Case Studies in Computational Heat Transfer

$95.00

Panagiotis (Takis) Sismanis.B.Eng, M.Eng, PhD – Retired, Metallurgical Engineer, Piraeus, Attica, Greece

Series: Mechanical Engineering Theory and Applications
BISAC: SCI003000; TEC009070
DOI: https://doi.org/10.52305/VNRL3688

Mathematicians, scientists, and engineers often look for simple and robust computer programs in order to solve the Fourier differential equation for heat transfer in a numerical way. Specifically, the 2D differential equation may take various initial and boundary conditions applicable in many practical applications. Simplifications to 1D differential equations may also be applied to the numerical solution of various practical problems in heat and mass transfer. The implicit scheme for the finite difference solution of the heat transfer equation, as formulated by Patankar, is mostly deployed throughout the book. It is worth noting that the method is proven to be stable and fast for most time scales. In fact, fast convergence is attained for steady state problems that have searched for solutions within a time span of more than the universe age. Furthermore, mathematical problems that finalized phenomena within a period of a few milliseconds have proven the remarkable stability of the numerical solution. A practitioner has thirty selected cases in order to grasp the salient and advanced features of the implicit scheme. A program core developed in C++ allows the introduction of the relevant initial and boundary conditions and solves various problems with only minor changes. About two-thirds of the selected cases were based on mathematical problems for which analytical solutions exist. In this way, analytical and numerical solution programs are given in order for the practitioner to compile, run the code, and plot the desired results. Orthogonal and cylindrical coordinates are deployed according to border geometry; Python programs were developed for the analytical solutions in cylindrical symmetry. Induction heating of billets and slabs, rebar production by quench and tempering, and steel billet solidification are some of the most practical examined cases.

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Table of Contents

List of Figures

Preface

Acknowledgments

Chapter 1. Introduction
1.1. The Implicit Finite-Difference Scheme
1.2. Derivation of the Implicit-Scheme for the 1D Differential Equation
1.3. Deployed Solution Procedure
1.4. Computer Code for the Solution of the Case Studies

Chapter 2. Case Study 1
2.1. Steady-State in a Rectangular Plate (Isothermal Borders)

Chapter 3. Case Study 2
3.1. Steady-State in a Rectangular Plate (Mixed BC-I)

Chapter 4. Case Study 3
4.1. Steady-State in a Rectangular Plate (Mixed BC-II)

Chapter 5. Case Study 4
5.1. Unsteady State in a Rectangular Plate (Isothermal BC)

Chapter 6. Case Study 5
6.1. Unsteady State in a Rectangular Plate (Mixed BC-I)

Chapter 7. Case Study 6
7.1. Unsteady State in a Rectangular Plate (Mixed BC-II)

Chapter 8. Case Study 7
8.1. Unsteady State in a Rectangular Plate (Heat Generation)

Chapter 9. Case Study 8
9.1. Unsteady State in a Rectangular Plate (Mixed BC-III)

Chapter 10. 2D Geometry in Polar Coordinates

Chapter 11. Case Study 9
11.1. Unsteady State in a Long Cylinder (Zero Surface Temperature)

Chapter 12. Case Study 10
12.1. Unsteady State in a Long Cylinder (Parabolic Temperature Distribution)

Chapter 13. Case Study 11
13.1. Unsteady State in a Long Cylinder (Surface Temperature Proportional to Time)

Chapter 14. Case Study 12
14.1. Unsteady State in a Long Cylinder (Constant Heat Flux at the Surface)

Chapter 15. Case Study 13
15.1. Unsteady State in a Long Cylinder (Convection at the Surface)

Chapter 16. Case Study 14 – Induction Heating of Steel Billets
16.1. Cylindrical Billets with Sections Equivalent to Square Ones

Chapter 17. Case Study 15 – Slab Reheating
17.1. Thin Slab Reheating

Chapter 18. Case Study 16 – Heavy Slab Reheating
18.1. Thick Slab Reheating

Chapter 19. Case Study 17
19.1. Quenched and Tempered (Q&T) Reinforced Concrete Steels

Chapter 20. Case Study 18
20.1. A Cylindrical Billet in a Soaking Pit

Chapter 21. Case Study 19
21.1. A Spherical Specimen in a Soaking Pit

Chapter 22. Case Study 20
22.1. A Refractory Wall under Preheating

Chapter 23. Case Study 21
23.1. Heat Conduction in a Slab

Chapter 24. Case Study 22
24.1. Temperature of a Bar (1D Case-I: Mixed IC and BC)

Chapter 25. Case Study 23
25.1. Temperature of a Bar (1D Case-II: Mixed IC and BC)

Chapter 26. Case Study 24
26.1. Temperature of a Bar (1D Case-III: Mixed IC and BC)

Chapter 27. Case Study 25
27.1. Temperature of a Bar (1D Case-IV: Mixed IC and BC)

Chapter 28. Case Study 26
28.1. Temperature of a Bar (1D Case-V: Mixed IC and BC)

Chapter 29. Case Study 27
29.1. Temperature of a Bar (1D Case-VI: Mixed IC and BC)

Chapter 30. Case Study 28
30.1. Temperature of a Bar (1D Case-VII: Mixed IC and BC)

Chapter 31. Case Study 29
31.1. Temperature of a Bar (1D Case-VIII: Mixed IC and BC)

Chapter 32. Case Study 30
32.1. Solidification of a Billet in a Continuous Casting Machine

Chapter 33. Some Published Works on the Topic
33.1. Special Reinforced NSC-Columns under Fire
33.2. Continuous Casting of Carbon Steels

Conclusion

Appendix 1
A1.1. Routines Used for Plotting

Appendix 2. Case Studies Borrowed from the Literature
A2.1. Steady State in a Rectangle (Case Study 1)
A2.2. Steady State in a Rectangle (Case Study 2)
A2.3. Steady State in a Rectangle (Case Study 3)
A2.4. Unsteady State in a Rectangle (Case Study 4)
A2.5. Transient State in a Cylinder (Case Study 9)
A2.6. Transient State in a Cylinder (Case Study 10)
A2.7. Transient State in a Cylinder (Case Study 11)
A2.8. Unsteady State in an Infinite Cylinder (Case Study 12)
A2.9. Unsteady State in an Infinite Cylinder (Case Study 13)
A2.10. A Refractory Wall (Case Study 20)
A2.11. Heat Conduction in a Slab (Case Study 21)
A2.12. Temperature of a Bar (1D Case-I: Mixed IC and BC) (Case Study 22)
A2.13. Temperature of a Bar (1D Case-II: Mixed IC and BC) (Case Study 23)
A2.14. Temperature of a Bar (1D Case-III: Mixed IC and BC) (Case Study 24)
A2.15. Temperature of a Bar (1D Case-IV: Mixed IC and BC) (Case Study 25)
A2.16. Temperature of a Bar (1D Case-V: Mixed IC and BC) (Case Study 26)
A2.17. Temperature of a Bar (1D Case-VI: Mixed IC and BC) (Case Study 27)
A2.18. Temperature of a Bar (1D Case-VII: Mixed IC and BC) (Case Study 28)
A2.19. Temperature of a Bar (1D Case-VIII: Mixed IC and BC) (Case Study 29)

Appendix 3. A3.1. Induction Heating of Steel Billets/Slabs

Acronyms

References

About the Author

Index

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