Chi Mei Jiemin Zhou Xiaoqi Peng Naijun Zhou Ping Zhou
Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering
Chi Mei Jiemin Zhou Xiaoqi Peng Naijun Zhou Ping Zhou
Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering With 132 figures
Authors Prof. Chi Mei School of Energy Science and Engineering Central South University, 410083, China E-mail:
[email protected] Prof. Jiemin Zhou School of Energy Science and Engineering Central South University, 410083, China E-mail:
[email protected] Prof. Xiaoqi Peng School of Energy Science and Engineering Central South University, 410083, China E-mail:
[email protected] Prof. Naijun Zhou School of Energy Science and Engineering Central South University, 410083, China E-mail:
[email protected] Prof. Ping Zhou School of Energy Science and Engineering Central South University, 410083, China E-mail:
[email protected] Based on an original Chinese edition:
lj᳝㡆ފ䞥♝づӓⳳϢӬ࣪NJ (Youse Yejin Luyao Fangzhen Yu Youhua), Metallurgical Industry Press, 2001.
ISBN 978-7-5024-4636-9 Metallurgical Industry Press, Beijing ISBN 978-3-642-00247-2 e-ISBN 978-3-642-00248-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009920461 ӄ Metallurgical Industry Press, Beijing and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Cover design: Frido Steinen-Broo, Estudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science+Business Media(www.springer.com)
Authors Prof. Chi Mei School of Energy Science and Engineering, Central South University, 410083, China E-mail:
[email protected] Based on an original Chinese edition: lj᳝㡆ފ䞥♝づӓⳳϢӬ࣪NJ(Youse Yejin Luyao Fangzhen Yu Youhua), Metallurgical Industry Press, 2001.
к⠜㓪Ⳃ˄$*1˅᭄ ᳝㡆ފ䞥♝づӓⳳϢӬ࣪˖㣅᭛ˋṙ⚑ㄝ㨫.ü࣫Ҁ˖ފ䞥ᎹϮߎ⠜⼒ˈ2010 ˄ފ䞥ডᑨᎹᄺϯк˅ ,6%1 đ᳝Ă ĒṙĂ ēķ᳝㡆ފ䞥♝ü䅵ㅫᴎӓⳳü㣅᭛ĸ᳝㡆 ފ䞥♝ü᳔Շ࣪ü㣅᭛ Ĕ7) Ё⠜ᴀк佚 &,3 ᭄Ḍᄫ ো ISBN 978-7-5024-4636-9 ISBN 978-3-642-00247-2 e-ISBN 978-3-540-00248-9
Metallurgical Industry Press, Beijing Springer Heidelberg Dordrecht London New York Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2009920461 ӄ Metallurgical Industry Press, Beijing and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, Estudio Calamar, Spain Printed on acid-free paper ܗ Springer is part of Springer Science+Business Mediaࠀwww.springer.comࠁ ҙ䰤Ё䰚ഄऎ䫔ଂ
Preface
Due to the tremendous variety of nonferrous metals and their processes of extraction, the furnaces and kilns used for nonferrous metallurgical engineering (FKNME) vary largely in terms of structure, heating mechanism and functionality. The incomplete statistics show that currently there are over one hundred types of FKNME around the world. Despite this wide variety, however, these FKNME share a few characteristics in common: first of all, most FKNME are heavily energy-consuming, with low energy utilization effectiveness usually ranging from 15% to 50%. The energy needed to extract nonferrous metals is approximated 2.5 to 25 times that for ferrous metals. China is facing an even bigger challenge in this area. The mean energy consumption rates in China are much higher than that of the most advanced indices in the world. Secondly, FKNME usually generate more toxic emissions such as sulfur dioxide, fluoride, chloride, arsenide,etc. Thirdly, the performance of the FKNME is often influenced by many factors, the effects of which are usually non-linear and considerable hysteresis can be found. These difficulties account for the relatively lower process controllability and lower automatization level of the FKNME. It is clear, from the three common characteristics described above, that the FKNME practices are challenging for the industry and therefore deserve more strenuous investigation. For the purpose of effectively upgrading FKNME technologies and improving performance, it is imperative that the following issues be addressed and resolved. Firstly, the output should be maximized by improving the efficiencies of both thermal and production processes. Secondly, the quality control of the production should be more stringent so as to minimize contaminations in the products and the losses of the useful elements. Thirdly, a longer service life of the FKNME can be achieved by reducing the consumption of the refractory and other construction materials. The fourth and the fifth issues are respectively the reduction of the energy consumption and the pollution emissions. The last two issues are highly correlated; reducing energy consumption normally leads to reduction of emissions (such as SOx, CO2, NOx, CO and soot). As a matter
VI Preface of fact, energy consumption can be similarly reduced by augmenting output rate, improving production quality and extending service life. These five issues, therefore, are linked to each other, and may be categorized as the three “highs” (output rate, quality and service life) and two “lows” (energy consumption and pollution emission). Essentially, achieving the objectives of technological upgrading and innovation for the FKNME are equivalent to achieving the overall systematic optimization among the “three highs and two lows”. Back to the 1950s, G. L. Giomidovskij, a researcher from the former Soviet Union, conducted a series of primitive but quantitative investigations into the fluid flows, fuel combustion, heat transfer and mass transport as well as the physico-chemical reactions in a number of most frequently used FKNME. His work has had a far-reaching impact on the researches on the FKNME. However, due to the unavoidably limited research tools available to himücomputation facilities, in particular ü the results of his work, at best, provided general information. It was not until the 1970s that the investigations of the FKNME evolved from being limited within macro-phenomena and lumped, averaged information to exploring the micro-mechanism and obtaining fields information. Such change was mainly a result of extensive and rapid development of the computational fluid dynamics (CFD), as well as heat transfer and combustion techniques, thanks to the unprecedented development of the modern information and computer technologies. As early as the 1980s, the author of this book began applying numerical simulation techniques to investigate the aluminum reduction cells that are among the most widely and frequently used FKNME. The optimized cell lining structures under different operation conditions and system setups were identified by carrying out numerical experiments, i.e., simulations. In the meantime, the research group led by the author used the same methodology to carry out a series of investigations, such as the optimization of the inner wall profiles of the resistance furnaces, and the temperature field prediction and the optimization of the soderberg electrode in the electro-thermal ore-smelting furnaces. The outcomes of these investigations have been proven to be much more effective and accurate than what could be achieved by using the “traditional” research methodology. As the computation capacity has been continuously improved, the research interests of the group have been extended to the investigations of electric furnaces, flame furnaces, muffle furnaces, bath smelting furnaces and boilers. The research scope has also broadened from single-process simulations and single-objective optimizations to multi-process coupling simulations and multi-objective optimizations. Besides mathematical modeling, artificial intelligence modeling has also been adopted to enable more powerful simulations. Thanks to this progress, the research group has been able to develop various tools for industrial applications. These tools range from the CAD packages for FKNME optimization
Preface VII
and decision-making support systems for operation optimization to the integrated FKNME operational management systems featuring unified platforms for monitoring, controlling and managing. Throughout decades of investigations, a new research methodology for FKNME has gradually taken shape and been consistently used in recent years in the research group. This methodology, called the “hologram simulation”, requires at first building up a mathematical or artificial intelligence model for the furnace or kiln concerned. Based on this model, a computer code can be developed so that comprehensive and detailed simulation can be performed for the furnace or kiln. Details on the hologram simulation will be covered in Chapter 3. With the help of this hologram simulation tool, multi-objective optimization of the furnace or kiln can be accomplished by systematically carrying out numerical experiments with or without human intervention. Many people have been involved in work reported in the book. The authors are much indebted to their colleagues and students who participated in the research activities. We would also thank colleagues and friends who contributed to the contents of this book with their ideas and suggestions. Moreover, our deep gratitude is given to Dr. Zhuo Chen for her great efforts in editing the whole book; and special thanks are given to Mr.&Mrs. Ames in Sheffield, U.K., Dr. Siow Yeow at Purdue University, U.S.A. and Prof. Chengping Zhang at Central South University for their careful proofreading of the manuscripts. Without all their work and help, this book would not have been accomplished. It is our sincere hope that this book would serve as a bridge, helping to exchange academic ideas among the FKNME fellow researchers and developers around the world. We hope the work we have done is useful for colleagues in relative fields. Finally, we wish further development and success in the FKNME research, and we should much welcome any comments on the book that readers may care to send. Mei Chi May 2009
Contents
1 Introduction......................................................................................................1 1. 1 Classification of the Furnaces and Kilns for Nonferrous Metallurgical Engineering (FKNME)........................................................1 1. 2 The Thermophysical Processes and Thermal Systems of the FKNME.....2 1. 3 A Review of the Methodologies for Designs and Investigations of FKNME......................................................................................................4 1. 3. 1 Methodologies for design and investigation of FKNME ..................4 1. 3. 2 The characteristics of the MHSO method..........................................5 1. 4 Models and Modeling for the FKNME......................................................7 1. 4. 1 Models for the modern FKNME........................................................7 1. 4. 2 The modeling process ........................................................................7 References............................................................................................................9 2 Modeling of the Thermophysical Processes in FKNME ............................ 11 2. 1 Modeling of the Fluid Flow in the FKNME ............................................ 11 2. 1. 1 Introduction...................................................................................... 11 2. 1. 2 The Reynolds-averaging and the Favre-averaging methods ...........13 2. 1. 3 Turbulence models...........................................................................15 2. 1. 4 Low Reynolds number k-ε models...................................................21 2. 1. 5 Re-Normalization Group (RNG) k-ε models ...................................25 2. 1. 6 Reynolds stresses model(RSM) .......................................................26 2. 2 The Modeling of the Heat Transfer in FKNME ......................................27 2. 2. 1 Characteristics of heat transfer inside furnaces ...............................27 2. 2. 2 Zone method ....................................................................................29 2. 2. 3 Monte Carlo method ........................................................................33 2. 2. 4 Discrete transfer radiation model.....................................................35 2. 3 The Simulation of Combustion and Concentration Field ........................38 2. 3. 1 Basic equations of fluid dynamics including chemical reactions....38 2. 3. 2 Gaseous combustion models............................................................42
X Contents 2. 3. 3 Droplet and particle combustion models .........................................48 2. 3. 4 NOx models ......................................................................................54 2. 4 Simulation of Magnetic Field ..................................................................60 2. 4. 1 Physical models ...............................................................................60 2. 4. 2 Mathematical model of current field ...............................................61 2. 4. 3 Mathematical models of magnetic field in conductive elements.....62 2. 4. 4 Magnetic field models of ferromagnetic elements ..........................66 2. 4. 5 Three-dimensional mathematical model of magnetic field .............69 2. 5 Simulation on Melt Flow and Velocity Distribution in Smelting Furnaces....................................................................................69 2. 5. 1 Mathematical model for the melt flow in smelting furnace ............70 2. 5. 2 Electromagnetic flow .......................................................................71 2. 5. 3 The melt motion resulting from jet-flow .........................................75 References..........................................................................................................80 3 Hologram Simulation of the FKNME ..........................................................87 3. 1 Concept and Characteristics of Hologram Simulation ............................87 3. 2 Mathematical Models of Hologram Simulation ......................................89 3. 3 Applying Hologram Simulation to Multi-field Coupling........................92 3. 3. 1 Classification of multi-field coupling ..............................................92 3. 3. 2 An example of intra-phase three-field coupling ..............................93 3. 3. 3 An example of four-field coupling ..................................................94 3. 4 Solutions of Hologram Simulation Models .............................................97 References..........................................................................................................98 4 Thermal Engineering Processes Simulation Based on Artificial Intelligence....................................................................................101 4. 1 Characteristics of Thermal Engineering Processes in Nonferrous Metallurgical Furnaces .......................................................101 4. 2 Introduction to Artificial Intelligence Methods .....................................102 4. 2. 1 Expert system.................................................................................103 4. 2. 2 Fuzzy simulation............................................................................104 4. 2. 3 Artificial neural network................................................................106 4. 3 Modeling Based on Intelligent Fuzzy Analysis .....................................107 4. 3. 1 Intelligent fuzzy self-adaptive modeling of multi-variable system ............................................................................................108 4. 3. 2 Example: fuzzy adaptive decision-making model for nickel matte smelting process in submerged arc furnace ......................... 111 4. 4 Modeling Based on Fuzzy Neural Network Analysis............................ 116 4. 4. 1 Fuzzy neural network adaptive modeling methods of multi-variable system .................................................................... 117
Contents XI
4. 4. 2 Example: fuzzy neural network adaptive decision-making model for production process in slag cleaning furnace .................120 References........................................................................................................123 5 Hologram Simulation of Aluminum Reduction Cells...............................127 5. 1 Introduction............................................................................................127 5. 2 Computation and Analysis of the Electric Field and Magnetic Field....131 5. 2. 1 Computation model of electric current in the bus bar ...................132 5. 2. 2 Computational model of electric current in the anode...................133 5. 2. 3 Computation and analysis of electric field in the melt ..................134 5. 2. 4 Computation and analysis of electric field in the cathode.............138 5. 2. 5 Computation and analysis of the magnetic field............................140 5. 3 Computation and Analysis of the Melt Flow Field................................146 5. 3. 1 Electromagnetic force in the melt ..................................................147 5. 3. 2 Analysis of the molten aluminum movement ................................148 5. 3. 3 Analysis of the electrolyte movement ...........................................149 5. 3. 4 Computation of the melt velocity field ..........................................150 5. 4 Analysis of Thermal Field in Aluminum Reduction Cells ....................152 5. 4. 1 Control equations and boundary conditions ..................................153 5. 4. 2 Calculation methods.......................................................................156 5. 5 Dynamic Simulation for Aluminum Reduction Cells............................158 5. 5. 1 Factors influencing operation conditions and principle of the dynamic simulation .............................................................159 5. 5. 2 Models and algorithm ....................................................................160 5. 5. 3 Technical scheme of the dynamic simulation and function of the software system ...................................................................161 5. 6 Model of Current Efficiency of Aluminum Reduction Cells.................163 5. 6. 1 Factors influencing current efficiency and its measurements........164 5. 6. 2 Models of the current efficiency ....................................................166 References........................................................................................................169 6 Simulation and Optimization of Electric Smelting Furnace ....................175 6. 1 Introduction............................................................................................175 6. 2 Sintering Process Model of Self-baking Electrode in Electric Smelting Furnace......................................................................176 6. 2. 1 Electric and thermal analytical model of the electrode..................178 6. 2. 2 Simulation software .......................................................................182 6. 2. 3 Analysis of the computational result and the baking process........183 6. 2. 4 Optimization of self-baking electrode configuration and operation regime ............................................................................190 6. 3 Modeling of Bath Flow in Electric Smelting Furnace...........................192
XII Contents 6. 3. 1 Mathematical model for velocity field of bath ..............................193 6. 3. 2 The forces acting on molten slag ...................................................194 6. 3. 3 Solution algorithms and characters................................................196 6. 4 Heat Transfer in the Molten Pool and Temperature Field Model of the Electric Smelting Furnace ................................................198 6. 4. 1 Mathematical model of the temperature field in the molten pool ....................................................................................199 6. 4. 2 Simulation software .......................................................................203 6. 4. 3 Calculation results and verification ...............................................203 6. 4. 4 Evaluation and optimization of the furnace design and operation .................................................................................208 References........................................................................................................210 7 Coupling Simulation of Four-field in Flame Furnace...............................213 7. 1 Introduction............................................................................................213 7. 2 Simulation and Optimization of Combustion Chamber of Tower-Type Zinc Distillation Furnace...................................................215 7. 2. 1 Physical model ...............................................................................216 7. 2. 2 Mathematical model.......................................................................217 7. 2. 3 Boundary conditions ......................................................................217 7. 2. 4 Simulation of the combustion chamber prior to structure optimization ....................................................................218 7. 2. 5 Structure simulation and optimization of combustion chamber..........................................................................................220 7. 3 Four-field Coupling Simulation and Intensification of Smelting in Reaction Shaft of Flash Furnace ........................................221 7. 3. 1 Mechanism of flash smelting process—particle fluctuating collision model ............................................................223 7. 3. 2 Physical model ...............................................................................224 7. 3. 3 Mathematical model—coupling computation of particle and gas phases ..................................................................225 7. 3. 4 Simulation results and discussion ..................................................227 7. 3. 5 Enhancement of smelting intensity in flash furnace......................229 References........................................................................................................232 8 Modeling of Dilute and Dense Phase in Generalized Fluidization.......... 235 8. 1 Introduction............................................................................................235 8. 2 Particle Size Distribution Models ..........................................................238 8. 2. 1 Normal distribution model.............................................................239 8. 2. 2 Logarithmic probability distribution model...................................240 8. 2. 3 Weibull probability distribution function ......................................241
Contents XIII
8. 2. 4 R-R distribution function (Rosin-Rammler distribution) ..............241 8. 2. 5 Nukiyawa-Tanasawa distribution function ....................................242 8. 3 Dilute Phase Models ..............................................................................244 8. 3. 1 Non-slip model...............................................................................245 8. 3. 2 Small slip model ............................................................................247 8. 3. 3 Multi-fluid model (or two-fluid model).........................................248 8. 3. 4 Particle group trajectory model......................................................251 8. 3. 5 Solution of the particle group trajectory model .............................256 8. 4 Mathematical Models for Dense Phase .................................................257 8. 4. 1 Two-phase simple bubble model ...................................................258 8. 4. 2 Bubbling bed model.......................................................................259 8. 4. 3 Bubble assemblage model (BAM).................................................261 8. 4. 4 Bubble assemblage model for gas-solid reactions.........................265 8. 4. 5 Solid reaction rate model in dense phase.......................................267 References........................................................................................................272 9 Multiple Modeling of the Single-ended Radiant Tubes ........................... 275 9. 1 Introduction............................................................................................275 9. 1. 1 The SER tubes and the investigation of SER tubes.......................276 9. 1. 2 The overall modeling strategy .......................................................278 9. 2 3D Cold State Simulation of the SER Tube...........................................279 9. 3 2D Modeling of the SER Tube ..............................................................283 9. 3. 1 Selecting the turbulence model......................................................283 9. 3. 2 Selecting the combustion model ....................................................286 9. 3. 3 Results and analysis of the 2D simulation .....................................289 9. 4 One-dimensional Modeling of the SER Tube........................................291 References........................................................................................................295 10 Multi-objective Systematic Optimization of FKNME............................ 297 10. 1 Introduction..........................................................................................297 10. 1. 1 A historic review ..........................................................................297 10. 1. 2 The three principles for the FKNME systematic optimization .................................................................................298 10. 2 Objectives of the FKNME Systematic Optimization ..........................299 10. 2. 1 Unit output functions ...................................................................300 10. 2. 2 Quality control functions .............................................................305 10. 2. 3 Control function of service lifetime.............................................306 10. 2. 4 Functions of energy consumption................................................308 10. 2. 5 Control functions of air pollution emissions ...............................309 10. 3 The General Methods of the Multi-purpose Synthetic Optimization ........................................................................................309
XIV Contents 10. 3. 1 Optimization methods of artificial intelligence ...........................309 10. 3. 2 Consistent target approach...........................................................312 10. 3. 3 The main target approach.............................................................314 10. 3. 4 The coordination curve approach ................................................315 10. 3. 5 The partition layer solving approach ...........................................315 10. 3. 6 Fuzzy optimization of the multi targets .......................................316 10. 4 Technical Carriers of Furnace Integral Optimization ..........................318 10. 4. 1 Optimum design CAD .................................................................319 10. 4. 2 Intelligent decision support system for furnace operation optimization.................................................................320 10. 4. 3 Online optimization system .........................................................327 10. 4. 4 Integrated system for monitoring, control and management .......330 References........................................................................................................334 Index .................................................................................................................... 337
Contributors
Chi Mei, Professor Central South University, China E-mail:
[email protected] Jiemin Zhou, PhD and Professor Central South University, China E-mail:
[email protected] Xiaoqi Peng, PhD and Professor Central South University, China E-mail:
[email protected] Naijun Zhou, PhD and Professor Central South University, China E-mail:
[email protected] Ping Zhou, PhD and Porfessor Central South University, China E-mail:
[email protected] Zhuo Chen, PhD and Associate professor Central South University, China E-mail:
[email protected] Feng Mei, PhD Central South University, China E-mail:
[email protected] XVI Contributors
Shaoduan Ou, PhD Central South Univeristy, China E-mail:
[email protected] Hongrong Chen, MD Central South University, China E-mail:
[email protected] Yanpo Song, MD Central South University, China E-mail:
[email protected] Junfeng Yao, PhD and Associate professor Xiamen University, China E-mail:
[email protected] Kai Xie, PhD and Associate professor Central South University, China E-mail:
[email protected] Hui Cai, MD Changsha Engineering and Research Institute of Nonferrous Metallurgy, China E-mail:
[email protected] List of Contributors Chapter 1 Chi Mei
Hongrong Chen
Chapter 2 Ping Zhou
Professor Central South University Email:
[email protected] Central South University Email:
[email protected] PhD and Porfessor Central South University Email:
[email protected] Feng Mei
PhD Central South University Email:
[email protected] Hui Cai
Changsha Engineering and Research Institute of Nonferrous Metallurgy Email:
[email protected] Chapter 3 Chi Mei Zhuo Chen
Chapter 4 Xiaoqi Peng
Yanpo Song
Chapter 5 Naijun Zhou
PhD and Associate professor Central South University Email:
[email protected] PhD and Professor Central South University Email:
[email protected] Central South University Email:
[email protected] PhD and Professor Central South University
Email:
[email protected] Chapter 6 Jiemin Zhou
PhD and Professor Central South University Email:
[email protected] Ping Zhou Chapter 7 Ping Zhou Zhuo Chen Kai Xie
Chapter 8 Chi Mei Shaoduan Ou
PhD and Associate professor Central South University Email:
[email protected] Central South Univeristy Email:
[email protected] Chapter 9 Feng Mei Chapter 10 Xiaoqi Peng Yanpo Song Zhuo Chen Junfeng Yao
Xiamen University Email:
[email protected] Introduction $IJ.FJBOE)POHSPOH$IFO
The furnaces and kilns for nonferrous metallurgical engineering (FKNME) are a big family. There were few reports on systematic research of FKNME, especially the accurate and quantitative analysis. In this chapter ideas and analysis methods are introduced for comprehensive and in-depth understanding of the FKNME system.
1.1
Classification of the Furnaces and Kilns for Nonferrous Metallurgical Engineering (FKNME)
In most industrial applications and research activities, metals are generally categorized into two groups: the ferrous metals and the nonferrous metals. Chromium, manganese and iron are ferrous metals while all the rest are the nonferrous metals. The nonferrous metals can be further classified by their applications as follows (Zhao, 1992): a) Heavy metals: copper, lead, zinc, tin. b) Light metals: aluminum, magnesium, titanium. c) Precious metals: gold, silver, platinum group. d) Metals in the iron and steel industry: nickel, chromium, cobalt, molybdenum, tungsten, vanadium, niobium, tantalum. e) Metals in the electronic industry: cadmium, gallium, germanium, mercury, indium, rhenium, selenium, tellurium, manganese. f) Metals in the nuclear industry: uranium, zirconium, hafnium, cesium, rubidium, beryllium, rare earth. g) Metals in the chemical industry: bismuth, antimony, lithium. The nonferrous metals family is much larger and plays an important role in industrial activities and people’s daily life. The furnaces and kilns built for extracting and processing nonferrous metals are generally called FKNME. They vary a lot in structure and function. Currently up to a hundred different FKNME
Chi Mei and Hongrong Chen can be named. For the convenience of investigation, the FKNME, are classified as shown in Table 1.1, by their intended functions, heating sources, heating methods, working mechanisms, furnace structures and thermal performances. Table 1.1 A classification of the FKNME (Mei, 2000) Criterion
Examples
Intended functions
Drying furnace (kiln), baking furnace, calciner, heating furnace, chlorinator, smelter, melting furnace, converting furnace, refining furnace, heat treatment furnace, reduction furnace, fuming furnace, volatilizing furnace, distillation furnace, diffusion furnace, anode furnace, cathode furnace
Heating sources
Autothermic furnace, fuel furnace, electrical furnace
Heating methods
Flam furnace, downdraft furnace (kiln), muffle, resistance furnace, inducting furnace, arc furnace, ore smelting electric arc furnace, electron bombard furnace, ion furnace, thermal plasma furnace
Working mechanisms
Fluidized furnace, cyclone furnace, blast furnace, flash furnace, top blown converter, side blown converter, bottom blown furnace, bath smelter, hot air recycling furnace, air cushion furnace, suspension roasting furnace
Furnace structures
Rotary kiln, reverberatory furnace, multiple hearth furnace, vertical well furnace, crucible furnace, chamber furnace, vertical distillation kettle, tower redistillation furnace, carbon tube furnace, tungstate rod furnace, molybdenum wire furnace, bell furnace, walking beam furnace, carbon particle furnace, muffle
Thermal performances
Simple-structured stoves (such as combustion chamber, air heating furnace) Heater (heating furnace, heat treatment furnace, melting furnace, downdraft kiln, tunnel kiln, etc.) High temperature reactor (converting furnace, bath smelter, molten salt electrolysis cell)
1.2
The Thermophysical Processes and Thermal Systems of the FKNME
Despite the large variety, most of the thermophysical processes taking place in the kilns and furnaces can be fundamentally disassembled into the following basic processes: a) Fuel combustion or thermo-electrical conversion. b) Confined gas flow, gas-particle multi-phase flow or general fluidized bed flow. c) Mechanical or electromagnetic movements of materials (loads). d) Heat and mass transfer. e) Physical-chemical reactions under high temperature.
1 Introduction
Note that these processes are: on one hand, the purpose of the furnaces (kilns) are designed for; on the other hand, the joint effects of the other processes listed in clusters a~d. The thermal system of the furnaces and kilns generally consists of the main body and the supporting parts as listed in Table 1.2. Table 1.2 The thermal system for the furnaces and kilns Item
Main body
Components
Description
The furnace foundation
Concrete
The hearth and refractory wall
Refractory brick and pounding materials
The insulation
Insulation materials
The holes and doors The shell reinforcement
and
external
Steel structure
Running machinery
Heat supply by fuel-firing or electrical heating
Fuel preparing and combustion facilities
Ventilation and exhausting
Voltage and frequency changing element, air heating device, vacuum system
Materials charging and discharging Supporting parts Cooling system
Water-cooling equipment
or
air-cooling
Heat recuperation
Heat exchanger, storage, waste heat steam system
Monitoring system
Measuring instrument control system
and
Even though separately described as the main body and the supporting parts, the thermal system should be considered inseparable as for delivering the performance. A malfunction of any element may result in failure, if not serious damage, of the whole system. Therefore, the developer of a furnace or kiln should make sure the main body and the supporting parts are integrately designed. A researcher should also keep the whole system in mind when investigating any individual part. This “integrate thinking and systematic approach” is actually a key rule applicable to the practice of investigations and design of the FKNME system.
Chi Mei and Hongrong Chen
1.3
A Review of the Methodologies for Designs and Investigations of FKNME
In past years,methodologies to design and investigate the FKNME were greatly progressed.With development of computer technology,mathematical models and numerical computations have become more and more important to research of the FKNME.
1.3.1
Methodologies for design and investigation of FKNME
Methodologies for design and investigation of FKNME can be put into following categories. 1.3.1.1 Empiric analysisüanalog estimationüapproximate quantification Even though this method is considered very “primitive” in collecting and developing knowledge about the furnaces and kilns, it is still quite efficient in some specific cases as well as for the SME’s (small-sized and middle-sized enterprises) in less sizable furnaces or kilns, because it is easy to understand, reliable and easy to use. Nevertheless, this methodology is quite limited in adapting modern research techniques and tools for better effectiveness (Mei, 1996). 1.3.1.2 Simulating experimentüsimilarity analysisüempirical formulation This method has been developed since the mid 19th century and is still being widely used nowadays. The theoretical foundation of this method is the Similarity Theory (the positive law, the reverse law and the ± law), based on which the similitude experiments can be designed and carried out. The results of these experiments are usually formulated into dimensionless numbers and empirical equations (Mei, 1996). The simulating experiments can be designed in two ways. One is called the similar model experiment, which is used when the model and the real application share some identical properties, such as the wing profile and high-speed train contour experiments in the wind tunnels. The other is called the analog model experiment (or heterogeneous model experiment), which means using a situation that is more easily observed or measured to analogize another one that is difficult to observe or measure. An example of it is using electrical phenomenon or hydraulic phenomenon to simulate heat transfer. This method can be very useful in optimizations of the furnace (kiln) structure and internal geometry as well as investigations of the fluid flow and heat (mass) transfer in the furnace (kiln). It becomes inappropriate, however, when the geometry becomes very complicated or when temperature, pressure or flow velocity goes too high or when toxic substances are presented.
1 Introduction
1.3.1.3 Mathematic modelingünumerical analysis Starting from the end of the 1960s and beginning of the 1970s, mathematic models and computer program packages have been developed in the western countries to simulate mono-process representing the magnetic field, temperature field or cell shell stresses field of the aluminium electrolytic cells. These packages have been soon used in assisting the engineering design of the cells. Similar works started in China in the 1980s with numerical studies and optimization of the resistance furnace and the inner liner structure of the periodic working furnaces (kilns). In recent decades, more and more FKNME mathematic models have been developed with higher reliability and better applicability as a result of the remarkable development of numerical techniques for fluid dynamics, heat transfer and combustion. 1.3.1.4 Mathematic modelühologram simulationüsystematic optimization Based on the accomplishment of the mono-process orientated mathematic modeling researches, engineers and scientific researchers started to develop more sophisticated mathematic models for optimizations of the FKNME structures and their operational conditions. However, focus changed from modeling mono-process to modeling multiple processes, and approaches also changed from ignoring or simplifying the coupling effects of the concurrent processes to modeling these effects with extensive details and high precision. This kind of simulations is called the “hologram simulations”, because they enable highly truthful and comprehensive reflection of the processes in the furnaces (kilns). The course of using hologram simulation techniques to find automatically the operational conditions for the optimized or expected furnace (kiln) performance is called the “hologram simulation experiment”. The collection of numerical simulation techniques and tools based on hologram simulation experiments targeting multi-objective system optimization is called the “numerical modelü hologram simulationüsystematic optimization” method (MHSO).
1.3.2 The characteristics of the MHSO method The MHSO method has following characteristics (Mei, 1996; Mei et al., 1999): a) The procedure of implementing the MHSO methodology includes maily three steps. Firstly, the studied objects should be understood and measured at macroscopic level, based on which the mathematic modeling and numerical simulation can be done to obtain the vector (scalar) fields information at the microscopic level. Secondly, the determining relationship of the microscopic variables to the macroscopic performances must be revealed and the influences of the building structure and operational conditions to the microscopic variables must be understood. After all these relationships have been well determined, the
Chi Mei and Hongrong Chen optimization of the macro performance can then be carried out by adjusting the microscopic variables by modifying the operational conditions and the structure. b) The research strategy of the MHSO method is first to build up qualitatively the physical models, then to make an approximate quantitative decision by solving equation set or by artificial intellectual techniques. This preliminary quantitative decision is then refined for higher reliability through a series of validating and correcting processes using information obtained from practice. This strategy combines the advantages of both qualitative and quantitative approaches. c) The effectiveness of the MHSO method is obvious. Application of the method will directly improve the performance of furnaces and kilns which is automatically verifiable in practice. Therefore, the MHSO method combines the theory-based investigations with facts and figures obtained in experiments and applications, which guarantees the practicability of this method on the one hand and allows continuous and systematic optimization of the furnaces and kilns on the other hand. Compared to other methods mentioned above, the MHSO method also enables obtaining information that is more comprehensive, reliable and easily understood. However, this method is more demanding regarding mathematical models or artificial intellectual models and requires support of efficient algorithm, advanced computer capacity and reliable online measuring technologies. Table 1.3 compares the classic methodologies and modern methodologies used in scientific and engineering investigations. Table 1.3 Comparison of modern and classic methodologies used in scientific and engineering investigations Comparison point
Traditional methodologies
Modern methodologies
Research perspectives
The dependency of system outputs on system inputs The determination of the “macro structure” to the “macro performance”
The dependency of the macro performance on micro mechanisms Currently more focus on acquiring information at microscopic level
Basis
Empiric observation Statistics Similarity model experiment Approximate analysis
Model style
Input-output “black box”
Information collection
Space-averaged and time-averaged info Momentary info at sampling points
Tools
Classic mathematics methods
Mathematic models based on theories of computational fluid dynamics, heat (mass) transfer, combustion, metallurgical reaction kinetics, and artificial intelligence, validated by experimental and practical data Hybrid “gray model” combining theoretical and empirical knowledge Artificial intelligence model Vector (scalar) fields info (such as fluid flow field,temperature field,concentration field and so on) Dynamic and quasi-dynamic information Computer-based simulation packages Artificial neural network (auxiliary analysis integrated system)
1 Introduction
1.4 Models and Modeling for the FKNME Models and modeling are important to research of the FKNME.Though forms of FKNME models are greatly different,the modeling consideration and progress share something in common.
1.4.1 Models for the modern FKNME From the viewpoint of practical exercises, any equation set, formulation, graph or data table etc. that reflects the interactions and dependencies among the functioning variables of a process can be regarded as the mathematic model of this process. The forms of mathematic models are enormously different due to the large variety of the ways to establish these models and the different characteristics of the mathematical physical equations. There is so far no classification of the existing models. The same model can be named differently depending on the preference of the users or developers. For example, a country, a region or an enterprise can use their own names to label their energy consumption model, pollution model, equipment capacity model, valuable elements recovery model, equipment service life model etc. Scientific researchers can name models by their mathematic patterns as linear models, parabolic models, normal distribution models, sinusoid models, periodic pulse models etc. Models can also be named by their theoretical foundation, such as the statistical models, mechanism analysis (or logic) model, system identification model, mechanism analysis-system identification model, etc. According to how much the processes have been quantitatively understood, they can also be named as “white box” models, “grey box” models and “black box” models. For the engineers and researches in the FKNME field, we propose a more generally applicable classification of the models as shown in Table 1.4 (Mei, 1996; Zuoteng, 1985; Smith, 1982). In practice, the models in different classes can be jointly used for better accuracy and effectiveness.
1.4.2 The modeling process The FKNME modeling process is schematically illustrated in Fig. 1.1. As indicated in Fig. 1.1, the modeling of the FKNME starts from comprehensive and in-depth understanding of the system by observing and measuring its structure, working mechanisms, operation conditions and performance. Based on these understandings, necessary and reasonable
Chi Mei and Hongrong Chen assumptions can be made so that the complex reality can be simplified and modeled by the most representative processes and variables. The reasonably simplified system is then described using mathematical language, so that the governing equation set can be established with conservation equations closed with other equations describing the properties, transfer rates and other constraining relations. The equation set is eventually resolved through a series of numerical processes. For those systems that cannot be easily formulated by mathematical methods, or those involving stochastic process or multiple processes with strong coupling and (or) lag effects, the artificial intellectual methods (such as expert system, fuzzy analysis, artificial neural network etc.) have to be used, together with qualitative and quantitative dynamic models and judge regulation with “symbol” add “searching” system.(Mei and Zhou, 1991a,b,c; Mei et al., 1994a,b; Peng and Mei, 1996) The validity of models must be checked before practical use. This is normally conducted using measurements specifically prepared for model validation or data collected in productions. The reliability of a mathematical model can only be proved after repeated checking-and-correcting processes or by building in self-learning and correcting functions in the model. Table 1.4 Mathematic models classification for FKNME Class Empirical statistical models
Examples
Applicable circumstances
Black box model
Regression equation Curve fitting expressions
For easy and brief calculation, valid specifically within the range where the data are collected
White box model with clearly understood mechanism
Heat conduction in solid medium Velocities of laminar flow in a tube
Infrequently used in engineering practice
Grey box model
Usual turbulence models Heat and mass transfer models in fluids Combustion models
Widely used in engineering practice
Dynamic experience model
Expert systems Fuzzy analysis model Artificial neural network model
Mostly applicable in cases involving unsteady state multiple processes with strong non-linear and (or) strong coupling effects
and
Analytic models with governing equation(s)
Hybrid models (theoretialempirical)
Artificial intelligence models
Description
1 Introduction
Fig. 1.1 General modeling of nonferrous metallugical furnaces and kilns
References Mei Chi (1996) Simulation and optimization of the nonferrous metallurgical furnace (in Chinese). Journal of Chinese Nonferrous Metals, 6(4): 19~28 Mei Chi (2000) Handbook of The Nonferrous Metallurgical Furnace Design (in Chinese). Metallurgical Industry Press, Beijing Mei Chi, Peng Xiaoqi, Zhou Jieming (1994a) An intelligence decision support system (IDSS) on the process of nickel matte smelter. Journal of Central South University of Technology, 1(1): 14~18
Chi Mei and Hongrong Chen Mei Chi, Peng Xiaoqi, Zhou Jieming (1994b) Fuzzy and adaptive control model for process in nickel matte smelting furnace. Tran. of Nonferrous Metals Society of China, 4(3): 9~11 Mei Chi, Ying Zhiyun, Zhou Ping (1999) “Hologram” simulation of the modern furnace (kiln) (in Chinese). Journal of Central South University of Technology, 30(6): 592~596 Peng Xiaoqi, Mei Chi (1996) An intelligent decision support system (IDSS) in the operation process of electric furnace for clearing slag. Journal of Central South University of Technology, 3(2):170~180 Smith J M (1982) Mathematical Model and Numerical Simulation for Engineers and Researchers. Wang Xingyong etc. translation. Nuclear Energy Press, Beijing Zhao Tiancong (1992) Handbook of Nonferrous Metals Extraction (in Chinese). Metallurgical Industry Press, Beijing Zhou Jieming, Mei Chi (1991) A mathematic model of electric smelting furnace used for improving design and operation. Extraction and Processing Division Congress’91, TMS: 331~352 Zhou Jieming, Mei Chi (1991) Computer simulation of electro-thermal field in nickel ore smelting electric arc furnace (in Chinese). Journal of Central South Mining & Metallurgical Institute, 22(1): 46~53 Zhou Jieming, Mei Chi (1991) Mathematic model and computer simulation of Soderberg electrodes in electric smelting furnace. Electro Warm International (B), 10(1): 210~215 Zuoteng Chilong (1985) Mathematic Model (in Japanese). Gong Rongzhang etc. translation. Mechanical industry Press, Beijing
Modeling of the Thermophysical Processes in FKNME Ping Zhou, Feng Mei and Hui Cai
The fluid (including molten mass) flow, heat transfer and combustion processes in the FKNME are collectively called as the FKNME thermal processes. The modeling of these processes is the foundation to simulate the FKNME as well as the key element. In this chapter we discuss the theories and applications of the modeling of the flow filed, temperature field, species concentration field and electro-magnetic field that are usually involved in the FKNME.
2.1 Modeling of the Fluid Flow in the FKNME As one aspect of the thermal processes, fluid flow in the FKNME is usually accompanied with combustion and heat transfer, thus it is quite complicated and difficult for modeling. As a result, different models have then been developed and proposed to describe and investigate the complex process in the FKNME.
2.1.1 Introduction In the FKNME applications large variety of fluids (gases, molten mass, gas-particle mix etc.) and flow patterns have been involved. For the convenience of investigation, we generally classified them per their fluid dynamics characteristics as shown in Table 2.1. Table 2.1 Patterns of fluid flow usually applicable to FKNME Examples Patterns
Pipe flow
Single phase
Two-phase (gas-particles)
Gas transportation pipes Molten mass transportation pipes
Pneumatic transportation pipes
Characteristics of the flow field 1D turbulent flow 1D laminar flow or turbulent flow
Ping Zhou, Feng Mei and Hui Cai Continues Table 2.1 Examples Patterns
Characteristics of the flow field
Single phase
Two-phase (gas-particles)
Simple jet
Straight flame burners
Pulverized coal burners
2D or 3D turbulent flow
Swirling jet
Flat flames burners
Swirling pulverized coal burners
3D turbulent flow
Jet flow
Recirculation flow
Gas-particle two-phase flow
Recirculation zones in the flame furnaces Stirring of the melting pool Electro-magnetic flow in the Al reduction cell
Upper part of the flash smelting furnaces
3D turbulent flow
Tangentially fired burners Dense phase fluidized bed Circulating fluidized bed
One of 3D or 1D turbulent flows
Table 2.1 illustrates that the fluid flows in the FKNME are dominated by complex turbulent flows. In this section we chiefly discuss the incompressible steady-state turbulent flows with Mach number far less than 1. The recirculation flows of the molten mass in the smelters are to be covered in Section 2.5 and the general fluidized beds are to be discussed in Chapter 8. Different from the laminar flows that are usually investigated at the molecular motion level, turbulent flows are mainly investigated at the molecular micro-conglomerations (called eddies) level with main interests on the generation, transportation, breaking up and interactions of these conglomerations. Up to date we have still not fully understood the turbulence phenomena. We still have difficulty to describe them in a perfect mathematic way. From a physical point of view, however, turbulence can be generally viewed as a set of eddies in a large range of sizes. The large eddies are usually situated in the middle of the flows to carry out the major part of the energy. The small ones are usually close to the confined boundaries where energy is dissipated. Larger eddies are strongly anisotropic whereas the smaller ones are more homogeneous. Smaller eddies are usually generated through the break down of larger eddies, in the meantime energy is transferred from larger eddies to the smaller ones. The energy containing in an eddy falls dramatically when the size of the eddy is below certain criterion level, which is measured by the Kolmogorov scale. Our major interest is the turbulence phenomena above the Kolmogorov scale level. The governing equations set of the viscous flows consists of the conservations of mass, momentum and other related scalars, as shown in Eq.2.1~Eq.2.6.
2 Modeling of the Thermophysical Processes in FKNME
The mass conservation (or called continuity equation): ∂ρ ∂ρu i + =0 ∂t ∂xi
where ui is the instantaneous velocity on the i direction. The momentum conservation equation: ∂ρui ∂ρui ∂p ∂tij + uj =− + + ρg i ∂t ∂x j ∂xi ∂x j
(2.1)
(2.2)
where p is the instantaneous static pressure; ρ gi is the gravity on i direction (this term is usually ignored when forced flow predominates); t ij is the viscous stresses tensor, which is defined as: tij = 2 μ 0 sij (2.3) where μ 0 is the molecular viscosity, sij is the strain rate tensor, which is defined as:
1 ⎛⎜ ∂ui ∂u j ⎞⎟ 1 ∂u k δ ij − + 2 ⎜⎝ ∂x j ∂xi ⎟⎠ 3 ∂xk is the so-called Kronecker number: ⎧0, i ≠ j δ ij = ⎨ ⎩1, i = j sij =
where δ ij
The scalar conservation equations are generally written as: ∂pϕ ∂ ∂ + ρu j ϕ = J + Sϕ ∂t ∂x j ∂x j ϕ
(
)
( )
(2.4)
(2.5)
(2.6)
where ϕ is the concerned scalar of any kind; Jφ is the diffusion flux of ϕ on the j direction; Sφ the source term of the scalar ϕ . The form of Eq.2.6 is principally applicable to all conservative scalars. This makes it possible to develop a general problem-solving scheme that first establish conservative equations for a selected number of scalars in the form of Eq.2.6 and then solve the equations set using the same numerical procedure. The difficulty of this scheme is, however, that the source terms of these equations are often difficult to be treated. The computational capacity of modern computers still does not allow us to solve the equations set with instantaneous variables. Even though large eddy simulation (LES) has been increasingly used in the recent years, the modeling approach stays to be the mainstream in most engineering applications. With the modeling approach, the instantaneous form of the equations set has to be transformed into a time-averaged form before it is numerically solved.
2.1.2 The Reynolds-averaging and the Favre-averaging methods The Eq.2.1, Eq.2.2 and Eq.2.6 are called the general Navier-Stokes equations set. This equation set is usually solved by the Reynolds averaging method is normally
Ping Zhou, Feng Mei and Hui Cai
applied. The Reynolds averaging method, which defines an instantaneous variable as a sum of a time-averaged part and a fluctuating part: Φ = Φ + Φ′ (2.7) where Φ is the time-averaged ϕ : 1 Δt Φ= ϕ dt (2.8) Δt 0 Being aware that ϕ ′ =0, we can simplify the production of the two instantaneous quantity into (2.9) ϕ1 • ϕ 2 = ϕ1 • ϕ 2 + ϕ1′ • ϕ 2′
∫
Substituting Eq.2.7 and Eq.2.9 into the equation set Eq.2.1~Eq.2.6 we come down with the Reynolds averaging Navier-Stokes equations set as shown in Eq.2.10~Eq.2.12 by assuming that the higher order fluctuation terms are ignorable and the fluctuation of density is not significantly correlated with the other variables. ∂ρ ∂ ρui + =0 ∂t ∂xi
(
(2.10)
)
∂ ρui ∂ ∂p ∂ + ρui u j = − + (2μ 0 S ij − ρui′u ′j ) ∂t ∂x j ∂xi ∂xi
(
)
(
)
∂ ρϕ ∂ ∂p ∂ + ρϕu j = − + J k − ρu i′u ′j + ρ Sϕ ∂t ∂x j ∂xi ∂x j
(2.11)
(2.12)
Removing the density-correlated terms should not bring in substantial errors so long as the density does not vary remarkably. This is apparently not the case if combustions and (or) chemical reactions involve. Theoretically speaking the Favre averaging (or called weighted averaging), is considered more appropriate. Different from the Reynolds averaging, the Favre averaging includes the density as a weighing factor into the definition of the averaged part of the variables:
ϕ~ =
ρu ~ , φ′′ = φ φ ρ
(2.13)
The Favre averaging enables the elimination of the fluctuation term in the continuity equation without simplification. The convective term turns to be: ∂ ρ u~ ∂x j The Favre averaging method leads to the following mass, momentum and scalar conservation equations set: ∂ ρui ∂ ρ + =0 ∂xi ∂t
(2.14)
2 Modeling of the Thermophysical Processes in FKNME
∂ ∂ ∂p ∂ ( ρ u~i ) + ( ρ u~i u~j ) = − + (2μ 0 Sij − ρui′′u′j′ ) ∂t ∂x j ∂xi ∂x j where Reynolds averaging Sij term: 1 ⎛ ∂u ∂u ⎞ 1 ∂uk S ij = ⎜ i + j ⎟ − δ ij 2 ⎜⎝ ∂x j ∂xi ⎟⎠ 3 ∂xk
∂ ⎛ ∂ρ ~ ∂ ⎛ ~ ~⎞ ϕ+ ⎜ J ϕ − ρϕ ′′u′j′ ⎞⎟ + ρSϕ ⎜ ρ uj ϕ ⎟ = ⎠ ∂t ∂x j ⎝ ⎠ ∂x j ⎝
(2.15)
(2.15a)
(2.16)
the transport flux of scalar ϕ
⎞ ⎟ (2.16a) ⎟ ⎠ Density-related correlation terms are eliminated by the Favre averaging but the Reynolds averaging viscous stresses terms in Eq.2.15 and the viscous scalar flux terms in Eq.2.16 stay. These terms are usually treated by one of the following two means in most engineering practices. The first means is to ignore the viscous stresses effects under the high Reynolds number assumption therefore Eq.2.15a and Eq.2.16a are entirely eliminated. The other means is to assume, in case that viscous stresses are not ignorable, that the results of Favre averaging is identical to that of Reynolds averaging, namely u = u~0 . There is no clear-cut rules that can decide whether the Reynolds averaging method or the Favre averaging method should be used in a specific circumstance. Generally speaking, Favre averaging method is more computationally “economical”, which is an important advantage when substantial density variation must be considered. The velocities measured by pitot tubes or hot wires are closer to the Favre averaging, meanwhile Laser Doppler Velocimeters technique is more likely to measure Reynolds averaging quantities. The temperature measured by thermal couples should be closer to the Favre averaging. This is because the measurements determined by contacting-natured measuring methods are closely associated to the kinetic energy of the fluid flow and therefore are more likely to be reflected by Favre averaging method. For practical reasons the discussion here-after will only use the Reynolds averaging form and the averaging bar will be also left out. Jϕ =
μ 0 ⎛⎜ ∂ϕ δ ϕ ⎜⎝ ∂x j
2.1.3 Turbulence models Applying the averaging process considerably simplifies the computation but also bring up new unknown quantities, which are:
ρui′′u′j′ or ρϕ ′u′j These two terms are called the turbulent stresses and the turbulent scalar flux.
Ping Zhou, Feng Mei and Hui Cai
The objective of modeling turbulence is to correlate these terms with any known variables so that the equations set can be closed. Starting from Boussinesq who proposed the first turbulence model, the investigation of turbulence modeling has gone through a history over 100 years. There have been a large number of turbulence models and turbulence theories. Mathematically there are algebraic models and differential models. Physically there are turbulent viscosity models and Reynolds stresses models. Influenced by the availability of computation capacity and inclinations towards certain theories, each age has its “favorable” models. The revolutionary development of computer technology has dramatically reshaped the landscape of CFD (computational fluid dynamics) from the 1990s. Many models that were developed mainly to simplify computation have gradually faded out. The winners are those either robust enough for wide range of applications or accurate in predicting some specific problems. The interest of this book focuses on a few k- ε models that have been widely used in engineering practice, followed by a brief introduction of the Reynolds Stresses models. The readers are referred to other textbooks and monographs for other classic turbulent models such as the Prandtl Mixing Length model from the Algebraic Turbulent Stresses model group, the single equation turbulence model from the Differential Turbulent Stresses model group or the algebraic stresses model from the Reynolds Stresses model group. Boussinesq assumed that the turbulent stresses be proportional to the mean velocity gradient. He defined the turbulent stresses in analogy to molecular viscous stresses: ⎛ ∂u ∂u 2 ∂u n ⎞ 2 τ ij = − ρ ui′u ′j = μT ⎜⎜ i + i − δ ij ⎟⎟ − ρkδ ij (2.17) ⎝ ∂xi ∂xi 3 ∂xk ⎠ 3 where μT is a hypothetic “turbulent viscosity”, which is not a physical property of the fluid but a local variable depending on the turbulent flow. Introducing this turbulent viscosity re-directs the investigation interest from how to determine turbulent stresses to how to determine turbulent viscosity, from which various k- ε models have been raised. A very nature idea when we are studying turbulence is to measure the turbulence by its kinetic energy. The definition of turbulent kinetic energy is: 1 k = u ′j′u ′j′ 2 Or written in the Cartesian coordinate system as 1 2 2 2 k = ⎛⎜ u′x + u′y + u ′z ⎞⎟ (2.18) ⎠ 2⎝ Turbulent kinetic energy measures the intensity of turbulence in the three directions. From the viewpoint of turbulence microstructure, the turbulent energy
2 Modeling of the Thermophysical Processes in FKNME
is mainly stored in large scale eddies. Therefore k1/2 is also an indication of the large eddies. Through dimension analysis the Kolmogorov-Prantl expression is obtained:
μT = ρCμ k 1/ 2l
(2.19)
where l is the characteristic length of the turbulence, C μ is a constant, ρ is the density. Eq.2.19 indicates that μT is obtainable so long as k and l can be determined. Kolmogorov and Prantl derived the accurate transport equation for k out of the Navier-Stokes equation. Under high Reynolds number condition, the mean turbulent kinetic energy equation becomes: ⎞ ∂ρ k ∂ρ k ∂ ⎛ ρ ui′u ′j u ′j + uj =− ⎜ + p ′u ′jδ ij ⎟ ⎜ ⎟ ∂t ∂x j ∂xi ⎝ 2 ⎠ (,) (II) (III) (IV)
− ρ ui′u ′j
∂u j ∂xi
+ p′
(V)
∂u ′j ρ ′u ′j ∂p ∂uk′ − τ ij + ∂xk ∂xi ρ ∂x j
(VI)
(VII)
(2.20)
(VIII)
The physical meaning of each term as follows: ( , ) The transitional effect ( II ) The mean velocity-based convective term ( III ) The diffusion transport of the fluctuation of turbulent kinetic energy ( IV ) The diffusion transport of the fluctuation of pressure ( V ) The contribution of shear stresses to the generation of k, which is the interaction of the local fluctuation and mean flow ( VI ) A source of turbulence noise, which is usually ignored under low Mach number ( VII ) The part of viscous dissipation that is transformed into internal energy ( VIII ) The generation of turbulent energy due to the density fluctuation caused by pressure gradient The above terms are modeled as follows: ( III + IV ) the diffusion term: ρu ′i u ′j u ′j μ ∂k + p ′u ′j δ ij = − T (2.21) 2 σ k ∂xk
( VII ) the dissipation term:
τ ij ( V + VIII ) the generation term:
∂u′j ∂xi
= ρε
(2.22)
Ping Zhou, Feng Mei and Hui Cai
⎡ ⎛ ∂u ∂u ⎞ 2 ⎛ ∂u ⎞ ⎤ ∂u μ ∂p ∂ρ (2.23) G = ⎢ μT ⎜⎜ i + i ⎟⎟ − ⎜⎜ μT k + ρk ⎟⎟δ ij ⎥ j − T2 ⎠ ⎦⎥ ∂xi ρ ∂xi ∂x j ⎣⎢ ⎝ ∂xi ∂xi ⎠ 3 ⎝ ∂xk Under minor density variation, Eq.2.23 is further simplified into: ⎡ ⎛ ∂u ∂u ⎞ 2 ⎤ ∂u (2.24) G = ⎢ μT ⎜⎜ i + i ⎟⎟ − ρkδ ij ⎥ j ⎢⎣ ⎝ ∂xi ∂xi ⎠ 3 ⎥⎦ ∂xi Substituting above correlations into Eq.2.20 gives the final k transport equation ∂ρk ∂ρu j k ∂ ⎛ μ T ⎞ ∂k ⎜ ⎟ + = + G − ρε (2.25) ∂t ∂x j ∂x j ⎜⎝ σ k ⎟⎠ ∂x j The next task is to determine the local characteristic length of turbulent energy. Many models have been proposed but the most successful one so far came from P. Y. Zhou (Zhou,1945), Davidor et al. with their correlation between turbulence dissipation¦ to l and k: l ~ k3/ 2 (2.26) μT = C μ ρ k 2 ε (2.27)
The interest is now redirected to determining ε . The derivation of the transport equation for ε is similar to that for k, namely we first derive the accurate equation followed by a number of modeling treatments to close the equaiton. What is different is that the accurate equation for ε is much more complicated than that for k. Under high Reynolds number and local balance assumptions, the transport equation for ε is written as: ∂ρε ∂ ∂ ⎛⎜ μ T ∂ε ⎞⎟ ε ( ρu j ε ) = + (Cε 1G − Cε 2 ρε ) (2.28) + ∂t ∂x j ∂x j ⎜⎝ σ ε ∂x j ⎟⎠ k The equations set Eq.2.25~Eq.2.28 is the most frequently used k- ε two-equation model or called the standard k- ε model. The constants employed in the model are listed in Table 2.2. Table 2.2 Constants for k-ε equations Constant
Cμ
Cε 1
Cε 2
σk
σε
Value
0.09
1.44
1.92
1.0
1.3
The equations for the turbulent transport of scalars can be closed in analogy to the way closing the turbulent stresses. The turbulent transport of a scalar is assumed proportional to the local gradient of the scalar, namely: ∂ϕ − ρϕ ′u ′j = φ (2.29) ∂x j where φ is the turbulent diffusion coefficient, which is assumed proportional to the turbulent viscosity:
2 Modeling of the Thermophysical Processes in FKNME
φ=
μT σϕ
(2.30)
where σφ is the turbulent Prandtl number of the concerned scalar. Same to the turbulent viscosity, the turbulent diffusion coefficient is not a physical property but a variable depending on the process. It is worthwhile to mention that the present analogy approach is not merely out of the need of simplification but is also backed by some experimental observations. As it is widely known, the temperature field and the flow field are indeed found similar under certain conditions. The transport equations for vectors and scalars can be uniformly written as ⎞ ⎛ ⎜ Γ φ ∂ϕ ⎟ + S ϕ (2.31) ⎜ ∂x j ⎟⎠ ⎝ where φ denotes any variables including velocities, k, ε , energy, components etc. Sφ denotes the source term andφ the diffusion coefficient. Eq. 2.31 is called the generalized turbulent Navier-Stokes equations set. Table 2.3 and Table 2.4 list the detailed settings of Eq. 2.31 in two-dimensional Cartesian coordinates and Cylindrical coordinates. Under the cylindrical coordinates, the generalized NS equations set is written as: ∂ ( ρϕ ) 1 ⎡ ∂ ∂ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞⎤ ⎤ 1⎡ ∂ ⎛ + ⎢ ( rρvzϕ ) + ( rρvrϕ )⎥ = ⎢ ⎜ rΓ φ ⎟⎥ + S φ ⎟ + ⎜ rΓ φ ∂t ∂r ∂z ⎠ ∂z ⎝ ∂r ⎠⎦ r ⎣ ∂z ⎦ r ⎣ ∂z ⎝ ∂ρϕ ∂ ∂ ( ρu jϕ ) = + ∂t ∂x j ∂x j
(2.31a) The standard k- ε model has been proven robust and widely applicable through long time and extensive tests by practical use, which makes it the so far the most widely recognized turbulence model. However, this model may result in unsatisfactory predictions if one or more following conditions is (are) applicable: a) Strong swirling flow. b) Buoyancy flow. c) Gravity separation flow. d) Curving wall boundary. e) Low Reynolds number flow. f) Non-homogenous turbulent flow. ķ
Table 2.3 The expressions of Eq.2.31 in two dimensional Cartesian coordinates Equations
ϕ
φ
Sφ
Continuty
1
0
0
x-momentum
ux
μ eff (= μ 0 + μT )
−
∂u y ⎞ ∂u ⎞ ∂ ⎛ ∂p ∂ ⎛ ⎟ + ⎜ μ eff x ⎟ + ⎜⎜ μ eff ∂x ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂x ⎟⎠
y-momentum
uy
μ eff (= μ 0 + μT )
−
∂u y ⎞ ∂p ∂ ⎛ ∂u ⎞ ∂ ⎛ ⎟ + ⎜ μeff x ⎟⎟ + ⎜⎜ μ eff ∂y ∂x ⎜⎝ ∂y ⎠ ∂y ⎝ ∂y ⎟⎠
Ping Zhou, Feng Mei and Hui Cai Continues Table 2.3 Equations
ϕ
φ
Turbulent kinetic energy
k
μ0 +
Turbulence dissipation
¦
μ0 +
Energy
h
Species
γ
ķμ
⎛
μT ∂T
⎝
σ T ∂x
Gb = − βρ ⎜ g x
μT σk
μT σε μ 0 μT + Pr σ h μ 0 μT + Pr σ r
= μ 0 + μ T , μ T = Cμ ρ
eff
Sφ
+ gy
k2
ε
Gk + Gb − ρε
ε
(C
k
Gk − Cε 2 ρε )
ε1
qr (heat effect of radiation or chemical reaction) ¹s (generation of chemical reaction or combustion)
⎧ ⎡⎛ ∂u ⎞ ⎛ ∂u y ⎞ ⎤ ⎛ ∂u ∂u y ⎞ ⎫⎪ x x ; Gk = μ T ⎪ + ⎨2 ⎢⎜ ⎟ ⎥+⎜ ⎟ ⎬; ⎟ +⎜ 2
2
∂x ⎠ ⎩⎪ ⎣⎢⎝
2
⎝ ∂y ⎠ ⎦⎥ ⎝ ∂y
∂x
⎠ ⎭⎪
μT ∂T ⎞
⎟;
σ T ∂y ⎠
where g is the gravity; £ is the gas volumetric expansion coefficient. The values of the constants are given as follows:
Cμ
σ ε1
Cε 2
σk
σε
σh
σy
σT
0.09
1.44
1.92
1.0
1.3
0.9
0.9
0.9
Note in some literatures, the value of σ ε is set as 1.11, and that of σ h is 0.7.
Table 2.4 The expressions of Eg.2.31 in two dimensional cylindrical coordinates Equations
ϕ
φ
Sϕ
Continuity
1
0
0
r-momentum
ur
μ eff
−
∂u ⎞ ∂p u ∂ ⎛ ∂u ⎞ ∂ ⎛ − 2μeff 2r + ⎜ rμeff r ⎟ + ⎜ μeff z ⎟ r r∂r ⎝ ∂r ⎠ ∂r ∂r ⎠ ∂z ⎝
z-momentum
uz
μ eff
−
∂u ⎞ ∂p ∂ ⎛ ∂u ⎞ ∂ ⎛ + ⎜ rμeff r ⎟ + ⎜ μeff z ⎟ ∂z ⎠ ∂z r∂r ⎝ ∂z ⎠ ∂y ⎝
Turbulent kinetic energy
k
μ0 +
μT σk
Turbulence dissipation
ε
μ0 +
μT σε
Energy
h
μ0 +
μT σT
−qr
Species
Y
μ0 +
μT σ 'Y
−ω s
ķ μ eff = μ 0 + μT ; μT = Cμ ρ
Gk + Gb − ρε
ε k
(C
Gk − Cε 2 ρε )
ķ
ε1
⎧ ⎡⎛ ∂u z ⎞ 2 ⎛ ∂ur ⎞ 2 ⎛ ur ⎞ 2 ⎤ ⎛ ∂u z ∂ur ⎞ 2 ⎪⎫ ; Gk = μT ⎪ + ⎨2 ⎢⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ⎥+⎜ ⎟ ⎬. ε ⎩⎪ ⎣⎢⎝ ∂z ⎠ ⎝ ∂r ⎠ ⎝ r ⎠ ⎥⎦ ⎝ ∂r ∂z ⎠ ⎭⎪
k
2
2 Modeling of the Thermophysical Processes in FKNME
2.1.4 Low Reynolds number k-ε models The low Reynolds number flows are not rare in FKNME applications, such as the recirculation zone of a sudden expansion flow at the outlet of the nozzle; the close-to- wall boundary flows, viscous molten mass and viscous molten slag flows etc. These cases should be better modeled by the so-called Low Reynolds Number (LRN) k-ε models. W.P.Jones and B.E. Launder (Jones and Launder, 1972) proposed the first correction to the standard k-ε model for low Reynolds number flows, in which the following phenomena have been considered: a) The viscous diffusion of k and ε. b) The changes of the turbulent viscosity and turbulent energy dissipation as functions of Reynolds number, the turbulent viscosity should decrease as the Reynolds number falls. c) The sensitivity of ε towards the direction in the area close to walls. Jones and Launder inversely deduced the correction functions for μ r , k and¦ based on experimental observations, which led to the first LRN k-ε model: ρk 2 μ T = Cμ f μ (2.32)
ε
μ ∂ ∂ ⎛ ⎜μ + T ( ρu j k ) = ⎜ ∂x j ∂x j ⎝ σk μ ∂ ∂ ⎛ ⎜μ + T ( ρu jε ) = ∂x j ∂x j ⎜⎝ σε
⎞ ∂k ⎟ ⎟ ∂x + G − ρε + Dk ⎠ j
(2.33)
⎞ ∂ε ε ⎟ ⎟ ∂x + k (Cε 1 f1G − Cε 2 f 2 ρε ) + Eε ⎠ j
(2.34)
⎛ ∂k 1 / 2 ⎞ ⎟⎟ Dk = −2μ ⎜⎜ ⎝ ∂n ⎠ ⎛ ∂ 2u ⎞ Eε = 2 μμT ⎜⎜ 2 ⎟⎟ ⎝ ∂n ⎠
where
2
n is the distance normal to wall.
f1 = 1.0
f 2 = 1.0 − 0.3 exp(− Ret2 ) Ret =
Boundary conditions: k=ε = 0 on walls.
ρk 2 με
Ping Zhou, Feng Mei and Hui Cai
where fμ, f1, f2 are the damping functions. The LRN k-ε model can be numerically solved in the way same to solving the standard k-ε model, except that no more wall functions are necessary for determining the variables on the walls. Instead, one needs to arrange 20 to 30 mesh points along the direction vertical to the wall. That is why sometimes LRN models are called as “direct approach for wall boundaries”. However, the large demand of computation in the vicinity of the walls brings a lot of difficulty to numerical convergence, which accounts for the imposing of k=ε =0 on the boundaries to simplify computation. The Dk term in the turbulent energy equation is particularly introduced to balance the dissipation term on the wall, which actually does not equal to zero. A lot of other LRN k-ε models have been developed following the ideas of Jones and Launder. A number of the most frequently used LRN models have been listed in Table 2.5. The LRN k-ε models have been extensively investigated and developed in 1970s and 1980s and have been increasingly applied to engineering practices. The LRN models are very useful to predict low Reynolds number turbulent flows and other flows where the wall functions are difficult to apply, e.g., flows through or over a narrow gap and flows with dominating body forces or substantially changing fluid properties. The disadvantages of the LRN models are also obvious. The damping functions are deduced from laboratory measurements, which are not generally applicable. Different corrections have been reported by a large number of researchers based on their own studies and interpretations to the problems. This leads to the emerging of a large variety of LRN models with relatively limited range of applicability. It is highly recommended to be clearly aware about the range of validity of a LRN model before using it for one’s own simulation. Experimental validation is sometimes quite necessary if no report can be found about the performance of the concerned LRN model for the specific application. The second weak point of LRN models is the requested computational capacity that is much larger than that of the wall function approach. This explains why the LRN models become popular only when the high-speed computers have been widely available. One more important issue that has been frequently overlooked is that the LRN models do not guarantee good performance on temperature field prediction, because they are developed based on flow field measurements. For example, in simulating the sudden expansion flow and heat transfer, the Nagano-Hishida model predicts a peak Nusselt number 100% higher than the measurements. Further more, this model predicts a second Nusselt number peak that physically does not exist. This peak persists after applying the correction proposed by C. Yap (Yap, 1987; Ramamurthy, et al., 1993). Who tried to improve the performance of the Nagano-Hishida model. Same problem exists with the Jones-Launder model and the Launder-sherma model in computing the same sudden expansion flow. The peak Nusselt numbers predicted by these two models have been found four times as high as the measurements.
2 Modeling of the Thermophysical Processes in FKNME
Ping Zhou, Feng Mei and Hui Cai
2 Modeling of the Thermophysical Processes in FKNME
2.1.5 Re-Normalization Group (RNG) k-ε models The afore-mentioned k-ε models are established based on the theory that the turbulent fluctuation can be considered as an extra viscous effect added on top of the molecular viscosity. This extra viscous effect is found proportional to ρk2/¦ and the coefficient Cμ can be determined via experimental data. This is a typical hybrid modeling approach that combines theory, compromising assumptions and empirical observations. Surprisingly, a very similar result can be obtained from a pure mathematic derivation starting from basic turbulent theories. This is the Re-Normalization Group (RNG) k-ε model developed by V. Yakhot and S.A.Orszag in 1986 (Smith and Reynolds, 1992; Lam, 1992; Yakhot and Orszag, 1998; Yakhot and Smith, 1992) with isotropic assumption to high frequency turbulence (small scale eddies). Yakhot and Orszag studied the turbulence phenomena using frequency spectrum analysis and reorganized the N-V equations in wave vector form. From this new perspective, the large eddies are equivalent to low frequency, long wave length oscillations and the small eddies are high frequency and short wave length oscillations. The low frequency sectors are the major energy holders whereas the high frequency sectors are of no importance in terms of energy contribution to the system. The approach of Yakhot and Orszag was to “cut-off” first an infinitesimal part of the spectrum from the high frequency end and then add back the equivalent effect of this lost part to the system by appropriately estimating the impact of the cut section in the spectrum. From the viewpoint of turbulent, this impact is reflected by the change of the viscosity. By repeating this process the high frequency waves have been gradually filtered out but the contribution of these sections to the turbulent effect has been preserved. In the early days of this model, the spectrum was renormalized each time after the cut off process. That is why the model is called the “Re-Normalization Group” model or RNG model. The more recent RNG model can be done through a so-called ε-expansion, in which ε does not stands for the turbulent energy dissipation but an assumed source of high frequency noise. After this renormalization process, the N-S equations set is transformed into the following RNG k-ε model, which is pretty similar to the standard k-ε model (Yakhot and Orszag, 1992): ∂k ∂ ∂ ∂ + uj k= α μT k −ε +G ∂t ∂x j ∂x j ∂x j
(2.35)
∂ε ∂ ∂ ∂ ε ε2 + uj ε= α μT ε + Cε 1G − Cε 2 + R ∂t ∂x j ∂x j ∂x j k k
(2.36)
μ T = Cμ′ ρk 2 / ε
(2.37)
ρCμη (1 − η / η0 ) ε 1 + βη 3 k 3
R=
2
(2.38)
Ping Zhou, Feng Mei and Hui Cai
η = Sk / ε
where
⎛ 1 ⎞ S = 2 S ij S ij = ⎜⎜ G ⎟⎟ ⎝ μT ⎠ η0 = 4.38
(
)
1/ 2
1/ 2
β = 0.015 C μ′ = 0.0837, α = 1.39, Cε 1 = 1.42, Cε 2 = 1.68
Comparing to the standard k-ε model, the major difference is an extra R term in the dissipation equation. It can be considered as the contribution from the strain rate of the flow. It has been indeed found in the laboratory research that the R term becomes significant at locations with large strain rate and substantial anisotropic effect, such as in the vicinity of the walls (Shea and Fletcher,1994). On the other hand, the R term decays remarkably at locations where the strain rate is small or the flow turns to be isotropic, which transforms the RNG model into a high Reynolds number k-ε model. From this point of view, the R term is pretty similar to the damping functions in the LRN models. Different from most other terms in the model that are through mathematic derivation, the determination of the R term was partially through physical and mathematical analogies. Even though the R term is not universally applicable, it is still true that the R term is much more widely applicable than most of the damping functions in the LRN models. The value of C μ′ applied in the RNG model is slightly lower than the empirically determined Cμ′ =0.09 which is used by the standard Reynolds number k-ε model. Note that we often take C μ′ = 0.085 instead of 0.0837 in practice. There are actually more than one RNG models. The turbulent Prandtl numbers in the k and ε equations can be expressed as functions of μT. And μT itself can also be modeled in differential or algebraic expression. These modifications can improve the performance of the model under low Reynolds number flows but also add extra difficulties to the numerical process. The RNG model introduced in this section is a very basic version. The readers are referred to Chapter 9 for further discussion on the applications of RNG k-ε model.
2.1.6 Reynolds stresses model(RSM) The turbulent models discussed in the previous sections are all based on the turbulent viscosity assumption. The advantage of these models is the simplicity of modeling whereas the disadvantage is the isotropic presumption that may not apply to anisotropic turbulence such as the nature convection, swirling and near-wall flows. The RSM tends to model the Reynolds stresses individually in each direction so that anisotropic processes are allowed. The weakness of RSM is the high demand to computation capacity. For a 3D problem with heat transfer
2 Modeling of the Thermophysical Processes in FKNME
process, a RSM may require solving 11 differential equations. Besides, the physical foundation of the modeling of the pressure-strain term is not sufficiently sound. Therefore the RSM actually does not necessarily predict more accurately than the standard k-ε model in cases of considerable variation of density. Too many constants involved is sometimes also a problem for applying RSM when there is no sound experimental data to validate these constants in a particular case. For these reasons RSM is not often the appropriate option in FKNME applications unless the users have developed sufficient experience in using RSM. Nevertheless, RSM should still be considered very promising in engineering applications in the future when noticing that both CFD techniques and computational capacity have been sufficiently developed.
2.2 The Modeling of the Heat Transfer in FKNME Heat transfer in the FKNME involves processes of conduction, convection and radiation; however, it is dominated by radiation and combustion in most cases. In this section, we will focus to introduce methods of radiation computations.
2.2.1 Characteristics of heat transfer inside furnaces The heating processes in the FKNME are mostly realized through the ways as listed in Table 2.6. Table 2.6 Heat transfer in the FKNME No.
Heating mechanism
Physical models
Mathematical models
Examples
1
Solids heated by electric current
Coupling of electrical and thermal conductions
Laplace’s equation for electrical conduction Poisson equation for thermal conduction
Temperatures computation in carbon electrode
2
By electrical heat source in molten mass
Coupling of electromagnetic flow and turbulent heat transfer
Electromagnetic Field equations Navier-Stokes equations
In bath of inductance furnaces and aluminium reduction cell
3
By chemical reaction heat released in molten mass
Coupling of turbulent flow and gas-solid chemical reactions
Navier-Stokes equations Chemical reaction rate equations
Converter process
By hot (towards objects)
Coupling of convection (on surface) and conduction (within the solid body)
General turbulent motion equations Poisson equation for thermal conduction Convective heat transfer empirical equations
Heating furnace with hot air recirculation
4
gases solid
Ping Zhou, Feng Mei and Hui Cai Continues Table 2.6 No.
Heating mechanism
Physical models
Mathematical models
Examples
5
By flames in direct contact (towards solid or molten mass)
Coupling of turbulent flow, combustion and heat and mass transfer
Single-fluid or two-fluid model General turbulent motion equations Radiation model Chemical reaction models
Smelting or heating Furnaces boilerfurnaces
By flames in separated chambers
Coupling of heat transfer processes a) Flames to wall b) Within the wall c) Wall to load
Single-fluid or two-fluid model General turbulent motion equations Radiation model Chemical reaction models Thermal conduction equation
Retort furnaces Carboneletrode baking furnace
7
In the fluidized bed system
Gases-particles heat transfer Fluidized beds to the closure wall and the immersed objects’ surface
Empirical equations
Sulphide concentrate roasting furnace Dilute-phase-type fludidigation furnace
8
By surface radiation
Coupling of radiations between enclosed surfaces
Zone method Monte Carlo method or radiation network method
High temperature electric-resistance furnaces Radiation rube
6
As shown in the Table 2.6, the heat transfer processes in the FKNME are remarkably complex. If we may still state that we know something about turbulence, we would probably have to admit that we do not have much clear idea about heat transfers in the fluid flows. This partially results from the extreme difficulty to measure temperatures in the flows with high accuracy. Most of our analysis work to the heat transfer processes in flows is actually via analogizing to velocity fields, which means we presume (with some reasons) enthalpy field is similar to the flow field. This assumption allous us to write the enthalply transport equation in the following way under the k-ε model: ∂ρh ∂ ∂ ⎡⎛ μ T μ 0 ⎞ ∂h ⎤ ⎟ ρhu j ) = + ( + ⎥ + Sh ⎢⎜ ∂t ∂x j ∂x j ⎣⎢⎜⎝ σ h Pr ⎟⎠ ∂x j ⎦⎥ where h denotes the enthalpy in the flow which is defined as: n Tref ⎡ T ⎤ h0 h= Ya ha = Ya ⎢ c p ,a dT + a + c p ,a d T ⎥ (2.39) T0 Ma a =1 ⎣ Tref ⎦ where a is chemical speciesˈa˙1ˈ2ˈ…ˈn˗ Ya is mass fraction of a; c p ,a is
∑
∑ ∫
∫
2 Modeling of the Thermophysical Processes in FKNME
specific heat at constant pressure of a; ha0 is enthalpy of formation of a; Tref is referential temperature to ha0 ; T0 is standard temperature, T0 ˙ 273K (0 ć ); M a is molecular mass of a. ěYa stays constant that should equal to 1 in case of no chemical reaction, which turns the enthalpy definition equation into a simplified version:
h=
∫
T Tref
c p dT
This definition is further simplified by assuming cp is independent of temperature: h = c pT The above enthalpy transport equation and the general turbulent equation can be used to solve the heat transfer field in most flows prevailed by turbulence such as molten mass flow with internal heating source or solid objects heated by hot air. To general fluidization system, however, empirical equations are still more frequently used for computing heat transfer (the readers are referred to Chapter 8 for details). In the high temperature flames furnaces, especially those fired by pulverized coal or heavy oil that are of high emissivity and absorptivity, the heat transfers are dominated by radiations and (or) combustion. In these cases, the major efforts should be put on properly handling the computation of radiation and chemical reactions, which are represented by the source terms in the enthalpy transport equation. Radiation can be computed by algebraic equation in the simplest cases such as radiations among surfaces in enclosure with approximately uniform temperature distribution on each surface. The zone method or the radiation network method (electric resistance analogy method) can be applied if uniform surface temperature cannot be assumed. If situation gets more complicated, such as in cases of non-uniform temperature field in gray medium, we may consider heat flux method, zone method, Mont Carlo method or discrete transfer method.
2.2.2 Zone method Zone method has been proved efficient for computing enclosures with simple geometry and minor temperature variation. The basic idea of zone method is to divide the enclosure into a number of surfaces and areas (or volume in 3D) elements. The physical properties are assumed staying constant within each element; the medium is assumed gray and the surfaces are assumed of diffusive and gray (Robert and John,1990;Mei et al., 1997; Wang, 1982; Zhu, 1989). Fig.2.1 illustrates the radiation between the volume Vγ and the surface Ak. The finite radiation heat flux from the volume dVγ within the wave length bandwidth
Ping Zhou, Feng Mei and Hui Cai
dλ is written as:
dQλ =
∫
w=4π
ε λ I λb d Vγ d λ d w
= 4 πε λ I λb dVγ dλ = 4ε λ Eλb dVγ dλ
(2.40)
Fig. 2.1 The radiation between surfaces and gray gases
where ε λ is the monochromatic emissivity of the gas, Eλb is the monochromatic energy density of black body in unit W/(m3 • ) (Wang, 1982) ; the radiation heat flux from dAk to dVγ in the solid angle within wave length range d λ equals to ε λ I λb dVγ dλ . The transitivity τ λ (λ , T , P, l ) of the heat flux through the distance ls − k is: τ λ (λ , T , P, l ) = 1 − α λ (λ , T , P, l ) = exp(−α λ l ) (2.41) where α λ is the monochromatic absorptivity of the gas. Integrating over volume Vγ and surface Ak results in the monochromatic radiation heat flux: dH λ ,l Ak = dλ ∫
Vr
∫
ε λ (γ ) • I λ b (γ ) cos θ k Ak
lγ2− k
• τ λ , l • dAk • dVγ
(2.42)
Further integrating Eq.2.42 over wave length leads to the total incident heat flux to the surface Ak :
2 Modeling of the Thermophysical Processes in FKNME
H γ − k Ak =
ε γ σ Tγ4 π
∫ ∫ Vγ
cos θ k • τ l • dAk • dVγ lγ2− k
Ak
(2.43)
By defining the direct exchanging area g r sk , which represents the overall influence resulting from geometry and physical properties: εγ cos θ k g r sk˙ ∫ ∫ • τ l • dAk • dVγ (2.44) π Vγ Ak lγ2− k We simplify Eq.2.43 into:
H γ − k Ak = g r S k σ Tγ 4
(2.45)
The above equation reveals that the heat flux from Vγ to Ak is the production of the black body radiation and the direct exchanging area. The total heat flux towards Ak from all N volume elements should be: Nγ k =
1 Ak
N
∑ g S σT r
k
γ
4
(2.46)
γ =1
On the other hand, all of the M gray diffusive surface elements with area Akj (j= 1, …, M) radiate to Ak : Jj cos θ j • cos θ k • dAj • dAk (2.47) H jk Ak = τl ∫ ∫ A A k j π l 2j − k where J i is the effective radiation of Aj . Eq.2.47 can be expressed in a way similar to Eq. 2.45:
H jk • Ak˙S j S k J i
(2.48)
where S j S k is the direct exchanging area between the surfaces j and k with definition out of Eq.2.47: S j Sk = ∫
Ak
∫
Aj
τl
cos θ j • cos θ k • dA j • dAk
πl 2j − k
(2.49)
The total heat flux from the M surface elements towards the surface element Ak is then: 1 M (2.50) H skj = S j Sk J i Ak j =1
∑
The big total of the heat flux towards Ak is obtained by summarizing contributions from both surfaces and bodies: M ⎞ 1 ⎛ N 4 ⎜ H k= H γ k + H jk = g S σ T + S j S k J i ⎟⎟ (2.51) γ r k ⎜ Ak ⎝ γ =1 j =1 ⎠ As the effective radiation from Ak is:
∑
∑
J k = ε kσTk4 + (1 − ε k ) H k The difference between incoming and outgoing radiation flux is:
(2.52)
Ping Zhou, Feng Mei and Hui Cai
qk = J k − H k= ε kσTk4 − ε k H k
(2.53)
In case of unknown gases temperature T¤, another N equations are needed to close the equations set. The following energy balance equation can be used to describe the relationship among volume elements. N 4ε γ σTγ 4∗ dVγ ∗ ε 4 τ l 2 γ dVγ + 4ε γ σTγ Vγ = Vγ Vγ ∗ Lγ ∗ −γ 4π
∑∫ ∫
εγ J k cos θ k dAkτ l 2 π l∗
M
∑∫ γ ∫ V
Ak
γ −k
τ l dVγ ∗ dV
N
= ε γ2
∑ σT ∫ ∫ 4
γ∗
γ ∗ =1
V∗
Vγ
M
εγ
∑J ∫ ∫ k
Vγ
Ak
γ
πlγ2∗ −γ
+
cosθ k τ l dAk dVγ πlk2−γ
(2.54)
Define sk g r as the direct exchanging area from a surface to a gaseous volume:
sk g r =
εγ
π ∫ ∫ Vγ
Ak
cosθ k τ l dAk dVγ lk2−γ
(2.55)
Knowing from Eq.2.44 that the direct exchanging area from a surface to a gaseous volume should be identical to that from a gaseous volume to a surface, namely: sk g r = g r sk (2.56) Similar relationships exist for gases to gases and surface to surface: si s j = s j si ; gi g j = g j gi ∗
The direct exchanging area between gaseous volumes gγ gγ is defined as:
gγ ∗ gγ =
ε γ2 π
∫γ∫γ V
V∗
τ l dVγ ∗ dV
(2.57)
l 2∗
γ −γ
Substituting all above definitions and relationships into Eq.2.54 leads to N
4ε γ σTγ4Vγ =
∑ γ ∗ =1
σTγ4∗ gγ ∗ gγ +
M
∑J
k
g r sk
(2.58)
j =1
With the help of Eq.2.58, N energy balance equations can be established for N gaseous volumes. Therefore the temperatures fields and the heat fluxes among the elements in any enclosure can be determined by the following procedure: a) Determining the direct exchange areas of the involved elements. b) Resolving MˇN heat balance equations for temperatures in the M surface elements and N gaseous volume elements. c) Determining the difference in fluxes balance of the M surface elements as
2 Modeling of the Thermophysical Processes in FKNME
well as the total heat exchange in the system. Obviously the accuracy of the zone method chiefly depends on how to define the zones and how many zones to be defined. Larger number of zones helps to improve accuracy but also exponentially raises the computational load. Automatic zone partition becomes difficult if the geometry is complicated. Basically the zone method requires large amount of human intervention for high computational efficiency and accuracy. It has also been found that serious inconsistency between the defined radiation zones and the mesh grids setup can also bring substantial difficulties to numerical resolving process when applying finite element or finite difference method.
2.2.3 Monte Carlo method The Monte Carlo method is a computational algorithm, which relies on repeated random sampling to simulate physical processes. In the case of simulating a radiation process, a number of rays of radiation are sampled and simulated along the entire process from emitting, traveling in and interacting with the medium to finally arriving on a surface and being absorbed or reflected. So long as the number of sampled radiation rays is sufficiently large (more than 100,000) the Monte Carlo method may satisfyingly predict the overall heat transfer process. The disadvantage of the Monte Carlo method is the demanding need to the computational capacity. A relatively simple problem may also require large amount of computation. Assuming the elementary volume Vγ emits N rays with ω Joule energy in each ray. These rays are “emitted” following the directions randomly decided by the random cumulative probability function Rθ (Siegel and Howell, 1990; Zhu, 1989) .
θ = sinˉ1 Rθ
(2.59)
We assume the ray concerned goes in a randomly determined direction Rθ and is absorbed after traveling over a distance of l. The energy intensity of the ray should be reduced to e-αs after traveling over a distance of s in a medium with constant absorptivity. The probability function p (s ) represents the probability of this ray traveling over a distance s before it is entirely absorbed. e −α s = α e −α s p(s ) = ∞ −αs e ds
∫
0
Adopting the cumulative probability function Rl = or
l
∫ p(s )ds = 1 − e 0
−α l
Ping Zhou, Feng Mei and Hui Cai
l=−
1
α
ln(1 − Rl˅
The above expression can be equivalently written as l = −
1
α
ln Rl because Rl is
a random value uniformly distributed between 0 and 1. Under thermal equilibrium condition, the medium emits an identical number of rays after having absorbed a certain quantity of radiation rays. The emission direction ( γ ) over a sphere around the medium is determined by a cumulative probability function Rγ
γ = cosˉ1 (1 − 2 Rγ )
(2.60)
Together with Rl, the emission direction given by Eq.2.60 determines where this ray will be absorbed. This emittingüabsorbingüemitting (diffusion) process is repeated until the ray reaches its destination. We then may move on for the next ray until all of the N rays have been “emitted”. The number of the rays that have been absorbed by the surfaces or the gaseous volumes will be then recorded. Under thermal equilibrium condition, the energy radiated by a medium volume should equal to the energy it absorbs. Giving Sdv as the number of absorbed rays by the medium finite volume dv and ω as the energy contained in each ray, we have ω ⋅ S dv = 4α pσTd4v dv (2.61) where α p is the Planck mean absorptivity. The average temperature in the finite volume is then determined by 1/ 4
⎛ ω Sdv ⎞ Tdv = ⎜ (2.62) ⎜ 4α σ dv ⎟⎟ ⎝ p ⎠ The heat flux through the surface i is obtainable from the balance between the N emitted rays and the number of absorbed rays. The dimensionless heat flux qi / σTi 4 can be written as
qi
σTi 4
=
qeff − ωS wi ω (N − S wi ) S = = 1 − wi σTi 4 ωN N
The readers are referred to references (Siegel and Howell, 1990; Zhu, 1989; Howell, 1964a, b; Perlmutter, 1964) for the expressions of the involved cumulative probability functions. Monte Carlo method may result in considerable error in cases of low optical thickness ( kλ = L lm , namely the ratio between the geometrical scale and the mean penetration distance of the photons). On the other hand, increased optical thickness may lead to rapidly rising demands of computational capacity. Therefore Monte Carlo method is considered efficient mostly in cases of complicated system with sensitively changing radiation properties.
2 Modeling of the Thermophysical Processes in FKNME
2.2.4 Discrete transfer radiation model DTRM (discrete transfer radiation model) is a method combining the zone partition idea from zone method and the ray tracing principle from Monte Carlo method. This makes DTRM generally applicable to wide range of cases with relatively lower demand to computational efforts. DTRM works especially effective with optically thin applications. The radiation intensity of a randomly selected ray changes as it travels through an absorptive-diffusible medium. The governing equation is: diλ′ σλ 4π = −ai iλ′ ( x ) − σ λ iλ′ ( x ) + aλ iλ′b ( x ) + iλ′ ( x, ωi ) ϕ ( λ , ω , ωi ) dωi (2.63) dx 4π ∫ 0 (I) (II) (III) (IV) Term ĉ in Eq. 2.63 denotes the absorption of the medium. Term Ċ represents the deterioration in x direction due to scattering. Term ċ is the emission in x direction from the medium itself. Term Č include all radiations in x direction scattered by all other rays. We usually ignore the scattering term when there are only minor particles presented in the furnace gases, which simplifying the equation into: σT 4 diλ′ = − ai iλ′ ( x ) + aλ iλ′ b ( x ) = − ai iλ′ ( x ) + aλb (2.64) π By assuming the radiation intensity is independent of wavelength and direction, we rewrite Eq.2.64 into:
σT 4 di = −ai + a (2.65) ds π The procedure of applying DTRM is illustrated below by a 2D model. A 2D space as shown in Fig. 2.2 is divided into i volume elements and j surface elements. In most practices the zones are divided following the mesh grids for computing the convection and diffusion processes of fluid flow. We name the center point of a surface element as Pj. The hemisphere around Pj is divided into k solid angles sections. For each of these angular sections an emitting direction can be defined, following which a radiation ray will be “emitted” and traced. The prediction accuracy rises if a larger number of sections (k) is selected. To carry out the computation, we assume as well: a) Thermophysical properties within an element stay constant. b) Surface elements are diffusive and gray. c) Radiation intensity of each ray is identical. d) All consequent effects from rays incident to surface element j are contributed to the center point Pj.
Ping Zhou, Feng Mei and Hui Cai
Fig. 2.2 A schematic diagram for 2D DTRM
The implementing of DTRM becomes pretty easy once the above assumptions having been established. a) Going through all angular sections and tracing back each incident rays until their emitting point. Along the tracing process, the variation of the radiation energy is updated for each volume element the ray has traveled through. The variation is considered as the contribution of this ray to this particular volume element, which is described by the restructured Eq.2.65: σT 4 iiˇ1 = ii e − aΔx + (1 − e − aΔx ) (2.66) π where Δx denotes the gain in traveling distance in this volume element; a denotes the absorptivity of the volume concerned. b) Once having traced back to the original emitting surface of the ray, the radiation intensity on this surface is calculated by Eq.2.67 (Fig.2.3):
Fig. 2.3 Radiation intensity of surface element (² is reflection coefficient; ¦ is radiation coefficient)
2 Modeling of the Thermophysical Processes in FKNME
iP+j =
q +j
=ε
σTϕ4
+ (1 − ε )
q _j
(2.67) π π π where q +j denotes total radiation energy; q _j denotes the obtained incident energy from elsewhere;¦ denotes the emissivity of the surface, which equals to the absorptivity. In case the boundary is not defined by surface temperature but by net radiation heat flux q net , Eq.2.68 is used: q +j
q _j q net + (1 − ε ) (2.68) π π π The above 2D case can be easily expanded into 3D (Fig.2.4) simply by rewriting Eq.2.67 into: i +p j =
=
(
)∫ ∫ + (1 − ε )∑ i
q +j = ε iσT 4 + 1 − ε j
= ε i σT 4
2π
π/2
ϕ =0
θ =0
j
_ j
i _j cosθdω
cosθΔω
(2.69)
Δω
Eq.2.66 is still applicable for determining the radiation intensity within a volume element.
Fig. 2.4 Solid angle θ and φ in a hemisphere
Before closing the discussion about the radiation models, we go back to the starting point of Section 2.2.1 where the source term in the energy conservation equation is discussed. To determine the radiation source term for each volume i, we add up contributions from all involved parts:
∫ ( ∇ • q ) dv = ∫ Δvi
J
Δsi
q • ds j = ∑ J j
(2.70)
j =1
where J j = (in+1 − in )cosθ j Δω j Δs j is the contribution of surface element j towards the concerned volume element i. The radiation intensities in+1 and in are
Ping Zhou, Feng Mei and Hui Cai
computable from Eq.2.66 with the starting point at i+.
2.3 The Simulation of Combustion and Concentration Field Combustion is a flow phenomenon with chemical reactions producing a lot of heat. So it also involves the characteristics of mass transfer and chemical reactions. Apart from the characteristics of turbulent flow and heat transfer discussed in the former sections. The interaction and strong coupling between turbulence and combustion make the differential equations of combustion very complicated. Generally, these equations can hardly be solved by analytical method but numerical simulation method. Knowing that the actual combustion processes in the FKNME applications are mostly in turbulence, we focus in this section the simulation of turbulent combustion, including gas phase and gas-particle two-phase combustions. The objective of combustion simulation is to gain better understanding about the temperatures, velocities, concentrations (species mass fraction) and heat release (chemical reaction rate) based on the conservation of mass, momentum and energy, and the rules governing the reaction rates as a function of temperature, pressure and reactant concentration.
2.3.1 Basic equations of fluid dynamics including chemical reactions For a chemical reaction system consisting of NS chemical species and probably involving NR basic reversible reactions, the jth reaction can be written in the form ar1 j X 1 + ar 2 j X 2 + ... = a p1 j X 1 + a p 2 j X 2 + ... (2.71) Xi (i=1, …, NS) refer to chemical species. Reactants are not distinguished from products in this equation because reactants and products are reversible in the reactions. arij and apij are the stoichiometric coefficients of species i at both sides of j th basic reaction equation. Their values are positive or zero. The difference: nij= apij arij (2.72) is the overall stoichiometric coefficient for species i in reaction j, and is positive for products and negative for reactants. The impacts of chemical reaction on turbulent combustion are mainly repnesented by the source terms in transport equations. Since the momentum equations are identical to what have been discussed in the previous sections, in this section we mainly discuss the conservation equations of chemical species and enthalpy and the reaction rates in connection with the source terms of these equations.
2 Modeling of the Thermophysical Processes in FKNME
2.3.1.1 Reaction rate
Chemical reaction rate means the species quantity produced or consumed at unit time, unit volume (mol/(m3 • s)) and is modeled as a function of reactant and product concentration (Fan and Wang, 1992; Fan et al., 1987), by NS
α ij
ωij
NS
R j = k fj ∏ [ xi ] − kbj ∏ [ xi ] i =1
(2.73)
i =1
or in case of reaction rates influenced by concentrations of third non-reacting third bodies (inert species), by α ij ωij NS ⎞ ⎛ NS ⎞ ⎛ NS R j = ⎜ ∑ γ ij [ xi ] ⎟ ⎜ k fj ∏ [ xi ] − kbj ∏ [ xi ] ⎟ (2.74) ⎟ i =1 ⎝ i =1 ⎠ ⎜⎝ i =1 ⎠ where [xi] is molar concentration of species i; α ij is forward rate exponent; ω ij is
backward rate exponent; γ ij is efficiency of species i as a third body (0İ γ ij İ1); kfj is forward rate constant; kbj is backward rate constant. Rate constant can be calculated as a function of temperature from the modified Arrhenius law. For the j th forward reaction, this is β
k fj = Afj T fj e
− E fj RT
β
= Afj T fj e
− T fj T
(2.75)
where Afj is pre-exponential factor or frequency factor; β fj is temperature exponent; Efj is the activation energy; Tfj is the activation temperature (or Efj/R). The backward rate constant can be defined similarly. 2.3.1.2 Species conservation and mass continuity equation
General mass fraction Yi (i=1, …, NS) represents chemical species concentration, then mass conservation equation for species can be written as (Fan et al, 1987) ∂ρ Yi + ∇ • ( ρ UYi + J i ) = Si (2.76) ∂t where U=˄u, v, w˅ is average velocity of the mixture (u, v, w stand for velocity components in three orthogonal coordinate directions, respectively ); ρ is the density of the mixture; t is time; Ji is the mass flux of species i relative to the mean (bulk) flow, kg/(m2 • s); Si is the product and consume rate of species i due to chemical reaction. Here J i ≡ ρYi (U i − U ) (2.77) where Ui is the velocity of species i. The first, second, third term on the left hard side and the term on the right hard side in Eq. 2.76 are called respectively the time variation rate term, convection term, diffusion term and reaction source term of species i. According to Fick’s law
40 Ping Zhou, Feng Mei and Hui Cai
J
μ j==-[rj+~:)v~
(2.78)
μ is the turbulent viscosity, oy is turbulent Prandtl numbers. The mass where IlT conservation equation of species i is defined by
a~; + V .{PU~ -[ r, + ~μ )v~ }== S,
(2.79)
the species source term due to chemical reaction (kg/nrr' • s)) is summed from the contributions of the reaction rates by NR
s, = MiLnijR
j
(2.80)
j=l
where M is the molecule weight for species i, Rj is the j th reaction rate.
2.3.1.3
Enthalpy equation
Enthalpy equation is the application of the first law of thermodynamics to fluid dynamics and can be expressed as that the increase of total energy per volume is equal to the sum of the net incoming stagnation enthalpy, the heat transfer and the work from the surroundings. How to choose the variables in the energy equation has a great influence on the equation solving process. For the combustion system with multi-species and chemical reactions, the four kinds of mixture enthalpy are usually used as dependent variable in energy equation: thermal enthalpy, thermal enthalpy + chemical enthalpy, total enthalpy (thermal enthalpy + chemical enthalpy + kinetic energy) and sensible enthalpy (thermal enthalpy + kinetic energy). If the work from external bulk force and other external heat sources apart from thermal radiation are neglected, the equation for total enthalpy is (Zhou, 1994; Carol, 1987)
~t [p(H, -~)]+ V• (pUH, )== cIJ+Qrad - V -s,
(2.81)
where (/J is viscous dissipation, or shear deformation work; Qrad is source term (radiative heat); qt is total energy flux (including the energy carried into by thermal conduction and species diffusion): Ns
qt =-AVT+ LJiHti i
A is thermal conductivity; H; is total enthalpy of species i and can be written by tt, == hfi + ~ (~U2 +k)
(2.82)
hfi is specific enthalpy of formation of species i, U is the averaged velocity of the
mixture, and k is the turbulent kinetic energy. The energy equation with sensible enthalpy (H) is more generally applied in the
2 Modeling of the Thermophysical Processes in FKNME
practice. If the dissipation term is ignored and the turbulent viscosity hypothesis ⎛ μT ⎞ ⎜ΓH = ⎟ is adopted to Eq.2.81, the sensible enthalpy equation of the mixture σH ⎠ ⎝ can be obtained in the form ⎛λ μ ⎞ ∂ρ H ⎪⎧ ⎪⎫ ∂ p (2.83) + ∇ • ⎨ ρ UH − ⎜ + T ⎟ ∇ H ⎬ = + Qrad + QR ⎜ ⎟ ∂t ⎪⎩ ⎪⎭ ∂ t ⎝ cp σ H ⎠ where ΓH is the turbulent diffusivity of enthalpy, μT is the turbulent viscosityˈσH is the turbulent Prandtl numbers. The sensible enthalpy is defined as T Tref 1 H = ∫ c p (T ′)dT ′ − ∫ c pB (T ′)dT ′ + U 2 + k 0 0 2 where cp is specific heat weighed on mass fraction, cpB is specific heat of species before reaction, Tref is the reference temperature of enthalpy. The source term from chemical reaction is NR
QR = −∑ R j ΔH Rj ( 0 )
(2.84)
j
The reaction heat ΔHRj˄J/kmol˅can be calculated based on species specific enthalpy of formation at the given reference temperature. For nij kilo-molar products or reactants, it is T NS ⎧⎪ ⎫⎪ (2.85) ΔH Rj (T ) = ∑ nij M i ⎨h fi (T f 0i ) + ∫ c p (T ′ ) dT ′⎬ i T f 0i ⎩⎪ ⎭⎪ A set of basic equations of chemical fluid dynamics in combustion are got in term of the above species conservation, enthalpy, momentum and continuity equations: a) Continuity equation: ∂ρ + ∇ • ( ρU ) = 0 ∂t b) Species conservation equation: ⎛ ⎞ ∂ρYi μ + ∇ • ⎜ ρUYi − (Γ i + T )∇Yi ⎟ = Si ⎜ ⎟ σy ∂t ⎝ ⎠ c) Momentum equation: ∂ρU + ∇ • ( ρUU ) = −∇p + Fs + ∇ • Π + Δρ g ∂t where Π refers to Reynolds stress tensor 2 2 T Π = μ eff ∇U + ( ∇U ) − μ eff ∇ • UI − ρ kI 3 3 where p is pressure; Fs is the volume force; μeff is the effective fluid viscosity (equal to the sum of molecule viscosity and the turbulent viscosity, i.e. μ+μT), I is the unit tensor. The dyad (a two-order of tensor) is defined by
(
)
Ping Zhou, Feng Mei and Hui Cai
( "# )ij
= Ai B j
d) The enthalpy equation refers to Eq. 2.83.
2.3.2 Gaseous combustion models There are very strong interactions between the turbulence and chemical reaction in the combustion. The chemical reaction has an effect on the density and viscosity due to heat release, which further influences turbulence. On the other hand, the turbulence influences the combustion by intensively mixing reactants and products. From the former section, in order to solve the basic equations of chemical fluid dynamics in the combustion, it is needed to solve the second order nonlinear partial differential equation with the source term including average chemical reaction rate. Therefore, the key of the turbulent reaction model is how to model the average chemical reaction rate. It is difficult to develop a general model because it is simultaneously influenced by turbulent mix, molecule transport and chemical reaction. So far, of the models mentioned above (Fan and Wang, 1992; Fan et al., 1987; Zhou, 1994; Carol, 1987; Zhao et al., 1994; Zhen and Zhou, 1996; CFX-4.2 Solver, 1997), the mixed-is-burnt and eddy-break-up models are most widely applied. To study combustion phenomena in the combustion devices, the heat effects caused by combustion (such as the distribution of temperature and heat flux) are mainly considered. Moreover, the influence of chemical reaction on flow is also caused by its heat effect. Therefore, “a simple chemical reaction system” is usually used to simulate complicated reaction dynamics processes. In the simple chemical reaction system, it is assumed: a) The turbulent transport coefficient of all species are all the same at each point of the flow field, i.e. ΓF˙ΓO˙Γ˄ P subscripts F, O and P stand for fuel, oxidant and product respectively˅. b) Fuel and oxidant are combined in a fixed ratio i, the stoichiometric ratio, such that: 1kg fuel + i kg oxidant ⎯⎯ → (1+ i ) kg product (2.86) Obviously, if any two of the three species concentrations are available in this system, the third one can also be solved. The mixture fraction f for the reaction can be defined by
f =
χ − χO χF − χO
(2.87)
YO i
(2.88)
where
χ = YF −
2 Modeling of the Thermophysical Processes in FKNME
where Y is mass fraction. So, χo and χF in Eq. 2.87 refer to χ value of oxidant and fuel respectively. Furthermore, χ O = − , ǂχF=1. 1 i
In definitionˈf is always positive, attaining its stoichiometric value fST when
χ = 0. Thus: 1 (2.89) 1+ i The mean value of the mixture fraction f satisfies the following conservative transport equation without source term and is a scalar with conservation feature. ⎛⎛ μ ⎞ ∂ρ f μ ⎞ + ∇ • ρU f − ∇ • ⎜ ⎜ T + ∇f ⎟=0 (2.90) ⎟ ⎜ σ ⎟ ∂t ⎝⎝ T σ L ⎠ ⎠ where ρ is the fluid density, U is the mean fluid velocity, σ L and σ T are the equivalent laminar and turbulent Prandtl numbers respectively and both equal to 0.9. fST =
(
)
2.3.2.1 Mixed-is-burnt model
The mixed-is-burnt model is mostly used to simulate the turbulent diffusion flame, which feature much higher chemical reaction rates comparing to the mixing rates between fuel and oxidant. The mixed-is-burnt model assumes that: a) The chemical reaction rate is infinite. b) Fuel and oxidant cannot coexist instantaneously. For the above diffusion flame, the instantaneous mass fraction can be calculated based on instantaneous mixture fraction by the following relationship. If f˚fST, the mixture is made of fuel and product. There are f − fST , YO = 0 , YP = 1 − YF (2.91) YF = 1 − fST If f˘fST, the mixture is made of oxidant and product. There are f YF = 0 , YO = 1 − , YP = 1 − YO fST
(2.92)
Although most combustions in real installations are taking place in turbulent flow, we generally pay attention to the distribution of the mean value of various variables, instead of their instantaneous value. However, we haven’t yet known the relationship between the mean mass fraction or the mean temperature and the mean mixture fraction. So, in order to obtain the mean mass fraction and the mean temperature, the concept of the probability density function (Fan and Wang, 1992; Fan et al., 1987; Zhou, 1994; Zhen and Zhou, 1996; Meng, 1997) has been proposed. For random mixture fraction “ f ā fluctuating with time from 0 to 1, its
Ping Zhou, Feng Mei and Hui Cai
probability existing in the region [f , fˇdf] can be defined as p(f)df, and p(f) is called the probability density function.
∫
1 0
p( f )df = 1
(2.93)
The mean mixture fraction f and variance of the mixture fraction f ′2 ˄for convenience, f ′2 is expressed as g˅are determined by 1
f = ∫ f p( f )df 0
1
1
0
0
g = ∫ ( f − f ) 2 p( f )df = ∫ f 2 p( f )df − ( f ) 2
(2.94) (2.95)
For any function ϕ (f), its mean and variance are: 1
ϕ ( x ) = ∫ ϕ ( f ) p ( f , x ) df 0
1
ϕ ′2 ( x) = ∫ ϕ 2 ( f ) p( f , x) df − (ϕ )2 0
(2.96) (2.97)
In order to determine p (f), apart from calculating f , it is necessary to get g. The modeled form of g equation used is ⎛⎛ μ ∂ρ g μ ⎞ ⎞ ε 2 + ∇ • ( ρUg ) − ∇ • ⎜ ⎜ T + ⎟ ∇g ⎟⎟ = Cg1 μT (∇ f ) − Cg 2 ρ g (2.98) ⎜ ∂t k ⎝⎝ σT σ L ⎠ ⎠ where k is the turbulent kinetic energy;ε is the turbulent dissipation rate; Cg1 and Cg2 are the empirical constants and their values are respectively 2.8 and 2.0. If the expression of p (f) is given, f and g are got from the corresponding equations, then the mean mass fraction of species can be obtained. This model is called as k-ε- f -g model. The mean mass fraction of fuel, oxidant and product are respectively: 1 ⎛ f − f ST ⎞ , 0 ⎟ p ( f ) df (2.99) Y F = ∫ max ⎜ 0 ⎝ 1 − fST ⎠ 1 ⎛ ⎞ f Y O = ∫ max ⎜ 1 − , 0 ⎟ p ( f )df 0 fST ⎠ ⎝
(2.100)
and Y P = 1− Y F − Y O (2.101) Therefore, the crucial problem is how to determine the probability density function (PDF) of the mixture fraction. At present, there are mainly three kinds of methods to determine PDF: a) p(f) is specified on the knowledges of the turbulent fluctuation (Fan and Wang, 1992; Fan et al., 1987; Zhou, 1994; Carol, 1987). b) The control equations of p(f) are constructed and solved. (Zhou, 1994; Zhen and Zhou, 1996) Moreover, the PDF transport equation, which includes the joint probability density function of velocity and chemical thermo-dynamics parameters,
2 Modeling of the Thermophysical Processes in FKNME
can be use to predict accurately any complicated chemical reaction mechanism. But this method needs lots of computing time, hence is limited in the engineering application. c) The ESCIMO (engulfment-stretching-coherence-interdiffusion-interactionmoving observer), which was proposed by Spalding, can analyze quantitatively the influence of the factors such as turbulence, molecule transport and chemical dynamics of turbulent combustion (Fan and Wang, 1992; Carol, 1987), in which the complicated molecule transport and chemical dynamics model is able to be used. However, when this method is used to analyze more complicated practical problem, there are a lot of problems to be solved, for example, back flow and unsteady process. The first method is more extensively applied in the engineering problems. In addition to the simple form of p(f), the velocities and temperatures predicted by this method are in better agreement with the practical result. Here, two simple PDF models are introduced. Double delta function Presented by Spalding, it assumes that: f takes only two values: f+ and fˉ ; if the time fraction is α when f = fˉ, then the time fraction must be (1α) when f = f+. So, the mixture fraction varies with time in a rectangular wave way, which indicates that fuel and oxidant cannot simultaneously present at the same point. A double delta function has the following form p ( f ) = αδ ( f − ) + (1 − α )δ ( f + ) (2.102) In the practical application, α is usually set to be 0.5, then f+ = f + g ˈ f− = f − g Beta function Compared to p (f) in a double delta function, the predicted result by adopting p(f) in a beta function is closer to the experimental data (Dong,1998). This PDF model has the following form: p( f ) =
f ( a −1) (1 − f )
∫f
( a −1)
(1 − f )
b −1
b −1
(2.103) df
⎡ f (1 − f ) ⎤ a= f ⎢ − 1⎥ g ⎣ ⎦
(2.104)
⎡ f (1 − f ) ⎤ b = (1 − f ) ⎢ − 1⎥ g ⎣ ⎦
(2.105)
2.3.2.2 Eddy break-up model (EBU)
The turbulent premix flames can be considered as many micro gaseous eddy groups which includes already burned eddies and going to be burned eddies in
Ping Zhou, Feng Mei and Hui Cai
the different extent. The chemical reactions take place on the interface of the two eddies. It is assumed that: the chemical reaction rate depends on the rate of the micro groups of gaseous fuel breaking into smaller micro groups in the turbulence; the break up rate is directly proportional to the dissipation rate of the turbulent fluctuant kinetic energy. Therefore, the source term (that is the mean chemical reaction rate) in the species conservation equation can be compoted in terms of k and ε (Fan and Wang, 1992; Fan et al., 1987; Zhou, 1994; Carol, 1987), this is called as eddy break-up model (EBU). In the eddy break-up model, an explicit equation is solved for the mass fraction of the fuel: ⎛⎛ μ ⎞ ∂ρYF μ ⎞ ε + ∇ • ( ρUYF ) − ∇ • ⎜ ⎜ T + ∇YF ⎟ = − ρ CR CA M lim (2.106) ⎟ ⎜ σ ⎟ σ k ∂t L ⎠ ⎝⎝ T ⎠ with 1 ⎧ 4 ⎛ ⎞ με ⎪⎪23.6 , viscous mixing model ⎜ ⎟ 2 CR = ⎨ ⎝ ρk ⎠ ⎪ collision mixing model ⎪⎩ 4.0 , infinite rate chemistry ⎧1.0, ⎪ CA = ⎨1.0, Daı ≥ Die , finite rate chemistry ⎪ 0.0, Da >1, the combustion is the finite reaction flow controlled by diffusion; if Da Wgl
a) Reaction heat is negligible b) Wqt<Wgl
Internal thermal resistance of materials is negligible
Remark
3cpq ρgl (1 − ε ) ⎛ W ⎞ aΣv ⎜1 − gl ⎟ ⎜ W ⎟ qt ⎠ ⎝ 3cpq ρgl (1 − ε ) ⎛ Wgl ⎞ −1 ⎜ Wqt ⎟⎟ ⎝ ⎠
αΣv ⎜
G (qD + cprt Δtgr ) Qq−L − Qss
Load heated to 95% of the gas exhaust temperature. Wgl is water equivalent of material flow; Wqt is water equivalent of gas flow
q D is heat needed for melting one ton materials, kJ/kg; cprt is melt specific heat, kJ/(kggK-1); Qss is heat loss of melting bath, kJ/h; Qq−L is heat transfer between gas and surface of melting bath, kJ/h
10 Multi-objective Systematic Optimization of FKNME Continues Table 10.1 Process
t
Reaction models
ķ
Remark
′′ ⎞⎟ ⎛⎜ Twl ′ ⎞⎟⎤ ρCV × 108 ⎡ ⎛⎜ Twl ⎥ ⎢ϕ −ϕ 4
Thin plane materials heated in furnaces at constant temperature
C
∑ FT
q
⎢⎣ ⎜⎝ Tq ⎟⎠
⎜ Tq ⎟⎥ ⎝ ⎠⎦
Internal thermal In which: resistance of ⎛T ⎞ 1 ⎛T ⎞ materials is ϕ ⎜ wl ⎟ = arctan⎜ wl ⎟ + ⎜ Tq ⎟ ⎜ Tq ⎟ 2 excluded ⎠ ⎝ ⎠ ⎝ ⎡ ⎛ ⎞ ⎛ ⎞⎤ 1 ⎜ Twl ⎟ ⎜1 − Twl ⎟⎥ ln ⎢ 1 + ⎜ ⎟ ⎜ 2 ⎢⎝ Tq ⎠ Tq ⎟⎠⎥ ⎝ ⎣ ⎦
Melt without Slag disturbance settling in motion in the smelting direction that bath is normal to the fluid flow
ρ , c, V, F are respectively the density, specific heat, volume and heated area of the materials to be heated; Tq is
the gas temperature; Twl ′ and ′′ are the initial and end Twl temperatures of materials hzc is the thickness of the slag layer; ν z is the kinematic
18hzcν z ρ z
gd m2 (ρ rt − ρ z )
viscosity of the slag melt; ρ z is the density of the slag melt; g is the gravity the acceleration; d m is diameter of the metal or alloy melt; ρ rt is the density of the metal or alloy melt
ķNomenclatures: k v , k c are the reaction rate constants based on particle volume and particle surface respectively, cm3/(molgs),cm/s; k f is the mass transfer coefficient between gas the and particles, cm/s; δ fa is the gas diffusivity in ashes, cm2/s; ρ sm is the molar density of solid reactants, mol/cm3.
The output rate function of rotary kiln is written as: G = um • A • ϕ (t / h) where um represents the mean axial speed of materials in the kiln, m / s ; A represents the cross-sectional area of the empty kiln, m 2 ; ϕ is the filling ratio of the materials to the kiln’s cross section. For usual drying and roasting kilns: um ≈ 5.78 Dr • β • n (m / h) (10.10) where Dr represents the diameter of the empty kiln, m ; β represents the inclined angle of the kiln’s central axis, (e) ; n denotes the rotate speed, r/min. As for high-temperature sintering kilns, materials are to be softened, aggregated and sintered in the high temperature zone. In such a case, we have D ni sin 3 θ um ≈ 2.32 r • (m / h) (10.11) sin α 2θ1 − sin 2θ1 where i is the inclines of the kiln, %; 2θ is the central angle of the sector occupied by materials in the kiln’s cross section; and α is the angle of repose of materials in the rotating kiln. For the periodically operated FKNME, the material unloading speed (in casting
Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao
period) is dependent on the capability of the casting machine. For continuous casting, the speed is determined by the cross-sectional area of the crystallizer and the forced cooling capacity for the crystallization. In this case, the output rate function usually has to be empirically determined. d) Output rate function defined by gas flux (or gas exhaust capability). In many furnaces and kilns, the gas flowrate are strictly restricted, which in consequence limits the output capacity of the furnaces. Assuming usy is the velocity of high temperature gases in the furnace, the output rate of the furnace should be: 3600usy A G= (10.12) Vdc where usy is the economical velocity of gas, m / s ; A is the cross-sectional area of the furnace chamber, m 2 ; Vdc is the gas volume needed to process unit mass of material or product, m 3 / t , which is determined through mass and heat balance calculations. The range of limits for usy is given in Table 10.2. Table 10.2 Limits of the proper gas speed usy in hearth or flue
FKNME type
usy/mgs−1
Remarks
Dense phase in fluidized bed
u mf < usy < u t
Refer to Section 8.1
Circulating fluidized bed
ucd < usy < u FD
Gas drying or conveying
usy < u FD
Shaft smelting furnaces and blast smelting furnaces Flame furnace (without powder in bath) (with powder in bath) Rotary kilns (for roasting and sintering) (lots of powder materials) Gas vent or wind pipe 1600 800 Density of conveying 160 −3 media /kggm : 16 0.16
usy < u mf 10̚16 6 ̚9 4̚8 2.5̚5.0 2.4 3.0 4.9 9.4 18.0
A usual way to intensify the process for higher output capacity is to raise the effective gases flow rate through the furnaces/kilns. This can be done by changing either the material properties or the composition of the gases. In the general fluidized beds, for example, granularizing the powder-formed load may effectively increase the size of the particles to raise the critical fluidizing velocity umf and the terminal velocity ut or uFD. Increasing oxygen concentration in air may also effectively reduce the quantity of inertia gases such as nitrogen.
10 Multi-objective Systematic Optimization of FKNME
10.2.2 Quality control functions The meanings of a good processing quality of the feeding in furnaces are twofold: a) High reactivity or high conversion rate. b) High direct recovery rate (or in other words, low burning or volatile loss) of the materials in processes. The control functions of materials’ conversion rate and recovery rate are largely variable, depending on different furnaces/kilns and different processes. There is no universal expression covering all situations. For gas-solid reactions, much work has been reported on correlations between the conversion coefficient and the factors including the residence time in the reaction zone, composition of gas phase, reaction rate constant etc. Some examples are given in Table 10.3. Table 10.3
Conversion coefficient in gas-solid reactions (Hetsroni, 1997)
Kinetic models
Average conversion
ķ
Remark
Uniform reaction model: 1
Single order
1−
N-order in series
1− ⎢
1 + k v CA t
⎡
⎤ ⎥ 1 + k C t ⎣ v A i ⎦
N
1
ti =
t N
Shrinking core model: a) Gas film diffusion control: Single order
(1 − e ) β
N-order in series
⎡1 − e−β N ⎢ + β ⎣β
1
β =
−β
1
N −1
∑ m=0
2
When
1
β
˚1
3
(
⎛1⎞ ⎛1⎞ ⎛1⎞ 3⎜⎜ ⎟⎟ − 6⎜⎜ ⎟⎟ + 6⎜⎜ ⎟⎟ − e − β β β ⎝β ⎠ ⎝ ⎠ ⎝ ⎠
)
1 − 0.25 β + 0.05 β 2 − 0.0083 β 3 1−
N −1
(N − m + 2) !
∑ (N − m + 1)!m! (β N )
m −3
e −β N
m =0
β -order in series
(N + m − 1) !3! ⎛⎜ − 1 ⎞⎟ ∑ ⎜ βN⎟ ⎠ m =0 ( N − 1) ! m!(3 − m)! ⎝ 3
−
m
c) Ash diffusion control: Single order
ķ
m is a natural number (mİN−1)
m( Nβ ) N −m−1 ⎤ ⎥ ( N − m)! ⎦
b) Chemical reaction control: Single order
t t
1 − 0.2 β + 0.045 β 2 − 0.0089 β 3 + 0.0015 β 4
ķ Definition of symbols in this table identical to that in Table 10.1.
Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao
The losses in smelting and melting processes include flying ashes, molten slag and discarded slag taken away from the furnace and burning loss etc. To estimate these losses, specific empiric correlations have to be developed for different furnaces and processes. Given xB as the conversion coefficient and R the recovery rate, the control function of the product quality of a furnace is: ϕ = xB • R (10.13)
10.2.3 Control function of service lifetime For most FKNME, the service lifetime under continuous working condition is usually determined by the erosion rate of the refractory linings. Generally, the causes of refractory linings erosion include: a) Molten slag permeating through the gaps between or holes in refractory bricks (static erosion). b) Strength reducing or melting of the refractory due to high temperature (high temperature erosion). c) Scouring and dissolving of the lining surface by molten slag (chemical and mechanical erosion). According to the research of F. Oeters et al. (Oeters, 1994), the erosion rates can be estimated as follows: a) Static erosion rate v jt : ⎛ d σ cos γξ ⎞ =⎜ 0 (10.14) ⎟ ⎝ 16η • t ⎠ where d0 represents the original diameter of a gap or hole; σ represents the surface tension at interface; γ represents the wetting angle between molten slag and the refractory lining; ξ represents the coefficient of labyrinthic degree ( 0 < ξ < 1 ); η represents the viscosity of molten slag; t represents the time that molten slag contacts the refractory surface; and s jt represents the thickness of the eroded refractory linings. b) High temperature erosion vgw : for electric arc furnace, giving the arc voltage v jt =
ds jt dt
U arc , current I arc , the erosion coefficient of the furnace’s lining REw at high temperature can be expressed as (Oeters, 1994), U2 I RE w = arc2 arc (10.15) r where r denotes the radius of the hearth. Taking the static erosion into account, the erosion rate of the linings of an electric arc furnace can be expressed as:
10 Multi-objective Systematic Optimization of FKNME
vgw = where K =
dsgw dt
=
K λ t −t (1 − fs ) REw − w a + v jt L L δ
(10.16)
ew is a ratio with ew representing the radiative heat flux through RE w
hearth wall surface. The shielding coefficient fs measures how much the arc is shaded by the solid charges in the furnace (0
