6, part II)
......Page 867
Literature......Page 869
Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, Part II)
......Page 870
Subprogram for solving linearalgebraic equations system using Gauss method
......Page 882
Subprogram SOR section appendix f subprogram,for solving a linear algebraic equations system by means of over-relaxation method
......Page 884
Subprogram for solving an ordinary differential equations system of 1st order using Runge-Kutta method of 4th order
......Page 885
Appendix H Determining inverse Laplace Transform (for Chap. 15, part II)
......Page 886
Literature......Page 890

✦ Subjects

Топливно-энергетический комплекс;Тепло- и массообмен;