HEAT TRANSFER ͳ MATHEMATICAL MODELLING, NUMERICAL METHODS AND INFORMATION TECHNOLOGY Edited by Aziz Belmiloudi Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Edited by Aziz Belmiloudi Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Iva Lipovic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Zadiraka Evgenii, 2010. Used under license from Shutterstock.com First published February, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology, Edited by Aziz Belmiloudi p. cm. ISBN 978-953-307-550-1 free online editions of InTech Books and Journals can be found at www.intechopen.com Part 1 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Part 2 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Preface IX Inverse, Stabilization and Optimization Problems 1 Optimum Fin Profile under Dry and Wet Surface Conditions 3 Balaram Kundu and Somchai Wongwises Thermal Therapy: Stabilization and Identification 33 Aziz Belmiloudi Direct and Inverse Heat Transfer Problems in Dynamics of Plate Fin and Tube Heat Exchangers 77 Dawid Taler Radiative Heat Transfer and Effective Transport Coefficients 101 Thomas Christen, Frank Kassubek, and Rudolf Gati Numerical Methods and Calculations 127 Finite Volume Method Analysis of Heat Transfer in Multi-Block Grid During Solidification 129 Eliseu Monteiro, Regina Almeida and Abel Rouboa Lattice Boltzmann Numerical Approach to Predict Macroscale Thermal Fluid Flow Problem 151 Nor Azwadi Che Sidik and Syahrullail Samion Efficient Simulation of Transient Heat Transfer Problems in Civil Engineering 165 Sebastian Bindick, Benjamin Ahrenholz, Manfred Krafczyk Applications of Nonstandard Finite Difference Methods to Nonlinear Heat Transfer Problems 185 Alaeddin Malek Contents Contents VI Fast BEM Based Methods for Heat Transfer Simulation 209 Jure Ravnik and Leopold Škerget Aerodynamic Heating at Hypersonic Speed 233 Andrey B. Gorshkov Thermoelastic Stresses in FG-Cylinders 253 Mohammad Azadi and Mahboobeh Azadi Experimentally Validated Numerical Modeling of Heat Transfer in Granular Flow in Rotating Vessels 271 Bodhisattwa Chaudhuri, Fernando J. Muzzio and M. Silvina Tomassone Heat Transfer in Mini/Micro Systems 303 Introduction to Nanoscale Thermal Conduction 305 Patrick E. Hopkins and John C. Duda Study of Hydrodynamics and Heat Transfer in the Fluidized Bed Reactors 331 Mahdi Hamzehei Particle Scale Simulation of Heat Transfer in Fluid Bed Reactors 383 Zongyan Zhou, Qinfu Hou and Aibing Yu Population Balance Model of Heat Transfer in Gas-Solid Processing Systems 409 Béla G. Lakatos Synthetic Jet-based Hybrid Heat Sink for Electronic Cooling 435 Tilak T Chandratilleke, D Jagannatha and R Narayanaswamy Turbulent Flow and Heat Transfer Characteristics of a Micro Combustor 455 Tae Seon Park and Hang Seok Choi Natural Circulation in Single and Two Phase Thermosyphon Loop with Conventional Tubes and Minichannels 475 Henryk Bieliński and Jarosław Mikielewicz Heat Transfer at Microscale 497 Mohammad Hassan Saidi and Arman Sadeghi Chapter 9 Chapter 10 Chapter 11 Chapter 12 Part 3 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Contents VII Energy Transfer and Solid Materials 527 Thermal Characterization of Solid Structures during Forced Convection Heating 529 Balázs Illés and Gábor Harsányi Analysis of the Conjugate Heat Transfer in a Multi-Layer Wall Including an Air Layer 553 Armando Gallegos M., Christian Violante C., José A. Balderas B., Víctor H. Rangel H. and José M. Belman F. An Analytical Solution for Transient Heat and Moisture Diffusion in a Double-Layer Plate 567 Ryoichi Chiba Frictional Heating in the Strip-Foundation Tribosystem 579 Aleksander Yevtushenko and Michal Kuciej Convective Heat Transfer Coefficients for Solar Chimney Power Plant Collectors 607 Marco Aurélio dos Santos Bernardes Thermal Aspects of Solar Air Collector 621 Ehsan Mohseni Languri and Davood Domairry Ganji Heat Transfer in Porous Media 631 Ehsan Mohseni Languri and Davood Domairry Ganji Part 4 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Pref ac e During the last years, spectacular progress has been made in all aspects of heat trans- fer. Heat transfer is a branch of engineering science and technology that deals with the analysis of the rate of transfer thermal energy. Its fundamental modes are conduction, convection, radiation, convection vs. conduction and mass transfer. It has a broad ap- plication to many diff erent branches of science, technology and industry, ranging from biological, medical and chemical systems, to common practice of thermal engineer- ing (e.g. residential and commercial buildings, common household appliances, etc), industrial and manufacturing processes, electronic devices, thermal energy storage, and agriculture and food process. In engineering practice, an understanding of the mechanisms of heat transfer is becoming increasingly important, since heat transfer plays a crucial role in the solar collector, power plants, thermal informatics, cooling of electronic equipment, refrigeration and freezing of foods, technologies for produc- ing textiles, buildings and bridges, among other things. Engineers and scientists must have a strong basic knowledge in mathematical modelling, theoretical analysis, experi- mental investigations, industrial systems and information technology with the ability to quickly solve challenging problems by developing and using new, more powerful computational tools, in conjunction with experiments, to investigate design, paramet- ric study, performance and optimization of real-world thermal systems. In this book entitled ”Heat transfer - Mathematical Modelling, Numerical Methods and Information Technology”, the authors provide a useful treatise on the principal concepts, new trends and advances in technologies, and practical design engineering aspects of heat transfer, pertaining to powerful tools that are modelling, computation- al methodologies, simulation and information technology. These tools have become essential elements in engineering practice for solving problems. The present book con- tains a large number of studies in both fundamental and application approaches with various modern engineering applications. These include ”Inverse, Stabilization and Optimization Problems” (chapters 1 to 4), which focus on modelling, stabilization, identification and shape optimization, with application to biomedical processes, electric arc radiation and heat exchanger sys- tems; ”Numerical Methods and Calculations” (chapters 5 to 12), which concern finite- diff erence, finite-element and finite-volume methods, la ice Boltzmann numerical method, nonstandard finite diff erence methods, boundary element method and fast X Preface multipole method, quadrature scheme and complex geometries, hermitian transfinite element, and numerical simulation with various applications as solidification, hy- personic speed, concert hall, porous media and nanofluids; ”Heat Transfer in Mini/ Micro Systems” (chapters 13 to 20) which cover miniscale and microscale processes with various applications such as fluidized beds reactors, flows conveying bubbles and particles, microchannel heat sinks, micro heat exchangers, micro combustors and semi- conductors; ”Energy Transfer and Solid Materials” (chapters 21 to 27) which concern heat transfer in furnaces and enclosures, solid structures, moisture diff usion behav- iour, porous media with various applications such as tribosystems and solar thermal collectors. The editor would like to express his thanks to all the authors for their contributions in diff erent areas of their expertise. Their domain knowledge combined with their enthu- siasm for scientific quality made the creation of this book possible. The editor sincerely hopes that readers will find the present book interesting, valuable and current. Aziz Belmiloudi European University of Bri any (UEB), National Institut of Applied Sciences of Rennes (INSA), Mathematical Research Institute of Rennes (IRMAR), Rennes, France. [...]... d − 1 )( 1 − θ d ) ⎤ ( 60 U ) 3 ⎣ ⎦ { (10 3a) 15 5 } 4 2 ⎡ 3θ d ( 5 − 2θ d ) + Z1 Z2 + 10 θ d + 1 ( 3φ0 + 2θ d − 2 ) ⎤ ⎣ ⎦ (10 3b) 35 where Z1 = ( 1 + bξ ) ( 1 − θ d ) 2 (10 4a) { Z2 = ( 1 − θ d ) ( 1 + bξ ) 5 ( 2φ0 + θ d − 1 )( 5 − 6φ0 − 2θ d ) + 5φ0 ( 4φ0 + 3θ d − 3 ) + 3 ( 1 − θ d ) 2 } (10 4b) 2.6.2 Optimum partially wet pin fin for both length and volume constraints The temperature distribution and. .. − ωa ) h fg } (21a) (21b) By using Eqs (2) and (3), Eq ( 21) is made in normalized form and it can be expressed as follows: 10 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology ⎡ d ⎛ dθ ⎞ ⎤ θ ⎤ for dry domain θ > θ d ⎢ dX ⎜ Y dX ⎟ ⎥ ⎡ ⎝ ⎠⎥ ⎢ ⎥ ⎢ = ⎥ ⎢ d ⎛ dφ ⎞ ⎥ ⎢ ⎢ ⎜Y ⎟ ⎥ ⎢( 1 + bξ )φ ⎥ for wet domain θ ≤ θ d ⎣ ⎦ ⎣ dX ⎝ dX ⎠ ⎦ (22a) (22b) The heat transfer through... Fin Profile under Dry and Wet Surface Conditions ( ) θ = 1 − 1 + θp X L (15 ) and ( 1 + bξ ) Y= 2 ( L − X )2 (16 ) The optimum fin length Lopt can be obtained from Eqs (9) and (16 ) The maximum heat transfer rate through the fin can be written by the design variables as follows: 13 ⎡ ⎤ ⎡ Lopt ⎤ ⎢ {6 U ( 1 + bξ )} ⎥ ⎢ ⎥=⎢ 1 3⎥ 2 ⎢Qopt ⎥ ⎢φ0 3U ( 1 + bξ ) 4 ⎣ ⎦ ⎥ ⎣ ⎦ { (17 a) } (17 b) 2 .1. 2 Optimum longitudinal... and (10 0) for the partially wet conditions as follows: φ = φ0 Yb Y ⎡ 2φdY 2 + (φd − θ d ) Y02 + θ dYt2 ⎤ , Y0 ≤ Y ≤ Yb ⎢ 2 2 2⎥ ⎢ 2φdYb + (φd − θ d ) Y0 + θ dYt ⎥ ⎣ ⎦ ( θ = θ d Y0 Yt 2Y 1 + 2Y ) (Y t 2 ) (10 5a) + 2Y0 2 , Y0 ≤ Y ≤ Yt (10 5b) ( 1 + bξ ) ( L0 − X ) (10 6a) and (2 Y Y ∫ Y0 (Y 2 − Y0 2 2 ) − Y0 2 dY ) (Y 0 2 ) + 2Y 2 Y =2 24 Heat Transfer - Mathematical Modelling, Numerical Methods and Information. .. (82) and (86), temperature distribution and fin profile for the optimum fin can be written as ( ) θ = 1 − 1 + θp X L (87a) and Y= ( 1 + bξ ) ( L − X )2 (87b) 2 The optimum fin length and heat transfer rate are determined as follows: { { } 15 ⎡ ⎤ 2 20 U ( 1 + bξ ) ⎥ ⎡ Lopt ⎤ ⎢ ⎥ ⎢ ⎥=⎢ 15 ⎥ ⎢Qmax ⎥ ⎢φ 12 5U 3 1 + bξ 4 8 ⎣ ⎦ ( ) 0 ⎢ ⎥ ⎣ ⎦ } (88a) (88b) 2.5.2 Optimum fully wet pin fin for both length and. .. volume for a partially wet pin fin is written in an integral form as U= V (h k) π 3 = L0 ∫ X =0 Y 2 dX + L ∫ Y 2 dX (95) X = L0 For the application of variational method, a functional F is constructed from heat transfer rate and fin volume expressions: 22 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology F = Q − λU = + L0 1 ∫ φ0 X = 0 2 Y ⎡Y ( dφ dX ) + 2 ( 1 + bξ )φ... X=L0 and X=L, δX is nonzero; thus, the location for both dry and wet surfaces coexist and the fin tip satisfies are the optimality conditions: ⎡ Y ( dθ dX )2 + θ 2 − λθ φ Y φ ⎤ ⎡0 ⎤ d 0 d ⎢ ⎥ = ⎢ ⎥ at X = L0 ⎢Y ( dφ dX )2 + ( 1 + bξ ) φ 2 − λφ Y ⎥ ⎣0 ⎦ 0 ⎦ ⎣ (32a) (32b) and Y ( dθ dX ) + θ 2 − λθ dφ0Y φd = 0 2 at X = L (33) 12 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology. .. where dry and wet sections live together L0 = L ( 1 − θ d ) (36) Here L is not a constraint L can be obtained from Eqs (27), (35) and (36) The optimum length and the maximum heat transfer rate through a fin can be written as Lopt = ( 6 U )1 3 ⎡θ 2 ( 3 − 2θ ) + ( 1 + bξ ) ( 1 − θ )2 ( 3φ + 2θ − 2 ) ⎤ d d d 0 ⎢ d ⎥ ⎣ ⎦ (37a) 13 and 2 Qopt = 2 ⎡θ d + ( 1 + bξ ) ( 2φ0 + θ d − 1 )( 1 − θ d ) ⎤ ( 6 U ) 1 ⎣ ⎦... or both fin volume and length The analysis was formulated for the dry, partially and fully wet surface conditions For the analytical solution of a wet fin equation, a relationship between humidity ratio and temperature of the saturation air is necessary and it is taken a linear variation The influence of 6 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology wet fin... optimum actual heat transfer rate which can be determined from the expression give below: ){ 1 − ( 1 − θd )2 } − 2Lopt { 1 − ( 1 − θd )3} 2 + ( 1 − θ d ) ( 1 + bξ ) L2 { 6 Riφ0 + 3 (φ0Lopt − Ri ) ( 1 − θ d ) − 2 Lopt ( 1 − θ d ) } ⎤ opt ⎥ ⎦ Qopt = Lopt ⎡ 6 R θ + 3 Lopt − Ri ⎣ i d 6 ⎢ ( ( 71) 2.4.2 Optimum annular fin for both length and volume constraints The temperature distribution and fin profile . HEAT TRANSFER ͳ MATHEMATICAL MODELLING, NUMERICAL METHODS AND INFORMATION TECHNOLOGY Edited by Aziz Belmiloudi Heat Transfer - Mathematical Modelling, Numerical Methods and Information. Sadeghi Chapter 9 Chapter 10 Chapter 11 Chapter 12 Part 3 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Contents VII Energy Transfer and Solid Materials. Taler Radiative Heat Transfer and Effective Transport Coefficients 10 1 Thomas Christen, Frank Kassubek, and Rudolf Gati Numerical Methods and Calculations 12 7 Finite Volume Method Analysis of Heat Transfer