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

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.

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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
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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 = Œφ (2.29) ∂x j where Œφ is the turbulent diffusion coefficient, which is assumed proportional to the turbulent viscosity:

2 Modeling of the Thermophysical Processes in FKNME 

Œφ=

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 andŒφ 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

Œφ

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

Œφ

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

Œφ

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
Mlim can be determined by solving transport equation or algebra expression. Here, it is modeled as ⎧ ⎛ YO ⎞ without a product term ⎪ min ⎜ YF , ⎟ , i ⎠ ⎝ ⎪ M lim = ⎨ ⎪min ⎛ Y , YO , YP ⎞ , with a product term ⎜⎜ F ⎟ ⎪ i (1 + i ) ⎟⎠ ⎝ ⎩ Damkohler number, Da, is defined by Da ≡ where τ e ≡

k

ε

τe τ CH

, it has the dimensions of time and is called as the turbulent

diffusion time. τ CH is the chemical reaction time: TA

τ CH = ACH e T ( ρYF ) ( ρYO a

2

)

b

(2.107)

where ACH is rate constant; TA is activation temperature; a and b are exponents for the fuel and oxidant density respectively; Die is the ignition/extinction value of D. If Da>>1, the combustion is the finite reaction flow controlled by diffusion; if Da<<1, the combustion is the finite reaction flow controlled by kinetics. The mass fraction of oxidant and product are given by

2 Modeling of the Thermophysical Processes in FKNME 

f − YF fST

(2.108)

YP = 1 − YF − YO

(2.109)

YO = 1 − YF −

2.3.2.3 Magnussen soot model

Soot is a kind of black solid particles taking formation when carbon fuel is burned in a poor oxygen cnciornment. The soot in gaseous flames can result in significantly enhancement of radiative heat transfer, but soot emission cause serious environmental pollution. Detail chemical kinetic models of soot formation are very complex, and there are many factors influencing formation and space distribution of soot, namely turbulent scale, chemical stoichiometric ratio and mixture fraction. There are a lot of models of soot formation, all of which include some empirical parameters (Shadman, 1989; Magnussen, 1989; Tesener et al., 1971). Here, we only introduce the mathematics model of soot formation in gaseous flame, which was suggested by Magnussen. The formation of soot particles occurs in two steps. In the first step radical nuclei are created. Radical nuclei are defined to be the active sites on particles from which the soot deposits will grow. The rate of formation of the radical nuclei is given by ⎛⎛ μ ∂ρ n μ ⎞ ⎞ + ∇ • ( ρUn ) − ∇ • ⎜ ⎜ T + ⎟ ∇n ⎟⎟ = n0 + ( f n − g n ) n − g0 nN (2.110) ⎜ σ ∂t ⎝⎝ T σ L ⎠ ⎠ where n is the concentration of radical nuclei , mol/kg; N is the soot particle concentration , kg/kg; n0 is the rate of spontaneous formation of radical nuclei; (fn − gn) and g0 are constants which in the case of an acetylene flame take the values: (2.111) ( f n − g n ) = 102 g0 = 10−15

(2.112)

Radical nuclei are spontaneously formed at the rate n0 = AYC Cfu e

−

E RT

(2.113)

where YC is the mass fraction of carbon in the fuel and Cfu is the mean concentration of fuel (kg/m3). In the case of an acetylene flame the constants A and E take the values: A = 13.5 × 1036 (2.114) E = 9 × 104 K (2.115) R The soot particle diameter d can be related to A, and there is the following form: (2.116) A • d 3 = constant

  Ping Zhou, Feng Mei and Hui Cai

The rate of soot particle formation is assumed to depend on the interaction between the active radical nuclei and the carbon radicals, which combined with the fact that the radical nuclei are destroyed on the surface of the soot particles: ⎛⎛ μ ⎞ ∂ρ N μ ⎞ + ∇ • ( ρUN ) − ∇ • ⎜⎜ ⎜ T + (2.117) ⎟ ∇N ⎟⎟ = ( a − bN ) n ∂t ⎝⎝ σT σ L ⎠ ⎠ where a and b are constants.For the case of an acetylene flame, take the values a = 105 , b = 8 × 10 −14 .

2.3.3 Droplet and particle combustion models In this section, combustion of coal and oil is to be taken as examples respectively for introduction of particle and droplet combustion models. 2.3.3.1 Coal combustion in gas phase

The combustion of a coal particle is a two-stage process: the devolatilization of raw coal particle and the oxidation of residual char. At the first stage, the coal particles give off volatile, which is burned around char particle and the gaseous flame in space is formed. At the second stage, the gas-solid two-phase combustion takes place between the chars and oxidants. So, two separate gases are given off by the particles, the volatiles and the char products, the latter come from the burning of carbon within the particle. The simulation of coal particles combustion is to solve jointly the gas-particle two-phase flow model and the volatile-char oxidizing reaction model. The coupling of the particle phase, the gaseous pahse and the concentration filed can be handled by the PSIC method (Zhou,1994;Zhen and Zhou,1996). In the gas phase, the description of the mixed-is-burnt ( k-¦-f-g) and the eddy break-up (EBU) models previously addressed are all applicable to the combustion of volatile as fuel, apart from the volatiles, oxidant and products form only the part of the gas phase that is not char products. If the mixed-is-burnt model is being used, the instantaneous mass fractions are given by: If f ı fST (1 − YPC ) : YF =

f − fST (1 − YPC ) 1 − fST

, YO = 0

(2.118)

If f < fST (1 − YPC ) : YF = 0 , YO = 1 − YPC −

f fST

(2.119)

If the eddy break-up model is used, and the equation for mass fraction of fuel is

2 Modeling of the Thermophysical Processes in FKNME 

unchanged, the mass fraction of oxidant is given by: f − YF YO = 1 − YF − − YPC fST For both models, the amount of products is given by: YP = 1 − YF − YO − YPC

(2.120)

(2.121)

where YPC is the mass fraction of char products, the definition of mixture fraction and stoichiometric mixture fraction can be seen in Section 2.3.2. 2.3.3.2 Coal pyrolysis and devolatilization

There are single-reaction model (Badzioch and Hawksley,1970) and two-reaction model (Ubhayakar et al., 1976) to describe the devolatilization of the coal. In the single-reaction model (Field et al., 1967), the coal is considered to have fixed fractions of volatiles, char and ash. The rate of production of the volatile gases is given by the first order reaction (Gibb,1985): dm = kV ( mfv − m ) (2.122) dt where m refers to the mass of volatiles which have escaped from unit mass of raw coal; mfv refers to the total yield of volatiles, the rate constant kV follows Arrhenius law: ⎛ E ⎞ (2.123) k V = AV exp ⎜ − V ⎟ ⎜ RT ⎟ p ⎠ ⎝ where Tp is the temperature of coal particle (assumed uniform); AV and EV is preexponential factor and activation energy respectively, and they are empirical constants and vary with the type of coal. Integration of Eq. 2.122 gives the fractional volatile yield as a function of time: t m (2.124) = 1 − exp − ∫ kV dt 0 mfv

(

)

Since the coal particles are heated by the furnace gases, Tp and kV varies with time. The single-reaction model is only appropriate for the devolatilization at the medium temperature. EV, AV and mfv are functions of temperature, and their values at the higher temperature are very different from that at the lower temperature or at the medium temperature. Therefore, the two-reaction model is more generally used. It assumes two-reaction with different rate parameters compete to pyrolyse the raw coal. Considering the raw coal (dry and ash-free) with the mass mc, then k1 raw coal mc k2

volatiles˄mv1˅+ residual char˄mch1˅ α1 (1ˉα1) volatiles˄mv2˅+ residual char˄mch2˅ α2 (1ˉα2)

  Ping Zhou, Feng Mei and Hui Cai

α1 and α2 are the equivalent percentages of the volatiles for the corresponding reaction. The first reaction dominates at lower particle temperatures and the second reaction dominates at higher temperatures. Generally, the volatile α1 is less than α2 during the devolatilization of coal. As a result, the total yield of volatiles will depend on the temperature history of the particle, and will increase with temperature. k1 and k2 is the rate constant following Arrhenius law: ⎛ E ⎞ kn = An exp ⎜ − n ⎟ (n=1,2) ⎜ RTp ⎟ ⎝ ⎠ At time t, assume that a coal particle originally with unit mass consists of mass mc of raw coal, mass mch of residual char after devolatilization has occurred, and mass mA of ash. The reaction rate constants k1 and k2 determine the rate of conversion of the raw coal dmc = − ( k1 + k2 ) mc (2.125) dt The rate of volatiles production is given by: dmv = (α1k1 + α 2 k2 ) mc (2.126) dt and the rate of char formation is: dmch = ( (1 − α1 ) k1 + (1 − α 2 ) k2 ) mc (2.127) dt The initial value of m c is equal to (1 − mA ); The values of α1, A 1 and E1 can be got from approximate analysis of the coal. α2, A 2 and E 2 can be got from  the pyrolysis analysis. Some references suggest: A 1=3.7×10 5 s 1 ,  A2 =1.46×1013s 1 , E1 =7.41×10 4kJ/mol, E 2 =2.525×105 kJ/mol (Smoot and Pratt, 1983;Zhou,1986). 2.3.3.3 Char oxidation (heterogeneous reaction)

Although there are many models for describing char oxidation, Field model (Field et al., 1967) and Gibb model (Gibb, 1985) are usually used in the simulation. The former is a simple reaction model and the latter takes into account the diffusion of oxidant within the pores of the chars. Since the combustion of the chars is much slower than the coal devolatilization, it will decide the time of coal particle burnt out. In the Field (Fu and Wei, 1984), a char particle is considered to be a spherical particle surrounded by a stagnant boundary layer (called stagnation film) through which oxygen must diffuse before it reacts with the char. The oxidation rate of the char is limited by the diffusion of oxygen to the external surface of the char particle and by the effective char reactive rate. The diffusion rate of oxygen is given by kd (pgˉps), where pg is the partial pressure of oxygen in the furnace gases far from the particle boundary layer and ps is the oxygen pressure at the

2 Modeling of the Thermophysical Processes in FKNME 

particle surface. The value of kd is given by. kd =

2.53 × 10−7 ⎛ Tp + Tg ⎞ ⎜ ⎟ Rp ⎝ 2 ⎠

0.75

pA p

(2.128)

where Rp is the particle radius; Tp is the particle temperature; Tg is the far-field temperature˗p is the local pressure and pA is atmospheric pressure. The char oxidation rate per unit area of particle surface is given by kcps. The chemical reaction rate coefficient kc is given by ⎛ T ⎞ kc = AcTp exp ⎜ − c ⎟ (2.129) ⎜ Tp ⎟ ⎝ ⎠ where Ac and Tc depend on the type of coal and their values can refer to the relative references (Smoot and Pratt,1983;Zhou,1986;Wall,1986). For this model, kd and kc are in units of kg/(m2 • atm • s). The overall reaction rate of a particle is given by: −1 (2.130) ( kd−1 + kc−1 ) pg 4πRQ pp A and is controlled by the smaller rates of the kd and kc. Gibb model (Gibb, 1985) takes into account the void fraction ε of the char particle, the particle volume /internal surface ratio a, the effective internal diffusion coefficient Dp of oxygen within the pores, and the molar ratio φ of carbon atoms/oxygen molecules. The oxidation mechanism of carbon can be characterized by the parameter φ , then the oxides are produced according to the equation: (2.131) φ C + O 2 ⎯⎯ → 2 (φ − 1) CO + ( 2 − φ ) CO2 The value of φ is assumed to depend on the particle temperature Tp ⎛ T ⎞ 2 (φ − 1) = As exp ⎜ − s ⎟ ⎜ Tp ⎟ 2 −φ ⎝ ⎠

(2.132)

where As and Ts are constants. Gibb suggests As=2500K and Ts=6240K. By solving the oxygen diffusion equation analytically, the following equation is obtained for the rate of decrease in the char mass mc dmc 3φ M c C∞ −1 −1 −1 (2.133) k1 + ( k2 + k3 ) =− dt 1 − ε M 0 ρ c

(

)

where the far field oxygen concentration C∞ is taken to be the time-averaged value

obtained from the gas phase calculation; ρc is the density of the char; k1 is the rate of external diffusion; k2 is the surface reaction rate; k3 is the rate of internal diffusion and surface reaction. These are defined as follows: D (2.134) k1 = 2 Rp

  Ping Zhou, Feng Mei and Hui Cai

where D represents the external diffusion coefficient of oxygen in the surrounding gas. k (2.135) k2 = (1 − ε ) c Rp where kc is the carbon oxidation rate, defined by the modified Arrhenius equation: ⎛ T ⎞ kc = AcTp exp ⎜ − c ⎟ (2.136) ⎜ T ⎟ ⎝ p⎠ where Ac and Tc are constants. Gibb recommends Ac=14m/s and Tc=21580K. Further k3 = kc ( β cothβ − 1) / β 2 a

(2.137)

0.5

⎛ kc ⎞ (2.138) ⎜ D ε a ⎟⎟ ⎝ p ⎠ Field model may predict sufficiently accurate when the temperature is not too high but correction on the reduction and combustion of CO2 must be considered if the temperature is high (Zhou, 1982).

β = Rp ⎜

2.3.3.4 Oil combustion

The simulation on oil combustion is similar to that on coal combustion. The mass fractions of fuel, oxidant and product in gaseous phase are predicted through computing mixture fraction, which is identical to the aforementioned gaseous combustion model (Guo, 1997). The difference is that both coal devolatilization and carbon oxidation need to be considered in coal combustion, which are the chemical reactions in solid and gaseous phase. However, for oil combustion, the evaporation of the drops needs only to be considered. Generally, there are no chemical reactions at the surface of oil droplets and the combustions only carry out in volatiles given off by the evaporation of the drops. Spherical liquid droplets are assumed to be heated up to their boiling point temperature and then to evaporate at a rate determined by the heat transfer to the droplet and the latent heat of the liquid. The decrease in droplet diameter is computed from the evaporative mass loss based on the assumption that the liquid density remains constant. In 1993, Barreiros suggested a mathematic model for oil combustion (Barreiros et al.,1993). At any instant t, the droplet is assumed to have diameter d p , a uniform temperature T p, while liquid density ρ p and specific heat capacity cp are assumed to be constant. According to the film theory for the evaporation of liquid droplets, once the gas temperature T g exceeds the

2 Modeling of the Thermophysical Processes in FKNME 

boiling point temperature Tb , the evaporation equation for the droplets is given by: d(d p2 ) 4 Nu* λg ln (1 + B ) =− (2.139) dt ρ p cp that is dd p dt

=−

2 Nu * λg

ρ p cp d p

ln (1 + B )

(2.140)

where λg is thermal conductivity; B is the mass transfer number defined by: B=

cg L

(T

g

− Tb )

(2.141)

where L is the latent heat of vaporization of the liquid and cg is specific heat of gas, Nu* is the modified Nusselt number and is given by: Nu * = C • NuRe = 0 where Nu = 2 + 0.55 Re0.5 Pr 0.3333

1.237 ⎞ ⎛ C = 1 + 0.278Re0.5 Pr 0.3333 ⎜ 1 + 1.3333 ⎟ ⎝ RePr ⎠

(2.142) −0.5

(2.143)

where Re is the Reynolds number of the liquid droplet, that is: ρg d p Re = vg − vp

μg

(2.144)

where ρg is gas density, vg and vp are gas velocity and droplet velocity respectively; Pr is the Prandtl number: Pr =

μ g Cg λg

(2.145)

Hence, the decrease rate of droplet diameter is given by: dd p dt

= −C

4λg

ρ p cp d p

ln (1 + B )

(2.146)

Heat transfer to the droplet is modeled by the equation: dTp dt

=

6λg Nu

ρ p cp d

2 p

(T

g

− Tp ) +

3 L dd p 6ε + I p − σ 0Tp4 ) ( cP d p dt ρ p cP d p

(2.147)

where Ip is the radiative heat flux through environment to droplets˗ε is emissivity of droplet˗σ0 is the Stefan-Boltzmann constant.

  Ping Zhou, Feng Mei and Hui Cai

2.3.4 NOx models NOx given off from combustion consist of mostly nitric oxide NO and nitrogen dioxide NO2. Besides, there is also minor nitrous oxide N2O. During the combustion, the concentration of NO is higher than that of NO2, and NO2 is produced from NO. Thus, the key to the reaction dynamic model of NOx is to research into the formation of NO. NOx is classified as thermal NOx, prompt NOx and fuel NOx, based on three distinct chemical kinetic processes that form NOx. Thermal NOx is formed by oxidation of atmospheric molecular nitrogen N2 at high temperature. According to Zeldovich mechanism, it is generally described by˖ N2ˇO

NOˇN

(2.148)

NˇO2

NOˇO

(2.149)

Prompt NOx is formed by a series of reactions and many possible intermediate species between hydrocarbon and N2. Based on Fenimore mechanism, it is generally simplified as: (2.150) CHˇN2 HCNˇN (2.151) CH2ˇN2 HCNˇNH 1 N2ˇO2 NOˇO (2.152) 2 NˇOH NOˇH (2.153) CNˇO2 NOˇCO (2.154) Fuel NOx is produced by oxidation of nitrogenous compound in devolatilized fuel. Although the reaction mechanism for the formation and reduction of fuel NOx is unclear, the following observations can be made based on the researches reported in the past a few years (Mao et al., 1998): a) In the normal combustion conditions, nitrogenous organic compound in fuel are heated and decomposed into HCN, NH3 and intermediate product CN, etc, which are emitted from fuel with volatiles. They are called volatile-N. Nitrogenous compound remained in char are called char-N. b) The percentages of HCN and NH3 among volatile-N depend not only on the types of fuel and the character of volatiles, but also on the chemical character such as combination state between N and hydrocarbon. Besides, they are related to the combustion conditions such as temperature. c) In oxidizing atmosphere, HCN is oxidized into NO; In reducing atmosphere, NO and HCN are reduced into N2. The simplified model representing the forming and reducing mechanism of fuel NOx is showed in Fig. 2.5.

2 Modeling of the Thermophysical Processes in FKNME 

Fig. 2.5

Main path for fuel NOx formation and reduction

For thermal and prompt NOx, only the transport equation for NO species is needed: ⎛⎛ μ ⎞ ∂ρYNO μ ⎞ + ∇ • ( ρUYNO ) − ∇ • ⎜⎜ ⎜ T + ⎟ ∇YNO ⎟⎟ = S NO ∂t ⎝⎝ σT σ L ⎠ ⎠

(2.155)

For coal and oil combustion, in addition to the above equation, the transport equations for HCN and NH3 species are needed: ⎛⎛ μ ⎞ ∂ρYHCN μ ⎞ + ∇ • ( ρUYHCN ) − ∇ • ⎜⎜ ⎜ T + ⎟ ∇YHCN ⎟⎟ = S HCN ∂t ⎝ ⎝ σT σ L ⎠ ⎠ ∂ρYNH3 ∂t

⎛⎛ μ ⎞ μ ⎞ + ∇ • ρUYNH3 − ∇ • ⎜ ⎜ T + ⎟ ∇YNH3 ⎟⎟ = S NH3 ⎜ σ ⎝⎝ T σ L ⎠ ⎠

(

)

(2.156)

(2.157)

where YNO, YHCN and YNH3 are respectively mass fractions of NO, HCN and NH3; SNO, SHCN and S NH3 are respectively source or sink term in the mass fraction transport equations for NO, HCN and NH3. Since the mass fractions of those pollutants are generally small (<10−3), they are calculated as passive combustion scalars, that is their influence on flow velocity and other scalars, such as temperature, pressure and the concentration of other species, can be neglected. 2.3.4.1

Thermal NOx formation

According to Zeldovich mechanism, the net rate of formation of thermal NO is

  Ping Zhou, Feng Mei and Hui Cai

given by d [ NO ] dt

= k1 [ O ][ N 2 ] + k 2 [ N ][ O 2 ] − k −1 [ NO ][ N ] − k −2 [ NO ][ O ]

(2.158)

In order to calculate the formation rate of NO, the steady state assumption for [N] is used (Mao, 1998), thus we obtain: d [ NO ] dt

2 [ O] (k1k2 [ O 2 ][ N 2 ] − k−1k−2 [ NO ] ) 2

=

k2 [ O 2 ] + k−1 [ NO ]

(2.159)

where [NO], [O2], [N2], [O] and [N] refer to the molar concentration of corresponding species respectively, and their units are mol/m3. Their reaction rate coefficients are respectively given by (Dong, 1998): (m3molgV k1=1.8×108exp(−38370/T) 7 k-1=3.8×10 exp(−425/T) (m3molgV 4 (m3molgV k2=1.8×10 Texp(−4680/T) k-2=3.8×103Texp(−20820/T) (m3molgV Based on the assumption that the dissociation reactions of [O] reach a partial equilibrium, we have:

[O] = 36.64T 0.5 [O 2 ]

0.5

exp(−27123 / T )

(mol/m3)

(2.160)

In the transport equation for mass fraction of NO, the NO source term due to thermal NOx mechanism is: d [ NO ] S thermal,NO = M NO (2.161) dt where MNO represents the molecule weight of NO. 2.3.4.2

Prompt NOx formation

Prompt NOx mainly result from the combustion circumstance with poor oxygen and rich fuel in which there are more CHi atomic groups. In hydrocarbon flame, the atomic groups of CH and CH2, via the reactions Eq.2.150 and Eq.2.151, form the intermediate products, and they are further oxidized into prompt NOx. Prompt NOx formation is proportional to the number of carbon atoms present per unit volume. Soete’s research results show that, for most hydrocarbon fuel, the control equation of prompt NOx formation rate can be described as (Soete,1975): d [ NO ]

⎛ −En ⎞ α (2.162) = fs kprn [ O 2 ] [ N 2 ][ Fuel] exp ⎜ α ⎟ dt ⎝ RT ⎠ where both kprn and Eαn are experimental constants and their values can refer to reference (Dupont and Porkashnian,1993). α is the order of oxidation reaction which depends on combustion conditions, as

2 Modeling of the Thermophysical Processes in FKNME 

⎧1.0, ⎪ ⎪⎪ −3.95 − 0.9 ln xO 2 , α =⎨ ⎪ − 0.35 − 0.1ln xO 2 , ⎪ ⎪⎩0 ,

xO2 <4.1× 10 −3 4.1× 10 −3 < xO2 <1.1× 10−2 1.1× 10−2 < xO2 <0.03

(2.163)

xO2 > 0.03

where xO2 is oxygen molar fraction; fs is Soete model’s correction factor which incorporates the fuel type, as f s = 4.75 + 0.0819n − 23.2φ + 32φ 2 − 12.2φ 3

(2.164)

where n is the number of carbon atoms per molar hydrocarbon fuel; φ is the equivalence ratio. Therefore, in the transport equation for mass fraction of NO, the NO source term due to prompt NOx mechanism is: d [ NO ] Sprompt,NO = M NO (2.165) dt In the common combustion, prompt NOx accounts for a very small part, namely, for coal combustion, prompt NOx formation is less than 5% of the total. 2.3.4.3 Fuel NOx formation HCN path

The main reaction paths of fuel NOx by the intermediate products HCN are shown as Fig. 2.6.

Fig. 2.6

The main reaction paths of fuel NOx by the intermediate products HCN

In terms of the NO reaction mechanism suggested by Lockwood and the assumption that all char-N can be directly converted into NO (Lockwood,1992), the source terms in the equations for mass fraction can be described as: S HCN = S pf,HCN + S HCN −1 + S HCN − 2

(kg/(m3 • s))

S NO = SC,NO + S NO −1 + S NO − 2 + S NO − 3

(kg/(m3 • s))

(2.166) (2.167)

The rate of HCN production is equivalent to the rate of liquid fuel released into

  Ping Zhou, Feng Mei and Hui Cai

the gas phase through evaporation or the rate of solid fuel released into the volatile through volatilization. That is: Spf,HCN = Spf mNF M HCN / M N / V

(2.168)

where Spf is the rate of fuel release into the gas phase through evaporation or volatilization; mNF is the mass fraction of nitrogen in fuel; MHCN and MN are respectively the molecule weight of HCN and N; V is the cell volume. The mass consumption rates of HCN which appear in Eq.2.168 are computed as: S HCN −1 = − R1 M HCN p / RT

(2.169)

S HCN − 2 = − R2 M HCN p / RT

(2.170)

where R1 and R2 are conversion rates of HCN in reactions ( ĉ ) and ( Ċ ) respectively, s-1; p is pressure, Pa, T is the mean temperature, K; R is the general gas constant. According to Soete’s research result, we have: R1 = A1 xHCN xOα 2 exp(− E1 /( RT ))

(2.171)

R2 = A2 xHCN xNO exp(− E2 /( RT ))

(2.172)

where α is the order of oxidation reaction and is the same as Eq.2.163, T is the instantaneous temperature; x is the molar fraction and 

A1=3.5×1010

˄s 1 ˅

A2=3.0×1012

˄s 1 ˅



E1=2.805×105

˄J/mol˅

5

˄J/mol˅

E2=2.512×10

NOx is produced in reaction (ĉ) but destroyed in reaction (Ċ), thus the source terms for Eq.2.167, i.e. SNO−1 and SNO−2, are evaluated as follows: S NO −1 = − S HCN −1 M NO / M HCN = R1 M NO p /( RT )

(2.173)

S NO − 2 = SHCN − 2 M NO / M HCN = − R2 M NO p /( RT )

(2.174)

The mass consumption rates of NO in reaction (ċ), SNO-3 (kg/(m • s)), can be 3

expressed as: S NO −3 = ABET Cs M NO R3 /1000 Ზ

(2.175)

2

where ABET is BET surface area (m /kg); Cs is the concentration of particles, kg/m3; ᲖBET surface area is a method for measuring specific surface area, which is suggested by Brunauer, Emmet and Teller and based on multi-layer absorption theory for the gas absorbed by the surface of solid particles (refer to “Chemical Engineering Handbook”(in Chinese),1989). It is called as BET method and the specific surface area tested by means of this method is called as BET surface.

2 Modeling of the Thermophysical Processes in FKNME 

and the heterogeneous reaction rate of NO reduction on the char surface R3 can be modeled by (Dong,1998): R3 = A3 xNO exp(− E3 /( RT )) p where A3=230mol/ (atm

•

m

2

BET •

(2.176) 5

s); E3=1.428×10 J/mol; T

is the mean

temperature, K; p is pressure, atm,latm=101325Pa; xNO is the molar fraction of NO. The formation rate of NO from char-N is given by: SC,NO = SC mNC M NO / M N / V

(2.177)

where SC,NO is the char burnout rate, kg/s; and mNC is the mass fraction of nitrogen in the char. For the liquid fuel, both SC,NO and SNO−3 are equal to zero. NH3 path

The main reaction paths of fuel NOx by the intermediate products NH3 are shown as Fig.2.7. The overall reaction are:

Fig. 2.7

The main reaction paths of fuel NOx by the intermediate products NH3 k1 NH3ˇO2 ⎯⎯ → NOˇH2Oˇ0.5H2

(2.178)

NH3ˇNO ⎯⎯→ N2ˇH2Oˇ0.5H2

(2.179)

k2

According to Soete’s research result, the reaction rates are respectively given by: k1 = 4.0 × 106 xNH 3 xOα 2 exp(−32000 /( RT ))

(2.180)

k 2 = 1.8 × 10 xNH 3 xNO exp(−27000 /( RT ))

(2.181)

8

where α is the oxidizing reaction order just as Eq.2.163. The source terms in the transport equations can be described as follows: S NH3 = Spf,NH3 + S NH3−1 + S NH3− 2 (kg/(m3 • s)) (2.182) S NO = SC,NO + S NO−1 + S NO−2 + S NO−3

(kg/(m3 • s))

(2.183)

  Ping Zhou, Feng Mei and Hui Cai

The calculating method of Spf, NH3 is similar to that in HCN paths, and there is: S pf,NH3 = S pf mNF M NH3 / M N / V (2.184) where

M NH 3

is the molecule weight of NH3. S NH3−1 = −k1 M NH3 p / RT

(2.185)

S NH3−2 = − k2 M NH3 p / RT

(2.186)

NO is produced in reaction (ĉ) but destroyed in reaction (Ċ) and (ċ), we have S NO−1 = − S NH3−1 M NO / M NH = k1 M NO p /( RT ) (2.187) 3

S NO−2 = S NH3−2 M NO / M NH3 = −k2 M NO p /( RT )

(2.188)

The calculating method of SC,NO and SNO-3 are similar to that in HCN paths.

2.4 Simulation of Magnetic Field Electric field and magnetic field widely exist in the power equipments in FKNME. The interaction between the magnetic field and the current flow in the melt generates electromagnetic forces causes the melt flow and interface wave thus effecting the production directly. So, to simulate the melt flow and its flow field, the current field must be first computed before the magnetic field can be determined.

2.4.1 Physical models The magnetic field in FKNME is a joint effect of the currents in the electrode (anode, cathode, bus bar), the electric heating elements and the melt. Meanwhile, because of the ferromagnetic materials such as the jack and steel structure of the furnace, influences of ferromagnetic shields should also be considered in order to simulate precisely the magnetic field in the furnace. Simplification has to be made if the electric field and the magnetic field are to be simulated in a complex FKNME. Besides simplifying the furnace structure and the shape of electrical conductor, the following simplification still has to be done. The time-averaged value is used to represent the current flow in the melt to eliminate the current fluctuation with the operation process. The fluctuation is actually the outcome of many factors and it is difficult to be described in a mathematical way. The current flow in the electric conduction elements is simplified into the

2 Modeling of the Thermophysical Processes in FKNME 

constant current, arithmetical progressive changing current and linear changing current. In cases when there are more than one metallurgical furnaces in the workshop, it is also necessary to take the effects of adjacent furnaces into account.

2.4.2 Mathematical model of current field The purpose to study the current field is to determine the current distribution in the melt of the furnace, so as to study the interactions between the current and magnetic fields. On the basis of previous experience and some proper hypothesis, the physical model is to be simplified into a two-dimensional section model. The boundary element method is used to simulate the current flow field. Basic equations for problem are: ⎫ Ω:∇2u =0 ⎪ ⎪ (2.189) S1:u =u 0 ⎬ ⎪ S2 :q =∂u / ∂n=q0 ⎪⎭ where u is the potential function; Ω is the solution domain; S1 is the first boundary condition; S2 is the second boundary condition, S = S1 + S2 is the boundary of Ω, and n is the exterior normal direction of S. According to Green theorem or weighted residual method, the boundary integral equation can be deduced by rendering the basic solution as a weight function: ∂u * ⎞ ⎛ ∂u Ciui = ∫ ⎜ u * − u ⎟dS ∂n ∂n ⎠ s⎝ 1 i∈Ω ⎫ ⎧ ⎪ ⎪ Ci = ⎨ 1/ 2 i∈S ⎬ ⎪ ⎪ 1 i S ∉ U Ω ⎩ ⎭ where u* is the basic solution. For three-dimensional isotropic dielectric: u* =

1 4πr

where r is the distance from the field point to the source point. In a two-dimensional field: u* =

1 1 ln 2π r

(2.190)

(2.191)

  Ping Zhou, Feng Mei and Hui Cai

2.4.3 Mathematical models of magnetic field in conductive elements Knowing current distribution in the electrodes (anode, bus bar), the electric heating element and the melt, the total magnetic field is computed using the Biot-Savart law and its mathematical model is as follows (Yang, 1998). 2.4.3.1 Mathematical model of equivalent axial current

The equivalent axial current model assumes the current through the conductors all concentrate along the axis so that can be represented by a linear current (Cai, 1992). Given a constant magnetic field, the magnetic induction intensity generated by the current flow can be computed with the Biot-Savart law. B ( x, y , z ) =

μ0

∫ J ( x′, y′, z ′) ×

4π v1

r0 dV r2

(2.192)

where μ 0 is the vacuum permeability, μ 0 = 4π×10-7H/m; dV is the elementary volume around the source point; J ( x′, y ′, z ′) is the current density; the integral domain V1 is the whole current carrying conductor domain (Fig. 2.8). For linear currents, J ( x′, y ′, z ′) dV =J • ds • dl=I • dl. So, Eq. 2.192 can be rewritten into: μ Idl × r 0 B ( x, y,z ) = 0 ∫ (2.193) 4π l r 2

Fig. 2.8

Magnetic field of current

2 Modeling of the Thermophysical Processes in FKNME 

If there is linear varying current running along axis s, of which the current equation is I=αs + β , where AǃB are the two ends of the linear current, and the corresponding coordinate is s1ǃs2 (s1<s2), P is field point (Fig. 2.9), according to Eq. 2.193, the magnetic induction intensity of point P is:

B=

μ0 4π

s2

∫

s1

Ids × r 0 r2

(2.194)

of which, its value can be obtained as:

μ 0 2 (α s + β ) sin θ s

B=

4π

∫

s1

r2

ds

(2.195)

Fig. 2.9 Magnetic field of linear varying current

Because r = a / cos φ , s = a / tan φ , the modulus of magnetic induction intensity vector at point P is: B=

⎛ 1 1 μ0 ⎡ − ⎢ −α • a ⎜⎜ 2 2 2 4π ⎣⎢ s1 + a 2 ⎝ s2 + a

⎞ β⎛ s2 s1 − ⎟⎟ + ⎜⎜ 2 2 2 s1 + a 2 ⎠ a ⎝ s2 + a

⎞⎤ ⎟⎟ ⎥ ⎠ ⎦⎥ (2.196)

where α and β are unknown constants, and their values can be determined by the current at point A and B. Supposing the current of points A and B are I1 and I2 respectively, then: I1 − I 2 α= s1 − s 2 I 1s 2 − I 2 s1 β= s 2 − s1 Replacing above α and β into Eq. 2.196, the magnetic induction intensity at point P (B), can be calculated with values of s1ǃs2ǃI1ǃI2. The components Bxǃ ByǃBz can then be determined by cosine of ds × r 0 .

  Ping Zhou, Feng Mei and Hui Cai

When the value of current I is a constant, there is: B=

μ 0I ⎛ s2 s1 − ⎜ 2 4 πa ⎜⎝ s 2 2 + a 2 s1 + a 2

⎞ ⎟⎟ ⎠

(2.197)

2.4.3.2 Mathematical model of equivalent column current

Mathematical model of equivalent column current flow is to replace the conductive element of a cross section in any shape with a conductive column. The equivalence follows three principles: a) the cross-sectional area stays the same, so the current density remains unchanged; b) the axis of the conductive column coincides with that of the real conductive element; c) the equivalent column conductor and the real one share the same ends, thus have the same length. For a column conductor with radius R0 and finite length (as shown in Fig. 2.10), the magnetic induction intensity at point P in the space around it is (Feng, 1985):

B=

μ0

Δz2

4 π Δ∫z1

Idz × r 0 r2

In Fig.2.10, I = α s + β , dz × r 0 = sin θ dz ˈso, the modulus of the magnetic induction intensity at point P is: B=

μ0

Δz2

(α s+β ) sin θ dz 4π Δz1 r2

∫

(2.198)

Fig. 2.10 Magnetic field of column current

Eq. 2.198 is only applicable to calculate the magnetic field outside the column conductor (RıR0). Its form is the same as Eq. 2.195. When the interior magnetic

2 Modeling of the Thermophysical Processes in FKNME 

field of the column conductor is calculated, the magnetic induction intensity is: R B= B0 (0İRİR0) R0 where B0 is the magnetic induction intensity when R=R0. The above two mathematical models are equivalent. When the computed field point is far away from the conductive element, the two models give the same results with enough precision. When the conductive element is a column and the point calculated is within the element, the results by the equivalent axis current model will predict larger error, and the equivalent column current model should be a better choice. When conductive element is rectangle (such as bus bar of aluminum cell ) and the point calculated is within the element or near it, neither of the models gives satisfactory result. In this case, a mathematical model for the magnetic field in rectangular bus bar should be used. 2.4.3.3 Mathematical model for calculation of magnetic field in rectangular bus bar (for aluminum cells)

In order to improve the precision of calculation of the magnetic field in aluminum cells, the problems of equivalent mathematical models mentioned above must be overcome, that is, rectangular cross section of the bus bar in aluminum cells must be taken into account. Hence, it would be better to compute the magnetic field in aluminum cells, with a model in which the bus bar is a rectangular, with finite length, and the current flowing though it is uniformly distributed. The model is called the rectangular bus bar model of the magnetic field in aluminum cells, and it is also applicable for magnetic fields of other rectangular conductive elements (Liang,1990). As shown in Fig. 2.11 (a), the bus bar is parallel to axis z, and the lengths of the sides of the cross-section of the bus bar are respectively 2a and 2b. Meanwhile, the length of the bus bar is L=z2z1, the current is I, and flows along axis z. Taking a filament, of which the cross-sectional area is dx 'dy ' , the length is L, the current is I ' and parallel to axis z (as illustrated in Fig. 2.11(b)). Then by Eq. 2.197, the magnetic induction intensity caused by current I ' at the point P (x, y, 0) is: ⎛ μ 0I ' z2 dB = ×⎜ − 2 2 2 ⎜ ( x '− x) + ( y '− y ) 2 + z 2 4 π ( x '− x ) + ( y '− y ) 2 ⎝

⎞ ⎟ ( x '− x) 2 + ( y '− y ) 2 + z12 ⎟⎠

z1

(2.199)

  Ping Zhou, Feng Mei and Hui Cai

Fig. 2.11 Magnetic field of rectangular bus bar

If the current in the rectangle is considered as uniformly distributed, then:

I′ =

I dx′dy ′ 4ab

Replacing above equation into Eq.2.199 and integrating it, the magnetic induction intensity at point P is: ⎛ a b z2 μ 0I B=∫ ∫ ×⎜ − − a −b 2 2 2 16πab ( x '− x ) + ( y '− y ) ⎜⎝ ( x '− x) + ( y '− y ) 2 + z22

⎞ ⎟ dx 'dy ' (2.200) ( x '− x) 2 + ( y '− y ) 2 + z12 ⎟⎠ With cosine of dz × r 0 , the components Bx, By, Bz can then be obtained with the result of Eq.2.200. When the magnetic induction intensity of any point P(x, y, z) is to be computed in the cell, the coordinate should be changed first, that is, to move the origin of the coordinate upwards (or down words) along axis z to point z, so that the new coordinate of point P can be changed into P(x, y, 0), before the magnetic field of the current carrying bus bar can be computed with the above method. z1

2.4.4 Magnetic field models of ferromagnetic elements Because of the existence of ferromagnetic materials, the influences of ferromagnetic shield must be considered as well so that the magnetic field in the furnaces can be simulated correctly. Two research techniques can be used to study the influence of ferromagnetic elements on the magnetic field: the analogy method and the numerical simulation method. The latter can be further classified as

2 Modeling of the Thermophysical Processes in FKNME 

differential equation method, integral equation method and hybrid method. There is another method called the magnetic shielding factor method, which is considered as an approach between the analogy method and the numerical simulation method. 2.4.4.1 Analogy method

Analogy model is based on the analogy principle. It is a model that transforms industrial data into laboratory model, with which the influence coefficients of ferromagnetic material on the magnetic field can be studied, thus the value of magnetic induction intensity of corresponding points in the cells can be computed. The analogy model is particular useful when new furance is to be developed but no reliable data is available to serve as development guidelines. This method requires, however, particular analogy model be built up for each new furnace development work and the model be physically identical to the prototype. These requirement makes the analogy method very expensive and time-consuming. 2.4.4.2 Magnetic shielding factor method

If there are magnetic materials between the conductive element and the point to be computed, the magnetic field will be weaker than that when there is no magnetic material. By introducing parameter MAF, the magnetic field B with magnetic materials can be computed from the magnetic field B0 that is without magnetic material: B=MAF • B0 where MAF is the magnetic attenuation factor, and usually MAFİ1, but for nonmagnetic material, MAF=1. Generally, MAF is relating to the shape, size and relative permeability of the magnetic materials. MAF can be determined by image method principle or model experiment. 2.4.4.3 Numerical method Differential equation method

The intensity of the magnetic field at one point in the space (H) comes from Hs and Hm, of which Hs is the field intensity produced by the current in the space, and Hm is the field intensity caused by the ferromagnetic material. H= Hs + Hm

If to describe the field by simplified scalar potential (Φ is scalar potential), then there is: Hm = −∇ Φ

If the field is described by Poisson equation, then there is:

  Ping Zhou, Feng Mei and Hui Cai ∇ • μ∇ Φ = −∇ • μ H s

(2.201)

With proper boundary conditions, Eq. 2.201 can be solved with finite-element method or finite difference method. This is the differential equation method.

Integral equation method

The field can be described with integral as (Fan and Yan, 1988): Hs =

Hm = −

1 J (r ′) × r dV 4 π ∫V r3

(2.202)

1 r ⎞ ⎛ ∇ • ⎜ M • 3 ⎟ dV 4π ∫V ⎝ r ⎠

(2.203)

where M is the magnetic polarization, M = qml / ∇V = pm /( sl ) , and pm is the magnetic moment. As in Eq.2.203, M =B / μ − H, M is a function of H, and the equation can be solved by discretization within the whole field. When the ferromagnetic material is simplified as a pair of magnetic dipole, it can only be polarized along the axis, and this method is called the equivalent magnetic dipole method. As shown in Fig.2.12, the magnetic density and magnetic induction intensity produced by the magnetic dipole at the point P is:

Fig. 2.12

H=

Drawing for magnetic dipole

1 ⎡ 3( pm • r )r pm ⎤ − 3⎥ 4π ⎢⎣ r5 r ⎦

B = μ 0H =

μ 0 ⎡ 3( pm • r )r

4π ⎢⎣

r

5

−

pm ⎤ r 3 ⎥⎦

where pm=SMl ˈ in which S is sectional area of magnetic dipole, and M is polarization of that.

2 Modeling of the Thermophysical Processes in FKNME 

The above equation must satisfy the condition that r (the distance from point P to the center of the magnetic dipole) is much longer than l. When the condition is not satisfied, there is: B=

SM ⎛ r20 r10 ⎞ ⎜ − ⎟ 4π ⎝ r22 r12 ⎠

Combination method

The hybrid method includes integral-differential equation method and differential-integral equation method. Both methods need to eliminate unknown variables by the boundary conditions at the interfaces. The major difficulty, however, is how to determine such boundary conditions.

2.4.5 Three-dimensional mathematical model of magnetic field In simulation of the magnetic field of aluminum reduction cells, there is a more reliable method, that is the three-dimensional model which solve the magnetic fields in both the bus bar and the cell with the Maxwell equation as: ∇× H = J ∇• B = 0 The above equation can be solved with double scalar quantity method. The three-dimensional model can be used to compute the distribution of the magnetic field of ferromagnetic materials including steel cell shell.

2.5  Simulation on Melt Flow and Velocity Distribution in Smelting Furnaces The extracted metals or ores often appear as molten state in the process of extracting and refining metal, such as the liquid aluminum in reduction cells, copper (or nickel) matte in electric smelting furnace and liquid steel in tundish. The smelting processes of these metals comprise heat transfer, momentum transfer and mass transfer as well as chemical reaction in smelting equipment. And there are inter-coupling actions among these transport phenomena. When reaction system is determined, it is necessary to study how to determine flow, heat transfer and mass transfer conditions of the fluid in order to improve metallurgical reactions or machining processes. Among others, the flow process of high temperature melts in the smelting furnace takes an important role in the efficiency, rates of metallurgical reaction and the life of equipments. It is not until recently that more and more attention has been paid to the

  Ping Zhou, Feng Mei and Hui Cai

concept of fluid flow on the research of metal smelting process. While the traditional research methods used in fluid dynamics are still widely applied in high temperature melts numerical simulation is becoming more and more popular in studying complicated fluid flow in the metallurgy applications all over the world.

2.5.1 Mathematical model for the melt flow in smelting furnace The melt flow can be described by means of Navier-Stokes equations and the tensor form in Cartesian coordinate system can be written in following. Continuity equation: ∂ρ ∂ + ( ρu j ) = 0 ∂t ∂x j

(2.204)

The melt in smelting furnace can be seen as incompressible fluid, then fluid divergence is equal to zero. Based on the generalized Newton’s viscosity law, the momentum equation of the melt can be got: ⎛ ∂u ∂u j ⎞ ⎤ ∂ ∂ ∂p ∂ ⎡ + ( ρ ui ) + ( ρ u j ui ) = − (2.205) ⎢ μ eff ⎜ i + ⎟⎟ ⎥ + Fsi ⎜ ∂x ∂t ∂x j ∂xi ∂x j ⎢⎣ ⎝ j ∂xi ⎠ ⎥⎦ The sensible enthalpy equation (refer to Section 2.3.1) can be given by: ⎡⎛ λ μ ⎞ ∂H ⎤ (2.206) ⎢⎜ + T ⎟ ⎥ + QR ⎢⎣⎜⎝ cp σ H ⎟⎠ ∂x j ⎥⎦ where ρ is melt density; t is time; p is pressure; uj˄j=1,2,3˅ is melt velocity components in three coordinate directions; μeff is efficient viscosity˄the sum of the molecule viscosity and turbulent viscosity, or μ + μT˅; Fsi is the body force components acting on the melt˄including gravity, buoyancy and electro-magnetic force˅; H is sensible enthalpy; λ is melt thermal conductivity; cp is melt specific heat; σH is Prandtl numbers of enthalpy; QR is chemical reaction heat. Above equations (including continuity, momentum and enthalpy equation) comprise the basic equations used in studying turbulence of high temperature melt. The fundamental methods to solve these equations are direct numerical simulation (DNS), large eddy simulation (LES) and the Reynolds-averaged simulation (RAS). But the former two need to take quite many computer capacity and CPU time, and there are definite gap from solving the practical problems. At present, RAS is the most economic and efficient method used in solving practical engineering problem. The basic thought of RAS is to use lower order relevant variables and feature of time-averaged flow to model the unknown higher order relevant variables, which make the time-average equations or correlation equations closed. In the engineering

∂ ∂ ∂p ∂ (ρ H ) + ( ρu j H ) = − + ∂t ∂x j ∂t ∂x j

2 Modeling of the Thermophysical Processes in FKNME 

application, what we need to know are only time-averaged velocity, time-averaged enthalpy and time-averaged turbulent features, and it is unnecessary to master the details about production and development of the turbulence. Therefore, RAS method has been widely applied in solving the practical engineering problems. In the Reynolds-averaged simulation, according to different Reynolds number (Re) , there are different models for closing time-averaged equations. For the turbulence with high Re, standard k-ε two-equation model and RNG model are often used. For turbulence with low Re or the flow in near-wall regions, low Re k-ε model is often used. In addition, RNG model is also used in model turbulence with low Re (refer to Section 2.1). In the case of the molten metal with high temperature, the fluid flowability varies greatly with the type of the metal. For example, the viscosity of liquid steel  is 8.7×10 7m2/s, the viscosity of nickel matte slag is 9.38×10-5m2/s, the viscosity of  nickel matte is 1.11×10 5m2/s (Kaiser and Downing,1977) . At the same time, the melt velocity greatly varies with the different equipments. For example, the velocity of liquid aluminum in the nozzles of high speed roll-casting is about 5m/s, while the velocity of high temperature melt in bath pool smelting is only of the  order of 10 1m/s. Therefore, when the flow of high temperature melt is simulated, it is necessary to choose the suitable turbulent model according to the characteristics of the melt or equipments.

2.5.2 Electromagnetic flow For most practical fluid motion, body force Fsi often appears in the form of gravity and buoyancy. But, when the electrical current goes through the molten metal or the external of furnace, it will induce an electromagnetic field and an additional body force called as electromagnetic force, such as in aluminum reduction cell, high-frequency smelting, electroslag remelting and ladle refining with electromagnetic stirring. Fig.2.13 is a sketch of equipment used in industrial production. In the equipment, electromagnetic stirring has an important influence on the dynamic conditions in the metal smelting process. For an induction furnace, when current goes through coil (Fig.2.13 (a)), induced current is produced in the crucible melt and further forms an electromagnetic field, which drives the melt to move. For an electric arc furnace (Fig.2.13 (b)) there is also induced current and melt circulation motion. When theoretically analysis is carried out to study the fluid motion driven by the electromagnetic force, it is necessary to couple the electromagnetic force governing equations (including Maxwell equations and Ohm’s law) and the fluid flow governing equation (Navier-Stokes equations).

  Ping Zhou, Feng Mei and Hui Cai

In the area of electromagnetic fuid dynamics, Maxwell equation is expressed by (Szekely, 1985; Li et al., 1985; Deng et al., 1997; Zhou and Mei; 1997a, b): ∇× E = −

∂B ∂t

∇ × B = μm J

˄Faraday’s law˅

(2.207)

˄Ampere’s law˅

(2.208)

∇ • B = 0 ˄Continuity condition of magnetic field˅ ∇• J = 0

˄Law of current continuity˅

(2.209) (2.210)

Fig. 2.13 The sketch of equipment with fluid motion driven by electromagnetic force

If there is a fluid motion, Ohm’s law can be written as J = σe (E +U × B )

(2.211) 2

where E is electric field intensity,V/m; B is magnetic flux density,Wb/m ; J is 2 current density, A/m ; U is melt velocity, m/s; μm is magnetic permeability, H/m; σe is electric conductivity, S/m. The second term at the right of Eq.2.211 stands for current induced by the convective motion of the melt. In most practical application related to induction stirring, the term is often neglected because of low melt velocity (Szekely,1985). Then Eq. 2.211 can be approximately expressed by J = σe E

(2.212)

Current density J and magnetic flux density B can be got by solving Eq. 2.207 ~Eq. 2.210. Therefore electromagnetic body force (Lorentz Force) can be obtained based on the following equation. (2.213) Fs = J × B

2 Modeling of the Thermophysical Processes in FKNME 

The major difficulty to solve Lorentz Force is that the boundary conditions of the electromagnetic field (or the value of B, E and J) are not easily obtainable unless through complex and repeated field computations. In practical application, above equations are often simplified according to the specific conditions of the problem. By eliminating variables E and J among Eq. 2.207̚Eq. 2.210, the governing equation of the magnetic flux density can be given by ∂B = ∇ × (U × B ) + ηm ∇ 2 B (2.214) ∂t where ηm =

1

σ e μm

. It is called as diffusion coefficient of magnetic field, m2/s.

The first term on the right hand side of the above equation represents the convection transport of B and the second term represents the diffusion transport of B. For most electromagnetic flow phenomena in smelting process, it is possible to assume that diffusion transport caused by magnetic flux density is far more than convection, that is Magnetic Reynolds Number Rm is quite small (Li et al., 1985). Then above equation can be simplified as ∂B = ηm ∇ 2 B (2.215) ∂t In this way, transport equation of magnetic flux density is independent of momentum equation, which simplifies the computation. Here, magnetic Reynolds number Rm is equal to the ratio of magnetic convection and magnetic diffusion, and there is Rm = U 0 L0 / ηm

(2.216)

where U0 and L0 are the characteristic velocity and characteristic length respectively. For static magnetic field, there is no eddy current, and for steady liquid metal with high electric conductivity, there is no free electric charge. In this case, it is convenient to introduce magnetic vector potential A which is defined as B = ∇× A ∂A  ∂t ∇• A= 0 as well as When above definition is combined with Maxwell equations, there is ∂A ∇ 2 A = μ mσ e ∂t

and  

    

E=−

(2.217) (2.218) (2.219) (2.220)

  Ping Zhou, Feng Mei and Hui Cai

or

∇2 A = −μm J



  

(2.221)

The computation of electromagnetic force field becomes simpler if B at the boundary is given. Otherwise, these boundary conditions must be computed by means of the current distribution in coils. There fore the determination of the additional body force (mainly refer to Lorentz Force) is always the key process for solving any magnetic fluid dynamic problem. ASEA-SKF is a kind of ladle refining equipment and used in deoxidation and composition last adjustment of special steel. To improve the effect, moving wave and low frequency coil current are adopted. Fig. 2.14 shows the simulated velocity field in ASEA-SKF, and major parameters in computation are listed in Table 2.7 (Szekely, 1985). Table 2.7 The major parameters used in simulating melt velocity field in ASEA-SKF Parameter

Value

Melt density

7200 kg/m3

Melt viscosity

6×10−3 Pa·s

Depth of bath pool

1.7 m

Radius of bath pool

1.13 m

Electrical resistivity

4×10−6 Ω·m

Magnetic permeability

1.26×10−6 H/m

Permittivity

8.85×10−2 F/m

Frequency

1.4 Hz

The maximum magnetic intensity in z direction measured at inner wall

3.69×104 A/m

Fig. 2.14 Sketch of fluid motion driven by electromagnetic force

2 Modeling of the Thermophysical Processes in FKNME 

2.5.3 The melt motion resulting from jet-flow Gas jet is a very popularly applied technique in modern metal extraction and refining, such as top-blowing BOF used in steel smelting, converter, Noranda furnace and Vanukov furnace used in smelting or refining copper, lead, zinc and nickel. When gas is jetted into these equipments, the processes, such as bubble formation, separation, rising and pumping up, result in the melt circulation, which intensify a series of metallurgical physical processes including mass transfer and chemical reaction at interface, engulfment of slag-metal interface, and removal of inclusion (Chen and Mei,1993). According to the contact mode between the gas phase and the liquid phase in the reaction, external gas jet can be divided into two types: a) Impinging gas jet. There is a certain distance from gas nozzle to solid or liquid surface and gas is only ejected to solid or liquid surface, such as LD converter. b) Immersed gas jet. A gas nozzles or orifice is immersed into liquid, as copper blow converter (Fig. 2.15).

Fig. 2.15 Sketch of copper converting furnace

Per installing location of gas nozzle, immersed gas jet can also be divided into bottom-blowing, horizon-blowing (side-blowing) and top-blowing. In extraction and refining of nonferrous metals, the former two are broadly applied. The influence of gas jet technique on the intensifying metallurgical physical chemistry and service life directly depends on the flow pattern and turbulent characteristics of fluid in bath pool. However, the flow pattern and turbulent characteristics of fluid depend on many factors, such as the bubble behavior, jetting mode, geometry of bath pool, physical properties of melt and gas volume

  Ping Zhou, Feng Mei and Hui Cai

jetted, etc. To understand better the heat and mass transfers in gas-liquid metallurgical reactors, metallurgical industries have carried out a lot of researches in the recent a few decades including the following areas: a) The formation, motion, mass transfer and reaction behavior of bubble in metallurgical melt. b) Bubbling, jetting and their conversion conditions for immersed nozzles in the melt. c) The behaviors of high pressure air jetting into free space and impinging liquid surface. d) The stirring effect of gas on the melt. e) The melt flow driven by gas stirring. f) Simulation on operation process in metallurgical gas-liquid reactor. When gas is jetted into bath pool by nozzle or immersed spray gun, continuous gas phase moves in the liquid and there is a gas-liquid interface. Because of the stability of the interface, jet-flow forms entrainment from ambient medium, which makes jet-flow expand and its velocity reduce. The gas flow behavior at the section of nozzle outlet has an influence on jet penetration, melt motion, nozzle erosion, solid particles penetration and the gas-liquid-solid inter-phase reaction. 2.5.3.1 Free jet characteristic in gas-gas jet system

The structure of a free jet flow is shown in Fig.2.16. In gas-gas jet system, gas is jetted at velocity u0 from the nozzle with diameter d0. It can be assumed that gas on section AA′ is of the uniform velocity. As jet-flow moves forward in x axis direction, ambient fluid is engulfed which makes jet-flow expand, but central gas still keeps the initial velocity, this region is called the potential core zone. Because of engulfment, the diameter of potential core zone gradually reduces until zero as the increase of x. The distance, from the nozzle outlet to the location where the velocity on the jet-flow axis starts to reduce, is called the length of potential core zone. The periphery of potential core zone is mixing zone where the velocity gradually decreases until the velocity at peripheral boundary is equal to zero. The width of mixing zone gradually increases with x. This area is a developing zone of jet-flow, and the gas flow is called the starting stage and is about (0̚6.4)d0. In this zone, expansion velocity of jet-flow can be described by the expansion angle, which is usually obtained from measurements. For gas-gas system, expansion angle is 20°̚26°. After jet-flow pass through starting stage, gas velocity on the axis begins to decrease and jet-flow goes into the second stage namely the transition stage. In this stage, gas velocity tends to be identical and the whole transition zone belongs to mixing zone, which is located in (6.4 ̚ 8.0)d0. Jet-flow has fully developed from section CC′, time-averaged velocities in axis direction on different Sections have the similar

2 Modeling of the Thermophysical Processes in FKNME 

distribution. When the central velocity of jet-flow is reduced to zero, the energy of jet-flow is completely consumed and the jet-flow vanishes into environment.

Fig. 2.16 The sketch of free jet configuration (I is starting stage; II is transition stage; III is basic stage)

2.5.3.2 Limited jet characteristic in gas-liquid system

Smelting process finds often jets of gas to high temperature melt. Fig.2.17 shows the experimental results of gas jet-flow in gas-liquid two-phase system. The results indicate that, the axis velocity starts to decline much earlier than that in the gas to gas jet. Measurements showed that the axis velocity started to decline at distance of d0 to 2d0 from nozzle. The pattern of velocity declines within the first (0.8 ~ 5) d0 is quite different from the pattern with distance >5d0. In the latter case, the declining accelerate remarkably and the axis velocity (u20d0 ) has dropped to 1/10 of the initial velocity. In contrast, in the case of gas to gas jet it takes about 60d0 distance for the axis velocity to reduce to 1/10. The buoyancy of gas bubbles in the liquid phase is the major factor affecting the behavior of jet-flow.

Fig. 2.17 The decline of axis relative velocity of gas jet in gas and liquid

2.5.3.3 Penetration depth of gas jet

The depth of penetration is an important characteristic parameter of gas jet in bath pool. Many researches have been reported on this topic (Hu et al., 1999; Devia, 1998).

  Ping Zhou, Feng Mei and Hui Cai

Kozlovszky and Rhys proposed an empirical correlation for vertical jetting of air or helium into water (Kozlowski, 1986): lp / d0=1.92FrĄ0.384

˄air-water˅

(2.222)

lp / d 0 = 2.23Fr ′ 0.367

˄helium-water˅

(2.223)

An empirical correlation of penetration depth in horizon-blowing jet-flow has been proposed by Hofield and Brimakeheim (Hoefele, 1979): lp / d 0 = 10.7 Fr ′0.46 ( ρg / ρl )

(2.224)

where lp is penetration depth; d0 is nozzle diameter; ρg and ρl are gas and liquid ρgu02 density, respectively; Fr′ is modified Froude number Fr ′ = . ( ρl − ρg ) gd0 2.5.3.4 Track of immerged horizontal gas jet

By experiments of immerged horizontal jet-flow it has been found that jet-flow is dispersed at a range of distance from the nozzle. This is because the engulfment effect greatly slows down the jet velocity. Once the horizontal velocity of the jet flow reduces to a level smaller than the rising velocity of the bubbles in the liquid phase, the vertical component velocity of the jet starts to dominate and the jet flow will be broken into bubbles (Szekely, 1985). Sehmreis and Szekely studied the influence of buoyancy and liquid engulfment in jet flow via mass and momentum conservation analysis. By assuming that both the gas volume fraction and the expansion of the jet flow are related with the horizontal distance from the nozzle, they worked out an equation describing the trajectory of the jet flow (refer to Fig. 2.18) (Szekely, 1985) as follows: 2 1/ 2

d 2 yr ⎛ θ ⎞ ⎡ ⎛ dy ⎞ ⎤ = 4 Fr ′−1 tan 2 ⎜ c ⎟ ⎢1 + ⎜ r ⎟ ⎥ xr2 c (2.225) 2 dxr ⎝ 2 ⎠ ⎢⎣ ⎝ dxr ⎠ ⎥⎦ where yr and xr stand for vertical and horizontal dimensionless distance of the jet-flow from the cone angle origin, which can be calculated by taking nozzle diameter as characteristic size; θc is the expansion angle where jet-flow pass the origin; d0 and d are respectively jet-flow diameter on the section of nozzle outlet and of horizontal distance x; c is gas volume fraction on the section of horizontal distance x, which is given by the following formula (Szekely, 1985).

d ⎛ ρ ⎟⎞ c = 0 ⎜⎜⎜c + l (1− c)⎟⎟ ⎟⎟⎠ d ⎜⎝ ρg

1/ 2

(2.226)

2 Modeling of the Thermophysical Processes in FKNME 

Fig. 2.18

Schematic of horizontal jet in liquid phase with buoyancy effect

Fig.2.19 shows the trajectories of the axis of a jet flow in water computed by Eq.2.225 and Eq.2.226 under the condition of θc=20eand different modified Froude numbers. Suggestions of modification of this model can be found in literature (Zhu et al., 1998).

Fig. 2.19 Calculations of the axis trajectory of a horizontal jet flow in water at different modified Froude numbers

2.5.3.5 Simulation on fluid flow in bath pool with immerged gas jet

Eq.2.204 and Eq.2.205 can also be used to describe the governing differential equations for fluid flow in gas stirring system. But here the body force Fsi represents the buoyancy of gas bubbles, or cρ1gi where c stands for gas volume fraction in gas-liquid two-phase region. Since the buoyancy resulting from temperature difference is much less than that resulted from bubbles, the movement of the melt is mainly driven by the gas buoyancy and therefore the thermal buoyancy can be basically ignored when the fluid flow is to be studied. Obviously, properly determining Fsi is the key in studying the flow phenomena of the melt in jetting bath pool. Generally, the rising region of immerged jet-flow˄or called two-phase region˅ the distribution of the gas phase in the radial direction can be described by Gaussian distribution. Compared to bottom-blowing jet side-blowing jet shows

  Ping Zhou, Feng Mei and Hui Cai

larger radius but lower maximal void fraction on the axis of the two-phase region. Different researchers reported their studies and empirical expressions on the void fraction in two-phase region (Szekely, 1985; Zhu et al., 1998; Castillejos, 1987). Fig.2.20 shows the predicted flows in a steel ladle with side-blowing and bottom-blowing. The side-blowing is found superior to bottom-blowing from the viewpoint of stirring and mixing effects. Interested readers are referred to the literature (Szekely, 1985; Zhan et al., 2003; Zhu et al., 2006; Pan and Zhu, 2007; Zhan and Wang, 2006) for fluid flows in bath pool with gas jets.

Fig. 2.20 Simulations on the melt flow in a steel ladle with side-blowing and bottom-blowing jet (a) Side-blowing; (b) Bottom-blowing

References Amano R S (1984) Development of a turbulence near-wall model and its application to separated and reattached flows. Numerical Heat Transfer, (7): 59~75 Badzioch S, Hawksley P G W (1970) Kinetics of thermal decomposition of pulverised coal particles. Industrial Engineering Chemistry Process Design and Development, (92):231 Bakewell H P (1969) Viscous sublayer and adjacent wall region in turbulent flow. Physics Fluids, 10 Barreios A et al (1993) Prediction of the near burner region and measurements of NOx and particulate emissions in heavy fuel oil spray flumes. Comb. Flame, (92):231 Cai Hui (1992) Research and Magnetic Fields Compute of Aluminum Reduction Cell: [academic degree article] (in Chinese). Central South University,Changsha Carol E E (1987) The Simulation of Burning house and furnaces (in Chinese). Science

2 Modeling of the Thermophysical Processes in FKNME  Press, Beijing Castillejos H, Brimacombe J K (1987) Measurement of physical characteristics of bubbleĉ gas-liquid plumes, PartĊ, Local properties of turbulent air-water plumes in vertically injected jets. Metal. Trans., 18B: 659~671 Cen Kefa et al (1990) The Computation and Theory of Engineering Gas-particle Multi-phases flow (in Chinese). Zhe Jiang University Press, Hangzhou,436~468 Cen Kefa et al (1991) Combustion hydrodynamics (in Chinese). Irrigation and Electric Power Press, Beijing CFX-4.2 Solver. Oxfordshire OX11 0RA. United Kingdom Cha Jinrong, Chen Jiayong (1997) The Principles and Application of transfer process (in Chinese). Metallurgical Industry Press, Beijing “Chemical Engineering Handbook”editorial committee (1989) Chemical Engineering Handbook,vol.5 (in Chinese). Chemical Machine Press, Beijing Chen Shiyu, He Zhihui (1987) The error analysis and mathematical model of computing magnetic field in aluminum reduction cell (in Chinese). Journal of Huazhong Institute of Science and Technology, (6): 85~92 Chen Wenxiu, Mei Chi (1993) The Metallurgical Handbook of nonferrous Metallurgy Extraction—Modern Equipment (in Chinese). Metallurgical Industry Press, Beijing De Soete G G (1975) Overall reaction rates of NO and N2 formation from fuel nitrogen. Technical report, Fifteenth Symp.(Int’l.) on Combustion, The Combustion Institute Deng Kang et al (1997) The electromagnetic field numerical analysis of levitation melting procession with water cold crucible (in Chinese). The Chinese Journal of Nonferrous Metals, (7)(supplement): 100~104 Devia M I (1998) The Phenomenon of Gas Spray in P-S Convertor, Copper Pyrometallurgy, International Conference Collection of Copper in 1995. Metallurgical Industry Press, Beijing, 217~232 Dong Wei (1998) Three-Dimesional Computer Simulation For Grate Combustion And NOx Emission With a New Air Supply Ecotubs SystemˈLicenitiate ThesisˈDepartment of Metallurgy Royal Institute of Technology. Sweden: Stockholm Dupont V, Porkashnian M (1993) Reduction of NOx Formation in Natural Gas Burner Flames. Fuel, 72(4):497~503 Dupuis M, Bojarevics V (2006) Busbar sizing modeling tools: comparing and ansys based 3D model with the versatile 1D model part of MHD-valdis. Galloway T J. Proceedings of the Technical Sessions (135th TMS Annual Meeting, Light Metals). Warrendale: AIME, 341~346 Fan Minwu, Yan Weili et al (1988) Electromagnetic Field Integral Equation Method (in Chinese). China Machine Press, Beijing Fan Weicheng, Chen Yiniang et al (1987) Computational combustion (in Chinese). Anhui Science and Technology Press, Hefei Fan Weicheng, Wang Yaopeng (1992) The Model and Computation of Fluid and Burning (in Chinese). University of Science and Technology of China Press, Hefei

  Ping Zhou, Feng Mei and Hui Cai Feng Cizhang (1985) Electromagnetic Field (in Chinese). High Education Press, Beijing Field M A et al (1967) Combustion of Pulverized Coal, British Coal Utilisation Research Association, Leatherhead. Surry, 175 Fu Weibiao, Wei Jingshang (1984) The Fundamental of combustion physics (in Chinese). China Machine Press, Beijing Gibb J (1985) Combustion of residual char remaining after devolatilization, Lecture at course of pulverised coal combustion. Mechanical Engineering Dept., Imperial College, London Grad Hetsroni (1993) Handbooks of Multi-phase flow and Heat transfer (in Chinese). China Machine Press, Beijing Grjotheim K, Welch B J (1997) Technology of aluminum reduction (translation). Light Metal Guo Yincheng et al (1997) Numerical modeling of spray of combustion in a dual-inlet dump combustor (in Chinese) . Journal of Engineering Thermophysics, 18(4): 502 Gusev A, Kriuokovsky V, Krylov L et al (2004) Busbar optimization of high-current reduction cells. Tabereaux A T. Proceedings of the Technical Sessions (133th TMS Annual Meeting, Light Metals). Warrendale: AIME, 467~472 Hoefele E O, Brimacombe J K (1979) Flow regions in submerged injection. Metal. Trans., 10B:631~648 Howell J R et al (1964a) Monte-Carlo solution of radiant heat transfer in a nongray onisothermal gas with temperature dependent properties. AICHE. J., 10(4) Howell J et al (1964b) Monte-Carlo solution of thermal transfer through radiant media between gray walls. J. Heat Transfer, 86(1) Hu Jun, Mei Chi et al (1999) Research and improvement strategy of furnace lining in continuous converters (in Chinese). Nonferrous Metals(Extractive metallurgy), (6):40~44 Jones W P, Launder B E (1972) The prediction of laminarization with a two-equation model of turbulent. Inter. J. Heat Mass Transfer, 15 Jones W P, Launder B E (1973) Predictions of low reynolds number phenomena with a two equation model of turbulence. Int. J. Heat and Mass Transfer: 16 Kaiser R H, Downing J H (1977) Heat transfer through the hearth of an electric smelting furnace and its moppet on operations. Electric Furnace Proceedings, 119~126 Klebanoff R S (1955) Characteristics of turbulence in a boundary layer with zero pressure grandient. NACA, TR, 1247 Kozlowski M A, Wraith A E (1986) Jet cavity at a submerged vertical lance. Iron-making and Steel-making, (13):190~194 Lan X R, Khodadadi J M (1997) Evaluation of six k- ε turbulents model predictions of a flow in a continuous casting rillet-mold water model using laser doppler velocitmetry measurements. Metallurgical and Materials Transactions B, April, 28B: 321~332 Lam S H (1992) On the RNG theory of turbulence. Phys. Fluids A, May; 4(5) Launder B E, Spalding D B (1972) Mathematical Models of Turbulence. Academic Press Launder B E, Spalding D B (1974) The numerical computation of turbulent flow, Comp.

2 Modeling of the Thermophysical Processes in FKNME  Math. In: Appl. Mech. And Eng., (3):269 Li Baokuan et al (1985) Computation Hydromechanics in Steel-making (in Chinese). Metallurgical Industry Press, Beijing Liang Xuemin (1990) Research the busbar design of modern large scale aluminum reduction cell (in Chinese). Light Metal, (1): 20~26 Liang Xuemin et al (1998) CFD research and mathematical model of physics field in aluminum reduction cell (in Chinese). Light Metal (Supplement), 145~150 Libby P A, Williams F A (1980) Turbulent Reacting Flows. Berlin: Springer Verlag Lien F S, Leschziner M A (1994) Assessment of turbulence-transport models including nonlinear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. Computers Fluids, 23(8): 983~1004 Lu Jiayu et al. Research articles of aluminum reduction cell “three fields” (in Chinese). (inner materials) Lockwood F C, Romi Millanes C A (1992) Mathematical modelling of fuel-NO emissions from PF burners. J. Int. Energy, (65):144~152 Mao Jianxiong et al (1998) Cleaning Burning of Coals (in Chinese). Science press, Beijing, 210~222 Mei Chi (1987) The Principle of Metallurgy Transport Process (in Chinese). Central South University Press,Changsha Mei Chi et al (1997) Development of radiation transfer in furnaces (in Chinese). Industrial Furnace, 19(1): 27~30 Meng Ning et al (1997) Numerical study of radial distribution of composition of methane turbulent jet flame by PDF method (in Chinese). Journal of Engineering Thermophysics, 18(6):759 Mischke C R (1980) Mathematical Model Building. IOWA: The IOWA State University Press Mohammed Zaoui (1996) Modelisation De La Combustion Dans UN Ecoulement Gaseux Turbulent Premelange, Thesis(Ph. D), Facult Polytechnique de Mons Pan Shisong, Zhu Miaoyong (2007) Motion characteristics of injected powder through porous brick mounted in the bottom of refining ladle (in Chinese). ACTA Metallurgical Sinica, 43(5):553 ~556 Patankar S V (1984) Numerical simulation of heat transfer and fluid flow (in Chinese). Science Press, Beijing Patel V C, Rodi W, Scheurer G (1985) Turbulence models for near wall and low Reynolds number flows: A review. ALAA J, Feb. Perlmutter M et al (1964) Radiant transfer through a gray gas between concentric cylinders using Monte-Carlo. J. Heat Transfer, 86(2) Ramamurthy H, Ramadhyani S, Viskanta R (1993) A study of low-Reynolds-number k- ε turbulence models for radiant-tube. J. of the Institute of Energy, Dec., 66 Robert Siegel, John R Howell (1990) Heat Transfer of Radiation (translation) (second edition). Science Press, Beijing

  Ping Zhou, Feng Mei and Hui Cai Shadman F (1989) Kinetics of soot combustion during regeneration of surface filter. Comb Sci &Tech, 63(3):4~6 Shea N O, Fletcher C A J (1994) Prediction of turbulent δ wing vortex flows using an RNG k- ε model. In: Proceeding of the 14th Int. Conference on Numerical Methods in Fluid Dynamics Siegel R, Howell J R. Thermal Radiation Heat Transfer (Third Edition). Hemisphere Publishing Corparation Smith L M, Reynolds W C (1992) On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models. Phys. Fluids A, Feb.; 4(2) Smoot L D, Pratt D T (1983) The Pulverized-coal Combustion and Gasification (in Chinese). The Tsinghua University Press, Beijing Szekely J (1985) The Fluid Flow Phenomenon in Metallurgy (in Chinese).Metallurgical Industry Press, Beijing Tesener A, Tsygankova E I et al (1971) Combustion and Flame, (17):279 Ubhayakar S J, Stickler D B et al (1976) Rapid devolatilization of pulverised coal in hot combustion gases. 16th Symposium (International) on Combustion, The Combustion Institute, 246 Wall T F et al (1986) The prediction of scaling of burnout in swirled pulverised coal flames. International Flame Research Foundation Report F388/a/3 Ijmuiden. The Netherlands Wang Buxuan (1982) Engineering Heat and Mass Transfer (volume one) (in Chinese). Science Press, Beijing Wang Ruliang, Yang Xiaodong et al (1998) The optimizing of busbar equipment and cell configuration in 280kA prebakeed (in Chinese). Light Metals (Supplement), 58~63 Wang Shaoting, Chen Tao (1986) Momentum, Heat and Mass Transfer (in Chinese). Tian jin Science and Technology Press, Tianjin Wang Yingshi, Fan Weideng, Zhou Lixing, Xu Xuchang (1994) Numerical computing in combustion process (in Chinese). Science Press, Beijing Wilcox D D (1993) Turbulence Modelling for CFD. California: Griffin Printing, Glendale Yakhot V, Orszag S A (1992) Development of turbulence models for shear flows by a double expansion technique. Phys. Fluids A, July, 4(7) Yakhot V, Orszag S A (1998) Renormalization group analysis of turbulence. Basic Theory, Journal of Scientific Computing; 1(1) Yakhot V, Smith L M (1992) The renormalization group, the ε-expansion and derivation of turbulence models. J. of Scientific Computing, 7(2) Yang Xiaodong et al (1998) Magnetic fluid design of large scale prebakeed cell (in Chinese). Light Metals(Supplement), 58~63 Yap C (1987) Turbulent Heat and Momentum Transfer in Recirculating and Impinging Flows, thesis(Ph. D). United Kingdom: University of Manchester Zhan Shuhua , Lai Chaobin , Xiao Zeqiang (2003) Gas stirring behavior in side-blown metallic bath (in Chinese). Journal of Central South University of Technology, 34(2):148~151

2 Modeling of the Thermophysical Processes in FKNME  Zhan Shuhua, Wang Jianjun, Qiu Shengtao et al (2006) Flow control and inclusion removal in continuous casting tundish with bottom gas blowing (in Chinese). Journal of Anhui University of Technology, 23(4):367~372 Zhao Jianxing et al (1994) Numerical and experimental study of turbulent combustion model (in Chinese). Journal of Engineering Thermophysics, 15(1):99 Zhen Chuguang, Zhou Xiangyang (1996) The PDF Model of Turbulent Reaction Flow (in Chinese). Huazhong University of Science and Technology Press, Wuhan Zhou Lixing (1982) Particles with phases changed—multi-phase hydrodynamics in gas system (in Chinese). Advance in Mechanics, 12(2) Zhou Lixing (1986) The Theory of Combustions and Chemical Hydrodynamics (in Chinese). Science Press, Beijing Zhou Lixing (1991) Numerical Modeling of Turbulent Two-phases Flows and Combustion (in Chinese). Tsinghua University Press, Beijing Zhou Lixing (1994) The Theory and Numerical Modeling of Turbulence Gas-particle Two Flows and Combustion (in Chinese). Science Press, Beijing Zhou P Y (1945) On the velocity correlations and the solution of the equations of turbulent fluctuation. In: Appl. Math., (3):38 Zhou Ping, Mei Chi (1997a) Flow field in molten bath of a round electirc cleaning furnace: numeric calculation and hydraulic model experiment (in Chinese). Mining and Metallurgical engineering, 117(3): 43~46 Zhou Ping, Mei Chi (1997b) Numerical simulation and analysis on the velocity field in bath of round electric cleaning furnace. Journal of Central South University of Technology, 4(1): 54~57 Zhu Gujun (1989) Engineering Heat and Mass Transfer (in Chinese).Aviation Industry Press, Beijing Zhu Miaoyong, Xiao Zeqiang et al (1998) The Mathematic and Physical Model of Steel Fining Process, Metallurgical Response Engineering Books (in Chinese). Metallurgical Industry Press, Beijing Zhu Miaoyong, Zhou Haibin, Chen Zhaoping et al (2006) Behaviors of jet and fluid flow in AOD converter for stainless steel refining (in Chinese). ACTA Metallurgical Sinica, 42(6): 653~656


Hologram Simulation of the FKNME $IJ
.FJ
BOE
;IVP
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In this chapter, the concept “hologram simulation” is introduced and elaborated. The multi-field coupling and equation-solving techniques are also discussed as application examples of the hologram simulation strategy.

3.1 Concept and Characteristics of Hologram Simulation The “hologram simulation” is a numerical simulation strategy aimed at the optimization of model development and simulation results analysis for the purpose of the best understanding of the engineering problems that are studied. The term “hologram simulation” is inspired by the hologram theory and information theory. The word Āhologram” originates from the Greek words “holos gramma”, which means “whole message”. The theory of holography was introduced by the Hungarian scientist D. Gabor in 1948, but it had no application until the emergence of the high intensity laser technology in the early 1960s. The principle of the holography is to record both amplitude and phase information of the light waves reflected by the objects, and to generate images with three-dimensional lifelike effects. In contrast, normal photography technique only records the amplitude of the light waves, which only carries the information about the intensity of the lights. In short, hologram technique enables more comprehensive and precise observation of the objects by

  Chi Mei and Zhuo Chen presenting more information from the objects. The information theory and its recent extensive developments are the other source of this technique. Shannon published his statistics-based grammatical information theory in 1948. As a rigorously deductive system, Shannon’s work is one of the important theoretical foundations which strongly backed the development of the modern communication technologies. However, the development in psychology and artificial intelligence areas in the past decades has made the grammatical information theory handicapped in dealing with the complex systems. In these cases, information is required to convey message about not only the forms (grammar information) of the objects but also the meanings (semantic information) and the functions (pragmatic information). As a result, new theories were raised. To be differentiated from the classic grammar information concept, the new concept is called “the comprehensive information”. The above-mentioned ideas developed from the holography theory and the information theory are equally applicable when carrying out numerical simulations of the engineering processes, including those involving furnaces and kilns for nonferrous metallurgy. A successful model should not be merely a mathematic approach to snapshoot a superficial, localized and stationary picture flatly reflecting the physical processes, but it should be rather more a powerful tool to reveal the comprehensive information, namely the fundamental, full-scaled and dynamic aspects describing the hologram of the processes. The modeling strategy that aims to obtaining “the comprehensive information” of the objects is called “the hologram simulation”. In contrast to the usual simulation strategy that is single parameter or single task orientated, the hologram simulation highlights more efforts in the following phases and aspects (Mei et al., 1999). a) When physical models are established, the fundamental integrity and coherency should be emphasized in studying the physical system. Approaches intending to separate or isolate the processes should be avoided as much as possible. b) When mathematical are established, the inter-dependency or interactive nature among the physical processes should be always primarily considered. The governing equation set should be able to reflect as much as possible all of the important processes and their interactions. Mathematical simplifications that may result in cutting off the interactive links between physical processes should be minimized. c) During post-processing and analyzing the simulation results, obtaining

3 Hologram Simulation of the FKNME 

comprehensive information is recommended to reflect all aspects of the objects instead of narrowly focus on one or two of the immediately relevant points. Take the study of the temperature field of the aluminum electrolytic cells for example. A hologram simulation means that not only the fluid flow and temperature fields should be investigated, but also the electric field, magnetic filed, current field, electromagnetic force field and the freeze profile of the cells should be analyzed. d) As to the scope of the simulations, attention should be paid to the entire domain but not just to obtaining localized information at a few controlling points or mean values in an area. The logic behind is that localized information or space-averaged information can be considered reliable only after the overall validity of the total information has been verified. e) Efforts to obtain dynamic or quasi-dynamic information of the main processes are encouraged. In fact, most processes that are considered as steady-stated at the macro-scale level could be time-dependent at the micro-scale level or when coupled with other processes. Based on this point of view, the overall reliability of the modeling would be best verified through examining at the dynamic or quasi-dynamic level. In short, the hologram simulation underlines higher integration and consistency regarding the model development process, and requires more comprehensive and detailed coverage regarding information revealing the objects being investigated.

3.2 Mathematical Models of Hologram Simulation As an information technology, hologram simulation is a continuous process in a relative sense that constantly approaches the truth and higher accuracy. Up to now, the state of the art of hologram simulation of the FKNME is still far from perfect. Most models are based on either “grey box” models, namely a mix of theoretical and empirical approaches, or hybrid models which combines “grey box” models and artificial intelligence models (refer to expert system, fussy analysis, artificial neural network, etc.) (Mei et al., 1996). For the convenience of model development, it is helpful to classify various engineering furnaces and kilns into groups by their thermal processes and operating mechanisms. Table 3.1 shows a possible way of classification, with their mathematical models listed in Table 3.2.

  Chi Mei and Zhuo Chen

3 Hologram Simulation of the FKNME 

  Chi Mei and Zhuo Chen

3.3 Applying Hologram Simulation to Multi-field Coupling Hologram simulation strategy requires a complete (or at least as complete as possible) coverage of all physical and chemical processes involved. Mathematically this requirement is satisfied by simultaneously establishing and solving the equation set governing all of these processes (refer to Table 3.2). Such an approach indeed brings the impression that the simulations are of “hologram-style”. This is, however, not enough to make sure that the simulations are capable of revealing the inherent mechanisms of the interactions among the processes, which is actually the true essential element making the simulation holographic. These interactions among different processes are often referred to as the coupling effects. The crucial task and the key indicator of hologram simulation are the insightful understanding and quantitative assessment of these coupling effects in the physical and chemical processes, which we call the “multi-field coupling”.

3.3.1

Classification of multi-field coupling

The multi-field coupling problems may be classified as follows (Table 3.3). Table 3.3 Overview of multi-field coupling problems Classification indicator

Class

Examples

Coupling in the same phase

Between velocity field and scalar fields in the same fluid phase

Coupling between two or among more phases

Between fluid and particles phase

Linear coupling (convection term, source term)

Simple coupling of time averaged (or steady-stated) variables

Fluctuation coupling (turbulent diffusion, source term)

Coupling terms

Bidirectional coupling (simple interaction)

Buoyancy vs flow field and temperature field

Loop-wise coupling

Refer to Fig. 3.1

Fluctuation coupling

The strong non-linear correlation of the second and(or) the third order fluctuations, chaotic phenomena

Multidirectional coupling

Refer to Fig. 3.2

The range of coupling

The mechanism of Coupling

The patterns of interaction

( ρ ui′ u′j ; c p n′k m&′j T ′ ) consisting of two or more fluctuation variables

3 Hologram Simulation of the FKNME  Continues Table 3.3 Classification indicator The formulation of the model

Class

Examples

Explicit coupling

Multivariable correlating terms

Implicit coupling

Model parameters such as turbulent viscosity or transport coefficients

3.3.2 An example of intra-phase three-field coupling The freeze profile of an aluminum reduction cell is a good example of the intra-phase coupling(Mei et al., 1986; Mei et al., 1997). The profiles are the direct result of the cathode temperature field, but also are strongly influenced by the melting mass flow field, the current field and the magnetic force field. The interactions of the fields are described by the energy equation and the momentum equation of the molten mass: Energy equation: ∂ ∂ ⎛ ∂T ⎞ q ′ (3.1) ( uiT ) = ⎜⎜ aeff ⎟⎟ + v ∂x j ∂x j ⎝ ∂x j ⎠ cp ρ Momentum equation: ∂ ∂ ⎡ ⎛ ∂ui ∂u j ⎞ ⎤ 1 1 ∂p + + (3.2) ui u j ) = − ⎢ν ⎜ ⎟⎥ + F ( ∂x j ρ ∂x j ∂x j ⎣⎢ eff ⎝⎜ ∂x j ∂xi ⎠⎟ ⎦⎥ ρ Bj where the magnetic force is: FBj = J i × Bk , in which Ji is the horizontal current in the cell, and Bk is the vertical magnetic induction intensity; aeff is the effective thermal diffusion coefficient, aeff =a+aT (3.3) veff is the effective kinematic viscosity, ν eff= ν + ν VT (3.4) aT and vT are respectively the turbulent diffusion coefficient and turbulent kinematic viscosity. The temperature filed in the cell determines the profile and the size of the cell, which influences the horizontal current. The horizontal current then influences the electro-magnetic force on the horizontal plane. Represented by the source term in Eq. 3.2, the horizontal electromagnetic force drives the horizontal movement of the melt, which affects not only the temperatures in the melt but also the turbulent thermal diffusion coefficient. In other words, the heat transfer and temperature distribution in the melt are impacted by the velocities through the parameter aeff. Note that aT in Eq. 3.3 and vT in Eq. 3.4 satisfy the equation:

  Chi Mei and Zhuo Chen

vT = PrT aT

(3.5)

where PrT is the turbulent Prandtl number (also referred to as σ T in literatures), which is an experimental constant. Usually vT is computed through the standard k − ε model (or low Reynolds number k − ε models): k2 k2 ν T = Cμ (or at small Re: ν T = Cμ f μ )

ε

ε

So: aT =

νT PrT

=

1 k2 1 k2 Cμ Cμ f μ or aT = ε PrT PrT ε

(3.6)

The turbulent kinetic energy k of the melt phase influences the flow field and the temperature field with the same order of magnitude. The parameters aeff andµ eff can be considered as two terms of intra-phase “implicit coupling”. The couplings of the flow field, the temperature field and the magnetic force field in the melt phase are realized through the convection term uiT, the diffusion ∂T and the source term FBj . Such an intra-phase three-field coupling term aeff ∂x j can be schematically illustrated in Fig. 3.1.

Fig. 3.1 The coupling of temperature field, flow field and magnetic force field in aluminum reduction cell

Note the source term qv′ in Eq. 3.1, which is the Joule heat generated by current. Apparently, all variables and processes that may influence the current distribution should have direct influences on the melt temperature. As mentioned above, the temperature field also influences the current distribution through the freeze profile. Therefore the current field and the temperature field is actually an example of bi-directional intra-phase coupling.

3.3.3 An example of four-field coupling In the generalized fluidization system with dilute-phase chemical reactions (such as the flash smelting, alumina hydroxide calcinations and pulverized coal combustion), the momentum equation of the gas phase is (Zhou, 1994; Cen and Fan, 1991):

3 Hologram Simulation of the FKNME  ∂ ∂t

( ρ ui ) +

∂τ ji

∂

( ρ u u ) = − ∂∂px + ∂x i

∂x j

+ Δρ g i +

j

j

∂

vi S + FMi −

∂x j

∑ τ ( n′ v′ mk

k ki

∑ρ

k

(vki − vi ) / τ rk +

( ρ v′ v′ + v ρ ′v′ + v ρ ′v′ + ρ ′v′v′ ) + j i

i

j

j

i

i

j

) ∑ v′ m&′ − ∑ n v′m&′ −

− n′k vi′ − vi

rk

k

k

k i

k

∑ m&′ n′ v′ − ∑ v′n′ m&′ k

k ki

i k

k

(3.7)

For the particle phase, the momentum equation is written as: ⎛ 1 m& ⎞ F ∂ ∂ ∂ ( nk • vki ) + ( nk vkj vki ) = ni gi + nk ( vi − vki ) ⎜ + k ⎟ + k ,Mi − (nk vkj′ vki′ + ∂t ∂x j ∂x j ⎝ τ rk mk ⎠ mk vkj nk′ vki′ + nk′ vkj′ vki′ ) + ( nk′ vi′ − nk′ vki′ ) τ rk + (vi nk′ m&k′ + nk vi′m&k′ + m&k nk′ vi′ + nk′ vi′m&k′ − vki nk′ m&k′ − nk vki′ m&k′ − m&k nk′ vki′ − nk′ vki′ m&k′ ) / mk −

∂ nk′ vki′ ∂t

(

) (3.8)

The energy equation for the gas phase is: ∂ ∂t

( ρh) +

∂ ∂x j

( ρ v h ) = ∂∂x

⎛ ∂T ⎜⎜ λ ⎝ ∂x j

j

j

∑n

k

∂ ∂x j

⎞ ⎟⎟ − qr + ∑ nk Qk + hs − h∑ n′k m&′k − ⎠

hk′ m&′k −

∂

∑ m& n′ h′ − h∑ n′ m&′ − ∂t ( ρ ′h′) − k

k

k

k

( ρ v′ h′ + v ρ ′h′ + h ρ ′v′ + ρ ′h′v′ ) j

j

j

j

(3.9) The energy equation for the particles phase is: ∂ ∂t

( nkCk Tk ) +

∂ ∂x j

( n v C T ) = n (Q k kj

k k

k

∂ ∂t

h

(

− Qk − Qrk ) mk + C pT − Ck Tk

( C n′ T ′ ) − ∂∂x × ( n C v′ T ′ + C v k

k k

k

k kj k

k kj

)

nk m&k mk

−

nk′ Tk′ +

j

) ( C p m& T ′n′ + C n′ m&′ T ′) m − ( C T n′ m&′ + C n T ′m&′ + C m& n′ T ′ + C n′ m&′ T ′ ) m

Ck Tk nk′ vkj′ + Ck vkj′ nk′ Tk′ + C p T nk′ m&k′ + C p nk′ m&k′ + k

k k k

k

k

p

k

k

k

k

k k

k

k

k k

k

k k

k

k

k

(3.10) The species equation for the gas phase is:

  Chi Mei and Zhuo Chen ∂ ∂t

( ρYs ) +

∂ ∂x j

( ρ v Y ) = ∂∂x j s

j

⎛ ∂Y ⎜⎜ D ρ s ∂x j ⎝

⎞ ∂ ρ v′j Ys′ + ⎟⎟ − ω s + α s nk mk − ∂ xj ⎠

(

)

Ys ρ ′v′j + v j ρ ′Ys′ + ρ ′v′j Ys′ − α s n′k m&′k −

∂ ∂t

( ρ ′Y ′ ) s

(3.11) where ωS is the source term of reactions and can be described by Arrhenius law: m m ⎛ E ⎞ Ys s ω S = kos ρ ∑ s exp ⎜ − (3.12) ⎟∏ s RT ⎝ ⎠ From Eq. 3.7 to Eq. 3.12, the subscript k denotes variables pertaining to the particles phase, subscript S denotes species components, and n denotes density measured by particle numbers. Qh, Qk and Qrk represent respectively the reaction heat flux, convection heat flux and radiation heat flux. The symbols m, m&k , S,

FM and Y stand for mass, mass change rate, mass source, Magnus force and species concentration (percentage). And α s is the excess air ratio, D is the diffusion coefficient, τji is the fluid viscous stress, and τ rk is the particle relaxation time. ρd 2 τ rk = k 18μ The set of Eq. 3.7 to Eq. 3.12 describes the velocities, temperatures, species, concentrations and reaction rates of both the gas and particles phases. The following conclusions can be drawn based on the equation set: a) The momentum, energy, species and reaction rates are intensively interacting between the gas phase and the particles phase (the reaction rates are referred specifically to the exothermic heat rate in sulfide roasting or pulverized coal combustion cases). Their coupling process is illustrated in Fig. 3.2.

Fig. 3.2 Four-field coupling mode

b) The coupling effects are chiefly represented by the source terms in the governing equations. c) Implicit coupling also exists. For instance, the term nk′ m&k′ from Eq. 3.7 to Eq. 3.11 represents the influences of gas turbulence on the flow, temperature,

3 Hologram Simulation of the FKNME 

concentration and endothermic reaction heat fields of the particle phase. In the case of vaporization of liquid drops and combustion of coke particles, this second-order correlation of fluctuation is modeled as: (Zhou, 1994; Cen and Fan, 1991) k 3 ∂n ∂T λ πd k Nu (3.13) nk′ m&k′ = Cm1 2 k cp ε ∂x j ∂x j or: nk′ m&k′ = Cm 2

k 3 ∂nk ∂YΔx πd k D ρ ε 2 ∂x j ∂x j

(3.14)

The above equations indicate that the mean particles reaction rates will be intensified by the increasing turbulent kinetic energy and dissipation, as well as the gas temperature and oxygen concentrations. Meanwhile the intensified particles reaction rates will in return more remarkably influence the flow, temperature and concentration field of the gas and particles phases.

3.4 Solutions of Hologram Simulation Models Coupling effects often bring about problems to numerical convergence. Usually the stronger the effects, the more difficult the numerical convergence. The equation set is usually solved either by a direct iteration or by a multi-level iteration. For a simple linearly coupling equation group, direct iteration is enough. In the complex situation with strong nonlinearly coupling effects, however, a multi-level iteration is needed. This iteration technique involves an “iteration-updating-resuming iteration” loop scheme (Mei et al., 1996; Zhou et al., 1990; Zhou et al., 1997; Mei et al., 1998). The solving process for the six-field coupling problem of an aluminum reduction cell is a good example of the multi-level iterations. As shown in Fig. 3.3, iterations have to be implemented not only within each particular field but also among different fields. The computation of dilute-phase reactions in the gas-particle two-phase flow is very complicated. J.J.Wormeck worked out a modulized, general-purpose software package consisting of four modules (Cen and Fan, 1990; Wormeck and Partt, 1976). The first three are Euler method based gas flow module (FLOW), gas combustion module (CHEM) and radiation module (QRAD). The last one is a Lagrangian method based module (PSIC) dealing with particle velocities, trajectories, concentrations and temperatures etc.(Fig. 3.4). The four modules communicate with each other during the iterations until the computation is convergent.

  Chi Mei and Zhuo Chen

Fig. 3.3 Solution of six-field coupling in an aluminum reduction cell (Mei et al., 1986; Mei et al., 1997)

Fig. 3.4

Flow chart of Wormeck’s four-module model

References Cen Kefa, Fan Jianren (1990) Engineering Theory and Computation of Gas-solid Multi-phase Flow (in Chinese). Zhejiang University Press, Hangzhou Cen Kefa, Fan Jianren (1991) Combustion Fluid Dynamics (in Chinese). China Water Power Press, Beijing Mei Chi, Li Xinfeng, Yin Zhiyun et al (1998) Dynamic simulation of anode baking furnace flue (in Chinese). Journal of Central South University of Technology (Natural Science), 29(5): 438̚441 Mei Chi, Tang Hongqing, Wei Wu (1986) Mathematical model and simulated experiments

3 Hologram Simulation of the FKNME  of electrical and heat fields in aluminum cell (in Chinese). Journal of Central South University of Technology, (6): 29̚36 Mei Chi, Wang Qianpu, Peng Xiaoqi et al (1996) Simulation and optimization of nonferrous metallurgical furnaces (in Chinese). Transactions of Nonferrous Metals Society of China, 6(4): 19̚23 Mei Chi, Yin Zhiyun, Zhou Ping et al (1999) The hologram simulation of modern industrial furnace and kilns (in Chinese). Journal of Central South University of Technology (Natural Science), 30(6): 592̚596 Mei Chi, You Wang, Wang Qianpu et al (1997) Study and development of the simulation software for the freeze profile in aluminum reduction cells (in Chinese). Journal of Central South University of Technology (Natural Science), 28(2): 138̚142 Mei Chi, Zhou Shuiliang, Wang Qianpu (1996) Mathematical simulation on electric and temperature fields in baths of a round electric cleaning furnace (in Chinese). Journal of Central South University of Technology (Natural Science), 27(5): 609̚612 Wormeck J J, Partt D T (1976) Computer modelling of turbulent combustion in a Longwell jet-stirred reactor. London: 16th Symposium on Combustion Zhou Jiemin, Zhao Tiancong, Mei Chi (1990) Mathematical model and computer simulation of Soderberg electrodes in electric smelting furnace. Elektrowarme international, 48B (4): 210̚215 Zhou Lixing (1994) Theory and Numerical Simulation of Turbulent gas-solid Two-phase Flow and Combustion (in Chinese). Science Press, Beijing Zhou Ping, Mei Chi, Zhou Shuiliang (1997) Numerical Simulation and Analysis on the Velocity Field in Bath of a Round Electric Slag-cleaning Furnace. Journal of Central South University of Technology, 4(1): 54̚57


Thermal Engineering Processes Simulation Based on Artificial Intelligence Xiaoqi Peng and Yanpo Song

Because of the complexity of nonferrous metallurgical processes, it is difficult to build accurate mechanistic models for them. While, artificial intelligence(AI) modeling method avoids the complex mechanism analysis and describes the object process by its historic data, therefore, it is very advantageous especially for complex industrial process in which historic process data have been accumulated plentifully. In this chapter, several important AI methods and their applications are introduced, based on these, two AI modeling methods for multi-variable systems are proposed: one is fuzzy adaptive modeling method, which has been applied to develop the fuzzy adaptive optimal decision model of the submerged arc furnace; another is fuzzy neural network adaptive modeling method, which has been applied to develop fuzzy neural network adaptive optimal decision model of the electric furnace for cleaning slag. Both of the models are self-learning and self-adaptive, and are able to avoid the disadvantage of the static decision-making model based on the calculation of material balance and thermal balance in a smelting process. They have achieved good performance in practice.

4.1  Characteristics of Thermal Engineering Processes in Nonferrous Metallurgical Furnaces There are many kinds of nonferrous metals, and their extraction processes are

  Xiaoqi Peng and Yanpo Song very complex. Nonferrous metallurgical furnaces, as key equipments, are mostly special furnaces without wide applications. Therefore, they have many characteristics which are different from some general industrial furnaces such as boiler, heating furnace, industrial furnace for steel making and so on: a) There are too many operational variables, including chemical components and physical characters of raw materials and fuel, technical conditions of furnace operation and others. Furthermore, they are changing randomly in different extent. b) There are serious coupling problems among multiple processes, such as heat transmission, chemical reactions, phase change, and flow of fluid or melt, which are often mutually conditional and interact fiercely. c) There exists time delay in getting information of process detecting or artificial monitoring. d) Since the scale and yield of nonferrous metals production are relatively small, basic research about it is not systemic enough, and some correlated thermal physical parameters, especially, the dynamical parameters about a process are deficient. Therefore, the mechanistic simulation models for these thermal engineering processes are not only difficult to develop, but also difficult to represent real processes. For example, to develop a mechanistic simulation model for the process of copper-matte converting, just the static model needs nearly 20 equations even if the influence of dynamical factors has been ignored. While, this analytical model could not be very precise because too many model parameters are lost. Furthermore, too complex computation would influence the real-time character of online control software. When real production data are enough, it is proposed to model using AI (artificial intelligence) methods, which can overcome the difficulties in modeling arose from above-mentioned characteristics of nonferrous metallurgical furnace in some satisfying extent.

4.2 Introduction to Artificial Intelligence Methods Since control theory was created by American scientist, Norbert Wiener, in 1940s, it has gained rapid progress and developed from the classic to the modern control theory. However, the controlled objects and processes are often extremely complex because of the problems such as complex nonlinearity, time-changing, serious coupling, random disturbances, complex reaction mechanism, besides, errors and delay exist in data measurement, the construction and dynamic characteristics are uncertain, high control performance are required, so neither the classic nor the modern control theory based on accurate mathematical model can work with good performance. Therefore, AI system emerged as the requirements of the times, which has good abilities of intelligent information processing,

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

intelligent feedback and intelligent control. AI means the abilities of some machines to execute some complex functions concerned on human intelligence such as judgment and decision-making, image identifying, learning and understanding etc. AI, which based on a symbol system and information processing, is an important branch of computer science. The main research fields include: natural language processing, logic deduction and automated theorem proving, intelligent data retrieval system, robot and its visual system, automated programming, expert system and so on. In 1965, Chinese American scientist K.S. Fu first proposed applying heuristic rules of AI theory to learning control systems (Fu, 1965). In 1971, after studying the relationship between intelligence technology and learning control, he put forward the concept of intelligent control (Fu, 1971), and pointed out that intelligent control is the cross of control theory and AI technology (that is the “binary elements theory” of intelligent control), which combines AI theory and technology with control theory and technology. In unknown environment, human intelligence is simulated so that system control can be realized effectively. In 1977, after proposing that intelligent control is the cross of control theory, operation research and AI technology (that is the “three elements theory” of intelligent control), G.N.Saridis proposed hierarchically intelligent control (Saridis, 1977), namely, the structure of intelligent control can be divided into three hierarchies from top to bottom: organization, coordination and control, the precision of control increases in turn, while intelligence degree decreases in turn. Since then, the research and application of intelligent control attracted more and more attention from many countries. Especially, fuzzy logic control, neural network control and expert control, as three typical intelligent control methods, have absolute superiority to traditional control methods, and can control effectively complex systems with the characteristics of nonlinear, multiple variables, long time delay, strong coupling and so on, therefore, they have been widely applied in engineering. At present, combining the abilities such as parallel learning, remembering and associating of neural network with fuzzy reasoning technology to form a self learning fuzzy controller (Zhang and Li, 1995), combining expert system theory and technology with fuzzy logic technology to form an expert fuzzy control and decision making system have been become the important trends in the field of intelligent decision making and control (Wang, 1994).

4.2.1 Expert system Intelligent control is essentially control based on knowledge. Special knowledge of some fields can be divided into two classes: one is quantitative knowledge which can be expressed by a precise mathematical model; another is empirical and

  Xiaoqi Peng and Yanpo Song qualitative knowledge which can not be expressed by a precise mathematical model but it can be expressed by heuristic rules in general. In dealing with difficult problems of some fields, it is usually the latter knowledge that the experts’ judgment, forecasting and decision-making lies on, therefore, the empirical and qualitative knowledge is the key of expert capability (Joseph and Gary, 2006). Converting this kind of knowledge into a computer program, the program can execute reasoning and decision making in the level of human experts, this program is called expert system (Cai and Xu, 2004). Combining expert system theory and technology with control theory and technology, simulating expert intelligence to control system, this control is called expert control, and the system is called expert control a system (Zhang and Sun, 1995). Its essence is that it can realize optimization decision making and control by simulating human experts. Expert system is a man-machine system with special knowledge to understand and solve problems in some fields, such as explaining, forecasting, diagnosing, designing, planning, monitoring, repairing, guiding, control and so on. Its performance lies on completeness and accuracy of special knowledge, and the key to develop expert system is knowledge expression and reasoning. The usual methods of knowledge expression include: predicate logic, temporal logic, framework, process, semantic network, Petri-Net, production rule, qualitative model, neural network production rule and metasymthesis, in which the latter four methods have significant value for expert control. While the usual reasoning methods include reasoning based on rules, fuzzy logic reasoning, reasoning based on neural network, qualitative reasoning, reasoning based on case and synthesis reasoning (Li, 2004), in which the former three methods have been widely applied in expert control systems. After the idea that introducing expert system technology into automatic control was proposed in 1983, K.J. Astron (Astron, 1986) proposed the concept of expert control in 1986. With the aid of expert system technology, traditional control engineering algorithms and heuristic logic reasoning were combined to develop an expert control system, which integrate many logic control functions and can adopt different control strategies according to different situations. Comparing with the traditional control, expert control has the characteristics as follow: a) It can collect knowledge and experience from many experts in some fields, so its ability may be better than individual experts. b) It can control effectively some complex industry processes which can not be expressed in precise mathematic models. c) It has self-learning ability and self-adaptive ability.

4.2.2 Fuzzy simulation In 1965, when studied on modeling for the process of thinking and judgment, L.A.

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

Zadeh (Zadeh, 1965, 1975) proposed using fuzzy set to quantify the process of thinking and judgment. Therefore, the fuzzy set theory was created, which provided a powerful tool for describing and processing fuzziness and uncertainties of systems and simulating human functions such as fuzzy logic thinking, judgment and decision-making. In 1973, L.A. Zadeh (Zadeh, 1973) gave the definition and theorem of a fuzzy logic controller, which laid the foundation for fuzzy control. In 1974, E.H. Mamdani (Mamdani, 1974) applied successfully fuzzy logic in the pressure control and the speed control of a boiler test equipment and a steam turbine engine for the first time, and opened the research field of fuzzy logic controllers based on the fuzzy language control rules. In 1979, he developed a self organizing fuzzy controller cooperating with T.J. Procky (Procky, 1979). This controller could continuously modify control rules to improve control performance in a control process, so it has good artificial intelligence. Since then, fuzzy control has been successfully applied in complex industrial processes, home appliance field, high-tech field and so on. Fuzzy technology has shown such a great application potentiality that it was called core technology in the 21st century (Li, 2004; Liu, 1997; Wang, 1987; Li, 1993). In 1992, Kosko Bart proved that an addition fuzzy system, as a kind of structural numeric estimator, could approach to any continuous function in compact domain with any precision (Kosko, 1992a, b); Lixin Wang proved that fuzzy system with production reasoning, central defuzzification and Gaussian- type membership function can also approach any real continuous function in any closed subset with any precision (Wang, 1992); Their research shows that fuzzy systems have strong ability to express human knowledge, and provide a strong theoretical foundation for building a fuzzy optimization decision-making and control system. Fuzzy set theory can quantify human qualitative thinking, judgment and decision-making. Therefore, when controlling or making decisions, besides the information measureo directly can be used, the information obtained by human sensory can be used also. Furthermore, these information are suitable for computer processing. As the research results around the world, comparing with the traditional control, fuzzy control based on fuzzy set theory, fuzzy language variable and fuzzy logic reasoning has the characteristics as follows: a) Because it uses language as expression method and does not need to build a precise mathematic model for the controlled process, it can control effectively some complex industry processes which could not be expressed by an exact mathematic model. b) For industrial fieldwork operators possessing certain experiences, the control and decision making methods described by the fuzzy rules are easier to master. c) Man-machine communication can be realized by the nature language, so process control can be realized effectively. d) Dynamic response characteristic of a fuzzy control system is usually better than conventional PID control, and has better robustness and self-adaptability.

  Xiaoqi Peng and Yanpo Song

4.2.3 Artificial neural network Human intelligent activities such as thinking, association, memory, judgment, decision-making and so on occur in brain, which is composed of huge amounts of neurons. Different neurons connect one another in some modes and form a biological neural network, and the connection strength between the neurons of the network can change adaptively according to external actuating signals. Neuron shifts the states between excitatory and inhibitory with the change in its received actuation signals. The learning process of brain is the process in which the connection strength between neurons change adaptively with external actuation signals. The results of information processing in brain are expressed by the states of neurons. ANN (artificial neural network) is an information processing system which simulates the structure and function of the biological neural network by engineering technologies. It is composed of a number of simple processing cells which simulate biological neurons and which are connected on another in some way, and its ability to process information relies on the cells’ dynamic response to the external input information. In 1943, psychologist W. Mcculloch and mathematician W. Pitts proposed cooperatively the earliest neuron model, which opens the era of neural science (Zhang, 1996; Hu, 1993). In 1949, D.O. Hebb (Hebb, 1949) proposed a hypothesis that synapse connection strength is changeable, and learning process is thought as occurring in synapses whose connection strength changes with the connecting neurons’ activities on its two ends. The learning rules based on this hypothesis laid a foundation for the learning algorithm of artificial neural network. In 1958, F. Rosenblat proposed a perceptron model to simulate human abilities of perception and learning for the first time, and designed the perceptron which was capable of self learning and self organizing. But in the book Perceptron published in 1969, the famous scientists M. Minsky and S. Papert proved theoretically that the monolayer perceptron had only limited ability. Furthermore, they speculated that the multilayer perception also had the limitation, which throw a damp over the research on neural network (Zhou, 1993). In early 1980s, after concentrating study, famous physicist J.J. Hopfield introduced the concept of energy function, proposed HNN model, presented criteria for the stability of neural network and pointed out that an electrical circuit composed by operation amplifiers could be used to simulate neurons. His outstanding results revived the research on ANN, and many researchers from the world were inspired to throw themselves into this field. Therefore, ANN theory and application research got rapid development, and new models and new theories constantly emerged (Hopfield, 1982, 1984). In 1983, G.E. Hinton and T.J. Sejnowski (Hinton and Sejnowski, 1986) combined the output function of a neuron with Boltzmann distribution of statistics, and proposed Boltzmann model. In 1987, B. Kosko (Kosko, 1987, 1988) proposed a

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

bidirectional associative memories model. Also in 1987, Hecht Nielsen (Hecht, 1987) proposed counter propagation neural network. Noticeably, a multi-layer back propagation neural network proposed by P.J. Werbos (Werbos, 1974) in 1974 revived and has been become one of the most widely applied networks at present. The research on the neural network theory and application is heading for the climax, and it has been widely applied in fields such as computer science, artificial intelligence, automatic control, robot, information procession and so on. ANN, as a simulation of human brain, is a kind of nonlinear dynamic system in essence. Comparing with other information process systems, its characteristics are as follows (Deng, 1994): a) It expresses special information by the connections among lots of neurons and their weights, that is, its information storage way is distributed, so it has strong robustness and fault tolerance, and can recover efficiently information even if some local neurons are damaged. b) It has the abilities of parallel and cooperation information processing and reasoning. c) It has the abilities of self-learning, associative memory and synthesis reasoning, and can process various signals and patterns which simulate human brain. d) It can approach any nonlinear function with any precision. Because of the advantages mentioned above, it has great potential in controlling the complex system with high nonlinearity and high uncertainty. Therefore, it has been widely applied in the field of process decision-making and control. Generally, the control based on ANN is called NNC (neural network control) or NC (neural control). The roles of ANN in NC are as follows: a) Serving as an object model in the structure of control based on a precise model. b) Serving as the controller in a feedback control. c) Serving for the functions of optimizing decision and computing in a control system. d) Combining with other intelligent control methods and optimization algorithms, and providing them with nonparametric models, reasoning models, optimized parameters and so on.

4.3 Modeling Based on Intelligent Fuzzy Analysis Most industrial processes are generally complex systems with characteristics such as multi-variable, non-linear and long time-delay. Therefore it is very difficult to make optimal decision by mathematical analysis, though the experienced spot workers can do it according to their abundant experience. This experience knowledge contained in huge amount of real production data and accumulated in operation is generally described as fuzzy rules, and it is very possible to extracted this usefue knowledge by fuzzy model. In this section, practical algorithms of fuzzy model identification and adaptive optimization for multi-variable system are proposed and applied to a typical

  Xiaoqi Peng and Yanpo Song complex industrial systemüa nickel-matte smelting submerged arc furnace. A fuzzy adaptive optimal decision-making model was developed as well to achieve the goals of energy-saving and consumption-reduction.

4.3.1 Intelligent fuzzy self-adaptive modeling of multi-variable system Providing a system with r inputs xi (t ) (i=1,2,…,r) and q outputs y j (t ) (j=1, 2, …, q), the value of any input xi (k ) at time k is decided by L fuzzy rules, these L rules constitute the fuzzy decision model of this input, in which the lth rule Rl can be expressed as follows (Tomohiro and Michio, 1985): if x1 ( k − 1) is A1l1 , x1 (k − 2) is A1l 2 ,…, x1 (k − n) is A1ln , M M M xr (k − 1) is Arl1 , xr (k − 2) is Arl 2 ,…, xr (k − n) is Arln , y1 ( k − 1) is B1l1 , y1 ( k − 2) is B1l 2 ,…, y1 (k − n) is B1lm , M M M l1 l 2 y q (k − 1) is Bq , y q (k − 2) is B q ,…, yq ( k − n) is Bqlm , then xil ( k ) = a0l + a11l x1 ( k −1) + a12l x1 ( k − 2) + K + a1ln x1 ( k − n) + l l a21 x2 ( k −1) + a22 x2 (k − 2) + K + a2l n x2 ( k − n) + K + arl 1 xr (k −1) + arl 2 xr (k − 2) + K + arnl xr (k − n) + b11l y1 (k −1) + b12l y1 ( k − 2) + K + b1lm y1 ( k − m) + K + l bql 1 yq (k −1) + bql 2 yq (k − 2) + K + bqm yq ( k − m ) + e l ( k ) (4.1) where xi (k ) and yi (k ) are the values of xi and yi at k time; el (k ) is the error of fuzzy rule; Ail1~ n and B lj1~ m denote the fuzzy subsets of xi and yi in the domains of input and output respectively, defining these fuzzy subsets needs to consider the special condition of an object, in this paper they are defined as shown in Fig.4.1, triangle subordination degree functions are chosen; aikl and b ljk are model parameters that need to be identified; xil (k ) is the input control variable decided by the lth fuzzy rule; n, m are the orders of a model.

Fig. 4.1

Fuzzy sub-sets Ai and Bj

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

Given a group of data [ x1 (k − 1) , x1 (k − 2) , K , x1 (k − n) , K , xr ( k − 1) , xr (k − 2), K , xr (k − n) , K , y1 (k − 1) , y1 ( k − 2) , K , y1 ( k − m) , K , yq ( k − 1) , yq (k − 2), K , y q (k − m) ], input control variable xi (k ) can be got by weighted averaging xil (k ) decided by L decision rules: L

L

xi (k ) = ∑ λl • xil (k )

(4.2)

μ Blji [ yi (k − j )]

(4.3)

l

l =1

where weight λl

∑λ l =1

can be got as follows: r

λl = I

n

I

i =1 j =1

μ Alji [ xi (k − j )]I

q

m

II

i =1 j =1

where “ Ι ” expresses “and” in fuzzy logic, namely minimizing operation; μ ljA [ xi (k − j )] expresses the membership function value of xi (k − j ) for fuzzy i subset Ailj , the mean of μ Blj [ yi (k − j ) is similar. i

4.3.1.1 Identification of fuzzy decision making model of multi-variable system (Peng et al., 1994)

By combining the mechanism analysis and the multiple linear stepwise regression method (Wu, 1987), n and m can be decided. If they are small, model parameters can be got by the following identification algorithm, otherwise, the input or output variables with weaker effect should be ignored in order to speed up the identification algorithm. Let l θ l = [ a0l a11l a12l K a1l n K arl 1 arl 2 K arnl b11l b12l K b1lm K bql 1bql 2 K bqm ]T (4.4)

ϕ = [1x1 (k −1) x1 (k − 2)]K x1 (k − n) K xr (k −1) xr (k − 2) K xr (k − n) xr (k −1) y1 (k −1) y1 (k − 2) K y1 (k − m) K T yq (k −1) yq (k − 2) K yq (k − m)˹

(4.5) (4.5)

then, Eq. 4.1 can be rewritten as: xil (k ) = ϕ T θl + el (k )

(4.6)

Substitute Eq. 4.6 into Eq. 4.2, then xi (k ) =

λ1 L

∑ λl l =1

ϕ T θ1 +

λ2 L

∑ λl

ϕ T θ2 + K +

l =1

λl L

∑ λl

L

ϕ T θL + ∑ λl el (k )

(4.7)

l =1

l =1

Let

ϕ lT =

λl L

∑ λl

ϕT

(4.8)

l =1

ϕ l = [ϕ1T ϕ 2T K ϕ LT ]T Θi = [θ1 θ 2 K θ L ]

T

(4.9) (4.10)

L

ei (k ) = ∑ λl el (k ) l =1

(4.11)

  Xiaoqi Peng and Yanpo Song

then Eq. 4.7 can be rewritten as: xi ( k ) = (ϕ L )T Θi + ei ( k )

(4.12)

This expression is called the control model of input variable xi (k ) , in which Θi is parameter vector, ϕ l is innovation vector and, it is a constant vector

converted from the first set of data according to Eq. 4.3, Eq. 4.8 and Eq. 4.9. According to a control objective, the quality of real production data can be judged by some performance indexes, while the data groups whose performance indexes meet the requirements are called optimal data groups. In order to get the optimal rules, optimal data groups should be used to identify the model parameters. Therefore, selecting N different optimal data groups from real production data (where N should not be less than the number of parameters), according to Eq. 4.3, Eq. 4.5, Eq. 4.8 and Eq. 4.9, innovation vectors ϕ 1 , ϕ 2 ,…, ϕ N can be found. Substituting N optimal data groups and their innovation vectors into Eq. (4.12), we have: [ xi (k1 ) xi ( k2 ) K xi ( k N )]T = [(ϕ 1 )T (ϕ 2 )T K (ϕ N )T ]T Θi + e ( k ) (4.13) where,

e (k ) = [ei (k1 )ei (k2 ) K ei (k N )]T

(4.14)

xi = [ xi ( k1 ) xi ( k2 ) K xi (k N )]T

(4.15)

φ = [(ϕ ) (ϕ ) K (ϕ ) ]

(4.16)

Let 1 T

2 T

N T T

then, Eq. 4.13 can be converted to: xi = φ Θi + e ( k )

(4.17)

By means of the least square parameter estimation method(Wu,1987), the best estimation for parameter vector Θi of fuzzy control model of xi (k ) is: Θˆ i = [φ T φ ]−1φ T xi (4.18) 4.3.1.2 Fuzzy adaptive optimization algorithm

The performance of a decision-making may be gradually declining due to the slowly-varying of external environments, production conditions and system structure. To make a model have the capability of self-learning and selfadaptation, obsolete rules in original model should be replaced dynamically by new fuzzy decision rules based on new optimal data groups, and then new Θi can be obtained. To reduce the computational complexity and speed up the identification, recursive least square method with moving range(Wu,1987) are usually used to revised the model adaptively. Identification algorithm of fuzzy a adaptive control model can be shown as Fig.4.2.

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

Fig. 4.2 Flow chart of identification algorithm for fuzzy adaptive decision-making model

4.3.2 Example: fuzzy adaptive decision-making model for nickel matte smelting process in submerged arc furnace If the slow-varying factors in a smelting process are neglected, the furnace can be considered as a multi-variable system as shown in Fig. 4.3. Sulfide nickel calcine is the most important raw material, quartzite is used as slag-making flux, coke powder is used as reducer, while, liquid hot converter slag is partially inputted to the electric furnace in order to increase the yield of nickel and cobalt. The produced slag is abandoned after breaking up by water, flue gas is sent to dust collection workshop, and low-nickel matte is sent to converter for converting.

  Xiaoqi Peng and Yanpo Song

Fig. 4.3 Submerged arc furnace

The characteristics of the process mentioned above is as follows: a) Composition of raw materials is complex and unsteady. b) It is difficult to realize real-time and accurate measurement of the technology parameters such as materials’ composition, temperature and so on. c) There exists serious random disturbances. d) Decision-making and operation rely on human experiences. e) Production indexes fluctuate acutely. 4.3.2.1 Modeling approach

The production indexes are influenced by many factors, such as the strong coupling among various parameters, the complex thermodynamic and dynamic problems, energy conversion problems and so on. All these factors influence one another in such a complex way that the smelting system in smelter is typical complex system with multi-variable, nonlinearity, long time-delay and so on. Therefore, only if the integrated balance point of all factors can be found out, it can be possible to realize the overall optimization of smelting process and achieve the goal of energy saving and consumption reducing. Traditionally, decision making of smelting a process relies on a material and heat balance model. This model plays an important role in guiding production, but it is not suitable for dynamic real production process because it is essentially static. Besides, there are big error and long time delay measurements because of the faultiness of measurement instruments, which limit the application of material and heat balance model. Therefore, in real production worksite, operation control mainly relies on human experience; production process is influenced seriously by human factors; smelter condition is not enough steady; techno-economical indexes and production indexes fluctuates acutely. The main controllable variables of a smelting process are input of nickel calcine, flux and converter slag, electric power of furnace, output of low-nickel matte and slag. Although the operating voltage is changeable in production, it

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

keeps constant under normal production conditions, so it is not regarded as a controllable variable in this chapter. If the composition and optimal input of calcine were estimated by some algorithm, then the optimal input of flux and producer could be estimated according to the furnace condition, composition and input of calcine, and then the optimal smelting power of electric furnace could be estimated according to the furnace condition and input of materials. Finally, the optimal output of low nickel matte and slag could be decided according to the estimated results mentioned above and the furnace condition. Thus, the integrated index of production process would be optimized, and the goal of energy saving and consumption reducing would be achieved. To realize these, an optimization decision-making model is necessary. Nickel smelter is a complex system with the characteristics of multi-variable, nonlinear and long time delay. It is difficult to build a precise mathematical model based on smelting mechanism, but the experienced spot workers and technicians can ensure the safety of a production process depending on their abundant experience. Fuzzy model has strong ability to approach a complex system with the characteristics of multi-variable, nonlinear and long time delay, and can describe human strategy of operation and control by way of language, while the huge amount of real production data accumulated in production practice contains human experience. Therefore, if the human experience was extracted from the data by using the strong ability of computer in data processing and computing, and expressed as fuzzy decision-making rules in the form of “If … and … Then …”, then these rules constitute the fuzzy decision-making model for a smelting process. If new decision-making rules were generalized constantly in a production process, that is, the model was revised constantly so that it would have good adaptability. Based on the above analysisˈit is proposed to build an adaptive fuzzy decision model for a smelting process(Peng,1998;Mei et al.,1994a,b) using the methods introduced in Section 4.3.1 and taking energy saving and consumption reducing as our goal. Composition of calcine is unstable and its measure data delays about two shifts, for different types of calcine, different decision-making models are need to obtain control variables. Therefore, we classify calcine according to its percentage contents of Fe, SiO2 and S firstly, then build a fuzzy optimization decision-making model for each type. Before making a decision, forecast the composition of the inputting calcine, then classify the calcine using the fuzzy clustering method(Li,1986), finally, finish the optimal decision-making for the production process using the corresponding model.

  Xiaoqi Peng and Yanpo Song 4.3.2.2 Objective function and optimal data

The wastage functions of the melting process in a smelting furnace was defined as y •z •d (4.19) Q = d b • wb + z n n xb where db is the price of electric energy (yuan/kW • h); wb is the electric energy consumption per ton of calcine (kW • h/t); yz is the output of the smelting slag (t/shift); zn is the percentage content of nickel slag (%); dn is the price of nickel (yuan/t); xb is the weight of the calcine inputted per shift (t/shift); Q is the wastage function of the melting process in a submerged arc furnace for nickel matte. The value of Q reflects synthetically the value of energy consumed and nickel lost, data with small value Q are selected as optimal data, a fuzzy model based on optimal data is the fuzzy optimization decision making model. 4.3.2.3 Calcine composition forecasting model

Percentage contents of Fe, SiO2 and S in calcine are unsteady, while measurement data delay several shifts in general, so it is necessary to forecast the composition of calcine, therefore, it is proposed to build calcine composition forecasting autoregressive model (AR model) as follows(Peng et al.,1993): a) Iron content in calcine, bFe ( k ) = c1FebFe ( k − 1) + c2 FebFe ( k − 2) + c3 FebFe (k − 3) + eFe (k ) (4.20) b) SiO2 content in calcine, bSi (k ) = c1Si bSi (k − 1) + c 2Si bSi (k − 2) + c3Si bSi (k − 3) + eSi (k )

(4.21)

c) Sulfur content in calcine, bS (k ) = c1SbS (k − 1) + c2SbS (k − 2) + c3SbS (k − 3) + eS (k )

(4.22)

where bFe(k) is iron content of calcine inputted in the kth shift; eFe(k) is iron content forecast error of the kth shift; bSi(k) is SiO2 content of calcine inputted in the kth shift; eSi(k) is SiO2 content forecast error of the kth shift; bS(k) is sulfur content of calcine inputted in the kth shift; eS(k) is sulfur content forecast error of the kth shift. The value of parameters should minimize error square sum, that is, the follows should be satisfied: N

∑ k =4

N

eFe2 (k ) → min

∑ k =4

N

eSi2 (k ) → min

∑ e (k ) → min 2 S

k =4

Therefore, given N (Nı6) groups of calcine composition data, the parameters of AR model can be obtained by means of the least squares method. To ensure the accuracy of forecasting result, the model parameters should be revised according new valid data before forecasting, that is, make the forecasting model can reflect

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Thermal Engineering Processes Simulation Based on Artificial Intelligence 

duly the change of calcine composition. 4.3.2.4 Fuzzy adaptive decision-making model for production process in submerged arc furnace

According to mechanism analysis of smelting process, considering of operation and control experience of spot technicians, and through stepwise regression analyzing the huge amount of real production data accumulated in production practice, the following relation functions can be obtained: a) Input of calcine at time k: xb ( k ) = Fb [ xb ( k − 1), xb ( k − 2), xS ( k − 1), xS ( k − 2), xW ( k − 1), xW ( k − 2), xf ( k − 1), xf ( k − 2), yd ( k − 1), yd ( k − 2), yz ( k − 1), yz (k − 2)]

(4.23)

b) Input of converter slag at time k: xf (k ) = Ff [ xb (k ), xb (k − 1), xb (k − 2), xS (k − 1), xS (k − 2), xW (k − 1), xW (k − 2), xf ( k − 1), xf ( k − 2), yd ( k − 1), yd ( k − 2), yz ( k − 1), yz (k − 2)]

(4.24)

c) Input of quartz sand at time k: xS (k ) = FS [ xb (k ), xb (k −1), xb (k − 2), xS (k −1), xS (k − 2), xW (k −1), xW (k − 2), xf (k ), xf (k −1), xf (k − 2),

(4.25)

yd (k −1), yd (k − 2), yz (k −1), yz (k − 2)] d) Electric energy consumption at time k: xW (k ) = FW [ xb (k ), xb (k −1), xb (k − 2), xS (k ), xS (k −1), xS (k − 2), xW (k −1), xW (k − 2), xf (k ), xf (k −1),

(4.26)

xf (k − 2), yd (k −1), yd (k − 2), yz ( k −1), yz (k − 2)] e) Input of coke powder at time k: xc (k ) = Fc [ xb (k ), xb (k − 1), xf (k ), xf (k − 1)]

f) Output of low nickel matte at time k: yd (k ) = Fd [ xb (k ), xb (k −1), xb (k − 2), xW (k ), xW (k −1), xW (k − 2), xf (k ), xf (k −1), yd (k −1), yd (k − 2)] + Δyd (k )

(4.27)

(4.28)

g) Output of slag at time k: yz (k ) = Fz [ xb (k ), xb (k −1), xb (k − 2), xS (k ), xS (k −1), xS (k − 2), xW (k ), xW (k −1), xW (k − 2), xf (k ),

(4.29)

xf (k −1), yz (k −1), yz (k − 2)] + Δyz (k ) where, ⎧( L − Lh ) / k l ⎪ Δyd (k ) = ⎨ 0 ⎪ (L − L ) / k l l ⎩ ⎧(Z − Z h ) / k z ⎪ Δyz (k ) = ⎨ 0 ⎪ (Z − Z ) / k l z ⎩

( L ıLh ) ( Ll < L < Lh )

(4.30)

( L İLl ) (Z ıZ h ) (Z l < Z < Z h ) ( Z İZ l )

(4.31)

  Xiaoqi Peng and Yanpo Song

where L is the height of liquid matte surface; Lh is the max allowable height of liquid matte surface; kl is correction coefficient for the height of liquid matte surface; Ll is the minimum allowable height of liquid matte surface; Z is the height of liquid slag surface; Zh is the maximum allowable height of liquid slag surface; kz is correction coefficient for the height of liquid slag surface; Zl is the minimum allowable height of liquid slag surface; Δyd (k ) is called matte surface height correction coefficient; Δyz (k ) is called slag surface height correction coefficient. According to Eq. 4.23 through Eq. 4.29 and the method introduced in Section 4.3.1, the fuzzy adaptive decision-making model for a smelting process can be built, the process of decision-making based on this model can be shown as Fig. 4.4.

Fig. 4.4 Fuzzy decision-making process of submerged arc furnace

4.4 Modeling Based on Fuzzy Neural Network Analysis For complex industrial systems that haven’t sufficient samples, it is difficult to develop optimal decision model directly using methods introduced in Section 4.3. In such a case, fuzzy neural network adaptive modeling methods are more suitable for multi-variable system. In this section, an optimal decision model for

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Thermal Engineering Processes Simulation Based on Artificial Intelligence 

slag cleaning arc furnace is developed with this modeling method.

4.4.1  Fuzzy neural network adaptive modeling methods of multi-variable system Providing a system with r inputs xi (t ) (i=1,2,…,r) and q outputs y j (t ) (j=1, 2, …, q), the value of any input xi (k ) at time k is decided by Eq. 4.7:

xi ( k ) =

λ1

L

∑λ

ϕ T θ1 +

l

λ2

L

∑λ

l

l =1

l =1

ϕ Tθ 2 + K +

λl

L

L

∑λ

ϕ T θ L + ∑ λl el ( k ) (4.32) l =1

l

l =1

According to Eq. 4.3 through Eq. 4.5 and Eq. 4.32, a fuzzy neural network (Peng et al., 1995) can be constructed as Fig.4.5, where ­, ě, ģ, h, / represent calculating the membership function value, summating, minimizing, multiplication, division for corresponding neuron, respectively. 4.4.1.1 Identification of fuzzy neural network decision making model of multivariable system

The learning algorithm of fuzzy neural network is inferred as follows. Rewriting Eq. 4.32 as: L 1 xi ( k ) = L [λ1θ 1T ϕ + λ2θ 2T ϕ + K + λLθ L T ϕ ] + ∑ λl el ( k )

∑λ

l =1

l

l =1

=

1 L

∑ λl

L

[λ1λ2 K λL ][θ 1T θ 2T K θ LT ]T ϕ + ∑ λl e l ( k )

(4.33)

l =1

l =1

Let Θ ' = [θ 1T θ 2T L θ L T ]T

(4.34)

Then 1 1 ⎤ ⎡ a10 a11 K a1r 1 K a1n1b111 K b11m K bq11 K bqm ⎢ ⎥ ⎢ a2a2 K a2 K a2 b2 K b2 K b2 K b2 ⎥ r1 n1 11 q1 qm ⎥ 1m Θ ' = ⎢⎢ 0 11 M M M M M ⎥⎥ ⎢ M ⎢ L L L L L L L L ⎥ ⎢⎣ a0 a11 K ar 1 K an1b11 K b1m K bq1 K bqm ⎥⎦ Let error function J be

J=

1 [ xˆ i ( k ) − xi ( k )] 2 2

(4.35)

(4.36)

where xˆi (k ) is the practical input in an optimal sample and xi (k ) is the output calculated by the fuzzy neural network. The study target of the fuzzy neural network is to minimize J. According to one order gradient algorithm:

  Xiaoqi Peng and Yanpo Song λ ∂J ∂J ∂xi (k ) = = − [ xˆi (k ) − xi (k )] • L l • ∂a0l ∂xi (k ) ∂a0l

(4.37)

∑ λl l =1

λ ∂J ∂J ∂xi (k ) • = = −[ xˆi (k ) − xi (k )] • L l ∂aiil ' ∂xi (k ) ∂aiil '

∑λ

− i ')

(4.38)

y j (k − j ')

(4.39)

• xi ( k

l

l =1

λ ∂J ∂J ∂xi (k ) = = −[ xˆi (k ) − xi (k )] • L l • ∂bljj ' ∂xi (k ) ∂bljj '

•

∑λ

l

l =1

where, iĄ=1, 2, …, n; jĄ=1, 2, …, m. Let Δxi (k ) = xˆi (k ) − xi (k )

λ = [λ1λ2 K λl ]

T

(4.40) (4.41)

Then, according to Eq.4.37 through Eq.4.39, we have: η • Δx (k ) T Θ ′ = Θ0′ + ΔΘ ′ = Θ0′ + L i • λ •ϕ

∑λ

(4.42)

l

l =1

where η is the learning speed of the neural network; Θ0 is the original value of Θ. Equation 4.42 is the learning algorithm of the neural network, it also can be regarded as an identification algorithm for a fuzzy decision-making and control model. The fuzzy neural network can be trained with optimal samples; the trained neural network is an optimal decision-making and control model; the decisions made by this model are optimal decisions; the control executed by it is optimal control. In order to simplify network structure and accelerate the learning process, firstly, n and m should be decided by a structure identification method on the basic of mechanism analysis and practical operation control experiences; then, the input or output variables with weaker effect should be ignored by some analysis; finally, the network with structure as shown in Fig. 4.5 can be decided. 4.4.1.2 Fuzzy adaptive algorithm

Slowly-varying of external environments, production conditions and the system structure can lead to the decline of the performance of decision-making model. To make the model have the capability of self-learning and self-adaptation, new optimal samples produced in real production should be used to “update” the neural network, that is, the decision-making model is revised dynamically to cater for current situation of the system and store new optimal decision-making experiences. Therefore, the decision-making ability of the model is improved increasingly.

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Thermal Engineering Processes Simulation Based on Artificial Intelligence 

Fig. 4.5 Fuzzy neural network control model of multi-variable system

4.4.1.3 Convergence of neural network

The major problems existing in modeling process are how to improve the convergence speed of a learning process and how to judge the learning result is global optimal or not. To accelerate the learning process, learning speed η is fuzzy adjusted: when error is big, a big η is used to accelerate the learning speed; while, when error is small, a small η is used to improve the approximation accuracy of the neural network. By this method, the learning process is accelerated. There is no mathematical method to judge whether the result is global optimal or not, so it is judged directly by practice: performing the real time simulation to the production process using a trained neural network model; if the simulation result is accurate, the model is considered as optimal model.

  Xiaoqi Peng and Yanpo Song

4.4.2 Example: fuzzy neural network adaptive decision-making model for production process in slag cleaning furnace If the slow-varying factors in slag a cleaning process are neglected, the furnace can be thought as a multi-variable system as shown in Fig. 4.6. Fuzzy neural network model can be build by using the methods introduced in Section 4.4.1. Since the input materials’ composition is steady in production process, there are no need to build different decision-making models for different types converter slag as in a smelting furnace.

Fig. 4.6 Electric furnace for cleaning slag

In order to recover cobalt and nickel from converter slag, liquid converter slag is smelted in a slag cleaning furnace. Sulphide is applied to sulphurize the valuable metals and make them go into matte; quartzite is slag-making flux; coke powder is reducer. The operation process is divided into melting stage and cleaning stage: input materials in the melting stage; matte and slag are separated in cleaning stage. This technological process is similar to a submerged arc furnace mentioned in Section 4.3. The main controllable variable of a slag cleaning process are input of converter slag, quartzite, coke powder, power, output of cobalt enriched matte and cleaned slag. In real production process, operation instructions are made in previous shift. Limited by converter process and other conditions, the real input of converter in this shift is often different from the amount given by instructions, while operators often do not adjust other controllable variables such as the input of quartzite, coke powder, the output of cobalt enriched matte and so on according to the difference, therefore, it is difficult to make the smelting process running under an optimum condition. If the optimal input of sulphide, quartzite and coke powder were estimated according to furnace state, output and composition of converter slag, then the optimal load current and its duration time in the melting stage and cleaning stage could be decided according to furnace state and input materials, finally, the

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

optimal output of cobalt enriched matte and slag could be decided according to furnace state and decisions mentioned above. That is, by optimizing the weight of input materials and load current and its duration time, electric consumption would be reduced, slag constitution would be improved, separation condition for matte and slag would be improved, valuable metals’ loss in slag would be decreased, thus, integrated index of production process would be optimized, and the goal of energy saving and consumption reducing would be achieved. To realize these, an optimization decision-making model is necessary. Slag cleaning process is a complex system with characteristics of multi-variable, nonlinear and long time delay (Peng,1998;Peng and Su et al.,1996;Peng and Mei et al.,1996). It is difficult to build a precise mathematical model based on smelting mechanism. Further more, because the optimal samples collected in real production process are relatively insufficient, it is difficult to build an optimization decision-making model using the fuzzy control model identification algorithm and adaptive optimization algorithm introduced in Section 4.3 for a slag cleaning process. Therefore, it is proposed to build an optimization decision-making model using fuzzy neural network adaptive modeling methods of multi-variable system introduced in Section 4.4.1. 4.4.2.1 Objective function and learning samples

The wastage functions of the slag cleaning process is defined as yp • ( z Ni • d Ni + zCo • d Co ) Qp = d b • wz + xz

(4.43)

where d b is the price of electric energy (yuan / kW • h); wz is the electric energy consumption per ton of converter slag (kW • h/t); y p is the output of the cleaned slag (t/shift); z Ni is the percentage content of nickel in slag (%); d Ni is the price of nickel (yuan/t); z Co is the percentage content of cobalt in slag (%); d Co is the price of cobalt (yuan/t); xz is the weight of the converter slag inputted per shift (t / shift). The value of Qp reflects synthetically the value of energy consumed and valuable metal lost is the slag cleaning process. Taking it as objective function, data with small value Qp are selected as learning samples from lots of real production data. The fuzzy neural network model trained by these samples is fuzzy neural network optimization decision making model. 4.4.2.2 Fuzzy neural network decision-making model for slag cleaning process

According to the mechanism analysis of slag cleaning process, considering of operation and control experience of spot technicians, and through stepwise regression analyzing the huge amount of real production data accumulated in production practice, the following relation functions can be obtained:

  Xiaoqi Peng and Yanpo Song

a) Input of converter slag at time k: xz (k ) = Fz [ xz (k −1), xs (k −1), xl (k −1), xg ( k −1), yg (k −1), yp (k −1), xWh (k −1), xWp (k −1), xTh (k −1), xTp ( k −1)]

(4.44)

b) Input of sulphide at time k: xl (k ) = Fl [ xz (k ), xz ( k − 1), xl ( k − 1), y g (k − 1), y p ( k − 1), xWh (k − 1), xWp (k − 1), xTh (k − 1), xTp (k − 1)]

(4.45)

c) Input of quartz sand at time k: xs ( k ) = Fs [ xz ( k ), xz ( k − 1), xl (k ), xl (k − 1), xs ( k − 1), y p (k − 1), yg ( k − 1), xWh ( k − 1), xWp ( k − 1), xTh ( k − 1), xTp ( k − 1)]

(4.46)

d) Input of coke powder at time k: xc (k ) = Fc [ xz (k ), xl (k )]

(4.47)

e) Load current in the smelting stage of the kth shift: xWh ( k ) = Fh [ xz ( k ), xl ( k ), xs (k ), xs (k − 1), y p (k − 1), yg (k − 1), xWp (k − 1), xTp (k − 1)]

(4.48)

f) Duration time of the smelting stage of the kth shift: xTh ( k ) = FTh [ xz (k ), xl ( k ), xs (k ), xs (k − 1), yp (k − 1), yg (k − 1), xWp (k − 1), xTp (k − 1)]

(4.49)

g) Load current in the cleaning stage of the kth shift: xWp (k ) = Fp [ xz (k ), xl (k ), xs (k ), xs (k ), xWh (k ), xTh (k ), y p (k − 1), yg (k − 1)]

(4.50)

h) Duration time of the cleaning stage of the kth shift: xTp (k ) = FTp [ xz (k ), xl (k ), xs (k ), xs ( k ), xWh (k ), xTh (k ), y p (k − 1), yg (k − 1)]

(4.51)

i) Output of cobalt enriched matte at time k: yg (k ) = Fg [ xz (k ), xz (k − 1), xl (k ), xl (k − 1), xs (k ), yg (k − 1), x Wh (k ), xTh (k ), x Wp (k ), xTp (k )] + Δy g (k )

(4.52)

j) Output of cleaned slag at time k: yg (k ) = Fg [ xz (k ), xz (k −1), xl (k ), xl (k −1), xs ( k ), xs (k −1), y p ( k − 1), xWh ( k ), xTh ( k ), xWp ( k ), xTp ( k )] + Δy p ( k )

(4.53)

⎧( L − Lh ) / kl ⎪ Δyg (k ) = ⎨0 ⎪ (L − L ) / k l l ⎩

(4.54)

where,

( LıLh ) ( Ll < L < Lh ) ( L İLl )

⎧( Z − Z h ) / k z ( Z ı Z h ) ⎪ Δyp (k ) = ⎨ 0 (Z l < Z < Z h ) (4.55) ⎪ (Z − Z ) / k ( Z İZ l ) l z ⎩ where L is the height of liquid matte surface at instant k−1; Lh is the max allowable height of liquid matte surface; kl is correction coefficient for the height

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

of liquid matte surface; Ll is the minimum allowable height of liquid matte surface; Z is the height of liquid slag surface at instant k−1; Zh is the max allowable height of liquid slag surface; kz is correction coefficient for the height of liquid slag surface; Zl is the minimum allowable height of liquid slag surface; Δyg (k ) is called matte surface height correction coefficient; Δy p (k ) is called slag surface height correction coefficient. According to Eq. 4.44 through Eq. 4.53 and the method introduced in Section 4.4.1, the fuzzy neural network decision making model for a slag cleaning process can be built, the process of decision making based on this model can be shown as Fig. 4.7:

Fig. 4.7 Fuzzy neural network decision-making process of cleaning slag furnace

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  Xiaoqi Peng and Yanpo Song Hopfield J J (1982) Neural networks and physical systems with emergent collective computational abilities. Proc. of the National Academy of Science, U. S. A., 79: 2554~2558 Hopfield J J (1984) Neural with graded response have collective computational properties like those of two-state neurons. Proc. of the National Academy of Science, U. S. A., 81:3088~3092 Hu Shouren (1993) Application technology of neural network (in Chinese). National University of Defense Technology Press, Changsha Joseph C G, Gary D R (2006) Expert Systems: Principles and Programming, Fourth Edition (in Chinese). China Machine Press, Beijing Kosko B (1987) Adaptive bidirectional associative memories. Applied Optics, 26(23): 4947~4960 Kosko B (1988) Bidirectional associative memories. IEEE Trans, SMC-18: 49~60 Kosko B (1992a) Fuzzy function approximation. IJCNN’92, Baltimore, 1: 209~213 Kosko B (1992b) Fuzzy system as universal approximators. IEEE Fuzzy’92: 1153~1162 Li Anhua (1986) Groundwork and Application of Fuzzy Mathematics (in Chinese). Xinjiang People Press, Xinjiang Li Shiyong (2004) Fuzzy Control, Neural Control and Intelligent Control Theory (in Chinese) (Second Edition). Harbin Institute of Technology Press, Harbin Li Youshan (1993) Fuzzy Control Theory and its Application in Process Control (in Chinese). National Defence Industry Press, Beijing Liu Zengliang (1997) Fuzzy Technology and its Application (in Chinese). Beijing University of Aeronautics & Astronautics Press, Beijing Mamdani E H (1974) Applications of fuzzy algorithms for control of simple dynamic plant. Proceedings of IEEE, 121(12): 1585~1588 Mei Chi, Peng Xiaoqi, Zhou Jiemin (1994a) An intelligent decision support system on the process of nickel matte smelter. Journal of Central South University of Technology, 1(1) Mei Chi, Peng Xiaoqi, Zhou Jiemin (1994b) Fuzzy and adaptive control model for process in nickel matte smelting furnace. Transactions of Nonferrous Metals Society of China, 4(3) Peng Xiaoqi (1998) The development and application of the intelligent decision technique for economizing electric energy and reducing consumption in nickel smelting and the integrated computer information network in smelting workshop (in Chinese). Thesis (Ph. D). Central South University of Technology, Changsha Peng Xiaoqi, Mei Chi, Zhou Jiemin (1996) An intelligent decision support system in the operation process of electric furnace for cleaning slag. Journal of Central South University of Technology, 3(2) Peng Xiaoqi, Mei Chi, Zhou Jiemin et al (1994) An identification method of multivariable fuzzy control model and its application in the decision support system of smelting furnace (in Chinese). Control Theory & Application, 11(5): 582~587 Peng Xiaoqi, Mei Chi, Zhou Jiemin et al (1995) A fuzzy neural network decision model on

4

Thermal Engineering Processes Simulation Based on Artificial Intelligence 

the operation process of electric furnace for cleaning slag and its application. Transactions of Nonferrous Metals society of China, 5(3): 21~24 Peng Xiaoqi, Su Daixiong, Mei Chi (1996) A dynamic fuzzy optimal decision model for the process of slag-cleaning electric furnace (in Chinese). Journal of Central South University of Technology (Natural Science edition), 27(6) Peng Xiaoqi, Zhou Jiemin, Mei Chi et al (1993) The fuzzy adaptive control of nickel matte smelting furnace (in Chinese). Journal of Central South Institute of Mining and Metallurgy, 24(6): 766~770 Procky T J, Mamdani E H (1979) A linguistic self-organizing process controller. Automatic, 15(1): 15~30 Saridis G N (1977) Self-organizing Control of Stochastic Systems. Marcel Dekker Inc, New York Tomohiro Takagi, Michio Sugeno (1985). Fuzzy identification of system and its application to modeling and control. IEEE Transaction on Systems, Man and Cybernetics, SMC-15(1): 116~132 Wu Guangyu (1987) System Identification and Adaptive Control (in Chinese). Harbin University of Technology Press, Harbin Wang Lixin (1992) Fuzzy system are universal approximators. IEEE Fuzzy’92: 1163~1170 Wang Xuehui (1987) Theory and Application of Micro-computer Fuzzy Control (in Chinese). Electronic Industry Press, Beijing Wang Zhenhuai (1994) Fuzzy control and expert system (in Chinese). Automation and Instrumentation, 9(1):1~3 Werbos P J (1974) Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Science. Thesis (Ph. D.). Appl. Math. Harvard University Zadeh L A (1965) Fuzzy Sets. Information and Control, 8: 338~353 Zadeh L A (1973) Outline of a new approach to the analysis of complex systems and decision process. IEEE Trans on Sys, Man and Cobern, 1: 28~44 Zadeh L A (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Information Sciences, 8: 199~249 Zhang Liangjie, Li Yanda (1995) The development and prospect on fuzzy neural network technology for intelligent control (in Chinese). Acta Electronica Sinica, 23(8): 65~69 Zhang Zaixing, Sun Zengqi (1995) On expert control (in Chinese). Information and Control, 24(3): 167~172 Zhang Zhenyu (1996) Groundwork and Application of Fuzzy Theory and Neural Network (in Chinese). Tsinghua University Press, Beijing Zhou Jicheng (1993) Artificial Neural Network (in Chinese). Science Popularization Press, Beijing


Hologram Simulation of Aluminum Reduction Cells Naijun Zhou

The “hologram simulation” concept is put forward to replace the traditional “Three-fields” (including magnetic field, thermal field and force field) notion in simulating aluminum reduction cells. The core of the hologram simulation is to simulate microstructures of the cells’ parameters distribution under various structural and operating conditions. In hologram simulation, the effect of various input information can no longer be treated as single or independent variables, because of the interacting they have with each other. The hologram simulation approach can be used for both static and dynamic analysis of the cells, and it as more suitable to describe the detail working of the cells. In this chapter, the models for several main physical fields including the current field, magnetic field, thermal field, and the molten metal flow field are respectively discussed; moreover, the dynamic simulation method and several calculation models of the current efficiency for aluminum reduction cells also are introduced.

5.1 Introduction As aluminum is widely used in industries, the need to maintain sufficient supply, therefore that the development of aluminum has very bright future. In 2006, the annual production of primary aluminum worldwide is about 23,869,000 t, of which the annual production of China is about 9,349,000 t, according to International Aluminum Institute. This makes China the first ranked producer in the world and it has an annual growth rate of 19.8% (Huang, 2007; Guo, 2007). Since its invention in 1886, the Hall-Heroult process has been the only method

  Naijun Zhou used in the aluminum industry. The aluminum production is a significant consumer of electricity. In China, the annual electricity consumption of aluminum production is 13.7 × 1010 kW·h, which accounts for 4.8% of the total industry consumption(Yang, 2007), and the expense in electricity is about 45% of the aluminum manufacturing cost. Therefore, to reduce the energy consumption and increase the current efficiency has being an important subject for researchers in aluminum industry. With the development and improvement of aluminum electrolytic technology, especially the application of computer technology in the design and operating of cells, the mean energy consumption per ton aluminum has greatly decreased from 41,900kW·h/t(Al) at the beginning of last century to present 12,900 kW·h /t(Al) in advanced aluminium reduction cells, the current efficiency has been improved to 95%, and the average cell life has been improved to more than 5 years. There are two main types of aluminum reduction cells: the self-baking anode cells and the prebaked anode cells. The self-baking anode cells are normally associated with small cell capacity, low current intensity which usually does not exceed 80kA, low current efficiency and bad production conditions. Hence, it is being gradually replaced by prebaked anode cells. Comparatively, the all production quota of the prebaked anode cells are better than that of self-baking anode cells. For example, the current intensity of modern large-size prebaked cells reaches 500kA, and annual potline output is about 390,000t (Vanvoren et al.,2001). Moreover, adopting the automatic point feeder, and production environment are better than that of the self-baking cells. At present in China, the productivity of the self-baking anode cells accounts for 3.5% of total production of aluminum (Huang,2007); while the prebaked cells are mainly in sizes of 200kA and 240kA, large-size cells such as 350kA and 400kA are in the development. Fig. 5.1 is a schematic

Fig. 5.1 Cross section sketch of a prebaked anode aluminium reduction cell

5 Hologram Simulation of Aluminum Reduction Cells 

of the cross section of a prebaked anode aluminum reduction cell. With development and application of many new technologies in the last decade, great progress has been achieved on technical parameters of prebaked cells in China. For example, the recent successful application of the fuzzy intelligent control technology in 160kA and 180kA aluminum reduction cells have respectively resulted in current efficiencies of 93.57% and 93.74%, and direct current power consumptions of 13,372 kWgh/t(Al) and 13,049 kW·h·/t(Al), which are close to the advanced level in the world(Fcrioncuot,1997). However, the aluminum electrolysis technology level in China is still behind that of the developed countries. For instance, the mean current efficiency in China is only 92%, and average DC power consumption is over 13,500 kW·h/t(Al), which indicates great potentials for technological progress. The three-fields technology plays an important role in the improvement of aluminum electrolysis technical and economic index. In China, the origin of the three-field technology can be tracked back to late 1970s, when 160kA central charging prebaked cells made by Japan Light-Metal Company were entirely imported to China. As the technical package consisted of three computer software, i.e. calculation programs for magnetic field, heat analysis of cathode and anode, and stress distribution of shell, it was called “three-fields technology” for short (refer to the magnetic field, thermal field and stress field) in the course of digestion and development of the software. Since 1980s, the three-fields analytic technique has been widely applied to the design of aluminum reduction cells, and made great contribution to the technical progress. It has also been applied to the design and rebuild of aluminum reduction cells as virtual methods in China since 1990s. It is now well known that there are various physical parameter fields in and around aluminum cells having considerable effect on the operation of cells, which include the current field, magnetic field, thermal field (temperature field), velocity field, concentration field, stress field and so on. Since the physical parameter fields inter-couple with each other (refer to Fig. 5.2), the analysis of any sole field becomes more and more difficult because it has to be relied on excess assumption which causes deviations between the predicted results and practical situations. As the traditional three-field technologies can hardly contain all the physical parameter fields and their interrelations, the authors are to use the concept of “hologram simulation” to replace the three-fields notion. The core of the hologram simulation is to simulate microstructures of parameters distribution with various structural parameters and operate conditions of the cells. The simulation process consists of four stages: model development, model solving, results output and test validation. Apparently, the traditional three-field technologies are still an important part of the hologram simulation. However, what makes them different is that in hologram simulation, effect of various input information on every physical

  Naijun Zhou

Fig. 5.2 Relation of various physical parameter fields in aluminum reduction cells

parameters and the parameters themselves can no longer be treated as independent or separated information. In fact, they are a unity inter-relating and interacting with each other. Therefore, compared to the three-field analysis technology, the hologram simulation technology is more suitable to describe the essence of the aluminum cell’s work. It can be used for both static analysis and dynamic action simulation for the cells, of which the latter is beyond the capability of the traditional three-field analysis technology. This is to be discussed in details in the Section 5.5. The principle of the hologram simulation technology for aluminum reduction cells is illustrated in Fig. 5.3. The keys to hologram simulation are: a) Comprehensive observation and analysis of cells’ working process, trying to find out the qualitative influence regularities among various information and influence of information on the results; b) Development of appropriate mathematical and physical model; c) Determination of solving conditions and algorithm; d) Test and modification of the simulation results. In this chapter, the first four sections are to discuss development of the models of several main physical parameter fields. The fifth section is to introduce an

5 Hologram Simulation of Aluminum Reduction Cells 

example on dynamic simulation for temperature field in an aluminum reduction cell. The sixth section will introduce calculation model of the current efficiency. Some typical simulation results will also be given in the chapter. It must be noted that hologram simulation of aluminum cells are not perfect, hence unremitting hard work is still required.

Fig. 5.3

Principle of hologram simulation technology for aluminum reduction cells

5.2 Computation and Analysis of the Electric Field and Magnetic Field The aluminum reduction cell is an electrochemistry reactor acted on high direct

  Naijun Zhou electric current. The current is the original cause of all the phenomena in aluminum reduction cells. On one hand, high electric current results in strong magnetic field and the electromagnetic force then causes melt motion and molten aluminum flowing and influences heat and mass transfer as well as the operating conditions. On the other hand, the electric current directly produces heat, which keeps cells at an appropriate temperature. The distribution of magnetic field in a cell depends on the electric current distribution in the cell and the adjacent cells. The electric current is usually divided in several sections such as bus bar, anode, melt Fig. 5.4 Conduction structure in an and cathode etc. and can be treated aluminum reduction cell respectively. Fig.5.4 is a schematic of the conduction structure in a typical aluminum reduction cell.

5.2.1 Computation model of electric current in the bus bar According to Kirchhoff’s law, current that flows through each part of the bus bar system can be calculated as:

∑E = ∑I • R ∑I = 0 j

(5.1)

(5.2)

j

where j is the node number. In calculation, all sections of the bus bar are replaced with equivalent resistance. Then on the basis of series-parallel connection relation among conductors of the bus bar, a network chart of electric circuit is obtained, and electric potential at each node and current along each section of the bus bar are computed using a program. In calculation, because of the complicated relation of the bus bar sections, computation is quite difficult and fussy. Fig.5.5 shows a typical bus bar calculation network chart. For details of the computational methods, program and the examples, please refer to references (Imery, 1989; Vao, 1990).

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.5 Computation current meshwork of bus bars

5.2.2 Computational model of electric current in the anode Based on the calculated current that flows through anode rod into anode block and appropriate assumption of boundary conditions, electric potential and current of each control volume (node) can be calculated with Kirchhaff’s law and resistivity of anode block by using three-dimensional finite difference method or finite element method. The procedure is the same as that of calculation of the bus bar. However, it must be noted that, since the resistivity is usually a function of temperature, the heat of the control element should be computed before working out the current distribution (electric field). Then the temperature distribution in anode block can be obtained by solving heat and conduction differential equations simultaneously. The control equations are: a) Electric conduction differential equation: ∇ • σ ∇V = 0 (5.3) b) Heat conduction equation: ∇ • λ ∇T + qvol = 0 (5.4) where σ is the electric conduction rate of the anode material; λ is the heat conduction coefficient; qvol is Joule heat in control volume that is given by: qvol ⇒ qi , j , k = qii,−j1,,kj , k + qii,+j1,,kj , k + qii,, jj,−k1, k + qii,, jj,+k1, k + qii,, jj,,kk −1 + qii,, jj,,kk +1 (5.5) where

qii,−j1,, k j , k = σ x • (Vi−1, j , k −Vi , j , k ) 2 • ΔΔyΔx z

And the rest may be inferred by analogy. The difference equations of Eq. 5.3 and Eq. 5.4 can be expressed as:

(5.6)

  Naijun Zhou Vi , j , k = [a1 • (Vi −1, j , k + Vi +1, j , k ) + a2 • (Vi , j −1, k + Vi , j +1, k ) + a3 • (Vi , j , k −1 + Vi , j , k +1 )] / 2(a1 + a2 + a3 )

Ti , j , k = [b1 • (Ti−1, j , k + Ti +1, j , k ) + b2 • (Ti , j −1, k + Ti , j +1, k ) + b3 • (Ti , j , k −1 + Ti , j , k +1 ) + qi , j , k ] / 2(b1 + b2 + b3 )

(5.7) (5.8)

where, Δy Δz ΔxΔz ΔxΔy , a2 = σ y • , a3 = σ z • Δx Δy Δz ΔyΔz ΔxΔz ΔxΔy , b2 = λ y • , b3 = λz • b1 = λx • Δx Δy Δz a1 = σ x •

Eq.5.7 and Eq.5.8 can be solved by Gaussian elimination or iteration. The boundary conditions are listed as following: a) The electric potential at molten aluminum surface is evenly distributed. b) The electric on the cathode carbon surface is evenly distributed. c) The temperature in the electrolyte is uniform everywhere, and the value can be obtained by measuring. d) The temperature at the joint of anode bus bar and rod should be given. e) The thermal boundary conditions on the carbon surface are defined as the type of heat convection boundary. The program chart of the computation is showed in Fig.5.6. This procedure was used to compute and analysis the electric field of anode, as introduced in references (Gao, 1991; JLMRI, 1980; Zen, 1996; Zen, 2004). Fig.5.7 gives a computation result.

5.2.3 Computation and analysis of electric field in the melt The melt in aluminum reduction cells is composed of electrolyte and molten aluminum. Because of the great difference of their electrical resistivity, the current distribution in the melt is evident. For instance, since the electrical resistivity in electrolyte is larger than in the molten aluminum, the current density in the electrolyte below the anode bottom is almost uniform, and pointing vertically downwards; however in the electrolyte at the side face of anode, the current density is much smaller. Haupin had proposed a sector coefficient method to approximate the current in these parts(Haupin,1990). Of course it can also be accurately computed by finite difference method or finite element method with nonlinear boundary conditions. The current distribution in the molten aluminum is complicated. The molten aluminum is a good conductor. Moreover, because of the existence of frozen ledge and deposits, there is usually large horizontal current found in the molten aluminum,

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.6 Computation procedure of current field for the anode system

which results in the fluctuation and flow of the melt, and causes some unwanted influences on the operation. Therefore, the current distribution in the melt must be carefully studied and analyzed. To compute the current distribution in the melt, there are many kinds of methods, such as finite difference method, finite element method, network graph theory, and boundary element method etc. (Arita, 1983; Tarapore, 1982; Richand, 1976; Chen, 1986). The basic control equations are Eq. 5.3 and Eq.5.4. However,

  Naijun Zhou

Fig. 5.7 Computational result of electric distribution in anode (Gao, 1991)

different from that in last section, because of the symmetrical layout of the current carriers, the current distribution in the melt is symmetrical along the longitudinal axis in aluminum reduction cells, which therefore can be treated as a two-dimensional problem. The computation is to be illuminated with method of network graph theory as below. The electrical potential drop occurs mainly in the electrolyte. In computation of the electric field in the electrolyte, the anode and molten aluminum can be regarded as equipotential volumes. In this case, the electric field in the electrolyte can be studied as a potential boundary value problem with given electric current. The potential drop is very small, so the potential difference at the electrolyte-metal interface and the metal-cathode blocks interface must be taken into account. With the current density at the interface of metal being determined, the electric field in it can be studied as a potential field problem with the second boundary condition. For an electric current field as shown in Fig. 5.8, the medium is homogeneously isotropic and the thickness is Δz . The potential of node A is Vi, j , and the conductance of branch AB is: Δy Δz (5.9) G =σ • Δx Current of branch AB is: I i , j = (Vi , j −Vi−1, j ) • G (5.10) According to Kirchhoff’s junction current law: ∑I = 0

(5.11)

For the inner node A(i, j ) :

G • [(Vi, j −Vi−1, j ) + (Vi, j −Vi, j−1 ) −(Vi+1, j −Vi, j ) −(Vi, j +1 −Vi, j )] = 0

(5.12)

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.8 Mesh nodes of current field

Namely: Vi +1, j + Vi , j −1 + Vi −1, j + Vi , j +1 − 4Vi , j = 0

Similarly, for the edge node B (i, j ) : 3Vi , j +1 + 4Vi +1, j + 3Vi , j −1 − 10Vi , j = 0

(5.13) (5.14)

Potential equations of other edge nodes can be obtained in the same way. For the boundary problem, the boundary condition of the electrolyte can be expressed as a matrix after discretization as: AV = B (5.15) For the node that is of the second boundary condition problems such as C (i, j ) , if the effluent current is I 0 (i, j ) , then: 1 1 G • [ (Vi , j −Vi +1, j ) + (Vi , j −Vi , j +1 ) − (Vi−1, j −Vi , j )] + I 0 (i, j ) = 0 (5.16) 2 2 That is: 2 Vi +1, j + 2Vi , j +1 + Vi −1, j − 4Vi , j = • I 0 (i, j ) (5.17) G A matrix can then be obtained with the potential equations at every boundary nodes as: ,7 = *0 (5.18) After the relative potential has been determined (for example, set the potential at the electrolyte-metal interface to be zero, i.e. V (i, j0 ) = 0 ), the potential of each node can be solved with matrix 5.15 by using elimination method or over-relaxation method, and the current density at the upper interface of the molten aluminum I 0 (i, j ) can be obtained as well. The current density at lower interface I 0 (i, j ) can be obtained in the same way. Substituting the current densities into matrix 5.18, the potential at each node of the molten aluminum can be solved using over-relaxation method, and the horizontal and vertical currents can thus be decided. A computed typical current distribution in the melt is shown in Fig. 5.9.

  Naijun Zhou

Fig. 5.9

Computational result of current distribution in the melt (a) Potential distribution; (b) Current distribution

5.2.4 Computation and analysis of electric field in the cathode The computation and analysis methods stated in the above two sections are also applicable for that of the cathode. Because of the existence of collector bar, the model of cathode is usually three-dimensional (or a mix of two dimensions and three dimensions). When taking the change of side ledge shape and deformation of the molten aluminum height into accounts, computation will be more convenient by using the boundary element method. The Boundary element method is to solve a group of integral equations with boundary conditions(Zhou,1983). The procedure is: a) To discretize the boundary into units with certain shapes. b) To construct interpolating function in each unit. c) To convert the boundary integral equations into a group of algebraic equations. d) To solve the equations. In Section 2.4.2, the integral equation and fundamental solution of liner constant current field boundary problem have been introduced. Here the solving process is to be discussed. Assuming V is the potential function and V * is the fundamental solution, Eq. 2.190 and Eq. 2.191 can then be expressed as:

5 Hologram Simulation of Aluminum Reduction Cells 

⎧ ∂V * * ∂V −V )dS ⎪CiVi = ∫ (V ∂n ∂n S ⎪ ⎪⎪ ⎧ 1 i∈Ω ⎨ ⎪ ⎪ C = ⎪ 1 i∈S ⎨ i ⎪ ⎪2 ⎪ ⎪⎩ 0 i ∉ S U Ω ⎩⎪

(5.19)

For three-dimensional isotropic medium: V* =

1 4 πr

(5.20)

where r is the distance from a field point to the source point. For two-dimensional field: V* =

1 1 ln 2π r

(5.21)

For a two-dimensional field, S can be divided into N units using liner interpolation function: ⎧⎪V = φ1V j + φ2V j +1 (5.22) ⎨ ⎪⎩ q = φ1q j + φ2 q j +1 1 2

1 2

where φ1 = (1 − ζ ) , φ2 = (1 + ζ ) , and ζ is the local coordinate. Eq. 5.19 can then be discretized as: N

CiVi =

N

∑G q − ∑ H V ij j

j =1

ij j

(5.23)

j =1

where:

φ2

∂V * ∂V * dS + ∫ φ1 dS Sj ∂n ∂n

Gij = ∫

φ2V *dS + ∫ φ1V *dS

H ij = ∫

S j −1

S j −1

Sj

For i =1, 2, …, N ( i ∈ S ), Eq. 5.23 can be written into a matrix as: )7 = (2

(5.24)

For the Dirichlet problem, V is a known variable, then: 2 = ( −17)

(5.25) Note that for the Neumann problem, V cannot be uniquely determined. Hence, a reference point must be decided before to solve V . For the hybrid problem, the unknown variables in Eq. 5.24 can be moved to the left of equation: Ax = F (5.26) where x is the column vector of V and q , the unknown variables at the boundary. After solving the unknown variables in S , the potential at each points in Ω can be obtained by solving Eq. 5.23. Then, with the following equation (Eq.5.27),current density Jx and Jy of each points in Ω can be solved.

  Naijun Zhou Jx =σ

∫ (V S

Jy =σ

where q* =

)

∂q * ∂V * −q V dS ∂y ∂y

∫( S

∂q * ∂V * −q dS ∂x ∂x

)

(5.27)

∂V * ; σ is conductivity. ∂n

In the model, there are following a ssumptions: a) The electric current filed is approximated to be two-dimension. b) The side ledge is an insulator. c) The outlet of the cathode block is isopotential. d) The cathode-electrolyte interface is isopotential. e) Temperature inside the cathode blocks and collector bar are uniform, and every subregions have the same resistivity. f) The cross sections of the cathode rod are the same, an equivalent height of the rod is used instead of the real value. It must be pointed out that, because the cathode carbon block is orthogonal anisotropic, the computation of the electric field in the cathode is a problem to solving a multi-media electric field in orthogonal anisotropic material. Lu Jiayu had proposed a method for this kind of medium(Chen et al., 1986). The computation procedure is given in Fig. 5.10, and a typical result is shown in Fig. 5.11.

5.2.5 Computation and analysis of the magnetic field The operation of aluminum reduction cells is ensured by high DC current. According to the electromagnetics, when current runs through a conductor, a magnetic field will be produced in the space around, and an electromagnetic force field will be produced under the interaction of the magnetic field and the current. It is the electromagnetic force that results in the melt movement, which has a negative effect on the process of aluminum production. Once an aluminum reduction cell is designed and installed, its magnetic field is basically fixed. Therefore optimization of bus bar collocation in design process is very important, so as to predict the magnetic field in the operation and to minimize the negative effect. With the results of the electrical fields as introduced in Section 2.4, computation and analysis of magnetic field can be carried out. In computation and analysis of the magnetic field in the cells. An equivalent mathematical model is usually employed. That is, to replace the rectangular bus bars (including the melt) with columnar bus bars that is of the same length and cross-sectional area, then for one

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.10 Program chart of computation of current field in cathode

Fig. 5.11 Calculation result of current field in cathode (Lu, 1986)

point in the magnetic field, to calculate magnetic vectors produced by each columnar bus bars with the Biot-Savart Law; the magnetic intensity at that the specified point can thus be obtained. In Section 2.4.3, the model and method of computation of magnetic field caused by electrified bus bars have been discussed in details. In previous research, it has

  Naijun Zhou

been found that the effect of iron shell is not neglectable in computation of the magnetic field distribution. There are several methods to calculate the additional magnetic field caused by ferro-magnetic materials, as discussed in Section 2.2.4, of which the magnetic dipole model method (Robl, 1978; Sele, 1974) is comparatively more popular and accurate. The principle and calculation procedure of this method are as follows: a) Using the equivalent column model to calculate the magnetic induction intensity (components) of one point produced by a current-bearing conductor. b) To simplify the iron structure into an equivalent magnetic dipole column rod with spherical ends (rotation ellipsoid), and to restrict the dipole to be along one of the three axes of the coordinates. Moreover, the length and the cross-sectional area of the dipole must be the same as that of the iron structure it represents. c) To calculate the magnetic effect of each end of the dipole at the specified point. The magnetic induction intensity at this point is then the superposition of the magnetic effects of the two ends of the dipole. A representative structure of the magnetic dipole of a shell is given in Fig. 5.12.

Fig.5.12 Structure of magnetic dipole in a steel shell (a) Main magnetic flux distribution in the shell; (b) Magnetic dipole of the shell

There are also other methods of calculating the magnetic field, such as differential method, integral method and two scalar potentials method etc. For details of these methods, please refer to references (Sneyd, 1985; Qiu, 1992; Li, 1993) It should be pointed out that a simpler method of equivalent potline current(JLMRI,1980) was used in original computation of the magnetic field at early time. However, for either the equivalent series currents method or the equivalent column method, errors are unavoidable because of the rectangular cross section of the bus bars, and larger errors may be found near the bus bars. To solve the problem, Chen Shiyu et al. developed a more accurate method (Chen, 1987) to directly derive the magnetic field with the actual shape of the bus bars. Meanwhile, many researchers carried out lots of computations of the magnetic field in aluminum reduction cells(Chen,1987;Zen,1996;Qiu,1992;Liu et al., 1996; Zeng et al.,1996;Liang et al.,1998;Severo,2005), and their results are all helpful for the optimization of the bus bars configuration.

5 Hologram Simulation of Aluminum Reduction Cells 

Here some examples are given to illuminate the application of methods mentioned above. Fig. 5.13 shows the computational result of the magnetic field in the middle part of the molten aluminum in a 160kA prebaked cell (Zen, 1996). In the result, the horizon magnetic field was found in a large vortical structure, leaning to current inlet side, while the vertical magnetic field is in a saddle shape.

Fig.5.13 Computational result of magnetic field in an aluminum reduction cell (160kA cell with four risers)(Zen,1996) (a) Horizontal magnetic field; (b) Vertical magnetic field

Fig. 5.14 is the computational result of the magnetic field in a 280kA test cell obtained by Wang Ruliang et al.(Wang et al.,1998) of Shenyang Aluminum & Magnesium Academe. Significant difference can be found in the result whether or not the magnetic screening action caused by ferro-magnetic material and the effect from the neighbor cells are taken into account. The magnetic field distribution was found to be optimized after some adjustment of the bus bars arrangement, according to the statement in the paper.

  Naijun Zhou

Fig.5.14 Vertical magnetic field distribution in molten aluminum with symmetrical bus bars (Wang et al., 1998) (a) Magnetic screening action is neglected;(b) Magnetic screening action is considered 1üCurrent inlet side (without effect by neighbor cells); 2üCurrent outlet side (without effect by neighbor cells); 3üCurrent inlet side (with effect by neighbor cells); 4üCurrent outlet side (with effect by neighbor cells)

Up to now ,it is still difficult to make an accurate theoretical computation of the influence of ferro-magnetic material on the magnetic field. Usually, physical simulation and experiments have to be employed for assistance (Shen et al., 1994). Fig.5.15 shows the magnetic distribution in the middle part of the molten aluminum of a 200kA prebaked cell computed with the commercial software ANSYS by the authors. In the results, two large vortices are found in the horizontal magnetic field, whilst the vertical magnetic field presents as anti-symmetric distribution in two sides of x-axis and y-axis which intersect in center of the cell. A research group at the Central South University measured the magnetic induction intensities in the molten aluminum of the same kind reduction cell using a three-dimensional gaussmeter. The measured values are listed together with the corresponding computational results in Table 5.1.

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.15 Computational result in a 200kA pre-baked cell (a)The horizontal magnetic field; (b)The vertical magnetic field Table 5.1 Comparison of measured values and simulated values of magnetic field of a 200kA pre-baked anode cell

Measured values of magnetic field /Gs

Measured point

Coordinates of measurement points/m x

y

z

Bx

Bz

Bx

By

Bz

1

1.13

0.25

1.30

107.5 4.4

3.3

105.45

3.17

5.87

2

2.53

0.25

1.30

108.5 5.3

6.4

103.27

5.06

11.86

3

3.93

0.25

1.30

124.4 20.2

4.1

101.22

10.43

6.34

4

5.33

0.25

1.30

131.4 32.2

13.7

113.86

28.83

7.30

5

6.73

0.25

1.30

123.4 23.2

13.0

108.72

11.61

7.13

6

8.13

0.25

1.30

114.5 14.2

12.2

102.60

10.15

13.68

7

9.53

0.25

1.30

106.0 8.2

11.4

92.97

7.53

6.41

8

1.13

3.65

1.30

103.6 22.3

14.9

102.30

20.25

13.30

9

2.53

3.65

1.30

102.3 24.0

1.3

98.08

11.63

5.04

10

3.93

3.65

1.30

90.3 6.5

5.5

86.33

3.42

2.16

11

5.33

3.65

1.30

83.8 5.3

9.5

80.03

2.27

5.56

12 13

6.73 8.13

3.65 3.65

1.30 1.30

74.5 10.9 105.2 6.5

17.9 16.2

75.95 98.18

9.54 6.73

8.05 5.04

By

Simulated values of magnetic field /Gs

  Naijun Zhou Continues Table 5.1 Measured point

Measured values of magnetic field /Gs

Coordinates of measurement points/m

Simulated values of magnetic field /Gs

x

y

z

Bx

By

Bz

Bx

By

Bz

14

9.53

3.65

1.30

114.8

17.2

11.2

109.07

20.33

8.40

15

0.35

1.10

1.30

25.4

14.4

1.1

29.18

10.14

4.46

16

0.35

1.95

1.30

4.7

16.5

28.5

2.33

3.90

24.50

17

0.35

2.70

1.30

66.4

5.3

22.9

54.12

0.82

17.23

18

10.3

1.00

1.30

37.1

17.6

11.7

44.26

11.55

8.57

19

10.3

1.95

1.30

14.4

13.2

10.6

8.34

3.41

6.59

20

10.3

2.70

1.30

68.4

6.4

31.2

57.75

7.93

28.91

As shown in Table 5.1, the values of measurement are in good agreement with most simulation results, and their changing trends are the same. The maximum deviations of the magnetic induction intensities are respectively 20Gs in x direction, 10Gs in y direction and 5Gs in z direction. The reasons for such deviations mainly are: a) Inaccurate positioning of the the measurement; b) Measurement errors; c) Fluctuation of currents and voltages during of the measurement.

5.3 Computation and Analysis of the Melt Flow Field The processes in the melt in aluminum reduction cells are very complex, which include electrochemical reactions, chemical reactions and physical processes such as dissolution, diffusion, fluid flow and heat transfer. In those processes, the physical processes are particularly important, having a critical effect on electrolytic reduction, of which the melt flow plays a decisive role. Therefore, computation and analysis of the melt flow field is a key part of the “three fields” analysis. Movement of the melt may be caused by many factors. Firstly, the electromagnetic force will cause the movement. Secondly, force produced when gases rising from the anode bottom in the electrolysis process will also result in the melt movement (Sun et al.,2004). Other factors such as mass transfer, heat transfer, pole changing, charging, aluminum discharging, edge processing and the effect treating will cause melt movement as well. The first two factors are the main reasons for the movement of the melt during the stable operation period.

5 Hologram Simulation of Aluminum Reduction Cells 

5.3.1 Electromagnetic force in the melt Current produces magnetic fields, thus an electromagnetic force (i.e. the Laplace force) will be caused and acted on the conductor in the magnetic field. The electromagnetic force is vertical to the plane that determined by vectors of current I and magnetic induction intensity B, and its direction can be decided by the right-hand rule. In the analysis, the horizontal cross-section of the bath is divided into four parts (i.e. section I to Č , as shown in Fig. 5.16). Fig. 5.16 Synthesis of horizontal Components of the magnetic induction electromagnetic forces in the melt intensity along axis Ox, Oy or Oz are respectively Bx, By, Bz. The Laplace force produced in the melt along axis Ox and Oy are correspondingly: f ( y ) = I z Bx − I x Bz

(5.28)

f ( x ) = I y Bz − I z B y

(5.29)

The sum of the Laplace force along y direction in sectionĉis: a

b

∫ ∫

FIy = dx f ( y )dy 0

(5.30)

0

where with the magnetic field distribution as shown in Fig. 5.16, the FIy is to be against the direction of y axis. The sum of Laplace force along x direction in sectionĉis: b

a

∫ ∫

FIx = dy f ( x)dx 0

(5.31)

0

where its direction is against x axis. The values and directions of the Laplace force in other three sections can be decided in the same way. As the horizontal resultant forces in the four sections all point to the origin of the coordinates, equations of force balance during the stable operation period of a reduction cell can be obtained: − Fĉ − FIy = FIVy , FIIIx = − FIVx , − FIIy = FIIIy x = F x, According to above equations, the electrolyte-molten aluminum interface should be in the shape of a curved surface. In each section, because bearing unbalanced electromagnetic forces, melt, especially molten aluminum, will move horizontally (Frederic et al., 2002; Wu et al., 2002; Sun et al., 2005).

  Naijun Zhou

5.3.2 Analysis of the molten aluminum movement Beside the electromagnetic force, there are many factors determining the flow speed and surface shape of the molten aluminum. Usually, Navier-Stokes equation and continuous equation are used to describe the movement of the molten aluminum. DU ρ = F − ∇P + μ ∇ 2U (5.32) Dt ∇•U = 0 where pressure gradient can be obtained by summing internal and external forces: ∇P = Fm + Fw + Fr + Fa (5.33) In which: External force is: F = Fm + Fw Electromagnetic force is: Fm = I × B Gravity force is: Fw = ρ g Flow resistance is: Fr = μ • ∇ 2U Flow inertia force is: Fa = ρU • ∇U It can be proved that(Liu et al., 1996): ∇ • F = ∇2 P (5.34) ⎛ DU ⎞ ∇× F = ∇×⎜ ρ − μ∇ 2U ⎟ (5.35) Dt ⎝ ⎠ The above two equations can be used to solve the pressure field and velocity field. As the rotation of the external force seldom equals to zero, the movement in aluminum reduction cells is a rotational flow. Basically, there are four kinds of flow fields as shown in Fig. 5.17. Of course, more structures of flow fields can be formed by superposing the four basic structures, as illustrated in Fig. 5.18.

Fig. 5.17 Four basic structures of velocity field in the molten aluminum

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.18 Superposition of basic movements

5.3.3 Analysis of the electrolyte movement The movement of electrolyte with certain speed and direction is good for the dissolution of alumina, the uniform concentration of electrolyte and the release of heat that produced under anode. However, it will also results in fluctuation of the molten aluminum and increasing of second reaction, thus makes the cells unstable and reduces the current efficiency. Hence, it is necessary to study the electrolyte movement. Beside the electromagnetic force, the bubble movement from chemical reaction is the other motive force of the electrolyte movement, which thus makes the process even more complex. Since the force from thermal heat pressure gradient is relatively less important, it is usually neglected. The movement caused by the electromagnetic force can be analyzed in the similar way to that of the molten aluminum. Actually, the horizontal current in the electrolyte can be neglected, so there is only horizontal electromagnetic force to be considered. The movement caused by bubbles is primarily horizontal under the bottom of the anode but vertical along the side of the anode. This movement are determined by the generation and motion of bubbles. The motion of bubbles can be divided into three stages: a) Bubbles are generated under the anode, and then growing up till depart from the surface of the anode. b) Under the anode, the bubbles movement are primarily caused by the pressure gradient and fluid viscousity. c) Buoyancy is the main force for the bubbles movement along the side of anode. The movements of the bubbles will cause the convection, or even turbulence in the electrolyte, thus leads to oscillation of the free surface. The bubbles can also produce electromagnetic effect. Because the gas is not a conductor, the effect will influence the distribution of current and magnetic field, and it may cut off the current and cause anode effect at worst (Xue et al., 2006).

  Naijun Zhou

The superposition of the motions that result from electromagnetic force and bubbles is the electrolyte movement. It is difficult to theoretically analyze the electrolyte movement. Many researchers have engaged in the study with physical models (Zhou, 2006). For example, Derneded et al. proposed firstly to simulate the flow of the electrolyte and molten aluminum in the possessing section at the sidepiece of the anode using water and organic liquid in a cold model. The average vertical velocity of the electrolyte at sidepiece of anode (Derneded, 1975) they determined is: −0.044

uc = 0.173 • ( gQ)

0.458

•h

0.467

•l

−0.258

⎛ μ ⎞⎟ ⎜ ⎟ •⎜ ⎜⎝ ρ ⎟⎟⎠

−0.165

⎛σ ⎞⎟ ⎜ ⎟ •⎜ ⎜⎝ ρ ⎟⎟⎠

(cm/s)

(5.36)

And the velocity of the electrolyte at the surface of the ledge is 0.25

⎛ h⎞ us = uc • ⎜ 2 ⎟ (5.37) ⎝ l⎠ where g is gravity, g=981cm/s2; Q is gas velocity per unit anode perimeter, cm/s; h is the depth of anode immersed in the electrolyte, cm; l is the distance between the anode and the ledge, cm; μ ρ is the kinematical viscosity of the electrolyte, cm2/s; σ is surface tension, N/cm; ρ is density, g/cm3.

5.3.4 Computation of the melt velocity field The movement of the melt in reduction cells is a three-dimensional turbulent motion. However, for the melt under the anode, because its motion is primarily forced by the horizontal electromagnetic force and the vertical component of the movement is far less than horizontal one, the computational model is simplified as a two-dimensional problem(Sun and Mei, 1989; Huang et al., 1994; Liang et al., 1998). The k-ε turbulent model is employed in the computation, in which only electromagnetic force is considered whilst effect of the bubble phase is neglected, and the two layers of the molten aluminum and electrolyte are considered as immiscible. To describe the movement of the melt, the Navier-Stocks equations is averaged along the depth of the layers. Because of the restriction of bottom surface, assumption of a “rigid cover” has to be made, and the control equations can be set up as following using square friction resistance: ∂u ∂v + =0 ∂x ∂y

(5.38)

∂u ∂uu ∂uv ∂u ⎞ ∂ ⎛ ∂u ⎞ ∂ ⎛ ∂v ⎞ 1 ∂p ∂ ⎛ + + =− + ⎜ν eff ⎟ + ⎜ν eff ⎟+ ⎟ + ⎜ν eff ∂t ∂x ∂y ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂x ⎝ ∂y ⎠ ρ ∂x ∂x ⎝ (5.39) ∂ ⎛ ∂v ⎞ 2 ∂k Cf 2 1 2 1/ 2 − (u + v ) u + Fx ⎜ν eff ⎟ − ∂y ⎝ ∂x ⎠ 3 ∂x H ρ

5 Hologram Simulation of Aluminum Reduction Cells 

1 ∂p ∂ ⎛ ∂v ∂uv ∂vv ∂v ⎞ ∂ ⎛ ∂v ⎞ ∂ ⎛ ∂u ⎞ + + =− + ⎜ν eff ⎟ + ⎜ν eff ⎟ + ⎜ν eff ⎟+ ρ ∂y ∂x ⎝ ∂t ∂x ∂y ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂x ⎝ ∂y ⎠ (5.40) 1 ∂ ⎛ ∂u ⎞ 2 ∂k Cf 2 2 1/ 2 − (u + v ) v + Fy ⎜ν eff ⎟− ρ ∂y ⎝ ∂y ⎠ 3 ∂y H where Fx and Fy are the electromagnetic forces of the unit volume; Cf is friction resistance coefficiency between the melt and the wall; H is the depth of the melt layer. To describe the turbulent flow, k-ε equations are used and averaged along the depth of the melt layer. ∂k ∂ ∂ ∂ ⎛ ν ∂k ⎞ ∂ ⎛ ν eff ∂k ⎞ + (uk ) + (vk ) = ⎜ eff ⎟+ ⎜ ⎟ + Pk + Pkv − ε (5.41) ∂t ∂x ∂y ∂x ⎝ σ k ∂x ⎠ ∂y ⎝ σ k ∂y ⎠ ∂ε ∂ ∂ ∂ ⎛ ν ∂ε ⎞ ∂ ⎛ ν eff ∂ε ⎞ ε ε2 (5.42) + (uε ) + (vε ) = ⎜ eff ⎟+ ⎜ ⎟ + Cl Pk + Pε v − C2 ∂t ∂x ∂y ∂x ⎝ σ ε ∂x ⎠ ∂y ⎝ σ ε ∂y ⎠ k k Cμ k 2 (5.43) ν eff = ν + ν T = ν +

ε

where C1 =1.44, C2 =1.92, σ k =1.0, σ ε =1.3, Cμ =0.09. The initial conditions are considered as static, and the boundary conditions are solid, non-slip walls. Solutions of above equations should be jointed with wall functions on the side walls. With the Navier-Stocks equations, the height of two molten liquid layers (as shown in Fig. 5.19) can be obtained by averaging the momentum equation of z direction: ( P − Pe ) − [( ρ m g − Fmz ) • h + ( ρe g − Fez ) • ( H − h)] (5.44) h′ = m ( ρ m g − Fmz ) − ( ρe g − Fez ) where Pm and Pe are the static pressure respectively at the bottom surface of the molten aluminum and the top of the electrolyte; ρ m and ρ e are the densities of the molten aluminum and electrolyte; Fmz and Fez are vertical electromagnetic force in the molten aluminum and electrolyte (the vectors Fig. 5.19 Interface of melt point to z direction, for solution of the vectors, please refer to Section 5.2); and g is the gravity. The above equations can be discretized with finite difference method. Using SIMPLE algorithm and staggered-grid, the equations can be solved by assigning velocity field and pressure field at different nodes. In references (Sun and Mei, 1989; Huang et al., 1994; Liang et al., 1998; Morris and Davidson, 2003; Gerbeau

  Naijun Zhou

et al., 2003; Sun et al., 2004), the velocity field of various of prebaked anode cells were computed and analyzed. Computational results of a 280kA cell in China cited from reference (Huang et al., 1994) are shown in Fig. 5.20 and Fig 5.21. Cai Qifeng et al. (Cai et al., 1993) had ever measured the velocity of the molten aluminum in the industrial aluminum reduction cells using iron bar dissolution method for many times. It should be noted that the computation of the velocity field is so hard to be highly accurate, because of the inaccurate analysis of the electromagnetic force field computation as well as the complicated movement of fluid in the cells.

Fig. 5.20 Horizontal velocity distribution in melted aluminum for a 160kA cell (a half cell) (Sun and Mei, 1989)

Fig. 5.21 Horizontal velocity distribution in electrolyte for a 280kA cell (Huang et al., 1994)

5.4 Analysis of Thermal Field in Aluminum Reduction Cells The aim of analysis of the static thermal field is to solve the voltage distribution and the temperature distribution inside the cell as well as the heat balance of the cell under certain structure and working current conditions, thus to optimize the design and structure of reduction cells. For such kind of analysis, steady-state conduction and thermal conductivity models are used, with given melt

5 Hologram Simulation of Aluminum Reduction Cells 

temperature, melt position and shape of the ledge. Hence it is a “static” problem. The original analysis of thermal field was made separately for three parts, i.e., the anode, the cathode and the bottom of slot, which may be limited by computer capacity and speed at that time. Nowadays, there is no need for such a separation. Because of the strong coupling between the electric field and the temperature field in reduction cells, the temperature field cannot be analyzed independently. Instead, it has to be solved simultaneously with the electric field, as introduced in Section 5.2. In fact, in Section 5.2.2, details of analysis of the thermal field of the anode have been described, and here will not be repeated any more. In this section, it is to treat the aluminum electrolysis cell as a whole to discuss models and solutions of the analysis of thermal field.

5.4.1 Control equations and boundary conditions Under the assumption of steady state, equations of electrical and thermal conduction in anode and cathode are the same as Eq. 5.3 and Eq. 5.4. What is different is that, at non-conductive part (i.e. the cell body outside ledge and below the cathode), the equation of thermal conduction can be transformed into the Laplace equation without solving the electrical conduction equation. ∇ • λ∇ T = 0 (5.45) Since the anode is symmetrical, it generally takes only half of the anode for a prebaked cell(or even a quarter) as the computational geometry; but for a self-baking cell, it is an area including two halves of conducting bar. It is assumed that no transverse current and heat flow through the cross section. Obviously, computation of thermal field for the anode must be treated as a three-dimensional problem. Those of the cathode and other parts of the cell is actually a mix of two and three dimensions, of which the cathode bar is to be treated as three-dimension, while others can be computed with two-dimensional models. Temperature field of the electrolyte and the molten aluminum are both homogeneous and their temperature are specified. It must be noted that the computational slices should be taken from the big and end faces of the cell. A schematic of the computational slice is given in Fig. 5.22. The boundary conditions for the electrical conduction equation are: a) Regarding the cathode block surface as basic potential surface, and the potentials in the molten aluminum are considered as equal. b) The ledge is not conductive and all currents pass through the cathode block. c) The currents passing through each anode and cathode rod are known and the value is an average.

  Naijun Zhou

Fig. 5.22 Schematic of the computational slice of an aluminum reduction cell

The boundary conditions of the thermal conduction equation are: a) The electrolyte and molten aluminum layers are an isothermal region, their temperatures are to be decided with experience and by request. b) The top of the cell is assumed to be adiabatic. c) The environmental temperature around the cell is certain. d) The coefficient of heat transfer between the external surface and the environment α is: 5.6 × 10−8 • ε w 4 α = C (Tw − Ta ) 0.25 + (Tw − Ta4 ) (W/(m 2 • K)) (5.46) Tw − Ta where C is an experimental coefficient, which is 2.6 for the side walls, 2.0 for the bottom, and 3.3 for the top plate; ε w is the radiance of the external wall, of which the value is usually taken as 0.82. e) Factors that influence the heat transfer between the melt and the internal faces are quite complicated, and no consist and reliable results can be referred (see Table 5.2). Generally, the coefficient can be calculated with following empirical formula (Mei and Tang, 1986; Mei et al., 1992; Zhou and Mei, 1992; Mei et al., 1996). Heat transfer from the melt to the ledge: Nu = 0.0365 Re0.8 • Pr 0.33 (5.47) Heat transfer from the molten aluminum to the cathode block: Nu = 5 + 0.025( Re • Pr )0.8

(5.48)

Big error may be found in Eq. 5.48. So in practice, it is usually corrected with additional thin film thermal-resistance method: 1 α′ = (5.49) δ L + B λm • Nu λB

where L is the inter width of the cell, m; Nu is the Nusselt number to be determined with Eq. 5.47 and Eq. 5.48; λm and λB are respectively the

5 Hologram Simulation of Aluminum Reduction Cells 

thermal conductivity of the melt and the boundary thinfilm, W/(m·K); and δ B is the thickness of the boundary thinfilm, m. The melt velocity used above equations are calculated with Eq. 5.36 and Eq. 5.37. Besides, the thermal properties of main materials used in the computation are given in Table 5.3, or refer to reference (Qiu, 1982). Table 5.2

Coefficient of heat transfer between melts and ledge (W/(m2gK))

Author

Aluminum-ledge

Remark

200

20~150 400

Effective heat transfer coefficient during side feeding (Ikenouchi et al., 1978); measured value(Ikenouchi et al., 1978)

[USA]W.E.Haupin

370

1500 1100

Calculated with criterion equation (Haupin,1971); measured value (Haupin, 1971)

[Japan]Arai et al.

232

793 (δB=0.3mm) 334 (δB=1.0mm) 183 (δB=2.0mm)

Arai and Ramazaki, 1975

[USA]J.G.Peacey

175~210

300~384

Peacey and Medlin, 1979)

[Norway]Perutne

200

[USSR] U.D.Dekopov et al.

Qmf = 14000

Qmf = 10500

W/m2

W/m2

[USSR] B.P.Romanov et al.

141~184

150~268

[Norway]A.Solhiem

For prebaked cells: 250~500 For self-baking cells: 300~650

[Japan]Haruhiko Ikenouchi et al.

Electrolyte-ledge

Perutne, 1982

Dekopov, 1978

Romanov, 1980

Solhiem et al., 1983

  Naijun Zhou Table 5.3 Thermal properties of materials

Material

Conductance /Wg(mgK)−1

Density /kggm−3

Semi-graphited cathode carbon block (Aparotsev, 1998)

8.81−0.072T −3

1573/(1+7.9h10−6 T)

Cathode iron rod (Aparotsev, 1998)

Cast iron (filled in iron rod and carbon block) (Aparotsev, 1998)

Side ledge (CSIOFMAM, 1977)

Bottom ledge (CSIOFMAM, 1977)

Insulation impervious material

58.85−0.0414 T −3

51.25−0.064 T −3

7830/(1+11.1h10−6 T)

7270/(1+12.1h10−6 T)

Specific heat capacity /Jgkg−1gK−1

307/(1+4.3h 10−4 T)

465/(1+12.1h 10−4 T)

419/(1+5.7h 10−4 T)

2.16−0.365h10−2 T +0.0558h10−4 T 2

4.970.869h10−2 T +0.098h10−4 T 2

0.319+1.37h10−4 T

5.4.2 Calculation methods In computation of the thermal field, finite difference method and Gaussian elimination method (or other methods) are usually used to solve the linear equations. The program chart of the computation is shown in Fig. 5.23. This algorithm was used in references (Mei and Tang,1986;Mei et al., 1992; Zhou and Mei, 1992; Mei et al., 1996; Mei et al., 1997; Mei et al; 1998; Zuo, 1996; You, 1997; You et al., 1998; Zhou et al., 1998) to analyze the thermal fields of various types of reduction cells, which provided examples of systematic research on the design of thermal fields and optimization of cell structures. A typical analytical result is shown in Fig. 5.24.

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.23 Program chart of static analysis of the thermal field (Mei and Tang., 1986)

  Naijun Zhou

Fig. 5.24 Analytical result of the temperature field in cathode (Mei et al., 1996)

5.5 Dynamic Simulation for Aluminum Reduction Cells An aluminum electrolysis cell is a complex high temperature electrochemical reactor, whose operation conditions are effected by many factors. Any changes of the factors will cause unstable state of the cells. A worse operation condition will even result in an increasing of power consumption, decreasing of production, and shorter cell life. Obviously, it’s difficult to accurately predict the effects of various technical parameters on the operation conditions with the static models discussed in previous sections. Hence, the dynamic simulation is needed and will be introduced in this section. The aim of the dynamic simulation is to simulate the process in the cell with realtime information obtained by computer and output (forecast) the results, in order for online monitoring the operation conditions and to instruct the operations. Equipped with automatic control system, it can then realize functions of online optimization and monitoring. Practices indicate that there are three critical factors for operations of aluminum reduction cells, that is, the electrolyte temperature, the cell freeze profile and the current efficiency. Since effective online measurements of these factors haven’t been found so far, it is hence of particular practical significance to simulate the parameters. At present, the primary task of the dynamic simulation of aluminum reduction cells is the dynamic forecasting of the electrolyte temperature and the current efficiency, as well as the online simulation of the cell freeze profile. The Central South University and some

5 Hologram Simulation of Aluminum Reduction Cells 

other institutes had carried out research in this fields for many years, and made great progresses (Mei et al., 1997; Mei et al., 1998; Zuo, 1996; You, 1997; You et al., 1998; Zhou et al., 1998). In this section, it is to be discussed the methods for online simulation of the electrolyte temperature and the dynamic forecasting of the cell freeze profile.

5.5.1 Factors influencing operation conditions and principle of the dynamic simulation You Wang (You, 1997; You et al., 1998) had studied factors influencing the operation conditions of the aluminum reduction cells. In his opinion, the factors can be put into two categories. One is the static factors, including bus bar configuration, cell structure, electro-thermal properties of materials, melt properties, and heat transfer conditions of the cell body etc. The other is the dynamic (transient) factors, including processing parameters such as series current, cell voltage, anode-cathode distance (ACD), molecular ratio, temperature of the electrolyte and molten aluminum, height of the as well as the electrolyte and molten aluminum, anode effect coefficient etc., as well as the routine operations such as alumina feeding, aluminum discharging, ACD adjusting, anode changing, edge treating, effect treating and so on. The static factors only influence medium and long term behavior of the cell, thus they can be used to establish the basic energy balance of the cell. The dynamic factors influence dynamic behaviors and they are the basis of dynamic simulation. All these factors must be quantitatively transformed into energy budget of the cell or corresponding disturbance parameters. The principle of the dynamic simulation is based on following facts. When a disturbance factor causes the change of energy inputting into the cell, it will firstly cause changes of the melt temperatures (especially, the change of the electrolyte temperature). As the liquidus temperature Tf is fixed corresponding to certain composition of the electrolyte, the change of the melt temperatures will result in melting or freezing of the electrolyte crust, or in other word, changing in the thickness of the side ledge. As a result, heat transfer through the ledge will also change until a new balance of energy is reached. Therefore, based on the contribution of every factor to the energy budget of the system, the energy balance of a cell can be continuously obtained to calculate the bath temperature with measured intensity and duration of the disturbance, thus to simulate the freeze profile by solving the unsteady heat conduction equation. The program chart of the process is given in Fig. 5.25.

  Naijun Zhou

Fig. 5.25 Program chart of the dynamic simulation for aluminum reduction cells

5.5.2 Models and algorithm When viscous dissipation of the melt flow is neglected, energy equation of the system is: r ∂H + div( ρ vH ) = div(k • gradT ) + qvol ρ (5.50) ∂τ The relation between the enthalpy and temperature is: For solid phase T < Tf : H = c p • T For liquid phase T > Tf : H = c p • T + λ At the interface T = Tf : H = c p • T + n • λ where Tf is the melting point of the electrolyte; c p is the specific heat capacity; λ is the latent heat of melting, n is the molten fraction (0İnİ1).

As the melt motion in reduction cells is complicated. Eq. 5.50 is actually hard to be solved. A simple method is: to replace the thermal conductivity k with the effective thermal conductivity k eff , so that the convection term can be eliminated: k eff = k + k t (5.51) where k is the molecular thermal conductivity; k t is the turbulent thermal conductivity. Furthermore, assuming cp is a constant, Eq. 5.50 can then be simplified as: ∂T = div(keff • gradT ) + qvol ρcp • (5.52) ∂τ This is an unsteady conduction equation with internal heat sources, in which the source term qvol is the heat produced when current passing through the melt, that

5 Hologram Simulation of Aluminum Reduction Cells 

is the “Joule heat”. It can be solved from the Laplace equation of electrical conduction, i.e. Eq. 5.3. Eq. 5.52 and the electrical conduction equation can be solved with the implicit finite difference format after the discretization. For a selected slice, except that the collector bar is treated as a three-dimensional part, others are all regarded as two-dimensional. The boundary conditions are the same as stated in Section 5.4.1. The solving and analysis of the unsteady heat conduction equation are based on “heat balance self-checking method”(Zuo, 1996). The procedure is: a) According to the static analytic method, to determine a basic heat balance of the aluminum cell under a given melt temperature. b) Supposing that a disturbance will bring the heat to increase ΔQ0 in one time step and make the melt temperature increase to T1 = T0 + ΔQ0 / W (W is the “water equivalent” of the melt). Under these conditions, the electric field and temperature field at next moment can be analyzed. c) To estimate the volume of melted ledge, meanwhile, check the heat balance and the melt temperature, then start the calculation of next time step. Melting or freezing of the ledge will result in changes of its edge position, which can be determined by the “moving boundary method”(You,1997). Take the melting of the ledge as an example. When disturbances starts and the calculated temperature of the boundary node exceeds Tf , the extra enthalpy ΔH was considered to melt the ledge in the control volume around the node. The melted ΔH . With the melted volume, a new position of the ledge can volume is ΔG =

λ

be decided. After several time steps, if the entire ledge in control volume of this node all melts, the boundary should move to the next node in next step of the computation. The same computation should be carried out at each node. After several time steps (which is set as the simulation time in the programme), the results of the freeze profile of the cell can be obtained and displayed on a WINDOWS interface. For the process that the ledge freezes and gets thicker, the computation is the same only the boundary moves in a reverse direction. As the computation goes on, error will accumulate, thus result in a distortion of the simulation results. It can be solved by periodically correcting the basic heat balance offline with the measured electrolyze temperature and freeze profile. Or it can be done by online correction with the cell wall temperatures (Zhou et al., 1998).

5.5.3 Technical scheme of the dynamic simulation and function of the software system According to above principles and models, in reference (Mei et al.,1998), the author put forward a technical scheme of the dynamic simulation (as shown in Fig. 5.26). The software system consists of six function modules, i.e., analysis of cell conditions, dynamic simulation, examination and correction of static parameters,

  Naijun Zhou

offline calibration, creation and print of report forms, and online help. The main functions of the system are listed below:

Fig. 5.26 Technique scheme of dynamic simulation (Mei et al., 1998)

a) Display of main parameters collected online (such as series current, voltage and wall temperatures etc.). b) Display of the graph of simulated freeze profile, and displaying temperature distribution of the cathode cross section. c) Display of the dynamic forecasted temperature of the electrolyte and the molten aluminum. d) Display of the simulated characteristic dimensions of the freeze profile. e) Examination and modification of the static structure parameters and physical parameters. f) Input and browse of the processing parameters offline. g) Recording and browse of the operation history. h) Indication and warning of the communication state. i) Creation, browse and output of report forms. Fig. 5.27 shows an displaying interface of the software. Application of the software in a 160kA prebaked anode cell were reported in references (Mei et al., 1997; Mei et al., 1998; Zhou et al., 1998).

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.27 Simulation software run in visualization windows and results (zhou et al., 1998)

For the precision of the simulation, error of electrolyte temperature is estimated to be less than 0.3%, and the ledge characteristic dimension is within 10%. The realtime response of the simulation has been tested, of which the analysis time of every slice is less than 5s at a PĊ300 micro computer. The result indicates the software has good engineering practical value.

5.6 Model of Current Efficiency of Aluminum Reduction Cells Current efficiency η in an aluminum reduction cell is defined as the percentage of actual aluminum production to its theoretical production, that is: Q Qt η = a ×100% = ×100% (5.53) Qt 0.3356 • I • t ×10−3 where Qa is the actual aluminum production of a cell; Qt is the theoretical production of the cell; I is the current intensity; t is time. There are two kinds of current efficiency: the “mean current efficiency for a long time” and the “transient current efficiency”. The current efficiency in electrolysis aluminum factories in China ranges from 87% to 92%. The advanced international index is about 94% to 95%, and the best record of tested cells approaches to 96%. The economic benefit resulting from every 1% increase of the current efficiency is that, not only the production can increase 1% corresponding, but also the power consumption will reduce 170kW·h

  Naijun Zhou

per ton aluminum. Study of the current efficiency has always been a topic of research of aluminum electrolysis technology since the Hall-Heroult aluminum reduction method was applied in industry. As a results, the current efficiency increased from original 60% to the present level, which reflected the great progress achieved in the aluminum electrolysis technology. The study of the current efficiency focuses on several aspects, such as the mechanism of reduction of the current efficiency, the factors influencing current efficiency, mathematical models of current efficiency and related factors, and measurements of current efficiency. The research method is mainly experimental methods including laboratory experiments and industrial experiments. Theoretical methods are also used in the studies. Theoretical methods usually make some simplification of various conditions and influencing factors with known theory and experiences; then on the basis of the process control steps, to deduce a calculation formula of the current efficiency according to fluid dynamic principles. The calculation formula, to some extent, simulates the relation among various factors and the current efficiency, and reflects a good correlation with actual measurements of the current efficiency. Hence it is proved to be applicable. This is also the point to be discussed in this section.

5.6.1 Factors influencing current efficiency and its measurements According to research up to now, the reasons of the reduction of the current efficiency in aluminum cells come from four aspects (Qiu, 1988; Huang et al., 1994): a) Dissolution of the aluminum. The precipitated aluminum dissolves back into the electrolyte, being brought to the space below anode by the circulating electrolyte and oxidized by anode gas. This is the main reason for the reduction of the current efficiency. b) Precipitation of sodium. The low concentration of alumina and the high temperature of cell, especially during the anode effect period, will result in sodium ions replacing with aluminum ions to discharge in the cathode, thus reduce the productivity of aluminum. c) Vain consumption of current. It includes vain consumption of current that caused by imperfect charge of polyvalent ions, electronic conduction, short circuit, current leakage and so on. d) Other losses. They include combination reaction of aluminum and inner lining materials, electrolysis of moisture content and impurity compounds, mechanical losses in aluminum discharging process and so on. The effects of various process parameters on current efficiency are summed up as follows:

5 Hologram Simulation of Aluminum Reduction Cells 

a) Temperature. All the research results indicate that increasing of the temperature causes reduction of current efficiency; and experimental results show that the current efficiency decreases about 2% when the electrolysis temperature rise 10ć. b) Anode-cathode distance (ACD). The characteristics of the effects of the ACD on current efficiency are that, the current efficiency improves with increasing of the ACDs, whereas high ACD will result in higher energy consumption perton aluminum. Therefore it is not a feasible way to improve the current efficiency by enlarging ACD. c) Molecular ratio. No consistent conclusion has been reached about the effects of the electrolyte molecular ratio on the current efficiency. Results of most research indicated that the current efficiency decreases with the molecular ratio. However, low molecular ratio can have impacts on electrolytic conductivity, solubility and volatilization loss, which should be considered synthetically. d) Alumina concentration. Research results indicated that there is an optimum alumina concentration. Exceeding or bellowing alumina concentration will all decrease the current efficiency. The optimum concentration value depends on the electrolyte composition. e) Chemical additives. Lots of experiments have been carried out on the electrolyte additives. The chemical additives with good influencing performance include CaF2, MgF2, Li2CO3 and so on. The action of the additives is mainly to decrease the electrolysis temperature and to control the dissolution of aluminum, so that the current efficiency can be improved. Proper amount of the chemical additive ranges from 3% to 5%. Too much or too less is unfavorable. f) Area current density. The influence of the area current density includes two aspects. One is that the current efficiency decreases with the anode area current density; and the other is the current efficiency increases with the cathode area current density. However, as the area current density cannot be changed easily during the operation of cells, it is generally taken into account only in the design. g) Melt motion. The melt motion strengthens the mass transfer process, thus accelerates the aluminum dissolution, and it may also cause current short circuit, which will decrease the current efficiency. In a word, there are many factors that have influences on the current efficiency. These factors interrelate with each other, and should not be isolated. As their effects depends on the control of test conditions, it is difficult to quantify their respective influences. There are two main measurements of the current efficiency: the dilution method and the eudiometry. The dilution method is to put known indicators (such as Cu, Mn, Ti, 60Co) into a cell and trace the variation of indicator concentration, thus to calculate out the current efficiency during this period. Or it also can be done by calculating the aluminum weight in the cell between two additions of the indicator into the cell. The

  Naijun Zhou

current efficiency can then be calculated through adding up and measuring the tapping aluminum quantity and power consumption in the period. The first method is called as the regression method, and the latter is the inventory method. The eudiometry is to calculate the current efficiency with the continuously measured CO2 concentration in the anode gas using an appropriate formula. Both methods have their own advantages and disadvantages, and they are often used by combine in utilization.

5.6.2 Models of the current efficiency The aim of developing a current efficiency model is to express quantitatively the relationship of parameters and the current efficiency with a mathematical formula, so as to provide a basis for the computer control of aluminum reduction cells. There are two methods to establish a current efficiency model. One is to make theoretical deductions with the reduction mechanism of the current efficiency. The other is to establish the model through regressing massive experimental data. The former reflects the internal connection of various parameters to some extents and has a certain physical meaning, however it is not well practical. The latter is a empirical formula, however, lacking of clear physical meaning, it is only applicable in some cases. A lot of research has been carried out on the calculation of the current efficiency. Here it is to list several representative models. 5.6.2.1 Theoretical model of the current efficiency

To illuminate the general method of theoretical modeling, a deduction is to be made of the relationship of the current efficiency, the area current density I Am , the ACD L and the concentration of metal ions in the anode region C3 by using the diffusion theory. Assumptions made in the model are as follows: a) The reduction of the current efficiency is caused completely by the metal aluminum solving into the electrolyte in form of Al+ and diffusing into the anode region where it is oxidized as Al3+. Effects of the electron transfer and the convective mass transfer are neglected in the model. b) Diffusion coefficient of Al+ and Al3+ are constants. c) The concentration of Al3+ at the cathode surface is certain value. d) At the beginning of electrolysis, the Al3+ concentration at the cathode surface is an equilibrium concentration, and that at the electrolyte surface is zero. e) Concentrations are allowed to be replaced with ionic activities with the unit of mol/cm3. f) Consumption and supplement of raw materials are close to a balance. The metal deposition reaction at the cathode is: Al3+ + 3e == Al (5.54) The metal solution reaction is:

5 Hologram Simulation of Aluminum Reduction Cells 

2Al + Al3+ == 3Al+ (5.55) Supposing that, C0 is the aluminum concentration at the cathode surface; C1 is

the Al+ concentration, the equilibrium constant in Eq. 5.55 is K = 1

C13 , then: C0

1

C1 = 3 K • C03 = bC03

(5.56)

If the solubility of aluminum in the electrolyte can be expressed as: 1

S = a + bC03

(5.57) +

3+

And when diffusing into the anodes, the oxidation of Al into Al in the anode region is: Al+ − 2e == Al3+ (5.58) Supposing reactions of Eq.5.57 and Eq. 5.58 are a quick process and it is controlled by diffusion, then, in accordance with the first Fick’s law, the rate of metal losses from unit of the cathode surface to the anode is: D ( S − 0) D1S = (5.59) Vloss = 1 L L where D1 is the diffusion coefficient of Al+. And the metal formation rate is: D (C − C0 ) (5.60) Vformation = 3 3 L where D3 is the diffusion coefficient of Al3+; C3 is the Al3+ concentration in the anode region. As the aluminum concentration at the cathode surface C0 is difficult to determine, it can be converted into a function of the area current density. Since: C − C0 I (5.61) = N e • D3 • 3 Am L It can be rewritten as: ⎛ ⎞ I L C0 = C3 ⎜1 − • ⎟ ⎝ Am N e • D3 • C3 ⎠ where Ne is the charge per gram ion. Then the current efficiency can be simply calculated as: V −V η = formation loss ×100% Vformation

(5.62)

1 (5.63) ⎡ ⎤ ⎛ N e • D1 • Am ⎢ I L ⎞3 ⎥ a + b ⎜ C3 − • = 100 − ⎟ • 100 % ⎢ I•L Am N e • D3 ⎠ ⎥ ⎝ ⎣ ⎦ Although in this formula, effects of temperature and electrolyte character on the current efficiency are not definitely indicated, and the convection is neglected, variables D1 , D3 , a and b are all related to these factors. Through analysis of the relationship, various formulas of the current efficiency have been obtained,

  Naijun Zhou

Several influential models are introduced below. a) Robl formula (Robl et al., 1977): ⎛ r ⎞ η = ⎜ 1 − m ⎟ × 100% (5.64) r ⎠ ⎝ where rm and r are respectively the metal formation rate and mass transfer rate in the electrolyte. Because factors influencing rm are complicated and difficult to be determined, the model is not practical. However, many researchers put forward their own models on the basis of it. b) Lillebuen formula (Lillebuen et al.,1980): η = 100 − 219.3I −1 Am Dm0.67 μ e−0.5u e0.83 L−0.17 ρ e1.5C m* (1 − f ) (5.65) where I is the current intensity, kA; Am is the surface area of the molten aluminum, m2; Dm is the diffusion coefficient of the metal, m2/s; μ e is the electrolyte viscosity, Pa·s; ue is the relative velocity of the electrolyte to the molten aluminum, m/s; L is the anode-cathode distance, m; ρ e is the density of the electrolyte, kg/m3; Cm* is the mass concentration of the metal in the electrolyte, %; f is the saturation fraction of aluminum at the interface of the electrolyte and the molten aluminum. Comparatively, parameters in this formula are easier to be determined, hence, it is used widely. c) Evans formula (Evans et al., 1981): Evans put forward his formula on the basis of Robl’s formula as:

⎛D ρV ⎞ (5.66) rm = ⎜ m e t ⎟ Cm* ⎝ σ ⎠ where Vt = k 0.5 is the pulsating component of the turbulent velocity; k is the turbulent kinetic energy; and σ is the surface tension coefficiency of the interface of the molten aluminum and the electrolyte. In addition, by considering two-dimensional turbulent mass transfer, Shen Hongyuan et al proposed a model, of which parameters are also easy to be determined (for details, please refer to reference (Shen et al., 1992)). All the models mentioned above are supported by themselves’ experimental results. 0.5

5.6.2.2 Statistic model of the current efficiency

Qiu Zhuxian et al (Qiu, 1998; Qiu et al., 1981) studied the relations of CO2 concentration in the anode gas and operating parameters in a 135kA prebaked anode cell with side processing and central feeding with automatic breaking shells. Using stepwise regression method, they obtained the following mathematic model: CO 2 % = 1297.2 − 89.67 X1 − 122.2 X 2 − 1.304 X 3 − 0.2183 X 22 −

0.1169 X 1 X 4 + 0.09552 X 1 X 3 + 0.2013 X 4 X 5 −

(5.67)

0.5951X 2 X 5 + 0.1341X 2 X 3 where X 1 , X 2 , X 3 , X 4 , X 5 denote respectively deviations of the anode current

5 Hologram Simulation of Aluminum Reduction Cells 

distribution (i.e. ADN value), the concentration of alumina, the electrolysis temperature, the electrolyte height, the concentration of additives (CaF2 and MgF2), and the molecular ratio. The current efficiency can then be obtained by choosing appropriate gas equation, such as: 1 η = (CO 2 % + 100%) + 3.5% (5.68) 2 Cai Qifeng et al. (Cai et al.,1991)obtained the current efficiency of a 160kA four risers prebaked anode cell with the multiple regression method: CO 2 % = 392.6 − 0.337 X 1 − 0.5204 X 2 + 0.1133 X 3 − 4.6392 X 4 + (5.69) 0.6148 X 5 − 0.5591X 6 + 12.3084 X 7 + 2.5105 X 8 where X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 are respectively the electrolyte temperature te, the molten aluminum height, the molecular ratio, the concentration of alumina, and the concentration of the additives (CaF2, MgF2 and LiF). Checked by t verification step by step, a optimum regression equation can be obtained, which is: (5.70) CO 2 % = 376.025 − 0.3301te + 3.4289LiF% Wang Hua and Wang Huazhang et al obtained their current efficiency models of 60kA side stud cells and 24kA self-baking cells in similar way, of which detail are given in references(Wang,1993;Wang et al.,1982). It should be noted that, present statistic models of the current efficiency are not universal, because they are usually based on a certain kind of aluminum reduction cells.

References Aparotsev B A (1998) Simulation for the carbon-cathode of aluminum electrolysis cells. Non-Ferrous Metals (in Russian), (1): 73 Arai K, Ramazaki K (1975) Heat balance and thermal losses in advanced prebaked anode cells. Light Metals, (1): 193 Arita, Ikenouchi (1983) Image of electrolyte and metal convection in aluminum reduction cells and numerical computation of the interface shape (in Japanese). Light Metals (Japan), (11) Cai Qifeng et al (1991) Mathematic model of current efficiency in 160kA central charging four terminals pre-baked cells (in Chinese). Light Metals, (5): 25 Cai Qifeng, Mei Chi et al (1993) Calibration experiment and industrial practice of molten velocity measurement by iron rod method (in Chinese). Light Metals, (9): 29 Cao Guofa (1991) Three dimensional numerical computation of potential fall in anode (in Chinese). Messages of Aluminum & Magnesium, (3): 1 Central South Institute of Mining and Metallurgy (1977) Reference Data of Nonferrous Metallurgical Furnace (in Chinese),Č

  Naijun Zhou Chen Shiyu et al (1986) Computation of current field in aluminum reduction cells with use of relative electrical parameters (in Chinese). In: Aluminum Reduction Cells Thesis Collection of Aluminum Reduction Cells “Three fields” , 9: 55 Chen Shiyu et al (1987) Mathematic model of magnetic field in aluminum reduction cells and error analysis (in Chinese). Journal of Huazhong Institute of Technology, (6): 85 Dekopov U D et al (1978) Non-Ferrous Metals (in Russian),(1):32 Derneded D (1975) Gas induced circulation in an aluminum reduction cell. Light Metals, (1): 111 Evans J W et al (1981) A mathematical model for prediction of currents, magnetic fields, melt Velocities, melt topography and current efficiency in Hall-Heroult cells. Metal. Trans.: 353 Jean Frederic, Tony Lelievre, Claude LE (2002) Metal pad roll instabilities. Light Metal: 483~487 Fuzzy Control Research Institute of North China University of Technology (1997) Development and extension of new control system for aluminum reduction cells (in Chinese). Journal of North China University of Technology, 9 (3): 85 Gan Yiren, Zhang Hong (1998) Impervious heat insulation material for aluminum reduction cells (in Chinese). Light Metals, (Supplement): 173 Gerbeau J F, Lelievre T et al (2003) Simulations of MHD flows with moving interfaces. Journal of Computational Physics,184(1): 163~191 Guo Jianjin (2007) Review of China’s aluminum market in 2006 (in Chinese). Shanghai Non-Ferrous Metals, 28(3) Haruhiko Ikenouchi et al (1978) Light Metals, (1): 59 Haupin W E (1971) Calculating thickness of containing walls frozen from melt. J. METALS. 23(7): 41~44 Haupin W E (1990) The 9th International Course of Process Metallurgy of Aluminum, Tronheim Huang Qianrui (2007) Status, problems and development: bauxite resource and aluminum industry in China (in Chinese). Conservation and Utilization of Mineral Resources, 3 Huang Yongzhong et al (1994) Production of Aluminum Electrolysis (in Chinese). Central South University of Technology Press, Changsha, 197 Huang Zhaolin et al (1994) Numerical simulation of turbulent flow and melt topography in aluminum reduction cells (in Chinese). Computational Physics, 11(2): 179 Imery Buiza J (1989) Electromagnetic optimization of the V-350 cell. Light Metals, 211 Japan Light Metals Research Institute (1980) Three Fields Computation Analysis Program (Internal Data) (in Chinese) Li Guohua et al (1993) Computation of magnetic field in aluminum reduction cells with use of double scalar potential method (in Chinese). Nonferrous Metal, 45(4): 55 Liang Xuemin et al (1998) Mathematic model of physical field in aluminum reduction cells and computer simulation (in Chinese). Light Metals, (Supplement): 145 Lillebuen B et al (1980) Variations of side lining temperature, anode position and

5 Hologram Simulation of Aluminum Reduction Cells  current/voltage load in aluminum reduction cells (in Chinese). Light Metals: 131 Liu Yexiang, Mei Chi et al (1996) Computer simulation of electromagnetic force field on the melt in alumina reduction cell (in Chinese). Transactions of Nonferrous Metals Society of China, 6(1):27 Lu Jiayu et al (1986) Computation of current field in large aluminum reduction cells with use of boundary element method (in Chinese). In: Thesis Collection of Aluminum Reduction Cells “Three fields”, 9: 73 Mei Chi, Tang Hongqing (1986) Computation analysis and simulation test of electric and thermal fields in aluminum reduction cells (in Chinese). Journal of Central South Institute of Mining and Metallurgy, (6): 29 Mei Chi, Wang Qianpu et al (1992) Study of thermal field in aluminum reduction cells (in Chinese). Light Metals, (1): 29 Mei Chi, Wang Qianpu et al (1996) Optimization of inner lining structure in 80kA self-baking aluminum reduction cells (in Chinese). Light Metals, (4): 21 Mei Chi, You Wang et al (1997) Study and development of the simulation software for the freeze profile in aluminum reduction cells (in Chinese). Journal Of Central South University of Technology, 28(2): 138 Mei Chi, Zhou Naijun et al (1998) Dynamic prediction of bath temperature and online simulation of freeze profile in aluminum reduction cells (in Chinese). Light Metals, (supplement): 16 Morris S J Sˈ Davidson P A (2003) Hydromagnetic edge waves and instability in reduction cells. Fluid Mech., 493(2):121~130 Peacey J G, Medlin G W (1979) Cell sidewall studies at Noranda aluminum. Light Metals, (1): 475 Perutne (1982) Freeze profile in side-break, cells-calculations and measurements. Light Metals: 359 Qiu Jie (1992) Computation of nonlinear magnetic field in aluminum reduction cells (in Chinese). Nonferrous Metal, (3): 55 Qiu Zhuxian (1982) Aluminum Electrolysis (in Chinese). Metallurgical Industry Press, Beijing, 292 Qiu Zhuxian (1988) Aluminium Metallurgy In Pre-Baked Cells (in Chinese). Metallurgical Industry Press, Beijing ,244 Qiu Zhuxian et al (1981) Mathematic model of current efficiency in industrial aluminum Reduction Cells (in Chinese). Light Metals, (9): 23 Richard M C (1976) A new approach for comparing the impact on magnetic fields from pot design alterations. Light Metals, (1): 109 Romanov B P et al (1980) Non-Ferrous Metals (in Russian),(1):66 Robl R F (1978) Influence by shell steel on magnetic fields within Hall-Heroult cells. Light Metals, (1):1 Robl R F, Haupin W E, Sharma D et al (1977) Estimation of current efficiency by a mathematical model including hydrodynamics parameters. Light Metals: 185

  Naijun Zhou Sele T (1974) Computer model for magnetic fields in electrolytic cells including the effect of steel parts. Metal. Trans: 2145 Severo D S et al (2005) Modeling magnetohydrodynamics of aluminum electrolysis cells with ANSYS and CFX (in Chinese). Light Metals: 475~480 Shen Hongyuan, Mei Chi, Cai Qifeng (1992) Mathematic model of current efficiency determined by two dimensional turbulent mass transfer in aluminum reduction cells (in Chinese). Journal of Central South Institute of Mining and Metallurgy, 23(6): 693 Shen Xianchun et al (1994) Physical simulation and experimental determination of magnetic field in aluminum reduction cells (in Chinese). Light Metals, (2): 26 Sneyd A D (1985) Stability of fluid layers carrying a normal electric current. Fluid Mech: 223 Solhiem A et al (1983) Heat transfer coefficients between bath and side ledge in aluminum cells. Light Metals: 425 Sun H, Zikanov O, Donald P et al (2004) Non-linear two-dimensional model of melt flows and interface instability in aluminum reduction cells. Fluid Dynamics Research, 35:255~274 Sun H, Zikanov O, Finlayson B A (2005) Effect of background melt flow and interface distortion on the stability of Hall-Heroult Cells. Magnetohydrodynamics, 41(3):273~287 Sun Ru, Mei Chi (1989) Numerical simulation of the circulation of molten metal In aluminum reduction cell. Journal of Central South Institute of Mining and Metallurgy (in Chinese), 20 (6): 618 Tarapore E D (1982) The effect of operating variables on flow in aluminum reduction cells. Metals, 34 (2): 541 Vanvoren C et al (2001) AP50: the pechiney 500kA cell. Light metals: 221~226 Vao Shihuan (1990) Selection of bus bar optimum section in high amperage reduction cells. Light Metals: 453 Wang Hua (1993) Mathematic model for optimization of current efficiency in 60kA side stud cells (in Chinese). Light Metals, (1): 24 Wang Huazhang et al (1982) Mathematic model of current efficiency in 24kA self-baking anode aluminum reduction cells (in Chinese). Light Metals, (8): 24 Wang Ruliang, Yang Xiaodong et al (1998) Optimization of bus bar configuration and cell structure in 280kA pre-baked cells (in Chinese). Light Metals, (Supplement): 58 Wu Jiangkang, Huang Ming, Huang Jun et al (2002) Finite element analysis of incompressible viscous flow with moving free surface by selective volume of fluid method. Light Metals: 511~514. Xue Yuqing et al (2006) Normal temperature analogue experiment of anode bubble’s behavior in aluminum electrolysis cells (in Chinese). The Chinese Journal of Nonferrous Metals, 10:1823~1828 Yang Jinxin (2007) Where is the right way for China’s non-ferrous metals industry to save energy and reduce consumption (in Chinese). Energy Saving of Non-Ferrous Metallurgy:3

5 Hologram Simulation of Aluminum Reduction Cells  You Wang (1997) Online dynamic simulation of freeze profile in large pre-baked aluminum Reduction cells (Docteral Dissertation, in Chinese). Central South University of Technology , Changsha You Wang, Wang Qianpu et al (1998) On line dynamic simulation of freeze profile in aluminum electrolysis (in Chinese). The Chinese Journal of Nonferrous Metals, 8(4): 695 Zeng Shuiping (1996) Computation of electro-magnetic field in aluminum reduction cells and continuous monitor of current efficiency (Docteral Dissertation, in Chinese). Changsha: Central South University of Technology Zeng Shuiping (2004) Mathematical model of current efficiency with anode current distribution in aluminum reduction cells(in Chinese). The Chinese Journal of Nonferrous Metals, 4:681~685 Zeng Shuiping, Cai Qifeng et al (1996) Study of bus bar configuration in vertical-stud aluminum reduction cells (in Chinese). Light Metals, (2): 32 Zhou Keding (1983) Basic theory of boundary element analysis of electro-magnetic field (in Chinese). Micro Motor, (3) Zhou Naijun (2006) Effect of electromagnetic force and anode gas on electrolyte flow in aluminum electrolysis cell. Journal of Central South University of Technology, 5:496~500 Zhou Naijun, Mei Chi et al (1998) Dynamic numerical simulation of melt temperature in large pre-baked aluminum reduction cells (in Chinese). Proceedings of China Engineering Thermophysics Annual Conference, Hefei Zhou Ping, Mei Chi (1992) Continuous monitor of freeze profile in aluminum reduction cells (in Chinese). Light Metals, (4):19 Zuo Jun (1996) Dynamic simulation of thermal field in aluminum reduction cells (Master Dissertation, in Chinese). Central South University of Technology, Changsha


Simulation and Optimization of Electric Smelting Furnace Jiemin Zhou and Ping Zhou

In this Chapter, the simulation models, including sintering process model of self-baking electrode, flow field model and temperature field model in molten pool of electric smelting furnace are built, and the cases in practice are employed to optimize them. Eventually, the computational software is developed to solve the physical fields and production condition of the electric furnace is improved.

6.1 Introduction Electric smelting furnace, also called submerged arc furnace or electric arc resistance furnace, is mainly applied in the smelting of nonferrous metals, ferroalloys, calcium carbide and yellow phosphorus. Ferroalloy furnaces are usually designed to be circular, and three electrodes insert into the charge forming a regular triangle. Most electric furnaces for smelting of nonferrous metal are rectangular, others are circular or elliptic. In these rectangular or elliptic electric furnaces, three or six electrodes are distributed in line. Current is conducted to the furnace slag or charge through electrodes, and the voltaic arc is produced at the interface of electrode and slag or charge, where most of the heat is released. When the currents flow through the slag or charge, the triangular and star-shaped circuits are formed, and Joule heat is produced. The resulting high temperature then melts the furnace charge, accompanied by various physical and chemical reactions, which enriches the valuable element. In nonferrous metallurgy, the electric smelting furnace is mostly used in the smelting of copper, nickel and tin, and the impoverishment of slag, or used as electrically-heated settler for separation of metal and slag.

  Jiemin Zhou and Ping Zhou Electric smelting furnace has the merits of high smelting temperature, relatively little fume, high recovery rate of metal, easy control and so on. However, it consumes electric energy that is converted from other kind energy, therefore, the high power consumption and high cost are its primary disadvantages. Meanwhile, large electric smelting furnaces commonly use self-baking electrode, which may result in accidents such as soft-broken and hard-broken, and the operation is often affected. In recent years, as the price of energy increases because of the short supply, investigation on saving energy and improving the quantity and quality of the production has become a hot topic for all technicians and theoretical researchers working in the field of electric smelting furnace. The performances, as well as technical and economic indices, of electric smelting furnace are affected by many factors, such as chemical compositions and physical characters of the charge, the way to prepare the charge, the electric condition, operation condition, the furnace configuration and so on; that is, the operation parameters and the configuration parameters of the furnace. These variables influence the furnace thermo-technical process parameters including current distribution, electricity-heat conversion, fluid flow, heat transfer, temperature distribution, etc. Such thermo-technical process parameters directly determine the furnace production indices, for instance, the production capacity, power consumption per unit production, and recovery of valuable metals, etc. The development of computer and calculation method makes it possible to study the heat transfer process, momentum transfer process and mass transfer process in the furnace quantitatively, which means we can simulate the correlation between operation parameter, configuration parameter and thermo-technical process parameter as well as various production indices, and further detect rules of the production process in electric smelting furnaces to improve the equipments, optimize the operations and improve the furnace performances and production indices. This chapter mainly investigates the sintering process of self-baking electrode in large scale electric smelting furnace, the melt flow process in slag impoverishment ˄depriving˅electric furnace and the heat transfer process in molten pool of ore smelting electric furnace. Then, relevant software is introduced to simulate and optimize the real electric smelting furnace, and specific approach is presented to improve the configuration parameters and operation condition.

6.2 Sintering Process Model of Self-baking Electrode in Electric Smelting Furnace The self-baking electrode(Fig.6.1), or Soderberg electrode, which has a lower cost than the prebaked electrode, is constituted of electrode paste and steel plate. It is the most important part of the electric smelting furnace and is baked to be a solid

6 Simulation and Optimization of Electric Smelting Furnace 

electrode during the smelting process inside the furnace. Having to conduct very huge current, and to endure conditions such as oxidization of high temperature in the furnace, sharp cooling of clamp water-cooled system, temperature variation during the start-up and shut-down of the furnace, and the mechanical impact of the charge, the electrodes must have excellent electrical conductivity, enough high-temperature mechanical strength and relatively high oxidizing temperature. To a great extend, the normal operation of the furnace depends on the electrodes’ working situation. In order to reduce the malfunction of electrodes and to improve the energy efficiency, a rational electrode baking rule, that is, a rational temperature distribution must be maintained. There are two ways to study the temperature distribution in the electrode: by measurement and by analytical modeling. Generally, measurements of the electrode temperature are carried out by inserting thermocouples into the electrode at different depth. During the descent of the electrode, the temperature from the thermocouple can be recorded continuously by the secondary meter, and credible data can be obtained. However, it is difficult to maintain a stable electrode temperature during several days’ measurements when the electrode is descending. Thus the testing error is unavoidable due to the variation of the processing conditions. As a result, when to exam the influence of different operating conditions on the temperature field of the electrode, a large amount of experiments have to be carried out, and it’s time-consuming. Especially, when the equipment configurations are to be changed, such as the dimension and the amount of the fin, or the distance between the clamp and the roof of the furnace, it is tedious and needs more human and financial resources. In condition that we adopt mathematical method to computer and display the temperature distributions which will appear in different design approaches, and then predict the electrode baking process as well as its operation performance, both the time and the cost of the experiments will be greatly reduced. This is what we called analytical modeling method. The electrode temperature field mainly depends on the power distribution in the electrode, i.e. the electrical field, therefore, the electric analysis and thermal analysis are bonded together. R. Innvaer et al. of Elkem Metal Company in Norway are the vanguards in the research of electric analytical and thermal analytical models of self-baking electrode in the electric smelting furnace. Since 1972, they have developed five self-baking electrode mathematical models, including two static models, one dynamic model, one stress model, and one three-dimensional analytical model. Of the static models, one is based on direct current and called Elkem-S, and the other is based on alternating current, called Elkem-X. These two models are generally used to compute the electrode temperature field and baking zone during the stable

  Jiemin Zhou and Ping Zhou operation in order to solve the electrode soft-broken problem. The dynamic model (Elkem-D) is used to compute the temperature field when operation parameters varying with time when to investigate the electrode soft-broken and hard-broken problem. The stress model (Elkem-T) can calculate the thermal stress in both steady and unsteady state to study the hard-broken problem for the baked electrodes. These models have become powerful tools in production and design (Olsen et al., 1972, 1976; Innvaer et al., 1976, 1980). However, they are all two-dimensional models, and did not take the electric and thermal uniformity along the perimeter of the electrode caused by the fins into account. As an improvement, R.Innvaer et al. developed a three-dimensional analytical model (Elkem-3X) of the self-baking electrode, in which the influences of electrode descent velocity, clamp current, furnace temperature, material and thickness of the electrode shell and the fin’s length on the temperature distribution of the electrode’s cross section and the electrode baking process were considered. They also used the Elkem-3X model to simulate a new electrode holder, and obtained some valuable data(Innvaer et al., 1986). This type of electrode holder was later sold to China (Vatland et al., 1987). Because in the solution procedure of the Elkem-3X model, the influence of alternating magnetic field on alternating electric field is considered, its result is more precise. However, as it needs to solve the electric field, magnetic field and temperature field simultaneously, the computing time and cost all increase. Therefore, Elkem-3X model is usually used only to calculate part of the self-baking electrode. There are some other literatures that reported research of the self-baking electrode, but they are not as systematic and consummate as the Innvaer’s Elkem-3X model. For such literatures, please refer to the references (Heiss, 1981; Bochmann et al., 1968 Cavigli, 1978; Ambrosio, 1981). There is no investigation on the analytical model of self-baking electrode before in China.

6.2.1 Electric and thermal analytical model of the electrode

Fig.6.1 Schematic of the self-baking electrode

With the three-dimensional model, multiple adjustable parameters, large amount of information and more precise result can be obtained. However, if we excessively pursue the precision of the solution, the calculation time and cost will be remarkably increased, which will limit the model’s application. Therefore, a

6 Simulation and Optimization of Electric Smelting Furnace 

relatively coarse grid is adopted to simplify the calculation of the electric field when the model is developed. Compared to literatures, further investigation and development have been carried out to improve the output form of the result (Zhou,1991). 6.2.1.1 Physical model The self-baking electrode of 16500 kVgA nickel electric smelting furnace in Jinchuan Nonferrous Metal Company was taken as an example in the analysis. Its schematic is shown in Fig. 6.1. In furnace No. 2, the electrode diameter is 1m, the electrode current is about 11̚18 kA, and the electrode consumption is 0.3 m/d. The compositions and properties of several kinds of domestic electrode pastes are shown in Table 6.1. Table 6.1 Component and properties of the electrode paste Producing area

Fixed carbon /%

Volatile /%

Ash /%

Moisture /%

Density /ggcm−3

Compression strength/Pa

Porosity /%

Jilin

69.26

19.23

11.07

0.45

1.50

260×105

20.0

Guiyang

70.25

18.38

10.52

0.85

1.61

135.5×105

25.0

Shanghai

79.23

11.99

8.78

1.57

270×105

23.9

Kunming

72.64

19.34

8.02

1.71

5

37.75×10

15.9

The tested electrical resistance of the electrode paste at different temperature are given in Table 6.2. Table 6.2

Electrical resistance of electrode paste

(¡gmm2/m)

Category

100ć

150ć

200ć

250ć

300ć

Standard paste

1.07×108

3.4×107

1.4×107

9.8×106

7.93×106

Inclosed paste

7.45×105

4.54×105

3.4×105

4.69×105

1.67×106

Category

350ć

400ć

450ć

500ć

650ć

Standard paste

5.22×106

1.72×106

9.30×104

1.37×104

2.4×103

Inclosed paste

5.44×105

9.45×104

1.86×104

4.72×103

2.49×103

The real electrode system is very complicated. To obtain a physical model that can be described in mathematical way, some simplifications are made as follows:

  Jiemin Zhou and Ping Zhou a) The electrode system is in a steady state, and its electric field and thermal field do not change with time. b) The electrode descends at a certain speed. c) The influence of skin effect on the electrode electric field is ignored. According to reference (Bochmann et al., 1968), the influence of skin effect on the current distribution is inapparent when the electrode diameter is less than 1.2 m. d) Assumption is made that when the electrode shell (casing) in the slag is burned out, it will be replaced by the electrode paste, which means the electrode diameter is unchanged, The burning loss occurs only at the tip of the electrode, and the speed corresponds to the electrode descent speed. This assumption simplifies the calculation without inducing big error (Olsen et al., 1972). Cylindrical coordinates are adopted in this model, and its computational domain is shown in Fig. 6.2. Due to the symmetry of the geometry, the electric field and the thermal field, only 1/2n of the cycle area of the electrode cross section needs to be calculated, where n is the number of the brass clamps. In z direction, the analytical domain is divided into five zones along the paste surface, the upper and downside edges of the clamp, the inner surface of the furnace roof, the slag surface and the tip surface of the electrode. While in the radial direction, the analytical domain is divided into two zones along the fin. Thus, suitable discretization can be applied to different zone according to the importance of each zone and the requirement for calculation precision, if necessary.

Fig. 6.2 Analytical domain and grids arrangement of self-baking electrode

6.2.1.2 Mathematical model In the cylindrical coordinates system, when the electrode descends at a constant speed, the differential equation of the thermal conduction that dominates the heat transfer process in the electrode is given as follows:

6 Simulation and Optimization of Electric Smelting Furnace 

1 ∂ ⎛ ∂T ⎞ 1 ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂T rλ + λ + λ + ρcV + p=0 (6.1) r ∂r ⎜⎝ ∂r ⎟⎠ r 2 ∂θ ⎜⎝ ∂θ ⎟⎠ ∂z ⎜⎝ ∂z ⎟⎠ ∂z where T is the temperature; λ is the thermal conductivity; ρ is the volumetric density; c is the specific heat; and V is average descent velocity of the electrode.

P =σ

⎡⎛ ⎢ ⎜ ∂ϕ ⎢ ⎜ ∂r ⎣⎝

⎞ ⎟⎟ ⎠

2

2 2 ⎛ ⎞ ⎛ ⎞ ⎤ + 12 ⎜⎜ ∂ϕ ⎟⎟ +⎜⎜ ∂ϕ ⎟⎟ ⎥⎥ r ⎝ ∂θ ⎠ ⎝ ∂z ⎠ ⎦

(6.2)

where σ is the electric conductivity, and ϕ is the electric potential. The equation for electric potential distribution is expressed as: 1 ∂ ⎛ ∂ϕ ⎞ 1 ∂ ⎛ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞ rσ + + =0 σ σ (6.3) r ∂r ⎜⎝ ∂r ⎟⎠ r 2 ∂θ ⎜⎝ ∂θ ⎟⎠ ∂z ⎜⎝ ∂z ⎟⎠ The boundary condition is given as follows: At the top of the analytical region, i.e. surface of the electrode paste: ∂ϕ ∂T = α1 (T1 −T ) = 0;λ ∂z ∂z On the two vertical surfaces of the analytical region (cross section 1 and 4 in θ ∂ϕ ∂T = 0; =0 direction in Fig. 6.2): ∂θ ∂θ ∂ϕ ∂T Along the central line of the electrode: = 0; =0 ∂r ∂r The boundary on the outer cylinder is very complicated and divided into several parts which have different boundary conditions. The boundary condition at the brass clamp is: ∂ϕ 1 ∂T = α 2 (T2 −T ) σ = (ϕ −ϕ ) ; λ ∂r ∂r R 1 For other electrode surfaces upon the furnace roof: ∂ϕ ∂T = 0; λ = α T1 −T 1 ∂r ∂r For electrode surfaces under the furnace roof and upon the slag level: ∂ϕ ∂T = α 3 (T3 −T ) + α 4 (T4 −T ) + α 5 (T5 −T ) = 0; λ ∂r ∂r For the exterior surface of the electrode under the slag level: ∂ϕ σ = i ( z ) ˗ T = T6 ∂r For the bottom of the electrode in the slag: ∂ϕ σ = i (r ) ˗ T = T6 ∂z where ϕ1 is electric potential of contact clamp, set to be zero;T1 is ambient temperature; T2 is temperature of cooling water,T3 is gas temperature in the furnace; T4 is temperature of inner surface of the hearth and the charge in the furnace; T5 is temperature of the slag surface;T6 is temperature at the interface of the electrode and the slag, i.e. temperature of the arc; R is contact resistance between the electrode and clamp; α1 is comprehensive heat transfer coefficient

(

)

  Jiemin Zhou and Ping Zhou

between the electrode and the ambient; α 2 is comprehensive heat transfer coefficient between the electrode and the cooling water ; α 3 is convection heat transfer coefficient between the electrode and the furnace gas; α 4 is radiation heat transfer coefficient between the electrode and the furnace chamber ; α 5 is radiation heat transfer coefficient between the electrode and the slag ; i ( z ) is current density on the side surface of the electrode in the slag; i ( r ) is current density on the end surface of the electrode. The distributions of i ( z ) and i ( r ) are shown in reference (struskij, 1982). Obviously it is impossible to solve this problem by using analytical method, so numerical method is adopted. Considering the shape of the electrodes, the finite difference method is used to accelerate the calculation. The computational grids are shown in Fig. 6.2.

6.2.2 Simulation software The calculation software consists of main program, 13 subroutines and 2 external functions, and the plotting software includes 3 data processing programs, 6 drawing programs and 15 subroutines. The electric power distribution must be known in advance when to calculate the temperature field, because the Joule heat originated from electricity is the main heat source in the electrode during the baking. However, as the electrical conductivity of the electrode material varies greatly with temperature, the thermal conductivity is also a function of temperature, thus iteration has to be used in the calculation, and the precision of solution depends on the iteration times. Proper acceptable error must be selected to ensure a desirable precision of calculation result as well as a reasonable computation time. To solve linear equations, direct solution method is adopted in every circulation of iteration. Coefficient matrix of the equation set is generally very big, from hundreds to thousands orders. In the development of the software, characteristics such as symmetry, positive definition as well as rarefaction of the coefficient matrix are fully employed, and the one-dimensional memory technics and the root-squaring method for linear equations set are adopted, to solve the large scale linear equations in a direct way at a relatively higher speed. To show the simulation result more illustratively, a plotting software was developed, by which various pictures can be conveniently drawn in the printer using FORTRAN language. The calculation result can be expressed as: a) The isopotential line and isothermal line of the vertical cut planes in the electrodes. b) The electric current field and heat flux field of the vertical cut planes in the

6 Simulation and Optimization of Electric Smelting Furnace 

electrodes. c) The stereo distribution graph of the electric potential, temperature and electric power of the vertical cut planes in the electrodes. d) The stereo distribution graph of the electric potential, temperature and electric power of the cross sections in the electrodes. e) The table of the heat balance of the electrodes. Besides, the needed information can also be printed out as tables. The simulation software can be applied in following situations: a) Optimal design of the electrode configuration. Some optional design schemes, such as different electrode diameter, different electrode shell material and thickness, different fin dimension and number, different brass clamp height and spacing, etc., can be calculated beforehand when the self-baking electrode for a new electric furnace is designed or the current electrode is improved. In this way, the electrode configuration corresponding to the most reasonable temperature distribution can be selected, and this configuration is the best design scheme. b) Selection of the most proper operation conditions. By importing various electrode operation parameters into the computer, such as electrode current, descent speed of the electrode, height of the paste surface, insert depth of the electrode into the slag, flow rate of the cooling water (or air), etc., the influences of the parameters on the temperature distribution during the baking can be examined, then the most reasonable operation condition can be obtained. c) The influence of the distance between the clamp and the furnace roof, the height of the furnace chamber, the nudity area of the melt slag, the variety (standard or inclosed) and property of the electrode paste and some other factors on the baking process of the electrode can be investigated, so that proper measures can be taken to improve the electrode baking condition, to strengthen the electrode intensity and to decrease the energy consumption.

6.2.3 Analysis of the computational result and the baking process Fig.6.3 to Fig. 6.7 show the simulation results of the self-baking electrode of the No. 2 furnace in Jinchuan Nonferrous Metal Company. Fig. 6.3 (a) is the electric potential distribution on the electrode vertical cut plane 1 (division of the vertical plane and cross section is shown in Fig. 6. 2), in which the electric potential is drawn as height of the plane. The electric potential above the clamp is almost constant, but it rises gradually from the lower margin of the clamp to the bottom of the electrode. The maximal electric potential drop (Vm) is about 3.5V.

  Jiemin Zhou and Ping Zhou

Fig. 6. 3 (b) is the three-dimensional temperature distribution on the electrode vertical plane 1. The acute temperature change at the slag surface and near the top of the furnace can be seen very clearly from this figure, and it can also be seen that the central temperature of the electrode is higher than the surface temperature in the furnace chamber. In this figure, the 300ć and 500ć isothermal lines and their position on the electrode have been drawn. Although the behavior of the paste varies with the composition, some assumptions can be made based on general conditions. That is, when temperature of the paste reaches 300ć, the state of the paste will change from fluid into plastic and the possibility of the paste flowing out of the cracks in the casing caused by arc is diminished greatly; when the temperature exceeds 500ć, the paste binder polymerizes and the risk of soft breakage is decreased. Even if the electrode breaks under the 500ć isothermal line, the electrode paste will not leak into the furnace, so such accidents can be easily dealt with (“Ferroalloy Production” composition group 1975). Under the present baking condition, the 300ć isothermal line of the No.2 electric furnace electrode is located between the brass clamp and the top of the furnace, while the 500ć isothermal line is near the top of the furnace.

Fig. 6. 3 Electric potential and temperature distribution on the vertical section of the self-baking electrode (a) Electric potential distribution;(b) Temperature distribution

Fig. 6.4 (a) shows the current field in the electrode plane section 1. It can be seen clearly the current’s magnitude and direction in the electrode. Because the electrode around the clamp has not been sintered yet, the electrode paste is almost electrically insulated. When current flows from the brass clamp into the electrode, it will all flow down along the electrode shell and fin. After the electrode entered the furnace, electric conductivity of the electrode paste will become stronger and stronger as the temperature increases. When the electrode shell is burned out, flow direction of the current will move towards the inner past of the electrode remarkably. Finally, the current will be distributed uniformly in the baked electrode.

6 Simulation and Optimization of Electric Smelting Furnace 

Fig. 6.4 Current field, heat flow field, equipotential line and isothermal line on the vertical cross section of self-baking electrode (a) Current field; (b) Heat flow field;(c) Equipotential line; (d) Isothermal line

Fig. 6.4 (b) is the heat flow field in the plane section 1. In this figure, it can be seen that heat is transferred from the lower side to the upper side of the electrode. Micro arc area is located on the interface of the electrode and the slag, where temperature is very high, and the heat transfers from the interface to the electrode. Above the slag surface, because temperature of the furnace chamber is much lower than that of the electrode, heat in the electrode is transferred by radiation acutely through the shell surface, whilst convection heat transfer also exists. The heat radiation intensity decreases with the increasing of the electrode position. Above the top of the furnace, temperature of the workshop is 500̚600

  Jiemin Zhou and Ping Zhou ć lower than that of the furnace chamber, and the temperature difference between the electrode and the ambient is also great, therefore, the heat loss from the electrode to the environment is remarkable. Near the brass clamp, heat flows from the electrode towards the clamp because there is a water cooling system here. Fig. 6.4 (c) is the isopotential line on the electrode plane section 1. The voltage drop on the interface of the clamp and the electrode is 0.6̚0.8V, which is about one fifth of the total voltage drop. The isopotential line near the top of the furnace has a relatively big slope, which means the electric conductivity of the electrode paste changes greater there. Fig. 6.4 (d) shows the isothermal line used to analyze the temperature distribution in the electrode. It can be seen that the isothermal lines in the electrode are very dense near the slag surface, which indicates an acute variation of temperature. The electrode isothermal lines are very sparse in the upper part of the furnace chamber, so the change of temperature is very smooth. Meanwhile, there is a relatively big temperature gradient near the top of the furnace because there is a dense distribution of isothermal line there. The electrode enters the clamp at a temperature of 60ć , and leaves the clamp at a temperature of 200ć . Its temperature is a little higher than 400ć when it is near the top of the furnace. Therefore, the electrode paste above the top of the furnace is generally in a state of fluid or plastic, and the paste will leak out if there is a hole on the electrode casing. Between the clamp and the top of the furnace, almost the whole weight of the electrode is born by the electrode casing and fins and all the current passes through the casing and fins, thus, an unsuitable operation or a bad welding of the electrode casing will easily result in a soft-broken accident. Fig. 6.5 shows the power density distribution of the electrode, whose four plane sections have evident difference from each other. For the electrode casing, at the clamp, there is a big contact electric resistance on the interface of the clamp and the casing, so the current’s power density is very big, and it increases with the descent of the electrode and the augmentation of the current (see plane 1 and 2). When the electrode leaves the clamp, its power density drops abruptly. However, with the slipping down of the electrode and the increasing of the temperature, the electric resistance of the casing increases, then the power density will increase continuously. With the increasing of the temperature, the paste starts to conduct electric current, the current in the casing decreases, the power density generally drops again. When the casing is burned out, the difference of power density in the electrode will be very small. Plane 4 is a section in the gap between the two clamps but not contacting either of them, so there is no peak in the power density of the casing near clamp. Inside the electrode, the paste under 400 ć has a poor electric

6 Simulation and Optimization of Electric Smelting Furnace 

conductivity and low power density, but the ascent of the temperature results in an increasement of the electric conductivity as well as the power density. In the slag, a part of current flows from the lateral surface, which makes a decreasing power density. Plane section 1 locates on the interface of the fin and electrode paste. With the slipping down of electrode and the ascent of temperature, power density in the fin increases firstly, then decreases, while the power density in the electrode paste increases smoothly. Hence the distribution curve of power density is very complicated.

Fig. 6.5 Power density distribution on the vertical plane section of the self-baking electrode (a) Plane 1; (b) Plane 2; (c) Plane 4

Fig. 6.6 (a) is the stereo electric potential distribution on electrode cross section (division of the cross sections is shown in Fig. 6.2, the area of the sectors is one eighth of the circle). Distribution of the electric potential is uniform on the cross sections except for (cross section 10) near the furnace roof. Fig. 6.6 (b) presents the power density distribution on the cross section. In the upper portion of the electrode where the paste has a poor electric conductivity, power density in the casing is much higher than that in the inner electrode, and the power density in the interior fin is also much higher than that in the electrode

  Jiemin Zhou and Ping Zhou

paste. Temperature distribution of the cross section is given in Fig. 6.6 (c). It can be seen that the central temperature in the electrode is higher than that on the surface above the slag (cross section 20).

Fig. 6. 6 Electric potential, power density and temperature distributions on cross section of self-baking electrode (a)Electric potential distribution on cross section 10; (b) Power density distribution on cross section 15;(c)Temperature distribution on cross section 20

Heat balance data of the electrode is given in Table 6.3. Calculation results indicate that most of the heat (more than 80%) is from the Joule heat of current. The heat from the melt is only a small portion, and the heat from the furnace gas is even less. The heat that consumes to bake the electrode is only 20% of all the energy, whilst most heat is lost through the surface of the electrode. Additionally, the heat that is carried away by the cooling water is also considerable.

6 Simulation and Optimization of Electric Smelting Furnace 

The plots and charts mentioned above give a lot of information. They illustrate the rules of electricity conduction, electricity-heat conversion, and heat transfer from various perspectives, including electricity and heat, potential field and flow field, microscopic and macroscopic scale, longitudinal and transverse direction, etc. This provides fundamental knowledge for the investigation and the analysis of electrode baking process. Table 6.3 Heat balance of the self-baking electrode of the electric smelting furnace Item

Heat income/kW

Joule heat from the electrode resistance

46.5

Joule heat from the contact resistance between the clamp and the electrode

10.1

Heat conducted from melt

7.9

Heat from hot gas

0.4

Heat expenditure/kW

Heat dissipated from the electrode surface

36.4

Heat consumed in baking electrode

12.1

Heat taken away by cooling water

16.5

Error

0.1

Total heat

64.9

64.9

The simulation results indicate that, for the self-baking electrode of nickel electric smelting furnace in Jinchuan Company under the configuration, dimension and operation condition introduced above, the electrode above the top of the furnace is in a serious shortage of heat, the electrode paste has little intensity and electric conductivity, and the electrode is in a danger of paste-leaking and soft-broken. As there is a big temperature gradient near the roof of the furnace, the too fast baking of the electrode will influence its baking quality, which is prone to result in a hard-broken of the electrode. In this case, the size of the casing and the fin should be chosen to ensure that they can bear the total current in the electrode and the weight of the electrode from the clamp to the bottom. It should also be done to make the welded seam to be smooth and regular to ensure a high welding quality. The way that the clamp is mounted should be improved as well to reduce the risk of paste-leaking and soft-broken of the electrode. Additionally, the baking condition should be reformed to obtain a smooth temperature gradient in the electrode to improve the electrode performance.

  Jiemin Zhou and Ping Zhou

6.2.4 Optimization of self-baking electrode configuration and operation regime In the following text, examinations are to be made on the influence of various parameters on the temperature distribution in the electrode through the analysis of simulation result to get some optimization schemes that aim for reducing the consumption of electrode and ensuring a safe operation of the electrode (Zhou, 1990; Zhou et al., 1990; Zhou et al., 1989). a) Influence of the electrode consumption (descent velocity) and current density on the baking process. When the electrode current density drops down to 1.25A/cm2, or even 0.2 A/cm2, the positions of the two isothermal lines (300ć and 500ć) change little, but central temperature of the electrode at the top of the furnace decreases. When current density increases up to 2.5 A/cm2, the 300ć isothermal line rises to the lower margin of the clamp, and it will continuously rise up to the intermediate section of the clamp when current density reaches 3.0 A/cm2 (Fig. 6.7 (a)). It has little influence on the temperature distribution when electrode consumption varies from 0.15̚0.6m/d. b) Influence of furnace temperature and electrode material physical properties on the temperature distribution in electrode. Fig.6.7(b) shows the temperature field when a lot of slag is exposed and the furnace temperature is 700ć. Under such a condition, the two isothermal lines rise a little bit. The isothermal lines will then fall down a bit when the slag surface is completely covered by the charge and the hearth temperature is 500ć. If the clamps are cooled by hot water or vaporization and the cooling medium is 100ć, there will be no evident temperature change on the electrode between the clamp and the top of the furnace. Previous researchers attempted to combine the baked electrode paste with the crude paste, then put them into the casing to improve the electric and thermal conductivity of the upper electrode paste˄Cavigli, 1978˅. However, simulation result shows that it is helpless to the electrode baking even if the electric conductivity of the upper paste increases to 1h104 S/m; but, when the paste’s thermal conductivity increases to 33.2 W/(mgK), the 300ć isothermal line rises up to the lower margin of the clamp. The similar effect can be obtained if the casing and the fin are made of metals with larger electric resistivity so as to double the resistance (Fig. 6.7 (c)). c) Influence of electrode configuration parameter and insulation measures on the upper electrode on temperature distribution in the electrode. It is found that joule heat will increase and the 300ć isothermal line will rise up to reach the inner of the clamp when the electrode diameter is 0.75m and the casing thickness is1.5mm (Fig. 6.7(d)). When the electrode diameter increases up to

6 Simulation and Optimization of Electric Smelting Furnace 

1.5m, the centre of the temperature curve at the top of the furnace forms a concave surface. If the fin is insulated and only the casing conducts electric current (Cavigli,1978), the temperature distribution in the electrode is similar to Fig. 6.7 (d). From the above analysis, it can be found that increasing the current density of the electrode, the electric resistance of the casing and the fin, as well as the thermal conduction of the paste will effectively make the 300ć isothermal line rise, whilst cause little changes for the 500ć isothermal line. The main reason for it is because a big temperature difference exists between the electrode above the furnace roof and the ambient, which causes great heat losses (Fig. 6.4 (b)). If a refractory fiber sleeve can be mounted on the outer surface (or on the interior

Fig. 6.7 Influence of various parameters on temperature field of electrode (Tm=1500ć) (a) Current density of electrode is 3Acm2;(b) Temperature in furnace chamber is 700ć; (c) Electric resistivity of electrode shell is doubled;(d)Electrode diameter is 0.75m; (e) Thermal insulation measures are taken between clamp and the top of the furnace; (f) Thermal insulation measures are taken, and electrode current is 25kA

  Jiemin Zhou and Ping Zhou

surface) of the electrode packing gland, to increase the thermal resistance between the electrode surface and the ambient, the heat dissipation through the electrode surface will be remarkably reduced, and the heat will be forced to flow upwards so that the baking condition of the electrode will be improved. The thermal insulation sleeve can be made of two semicircle rings for convenient mounting and uninstall. The temperature field in the electrode with this reform is shown in Fig.6.7(e). It can be seen that the 500ć isothermal line has been located at the lower margin of the clamp, and the danger of soft-broken of the electrode between furnace roof and clamp will be greatly reduced. In order to reduce the heat carried away by cooling water and to improve the baking condition, the vaporization cooling method can be considered. If the electrode can still bear a relatively big current as No. 3 furnace when it is insulated, the temperature curves on the electrode plane will be distributed as shown in Fig. 6.7 (f). Under this kind of electric regime, the 300ć isothermal line is approaching to the upper margin of the clamp, which decreases the flowing of the paste, and the 500ć isothermal line is fully in the clamp, so that the electrode has relatively high mechanical intensity when it leaves the clamp. The smooth and uniform ascent of temperature in the electrode is helpful to improve of the baking quality. However, because the slag around the electrode is exposed to gas, the temperature in the furnace chamber is very high, and the seal of the furnace is poor, the baked electrode will be quickly oxidized at conditions of high temperature and sufficient oxygen. Or even worse, as the electrode falls very slowly and stays for several days inside the furnace hearth, it will not only consumes more electrodes, but also results in the hard-broken of the electrode. Therefore, more modification of the furnace should aim for the fully covering of slag with furnace charge, decreasing furnace temperature, improving the air lightness of the electrode hole or painting antioxidation coating on the electrode surface. In this situation, improving the baking of the electrode above the furnace roof may remarkably enhance the electrode intensity, reduce accidents of electrode and decrease the consumption of energy and material.

6.3 Modeling of Bath Flow in Electric Smelting Furnace The bath movement in the electric smelting furnace has a great influence on the smelting process. So many scholars in the world have carried out a lot of researches into the movement rules of the bath. Giomidovskij systematically observed and analyzed the movement of the molten slag in the electric smelting furnace by substantial model (Giomidovskij, 1959). Moreover, he studied the quantitative relationship between the flow path and the heat transfer in the furnace, based on the simplified flow path of the molten slag. Although these empirical formulas have taken some guide roles in

6 Simulation and Optimization of Electric Smelting Furnace 

practices, they could not be used to understand the movement of the molten slag in the furnace. In 1980s, Jia Di et al. established mathematical models of steady state electric field, temperature field and velocity field for the single electrode smelting furnace, taking account of the coupling influence of velocity field on the temperature field. These researches became the base of numerical simulation on the velocity field in the furnace. Here, the electric cleaning furnace, which was a type of the electric smelting furnace, was taken as an example, and the numerical simulation for the velocity field in the furnace was introduced. The object chosen for analysis was the cleaning electric furnace that was used to tackle the slag in Noranda’s copper smelting furnace. Its rated electrical power is 6,300kV • A and its frequency of the alternating current is 50Hz. Three cylindrical carbon electrodes with 1,000mm diameter were fixed in the furnace with 9,200mm diameter.

6.3.1 Mathematical model for velocity field of bath The controlling equations and boundary conditions to solve the bath velocity field in aluminum cell are introduced as follows. 6.3.1.1 Basic equations

The following assumptions were used: a)The electrodes were in a stable work condition; b)The turbulent viscosity of the molten slag was isotropic; c)The influences of the molten slag dumping into the furnace on the fluid flow and of magnetic field on the fluid fluctuating were neglected; d)The matte was in a relative stationary state. The equations for the molten slag flow in furnace are expressed by: ∇ • (ρ U ) = 0 (6.4) ∇ • (U × U ) = F −

1

ρ

∇p +

1

ρ

μeff [∇U + (∇U )T ]

(6.5)

where variables are defined the same as in Section 2.3.1. Generally, the bath velocity in the electric smelting furnace was small, so the turbulent model with low Reynolds number was suitable for the simulation. In the computation of bath velocity field in the cleaning electric furnace, the k-ε two-equation model with low Reynolds number supposed by Launder (Launder, 1974) was adopted. There is: μt = cμ f μ ρ

k2 ε

(6.6)

  Jiemin Zhou and Ping Zhou

The k-ε equation were modified by: ⎡⎛

⎤

∇ • ( ρUk ) = ∇ • ⎢ ⎜ μ + μt ⎟∇k ⎥ + G − ρε − D σ k ⎠ ⎦⎥ ⎣⎢ ⎝

(6.7)

⎡⎛ ⎤ ε ⎞ ∇ • ( ρU ε ) = ∇ • ⎢ ⎜ μ + μt ⎟∇ε ⎥ + (C 1 G − C2 f 2 ρε ) + E σε ⎠ ⎥⎦ k ⎢⎣ ⎝

(6.8)

⎞

G = ( μ + μt )∇U [∇U + (∇U )T ]

(6.9)

where f μ , f2, D and E are respectively defined by

f 2 = 1 − 0.3exp(− ReT2 )

E=2

μμT (∇∇U ) 2 ρ

ReT =

ρk 2 με

The values of the empirical constants refer to Table 2.5. 6.3.1.2 Boundary conditions

The computational region is the molten slag layer in the electric cleaning furnace, which is shown in Fig.6.8. The boundary conditions are listed as following: a) The top surface (W2) in molten slag was taken as free surface. i.e. ∂U ∂V ∂k ∂ε = =W = = =0 ∂Z ∂Z ∂Z ∂Z b) The velocity of molten slag at the solid wall surfaces (W1) was zero. i.e. U=V=W=0 c) The velocity of molten slag at the electrode surfaces (W4) was assumed to be zero too. d) The velocity of matte at the bottom boundary (W3) was motionless. i.e. U=V=W=0 e) k and ε near the wall were treated as wall function (Fig. 6.8).

6.3.2 The forces acting on molten slag The flow pattern of the molten slag is complex as it is affected by a number of forces simultaneously. By analyzing the physical and chemical process occurring in the electric cleaning furnace, it can be concluded that four forces acting

6 Simulation and Optimization of Electric Smelting Furnace 

mutually on molten slag are electromagnetic force, buoyancy force, gravitational force and drag force caused by bubbles. The bubbles in the furnace mainly result from reduction reaction. As the quantity of reactants is 3%∼4% of molten slag and most reactions are carried out at the top layer of the bath, it is reasonable to neglect the drag forces of bubble in this model.

Fig. 6.8

Schematic of Analytical Zone and its Corresponding Cross Section of the Electric Field (a)Transect; (b)Vertical section of plane a-a.

6.3.2.1 Computation of electromagnetic force

The electromagnetic forces acting on the molten slag with high electric conductivity take the form: F=ρE+jhB (6.10) where F is the electromagnetic force per volume; E is the electric field intensity vector; ρ is the residual charge density; j is the current density vector; B is the magnetic induced intensity vector; Although there are a lot of positive and negative ions and free charges in the electric cleaning furnace, the whole number of positive charges balances with that of negative charges, that is ρ = 0. Then Eq. 6.10 can be written as: F=jhB (6.11) The magnetic field is induced from the electrode current and distributing current in the slag. The latter has only an effect on compressing the molten slag. As, it is much smaller than the former. It can be neglected in computation of the electromagnetic field. At time t, the electromagnetic force induced by electrode A at space point D (x, y, z) is given by: FAxt = j z ( x, y, z,t ) × B Ay ( x, y, z,t ) (6.12)

FAyt = j z ( x, y, z,t ) × B Ax ( x, y, z,t )

(6.13)

  Jiemin Zhou and Ping Zhou

FAzt = − j y ( x, y, z,t ) × B Ax ( x, y, z,t ) + j z ( x, y, z,t ) × B Ay ( x, y, z,t )

(6.14)

where FAxt , FAyt and FAzt refer to electromagnetic force components in the three coordinate directions respectively. BAx(x,y,z,t) and BAy(x,y,z,t) are the magnetic induced intensity components in x, y directions, respectively (their magnitude and direction are determined by Biot-Sevart law(Wang and Mei,1996)). Similarly, j x ( x, y, z,t ) , j y ( x, y, z,t ) and j z ( x, y, z,t ) are current densities, which are given by the results of the numerical simulation of the electric field. During a period, the magnetic forces from electrode A are given by: 1 (k +1)T FAi = ∫ FAit dt (6.15) T kT where i is taken as x, y and z, respectively. T is the periodic time and k is an integer. Similarly, at space point D, the magnetic field and electromagnetic force produced by the electrode B and C can be computed. At an arbitrary point in the furnace, the electromagnetic force acting on the molten slag is the sum of that produced by the electrode A, B and C. 6.3.2.2 Buoyancy forces and gravitational forces

As mentioned above, the buoyancy forces will be generated by temperature gradients in the furnace. The joint force including the buoyancy and gravitational force is described as follows: F ′ = β • ΔT • ρ • g (6.16) where β is the expansion factor of the molten slag; ΔT is the temperature difference between local and down nodes; g is the gravitational acceleration.

6.3.3 Solution algorithms and characters The method of the staggered grid and the control volume was used to establish the discrete equations and SIMPLE method was applied. Compared with a general SIMPLE algorithm, the program has the following characters: a) A cylindrical coordinate system and uneven grids were used. b) A program for TDMA algorithm to the non-tridiagonal matrix was developed. c) The pressure fields on the horizontal section were revised with that on the vertical section. Fig. 6.9~ Fig. 6.13 show the computational results of the velocity fields in the typical sections. It can be found that, to the top layer (section W2) of the molten slag, two eddies are formed near each electrode due to the influence of electromagnetic forces. Compared with section W2, the velocity field in section d-d (including electrodes) has similar flow pattern, while the eddies have different sizes and directions. The reason for the differences is that the horizontal fluid flow

6 Simulation and Optimization of Electric Smelting Furnace 

is affected by the vertical flow in the bath depth that electrods were inserted. In section f-f (without including electrodes), a large circulation occurs in the lower level of the molten bath, as electromagnetic forces drive the fluid flowing. This type of the velocity field is advantageous to increasing collision among matte particles in melting period, but it is disadvantageous in cleaning period to settling matte particles and to forming an efficient gradient concentration field between the inlet and the outlet of the molten slag. These are disadvantageous to improving the recovery of valuable metals in the molten slag. In section a-a (including an electrode) three distinct circulations are produced. In section b-b (without including electrodes), four circulations are produced and two circulations are formed near each electrode.

Fig. 6.9 Velocity at cross section W2

  Fig. 6.10 Velocity at cross section d-d    Fig. 6.11 Velocity at cross section f - f

  Jiemin Zhou and Ping Zhou

Fig. 6.12 Velocity at cross section a-a

Fig. 6.13 Velocity at cross section b-b

The results mentioned above indicate that the fluid in longitudinal direction near electrode is of large velocity. The material that is dumped into the region near the electrode can be molten more quickly, which reduces the melting time of material, increases the time for cleaning material and accelerates chemical reaction in the furnace, which is helpful for cleaning the slag.

6.4 Heat Transfer in the Molten Pool and Temperature Field Model of the Electric Smelting Furnace Various technical and economic indices of the electric smelting furnace depend on the electricity-heat conversion process, heat transfer process and temperature distribution, which directly influence the smelting process. With the development of productivity and the enlarging of furnace volume, it becomes more and more important to understand the electricity and heat transfer in the smelting process, and much work has been done to study mathematically the electric and heat distribution in the electric smelting furnace. W. D. Heiss (Heiss,1978a,b,1980) published several papers from 1975 to 1979 to introduce his research. For the circular three-electrode furnace, he assumed that no furnace wall existed, and the electrodes were simplified to be conducting wires without diameter. Besides, physical parameters were set be constants. In this way, he got the analytic solution of the current density. For the rectangular six-electrode furnace, an electric field function was constructed and the distributions of electric field and power were also calculated under similar assumptions. Then he obtained some useful conclusions agreeing well with the real furnace. However, as he emphasized, such simplifications differed greatly from the real situation, so its application was limited, and it could be only used as a basis of other more advanced numerical models. J. H. Downing et al. (Kaiser and Douning, 1978; Douning et al.,1978,1980) carried out numerical calculations of the heat transfer process at the bottom of the circular electric smelting furnace with the help of computer in 1977. Their data from the two-dimensional unsteady heat transfer model was then used as criterion for design of the furnace hearth, and also used to predict the damage of the hearth

6 Simulation and Optimization of Electric Smelting Furnace 

and to determine the maintenance period. In 1978, they built a electric and thermal analytical model of the furnace. They divided the circular three-electrode furnace into six symmetric parts as analytical domain. Although the convection problem of the slag could not be solved yet, and some physical parameters could not be determined precisely either, the model calculated the electric and the thermal fields in the circular furnace successfully, and predicted the furnace operation character. Furthermore, it gave reasonable interpretation to some experiential formula obtained by former researchers. In the following two years, J. H. Downing et al. used this model to investigate the influence of various design and operation parameter on the production process, and the result agreed with the practical data very well. For the circular single-electrode furnace, Esko Juuso (Esko, 1986) built a similar model, but the study of A. Jardy et al. (Jardy et al., 1986) was more unique. They solved the steady two-dimensional Maxwell, Navier-Stokes and Fourier equations set, and analyzed the electric, flow and temperature fields in the single-electrode furnace to make the calculation more perfect. There are still no very good models reported for the three-electrode and six-electrode electric smelting furnaces. This section is to systematically introduce the research of heat and temperature distribution in the molten pool of rectangular six-electrode nonferrous metal sulfide smelting furnace. Later in 1998, Y. Y. Sheng et al. (Sheng et al.,1998a, b) investigated this kind of furnace, adopting a similar method to that introduced in this book, to study the electric field of the molten pool. They studied the electric field, temperature field and flow velocity field simultaneously, which furthered the research on this subject.

6.4.1 Mathematical model of the temperature field in the molten pool In the rectangular copper and nickel sulfide smelting furnace, the six electrodes are mounted in a line, its configuration sketch is shown in Fig. 6.14. As a typical example, the No. 2 furnace in Jinchuan Nonferrous Metal Company has a power of 6500 kV噝A, a dimension of 21.5mh5.5mh4m, and a electrode diameter of 1m (JNMC, 1981). The composition of some charge (mainly sulfide calcine) and products (nickel matte and slag) of the furnace is given in Table 6.4. Table 6.4

Composition of charge, nickel matte and slag

(%)

Component

Ni

Cu

Co

S

Fe

MgO2

SiO2

Ca

Charge

5~6

2.5~3.0

0.15~0.25

15~18

27~32

10~14

13~19

1~2

Nickel matte

12~15

6~8

0.4~0.6

25~28

46~50

Slag

İ0.2

0.1~0.2

0.8~1.0

İ18

40~42

1.5~2.5

  Jiemin Zhou and Ping Zhou

Fig. 6.14 Configuration sketch of rectangular six-electrode electric smelting furnace (a) Longitudinal cross section; (b) Lateral cross section

Some physical properties of the slag, copper and nickel matte are listed in Table 6.5 and Table 6.6. Table 6.5 Physical properties of slag and copper matte (Kyllo and Richards, 1998) Physical properties

Slag

Copper matte

Density/kggm−3

3297+128.5[FeO]s−0.115Ts[SiO2]s

3880+404[Cu]M+4590[Cu]2M −3750[Cu]3M

Viscosity/Pags

⎡ 9380 ⎛ [Fe]s ⎞ ⎤ 2.06 × 10−3 exp ⎢ − 0.714 × ⎜ ⎟⎥ ⎝ [SiO2 ]s ⎠ ⎦⎥ ⎣⎢ Ts

⎛ 5000 ⎞ 3.36 ×10−4 exp ⎜ ⎟ ⎝ TM ⎠

Thermal conductivity 2.09 /Wgm−1gK−1

1.34−17.9[Cu]M+25.6[Cu]2M −13.7[Cu]3M

Note: The subscript S in the above table represents the slag phase; M represents the copper matte phase; the value in [ ] represents the component mass percentage; and T is temperature, K.

6 Simulation and Optimization of Electric Smelting Furnace  Table 6.6

Some properties of slag and nickel matte (1250ć)˄Kaiser and Downing 1978˅

Density Specific Kinematic Physical ρ /kg噝 heat c/J 噝 Viscosity viscosity μ /Pa 噝 s properties −3  v/m2噝s kg−1噝K−1 m 9.38h

Slag

3200

1250

0.3

10−5

Nickel matte

4500

720

0.05

10−5

1.11h

Thermal conductivity  λ /W 噝 m 噝  K

Thermal Electric diffusivity conductivity −1 a/m2噝s−1 σ /S噝m

8.0

2.0h10−6 5.25h

17.0

10−6

30.0 93.0

Some simplifications are made in order to construct a physical model that can be manipulated by mathematical method. These include: the electric and thermal fields of the furnace do not change with time; slag surface and matte surface stay steadily; and all the electrodes inset to the same depth into the slag. As the rectangular coordinates are used, the circular electrode can be simplified to be square electrode with the same conductive area (this will not result in a big calculation error). The influence of alternating magnetic field on electric field is neglected (Bochmann et al.,1968). With the consideration of the symmetry of the electricity and thermal distributions as well as their influence on the smelting process, the analytical zone is selected to be either side of the lateral cross sections of the furnace (Fig. 6.14), and the upper boundary is the slag surface. Under the rectangular coordinates, the differential equation of the heat transfer in the furnace is: ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ (6.17) ⎟+ ⎜λ ⎜λ ⎟+ ⎜λ ⎟+P = 0 ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ where T is temperature; λ is thermal conductivity; P is Joule heat, and P=0 in the lining. 2 2 2 ⎡ ⎤ ⎛ ∂ϕ ⎞ ⎛ ∂ϕ ⎞ ⎛ ∂ϕ ⎞ ⎥ ⎢ P=σ ⎜ + + ⎜ ⎟ ⎢⎝ ∂x ⎟⎠ ⎝ ∂y ⎠ ⎜⎝ ∂z ⎟⎠ ⎥ ⎣ ⎦

2

(6.18)

where σ is electric conductivity; ϕ is electric potential. The boundary conditions are as follows. On the central cross section which divides the furnace symmetrically along its width (y direction), there is no normal heat flow. The heat convection and radiation exist between the furnace lateral wall, as well as the exterior bottom surface, and the environment. Temperatures of the electrode, furnace chamber, cooling water and furnace raw charge are known. Other boundary surfaces of the computational domain are adiabatic.

  Jiemin Zhou and Ping Zhou

For the moving molten slag, its thermal conductivity can be expressed as: λ = λl + λt where λl is the thermal conductivity when the slag is immobile, which represents the heat transfer originated from diffusion and collision of molecules. λt is the turbulent thermal conductivity when the slag is moving, which represents the heat transfer originated from micelle mixing and spiral vortex. According to the similarity between heat transfer and momentum transfer when the fluid is flowing (Mei, 1987; Zhou, 1991), it can be assumed that Prandtl number is approximately a constant and irrelevant to the flow pattern of the continuous medium, that is: PrĬPrt vl vt ≈ al at

c p μl

Therefore,

≈

c p μt

λl λt λt μ t ≈ λl μ l

where Pr is Prandtl number; Prt is turbulent Prandtl number; v1,vt are molecular and turbulent kinematic viscosity respectively; al , at are molecular and turbulent thermal diffusivity respectively; μl , μ t are molecular and turbulent dynamic viscosity respectively; cp is specific heat. In the research of turbulent flow, there are many ways to evaluate μt , with which an approximate value of¬t can then be obtained. The electric conduction equation is: ∂ ⎛ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞ (6.19) ⎟ + ⎜σ ⎜σ ⎟ + ⎜σ ⎟=0 ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ Boundary conditions are as follows. The electric potential of electrode is known. The normal components of current at the central vertical surface, at the interface of slag and furnace wall, and at the interfaces of slag and charge as well as furnace gas are all zero. Other boundaries are zero-potential surfaces. Because the six electrods are connected to three transformers and the voltages of the two electrodes in the computational domain are not in the same phase, the voltage should be decomposed into a horizontal component and a vertical component firstly, then they will be combined at the end of the calculation in a geometric way. Finite difference method is adopted to solve the problem, and the calculation zone is divided into two parts, using different grids in the analysis of electric and thermal field respectively.

6 Simulation and Optimization of Electric Smelting Furnace 

6.4.2 Simulation software The electric and thermal field simulation software of electric smelting furnace consists of electric field calculation component, temperature field calculation component and drawing component. The circular iteration method is adopted to solve such problems as electric conductivity of slag and thermal conductivity of various refractory and insulation materials varying with temperature. The unidimensional compression memory technology and root-squaring method for linear equations are used to solve large scale differential equation set. The calculation results can be presented as isopotential lines, isothermal lines, electric current field, heat flux field on the cross sections and longitudinal section of the furnace, and can also be plotted as stereo distribution charts of electric potential, temperature and electric power. The heat balance table, the star current, delta current and total current of the electrode, the electrode -to-ground resistance, the electric power, the slag average temperature, the nickel matte average temperature, the charge consumption per day, and the electricity consumption of calcine etc. can also be printed out. Sufficient useful information can be obtained from the simulation.

6.4.3 Calculation results and verification Simulation is implemented based on the No.2 furnace in Jinchuan Nonferrous Metal Company, for conditions when the secondary voltage is 400V, electrode insertion depth is 0.4m, height of nickel matte is 0.75m, slag height is 2.2m, slag electric conductivity is 30 S/m (Si/Fe ratio in the slag is 1.2, and content of magnesia is 10.6%), and the distance between the charge and the electrode is 0.4m. Parts of the calculation results are shown in Fig. 6.15 to Fig. 6.22. Fig. 6. 15 is the current distribution on the vertical cross section passing the electrode (sketch of this cross section is shown in Fig. 6.14 (b)), which presents the relative magnitude and direction of the current. Fig. 6.16 is the isopotential line on this cross section. It can be seen that the potential gradient is big around the electrode, but drops smoothly in the region that is far away from the electrode. Fig. 6.17 is the stereo potential distribution on this cross section, which shows the variation of electric potential on the whole section more visually. Fig. 6.18 is the heat flux on the cross section. The heat mainly comes from electricity near the electrode, and most of the heat flows towards the charge, while some dissipates through slag surface, cooling water and furnace wall.

  Jiemin Zhou and Ping Zhou

Fig. 6.15 Current distribution on the cross section of furnace

Fig. 6.16 Equipotential lines on the vertical cross section

Fig. 6.17 Electric potential distribution on the cross section of furnace (9max=200V)

6 Simulation and Optimization of Electric Smelting Furnace 

Fig. 6.18 Heat flux distribution on the transversal section of furnace (Vector module of heat flux: 1cm represents 200kWm2)

Fig. 6.19 is the isothermal line, which illustrates the variation of temperatures in the molten pool and lining clearly. By drawing the isothermal lines of the slag and nickel matte melting points, it can also be examined whether or not the slag is frozen at furnace wall, or the matte is frozen at the furnace bottom. Fig. 6.20 is the distribution of the electric power on the cross sections of the furnace. It is obvious that the power liberated near the electrode is much larger than that in other regions in the pool, and the power density at the centre of the molten pool is also larger than that near the lateral wall. Besides, as cross section 1 is a central cross section between two electrodes which connect to the same transformer, while cross section 2 is a central section between two electrodes connected to different transformers, their distributions of power density are different, which is a character of six-electrode furnace. Fig. 6.21 is the current distribution on the longitudinal section near No.2 electrode (between cross section 1 and 2). Because its voltages and phases to the left and right neighbor electrodes are different, the current distributions on the two sides are also different. So there are two different delta currents in the later calculation. This problem has not been mentioned in previous literatures. Fig. 6.22 is the distribution of stereo electric potential and power density on the longitudinal section. It can be seen that the potential distributions are unsymmetrical on the two sides of the electrode. And it can also be found that the high power density is located in the region near the bottom of the electrode.

  Jiemin Zhou and Ping Zhou

Fig. 6.19 Isothermal lines on the cross section of furnace

Fig. 6.20 Power density distribution on the cross section of furnace (a) Cross section (passing the electrode center), PDmax=466.4kWm3; (b) Cross section 1, PDmax=46.6kWm3; (c) Cross section 2, PDmax=21.4kWm3.

The heat balance of the computational do main that is below the slag surface is shown in Table 6.7. In this calculation, apparent heat of the calcine, chemical reaction heat, heat taken away by gas and heat losses through the furnace wall above the slag are all represented as the heat that is needed to melt calcine and the heat radiated from the slag surface.

6 Simulation and Optimization of Electric Smelting Furnace 

Fig. 6.21 Current distribution on the longitudinal section of furnace

Fig. 6.22 Electric potential and power density distribution on longitudinal cross section of furnace (a) Electric potential distribution, Vmax=200V; (b) Power density distribution, PDmax=436.8kWgm-3

Table 6.7 Heat balance of the electric smelting furnace

Item

Heat income/kW %

Heat supplied by electricity

12369.5

Heat expenditure/kW

%

8851.0

69.1

100

Heat used to melt calcine Heat radiated from slag surface

3309.3

26.80

Heat transferred from electrode

5.9

0.05

Heat taken away by cooling water

229.9

1.9

Heat dissipated from furnace wall

303.2

2.5

(Bottom of furnace)

(117.9)

(1.0)

(Furnace wall)

(185.4)

(1.5)

−29.3

−0.2

12369.5

100

Error Total heat

12369.5

Other calculation results are:

100

  Jiemin Zhou and Ping Zhou Current in electrode/kA

10.3

Star current/kA

8.7

Delta current (between the two electrodes of the same transformer)/kA

1.3

Delta current (between the two electrodes of different transformers)/kA

0.6

Resistance (electrode to ground) /m Ω

19.4

Average temperature of slag/ć

1354

Average temperature of nickel matte/ć

1169

Charge consumption in every shift/t

171 −1

Electricity consumption per ton calcine/kW噝h噝t

578.6

To verify the calculation result, a water model test apparatus was made, of which the size is one fiftieth of the real furnace. KCl solution is used to simulate the slag, a bottom plane of graphite is used to simulate the nickel matte, and a wood board is used to simulate the charge. A double-sided compound copper plate and a bakelite electrode pasted with aluminium foil are used to make an instrument to measure the current densities in molten pool and electrode. The influences of pool depth, electrode size, electrode insertion depth and feeding mode on electric distribution in the liquid pool are measured. In this way, the electric characteristics of the rectangular six-electrode furnace are revealed (Zhou,1991). Temperatures from the simulation are verified with the real data taken from No.2 furnace in Jinchuan Nonferrous Metal Company. Calculation values of the electric and temperature fields agree well with the measured results.

6.4.4 Evaluation and optimization of the furnace design and operation Influences of the secondary voltage of furnace, the insertion depth of electrode, the way of feeding, the slag thickness, the width of molten pool, the spacing between electrodes, the electrode diameter, the lining thickness and the electric conductivity of slag on the furnace production capacity and electricity consumption are examined by using the electric-thermal analytical model. The quantitative relationship between the operation condition, design parameters of the nickel electric smelting furnace and the electricity and heat distributions of the furnace, as well as the main technical economic indices are established for the first time (Zhou et al., 1991a, b, 1993). It is then possible to further enhance the furnace productivity and to reduce the energy consumption by finding the optimum configuration parameters and operation technical conditions (manual or automatic optimization) according to the simulation of the software. For No. 2 furnace, there are such optimizations to be made, which include: improving of feeding way to make the charge cover the high-temperature slag near the electrode completely so that heat loss can be reducedˈmagnifying the

6 Simulation and Optimization of Electric Smelting Furnace 

dimension of furnace body or the depth of charge submerged in the slag to increase the heat exchange area between the molten slag and charge ˈ ensuring high temperature and velocity of the slag to improve the heat transfer intensity on the interface of molten slag and charge, creating the best temperature field to ensure a safe production and a long furnace body life while having a high-production and low-consumption. Referring to the production experience in this factory, some feasible schemes are also presented for the improvement of production condition of the furnace. According to the primary optimization that aims for the lowest energy consumption by using the analytical model, the following schemes can be chosen. These are, for smelting powder calcine, a uniform and continuous charge distributing is adopted, so a thin layer charge can cover the exposed slag surface near the electrode; the length and width of the molten pool is enlarged and then the area for melting charge can be increased. The specific configuration parameters and electric regime are as follows : Width of the molten pool/m

6.20

Spacing between the electrodes/m

3.30

Height of the furnace chamber/m

4.00

Power per square meter /kVgAgm−2

113

Secondary voltage/V

360

Insertion depth of electrode/m

0.57

Thickness of slag/m

1.45

Thickness of nickel matte/m

0.75

Thickness of charge/m

0.30

Under the production condition mentioned above, indices as following can be obtained when the electricity and the charge supply is sufficient: Electricity consumption per ton calcine/ kWghgt−1 Calcine melted per shift/t

502 245

Average temperature of slag/ć

1,334

Average temperature of nickel matte/ć

1,198

Compared with the production indices of No.2 furnace in 1987, the electricity consumption is 15% lower, and the charge melted per day increases 19%. The consumption of such materials as paste and casing are also reduced because the furnace chamber temperature decreases. This three-dimensional electric-thermal numerical model and software of the electric smelting furnace provide a new tool to reasonably choose electric regime and operation conditions of the smelting process, and to optimize the furnace configuration design. The model is more practical than that ever reported in previous research, and to some extends, the information obtained from it covers

  Jiemin Zhou and Ping Zhou

and even surpasses the previous investigations. This simulation software can be further improved to be a standard program for testing the design and operation of electric smelting furnace, and provides systematical data for online optimization control of the furnace (Warczok and Riverosa, 2007; Guo et al., 2002; Eric, 2004; Pan et al., 2006; Wang et al., 2006).

References Ambrosio P D et al (1981) Temperature and internal stress distribution of carbon electrodes used in an electric arc furnace for the production of silicon metal. High Temperatures-High Pressures, 13:307~311 Bochmann O et al (1968) Skin and proximity effects in electrodes of large smelting furnaces. In: 6th International Congress of Electro-heat, Brighto, 127 Cavigli M J (1978) Trend in technology of self-baking electrode. Electric Furnace Proceedings, 205̚208 Downing J H et al (1978) Mathematical model of an electric smelting furnace. Electric Furnace Proceedings, 209~216 Downing J H et al (1980) The mathematical model of a resistive electric smelting furnace. Proc. of INFACON, 83~108 Eric R H (2004) Slag properties and design issues pertinent to matte smelting electric furnaces. South African Institute of Mining and Metallurgy, Marshalltown, 2107, South Africa, 104(9) :499~510 Esko Juuso (1986) Computational analysis on temperature distribution and energy consumption in submerged arc furnace for the production of high carbon ferro-chrome. Translated by Yan Huijun. Iron Alloy, 32~47 “Ferroalloy Production” composition group. Ferroalloy production (in Chinese). Metallurgical Industry Press, Beijing, 157~168 Giomidovskij G L (1959) Nonferrous Metal Smelting Furnace. Translated by Smelting Furnace Education-Research Center of Central South Institute of Mining and Metallurgy. Metallurgical Industry Press, Beijing ,253~266 Guo D C, Gu L, Irons G A (2002) Developments in modelling of gas injection and slag foaming. Applied Mathematical Modelling, 26(2): 263~280 Heiss W D (1978a) Mathematical model of an electric smelting furnace ĉ. Elektrow噀rme International, 36(B2): 111~117 Heiss W D (1978b) Mathematical model of an electric smelting furnace Ċ. Elektrow噀rme International, 37(B6): 304~309 Heiss W D (1980) Modeling and simulation of electric smelting furnace. Electric Furnace Proceedings, 197~202 Heiss W D, Fields (1981) Power density and effective resistance in the electrode and furnace of an electric smelter. Elektrow噀rme International, 39(B5): 243~249

6 Simulation and Optimization of Electric Smelting Furnace  Innvaer R et al (1976) Temperature in SRderberg electrode in unsteady state conditions. In: 8th International Congress of Electro-heat. Liege : Id 6 Innvaer R et al (1980) Practical use of mathematical model for SRderberg electrodes. Electric Furnace Proceedings: 40~47 Innvaer R et al (1986) Three dimensional calculations on smelting electrodes. Electric Furnace Proceedings: 265~272 Jardy A et al (1986) Modeling of slag behavior in a nonferrous smelting electric furnace. In: The Reinhardt Rchuhmann International Symposium on Innovative Technology and Reactor Design in Extraction Metallurgy. Proc.of Conf. Colorado Springs, 419~431 Jinchuan Nonferrous Metal Company, Beijing General Institute of Nonferrous Metallurgy Design and Research (1981) Electric-furnace Smelting of Copper and Nickel Ore (in Chinese). Metallurgical Industry Press, Beijing, 5~8 Kaiser R H, Downing J H (1978) Heat transfer through the hearth of an electric smelting furnace and its impact on operations. Electric Furnace Proceedings, 119~126 Kyllo A K, Richards G G (1998) A kinetic model of the Peirce-Smith converter: Partĉ. Model formation and validation. Metallurgical and Materials Transactions, 29B: 239~248 Launder B E (1974) Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat and Mass Transfer 1: 131~138 Mei Chi (1987) Principle of Metallurgical Transfer Process (in Chinese). Press of Central South University of Technology, Changsha, 214~232 Olsen L et al (1972) Temperature distribution in SRderberg electrode. In: 7th International Congress of Electro-heat. Warsaw, 127 Olsen L et al (1976) Temperature dans les electrodes Soderbergen fonctionnement discontinue. In : 8th International Congress of Electro-heat. Liege : 1976 Pan Yuhua, Sun Shouyi, Jahanshahi et al (2006) Efficient and portable mathematical models for simulating heat transfer in electric furnaces for sulphide smelting. Minerals, Metals and Materials Society, Warrendale, PA 15086, United States, 10:599~613 Patankar S V (1984) Numerical Calculation of Heat Transfer and Fluid Flow. Metallurgical Industry Press, Beijing, 136~138 Sheng Y Y, Irons G A, Tisdale D G (1998a) Transport phenomena in electric smelting of nickel matte: Part ĉ . Electric potential distribution. Metallurgical and Materials Transactions, 29B: 77~83 Sheng Y Y, Irons G A, Tisdale D G (1998b) Transport phenomena in electric smelting of nickel matte : Part Ċ. Mathematical modeling. Metallurgical and Materials Transactions, 29B: 85~94 Struskij B M (1982) Electric Smelting Furnace. Translated by “Electric Smelting Furnace” Translation group. Metallurgical Industry Press, Beijing,62~66 Tao Wenquan. Numerical Heat Transfer (in Chinese). Xian Jiaotong University Press, Xian,420~431 Vatland A et al (1987) Data registration and calculation on SRderberg electrodes with

  Jiemin Zhou and Ping Zhou Modular holder. Electric Furnace Proceedings, 195~200 Wang Qianpu, Mei Chi (1996) Numerical simulation on the electric and temperature fields in bath of a round electric cleaning furnace. In: The International Conference on Modeling and Simulation in Metallurgical Engineering and Materials Science. Yu Zhongshen, et al eds. Metallurgical Industry Press, Beijing,428~485 Wang Yongning, Li Heming, Xu Boqiang, Sun Liling (2006) Simulation research of harmonies in electric system of arc furnace .Institute of Electrical and Electronics Engineers Inc., New York, NY 10016-5997, United States. 1: 902~906 Warczok A, Riverosa G (2007) Slag cleaning in crossed electric and magnetic fields. Minerals Engineering, 20(1): 34~43 Zhou Jiemin (1990) Mathematical Models and Computer Simulation of Nickel Metal Smelting Electric Furnace and its SRderberg Electrode: [Dissertation for Doctor Degree]. Central South University of Technology, Changsha Zhou Jiemin (1991) Experimental research on characteristics of rectangular six-electrode electric smelting furnace (in Chinese). Mining and Metallurgy Engineering, 11(2): 58~61 Zhou Jiemin et al (1991) Drawing software and its application in printer based on FORTRAN (in Chinese). Journal of Central South Institute of Mining and Metallurgy, 22(6): 682~686 Zhou Jiemin et al (1993) A quantitative analysis of the performance of a nickel matte smelter. In: Proceedings of the International Conference on Mining and Metallurgy of Complex Nickel Ores. Fu Chongyue et al. eds. International Academic Publishers, Beijing,440~443 Zhou Jiemin, Mei Chi, Zhao Tiancong (1989) Computational simulation on electric-heat distribution of self-baking electrode of 16500kVgA electric smelting furnace in Jinchuan Company (in Chinese). Nonferrous Metal (Part of Smelting), (6): 9~12 Zhou Jiemin, Mei Chi, Zhao Tiancong (1990) Mathematical model and computer simulation of SRderberg electrode in electric smelting furnace. Elektrow噀rme International, 48(B4): 210~215 Zhou Jiemin, Mei Chi, Zhao Tiancong (1993) Investigation on baking condition of SRderberg electrode (in Chinese). Journal of Central South Institute of Mining and Metallurgy, 24(1):58~63 Zhou Jiemin, Mei Chi, Zhao Tiancong (1991a) A mathematical model of electric smelting furnace used for improving design and operation. In: EPD congress ’91, Gaskell DR ed. West Lafayette: TMS, 321~330 Zhou Jiemin, Mei Chi, Zhao Tiancong (1991b) Computational simulation on electric-heat field in nickel smelting electric furnace. Journal of Central South Institute of Mining and Metallurgy, 22(1): 46~53


Coupling Simulation of Four-field in Flame Furnace Ping Zhou, Zhuo Chen and Kai Xie

A tower-type zinc distillation furnace and a copper flash smelting furnace are taken as the typical examples of single-phase combustion and two-phase combustion, respectively. The applications of the coupling multi-field numerical simulation are also described.

7.1 Introduction In metallurgical engineering, chemical engineering, mechanical manufacturing, electric power industries, etc., various industrial furnaces, in which fuels are the heat sources, are generally called the flame furnace. In these furnaces, flame is generally in direct contact with material. However, in some situations, flame has to be isolated from the material to prevent oxidation of charges or workpieces. In such cases, heat is transferred from the flame to the material via a partition wall. Flame furnaces are used extensively in roasting, drying, melting, smelting, heating and heat treatment of material or workpieces, since they have the following features (Lu, 1995): a) Flame furnaces are quite flexible to fuels. Various furnaces of different configurations can be constructed according to the types and specifications of the fuel, so as to meet the needs of the production processes. b) In general, the fuel supply of flame furnaces is sufficient and the cost is low. c) Flame furnaces have few limitations on the shape, size and specification of charges or workpieces. All materials, from big metal ingot in hundreds of tons to small particle, can be processed in flame furnaces.

  Ping Zhou, Zhuo Chen and Kai Xie d) Flame furnaces can be opevated at a wide range of femperatures. Given the same fuel, different furnaces can be operated at low, mild or high temperatures determined by the features and requirements of the production processes. e) Flame furnaces often use direct heating means, but are also able to adopt indirect means. Flame furnaces play an important role in various industrial processes. However, they consume a large amount of fuel, and are a major source of pollution to the atmosphere in China. Today, the major problems of flame furnaces in China are: a) Low energy-utilization ratio: 15%̚30% in reverberatory smelting furnace, 50%̚65% in rotary kiln and 55%̚70% in boiler. b) Serious atmospheric pollution. Among all pollutants to the atmosphere, 99% NOx, 99% CO, 91% SO2, 78% CO2, 60% dust and 43% hydrocarbons are produced by combustion of fossil fuels (Bao, 1997; Mao et al., 1998). c) Low automation level. To improve combustion efficiency, design new types of furnaces, optimize their operation parameters, prevent accidents, and enhance the equipment reliability and economy, It is imperative to thoroughly understand the velocity and temperature distributions of gas-flow in the furnaces, so as to develop reasonable calculation methods and design schemes, as well as establish optimal operation modes. In other words, thermal engineering theories in flame furnaces must be further studied. Thermal engineering theory in flame furnaces is an interdisciplinary subject. It correlates many subjects such as chemistry, fluid dynamics, combustion, heat and mass transfer, metallurgical principles, numerical simulation methods, measuring technique and the principles of furnace and kiln. What takes place inside a flame furnace involves complet gas-solid multi-phase flow with chemical reactions, non-isothermal conditions, and variable masses and diameters of particles. Historically, research of flame furnaces follows the “empirical analogy method → model experimental method → static simulation of single parameter → hologram simulation” method. These days most researchers adopt the isothermal model experiment or simulation of a single parameter to study the performance of flame furnaces. The designs of the furnaces, however, are still limited to using lumped or averaged parameters, and their operation controls are usually performed with the fixed-point monitoring method. Flame furnaces are complicated, multi-variable, nonlinear and strong coupling systems. The distributions of various parameters are highly complex. The velocity field, heat-releasing field, temperature field and concentration fieldü known as Āthe four-fieldā are correlated and mutually-restrained. Hence, methods that integrate the “three transfer principles and two reaction dynamics

7 Coupling Simulation of Four-field in Flame Furnace 

(i.e., the momentum, heat and mass transfer, and the dynamics of chemical and high-temperature reactions) must be employed, to develop mathematical models of the thermal and production process, and to carry out four-field hologram simulations in flame furnaces. Based on the simulation results, optimization experiments are performed to further exploit the equipment production potential, and to intensify the production process. With these methods, both construction and thermal processes of the entire equipment can be optimized to achieve high system productivity, good product quality, long equipment life, low energy consumption, and reduced pollutant emissions (Mei et al., 1999; Ou et al., 2007; Ma et al., 2006).

7.2 Simulation and Optimization of Combustion Chamber of Tower-Type Zinc Distillation Furnace Tower-type zinc distillation furnace is a key equipment in zinc pyrometallurgy, and one of the most complicated equipment in nonferrous metallurgical industry. Its equipment efficiency directly influences the refined zinc yield and energy consumption. In the tower-type zinc distillation furnace, coal-gas and air are supplied into the combustion chamber in a fractional mode, to achieve delayed burning and uniform temperature distribution, thus ensuring normal operation of tower trays. Recent innovations of the distillation furnace include the adoption of large tower trays, large furnace space and no column supporting structure. Although it has been proven that the innovations are helpful for higher productivity, some problems are encountered in practice. For example, the temperature in the upper combustion chamber was found higher than that in the lower chamber. The uneven temperature distribution, to some extent, has resulted in problems in operations. To rectify the problems, series of attempts has been made, such as increased flue draft, use of coal gas of higher heating value, and increased mass flux of the secondary coal gas. Although certain improvements were achieved in the operation of the furnaces, the problem of uneven temperature distribution remained ansolved. In order to achieve a uniform temperature distribution in the combustion chamber, in-situ measurements and numerical simulations were combined to study the velocity field, concentration field, heat release field and temperature field in the combustion chamber of different structures. Based on the measurements and simulation results, it is possible for the researchers to propose an optimal structure of the combustion chamber, so that higher productivity and longer equipment life can be achieved (Xie et al., 2003).

  Ping Zhou, Zhuo Chen and Kai Xie

7.2.1 Physical model As a large system, a tower-type zinc distillation furnace is made up of combustion chambers, a set of tower trays, a condenser, an air preheater, a coal gas (combustible gas produced from other industrial furnace) preheater, a flue and a chimney. Every part influences and imposes restraints on each other. Two combustion chambers are located on the east and west sides of the tower tray. Due to symmetry, the computational domain consists of, only the west combustion chamber, which includes air ducts and coal gas ducts away from preheaters, as well as a portion of the flue. For the computational domain, the south end is a wall dissipating heat into the environment; the north is adjacent to the preheaters and the ascending flue, and the east is the tower tray set. On the west wall, five levels of inlets are arranged top to bottom, in the order of primary coal gas, primary air, secondary air, secondary coal gas and tertiary air, A flue gas outlet is located on the bottom. There are four ports (nozzles) for each level of inlets and outlets (refer to Fig.7.1).

Fig.7.1 The sketch of the combustion chamber in a zinc distillation furnace

7 Coupling Simulation of Four-field in Flame Furnace 

Fresh air from the global air inlet is redirected into three horizontal channels, of which two streams flow up and down separately, and enter the combustion space through the primary and secondary air nozzles; the third flows down along the southern channel, then enters the combustion space through tertiary air nozzles. The coal gas from the global coal gas inlet is divided into two paths. One flows up and enters the combustion space through the primary coal gas nozzles. The other flows down into the northern channel and enters the combustion space through secondary coal gas nozzles. The coordinate system is shown in Fig.7.1. The origin is located at center of the combustion chamber.

7.2.2 Mathematical model In the tower-type zinc distillation furnace, coal gas is preheated to 600ć, air is preheated to 700ć, and the temperature of the gas mixture in the combustion chamber is over 1,000ćühigher than the ignition temperature. Hence in the combustion chamber, reactions take place at a rather high rate once the fuel comesin contact with air. Here, fuel and air are fed separately into the chamber, so there is no premixing prior to combustion. This means the reaction rates between the fuel and air are determined by their mixing and diffusion process, i.e., the gas-phase combustion is i.e. diffusion-controlled process. So, a mixed-is-burnt rapid chemical reaction model can be adopted. The definition and equations describing the mixed-is-burnt model can be found in Section 2.3.

7.2.3 Boundary conditions In order to determine the boundary conditions of the combustion chamber, some smelters carry out heat balance measurement for the tower-type zinc distillation furnace. The major parameters are shown in Table 7.1. Table 7.1 Inlet boundary conditions of coal gas and air TTemperature Flow rate Area Item /K /m3gh−1 /m2

Velocity Mass fraction / % /mgs−1 N2 O2 CO CO2

H2

Coal 873 gas

751

0.1666

4.004

0.78 0.6

3.74

Air

943

0.1123 8.311

0

0

973

58.28 0 76.8

29.34 7.26

23.2 0

0

CH4 H2O

0

According to the test results, the flue pressure at the outlet of the computational

  Ping Zhou, Zhuo Chen and Kai Xie domain is -83.4Pa. The heat fluxes through the south and north walls are identical 180W/m2. Because the north wall is adjacent to preheaters and the ascending flue, heat flux from the wall may be ignored. Therefore, adiabatic boundary is assumed on the north wall. The east wall of the tower tray set is of mixed boundary condition. The tower trays are always filled with liquid and gaseals zinc. Liquid zinc has high heat conductivity and latent heat of vaporization, whereby the temperature of liquid zinc in the tower tray set generally does not vary greatly. Therefore, the temperature T0 at the inner wall of the tower tray is assumed to be a constant and equal to the vaporization temperature of zinc, i. e., 920ć. The heat transfer from the combustion chamber to liquid zinc depends on the temperature in the combustion chamber and can be calculated by means of the following steps. The tower tray is made of silicon carbide (SiC) and refractory coating. The total thermal resistance of tower trays is given by: δ δ R= 1 + 2 (7.1) λ1 λ2 where δ1 and δ2 represent the thickness of the tower tray and refractory coating respectively; λ1 and λ2 are the thermal conductivity of the tray and coating respectively. Therefore, the relationship between the heat flux q through the tray wall and the temperature T outside the tray wall can be expressed as: 1 q = (T0 − T ) (7.2) R

7.2.4 Simulation of the combustion chamber prior to structure optimization The computational Combustion chamber simulation is performed prior to structure optimization. The computational grid is shown in Fig.7.2. Using the commercial CFD software CFX4, multi-field simulation of the combustion chamber is carried out. The standard k-ε turbulence model and the mixed-is-burnt gas-phase combustion model are used. The primary results are given in Fig.7.3 and Table 7.2.

Fig. 7.2 Grid on vertical section of combustion chamber

7 Coupling Simulation of Four-field in Flame Furnace 

Fig. 7.3 Temperature distribution on the vertical section of x=−0.68m Table 7.2 Predicted flow of air and coal gas nozzles prior to structure optimization Item

The ratio of nozzle flow to total flow /% No.1(south)

No. 2

No. 3

No. 4(north)

Σ

Primary coal gas

14.7

15.0

19.9

27.7

77.3

Secondary coal gas

3.4

5.3

7.0

7.0

22.7

Primary air

12.0

8.9

7.8

4.2

32.8

Secondary air

10.3

9.2

8.2

6.0

33.8

Tertiary air

9.3

10.4

8.1

5.6

33.4

Total coal gas

18.1

20.3

26.9

34.7

100

Total air

31.6

28.5

24.1

15.8

100

Fig.7.3 shows that the temperature distribution in the combustion chamber tends to be triangular, that is, high temperature flames exist near the four primary air nozzles and the maximum temperature is up to 2058.6K As the flue gas flows downward, the temperature greatly decreases while the temperature remains relatively high on top. In addition, the flames appear to extend into the flue. The triangular temperature distribution indicates that a portion of the trays operates at high loads while others at low loads, implying negative effects on yield and longevity of the trays.

  Ping Zhou, Zhuo Chen and Kai Xie Table 7.2 clearly reveals that the ineffective allocation of air and coal gas passages is the major reason that the triangular temperature distribution in the combustion chamber is formed. Furthermore, the measurement results of the tower-type zinc distillation furnace in operation indicate the temperature distribution outside the wall appears to have the same characteristics as the computational results. For the temperature outside the wall, the maximum deviation between simulation results and measured data is less than 3.5%.

7.2.5 Structure simulation and optimization of combustion chamber In order to reallocate the air and coal gas passages, the sizes of air and coal gas nozzles are adjusted and dozens of designs are obtained. For every design, multi-field coupling simulations of the combustion chamber are performed. By analyzing the simulation results in detail, the optimal design is identified where temperature is uniformly distributed in the combustion chamber. The simulation results of the optimal design are shown in Fig. 7.4.

Fig. 7.4 Temperature distribution on the vertical section of x=−0.68m after structure optimization

7 Coupling Simulation of Four-field in Flame Furnace 

Fig.7.4 shows that the uniformity of temperature distribution in the optimal combustion chamber has been greatly improved. The values of several parameters before and after optimization are listed in Table 7.3. From this table, it can be found that the optimization of the chamber structure not only improves the yield of refined zinc, but also prolongs the life of the trays. If the entire tower-type zinc distillation furnace is analyzed and simulated, further improvements can be achieved. Table 7.3 Calculated parameters before and after optimization Parameter name

Before

After

Difference

Average temperature outside the wall of tray/ć

1156.7

1191.9

+35.2

Temperature standard deviation outside the wall of tray/ć 136.4 Heat transferred from chamber to tray/W

7.141×10

Maximum temperature of gas phase/ć

1785.6

−31.9

104.5 5

5

7.563×10 +4.22×104 734.6

−51.0

7.3  Four-field Coupling Simulation and Intensification of Smelting in Reaction Shaft of Flash Furnace Flash smelting is one of the major modern copper pyrometallurgical techniques. Compared with traditional smelting techniques, flash smelting carry many advantages, including thorough use of the specific surface area of the powder concentrate, which greatly reduces energy consumption, promotes sulfur utilization and is friendly to the environment. Since the birth of the first Finland Outokumpu flash furnace in 1949 and operation of the Canada INCO flash furnace in 1953, the flash furnace has been rapidly developed, and the production of flash-smelted copper now accounts for nearly 50% in pyrometallurgy of the world. The flash furnace is not only an important smelting equipment, but is also replacing traditional P-S converter as a continuous blow-smelting equipment. There are two basic types of flash furnace: The Outokumpu type and the INCO type. For the former, concentrate is fed vertically into the furnace from the roof of the reaction shaft (refer to Fig.7.5) (Ren, 2001), and for the latter, concentrate is fed horizontally from the side of the furnace. In this book, flash smelting furnace refers to the Outokumpu type. In flash smelting powder concentrates, which has been deeply dehydrated (water content less than 0.3%), and oxygen-enriched air are rapidly jetted into the reaction shaft through a concentrate burner and mixed in the shaft. At high

  Ping Zhou, Zhuo Chen and Kai Xie temperatures, the suspended particles complete oxidation, desulfurization, melting and slagging reactions in a very short time. The melt then deposits on to the settler where further slagging occurs, the matte settled and separated from the slag.

Fig. 7.5 The sketch of Outokumpu flash furnace

In order to increase the output, reduce energy consumption and improve economic benefits, the “Four Highs” techniqueühigh concentrate loading, high matte grade, high oxygen enrichment of the process air, and high thermal intensityühas been broadly put into practice. However, to adopt the “Four Highs” technique, it is essential to ensure uniform mixing between the concentrate and oxygen, and an effective protection of furnace lining for high-load production. Major requirements ofĀ Four Highsāare as follows: a) Metallurgical chemical reactions are fully completed and no raw concentrate exists. b) No erosion on the linings of the reaction shaft occurs. c) The limited space in the reaction shaft is fully used. Therefore, research in some fundamental are as of the flash furnace is necessary in order to understand the smelting process and develop measures to improve copper recovery in the intensified smelting. Meanwhile, appropriate mathematical models are also needed to optimize both the operation and structure of the furnace in order to meet the requirement of intensive smelting process in the flash furnace (Chen et al., 2004; Li et al., 2003).

7 Coupling Simulation of Four-field in Flame Furnace 

7.3.1

Mechanism of flash smelting processüparticle fluctuating collision model

To understand the mechanism of the flash smelting process, particle samples are taken from a reaction shaft in operation and are analyzed. The following conclusions are obtained: a) The particle distribution in the reaction shaft is quite uneven, with most particles concentrated in the center of the reaction shaft. b) Particles are oxidized to different extents in the reaction shaft. Some react rapidly, and are over-oxidized where by magnetic iron oxide and metal copper or copper oxides were formed; some are deficiently oxidized or even nonreactive. The over-oxidized particles mostly appear in the outer region of the particle flow and consum excessive oxygen, while the under-oxidized and nonreactive particles are found in the center of the particle flow. c) Many free silica are found in the reaction shaft indicating that the slagging reactions are not fully completed and will continue in the settler. d) In microscopic observation of the slag, particles in the shape of strawberry (or honeycomb) are found. Some of the particles are found with pores, and some with oxidized layers. These information of the particles’ microstructure somewhat reveal that the particles split during the flash smelting process. e) The average diameter of particles increase approximately seven times as they flow along the reaction shaft, indicating that the particles collide and aggregate in the flash smelting process. With these results, it can be concluded that the splitting and aggregation of particles occur simultaneously in the smelting process in the reaction shaft. In fact, the splitting and aggregation of particles are not mutually exclusive; they are two phenomena occurring in different stages of the smelting process. Particle splitting is an inevitable result of quick ignition and combustion reactions of copper concentrate; it is also the prelude to particle aggregation. Particle aggregation then ascertains the continuation of oxidization and desulfurization reactions, and is an important process in ensuring oxygen be transferred between the solid and liquid phases. In gas-particle two-phase jet flow with intensive turbulence and high concentration of particles, violent reactions occur between the concentrate particles and oxygen, releasing large amount of gas. The interaction of particles and gas then induces particle collision and aggregation. When settling along the

  Ping Zhou, Zhuo Chen and Kai Xie reaction shaft, particles may exhibit five collision modes: a) Collision resulted from the velocity differences of particles. b) Collision induced by the intersections of gas-flow. c) Collision caused by the horizontal random motions of particles under the actions of Magnus force, Saffman force and counterforce induced by the gas released from chemical reactions. d) Mutual collision of particles and gas caused by fluctuations of gas flow and particles. e) Collision of particles, resulting in splitting and aggregation of particles. Here, the particle fluctuation collision (PFC) model is proposed with the following assumptions: a) The splitting of concentrate particles and collisions within the particle groups exist simultaneously at different heights of the reaction shaft. b) The fluctuation of concentrate is the main reason for the particles’ collision and aggregation. c) As oxidation progresses, molten sulfides and SO2 bubbles are formed in the center of a particle, while porous iron-oxides are formed as an outside cover of the particle. The gas phase forming in the molten core then causes the particles to split and collide. The higher the oxygen concentration and local temperature, the more intensive the process. d) Due to the different sizes, compositions, local oxygen concentration and local temperature, concentrate particles are oxidized to different extents. e) Along the reaction shaft, over-oxidized particles collide with each other or with under-oxidized particles, then aggregate to form larger particles. Meanwhile, the over-oxidized particles are reduced by the under-oxidized particles.

7.3.2 Physical model A model of a practical flash furnace including the reaction shaft and a portion of the settler (only the gas-phase region) is simulated for this research. The inner space of the reaction shaft is a cylinder with a diameter of 5m and a height of 6.640m. The settler is a rectangle with a length of 18.500m, a width of 6.700m, and height 1.590m (refer to Fig.7.6). The concentrate burner has the same structure as shown in Fig.7.7, and consists of four parts. The area of the inner ring is 0.0274922m2, the outer ring 0.0414478m2, the distributor 0.0029452m2, and the central oxygen pipe 0.0037285m2.

7 Coupling Simulation of Four-field in Flame Furnace 

Fig. 7.6 Computational grid of the reaction shaft

Fig. 7.7

7.3.3

The structure sketch and grid of the concentrate burner

Mathematical modelücoupling computation of particle and gas phases

The existence and motion of particles have influences on the flow field of the gas phase, while gas flow also affects the motion and the track of particles. Therefore, coupled computation of gas flow and particles is necessary to correctly describe their motion and behaviors. Here, gas phase is described by Euler’s equations and particle phase by Lagrange’s equations. The coupling between the gas and

  Ping Zhou, Zhuo Chen and Kai Xie particle phases is achieved by introducing the source terms of particle phase in each computational cells (PSIC, particle source in cell). When Lagrangian method is applied to predict the particles tracks, each track represents the behaviors of a particle group in the same diameter. If i denotes a particle group of diameter dpi, then the mass of particles moving along the track j is given by: M ijp = M 0p X j 0Yi 0 ˄7.3˅ where M 0p is the global mass flow rate of particles at the inlet; Xj0 is the mass fraction of the track j at the inlet ; Yi0 is the mass fraction of particle group i at the inlet. The number of particle group i moving along the track j is: M ijp nij = p ˄7.4˅ mij where mijp is the mass of a single particle. When particle group i moves through a computational cell along the track j, mass source term of the particle can be expressed as : [ΔM ijp ]kcell = nij e[(mijp )out − (mijp )in ]kcell ˄7.5˅

The subscripts Ā inā and Ā outā represent the time that the particle enters and leaves the computational cells, respectively. Total mass source term of particle in kth computational cell can be written as: ⎡ Σ Σ ΔM ijp ⎤ i j p ⎥ ( Sm ) kcell = ⎢ ˄7.6˅ ⎢ ΔV ⎥ ⎣ ⎦ kcell where ΔV refers to the volume of kth computational cell. Similarly, the momentum source term of particles in kth computational cell can be written by: ⎡ Σ Σ nij × [(uijp mijp )out − (uijp mijp )in ] ⎤ i j p ⎥ ( Su ) kcell = ⎢ ˄7.7˅ ⎢ ⎥ ΔV ⎣ ⎦ kcell p v kcell

(S )

p w kcell

(S )

⎡ Σ Σ nij × [(vijp mijp ) out − (vijp mijp )in ] ⎤ i j ⎥ =⎢ ⎢ ⎥ ΔV ⎣ ⎦ kcell

˄7.8˅

⎡ Σ Σ nij × [( wijp mijp )out − ( wijp mijp )in ] ⎤ i j ⎥ =⎢ ⎢ ⎥ ΔV ⎣ ⎦ kcell

˄7.9˅

Energy source term of particles includes the released or absorbed heat from their reactions, as well as the heat transferred by convection and radiation between the particle and gas phase. Because radiant heat is related to the time t when particles pass through a computational cell, energy source term can be written as:

7 Coupling Simulation of Four-field in Flame Furnace 

p h kcell

(S )

⎡ Σ Σ nij × [( hijp mijp − Qcp − tQrp )out − ( hijp mijp − Qcp − tQrp )in ] ⎤ i j ⎥ =⎢ ⎢ ⎥ ΔV ⎣ ⎦ kcell

˄7.10˅

It should be noted that the source term of the turbulence kinetic energy and that of the turbulence dissipation rate

(S ) p

ε

kcell

(S ) p

k

kcell

are often assumed to be zero.

7.3.4 Simulation results and discussion For the reaction shaft modeled in the simulation, the concentrate is fed at the rate of 93t/h, and heavy oil is supplied at the rate of 390kg/h. The major operating parameters of the reaction shaft are listed in Table 7.4. Table 7.4 The major operating parameters in reaction shaft Item

Flow rate Temperature /m3gh−1 /K

Mass concentration of Velocity of gas oxygen /% flow /mgs−1

Gas flow through inner ring 15,100

298

61.6

166.4

Gas flow through outer ring 3,233

298

61.6

23.6

The central oxygen

733

298

97.1

56.9

Gas flow for dispersion

1,233

313

23.3

180.2

Gas flow for combustion

3,066

298

39.2

8.8

Results of coupled stimulation of the reaction shaft are shown in Fig.7.8 and Fig. 7. 9. The streamlines of the gas phase are shown in Fig.7.8, from which the following conclusions can be drawn: a) Once jetted from the concentrate burner into the reaction shaft, the highspeed gas flows directly down and concentrates in the central region of the shaft. It then scatters upon reaching the slag surface in the settler, and rebounds when it hits the south, north and east walls of the settler, forming a series of small eddies. With the gas flowing to the uptake shaft, the energy of the eddies is enhanced and their sizes increase. As a result, two large eddies are ultimately formed. One is located in the east side of the reaction shaft, filling nearly all the eastern space. This large eddy then causes a portion of the flue to rise to the top of the reaction shaft, where it joins the high-speed gas jetted from the burner. The other large eddy is formed in the settler’s upper space, and is the confluence of small eddies from upstream of the settler. As it moves toward the uptake shaft, this eddy expands in size, keeping its direction unchanged.

  Ping Zhou, Zhuo Chen and Kai Xie

Fig. 7.8 Gas streamline

b) The streamlines cluster mainly in the central region of reaction shaft, indicating that the gas flow has very high speed. The streamlines in the west side of the reaction shaft is quite sparse, indicating a cower-speed gas flow in this region, called the “stagnant region”.

Fig. 7.9 Particle traces

Fig. 7.9 shows the particle traces and illustrates that the flow field of the gas phase plays a key role in the particle motion, especially the two eddies of in the gas phase mentioned above. Some particles move to the bottom of the reaction shaft, then rise to the top through the east space. The eddy downstream of the settler benefits from the separation of the particles, causing bigger particles to settle and smaller ones to enter the rising flue. When settling along the reaction shaft, the average speed of the particles is up to 25~30m/s, and up to 106m/s at the exit of concentrate burner. When approaching the molten slag surface, the particle speed rapidly declines to approximately 3m/s. Therefore, the residence time of particles in the shaft is very shortüless thanls, on average. Particles suspending in the large eddies have lower speed and thus longer

7 Coupling Simulation of Four-field in Flame Furnace 

residence time. In addition, the simulation results also include species distribution of the particle phase, such as CuFeS2 , Cu 5FeS 4 , Cu 2S, FeS 2, FeS, FeO, Fe 3O 4, SiO 2 , Fe2 SiO4 and C, and the gas phase, such as SO2 , O2 , CO2 and H2 O. The results provide a wealth of micro-information and distributions of 18 parameters such as the flow field, temperature field, releasing heat field and concentration field.

7.3.5 Enhancement of smelting intensity in flash furnace As the flash smelting process becomes more and more intensive, technical problems are confronted in practice. In particular, operation difficulties are considered to be related to the performance of the concentrate burner. Simulation is then carried out to study the influences of burner structure and operating parameters on the smelting process. 7.3.5.1 Problems of the concentrate burner faced in intensified smelting

The central jet distributor is a type of jet burner, where the central oxygen pipe, concentrate loading tube, inner ring and outer ring are all coaxial. Characteristics of the jet burner in a clude simple structure and small flow resistance. However, since the concentrate and gas are injected in parallel, the mixing is slow and the flame is long. Therefore, a generous space is needed to ensure complete reactions of the concentrate. As a result, why the height of a typical flash furnace is generally designed to be over 7m. When the process air flow, loading amount, and, consequently, copper output increase, the distribution air in the central. Jet distributor design is ineffective, and the mixing between the concentrate and gas becomes move uneven. As the process air increases, the residence time of concentrate particles in the reaction shaft greatly decreases, which results in incomplete reaction and existence of raw concentrate in the reaction shaft. To solve the problem of uneven mixing between the concentrate and gas in the central jet distributor, particulary in the burner design from Finland’s Outokumpu®, a horizontal distributor was added to enhance the horizontal movement of particles and their mixing with the gas. The effect of the distribution air is characterized by the ratio of its horizontal momentum to the vertical momentum of the mixture of the process air and concentrate particles. To increase copper output and intensify the flash smelting process, it is necessary to improve the burner design: to achieve uniform mixing and complete reactions between the concentrate and oxygen; to enhance collision among particles and

  Ping Zhou, Zhuo Chen and Kai Xie

reduction reactions; and to constrain the shaft lining erosion caused by high temperature particle jet-flow. Intensifying flash smelting requires that the smelting reactions be carefully controlled in a limited space, that is, in the highly efficient reaction region. This region has the characteristics of high temperature, high oxygen concentration and high particle density, and is also referred to as the “three centralizations” concept (Mei et al., 2003): a) High temperature centralization. High temperature is the essential condition for particle ignition, melting, collision and reduction reactions. Generally, complete combustion reactions for concentrate particles should occur 0.5~3.0m away from the burner outlet, where a high temperature centralization region is formed. In this region, high temperature speeds up oxidation-reduction, melting and slagging reactions. With further intensification of smelting, this highly efficient region becomes more crucial, and is very helpful to achieve self-heated smelting process. b) Relative oxygen centralization. The controlling factors of the reaction rate of particle at high temperatures are the oxygen concentration and the rate of oxygen diffusion into the particle core. Increasing the oxygen concentration and relative oxygen centralization are the pre-requisite to forming the high temperature center in the reaction shaft. c) Relative particle centralization. Relative particle centralization provides higher probabilities for collision and reactions among particles. Moreover, it can reduce shaft lining erosion caused by particles, thus prolonging the lining life. The concept of relative partide centralization is consistent with that of relative oxygen centralization. 7.3.5.2 Research of the concentrate burner in intensified smelting (Mei et al., 2003)

Using the current flash furnace design as a physical model, the intensified smelting processes are numerically simulated where the copper output is doulled. The influences of the burner structure and operating parameters on the smelting process are studied by simulating the particle motion, degree of mixing between the gas and particle, and chemical reactions at different operating conditions (such as the inner ring area, outer ring area, flow rate of central oxygen, flow rate of distribution air, flow rate of process air, etc.). This research can provide a helpful guidelive for the design of concentrate burner for intensified smelting. The main conclusions are as follows: a) The swirling process air can form swirling flow in the reaction shaft and drive particles to revolve around the central region (Fig.7.10). The revolving intensity has a close relationship with the initial velocity and direction of particles, as well as the swirl number of the process air. The traces of particles from the swirl burner are shown in Fig.7.11.

7 Coupling Simulation of Four-field in Flame Furnace 

Fig. 7.10 A sketch of flow field of swirling jet-flow

Fig. 7.11 computed particle traces in reaction shaft in the case of employing swirling nozzle

b) To achieve particle dispersion in the air, the swirl inducing design is superior to the old design having parallel distribution on air inlets. Some characteristics of the swirling design are: ü The irregularity of particle motion has decreased, which is helpful in reducing the dust generation ratio. üRegulating the swirl intensity of particles is relatively easy to achieve in practice.

  Ping Zhou, Zhuo Chen and Kai Xie üThe incoming swirl velocity of particles is greatly affects the swirling of particles in the reaction shaft. c) The axial velocity of the process air is an important factor for controlling and sustaining the cone shape of the suspension mixture. Increasing the axial velocity of the process air will decrease the trend of centrifugal motion of the falling particles. Moreover, the velocity of the process air at the burner exit has a great impact on the average axial velocity of particles. d) The irregular motion of particles and their erosion effect on the lining are influenced by the recirculation of fluids in the reaction shaft. Most particles having irregular motion return from the bottom to the top of the reaction shaft. In-depth research of the recirculation and its size is necessary for reduction of the irregular motion of particles. e) It is found that an increase of swirl number of the process air results in an increase of dust generation ratio. The reasons are: üAfter leaving the central swirling region, the particles are free from the constraints of the process air and drift in the reaction shaft. üSome particles, especially those of medium sizes, move along the periphery of the suspension mixture. When the swirl number increases, these particles can easily enter the stagnant region at the interface of the shaft and the settler, and flow foward the uptake shaft with the gas. f) It is proposed that the burner structure for intensified smelting can be modified in the following aspects: üThe pneumatic conveying method can be used to transport the concentrate particles into the shaft. The conveyed air flow should be controlled to meet the needs of burning decomposed sulfur vapor while avoiding backfiring. The swirl number of the conveyed air should be in the range of 1.0~1.5. üThe central oxygen pipe may continue to be used, or be replaced with a high-speed jet nozzle which can also double as a combustion stabilizer. üThe distribution air should be removed to simplify the structure of the concentrate burner. üThe two-ring structure of the burner is suggested to be replaced by a single ring. Oxygen enrichment of the process air is allowed to be adjusted and its velocity is limited around 150m/s. The swirl number should be within the range of 0~0.5 tangential velocity of the incoming process air. üThe volume of flue gas can be reduced by increasing oxygen enrichment of the process air, which can also decrease the average flue velocity.

References Bao Yunran (1997) Strategy of intensive economy incensement and energy development

7 Coupling Simulation of Four-field in Flame Furnace  (in Chinese). Energy Research and Information, 13(1): 1 Chen Hongrong, Mei Chi, Xie Kai et al (2004) Operation optimization of concentrate burner in copper flash smelting furnace (in Chinese). Transactions of Nonferrous Metals Society of China, 14(3):631̚636 Li Xinfeng, Mei Chi, Zhou Ping et al (2003) Mathematical model of multistage and multiphase chemical reactions in flash furnace (in Chinese). Transactions of Nonferrous Metals Society of China, 13(1):204̚207 Lu Zhongwu (1995) Flame Furnaces (in Chinese). Metallurgical Industry Press, Beijing Ma Aichun, Zhou Jiemin, Ou Jianping et al (2006) CFD prediction of physical field for multi-air channel pulverized coal burner in rotary kiln (in Chinese), 13(1):75̚79 Mao Jianxiong et al (1998) Clean combustion of Coal (in Chinese). Science Press, Beijing Mei Chi et al (1999) Hologram simulation of modern furnaces and kilns (in Chinese). Journal of Central South University of Technology, 30(6): 592̚596 Mei Chi et al (2003) Form and application of high effective reaction zone in copper flash smelting (in Chinese). Nonferrous Metallurgy, 55(4):85̚88 Ren Hongjiu (2001) Bath Smelting of Non-ferrous Metallurgy (in Chinese). Metallurgical Industry Press, Beijing Ou JianpingˈMa Aichun, Zhan Shuhua et al (2007) Dynamic simulation on effect of flame arrangement on thermal process of regenerative reheating furnace (in Chinese). Journal of Central South University of Technology, 14(2):243̚247 Xie Kai (2005) Several Theory and Operation Optimization Challenges in the Development of Modern Copper Flash Smelters (in Chinese). Thesis (PhD), Central South University Xie Kai, Mei Chi, Zhang Quan et al (2003) Research of uneven temperature field in the zinc fractionating tower furnace chamber (in Chinese). Journal of Central South University of Technology(Natural Science), 34(1):32̚35


Modeling of Dilute and Dense Phase in Generalized Fluidization Chi Mei and Shaoduan Ou

The gas-particle two-phase flow has been widely and intensively applied and investigated in recent decades. It is very difficult to describe such a non-linear and stochastic industrial system using accurate mathematical language. In this chapter, the principles and its applications are introduced and discussed regarding the modeling of the gas-particle two-phase flow.

8.1 Introduction In this book, the term “gas-particle generalized fluidization” refers to: a) Not only the processes of gas passing through bulk layer of powders to form a fluidized bed, but also the dilute phase fluidization of the gas-particle system and the suspended motion of the gas-particle two-phase flow. b) Not only the conventional fluidization of motion in the vertical direction, but also the two-phase flows in the horizontal or an arbitrary direction. Many FKNME such as fluidized roaster, air-flow drying system, pulverized coal combustion system, flash smelting and flash converting furnaces of sulfide concentrate, etc., all belong to the category of the generalized fluidization. Table 8.1 shows the classification of the generalized fluidization systems based on the gas-particle interactions and the thermal engineering characteristics of FKNME. The range of the characteristic parameters for each class can be found from Fig. 8.1 to Fig. 8.3.



 Chi Mei and Shaoduan Ou Table 8.1 Operating conditions of the gas-particle generalized fluidization systems Item

Dense phase

Fluidized condition

Fixed or moving bed

Dilute phase Bubbling fluidized bed

<45

45~70

Turbulent fluidized bed

Circulating fluidized bed

Suspension fluidized bed

80~95

>95

Linear velocity of gas flow / mgs−1 Porosity of the mixed system/%

Solid fuel; Stratified burning; Engineering vertical applications shaft furnace; blast smelting furnace

70~80

Pneumatic conveying system; Carrier Drying system; Fluidized Dilute Phase pulverized fluidized roaster; roaster; coal or Rapid fluidized fluidized heavy oil bed burning boiler burner; flash smelting and converting furnace

Fluidized roaster; fluidized heat treatment furnace; powder dense phase transport; fluidized burning boiler

ķ umf is the critical fluidization velocity of the bulk layer. For materials with wide size distributions,

Ar = ( 2 ~ 700 ) × 10 4 , characteristic size is taken as d p

ϕp

= ϕ p ∑ xi d i .

is sphericity, with the following experimental correlation (Cen et al.,1997) umf = 0.294

dp vg

0.584

0.056

⎛ ρp − ρg ⎞ ⎜⎜ ⎟⎟ ⎝ ρg ⎠

0.528

(8.1)

where vg is kinematic viscosity of fluidized gas, m2/s. ĸ uch is the velocity at the chocking point.If the flow rate is lower than uch the gas flow can no longer carry particles (average characteristic size) out from the bed layer; meanwhile, the pressure drop in the dense phase bed layer will increase dramatically. The formula for estimating uch is

uch = 1.463 gDb

0.288 −0.69 ⎡ ⎛ ρ − ρ ⎞ ⎤⎛ D ⎞ −0.2 ⎢( Gp • Dp / μ ) ⎜ p g ⎟ ⎥ ⎜ b ⎟ Ret ⎜ ρ ⎟ ⎥⎜ d ⎟ ⎢ g ⎝ ⎠ ⎦⎝ p ⎠ ⎣

(8.2)

where Db is the equivalent size of the bed layer, m; Gp is the solid phase mass flow, kg/s; Ret the Reynolds number corresponds to the terminal velocity. Ĺ uck is the converting velocity of the turbulent fluidized bed to the dilute phase. The characteristics (such as larger bubble disappearing, linguiform gas flow, particles dispersing, upward ejection along the bed height with zigzag form) of turbulent fluidized bed are gradually disappearing, the interface between the dense and dilute phase starts becoming fuzzy, the fluctuation of pressure drop in the bed layer becomes smaller, and the

8 Modeling of Dilute and Dense Phase in Generalized Fluidization
 frequency becomes higher. The estimation value of uck is (G.S.,1993)

(

uck ≈ 7.0 ρ s • d p

)

0.5

− 0.77

(8.3)

where ρs • d p is the product of particle material density and average size. ĺ uFD is the velocity at the converting point when the interface between dense and dilute phase starts disappearing. When the gas flow rate exceeds uFD, the system becomes a fairly uniform dilute phase. The formula for estimating uFD is as follows:

uFD ≈ 0.684 ⎡⎣( Gp • Dp / μ )( ρ p • ρ g / ρ g )⎤⎦

0.422

(D

b

/ dp )

−0.96

Ret −0.344

The physical meanings of symbols in this equation are the same as those in ĸ.

Fig. 8.1 Relation between the flow condition and porosity of bed layers and the velocity of the gas flow (Cen et al.,1997)

Fig. 8.2 Particle concentration of the fluidized system (Cen et al.,1997)

(8.4)



 Chi Mei and Shaoduan Ou

Fig. 8.3 Particle distribution characteristics of the vertical fluidization system (Cen et al.,1997)

8.2 Particle Size Distribution Models Most bulks used for industrial purposes are composed of particles (powders) of different sizes. There are great differences between particles with different sizes in the fluidized devices, in terms of dynamics, diffusion behavior and interface reaction rate, etc. Therefore, in order to develop a model for a generalized fluidization system, first of all, it is necessary to model accurately the size distribution of the particle groups. For any given particle out of a group of particles ( d = δ ), it is defined that f (δ i ) is the probability distribution density (or distribution density, in short) of

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


d at δ i .

f ′i p(δ i < d < δ i + Δδ ) = lim (8.5) Δδ Δδ → 0 Δδ The generalized integration of distribution density in the range of zero to δ i is the “distribution function” of the random variable d . f (δ i ) = lim

Δδ → 0

F (δ i ) = P(0 < d < δ i ) =

∫

δi 0

f (δ i ) dδ

(8.6)

The physical interpretation of the distribution function is the existence frequency of all particles with sizes smaller than δ i . It is also called the cumulative distribution passing sieve:

∫

Di =

δi

f (δ i ) dδ i

0

(8.7)

The existence frequency of particles with diameter larger than δ i is defined as the cumulative distribution on sieve: Ri = Obviously,

∫

Di + Ri =

∞

f (δ i ) dδ i

di

∫

∞

0

f (δ i ) dδ i = 1

(8.8) (8.9)

In addition,

dD dR =− (8.10) dδ dδ Several common distribution functions will be introduced in the following(G.S.,1993;The editorial committee of “Chemical Engineering Handbook”,1989). f (δ i ) =

8.2.1 Normal distribution model Normal distribution model is expressed as follows: ⎡ (δ − a )2 ⎤ 1 (8.11) f (δ i ) = exp ⎢− i 2 ⎥ 2 δ 2 πσ ⎣ ⎦ where a is a model constant, which is the particle size of maximal distribution density, or the mathematical expectation ( M d ) of a random variable d . ∞

M d = ∑ δ i ρ (δ i ) = ∫ i =1

+∞ −∞

δ f ( δ ) dδ

The arithmetic average of the measured values of d usually oscillates around its mathematical expectation M d , and therefore a = M d = δ m (i.e., the particle size when D = 0.5 ) where σ is the standard deviation

σ2 =

∫ (δ +∞

−∞

i

−δ

)

2

f (δ i ) dδ



 Chi Mei and Shaoduan Ou

The particle size distribution will be wider as σ increases; and the particle size distribution will be more focused around a if σ decreases. The cumulative distribution passing sieve, as ⎡ δ −δ ⎤ δi δi 1 exp ⎢ − i 2 ⎥ dδ Di = ∫ f (δ i ) dδ = (8.12) ∫ 0 0 2πσ ⎣⎢ 2δ ⎦⎥ or by approximation ⎞⎤ ⎟⎥ (8.13) ⎟ ⎠⎥⎦ erf (x ) is the error function of x , which can be found in many mathematical or statistical handbooks. Di ≈

⎛δ −δ 1⎡ ⎢1 + erf ⎜⎜ i 2 ⎢⎣ ⎝ 2δ

8.2.2 Logarithmic probability distribution model Many industrial bulks are close to this type of distribution with probably some deviations at the two extremes. In the equal sectional coordinates, f (δ i ) is asymmetrically distributed (as shown in Fig. 8.4). A coordinates transformation must be carried out if normal distribution transform is needed.

Fig. 8.4

Logarithmic probability distribution of particle group A and B

Assuming ζ = lgδ , then

(

)

⎡ ζ −ζ 2 ⎤ ⎥ (8.14) exp ⎢− ⎢ 2σ ζ 2 ⎥ 2πσ ζ ⎣ ⎦ ⎛ ζ − ζ ⎞⎤ ζi 1⎡ ⎟⎥ Di (ζ ) = ∫ f (ζ ) dζ = ⎢1 + erf ⎜ i (8.15) 0 ⎜ 2σ ⎟ ⎥ 2⎢ ζ ⎠⎦ ⎝ ⎣ As the logarithmic probability distribution functions, the particle number, surface area and weight distribution of the particle group follow the same rule and have the same standard deviation σ ζ . f (ζ ) =

1

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


8.2.3 Weibull probability distribution function The Weibull distribution function can be expressed as follows: ⎡ ⎛ δ − δ ⎞C ⎤ −XC min D = 1 − exp ⎢ − ⎜ ⎟ ⎥ = 1− e ⎢⎣ ⎝ δ m ⎠ ⎥⎦ Its more general form is dD C ⎛ δ − δ min ⎞ = f (δ ) = ⎜ ⎟ dδ δ m ⎝ δ m ⎠

C −1

(8.16)

⎡ ⎛ δ − δ ⎞C ⎤ CX C −1 C min exp ⎢ − ⎜ e− X ⎟ ⎥= δm ⎢⎣ ⎝ δ m ⎠ ⎥⎦

(8.17) where δ min is the minimal particle size; δ m is the specific particle size in the group; C represents the dispersivity of the particle group; X = C=

lgb ; b is a constant, b = δ lg δ m

δ − δ min ; δm

C m.

In order to obtain C , D − δ curve can be linearized, so that 1 ⎞ ⎛ lg ⎜ ln ⎟ = C lg X = Clg ( δ − δ min ) − Clgδ m ⎝ 1− D ⎠ 1 , and the 1− D horizontal axis represents δ − δ min . Since δ min is assumed to be a known value, it can be drawn as a straight line; and the constant C can therefore be obtained. This type of distribution function is not applicable to the particle group if an appropriate value of δ min cannot be given to make it linearized. Fig.8.5 is the diagram of Weibull probability distribution function, in which the value of C ranges from 1 to 3 for general industrial bulks. In the double log coordinates, the vertical axis represents ln

8.2.4 R-R distribution function (Rosin-Rammler distribution) R-R distribution represents an extreme case of Weibull distribution, that is, when δ min = 0 Di ( δ ) = 1 − exp ( − βδ n )

f (δ i ) =

(

dD = nβδ n −1exp − βδ n dδ

(8.18)

)

(8.19)



 Chi Mei and Shaoduan Ou

Fig. 8.5 Schematic of Weibull distributions

(a) Relationship among D, X and C ; (b) Relationship among f (δ ) , X and C

where β = 1/ δ Rn , therefore greater value of β indicates smaller size of the mean bulk 1 = 0.632 μm; n is the e distribution index, and the greater the value of n , the narrower the distribution of the particle size. To obtain the constants β and n , linearization can be tried with the D -δ curve by applying logarithm on both sides particles; δ R is equivalent to the particle size when Di = 1 −

1 ⎞ ⎛ lg ⎜ ln ⎟ = nlgδ − lgβ ⎝ 1− D ⎠ 1 1− D against δ in the double log coordinates. If such a line is not available in some specific cases, this specific bulk should be considered not applicable to R-R distribution.

so that β and n are obtained by drawing a straight line with ln

8.2.5 Nukiyawa-Tanasawa distribution function The distribution pattern of the particle group formed by atomization is close to the pattern described by the following model.

nb (a +1) / n n δ exp − bδ n (8.20) ⎛ a +1⎞ Γ⎜ ⎟ ⎝ n ⎠ where, the three constants, a , b and n depend on the atomization condition. Generally we take n < 1 , a = 2 . f (δ ) =

(

)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


Γ(z ) is a Γ function, and Γ( z ) =

∫

∞

0

δ z −1e −δ dδ

Γ(z ) has the following characteristics: Γ( z + 1) Γ( z ) = = (z − 1)Γ(z − 1) z Γ(1) = Γ(2 ) = 1 ⎛ a +1⎞ Γ0δ ⎜ ⎟ n ⎠ D= ⎝ and ⎛ a +1⎞ Γ⎜ ⎟ ⎝ n ⎠ where n and b are determined by Eq. 8.21and Eq. 8.22 using the arithmetic average particle size δ nl and the average volume of statistical surface area δ sv measured by experimentally collected N particles, n

⎡ ⎛ 6 ⎞⎤ ⎢ 1 Γ⎜ n ⎟ ⎥ ⎝ ⎠⎥ b=⎢ (8.21) ⎢ δ sv ⎛ 5 ⎞ ⎥ Γ⎜ ⎟ ⎥ ⎢ ⎝ n ⎠⎦ ⎣ ⎛ 4⎞ ⎛5⎞ Γ⎜ ⎟ Γ⎜ ⎟ δ nl n n = ⎝ ⎠ ⎝ ⎠ (8.22) δ sv Γ⎛ 3 ⎞Γ⎛ 6 ⎞ ⎜ ⎟ ⎜ ⎟ ⎝n⎠ ⎝n⎠ [Appendix] Statistical formula to calculate the average particle size The formulas commonly used to determine the average particle size of the particle group may have different definitions, according to the differences of usage and applicable conditions. Their definitions and calculation formulas are listed in Table 8.2. Table 8.2 Calculation formulas of average particle size in particle groups Name

Symbol

Definition

Formula

Arithmetric mean diameter

δ nl

∑l ∑n

∑δ n ∑n

∑v ∑s

∑nδ ∑nδ

i i

i

i

i

Surface-weighted mean diameter

δ sv

i

i

i

Mean surface diameter

δ ns

⎛ ∑ si ⎜⎜ ⎝ ∑ ni

i

1/ 2

⎞ ⎟⎟ ⎠

3 i 2 i

⎛ ∑ niδ i2 ⎞ ⎜⎜ ⎟⎟ ⎝ ∑ ni ⎠

1/ 2



 Chi Mei and Shaoduan Ou

Continues Table 8.2 Name

Symbol

Definition

Formula

Mean volume diameter

δ nv

⎛ ∑ vi ⎜⎜ ⎝ ∑ ni

⎛ ∑ niδ i3 ⎞ ⎜⎜ ⎟⎟ ⎝ ∑ ni ⎠

Length-weighted mean diameter

δ ls

∑s ∑l

Length-weighted mean surface diameter

δ lv

⎛ ∑ vi ⎜ ⎜ ∑l i ⎝

Mass mean diameter

δ vm

∑M ∑v

1/ 3

⎞ ⎟⎟ ⎠

1/ 3

∑nδ ∑nδ

i

i

i

i

⎞ ⎟ ⎟ ⎠

1/ 2

i

2 i

i

i

⎛ ∑ niδ i3 ⎞ ⎜⎜ ⎟⎟ ⎝ ∑ ni li ⎠

1/ 2

∑nδ ∑nδ i

i

4 i 3 i

Note: in this table, ni is the number of particles with size δ i in the sample; l , s and v are length, area and volume, respectively.

8.3 Dilute Phase Models The study of the gas-particle two-phase flow is focused on analysis of the movement of particle phase and the interactions between particle phase and gas phase. Currently two major analytical methods are used (Cen and Fan, 1990; Zhou, 1994). The first method is to take both gas and particle phases as continuous media, and to investigate the flow behavior of the mixture phase in Euler coordinates.The other method is to separate the particle phase from the gas phase (continuous media), and to track the particle movement with Lagrangian method. The most frequently used gas-particle two-phase flow models are summarized in Table 8.3.

8 Modeling of Dilute and Dense Phase in Generalized Fluidization
 Table 8.3 Type

Continuous media

Discrete media

Common gas-particle two-phase models

Model name

Action of particles on the fluid

Coordinates system

Inter-phase movement

Non-slip model

Partially considered

Euler

No

Small-slip model

Neglected

Euler

Particle diffusion relative to the fluid

Multi-fluid model

Considered

Euler

Yes

Particle group trajectory model

Considered

Lagrangian

Yes

8.3.1 Non-slip model Non-slip fluid model is the simplest multi-phase fluid model. The following are the major assumptions (Zhou,1994): a) Time-averaged velocities of all particles are equal to the local fluid phase velocities. b) The particle phase is assumed to be continuous media with turbulent diffusion, and the turbulent diffusivity is assumed to be equal to the diffusivity of the fluid phase. c) The interactions (momentum, heat and mass exchange) of particles between different size levels and between the particle phase and fluid phase are the same as the interactions between individual components in the fluid mixture. These assumptions indicate that the gas-particle two-phase flow is generally considered as a single-phased fluid. For the gas phase, the general conservation equation is: ∂ (ρϕ ) + ∂ ρ v jϕ = ∂ ∂t ∂x j ∂x j

(

)

⎞ ⎛ ⎜ Γϕ ∂ϕ ⎟ + Sϕ + S pϕ ⎜ ∂x j ⎟⎠ ⎝

where ϕ is a generally applicable variable; Γϕ

(8.23)

is the variable transfer

coefficient; Sϕ is the source term of gas phase; S pϕ is the source term from interactions between gas and particle phases. Parameters for different equations are listed in Table 8.4.



 Chi Mei and Shaoduan Ou Table 8.4 Parameters in the gas-phase equations of the single-phase model Equation

ϕ

Γϕ

Gas phase continuity

l

Gas phase momentum

Sϕ

S pϕ

0

0

S = −∑ nk m&k

vi

μ eff

−

Gas phase turbulent energy

k

μ eff σk

Gk − ρε

Turbulent dissipation rate

ε

μ eff σε

Gas phase species

Y

μ eff σY

−ω s

as s

Gas phase enthalpy

h

μ eff σh

− qr

∑n Q

In Table 8.4,

ε k

∂v j ⎞ ∂p ∂ ⎛ + Δρ g i + ⎜ μ eff ⎟ ∂xi ∂x j ⎝ ∂xi ⎠

vi s

0

( Cε1Gk − Cε 2 ρε )

0

k

k

+ hs

μ eff = μ 0 + μ T μ T = Cμ ρ

k2

ε

⎡ ⎛ ∂v ⎞ 2 ⎛ ∂v j Gk = μ T ⎢ 2 ⎜ i ⎟ + 2 ⎜ ⎜ ∂x ⎢ ⎝ ∂xi ⎠ ⎝ j ⎣

2 2 ⎞ ⎛ ∂vi ∂v j ⎞ ⎤ + ⎟⎟ + ⎜⎜ ⎟⎟ ⎥ ⎠ ⎝ ∂x j ∂xi ⎠ ⎥⎦ Particle continuity equation or diffusion equation is: ∂ρ k ∂n ⎞ ∂ ∂ ⎛ ν eff (8.24) + ρk v j ) = mk k ⎟ + nk m&k ⎜⎜ ( ∂t ∂x j ∂x j ⎝ σ kp ∂x j ⎟⎠ ∂nk ∂ ∂ ⎛ ν eff ∂nk ⎞ (8.25) + nk v j ) = ⎜ ⎟ ( ∂t ∂x j ∂x j ⎜⎝ σ kp ∂x j ⎟⎠ where ρ k , nk and mk represent the density, particle number and mass of the particle phase in group k , respectively; m&k is time changing rate of mk ;

ν eff is the effective kinematic viscosity of gas phase, ν eff =

1

ρ

( μ0 + μ T ) ; σ kp

is an empirical constant (Prandtl number) of particle diffusion, and is set at 1.0 here. The solution method of the single fluid model (non-slip model) for gas-particle two-phase flow is the same as that of the single-phase fluid equation, but with additional continuity equations (similar to the diffusion equation of the gas species)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


for the particle phase, and additional particle source terms in the gas phase equations. Simplicity is the major advantage of this model. However, the assumptions are remarkably far from the realistic condition. Therefore, the differences between the numerical predictions and the measurements are relatively large. That is why this model is not widely used in engineering area hitherto.

8.3.2 Small slip model The following are the major assumptions of this model (Cen and Fan, 1990; Zhou, 1994): a) The particles are regarded as continuous media and are grouped by size. Each group is a different phase possessing the same velocity vpk , temperature Tpg , density ρ pk , and particle size d pk . b) The velocities of the particle groups are different, and are not equal to the local gas velocity. c) The time-averaged velocity of the particle group is the same as that of gas. Because the velocity of slip is caused by turbulent diffusion of particles, the small slip is also called turbulent drift. d) The relationship between the multi-phase mixture and individual phase is similar to that between multi-component fluid mixture and individual component of the fluid. The continuity equations of fluid phase and particle phase, the momentum, k , ε and energy equations and the expressions of Sϕ and S pk in these equations are the same as those for the non-slip model. The difference is the fluctuating velocity component of the particle phase, which, according to the third model assumption, should be the sum of the time averaged mixture velocity and the particle turbulent diffusion component, namely vpk = vmj + vpkj (8.26) where vpkj is the turbulent diffusion drift velocity of group k in the particle phase. Then, the particle turbulent diffusion flux is J pkj = ρ pk vpkj = − Dpk ρ m

∂Ypk

(8.27)

∂x j

Substitute it into the continuity equation of particle phase, ∂Ypk ∂ ∂ ∂ ⎛ ρ mYpk ) + ρ mYpk vmj ) = ⎜⎜ Dpk ρ m ( ( ∂t ∂x j ∂x j ⎝ ∂x j where Ypk is the mass concentration fraction of particle k . In the non-slip model, v j represents the convection

⎞ ⎟⎟ + S pk ⎠

(8.28)

velocity, which is



 Chi Mei and Shaoduan Ou

represented by vmj here. The momentum equation of the fluid phase is: ∂pg ∂ ∂ ∂ pg vgi ) + pg vgi vgj ) = − + τ gji + pg Fgi − ∑ f rpki + vgi S ( ( ∂t ∂x j ∂xi ∂x j k (8.29) The momentum equation of the particle phase is: ∂p ∂ ( ppk vpki ) + ∂∂x ( ppk vpkj vpki ) = − ∂xpk + ∂∂x τ pkji + ppk Fpki + f rpki + vgi Spk ∂t j i j (8.30) where pg and ppk are the partial pressure of gas and particle kth phase; τ gji and τ pkji are viscous stress tensor; Fgj and Fpki are the volumetric force

component of the phases; f rpki is the resistance of the kth phase particles to the fluid f rpki =

ppk

τ rk

(v

gi

− vpki )

(8.31)

where τ rk is the residence time of the k th phase particle

τ rk =

ppk d p2k

18vg pg

(8.32)

where vg and ρg are the kinetic viscosity and density of gas phase; ρ pk and d pk are the density and diameter of the k th group particle. The energy equation of the fluid is: ∂ ∂ ∂ ⎛ ∂Tg ⎞ ρgj cpgTg ) + ρg vgj cpgTg ) = ⎜ λg ⎟ + ScpgTg − ∑ npk Qk − Qrg + Qcg ( ( ∂t ∂x j ∂x j ⎜⎝ ∂x j ⎟⎠ k (8.33) where npk is the particle number density of kth particle phase, number/m 3; Qk is the heat removed by the k th phase particles from the fluid through

convection; ScpgTg is the enthalpy change caused by the inter-phase mass changes (S); Qrg and Qcg are the radiant heat and chemical reaction heat of the gas phase, J/m3 .

8.3.3 Multi-fluid model (or two-fluid model) Unlike the previous two models (non-slip, small-slip), this model takes into account not only the draft (turbulent diffusion) perpendicular to the main flow direction of fluid, but also the time-averaged slip along the trajectories of the fluid in the particle groups. The following are all expressed in its physical model.  a) The particles are grouped by the initial size to form a number of particle

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


phases differing from each other in size. Each phase has its own velocity, temperature and volumetric fraction. b) Particle phases and fluid phase have their specific distributions of velocity, temperature and volumetric density, etc. (field properties). c) The differences between phase velocities are firstly due to the difference of the initial momentums and secondarily due to the drifts caused by turbulent diffusion. d) The particles collision frequency rises as a result of enlarged volumetric particle number concentration, which leads to the additional viscosity and increased heat and mass transfer among particles. The equations set for multi-fluid model is as follows: a) Continuity equation of gas phase: ∂ρ ∂ + (8.34) ( ρv j ) = S ∂t ∂x j b) Continuity equation of particle phases: ∂ρ k ∂ ∂ + ρ k v%kj ) = − ρ 'k v 'kj + ( Sk ) ( ∂t ∂x j ∂x j

(

)

(8.35)

or ∂nk ∂ ∂ + nk v% n 'k v 'kj ( kj ) = − ∂t ∂x j ∂x j

(

)

(8.36)

where v% kj is the instant velocity of the kth phase, which can be represented as the sum of the time averaged velocity vkj and the fluctuation velocity v 'kj : ' v% kj = vkj + vkj

According to Fick’s first law, the diffusion flux of particles is ∂f k J kj = ρ k vkj' = ρk ( v% kj − vkj ) = − Dk ρ m ∂x j

(8.37)

(8.38)

where f k is the particle concentration fraction of the kth phase. Therefore, Eq. 8.35 can be written as ∂ρ k ∂f ⎞ ∂ ∂ ⎛ + ρ k v%kj ) = − (8.39) ⎜⎜ Dk ρ m k ⎟⎟ + nk • m&k ( ∂t ∂x j ∂x j ⎝ ∂x j ⎠ c) Momentum equation of the gas phase: ⎡ ⎛ ∂v j ∂vi ⎞ ⎤ + ⎢ μ eff ⎜ ⎟⎟ ⎥ + ρ gi + ⎜ ⎢⎣ ⎝ ∂xi ∂x j ⎠ ⎥⎦ vi S + FMi + ∑ ρ k ( vki − vi ) / τ rk

∂ ∂ ∂ ∂ ρgvi ) + ρ v j vi ) = − p+ ( ( ∂t ∂x j ∂x j ∂x j

k

(8.40) d) Momentum equation of the particle phases:



 Chi Mei and Shaoduan Ou ⎛ 1 m& ⎞ ∂ ∂ ( nk vki ) + ( nk vkj vki ) = nk gi + nk ( vi − vki ) ⎜ + k ⎟ + ∂t ∂x j ⎝ τ rk mk ⎠ ⎛ ∂vkj ∂vki ∂ ⎡ + ⎢ nk vk ⎜ ⎜ ∂x ∂x j ⎣⎢ ⎝ i ∂x j

⎞⎤ ∂ ⎟⎟ ⎥ + ⎠ ⎦⎥ ∂x j

⎡v ⎢ k ⎣⎢ σ k

⎛ ∂nk ∂n ⎞ ⎤ + vki k ⎟ ⎥ ⎜⎜ vkj ∂x j ⎟⎠ ⎦⎥ ⎝ ∂xi

(8.41) e) Energy equation of the gas phase: ∂ ∂ ∂ ⎛ μ eff ∂hg ρg h j ) + ρg v j hg ) = ⎜ ( ( ∂t ∂x j ∂x j ⎜⎝ σ h ∂x j

⎞ ⎟⎟ − qr + ∑ nk Qρ + hg S k ⎠

(8.42)

f) Species equation of the gas phase: ⎛ ∂ ∂ ( ρgYs ) + ∂∂x ( ρg v jYs ) = ∂∂x ⎜⎜ D ρg ∂xYs ∂t j j ⎝ j

where ωs is the time averaged reaction rate.

⎞ ∂ ρg Y 's v ' j ⎟⎟ − ωs + α s s − ∂ xj ⎠

(

)

(8.43)

g) Energy equation of the particle phases: n m& ∂ ∂ nk cpk Tk ) + nk vkj cpk Tk ) = nk ( Qh − Qk − Qrk ) / mk + ( cpgTg − cpk Tk ) k k − ( ( mk ∂t ∂x j ∂ nk cpk v 'kj T 'k + cpk vkj n 'k T 'k + ck Tk n 'k v 'kj ∂x j

(

)

(8.44)

where Qh , Qk and Qrk are chemical, convection and particle radiation heat. The above equation set contains such unknown variables as v 'k T ' , n 'k T ' ,

n 'k v 'k and Y 's v ' j , etc., which has to be closed through

simplification and modularization. The closed equation set can be solved using IPSA method, a two-phase flow solver expanded from the SIMPLE algorithm proposed by Patankar and Spalding. In the case of suspension flow with low particle concentration, the PSIC method, as illustrated in Fig. 8.7, can be used. Readers can refer to references (Cen and Fan, 1990; Zhou, 1994) for more detailed discussions on continuous media multi-fluid models. The advantages of the multi-fluid model is that the equations of both gas and particle phases can be solved using the same numerical process, which results in more detailed information about the particle phases without making extra computational efforts. This model has been proven successful in predicting pulverized coal combustion, gasification, 3D turbulence and swirling gas-particle two-phase flow. Further investigations, however, are needed to apply this model in breaking and aggregation processes where evaporation, volatilization and phase change occur.

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


8.3.4 Particle group trajectory model Crowe and Smoot et al. brought up the conceptual model of particle groups in 1979 (Smoot and Partt, 1979). In this model a control volume with size CV and interface area CS is defined as containing fluid and dispersed particle groups (also called unit package), as shown in Fig. 8.6. Given that the surface area of a particle S p and the total interface area of the particle group C ' S ' , the model assumes (Cen and Fan, 1990): a) The gas phase is continuous whereas the particle phase is discrete. The velocities and temperatures of the gas and particle phases are different. b)The particles are grouped according to their initial sizes and within the same group all particles possess the same size, velocity and temperature. c) Every particle group moves along the same trajectory. The model follows up the changes on particle size, mass, velocity and temperature along the trajectory.

Fig. 8.6 Control unit of fluid and dispersed particle group

Take the cylindrical reaction space in the presence of swirl-flow for example, and the equation set governing the gas phase of the dispersed particle groups can be written in a unified form as follows: ∂ (ρuϕ ) + 1 ∂ (rρvϕ ) = ∂ ⎛⎜ Γϕ ∂ϕ ⎞⎟ + 1 ∂ ⎛⎜ rΓϕ ∂ϕ ⎞⎟ + Sϕ + Spϕ (8.45) r ∂r ∂r ⎠ ∂x ∂x ⎝ ∂x ⎠ r ∂r ⎝ where ϕ , Γϕ , Sϕ are listed in Table 8.5.



 Chi Mei and Shaoduan Ou Table 8.5 Parameters in the gas-phase equations of the particle group trajectory model Equation

ϕ

Γϕ

Sϕ

S pϕ k

Gas phase continuous

1

0

0

Axial momentum

u

μeff

−

−

Radial momentum

v

μeff

∑n ∂p ∂ ⎛ ∂u ⎞ 1 ∂ ⎛ ∂v ⎞ + ⎜ μ eff ⎟+ ⎜ rμ eff ⎟ ∂z ∂z ⎜⎝ ∂z ⎟⎠ r ∂r ⎜⎝ ∂r ⎟⎠

∂p ∂r

+

2 μ eff

ρvω

∂u ⎞ 1 ∂ ⎛ ∂v ⎞ ⎛ ⎜ μ eff ⎟ + ⎜ r μ eff ⎟ − ∂z ⎝ ∂z ⎠ r ∂r ⎝ ∂r ⎠

(

&k = S pm k ⋅m

k

∑f

k ⋅ z + uS pm

∂

2 v ω +ρ 2 r r

ω

ω ∂μ eff

k

∑f

k ⋅z + vS pm

k

∑f

Tangential momentum ω

μeff

−

Turbulent kinetic energy

k

μeff σk

Gk − ρε

Dissipation rate of turbulent kinetic energy

ε

μeff σε

ε

Fraction of mixture

f

μeff σf

0

0

Fluctuation value of f

gf

μeff σg

2⎤ ⎡ ∂f 2 gf ⎛ ⎞ ⎛ ∂f ⎞ C g1μ eff ⎢⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ ⎥ − C g 2 ρε ⎢⎝ ∂z ⎠ ⎥ r k ∂ ⎝ ⎠ ⎣ ⎦

0

Fraction of the gas mixture from particles

η

μeff ση

0

Spm

Fluctuation value of η gη

μeff σg

2⎤ ⎡ ∂η 2 gη ⎛ ⎞ ⎛ ∂η ⎞ ⎥ C g1μ eff ⎢⎜⎜ ⎟ + ⎜⎜ ⎟ − C g 2 ρε ⎟ ⎟ ⎥ ⎢⎝ ∂z ⎠ ∂ r k ⎝ ⎠ ⎦ ⎣

0

Gas component

Yj

μeff σY

−ω j

ajSpm

Gas energy

h

μ eff σh

−qr

∑n Q

k

r

− μ eff

r2

−

r

∂r

(Cε1Gk − Cε 2 ρε )

k ⋅θ + ω S pm

0

0

k

k

k

+cpTSpm

)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


In Table 8.5, 2 2 ⎡ ⎛ ∂u ⎞ 2 ⎛ ∂v ⎞ ⎛ ∂u ∂v ⎞ ⎤ Gk = μ eff ⎢2⎜ ⎟ + 2⎜ ⎟ + ⎜ + ⎟ ⎥ ⎝ ∂r ⎠ ⎝ ∂r ∂z ⎠ ⎦⎥ ⎣⎢ ⎝ ∂z ⎠

f kz , f kr and f kθ in S pϕ are the three components of the external force imposed on the particle groups in unit volume. If the pressure gradient force, virtual mass force, Basset force, Magnus force, Safman force, thermophoresis force and gravity of the particles imposed by the gas flow are neglected, and only the gas flow resistance is considered, then: 1 f kz = ( ug − upk )

τ rk

f kr = f

kθ

=

1

τ rk 1

τ rk

(v

g

(ω

g

− vpk ) − ω pk )

where τ rk is the particle relaxation time determined by Stokes’ resistance law, τ rk = ρ k d k2 / 18μ Table 8.6 lists the values of the constants obtained by numerical experiments. Table 8.6 Values of the constants in Table 8.5 Constant

C­

Cε 1

Cε 2

C g1

Cg 2

σk

σf

σg

ση

σY

σh

σε

Value

0.09

1.44

1.92

2.8

2.0

0.9

0.9

0.9

0.9

0.9

0.9

1.3

Eq. 8.45 actually presents the generalized Navier-Stokes equation set, and can be solved in the way of solving any simple fluid flow. 8.3.4.1 Trajectory of particle groups

Considering the trajectory of the particles in Lagrange coordinates, the motion equation of a single particle is as follows: d ( mp vp )

where ∑ Fp

= ∑ Fp + mp g + vp

dmp

(8.46) dt dt is the sum of all external forces imposed on this particle; mp g is the

gravity of the particle. The last term on the right side represents the inertia force caused by the mass change of the particle. It is actually rather difficult to consider in the model all the external forces imposed on a particle. For industrial bulks, such as pulverized coal and concentrate particle, a more practical approach is to start from modeling the most important facts and gradually add up the complexity to get closer to the reality step by step. If only the resistances of the particle are considered (including the inertial force due to the mass change), particle motion equations in cylindrical coordinates will



 Chi Mei and Shaoduan Ou

be stated as: ⎫ ⎪ dt τ rk ⎪ ⎪ 2 ω dvp 1 ⎪ p = vg − v p + ⎬ rp dt τ rk ⎪ ⎪ ω v dω p 1 p p⎪ = ωg − ωp − τ rk rp ⎪⎭ dt du p

=

1

(ug − u p ) + g

(

(

)

(8.47)

)

where rp is the radius of the particle position, u p , vp and ω p are the axial, radial and tangential velocity components respectively Eq. 8.47 can be solved numerically. The particle trajectory can also be obtained by Runge-Kutta integration method: dz p ⎫ = up ⎪ dt ⎪ drp ⎪ = vp ⎬ (8.48) dt ⎪ dθ p ⎪ r = ωp ⎪ dt ⎭ 8.3.4.2 Computation of the particle source term

The particle groups may affect the gas-phase flow and in the mean time the gas phase flow may also influence the trajectory and motion of the particle groups. The coupling effect between the gas flow expressed by Euler’s equation and the particles trajectory expressed in Lagrange equation is realized by the source term ( S pm , S pu , S pv , S pw , S ph ). When solving the particle motion by Lagrange method, every particle trajectory represents the motion of a group of particles with the same size. The flow rate expressed as the particl number of a group moving along the jth trajectory with size of d pi is nij nij = m&pij / mpij

(number/s)

(8.49)

where m&pij is the total flow rate of the particles; mpij is the mass of a single particle. When a particle passes through a computational cell k following trajectory j , a particle mass source term is generated: (8.50) ( Δm&pij )k = nij ⎡⎣( mpij )out − ( mpij )in ⎤⎦ k The “out” and “in” in the subscripts represent the particle mass leaving and entering cell k. nij keeps constant along the trajectory. The total particle mass source term in cell k is:

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


where

∑

⎡⎛ S pm,k = ⎢⎜ ⎢⎜ ⎣⎝

∑∑ m&

pij

i

j

⎤ ⎞ ⎟ / ΔV ⎥ ⎟ ⎥ ⎠ ⎦k

(kg/m3)

is the sum of the whole size group passing through cell k ;

i

(8.51)

∑

is

j

the sum of all particles with the same size but different initial position or trajectory j. Similarly, the source term in the momentum equation can be written as: ⎧ 1 ⎫ S pu , k = ⎨ nij ⎡( upij mpij ) − ( upij mpij ) ⎤ ⎬ (kg/(m2gs)) ∑∑ out in ⎣ ⎦ ⎩ ΔV i j ⎭k

⎧ 1 S pv , k = ⎨ ⎩ ΔV

∑∑ n ⎡⎣( v ij

i

j

pij

mpij )

out

⎫ − ( vpij mpij ) ⎤ ⎬ (kg/(m2gs)) in ⎦ ⎭k

(8.52) (8.53)

⎧ 1 ⎫ S pw, k = ⎨ nij ⎡( w pij mpij ) − ( wpij mpij ) ⎤ ⎬ (kg/(m2gs)) (8.54) ∑∑ out in ⎣ ⎦ ⎩ ΔV i j ⎭k The source terms in the energy equation of the particles include the heat effect of reactions of the particles, the convection between particles and gas and the radiation. ⎧⎪ 1 ⎫⎪ (8.55) S ph ,k = ⎨ nij − Q + C pT mpij out − mpij in ⎬ ⎪⎩ ΔV i j ⎪⎭ k where Qk is the convection and radiation heat transfer between particles and gas in cell k. The source terms for the turbulent energy and dissipation equations are usually taken as zero. Once the trajectories and the velocities on the trajectories of the particle groups are determined by the Lagrange method, the particle concentration and velocity fields can be solved by the volume average method. For each computational cell, the total density of particle number can be computed by the following equation: ⎫⎪ ⎧⎪ nk = ⎨ nij Δτ rij / ΔV ⎬ (number/m3) (8.56) ⎪⎭ k ⎪⎩ i j where Δτ rij is the residence time of the particle in the cell. Then the particle

∑∑ [

∑∑ (

( )

( )]

)

volumetric concentration (or volumetric fraction) is: ⎡ ⎤ C v ,k = ⎢ nij Δτ rij vpij / ΔV ⎥ (8.57) ⎢⎣ i j ⎥⎦ k The particle velocity in this cell can be obtained by the momentum average method:

∑∑ (

)



 Chi Mei and Shaoduan Ou

⎡ v p ,k = ⎢ ⎢⎣

⎤

∑∑ (nij Δτ rij vpij )/ ∑∑ (nij Δτ rij )⎥ i

j

i

j

⎥⎦ k

(m/s)

(8.58)

8.3.5 Solution of the particle group trajectory model A coupling method, called PSIC (particle-source-in-cell) for computing gas-particle two-phase flow, was introduced by Crowe in 1977. The method can be used to predict the relatively large slip between particle groups and the gas phase, as well as the changes in size, velocity and temperature of the particle groups along the trajectories. PSIC is simpler than IPSA. The computation procedure is shown in Fig. 8.7.

Fig. 8.7 PSIC computation scheme

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


8.4 Mathematical Models for Dense Phase Generally, when the velocities of the gas flow passing through the bulk bed are between the critical fluidization velocity ( umf ) of the particle and the terminal velocity (or carry out velocity, u t ), the bulk layer is considered to be a dense fluidized bed. The Reynolds number Remf with respect to u mf is determined by the following experimental equation: Remf =

where

Arε 4.75 18 + 0.61 ⎡⎣ Ar • ε 4.75 ⎤⎦ Remf ≡

Ar ≡

gd p3 vg2

d p • umf vg

(ρ p − ρg )

0.5

(8.59)

(8.60)

(8.61)

where ε is the void fraction of the fluidized system. For critical fluidized bed, ε = 0.40 ~ 0.45 in general. Vg is the dynamic viscosity of gas phase Given ε = 0.40 , then Ar Remf = (8.62) 1400 + 5.38 Ar 0.5 The gas flow velocity reaches the terminal sedimentation velocity when ε = 1 , Ar Ret = (8.63) 18 + 0.61Ar 0.5 where d p • ut (8.64) Ret ≡ vg In the fluidized reactor, the overflow outlet divides the dense and dilute phase; the fluid below this position is regarded as dense phase whereas that above is regarded as dilute phase. Even though from the point of view of reactor designing, the dense phase part and the dilute phases part should be considered as a unity because they are both within the reaction zone, there are still large differences between the dense and dilute phases in terms of fluidization condition, structural features, transport properties and the major interests of researching. The transport phenomena, either between the gas and particles in the dense phase or between different bed layers, are very complicated, featuring high fluctuation, randomicity and nonlinearity. The investigations in this area are still far from enough and consensus still has to be reached regarding the understanding and conclusions of the involved phenomena. Most of the published experiments



 Chi Mei and Shaoduan Ou

stay within the range of ρddp İ2.0kg/m2. Attempts to apply these data in practice are subject to some limitations. In this book only the frameworks of a few widely used physical models and numerical solutions regarding dense phase reaction zone will be discussed. Most of the currently available dense phase models focus on modeling the bubbles’ structure, motion and change of sizes as well as correlating these phenomena to variables about the fluidized beds. The major difference of these models is basically the degree of complexity of the models and correlations.

8.4.1 Two-phase simple bubble model The following are the basic assumptions: (Chemical Engineering Handbook, 1989): a) Gases flow through the bed layers either in bubble phase or in emulsion phase. The gas velocity in emulsion phase equals the critical fluidized velocity ( u mf ); the parts exceeding u mf are considered in bubble phase. b) The difference between the actual height of the bed layer ( Lf ) and the height of critical bed layer is considered as filled with bubbles. c) The change of the bubbles sizes and the influence of this change on other variables in bed layer are negligible. d) Chemical reactions only take place in the emulsion phase. e) The exchange rate between the bubble phase and the emulsion phase is the sum of penetration flow rate and the diffusion rate. Based on the above assumptions and under the first-order chemical reaction condition, the composition of the gas phase leaving the bed layer is as follows. When emulsion phase is assumed to be fully mixed flow:

(1 − Ze ) Co = Ze − x + Ci k '+ (1 − Ze − x ) −x 2

(8.65)

where C o and Ci are respectively the concentrations of gas phase at the outlet and inlet of the bed layer, ⎫ u Z = 1 − mf ⎪ u ⎪⎪ k ' = kc Lmf / u = kr pW / F (8.66) ⎬ ⎪ 6.34 Lmf ⎡umf + 1.3Dg0.5 ( g / d b )0.25 ⎤⎦ ⎪ X = QLf / ub vb = d b ( gd b )0.5 ⎣ ⎪⎭

where kc and kr are the first-order reaction rate constants respectively based on volume-concentration and weight-partial pressure of solid particles; p is the

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


total pressure; W is the catalyst weight; F is the total gas mole flow rate; Dg is the molecular diffusion coefficient of reactant gas; db is the equivalent bubble size (diameter). The relative velocity ubr can be determined by the empirical equation reported by Grace and Clift: ubr = 0.711( gd b )

0.5

(8.67)

On the other hand, the assumptions of the present model indicate that the relative velocity can also be expressed as: Lmf ubr = (uf − umf ) / ε mf = (uf − umf ) (8.68) Lf − Lmf Joining the above two equations, the equivalent bubble size ( db ) can be estimated as follows: Lmf ⎞ 1 ⎛ u − umf • db = ⎜ f ⎟ g ⎝ 0.711 Lf − Lmf ⎠

2

(8.69)

This is a model proposed in the early time for rough estimation of the bubbles in a steady state bubble. Subsequent studies (Grace and Clift, 1974) reported that the part of the gases flowing through the bed layer in the emulsion form ( umf • A ) is much greater than that predicted by this model (Cen et al.,1997).

8.4.2 Bubbling bed model Based on the simple two-phase model, further assumptions are made that the bubble phase contains bubble holes and bubble clouds (Fig. 8.8) surrounding the bubbles, indicating. that the bed layer is composed of bubbles, bubble clouds and emulsion phase. Similar to the simple two-phase model, it is still assumed that the sizes of the bubbles do not change with the height of the bed layer. The mass transfer between phases takes place only between bubble clouds and emulsion phase. The transfers within gas phase take place between the bubble holes and clouds as well as between the bubbles and the emulsion phase (Kim and Han, 2007). Under the first order reaction condition, the ratio between the compositions of the bubble phase leaving and entering the bed layer is:

Co = e−kf Ci

(8.70)

where kf is the mean overall mass transfer coefficient. Based on the above assumptions, kf is expressed as(Kunii and Levenspiel,1969):



 Chi Mei and Shaoduan Ou

Fig. 8.8

Structure schematic of the bubble and bubble cloud

kf =

⎛ yϕs ds2 Fbc ⎞ Dε + j Sh ⎜ b t ⎟ 6D ⎠ yds (1 − ε ) ⎝

(8.71)

where D is the mass diffusivity; Fbc is the gas exchange coefficient between the bubbles and the clouds (Davison and Harrison,1963):

Fbc = 5.85

D0.5 g 0.25 4.5umf + (s−1) db db1.25

(8.72)

where jb is the ratio of the particle volume in the bubble holes to the volume of the bubble holes; ε is the ratio of the total volume of the holes to the total volume of the bed layer; Sht is the Sherwood number, which is calculated based on the terminal velocity of particles: Sht = 2.0 + 0.6Ret0.5 Sc0.33 (8.73) where, Sht =

kf dsy D

, Ret =

uf d s vf , Sc = . vf D

where y is the logarithmic mean fraction of the non-diffusion component. The bubbling bed model further assumes that the particles in the bubble clouds are drawn into the wakes of the bubbles when the particles are moving downward. Within the wakes, the particles are homogenously mixed with equal flow rates while entering and leaving the wakes. This assumption allows us to estimate the particle exchange coefficient

( Fec )bs

bubbles and the emulsion phase:

( Fec )bs ≈

among the bubbles, the wakes of the

3 (1 − ε mf ) umf • (1 − δ ) ε mf d b

(s-1)

(8.74)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


where d b is the equivalent diameter of the bubbles; δ is the volume fraction of the bubbles in the dense phase. Although this model has made one step closer to the reality by adding the independent bubble cloud phase, the major weakness of this model is obvious, because neither the changes of the bubbles nor the impacts of these changes along the bed height have been taken into account.

8.4.3 Bubble assemblage model (BAM) The bubble assemblage model has considered the growth of the bubbles in the fluidized bed. Therefore it is even closer to the practical situations compared to the previous two models. The basic assumptions of BAM are as follows(G.S.,1993): a) From the distribution board upward, the fluidized bed layer is divided into several consecutive sublayers. The height of each sublayer equals the size of the local bubbles (Fig. 8.9).

Fig. 8.9 Schematic of parameters in the bubble assemblage model(BAM)



 Chi Mei and Shaoduan Ou

b) Each sublayer contains bubble phase and emulsion phase, and the gas is well mixed in each phase. c) The void volume in the emulsion phase equals the value when the bed layer starts fluidization, namely ε e ≈ ε mf . Meanwhile, it is assumed that the velocity of the gas in the void of the emulsion phase is so low that it can be neglected (namely when uf / umf ≈ 6 ~ 11, ue ≈ 0 ). d) The bubble phase is composed of spherical bubbles and their surrounding spherical clouds. e) The total volume of the bubbles in the bed layer is ( Lf − Lmf ) A , where A is the cross section area of bed layer. f) The gas exchange coefficient between the bubbles and the emulsion phase is related to the bubble size: Fbe = 11/ db (8.75) g) The voidage (¦) measured from the bottom to the height Lmf of the bed is a constant, and then increases linearly with the height. Assuming this value reaches 1 at the height of Lmf + 2 ( L − Lmf ) , that is to say, at Zİ Lmf ,

1− ε = at Lmf İZİ L

mf

Lmf (1 − ε mf )

+ z ( L − Lmf ) 1− ε =

(8.76)

L

Lmf (1 − ε mf ) Lmf (1 − ε mf )( L − Lmf ) − L 2 ( L − Lmf )

(8.77)

The BAM can be solved by making material balance within each sublayer. For the bubble phase in the n th layer, the material balance is Au ⎣⎡( Cb )n −1 − ( Cb )n ⎦⎤ = ⎡⎣ F 'be Vb • ( Cb − Ce ) ⎤⎦ n + ( rbVe )n (8.78) where F 'be is the gas exchange coefficient per unit bubble volume.

F 'be = r=

Fbe r

(8.79)

Total volume of bubble phase (void + cloud) Volume of bubble void

ubr + 2umf ε mf ⎛ Diameter of cloud ⎞ =⎜ ⎟ = ubr − umf ε mf ⎝ Diameter of bubble ⎠ 3

(8.80)

where, ubr = 0.711( g ΔLn ) ı 0.5

umf

ε mf

(8.81)

In Eq. 8.78, rb is the volumetric reaction rate of the solid material (with voidage ε mf ) in the cloud phase. Under the first order reaction condition,

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


rb = kCb

(8.82)

where k is a constant; Vb and Vc represent the total bubble volume (void + cloud) and the cloud volume in the sublayers, respectively. The total volume of the bubble phase is: Nπ ( ΔLn ) r 3

Vb ( void + cloud ) =

= Vbv • r

6

(8.83)

The volume of the cloud phase is: Nπ ( ΔLn ) ( r − 1) 3

Vc =

6 Material balance of the emulsion phase (for the n th layer) is: [ F 'be Vb × (Cb − Ce )]n = ( rbVe )n where, under first order reaction condition of the gas phase, re = kCe Ve is the volume of the emulsion phase on the n th layer: Ve = AΔLn − Vbn

(8.84)

(8.85) (8.86) (8.87)

The assumptions from a) to g) mainly apply to the gas phase reactions in the fluidized beds of catalysts particles. Modifications and corrections have been reported as follows: a) Kefa Cen et al. proposed an empirical bubble size correlation applicable to wide-range bed material(Cen et al.,1997 ): −0.576

⎛ ρp ⎞ ⎛ L ⎞ ⎛ u − umf ⎞ d bm = 30.576 ⎜ H0 ⎜ ⎟ (8.88) ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ umf ⎠ ⎝ H0 ⎠ ⎝ ρg ⎠ where H 0 is the static height of the bed, m; L is the distance to the bottom plate, m. 0.346

0.711

b) Maximal bubble size(Movi and Wen,1976): d bm = 0.652 ⎡⎣ A ( u − umf ) ⎤⎦

0.4

(8.89)

c) Height of the sublayer: ΔLn =

d 'bn 1 + 0.15 ( d 'bn + d bm ) / D

(8.90)

where d 'bn is the d b at the interface between the (n − 1) th sublayer and the nth sublayer. D is the equivalent diameter of the bed. The computation procedure of BAM is as follows: a) Calculating the jet height L j from the distributing plate. The correlation proposed by Kono (the editorial committee of Ā Chemical Engineering Handbookā,1989) is recommended: 0.187

⎛ ρ u2 ⎞ ⎛ u = 7.5 ⎜ • or ⎟ ⎜ d or ρ gd ⎝ umf s ⎠ ⎝ s b) Calculating the bubble size d b at the height of Lj

0.2

⎞ ⎟ ⎠ L using Eq. 8.88.

(8.91)



 Chi Mei and Shaoduan Ou

c) Determining the height of the d) Determining the height L :

n th sublayer with Eq. 8.90.

Ln = L' n + where L'n =

ΔLn 2

(8.92)

n −1

∑ ΔL , which is the distance from the bottom plate to the i

n th

i =1

sublayer. The bubble size ( d b ) at this sublayer can be determined by e) Determining the height of the last sublayer: ΔLn = Lf − L'n f) The bubble fraction δ bn at the following equation:

δ bn =

Ln . (8.93)

n th sublayer can be determined by the u − umf ub

(8.94)

ub = ( u − umf ) + 0.711( gd b )

0.5

The bubble volume in the n th sublayer is δ bn AΔLn . g) The total volume of the bubble and bubbles cloud Vbn in the Vbn = δ bn AΔLn (1 + β cn )

n th sublayer: (8.95)

where

β cn =

3umf / ε mf volume of bubble cloud Vcn = = volume of bubble Vbn ubr − umf / ε mf

ub = 0.711( gd b )

(8.96)

0.5

h) Determining the height of the first layer, which is usually set at the distance d from zero to Lj + d bo . The bubble size at L = bo is used to determine ΔL1 ; 2 The initial bubble size d bo can be estimated by the following equation (Darton et al.,1977): 0.4

⎡ ⎤ ⎛ A⎞ d bo = 1.63⎢(u − u mf )⎜ ⎟ / g 0.5 ⎥ (cm) (8.97) ⎝n⎠ ⎣ ⎦ where ( A / n ) is the average cross section area of the holes in the distribution plate, cm2. i) Determining the total volume of the emulsion phase in the n th sublayer: Ven = AΔLn − Vbn (8.98) j) Determining the interphase exchange coefficient volume of the bubbles and bubble clouds:

⎛ u − u mf / ε mf k '0 = k0 ⎜ br ⎜ u br − 2u jf / ε mf ⎝

⎞ ⎟ ⎟ ⎠

k ' 0 based on the total

(8.99)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


Using the material balance equations( Eq. 8.78 and Eq. 8.85), the sublayers can be computed from bottom to top following the above steps. The flow chart is shown in Fig. 8.10.

Fig. 8.10 Computation scheme for the BAM of gas-phase reactions

8.4.4 Bubble assemblage model for gas-solid reactions The BAM discussed in the previous section is developed for gas-phase estimation but this model is also applicable to the gas-solid reactions after making some extra assumptions in addition to those mentioned in section 8.4.4. a) Defining Jc as the ratio of the particle volume in the cloud phase to that in the bubble wake,



 Chi Mei and Shaoduan Ou volume of particles in bubble cloud and wake volume of bubble ⎛ 1− εb ⎞ = γ s (1 − ε mf ) ⎜ ⎟ ⎝ εb ⎠ is the share of the particles in the bubble phase. jc =

where γ s

γs ≈

(8.100)

δ ωε b 1− εb

where δ ω is the ratio of the volume of the bubble wake to that of the bubble. Generally, we have δ ω ≈ 0.12 ~ 0.5 for industrial fluidized beds. This value declines as Ar rises (Rowe and Partridge,1965 ). In Eq. 8.100, ε b is the share taken by the bubbles in bed layer. For most engineering applications, we have ε − ε mf (8.101) εb ≈ 1 − ε mf In the emulsion phase, we define J as the ratio of the volume of the particles to e

that of the bubbles:

⎛1− εb ⎞ & ⎟⎟ − J e (8.102) J&e = (1 − ε mf )⎜⎜ ⎝ εb ⎠ b) The particles that have been dragged up by the bubbles and bubble wakes are compensated by those brought down in the emulsion phase. The overall flowing direction of the particles is dependent on whether it is forward feeding (bottom feeding) or backward feeding (top feeding). For the n th sublayer, the total flow rate of particles moving upward is: ⎛ω δ A u ⎞ (8.103) ω bn = ⎜⎜ s + ω bn bn ⎟⎟ Abn A Abn ⎠ ⎝ where Abn is the cross section area of the bubbles in the n th sublayer: 3 (L − Lmf ) Abn = A (8.104) 2 L ωs is the volumetric feeding rate (being positive for forward feeding and negative for backward feeding). The total volumetric flow rate ω e (n +1) of the particles flowing into the next sublayer is: ⎛δ A u ω ⎞ (8.105) ω e (n +1) = ⎜⎜ ω bn b − s ⎟⎟( A − Abn ) A − A A⎠ bn ⎝ c) Particles exchange between the bubble phase and the emulsion phase with the exchanging coefficient estimated by the following correlation equation: (1 − ε mf ) umf • ub −1 (s ) (8.106) ( K be )bs ≈ 3 ε mf ubr d b

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


where (Kbe )bs is the particle exchange coefficient, s-1. An isolated bubble rises in velocity u br = 0.711(gd b )0.5 whereas in the bubbling bed the bubbles rise in velocity ub = ubr + u − u mf . A simplified approach in engineering applications is to assume infinitively large particles exchanging coefficients in each sublayer, that is to say, the particles in each sublayer are perfectly mixed. As discussed in the previous section, similar material balance relations as expressed in Eq. 8.78 and Eq. 8.85 have to be applied in each sublayer. For the bubble phase:

Au ⎡⎣( Cb )n−1 − ( Cb )n ⎤⎦ = ⎡⎣ FbeVbv × ( Cb − Ce ) ⎤⎦ n + [ jeVbv Kr Cb ]n

(8.107)

where Fbe (see Eq. 8.75) and je (see Eq. 8.91) are based on the bubble void volume Vbv ; Vbv is calculated by Eq. 8.83. Regarding the emulsion phase, we have: FbeVb v ⎡⎣( Cb − Ce ) ⎤⎦ n = [ jeVbv K r Ce ]n

(8.108)

Given that particles are perfectly mixed, the effective gas concentration experienced by the particles can be calculated by the following equation: ( j C ) + ( jeCe )n CAn = c b n (8.109) jc + je With all these parameters well determined, the time t* for a complete particles conversion can be calculated (refer to Section 8.4.6). Under the ash diffusion control condition, we have: ds2 ρs m t* = (8.110) 24bD • C A where d s is the average particle diameter; ρ s is the mole density of particle reactant; b is the mole number of solid reactant per unit mole gas; D is the mass diffusivity of gas in the ash layer, cm2/s. The mean residence time of particles in the n th sublayer is given by: Vb v ( jc + je ) (8.111) tn =

ωs

The particle conversion rate of each sublayer in the dense phase can be calculated using Eq. 8.109 to Eq. 8.111 together with the conversion rate equations that will be introduced in the next Section 8.4.6.

8.4.5 Solid reaction rate model in dense phase There are two major factors determining the reactions of the solid in the dense phase, namely the chemical reaction dynamics and the particles residence time in



 Chi Mei and Shaoduan Ou

the reaction zone. The analysis of the dynamics of gas-particle reaction is based on the understanding of the conversion rate of a single particle. There are two reaction models, which are the homogeneous reaction model applicable to porous particles (reactions take place throughout the particle) and the shrinking core model applicable to nonporous particles (reaction takes places on the shrinking surface of the unreacted core). 8.4.5.1 Simple homogeneous reaction model

Basic assumptions: a) Gaseous reactants penetrate through the whole particle. b) The particle is isothermal and homogeneous. Under the condition of the first order reaction that depends on the gas concentration, the reaction rate of the solid is dCS (8.112) = −k v • CA • CS dt The solid reaction rate is defined as X B C − CS X B = SO (8.113) CSO or

CS = CSO (1 − X B )

(8.114)

Therefore dX B = ρ ev • CA (1 − X B ) (8.115) dt where CA , CS and CSO are respectively the mole concentration of the gaseous, solid reactant species and initial solid reactant species; k v is the reaction rate constant of unit solid volume. Assume CA and k v are constants, then

t → ∞ as X B → 1 .

X B = 1 − exp ( − k v CA t )

(8.116)

We define t0*.999 as the time needed to complete 99.9% of the reaction for the homogeneous reaction model: 6.908 * t0.999 = (8.117) pev • CA This model maybe used to estimate approximately the processes such as gasification and burning of porous carbon and the reduction and chlorination-evaporation of the pellets, etc. The readers are referred to literatures (Sohn and Wad- sworth, 1979; Xiao and Xie, 1997) for more accurate models.

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


8.4.5.2 Shrinking core model for nonporous particles

Reactions of this kind can be generally expressed as: A ( gas phase ) + bB ( solid phase ) = pP ( gas phase ) + rR ( solid phase )

(8.118)

The reaction process of a single spherical nonporous particle is shown in Fig. 8.11.

Fig. 8.11 Reaction process of the shrinking core model

This model assumes: a) Reaction only initializes on the surface of the particle, which results in a layer of solid product (also called ash layer). b) Gaseous reactant diffuses through the ash layer into the unreacted core where reaction goes on; the thickness of the ash layer equals the thickness of the reacted layer (the solid particle size keeps unchanged). c) Gaseous product diffuses outwards through the ash layer with mass diffusivity and mass fluxes equal to that of the gaseous reactant. d) The instantaneous rate of mass transfer between gas and particle surface, gaseous reactant diffusion through ash layer and gaseous product diffusion outward are equal (quasi-steady-state). Levenspiel(Levenspiel,1962) and Wen(Wen,1968) reported their work on the solution of this model, which is summarized as follows: Outwards gas diffusion control condition: t = X B (Equivalent to zero order reaction) (8.119) t* d ρ t * = s sm (8.120) 6bkf CA Chemical reaction control conditions: t = 1 − (1 − X B )1 / 3 t* d ρ t * = s sm 2bkcCA Ash diffusion control conditions:

(8.121) (8.122)



 Chi Mei and Shaoduan Ou

t = 1 − 3(1 − X B )2 / 3 + 2(1 − X B ) * t d s2 ρ sm 24bδ fa CA

t* =

(8.123) (8.124)

In the above equations, δ fa is the gas diffusivity in the ash layer, cm2/s; kf is the mass transfer coefficient between gas and the surface of particles, cm/s; k v is the rate constant of chemical reaction based on the particle volume, cm3/(mol⋅s);

ρ sm is the mole density of solid reactant, mol/cm3. When the chemical reaction rate is at the same order of magnitude of ash diffusion, k can be used to replace kc in Eq. 8.122, and k is

⎛1 d k = ⎜⎜ + s ⎝ kc 12δ fa

⎞ ⎟⎟ ⎠

−1

(8.125)

If there is no more solid product around the shrinking core (for example, the product is gaseous or the ashes have fallen), the equation of chemical reaction control condition (Eq.8.122) can be used. If the diffusion resistance through the surrounding gas membrane is comparable to that of the chemical reaction resistance, the following equation can be used:

⎛ 1 1⎞ k = ⎜⎜ + ⎟⎟ k k f ⎠ ⎝ c

−1

(8.126)

The shrinking core model can be used to estimate the oxidations of metal particles, the oxidations or sulfating roasting of sulfide concentrate powders. It can also be used to assess gaseous reduction of oxide powder and burning of ash-containing coal, etc. In practice it is unrealistic for the solid particles to remain spherical or keep the total volume of particles unchanged. Readers may refer to reference(Sohn and Wadsworth,1979;Xiao and Xie,1997) for corrections in case that the real conditions are considerably different from the assumptions of this model. 8.4.5.3 Average reaction rate ( X B ) of the particle groups in the dense phase

The previous sections have covered the estimations of conversion reaction rates and reaction times, and the average reaction rate can be obtained if the dense phase particles distribution function at the outlet is available 1− XB =

t*

∫ (1 − o

X B ) E (t ) d t

(8.127)

where E (t ) is the age distribution function of the fluidized bed at the outlet; t*is the complete reaction time of a single particle. For a single-layer fluidized bed, it

8 Modeling of Dilute and Dense Phase in Generalized Fluidization


is assumed that particles are perfectly mixed, that is 1 (8.128) E (t ) = e − t / t t where the average residence time t is M t = &s (s) (8.129) Ms where M&s is the mass flow rate of the solid, kg/s; If the mass of the particles has largely diminished in the dense phase, the mass flow rate can also be estimated by averaging the feeding flow rate and the discharging flow rate; M s is the total mass of the solids in the dense phase, kg. For N dense phase bed layers in series, the age distribution function at the outlet is E (t ) =

⎛t⎞ ⎜ ⎟ ( N − 1)!t i ⎝ t i ⎠ 1

N −1

e−t / ti

(8.130)

where ti = t / N . Usually Eq. 8. 127, Eq. 8. 128, and Eq. 8. 130 should be solved simultaneously from N=1 to N=N. The computation procedure is quite complex. In reference (G.S.1993) the correlation diagram of X B against dimensionless residence time ( t / t * ) is listed based on the analysis results (referr to Fig. 8.12), which will be useful in engineering calculation.

Fig. 8.12 Relationship between solid average reaction rate X B and dimensionless time

The various dense phase models discussed above, to some extent, enable us to understand the physics and the reaction processes in the fluidized bed. Researches have also been reported on using chaos theory to analyze and predict the instantaneous temperature, pressure and heat transfer coefficient at various points in the dense phase (Karamavruc and Clark, 1996; Yong et al., 1999; Himmelblau, 2000). This opens new doors for the investigations in this area.



 Chi Mei and Shaoduan Ou

References Cen K F, Fan J R(1990) Theory and Computation of Gas-Solid Multi-Phase Flow in Engineering (in Chinese). Zhejiang University Press, Hangzhou Cen K F, Ni M J et al (1997) Theory, Design and Operation of Circulating Fluidized Bed. (in Chinese). China Electric Power Press, Beijing Davison J F (1961) Symposium on Fluidization-Discussion. Trans. Inst. Chem. Eng. 39: 230~232 Davison J F, Harrison D (1963) Fluidized Particles. Cambridge: Cambridge Univ. Press Darton R C, Lanauze R D et al (1977) Bubble growth due to coalescence in fluidized beds. Trans. Inst. Chem. Eng., 55: 274~280 Grace J R, Clift R (1974) On the two-phase theory of fluidization. Chemical Engineering Science, 29: 327 G S (1993) Handbook of Multi-phase Flow and Heat Transfer (in Chinese). Mechanical Industry Press,Beijng Himmelblau D M (2000) Applications of artificial neural networks in chemical engineering. Korean Journal of Chemical Engineering, 17(4). Karamavruc A I, Clark N N (1996) Application of deterministic chaos theory to local instantaneous temperature, pressure and heat transfer coefficients in a gas fluidized bed. J. of Energy Resources Technology. (Transactions of the ASME), Sept, 118: 214~219 Kim Jimin , Han Guiyoung (2007). Simulation of bubbling fluidized bed of fine particles using CFD. Korean Journal of Chemical Engineering, 24(3). Kunii D, Levenspiel O (1969) Fluidization Engineering. New York: Wiley Levenspiel O (1962) Chemical Reaction Engineering. New York: Willey Movi S, Wen C Y (1976) Simulation of Fluidized Bed Reactor Performance by Modified Bubble Assemblage Model, in Fluidization Technology. Washington: Hemisphere, 1: 189~203 Murray J D (1965) On the mathematics of fluidization: steady motion of fully developed bubbles. J. Fluid Mech, 21: 465~493 Rowe P N, Partridge B A (1965) An X-ray study of bubbles in fluidized beds. Trans. Inst. Chem. Eng., 43: 157~175 Smoot L D, Partt D T (1979) Pulverized Coal Combustion and Gasification. Plenum, New York Sohn H Y, Wadsworth M E (1979) Rate Processes of Extractive Metallurgy. New York: Plenum The Editorial Committee of “Chemical Engineering Handbook”(1989). Chemical Engineering Handbook (6) (in Chinese). Chemical Industry Press, Beijing Wen C Y (1968) Noncatalytic Heterogeneous Solid-Fluid Reaction Models. Ind. Eng.Chem.,(9):34~54

8 Modeling of Dilute and Dense Phase in Generalized Fluidization
 Xiao X G, Xie Y G (1997) Metallurgical Reaction Engineering Series: Fundamental of Metallurgical Reaction Engineering (in Chinese). Metallurgical Industry Press,Beijing Yong Kang et al (1999) Particle flow behavior in three-phase fluidized beds. Korean Journal of Chemical Engineering, 16(6) Yoshida K, Kunii D (1986) Stimulus and response of gas concentration in bubbling fluidized beds. J. Chem. Eng. Jpn. 1: 11~16 Zhou L X (1994) Theory and Numerical Simulation of Gas-Particle Two-Phase Flow and Combustion in Turbulence (in Chinese). Science Press, Beijing



Multiple Modeling of the Singleended Radiant Tubes Feng Mei

In this chapter the modeling of a gas-fired single-ended radiant (SER) tube is presented in full detail. The physical processes and models widely include turbulence, flow relaminarization, partially premixed combustion, radiation heat transfer, near-wall phenomena, dimension reduction modeling etc. The focus of this chapter, however, is not the SER tube modeling but the demonstration of the methodology, modeling effect assessment and problem-solving techniques that can be widely applied in numerical simulation. This chapter also demonstrates a few possible means to simplify or decompose a complex modeling problem into a number of smaller but easier modeling tasks as well as possible means to make the best use of the available computational resources for achieving optimized modeling results.

9.1 Introduction In the foregoing chapters a lot of numerical modeling techniques have been discussed for analyzing and controlling the FKNME and their thermal engineering processes. However, simulation results themselves can also be the objects of simulation. In this chapter we are going to discuss such a case, which is the modeling of the single-ended radiant (SER) tubes with a three dimensional burner. The objective of this study is twofold to optimize the operation condition of the SER tubes and to develop a 1D model so that quick online simulation and control can be realized. Given that the involved fluid dynamic and thermal physical

  Feng Mei processes are very complex and three-dimensional; the major effort for this study is to find a numerical solution to simplify the complexity and reduce the dimensions before the 1D model can be properly established. Baring all these in mind we focus no longer on the accuracy of any individual model employed but the overall effectiveness of the simulation and the compatibility among all employed models and numerical techniques.

9.1.1 The SER tubes and the investigation of SER tubes In applications such as heating, melting, sintering and heat treating, the workload often has to be stay free from contamination. For this purpose we can either use electrical heating elements, which are easily controllable for the heating power, or use radiation heat from tubes that are internally heated by gas-fired burners. This kind of tubes is called gas-fired radiant tubes. The single-ended radiant (SER) tube is one of the radiant tube family members with double tube structure and more uniform temperature distribution over the outer tube. The SER tube also shows advantages such as easy installation and high efficient heat recovery(Harder,1987). The working mechanism of the SER tube is as follows (Mei, 1999; Lisienko et al., 1986; Harder, 1987): as illustrated in Fig. 9.1, the partially premixed fuel (nature gas + air) is introduced in from inlet 1 while the rest of air flows in from inlet 2. The supplying air is preheated in the heat recuperator before it meets the partially premixed fuel at the burner outlet where combustion takes place. The flames are confined within the inner tube but the combustion products flow through down to the opening end and turn 180° into the annular channel enclosed by the inner tube and outer tube. The outer tube is heated by the combustion products (by convection) and the inner tube (by radiation) on the one side and radiates heat to the workload in the furnace on the other side. To ensure a uniform temperature distribution along the moving direction of the workload or to optimize the heat flux distribution, a number of radiant tubes are usually installed in rows in the furnace.

Fig. 9.1 A scheme of SER tube 1üInlet of nature gas and premixed air; 2üInlet of air; 3üHeat exchanger; 4üBurner;

5üOuter tube; 6üInner tube; 7üOutlet of combustion products

9 Multiple Modeling of the Single-ended Radiant Tubes 

The combustion and fluid flow in the SER tubes are three-dimensional (see Fig.9.2 and Fig. 9.3). As a matter of fact, 60% of total heat is released within the area close to the burner head where the fluid flow and combustion show strong 3D pattern. As the gases flow further downstream, 2D pattern gradually dominates until the gases enter into the annular channel. Due to the narrow space and high temperature the flow turns to be 1D plug flow in the annular channel. The continuous change of flow pattern brings us a problem: how many dimensions should we model the flow? A 3D model seems appropriate from the viewpoint of precisely simulating the combustion processes but apparently consumes too much computation resources and may lead to serious convergence difficulty. A 2D or 1D model seems more practical in terms of numerical approach but has to compromise remarkably in terms of prediction accuracy. One more difficulty comes from the so-called “relaminarization” phenomenon: once the gases are heated up the viscosity substantially rises and the turbulent flow will be transforming back into laminar flow. This is particularly true in the annular channel because of the substantial reduced characteristic length. These changes lead to the rapid falling of the Reynolds number, which causes the reverse transition from turbulent flow to laminar flow. Together with many other concerns, the above two problems bring up serious challenge to the simulation work.

Fig. 9.2 The exterior of burner in SER tube

Lisienko et al. (Lisienko et al., 1986) proposed a solution to overcome this situation: in their study of straight-through radiant tube, they suggested first simulating the fluid flow and combustion in a single radiant tube and then apply a simple numerical fitting to the computed “fuel consuming rate curve”, which was a indication of the intensity of heat release along the flow direction. By applying this consuming rate curve to represent the complex combustion process they managed to establish a new one-dimensional model for the fluid flow and heat transfer processes. This approach was approved successful because the fuel consumption rate had been found very efficient reflecting the major characteristics

  Feng Mei of the complicated combustion processes, which largely determined the heat transfer and fluid flow processes. Lisienko et al. published their study of 2D to 1D reduction based on this approach in 1986. They applied a laminar flow model in the 2D simulation and introduced a correction for the turbulent combustion. A correlation between the fuel consuming rate and the Reynolds number had been developed and then applied in the 1D model. This correlation was also found effective in predicting structure as simple as the straight-through radiant tube. Viskanta and his students (Harder et al.,1987; Ramaurthy,1994) proposed using low Reynolds number turbulent 2D model based on their wide review on the then published studies. Their fuel consuming rate correlation has taken the burner structure into account. A one-dimensional model has developed by applying this fuel consuming rate correlation. The above investigations focused on the simple-structured straight through radiant tube whereas in this chapter we present more complicated single-ended radiant (SER) tube. The extra difficulties for SER tube include:1. 3D burner structure, which can no longer be simulated by 2D model; 2. Partially premix combustion, which represents characteristics of both premix combustion and diffusion combustion; 3. Double channels, which force a U-turn of the flow. This U-turn is a challenge to many low Reynolds number (LRN) turbulent models because most of these models were not developed to consider sudden change of flow direction.

9.1.2 The overall modeling strategy The two major issues in this modeling were to reduce dimension and to select appropriate turbulence and combustion models. The burner head shown in Fig. 9.2 implies the flow will be three-dimensional. The modeling of three-dimensional require tremendous turbulent flow was therefore indispensable but would computational effort if a full-scale simulation has to be carried out. This direct approach is sometimes not feasible if the required computation goes beyond available computer capacity. A better solution would be to start with a study of the cold flow to understand the turbulence in the neighborhood of the burner head before we develop an appropriate full scale 2D model in using the information obtained from the 3D model as boundary conditions (see Fig.9.2). The objective of the 2D modeling is to determine the influence of the operational conditions, i.e. premix ratio, excess air ratio and mass flow rate, to the combustion process so that the correlation between the fuel consuming rate and the operational conditions can be established. So long as this correlation can be made, the 1D modeling becomes possible.

9 Multiple Modeling of the Single-ended Radiant Tubes 

Selecting appropriate models were the second key issue. Note that in most practical cases the optimization of the overall system is superior to that of any part of the system. As discussed previously, the modeling of the turbulence was found of the major technical difficulty for modeling of the SER tubes. We therefore ranked the turbulence model as first priority, followed by the combustion model as the second priority. We need a reliable combustion model because our two major interests in this application, namely the temperature distribution along the tubes and the fuel consuming rate, are both closely associated with the combustion process. The radiation model ranksproirity No.3 on top of the wall functions due to the fact that most heat is transferred by radiation at high temperature whereas convection (estimated by the wall functions) is relatively less important. The priorities have been ranked and now we may move on to select specific model for each process. To do so it is necessary to have a good understanding about the model performance and the physical processes. In the upcoming sections we are going to discuss how the models are accessed and selected against the physical phenomenon. Apart from the pure theoretical concerns in this selecting process, we also have to make necessary compromise with the available CFD code. Generically speaking, self-development of CFD code was less advised because it often consumes a very large part of research resources but not necessarily guarantee better overall results. It is actually more advisable to use commercial CFD packages as much as possible even though adaptation has to be made from time to time. The CFD package used for this study was Fluent 4.0. All the employed models, except the turbulence model and the combustion model, are the built-in standard models of Fluent 4.0.

9.2 3D Cold State Simulation of the SER Tube Fig. 9.3 illustrates the burner head structure of the SER tube. As stated in the foregoing section, the purpose of this modeling was to understand the separating and mixing patterns of the turbulent flow within the burner head structure. Note that the tube is much longer than 10 times tube diameter that we may reasonably assume the dead end of the SER tube has no influence to the flow field from the entry side where the burner head is located. Therefore we remove the outer tube from the computation domain in order to reduce computation load. The standard RNG k-ε model is applied (see next section for the reasoning of this selection) with a finite difference mesh grid of 17h27h150 and the SIMPLE scheme. The computation domain is 1/4 of the flow field.

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Fig. 9.3 A 3D model of nature gas burner (a) Radial distribution of meshes;(b) Axial distribution of meshes

Fig. 9.4 shows the 3D flow field structure at a characteristic Reynolds number (based on the inner tube diameter) of 9658. The physical positions of the cross sections are marked in Fig.9.5. Four recirculation zones are identified in Fig.9.4. One of these recirculation zones is located inside the burner head where the incoming fuel flow is split into an axial flow and four radial flows. The other three recirculation zones are all in the area where the central flow (partially premixed fuel) and the annular flow (air) are mixing. The sizes of the recirculation zone I and III are found different in Fig.9.4(a), (b), which strongly indicate their three-dimensional characteristics. By contrast, the recirculation zone II is basically independent of the sections, implying that the flow has been gradually reduced into 2D structure. Fig.9.4(c), (d) illustrate the recirculation of the radial injections. No strong entrainment is identified before the radial injections meet the annular injection. Fig. 9.6 compares the radial flow velocities in a range of 90 degrees along the circumference direction at J=13 and J=21. About 95% of the momentum is retained within 1/2 of the area even though the peak velocity has reduced 60% from section J=13 to J=21. This analysis can be used to measure how much the flow is “three-dimensional” because the velocities should be identical along the circumference direction in 2D flow. This means that under the condition of identical total flow rate and same J level a 3D injection flow should be much more concentrated than a 2D injection. The mixing effects in these two situations are

9 Multiple Modeling of the Single-ended Radiant Tubes 

quite different: This is the fundamental difference that we have to overcome for model dimension reducing. Note that in this case the driving force of the mixing process is the momentum transfer. The dimension reducing should be then based on a momentum similarity instead of a geometry similarity.

Fig. 9.4 3D flow field of SER tube (a) I=2; (b)I=9; (c)K=22; (d)K=26

Fig. 9.5 The SIMPLE scheme of 3D computation

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Fig. 9.6 The radial flow velocities along the circumference directions

Defining V # as the mean velocity of 95% of the total mass flow as showing in Eq. 9.1 and defining V0 as the initial velocity at the burner mouth(J=13), we may use V # /V0 to represent the attenuation of the radial injection before mixing. If the model is reduced from 3D to 2D, V # will be found diminished comparing to that in the 3D case at the height close to the mixing zone. This is because in 2D modeling the same mass flow is forced to spread equally in the circumferential direction, which results in weaker mixing effect. A possible solution to retain the mixing intensity is to enlarge the diameter of the burner head in the geometry model so that the distance between the injection mouth and the mixing interface is reduced, therefore the injection flow is more concentrated and V # /V0 is less diminished. 1 V# = Vds (9.1) S0.95 ∫S0.95

The above 3D analysis of cold flow helped us to set up the guideline of simplifying the model from 3D to 2D. Note that the transforming of the 3D geometry to 2D geometry cannot be fully based on the 3D simulation because the cold flow is still quite much different from the combusting flow. The V#/V0 ratio is not a constant either (varying from 0.22 to 0.436 over the range operation condition). In practice the diameter of the 2D burner geometry was actually determined by compromising all these factors based on information obtained from numerical experiments using both the 3D and 2D models. The second objective of 3D modeling was to move forward the boundary conditions to the line where the fuel and air enter into the combustion chamber. This was to avoid modeling the separating flow inside the burner head. The results of 3D simulation indicated that the ratio of the axial flow rate to the radial flow rate is quite close to a constant around 0.56 ± 0.02. This information enables us to estimate the axial and radial flow rates according to the total incoming flow rate.

9 Multiple Modeling of the Single-ended Radiant Tubes 

The geometry model used for the 2D simulation is illustrated in Fig. 9.7.

Fig. 9.7 The geometry model for the 2D simulation

9.3 2D Modeling of the SER Tube In this section, we talk about how to assess and validate all of the necessary models to be used for the involved thermal physical processes. Specific thoughts on the selecting of these models have been presented as well.

9.3.1 Selecting the turbulence model The turbulence models we were evaluating were the high Reynolds number k- ε model (or called the standard k- ε model), the low Reynolds number (LRN) k-ε model, the RSM(refer to Section 2.1.6) and the RNG(refer to Section 2.1.5) model. From the foregoing discussion, we understood that a turbulence model suitable for this application on the one hand should reliably predict low Reynolds number flow and transition processes, and on the other hand should be robust in solving complex and highly coupling problems. Form the concern of prediction accuracy, the LRN model and the RNG model should be superior to the standard k- ε model. In terms of numerical robustness, however, the standard k-ε model should be the best choice, followed by the RNG model, then the RSM and the LRN model. Compromising the needs from both sides, we ranked the RNG model as the

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most appropriate one. The RSM seems to be less attractive because this model not necessarily results in higher accuracy under varying density condition but it is surely more difficult to converge in the present case. To ensure we made the right choice performances of different models should still be compared against actual experimental data. First we compared the LRN model and the standard RNG model. The experimental data employed here were the measurements reported by C.T. Bowman and L.S.Cohen (Bowman and Cohen, 1975) regarding straight-through radiant tubes. These measurements had also been used by Viskanta et al. in their studies of 2D radiant tube model (Ramaurthy et al., 1994,1995) that employed Nagano-Hishida LRN model. Here we compared the predictions of Viskanta et al. with those predicted by the standard RNG model developed by the commercial CFD package Fluent 4.0. The combustion process was modeled by both £-PDF and double-α-PDF models (Borghi,1998;Libby and Williams, 1980; Kreinin and Kafiren,1980;Chan and Kumar,1990;O’Brien,1980). The DTRM model was employed for radiation. The nonequilibrium assumption wall function was applied to predict the near-wall phenomena. The predicted temperature and velocity fields are shown in Fig. 9.8. The comparison indicated that the RNG model and the Nagano-Hishida LRN model perform quite similarly even though the RNG model seemed to be slightly better. The predictions in the starting section were found remarkably different from the measurements, which could be explained by errors caused by the measuring devices. The LRN model converged to 1E-2 level whereas the RNG model could go down to 1E-3. Understandably, more serious convergence problem might happen with this LRN model in computing the SER tube that is more complicated developed geometry and physical processes. The performances of two RNG models namely the RNG model developed by Fluent 4.0 (here-after named as Fluent-RNG model) and the standand RNG model (see section 2.1.5 for detail), were also compared. Eq.9.2.is the estimation of the Fluent-RNG model for the effective viscosity (turbulent viscosity + laminar viscosity). The Fluent-RNG model estimated the Prandtl numbers of the scalars by the turbulent viscosity. These treatments indeed faithfully reflected the original idea of the Re-normalization Group process. This model, however, was found difficult to reach convergence under complex flow in numerical experiments. Fig.9.9 compares the predicted temperatures along the SER tube at two Reynolds numbers. The high Reynolds number k-ε model has basically failed to predict the detail of the temperature profile, the performance of the Fluent-RNG model was found to deteriorate at lower Reynolds number. The standard RNG model (refer to Section 2.1.5) demonstrated satisfying prediction in both cases.

μ eff

⎡ Cμ k ⎤ ⎥ ≈ μ ⎢1 + v ε⎥ ⎢⎣ ⎦

2

(9.2)

9 Multiple Modeling of the Single-ended Radiant Tubes 

Fig. 9.8 A comparison of the simulated radial velocity fields (a)and the simulated radial temperature fields (b) with the measurements in reference(Bilger,1987)

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Fig. 9.9 A comparison on performance of the different RNG models (In the figure, D is equivalent diameter of the outlet of SER)

9.3.2 Selecting the combustion model A combustion model actually consists of two submodels, namely a chemical reaction model that estimates the reaction rate and a mass transport model that governs the diffusion and convection of the reaction components. The combustion model should properly reflect the interaction between the chemistry and fluid dgnamics. This interaction is usually measured by the deteriorate Damkholer number: Da =

τt τC

(9.3)

where τ C represents the chemical time scale and τ t represents the turbulent time scale. There is large number of chemical reactions models which either assume infinitively large reaction rate or defined reaction rate. In the infinitively large reaction rate class find we the one-step reaction models (mixed-is-burned) and

9 Multiple Modeling of the Single-ended Radiant Tubes 

multiple-step reaction models. The later take into account intermediate products and equilibrium effects. There are also a large variety of models with defined reaction rates, among which the most frequently used one is the so-called Arrhenius model(Kreinin and Kafiren,1980). Regarding the mass transfer process, the eddy break-up (EBU) model is among the earliest ones but the probability density function models are more widely applied with new model development continuously coming out(O’Brien,1980). A complete combustion model is basically an assembly of a chemical reaction model and a mass transport model. Even though the EBU model is usually coupled with the Arrhenius reaction rate law model, theoretically other reaction models can also be “plugged-in”. For example, the mixed-is-burned model or the locational equilibrium model can be jointly used with the presumed PDF model such as the double-¢-PDF model or the£-PDF model which make up various quick reaction combustion models for diffusion flames. Meanwhile the Arrhenius reaction rate law model or other defined reaction rate models can be assembled with the presumed PDF models to make up premixed combustion models. In the present case the partially premixed combustion mode was used and the partial premix ratio (defined as the volume percentage of the premix air against air equivalent) varied from zero to 20%. Under different premix ratio, the combustion process might shift from pure diffusion mode to premixed mode. The fuel consisted of chiefly methane with minor ethane, propane and butane. Apparently this was not a typical fast reaction such as the one between hydrogen and. oxygen. As a matter of fact, many investigations reported remarkable error of prediction with instant locational equilibrium assumption for methane combustion due to the fact that the actual reaction rates were too far away from the required “fast chemistry” assumption. Therefore the best approach should be the definite reaction rate model plus a species components transport model that should be applicable to both diffusion and premixed flames. The above analysis allowed us to make proper assessment of the available combustion models in Fluent 4.0. The EBU (eddy break-up) model was proved, by numerical experiments, to be a failure for predicting methane combustion. Note also that Fluent 4.0 did not provide any partially premixed combustion model. Under this situation a so-called “pseudo-partial-premix-PDFā(or 4P’s) model was developed by adapting the available functions in Fluent 4.0 without carrying out expensive model developing or programming work. This 4P’s model enabled a very economical approach with fairly satisfactory simulation performance (Mei, 1999). As shown in Fig. 9.7, the 4P’s model clustered the mesh points on the inlet of the partial premixed stream into a number of artificial sub-jets that were defined either as pure natural gas jet or pure air jet. The total mass flow rates of the radial

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and axial jets were determined by the results of the 3D model and the jet velocities were calculated based on the opening area of the burner mouths. The computation was carried out using the Fluent β-PDF model with instant locational equilibrium assumption. The idea was to approach the premixing effect using very intensive turbulent diffusion process in order to cover the “blind point” of the commercial CFD package. For premixed combustion, the 4P’s approach forced to slower down the reaction rates that were overestimated by the instant locational equilibrium assumption. The weakness of the 4P’s approach was obvious. It was a ad hoc empirical treatment that could hardly be applied in wide range of application. Numerical experiments proved, however, that the 4P’s model were quite robust and the simulations were not the sensitive to the mesh points cluttering scheme or the velocity boundaries. Understandably, the flow velocities would be rapidly soaring high once the heat release started and the gases volume expanded. As a result, the turbulent effect downstream the inlet boundaries became the overwhelming factor dominating the diffusion process. Fig. 9.10 compares the predictions of the Fluent EBU model and the 4P’s approach. The later showed better simulation to the relaminarized combustion at low Reynolds number flow, in which the flame length is sensitive to the to operation conditions.

Fig. 9.10 A comparison of temperature fields predicted by EBU and the 4P’s approach

For carrying out the 2D simulation, the radiation model, the gases emissivity model and the wall functions had also been selected. These models are summarized together with aforementioned models in Table 9.1. Note that the standard wall function was selected instead of the nonequilibrium wall function was selected even though the latter normally gives higher accuracy.

9 Multiple Modeling of the Single-ended Radiant Tubes 

This decision was made to trade-off local accuracy with overall accuracy. The nonequilibrium wall function required that the near-wall mesh points must be arranged in the fully-developed turbulent region. Due to the relaminarization, the location of the turbulence region varied remarkably as a function of the flow rate, the premix ratio and the excess air ratio. This fact made it difficult to decide where and how to arrange the near-wall mesh points. Table 9.1 2D model structure of SER tube Process

Model

Turbulence

Standard RNG k-ε model

Near wall flow and heat transfer

Standard wall function

Combustion

4P’s approach

Radiation

DTRM

Discrete scheme

Quick for velocities, power-law for the others

Mesh

Axis-symmetric finite difference 290h49

9.3.3 Results and analysis of the 2D simulation Fig. 9.11 to Fig.9.13 demonstrate the predicted temperatures on the inner tube and outer tube under various premix ratios, excess air ratios and fuel flow rates as listed in Table 9.2. The dash-lines represent measurements. Table 9.2

Practical operational conditions and computational cases of SER tube

Test code

Fuel supply / kW

Premix ratio / %

Excess air ration / %

1*

16.3

55.0

11.0

1

16.3

25.1

11.0

2

16.3

14.4

11.0

2*

17.23

0.0

22.7

3*

17.23

20.0

60.0

3

17.23

20.0

22.7

4

17.23

20.0

5.4

4*

17.23

20.0

0.0

5

11.52

22.6

26.1

6

24.8

15.9

8.9

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Fig. 9.11 A comparison of tube wall temperatures simulated with the measurements at different premix ratio (the numbers of curves are related in Table 9.2) (a)For inner tube; (b)For outer tube

The premix ratio was found to be the major factor determining the overall fuel consuming rate of the flames. The temperatures on the inlet side of the tubes were a good reflection of the different fuel consuming rate. Fig. 9.11(a), (b) compared the temperatures along the outer and inner tubes. The temperatures demonstrated a wave-like pattern at large premix ratio, indicating that the fuel had been consumed rapidly before the flow reaching the dead end of the SER tube. By contrast, the temperatures rose steadily at lower premix ratio, indicating a moderate but continuous heat release along the tubes. Unfortunately the model failed to predict the overheating at the dead ends (Fig. 9.11(a)), which was probably due to the DTRM radiation model that underestimated the radiation contributed by the directional emissivity effect on metal surface.

Fig. 9.12 A comparison of predicted tube wall temperatures against the measurements at different excess air ratio (a)For outer tube; (b)For inner tube

The impact of excess air ratio was found relatively moderate but far-reaching. More access air might drag down combustion temperature at the initial stage but

9 Multiple Modeling of the Single-ended Radiant Tubes 

might also drive faster temperature increase due to better diffusion effect. However, the temperature would eventually slide down once all fuel had been consumed. On the other hand, combusting under lower excess air ratio showed flatter temperature profile. The model failed to predict the temperature fluctuation along the tubes that we had observed in experiment under high excess air ratio, which could be caused by the 3D to 2D simplification process. Note that there were recirculation zones vertical to the axial flow as shown in Fig. 9.4(c) , (d). These recirculations brought a part of oxidant from the annular air stream down to the central axial fuel streams, leading to better mixing and quicker reaction. In the reduced 2D model, however, the radial jets acted like a barrier between the annular air stream and the central fuel stream, leading to slower mixing process. The impact of total fuel supply to the temperature was primarily the overall temperature level. Fig. 9.13 shows the good agreement between the measurements and the prediction. The model was validated by the 6 experimental cases. The results were quite satisfactory, which made it possible for us to continue the dimension reduction process to 1D modeling of the SER tube.

Fig. 9.13 A Comparison of predicted tube wall temperatures against the measurements at different fuel supply(the numbers of curves are related in Table 9.2) (a)For inner tube; (b) For outer tube

9.4 One-dimensional Modeling of the SER Tube The slim geometry of the SER tube, especially the annular channel enclosed between the inner tube and the outer tube, physically justified the 1D treatment of the flow in a remarkable part of the computational domain. However, the following assumptions were still needed: a) The 3D-structured burner head could be simplified into a single entry. b) The gases could be treated as perfectly mixed from the very beginning of the

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process. c) The chemical reactions could be considered as “fuel + air = products”. d) The flow and heat transfer processes could be considered as one-dimensional. The 1D model of combustion is shown in Fig.9.14.

Fig. 9.14 1D model of combustion

The governing equations set under above assumptions can be written as: Energy conservation within the inner tubes: .

dTg1

d mf = α g1 −ω • Pω1 (Tω1 − Tg1 ) −Qf − Pω1 qg −ω1 m cp dx dx Heat transfer across the inner tube wall: R1 ⎡α g1 −ω1 Tg1 − Tω1 + qgr1 −ω1 ⎤ ⎣ ⎦ = ( R1 + δ ) ⎡α g2 −ω1 Tω1 − Tg2 + qωr 1 −ω2 − qgr 2 −ω1 ⎤ ⎣ ⎦ Energy conservation within the annular channel: . dTg m cp = α g2 −ω1 P1 Tω1 − Tg2 + α g2 −ω2 P2 Tω2 − Tg2 dx .

(

(

)

(

)

)

(

− P1q gr 2ω1 − P2 q gr 2 −ω 2

(9.4)

(9.5)

)

(9.6)

Note that chemical reactions had been finished before entering into this region. Therefore there is no more heat release term in Eq.9.6. Heat transfer across the outer tube wall: R2 ⎡α g2 −ω2 Tg2 − Tω2 + qgr 2 −ω2 + qωr 1 −ω2 ⎤ = ( R2 + δ ) qωr 2 − s (9.7) ⎣ ⎦ The radiation of the gases to the inside wall of the inner tube: εω σ ⎡⎣ε g1 Tg41 − ag1 Tω41 ⎤⎦ (9.8) qgr1 −ω1 = 1 − 1 − εω1 1 − ag1

(

)

(

)(

)

The radiation of the gases to the annular enclosure made by the inner and outer tube: qgr 2 = ε g 2 Ebg 2 − ag2 Ebω = σ ⎡⎣ε g2 Tg42 − ag2 Tω4 ⎤⎦ (9.9) The radiation of the inner tube to the outer tube: Ebω1 − Ebω2 qωr 1 −ω2 = ⎞ 1 R1 + δ ⎛ 1 − 1⎟ + ⎜⎜ ⎟ εω R2 ⎝ εω 2 1 ⎠

(9.10)

9 Multiple Modeling of the Single-ended Radiant Tubes 

The radiation of the outer tube to the environment (here referring to the larger cylindrical enclosure of the experimental setup): Ebω 2 − Ebs qωr 2 − s = (9.11) ⎞ 1 R2 + δ ⎛ 1 ⎜⎜ − 1⎟⎟ + Rs ⎝ ε s ⎠ εω 2 where Eb is the blackbody emissivity. In the equations above, α denotes the convective heat transfer coefficient, W/(m2 • K); P denotes the perimeter of the tube, m; Qf denotes heat release, kW/kg ; qr denotes specific radiation heat flux , W/m2; R1 is inner tube diameter , m ; R2 is outer tube diameter , m ;¥ is tube thickness , m; T is temperature, K ; m is mass flow rate , kg/s ; cp is specific heat capacity; σ is Stefan-Boltzmann constant, σ = 5.67h10ˉ8 W/(m2 • K4). The emissivities of the gases as function of partial pressure, total pressure and temperature were determined using separate subroutine. The heat transfer coefficients were calculated by empirical equations and were fixed to 20 W/(m2gK) for locations where Reynolds number is below 2,300. The radial and convective heat transfer estimated by the above equations were found lower than measurements, which was largely considered as an accumulated error due to the continuous dimension reduction process. Corrections were therefore introduced based on general experience and the 2D simulation results. The corrections were either formulated as constants or expression with values varying from 1.0~1.3. On top of the above equations, we still need two equations for the continuity and pressure drop: d ( ρv) =0 (9.12) dx dv dp ρv2 =0 (9.13) ρv + + f dx dx 2D where v denotes the velocity;p denotes pressure; f denotes the coefficient of flow resistance ; D denotes the equivalent diameter of the flow channel. So far all necessary equations had been ready except a governing equation for the fuel consuming rate in Eq. 9.4. This was the crucial information we were going to obtain from the 2D modeling results. By running the 2D model we managed to map out the fuel consuming rate curves at various operation conditions as shown in Fig.9.15 and Table 9.2, where the cases 1 to 6 were simulations to the actual experiments with temperature measurements whereas the case 1* to 4* were results of numerical extrapolation to extreme operational conditions. Higher resolution mapping of fuel consuming rates could be obtained by running simulations at more operation points.

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Fig. 9.15 The fuel consuming rate simulated by 2D model (The curves were numbers as in Table 9.2)

The fuel consumption rates could be structured either by algebraic expressions or by a lookup table. Both lead to a correlation between the fuel consume rate and operation parameters. This correlation could be generally expressed as: m&f = f (ϕ ,ψ , P, x ) (9.14)

where φ, ψ , P respectively denote premix ratio, excess air ratio and quantity of fuel supply. Given these parameters, the fuel consuming rate could be solely determined by the physical location x defined as the distance to the burner head (entrance). The combustion heat release would be then computed by substituting Eq.9.14 into Eq.9.4. The thermophysical properties of the combustion gases were calculated with the help of a component ratio index k(x): m& (9.15) k ( x) = 1− f m&f 0

where m&f0 denotes the initial fuel flow rate. The local air flow rate (m&a ) and the combustion products flow rate ( m&p ) were respectively: m&a = m&a 0 ⎡⎣1 −ψ k ( x ) ⎦⎤ m&p = m&a0 + m&f0 − m&a − m&f

(9.16) (9.17)

The equations set from Eq.9.4 to Eq.9.17 makes up the 1D SER tube model. The initial temperatures of the supplying fuel and air were determined by separated subroutine computing the performance of heat recuperator.

9 Multiple Modeling of the Single-ended Radiant Tubes 

The predictions of this 1D model are shown in Fig. 9.16 with good agreement to the measurements. Note that the better predictions at the dead end were resulted from the introduction of the experimental correction factors into the heat transfer equations.

Fig. 9.16 A comparison of the 1D simulation with measurements

References Bilger R W (1987) Turbulent jet diffusion flames. Prog. Energy Combustion Sciences, Jan. Borghi R (1998) Turbulent combustion modeling. Prog. Energy Combust. Sci., 14: 245~292 Bowman C T, Cohen L S (1975) Influence of aerodynamic phenomena on pollutant formation in combustion ĉ: Experimental results. EPA-650/2-75-061a, U Environmental Protection Agency Chan S H, Kumar K (1990) Analytical investigation of SER recuperator performance. 13th annual energy-sources technology conference and exhibition, New Orleans: Jan., 14~18 Harder R F, Viskanta R, Ramadhyani S (1987) Gas-Fired Radiant Tubes: a review of literature. Topical Report Kreinin E V, Kafiren Yu P (1980) Combustion of Gases in Radiant Tubes. Nerda, Leningrad

  Feng Mei Libby P A, Williams F A eds. (1980) Turbulent Reacting Flows, Springer-Verlag, Berlin Lisienko V G, Sklyar F R, Kryuchenkov Yu V, Toritsyn L N, Volkov V V, Tikhotskii A I (1986) Combined solution of the problem of external heat transfer and heat transfer inside a gas radiantion tube for thermal furnaces with a protective atmosphere. Journal of Eng. Physics, 50 Mei F (1999) Experimental and numerical investigation of a single-ended radiant tube. Thesis(ph. D), Facult Polytechnique de Mons, Belgium O’Brien E E (1980) The probability density function approach to reacting turbulent flows, Turbulent reacting flows. P. A . Libby and F. A. Williams (Eds.), Topics in Applied Physics 44, Springer-Verlag, Heidelberg, 184~207 Ramaurthy H, Ramadhyani S, Viskanta R (1994) A two-dimensional axisymmetric model for combusting. Reacting and radiating flows in radiant tubes, J. of the Institute of Energy, Sept. Ramamurthy H, Ramadhyani S, Viskanta R (1995) A thermal system model for a radiant-tube continuous reheating furnace. J. of Materials Engineering and performance, Oct., 14(5) Ramamurthy H, Ramadhyani S, Viskanta R (1997) Development of fuel burn-up and wall heat transfer correlations for flows in radiant tube. Numerical Heat Transfer, 31(6) Spalding D B (1997) Combustion Theory applied to engineering. Imperial College Report HTS/77/1


Multi-objective Systematic Optimization of FKNME Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

As most engineers may agree, it is not very difficult to improve the performance of an operating furnace or kiln by some means. The difficulties lie in finding the most effective strategy to optimize the system with a well-developed practical approach and bringing the ideas into practice. There fore, theoretically sound methodologies supported by necessary analyzing tools are needed. In this chapter, the optimization methodology and the objective functions are discussed for the FKNME optimization.

10.1 Introduction The optimization of engineering processes and production operations is to improve technical level and economic benefit through refining and adjusting system structures and operating conditions. The needs of optimization never end as the requirement to performance is always rising. Due to this nature, optimization process in general is regarded as an important source of technological developments and innovations in industries.

10.1.1 A historic review The methodology of optimization has been remarkably evolved. At the preliminary stage, only empirical optimization could be carried out by intuitions and common senses. This was later replaced by a more efficient analytical optimization approach

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao based on experimental inductions, which was usually single-objective oriented. It is not until the recent decades that a large variety of researching tools have been developed so that the analyzing, implementing and result validating in the optimization process can be carried out in a more systematic and efficient way. This has made it possible for multi-objective optimization, which is also called the “systematic optimization” (Liu, 1994; Liu and Bao, 1999). As new theories keep emerging over time, more and more new tools have been developed and used for optimization. As a result, qualitative analysis was firstly replaced by quantitative analysis tools such as mathematic programming and computer simulation. After that, more complicated artificial intelligence theories (such as expert system, fussy analysis, neural network, and genetic algorithm etc.) introduced, which enable us to carry out optimization in a broader and deeper range jointly using quantitative and qualitative analyses, accurate and fuzzy methods and algorithm-based mathematic modeling and searching techniques based on database of recognizable patterns. Optimizations were originally more passive-natured because they were usually actions consequently taken when the system was sufficiently understood after long time practice. Nevertheless, it has been more like an action taken aiming at proactively improving performance of the system nowadays. The latest trend is to launch optimization process as early as the prototyping stage for new system development in order to maximize the benefits of system operation at the best performing conditions. Before the environmental issues had aroused global concerns, the objective function set for optimization were usually those measuring economic interests or technical performance. Today, optimization may target objective functions that reflect the interests of the society, environment or the balanced development between technology, society, economy and environment (Hu, 1990).

10.1.2 The three principles for the FKNME systematic optimization Generically speaking, the idea of systematic optimization is reflected by the following three principles (Mei et al., 1996): a) Attention should be paid equally to the optimizations of the FKNME structure and its operational processing. Here “structure” means the main body of the FKNME (including their geometries, liner materials and technical configurations) and their thermal systems (including heat/electricity supplying, ventilation and gas exhausting, materials loading and unloading mechanisms). The “operational processing” includes the compositions of the loading materials, the combustion conditions and the procedures for material feeding/ discharging, air blasting and heat/electricity supplying institutions. b) The optimization should integrate needs at different levels, namely the working mechanisms ˄at the micro level˅, the FKNME structures ˄at the middle

10 Multi-objective Systematic Optimization of FKNME 

level˅ and the FKNME performances ˄at the macro level˅. The working mechanisms include the spatiotemporal characteristics of the fields of temperatures, component concentrations, heat-releasing rates and magnetic force etc. This information not only reveals the microdynamics of various transfer and physical chemistry processes in the FKNME, but also reflects the coupling effects between these processes. The optimization of macro performances should be realized through changing the microdynamics, which can only be observed by measuring the spatiotemporal characteristics of the relevant vector/scalar fields. On the other hand, the change of the microdynamics has to be done by modifying the FKNME structures or the operational processes. Fig.10.1 illustrates the inter-dependency and/or determining relationship among the three levels.

Fig. 10.1 Full-scale optimization of furnaces and kilns

c) A balance must be made among technological advancement, economical profitability and social interests. Pursuing economical profitability and technical excellence but ignoring its impacts to the environment is shortsighted and local-minded. From the sustainable growth point of view, we should encourage a systematic optimization that harmonizes the interests from all of the social, economical and technological aspects.

10.2 Objectives of the FKNME Systematic Optimization Even though the objectives of optimization are widely variable depending on different furnaces and kilns, there are a few aspects in common, such as maximizing the output rates, the product quality and the furnace/kiln’s service

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao lifespan. Particularly in the metallurgical engineering area, the furnaces and kilns are mostly well known for their remarkable energy consumption and pollution. Therefore, the objectives of optimization for these installations should also include minimizing the energy consumption and pollutive emissions in addition to the three aforementioned objectives. Mathematically, the multi-objective optimization problem can be expressed as:

x = [ x1 , x2 , K , xn ]

To solve: min f1 (x )

T

maxϕ1 (x )

min f 2 (x ) or : maxϕ 2 (x ) M min f q (x ) maxϕ q (x )

(10.1)

gu(x)İ with constraints: For most of the FKNME, f1(x), f2(x), …, fq(x) or ϕ1 ( x ), ϕ 2 ( x ),K , ϕ q ( x ) must cover functions as addressed in the following sections.

10.2.1 Unit output functions The unit output rate (or bed capability) is defined as the producing capability per unit bed area or per unit volume of the furnace (t/(m2gh) or t/(m3gh)). The rising of unit output rate is the ultimate way to lower down the consumption and increase the profitability over unit cost: The definition by unit bed area is: G (10.2) aF = ǂǂ (t /(m 2 • h)) F •τ The definition by unit volume is: G (10.3) aV = ǂǂ (t /(m3 • h)) V •τ where G is the mass processed or produced in τ hours by the furnace; F and V are respectively the bed area and the working volume of the furnace. The forms of a and G, although vary with different furnaces and kilns, should generally fall into one of the following: a) Output rate function defined by the heat transfer capability inside the furnace: n

ε m H m − ε miσTm4 + G=

∑α (T i

i =1

g

− Tm

)

(t/h)

(10.4)

qm

where ε m and Tm are respectively the surface emissivity and surface temperature (in Kelvin) of the furnace loads; Hm is the total incident radiation on the mth

10 Multi-objective Systematic Optimization of FKNME 

subarea Am: n N ⎞ 1 ⎛⎜ −1 g g s mσTg4 + s j s m J j ⎟ (kJ ⋅ m −22g ⋅ hh)) ) (10.5) ⎟ (kJ/(m Am ⎜ i =1 j =1 ⎝ ⎠ in which, g g sm and s j sm are respectively the gases-surface and surfacesurface direct exchange areas calculated by the zone method; Jj is the effective radiation at subsurface j: 2 −1 (kJ/(m J j = ε jσT j4 + 1 − ε j H m (kJ (10.6) ⋅ m −2g ⋅ hh)) )

Hm =

∑

∑

(

)

α i is the convection heat transfer coefficient between gas and material in the furnace. Many empirical formulations are available in various handbooks to calculate α i . The parameter qm (kJ/t) denotes the heat (absorbed from the furnace) required to process unit load or unit outputting products (usually measured in ton). b) Output rates function defined by the time for material transforming: for continuous process, the average residence time of loads staying in the reaction zone is: M t = wl (h) (10.7) G where Mwl is the capacity of the reaction zone measured in ton or kilogram; G is the mass flow rate of the loads passing through the reaction zone, t/h. The furnace’s processing capability G is calculated by rewriting Eq.10.7 as: M G = wl t Herein t is the time to complete all physical and chemical processes necessary for processing or transforming the loads. Assuming t* is the complete reaction time determined either by the chemical reaction dynamics model or by the experiments, then there must be: (10.8) t ı t* The complete reaction times predicted by simulations are listed in Table 10.1 for different processes. For the periodic operational FKNME: M G = n wl (10.9) ti

∑ i =1

where M wl is the mass of the loads per working session measured in ton; ti is the time needed for the process i. Taking the copper anode reverberating furnace as an example, ti represents the times for the processes of material loading, melting, oxidizing, reducing and casting. These times are normally determined empirically. In order to improve the productivity of the furnace’s productivity,

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

each individual process should be investigated so that the processing time can be minimized and the processes can be intensified. c) Output rate function defined by the conveying capability of the furnaces. The conveying capacity is often an explicit indication of the output rate of the FKNME that mechanically load/unload materials. Examples are the rotary kiln, mechanical roasting shaft kilns, copper refinery furnace, cathode zinc melting furnace, and aluminum melting furnace etc. The failure of properly matching the functions among different parts in the conveying mechanism would be an important factor limiting the enhancement of the output rate. The conveying capacity of the rotary kiln is relatively more complicated to be determined, of which the influencing factors include the rotating speed, the slope of the kiln, the materials angle of repose and the cross-sectional filling ratio in the kiln. Table 10.1 Estimation of time to complete a process (Xiao and Xie, 1997; Mei, 2000) Process

Transformation of the solid particles in gas-solid reactions

Loads heated by air flow in the shaft furnaces (fixed beds or mobile beds) Thin plane materials melted in furnaces under constant temperature

ķ

Reaction models

t

Homogenous reaction

6.908/kvgCA

This is the time needed for 99.9% transformation

Membrane diffusion control in shrinking core model

ds ρsm 6bkf CA

b denotes the quantity of solid reactants in mole reacting with 1mol gas

Chemical reaction control in shrink core model

d s ρ sm 2bk f CA

Diffusing control through ash layer in shrinking core model

ds2²sm /24b¥faCA

a) Reaction heat is negligible b) Wqt > 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.

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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
  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

furnaces is around 1.2 ×10 −8 m 2 / s ); l denotes the distance of the molten slag flowing around the lining; Rez is the Reynolds number of the molten slag flow close to the wall; Scz is the Schmidt number of molten slag. The chemical and mechanical erosion rate of the molten slag can be expressed as: dshj β c − z (Ci − C∞ ) • M rj × 10−6 vhj = = (10.22) (cm / s) dt ρ c

where Mrj is the molecular weight of the soluble compositions of the lining, and ρ c is the density of the lining, kg/m3. The overall erosion rate of a furnace can be estimated by adding up the two individual erosion rates: (10.23) v =v +v Σ

gw

hj

To maximize the service lifetime of a furnace, vΣ should be maintained as low as possible. Depending on the conditions in each specific case, the Eq. 10.14̚Eq. 10.20 may serve as a guide to develop a strategy of minimizing vΣ through optimizing the operations and/or the structure.

10.2.4 Functions of energy consumption Every type of furnace or production process has its own fuel consumption (or electricity consumption) function. For combustion furnaces, the fuel consumption is defined as: Gqyx + Qyq + Qhs + Qjs + Qsr + Qsl − (Qhr + Qkq + Qmq + Qw1 ) B= (kg/h or m3/h) QDW (10.24) where q yx is the heat to produce unit product; G is the output rate or processing capability of the furnace; Qyq , Qhs , Q js , Qsr and Qsl are respectively the heat brought away by gas, heat lost due to chemically incomplete burning, heat lost due to mechanically incomplete burning, heat lost through dissipation and heat carried away by coolant; Qhr , Qkq , Qmq and Qwl are respectively the chemical reaction heat (positive for exothermic reaction and negative for endothermic reaction), and physical heat carried in by air, coal gas and materials; QDW is the average heat value (lower) of fuel, kJ / kg or kJ/m3. For electrically heating furnaces, the electricity consumption function, DRF (dissipation rate function) is defined by: Gq yx + Qyq + Qxs + Qsr + Qsl − (Qhr + Qwl ) (10.25) DRF = (kW) 3600 where Qyq and Qxs are the heat losses in exhaust and electric circuits, kJ/h; the definitions of other symbols are the same as those in Eq. 10.24. The minimization

10 Multi-objective Systematic Optimization of FKNME 

of B and DRF is often achieved through optimized arrangement of the terms in Eq.10.24 and Eq.10.25.

10.2.5 Control functions of air pollution emissions Pollutants from the FKNME usually include SOx, NOx, smoke, dust, chloride, hydrogen chloride, fluorine, fluoride, silicon dioxide, asbestos dust and the compounds/vapor of metals such as Pb, Be, Ni, Sn and Hg etc. These emissions have been subject to strict regulation specified by national standards. Particularly in China, the following criteria must be met in the development and operation of the FKNME: [WRWi ] İ1 (10.26) WRF = [WRWi ]GB where [WRWi] is the emission rate (kg/h) or density (mg/m3) of the ith pollutant discharged into air. For particulates in the smoke, [WRWi] can also be smoke emissivity measured by Lingman’s fume meter. For any possible pollutant produced by the furnace, its specific [WRWi] must be measured. [WRWi]GB denotes the allowed upper limit of the emission of pollutant i ruled by Chinese National Standard (GB). Interested readers can refer to “The Comprehensive Emission Standard of Air Pollutants ü 1996 (GB16297)”, “The Emission Standard of Air Pollutants for Industrial Furnaces and Kilnsü1996 (GB9078)” and “The Emission Standard of Air Pollutants for Boilersü2001 (GB13271)” for more information.

10.3 The General Methods of the Multi-purpose Synthetic Optimization The optimization of each subobject f1(x), f2(x), …,fq(x) or ϕ 1(x), ϕ 2(x),…, ϕ q(x) may collide with each other in the Multi-purpose synthetic optimization. It is impossible to have their minimum point overlap together, which means, the optimum solution of each controlling functions can’t be obtained simultaneously. Decision favorable to the overall optimization then should be made by harmonizing all the optimum solution of each object. Various optimization methods are introduced as follows.

10.3.1 Optimization methods of artificial intelligence The artificial intelligent optimization methods include (Liu and Bao, 1999):

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

a) To converse multi-object optimization into single-object optimization. The comprehensive score is given to each sample according to some standards. The samples are then classified in terms of a score limit, or thinking of multi-object comprehensively, and they can be optimized by classes. b) To find out the common optimization zone. The optimization zone of a single object, in which optimum samples are situated, is searched by some spacial transmission. The overlapped optimization zones of individual objects are then the common ones of the multi objects, in which many targets are considered. The samples appeared in each common optimization zone have common characteristics of optimization. c) To optimize by combining neural network with multi-outputs and genetic algorithm (GA). In this method, the multi objects are regarded as output parameters, the original industrial parameters act as input ones, and the network is trained by iterations so that the functions of the input and output are constructed. The network function is consisted of the values of the multi optimization objects, which is the fitness function of GA, and the optimization values could be found by iterations. The process of the optimization can be divided into three steps (Nanjing University, 1978): a) Optimization of a single target variable: üSample classification: the training samples of the industry production are classified according to a single target variable, i.e. each variable corresponds to a certain standard. üMapping information: firstly, the noise samples are filtrated by applying belonging degree. Then, the best two-dimensional mapping graph of each of the good samples are obtained by applying the primary component analysis (PCA), the optimum decision plate (ODP), and the partial least square (PLS) separately. Thirdly, the best mapping graph is selected. The optimum industrial parameters of any single variable could be gotten in details from the best mapping graph of each of the single target. b) Comprehensive target optimization (Qin et al., 1980; Gong, 1979): üComprehensive classification of samples: the samples that met the indices of multi targets simultaneously are regarded as the first level; otherwise they belong to the second level. The classification pattern recognition research is conducted by applying belonging degree of samples and the back-propagation neural network (BPN). üOptimization direction: the target values of the sample pattern of the two class centers are forecasted with BPN. The two class centers can show the features of each of the sample class if the two target values differ greatly. Thus, the first class center corresponds to the stable and optimized sample pattern, which is the center of a high dimensional optimization space. The typical variable parameters

10 Multi-objective Systematic Optimization of FKNME 

of the first class center are the desired optimization parameters. The sample should be classified again if the difference between the two target values is small. To assure the reliability of the optimized class center, the self organizing feature mapping (SOFM) raised by Keinan is applied to find the class center clusters (CCC). The optimization result can be assured further if the mapping results can distinguish the two samples clearly. The optimization result of the characteristic variables that is centered in the class can be made by applying PCA, PLS or ODP, which can be verified further with the result gotten by SOFM. üThe optimization using GA and BP: the GA and BPN algorithms are applied to get better patterns. The fitness function is the trained neural network. The decimalization coding is used, the value scopes of the character variables are limited to ±5%, by which the optimum pattern could be obtained. Finally, the artificial neural network mathematical model is constructed with multi targets, in which the original character variables are the input parameters, the multi target variables are the output parameters and have a number of hidden nodes. The network structure can be obtained after being trained, and it can describe the training samples variables and the targets. The sample patterns of the two class centers and the performance indices of the optimum samples can be forecasted. c) Intelligent decision system (Wang and Dai, 1990): A key optimum parameter (variable A) that can be controlled strictly should be found out from the optimization industry parameter. As mentioned above, if necessary, an expert system can be constructed to forecast the optimum parameter according to other component parameters. üKnowledge express: Expressing of network knowledge: the target variables and other characteristic variables are the input parameters. A number of hidden nodes are constructed. The variable A is regarded as the output parameter. The neural network is trained by the collected sample data until the fitting precision reaches ±5%. Then the target variables are assured and the input variables is also fixed by the users. The value of A can be given when the users input the character variables which can be transferred to the neural network in graphical user interface (GUI). Knowledge expressing assisted by experience: the above neural network has such shortcomings as: the learning knowledge is too limited, as the learning sample numbers are not large enough; the industry parameter of the training samples has a value limit, if the value of the input parameters exceeds the scope, the neural network will give a less credible forecasting value because the network cannot extrapolate. Considering such a condition, the knowledge of the expert system depends on human beings. The rules summarized by people should be added into the rule base. Knowledge expressing by pattern recognition: whether the target parameter

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

forecasting by neural network is correct or not can be judged qualitatively by pattern recognition of classification. The force of the sample is calculated by regarding variable A and other character variables as the character parameters. The sample belongs to a good sample if its force is less than zero and an output parameter is gotten when the sample is considered as the input parameter, by which the sample can be tested whether it is good. üRunning process: the input parameters must be given when the system is running. The input parameters and sample character parameters are compared by the system. Variable A will be estimated with the experience rule if the input parameters exceed the scope of the later. The value of variable A will be gotten if the input parameters are in the scope of the later. The value of A is tested by the classification pattern recognition. The value of A is estimated by experience if A belongs to a bad sample. Otherwise it is directly showed in GUI if it was a good sample.

10.3.2 Consistent target approach The essence of the consistent target approach is that the target function f1(x), f2(x),…, fq(x)of Eq. 10.1 is united into a overall consistent target function f (x), i.e. (Liu, 1994) f (x)=f{f1(x), f2(x),Ă, fq(x) } (10.27) The problem of Eq. 10.1 is translated into such a pattern: min f (x) gu(x)İ0 (u =1, 2,Ă, m)

(10.28)

The optimum problem of a multi-object function can then be solved by converting it into a single target function. Such approaches can be taken in the process of minimizing the consistent target function f (x), so that each of the partial target function can approach its optimum value: a) The weight combined approach, also called linear combined approach or weight factor approach, i.e. a weight factor is introduced into the consistent target function in the process of combining the partial target function into the overall consistent target function, so that the difference of importance among the partial target functions, and the difference in the magnitudes and dimensions are taken into account. Therefore, Eq. 10.27 can be rewritten as. f (x ) =

q

∑w

i

f j (x )

(10.29)

j =1

where wi is the weight factor of the jth partial target function f j( x ) and it is larger than zero. Its value is decided by the magnitude and the importance of each target. The key to the approach is the choice of the weight factor.

10 Multi-objective Systematic Optimization of FKNME 

b) The target layout approach. The optimization value f j( x *) of each target function is obtained respectively. It is then adjusted properly according to the overall demands of the optimization design of the multi-object function, and ( 0) obtain the ideal optimization value f j . The consistent target function can be constructed as follows:

⎡f f ( x) = ∑ ⎢ j =1 ⎣ q

j

( x)

f

−f j

j

(0)

(0)

⎤ ⎥ ⎦

2

(10.30)

By Eq. 10.30, it means that the consistent target function f (x) gets the minimum value f j(0) when each of the partial target function reaches its ideal optimization value. The key lies in choosing the proper value of the f j( 0 ) ( j = 1,2,..., q ) . c) The efficacy coefficient approach. Here, each of the partial target function f j j ( = 1, 2,K , q ) is expressed in a proper efficacy coefficient η j ( j = 1,2,..., q ) , 0İη j İ1 (by η j =0, it means the worst; η j =1 is the best). The overall efficacy

coefficient is the average value of the coefficients η =

q

η

1

•

η 2 . . .η

q

That is, it can be used to evaluate the quality of the design scheme. Therefore, the optimum designing scheme is: η = q η 1 • η 2 . . .η q → max Thus, when the coefficient η =1, it means an ideal scheme is obtained. Otherwise, when η =0, it means the scheme can not be accepted. In the latter, there must be one efficacy coefficient of the partial target that equals to zero. Usually, 0.7< η j İ stands for good conditions. The overall efficacy coefficient η can be regarded as the consistent target function f(x) → max (10.31) f ( x ) = η = q η • η . . .η 1

2

q

The approach is very effective as it is intuitionistic and can be easily adjusted, although its calculation is a little complicated. Moreover, no matter what the magnitudes and the dimensions of the individual partial targets are, their values can be converted into the numerical values between zero and one. If there is any partial target function that is not good enough, its overall efficacy coefficient η  will be surely zero, which indicates the designing scheme cannot be accepted and the restricted condition or the critical value of the target function should be adjusted. Such an approach is especially applicable for conditions when the target functions are neither too big nor too small. d)The relative cost approach. With this method, all the q targets in the optimization problem of the multi-targets are divided into two kinds. The first includs variables such as consumption of raw materials, working hours, cost and weight, for which, the less the variables, the better the target value. The second are

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

variables including turnout, production values, profits and benefits. For such variables, however, the more the variables, the better the target value. If for the ⎤ ⎡ s first kind target, there are s terms ⎢ f j ( x )⎥ , and for the second kind there are ⎦⎥ ⎣⎢ j =1

∑

⎤ ⎡ q ˄q̢s˅terms ⎢ f j ( x )⎥ , the consistent target function can be expressed as ⎦⎥ ⎣⎢ j = s +1

∑

s

f ( x) =

∑f

j

( x)

j =1 q

∑

(10.32)

f j ( x)

j = s +1

It shows that the minimum value of f ( x ) is then the optimization value.

10.3.3 The main target approach The importance of individual targets is different in the multi-target optimization. As the key idea of the main target approach, the main targets should be considered first and the secondary targets must be considered simultaneously. With this approach, firstly, all the target functions of the multi-target optimization are rearranged according to their importance. The most important one should be put in the first. Secondly, the restriction optimization value of each target function is solved. The estimated proper values of all the other target functions, which are considered as accessorial restrictions, should be given based on the primary design. In this way, the restriction optimization of multi target functions is transformed into that of single target functions. The relative optimization values of entire problem can then be achieved (Liu, 1994). The mathematical model of restriction optimization value of the kth target function for the Eq. 10.1 is (1İkİq) min f k ( x ) = f k ( x * )

 u = 1, 2, ..., m ) g m + j ( x ) = f j ( x ) − f j ( x * ) İ

s .t . g u ( x ) İ

(10.33)

( j =1, 2, …, k̢1, k, k+1, …, q) where, f j ( x * ) is the estimated optimization value of other target functions besides minimizing, before getting the optimization value; it is the optimization value of the target function, after getting the optimization value. s.t.gu ( x ) is the accessorial restriction condition. In the situation with many target functions, when the target functions located in the front of the multi-target function arrangement have been reached the

10 Multi-objective Systematic Optimization of FKNME 

optimization values, the optimization process may be interrupted because no optimization value can be obtained for the next target functions. In such a case, an * excessive function Δ f j ( x ) could be added to the optimization value f j ( x * ) , by which the optimization precision will be decrease, to avoid the interruption. The accessorial restriction condition is then changed into g m + j ( x ) = f j ( x ) − [ f j ( x * ) + Δ f j ( x * )] İ ( j =1, 2, …, k-1, k, k+1, …, q) (10.34) In the practical optimization design, a right estimation and decision of all the designing indices can be made according to the basic requirements, and be rearranged in the light of importance. It is therefore not very difficult to apply the main target approach to practice.

10.3.4 The coordination curve approach In this method, the optimization scheme of the multi-target function optimization is searched in the whole designing space according to the coordination relation between the isoline and the restriction surface of the individual target functions. In solving the n-dimensional optimization of over two targets, a winding surface of coordination can be formed according to the coordination relations among the isolines of all the target functions. However, it is impossible to express the super winding surface by graphs. Only the variation scope of the target function can be given by the following mathematical model (Liu, 1994). min f j ( x ) (j =1, 2, …, q) s .t . h

v

( x ) = f v ( x ) − f v * İ (v=1, 2, …, q̢1; vĮj) g u ( x ) İ (u =1, 2, …, m)

(10.35)

where f v* is the expectation value of f v ( x )(v = 1,2,..., q) . The coordination curve approach is very useful for solving the target function with two contradictive target functions. The relation between the targets and the overall design scheme can be analyzed clearly by the coordination curve, by which the ideas to improve the design can be got, and the satisfactory result can be obtained.

10.3.5 The partition layer solving approach Differing from the minimum mathematical model of the common multi-target optimization, each target in Eq. 10.1 is not at the equal level; on the contrary, they has different preferential layer. As the model has the characteristics by considering target in each layer, the optimization is carried out by obtaining the optimized

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

solution for each layer according to the turns of preferential layer given in the model. The solution of the last layer is then the solution of the problem. This approach is called the complete partition approach. The simplest complete partition approach is (Hu, 1993): the target function f 1 ( x ) in the first preferential layer is minimized in the feasible solution field x, then the target function f 2 ( x ) in the second preferential layer is minimized in the optimization solution collection of the first preferential layer, and so on. Usually, the target function f s + 1 ( x ) in the s+1 preferential layer is minimized in the optimization solution collection of the sth preferential layer.

10.3.6 Fuzzy optimization of the multi targets The restriction conditions and the target functions of the usual optimization problems are very clear. However, they may be fuzzy in many practical problems. for example, the restriction conditions may be flexible and the restriction target functions are of a fuzzy data collection etc. The mathematical optimization problem under the fuzzy conditions, i.e., fuzzy restriction conditions or fuzzy targets is called fuzzy optimization (Kunii and Levenspied, 1969). It is impossible to get the maximum or minimum values for all the target functions at one point when there are many target functions. Hence, a compromise scheme has been adopted to get either the maximum or the minimum value of each target function as much as possible. To achieve this, the target function can be made fuzzy with the fuzzy mathematical approach. The details of the approach are as follows. Firstly, to solve the maximum value zi* of the single target zi (i=1, 2, Ă, r)

under the restriction condition Axİb xı n ⎧⎪ ⎫ z i* = max ⎨ z i | z i = c ij x j , Axİb xı ⎬ , ⎭ ⎪⎩ j =1 i=1, 2, …, r (10.36) This is the typical linear optimization problem of the single target. Secondly, the flexible index di is given for each single target zi, i=1, 2, Ă, r, di>0. The more important is the target, the smaller is the flexible index. In this way, each target will be made fuzzy. Supposing the fuzzy target Gi corresponds to the ith target zi, the subject function of Gi is defined as

∑

⎛ Gi(x) = gi⎜ ⎜ ⎝

n

∑

j =1

⎞ c ij x j ⎟ ⎟ ⎠

(10.37)

10 Multi-objective Systematic Optimization of FKNME  n ⎧ * ⎪ 0, ∑ c ij x j < z i − d i ⎪ j=i n ⎪⎪ ⎞ * 1 ⎛ * = ⎨1 − ⎜ z i − ∑ c ij x j ⎟ , z j − d i İ di ⎝ j =1 ⎠ ⎪ ⎪ n ⎪1, z *i İ ≤ ∑ c ij x j ⎪⎩ j =1 i =1, 2, …, r

n

∑c x ij

j

< zi*

(10.38)

j =1

r

Let G = Ι Gi , which is the fuzzy target of the typical linear problem; and D={ x|Axİ i =1

E xı}, which is the typical collection, i.e., the possible solution collection that

meets the restriction conditions, or called the feasible solution field. Thus, the fuzzy optimization solution of the typical linear optimization of the multi-targets can be obtained by the fuzzy discriminant. The fuzzy discriminant is Df =DģG. The best discriminant is x*, and x* is defined as the fuzzy optimization solution. x* satisfies

D f ( x * ) = max(u D ( x ) ∧ u G ( x )) = max G ( x ) xı ≥0

x∈ D

(10.39)

It can be concluded that the fuzzy optimization value x is the maximum point in the feasible solution field D. The above problem can be transformed into a typical linear optimization problem. Suppose: r

λ = G ( x ) = ∧ Gi ( x ) i =1

The typical linear optimization of multi- targets can be transformed as max λ with the restriction conditions of: n ⎧ ⎞ 1 ⎛ * ≥ λ , i = 1, 2, K , r ⎪1 − ⎜ z i − ∑ c ij x j ⎟ ı di ⎝ j =1 ⎠ ⎪ ⎪⎪ n ≤ b k , k = 1, 2, K , m ⎨ ∑ a ij x j İ ⎪ j =1 ⎪λ İ ≤1 ⎪ ı ≥0 ⎪⎩ λ ≥ 0, x ı

Or it could be expressed as with the restriction condition of

max λ

(10.40)

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

⎧ n ≥ z i* − d i , i = 1, 2, L , r ⎪ ∑ c ij x j − d i λ ı ⎪ j =1 ⎪⎪ n ≤ b k , k = 1, 2, L , m ⎨ ∑ a ij x j İ 1 = j ⎪ ⎪λİ ≤1 ⎪ ≥ 0, x ı ≥ 0 ⎪⎩ λ ı

(10.41)

Eq. 10.38 is a typical linear scheme problem to get the optimization solution ( x 1* , x 2* , L , x m* , λ * ) Eq. 10.38 is equivalent to the following formula maxλ of which, the restriction conditions are

⎧ n ≥ z i* − (1 L λ ) d i , i = 1, 2, K , r ⎪ ∑ c ij x j ı j =1 ⎪ ⎪⎪ n ≤ b k , k = 1, 2, K , m ⎨ ∑ a ij x j İ ⎪ j =1 ⎪λ İ ≤1 ⎪ ≥ 0, x ı ≥0 ⎪⎩ λ ı

(10.42)

n

It indicates that maxλ make the individual target functions

∑c

ij x j

(i = 1,2,Λ , r )

j =1

approach zi* the most closen. By solving Eq. 10.38, ( x 1* , x 2* , K , x n* , λ * ) can be obtained, in which λ* is the maximum value of λ , and ( x 1* , x 2* , K , x n* ) is the fuzzy optimization solution of the original typical multitarget linear optimization. z** = cx*, is the optimization solution of the target.

10.4 Technical Carriers of Furnace Integral Optimization The main forms of the furnace integral optimization technology applied in engineering are: a) CAD software for optimum design of engineering. b) Offline instruction system (or optimum decision support system) for furnace operation optimization. c) Online optimum control system or integrated optimum system integrating monitoring, control and management for furnace.

10 Multi-objective Systematic Optimization of FKNME 

10.4.1 Optimum design CAD In view of a given furnace, hologram simulation software packages could be developed by means of multifield coupling hologram simulation, which could be used to execute the simulation test after being verified by production practice or in situ measured data. Therefore, the best design parameters and design scheme could be found out by artificial optimizing or auto-optimizing program. As an example, the content and structure of six-field coupling simulation software package of aluminium electrolysis cell is shown in Fig. 10.2.

Fig. 10.2 Coupling simulation software package of aluminium electrolysis cell (Mei et al., 1996) (A, B, C, D: Simulation data and optimization information)

The principle of auto-optimizing or artificial optimizing (selecting artificially the optimal design scheme) using hologram simulation software is shown in

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

Fig.10.3. This is the operation process of optimum design aided by a computer (optimum design CAD).

Fig. 10.3 Optimizing by simulation experimentation

10.4.2 Intelligent decision support system for furnace operation optimization Intelligent decision support system has advantages such as low developing cost, small investment, fast effect and high efficiency, so it has been widely applied in assisted management, optimal decision making and process control. It is especially fit for multi-variable nonlinear system such as a complex production process with big random interference and accurate mathematical model being difficult to establish. 10.4.2.1 Introduction of intelligent decision support system

Decision support system is an interactive compute software system which is used to help decision makers to solve the unstructured or semistructured problems. An intelligent decision support system is an integration of expert system and decision support system, which can accomplish qualitative knowledge reasoning, quantitative model calculations, mass data processing and so on. It usually includes four components as shown in Fig. 10.4. Human computer interface component is used to receive, check and transform

10 Multi-objective Systematic Optimization of FKNME 

the users’ input information, harmonize communication in each component, and outputs the decision results and relates information according to users’ requirements. Data component is made up of a database and a database management system, and all needed data of decision-making are lodged in the database; the database management system is used to edit, amend and organize data, and connect database with other components. Data component supplies the data service for a decision making.

Fig. 10.4

Structure of decision support system

Model component is made up of a model base and a model base management system. Modal base is used to lodge the models for a system decision; the model base management system connects knowledge reasoning components and the model base, and manages model bases by establishing, adding and deleting, amending and calling models in it. Knowledge reasoning component is composed of the knowledge base, a knowledge base management subsystem, reasoning machines. It is the core of a decision support system. Knowledge base stores knowledge needed in forms of production rule, predicate logic, semantic network, frameworks, units and script and so on. As the main control unit of system, knowledge base management subsystem controls the synthetical application of all kinds of data, knowledge, model and method, and unifies all components. Reasoning machine can accomplish searching and matching for knowledge. 10.4.2.2 Example: intelligent decision support system for nickel smelting process

The technological process of some smelter is “submerged arc furnace-converter-slag cleaning furnace” (Xiao and Xie, 1997), as shown in Fig. 10.5. Obviously, it is a typical complex system with multi-variables, nonlinearity and big random interference. In order to realize energy saving and consumption reducing of production process, in this paper, an intelligent decision support system of nickel smelting process is developed based on the optimal decision

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

making model for a smelting process (Peng et al., 1994) introduced in Sections 4.3 and 4.4 and endpoint prediction model for the converting process (Peng, 1998), according to the technical route of “mathematical modelühologram simulationü synthetical optimization”. It provides decision support for various posts and various processes in full duration time from many aspects such as theory, calculation, scheduling, operation experience and so on. Practical applications showed that this system could guide production operation effectively, improve production performance indexes, obviously decline the unit production energy consumption, reduce the valuable metal loss in slag and auxiliary raw material consumption, promote metals resource utilization ratio. As a result, significant economic and social benefit were obtained. Besides, the system could also offer lots of statistical analysis charts on smelting processes, which provided decision makers with strong assistant tools for knowing production performance, analyzing influence factors of production indexes, and taking effective measures to improve production conditions (Peng and Mei, 1996; Mei et al., 1994; Peng et al., 1994, 1995).

Fig. 10.5 Technological process in smelting workshop

Comparing to the production indexes before the application of the system, the improvements achieved are as follows: a) The electric energy consumption per ton of calcine of submerged arc furnace was reduced by 14 kWgh, 2.3% lower than before; the nickel in smelting slag reduced 0.014, 6.4% lower than before. b) The weight of furnace charge smelt in converter was increased by 111%. c) The electric energy consumption per ton of converter slag was reduced by 182kWgh, 39.6% lower than before; the cobalt in smelting slag of electric furnace for cleaning slag was reduced by 0.022%, 23.7% lower than before; the nickel in smelting slag of electric furnace for cleaning slag was reduced by 0.043%, 61.4% lower than before. The results mentioned above showed that energy saving and consumption

10 Multi-objective Systematic Optimization of FKNME 

reducing effects of the system were obvious, and it reached the aim of the design. Besides, because of the application of the system, the furnace conditions were stabilized and its lifetime was prolonged, the accident rate was reduced, the consumption of raw materials such as quartz, electrode paste and refractories were reduced also. The overall structure of the intelligent decision support system for nickel smelting process is shown in Fig. 10.6.

Fig. 10.6 Intelligent decision support system for nickel smelting process

Man-computer interface communicates with users in session manner by multilevel menu driving. In database, all the data for production decisions and technological managements are stored. In knowledge base, the decision knowledge in forms of production rules, fuzzy decision rules and optimal samples is stored, which can be modified automatically by the system. In model base, the decision making models are stored. Model base management subsystem has self-learning and self-adaptive ability. Method base and its management system constitute the method base subsystem. All kinds of algorithms needed by the system operation are stored in the method base; method base management system links the knowledge base subsystem and the method base, and controls addition, deletion, modification and calling of methods in method base. Technology management subsystem analyzes technological parameters and manages the incoming material testing files by using the data in database, and accomplishes file management for process equipments, process technologies,

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

process accidents, process repairing, quality management and technology reforms. The structure of the model base subsystem is shown in Fig.10.7:

Fig. 10.7 Model subsystem

a) The material composition forecasting models are used to forecast the composition of materials in production, which overcomes the time-lag of the composition test data and supplies the necessary composition data for a optimal decision making. According to the prediction error and the recent composition change, the system can modify the forecasting model automatically and makes the forecasting results credible and accurate. b) The fuzzy adaptive optimization decision model of the submerged arc furnace production is used to optimize the input materials and the operation parameters for ore-smelting furnace. Because the modeling data come from a knowledge base, while the knowledge base is self-learning and self-adaptive, the model is a fuzzy adaptive optimal decision making modal. c) The fuzzy neural network optimization decision model of the slag cleaning electric furnace production is used to optimize the input materials and operation parameters of a cleaning furnace. The model is self-learning and self-adaptive because the final learning results are always the last decision-making model in system modeling. d) The converting endpoint prediction and control model (Peng, 1998) is used to predict converting endpoint and optimize converting parameters. e) The production materials balance and heat balance calculation model

10 Multi-objective Systematic Optimization of FKNME 

calculates the materials balance and the heat balance of furnaces, which can simplify users’ analyzing the flowing direction of materials and heat, and take the measures to improve production conditions and technology. The structure of a method base subsystem is as Fig. 10.8, the related algorithms that system running needs are stored in the method base:

Fig. 10.8 Method base subsystem

a) Fuzzy decision model recognition algorithm. It is used to establish the optimization decision model of proportioning and operation parameters in submerged arc furnace production. b) Fuzzy neural network weights learning algorithm. It is used to establish the optimization decision model of proportioning and operation parameters in slag cleaning electric furnace production. c) Time-series filtering prediction algorithm. It is used to establish the materials composition prediction model and the converting endpoint prediction and control model. d) Algorithms for solving linear equations. They are used to establish the material balance and heat balance calculation model of furnaces. e) Regression analysis algorithms, including generally used regression analysis algorithms such as multivariate linear regression, power series regression, exponent regression and logarithm regression. They are applied in the regression analysis and related analysis for all kinds of production data, which makes the users understand the production more easily and quickly. The structure of technology management subsystem is as Fig.10.9, the main

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

functions are as following:

Fig. 10.9 Technology management subsystem

a) Parameters control analysis. It provides a mass of statistic and analysis figures and tables about nickel smelting production of the whole smelting workshop, and gives all sorts of composition fluctuation figures of materials (calcine, quartz, vulcanizing agent, etc.), the relation curve of valuable metals content in slag (content of cobalt in slag, content of nickel in slag) and other content of slag compositions, and the power utilization curve of the electric furnace in the given time range, which helps the users know the dynamic production process directly and visually, and improve the product condition and technology in time. b) Balance calculation. It accomplishes the material balance and heat balance calculation of a nickel smelting submerged arc electric furnace, converter and slag cleaning electric furnace productions, and can print the material balance table, heat balance table, the chemical elements balance table and the composition tables of all sorts of materials for product plan and decision makers’ reference. c) Archives of incoming material inspection. It manages the information such as incoming data, product batch, quality, composition of material from other workshops by a computer. d) Archives of quality management. It manages all kinds of quality management information such as members in a quality management team of each sections and process, their work performance and production quality change by a computer. e) Archives of technology equipment. It manages information of technology equipment in workshop by a computer. f) Archives of technology. It manages information of all technologies in workshop and information of new technology and new equipment in the same domestic industry by computer. g) Archives of technology improvement. It manages technology improvement

10 Multi-objective Systematic Optimization of FKNME 

information of all sorts of equipments and technologies in workshop by a computer. h) Archives of technology accident. It manages all technology accident information in workshop by computer. i) Archives of technology check. It manages all technology check information in workshop by computer. Therefore, the main functions of the intelligent decision support system are as following: a) It collects formatively production data, statistically analyzes them and prints production management and statistic reports. b) Basic technical analysis of heat engineering process: calculate the elements distribution, material balance, heat distribution and heat balance in the smelting process, provide the material balance table, heat balance table, chemical element balance table and phase composition table of all sorts of materials so as to analyze flow direction of material and heat in smelting and improve the production condition and technology. c) Dynamic analysis of production index: perform dynamic analysis, regression analysis and related analysis to the main technology-economy indexes of individual furnaces, draw production index fluctuation figures, control figures, distributing figures and histograms so as to make decision makers realize production condition in time, analyze the factor which impacts the production indexes and improve the production condition and technology. d) Man-machine interactive optimizing decision-making in production with aim to save energy and reduce consumption: determine the optimized material proportion, production power of electric furnace and output amount of low-nickel matte and slag according to the submerged arc furnace production fuzzy self-adaptive optimization decision-making model; predict converting endpoint according to product condition of low-nickel and provide optimized converting strategy; determine the optimized material proportion, the load current and its duration in the smelting stage and cleaning stage, and output amount of cobalt matte and cleaned slag according to slag cleaning furnace fuzzy neural network adaptive optimal decision making model. The whole decision-making process is man-machine interactive, users can discretionarily change any decision-making result according to actual production condition, and system will make new optimal decision stages for reference automatically according to users’ requirements.

10.4.3 Online optimization system Because of the changes in the environment, the aging of equipment or catalyst, the fluctuation of raw material composition and so on, it is impossible for offline optimization to keep industrial process running optimally all the time. Hence,

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

online optimization system is needed. 10.4.3.1 Optimal control problem

Optimal control problem can be described as follows: Given a controlled system as: x&= f (t , x, u)

(10.43)

Initial state of the system at time t0 is: x (t0) = x0 (10.44) where, x is the state vector of the system; x&is the derivative of the state vector; u is the control vector; t is the time; the system usually is nonlinear and time-varying. Given the final state or goal set at the end time t1 is: x(t1)=x1 or g(t1,x(t1))=0 (10.45) Therefore, there needs searching an admissible control u(t ) ∈ U , t0İtİt1, where U is admissible control set, driven by an admissible control, by which the controlled system starts from initial state x0 at time t0 and reaches to the target state x1 or target set at time t1; meanwhile, the performance evaluation function (performance index) J should be minimum. This kind of problem is the optimal control problem. 1

J = S (t1 , x (t1 )) + ∫ L (t , x (t ), u(t ))dt 0

(10.46)

The performance evaluation function J reflects the quality and performance of the control system, and the first term S reflects the allowable deviation from the target, called the terminal-value performance index; the second term reflects the dynamic performance of the control system, called the integral performance index; the control function u * (t ) that makes J get the minimum value is called the optimal control, the roots track x * (t ) of the corresponding system state equation is called the optimal track. Therefore, the so-called optimal control is in fact the best control according to a performance evaluation criterion, the process of solving optimal control problems is the process of choosing an optimal control vector(or control project) from all optimal control vectors (or control projects). The performance index confirms the solving of the optimal control problem. 10.4.3.2 Introduction for common optimal algorithms

Researchers in math field and control field have carried out lots of studies on the solutions of the optimal control problems, and have obtained great achievements. Many solution methods have been used broadly in practice. For a controlled system with an accurate mathematic model, the main math methods to optimize it are classical differential method and variational method. In 1953~1957, when studying the multistage decision process optimization problem, Bellman found that the optimization decision sequence in the multistage decision

10 Multi-objective Systematic Optimization of FKNME 

process has properties as follows: no matter what the initial stage, the initial state and the initial decision are, taking the stage and state formed by the first decision as the initial condition, the subsequent decisions must constitute optimal decision sequence for the subsequent problems, which is the famous Bellman Optimal Principle; based on this, Bellman proposed dynamic programming solving for the optimal control problems. In 1956~1958, enlighten by Hamilton Principle of mechanics, Pontryagin proposed maximum principle on the foundation of classic variational method. Dynamic programming and maximum principle play important roles in classic optimal control theory (Peng and Mei, 1996). There are different solutions for different types of optimal control problems, the commonly used classic optimal algorithms include: variational method, Pontryagin maximum principle, gradient method, conjugate gradient method, Bellman dynamic programming, least square method, simplex method and so on (Gao, 1991; Xiao and Xie, 1997; Peng et al., 1994). Classic optimal control algorithms are mainly suitable for the deterministic control system with an accurate mathematic model, while the structures and parameters of most actual controlled objects are uncertain in some extent, and the change of a state possesses some randomness because of the effect of a disturbance, so it is difficult to solve this actual problems using the classic optimal control algorithms directly, therefore, the idea of developing an adaptive control system is proposed. So-called an adaptive control system is a control system which can measure continuously or periodically dynamic performance of an object and performance index of a system in the control process, then compare them with anticipant dynamic performance and given performance index, and make decision according to comparison result, that is, by changing structure and adjustable parameters or producing a control signal, make the system run in an optimal or suboptimal state no matter how the environment changes. The goal of the adaptive control system is to eliminate the effects of structure and parameters’ disturbance on performance of the system. The structure of an adaptive control system is as shown in Fig. 10.10.

Fig. 10.10 Structure of adaptive control system

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

Since 1970s, model reference adaptive system (Landau, 1985), self-tuning control system (Feng, 1986), learning control system (Chang, 1982; Katsuhiko, 1980) and other adaptive systems have gotten significant development both in theory and in practice. For complex controlled process of high orders, nonlinearity, long time delay, time varying and strong random disturbance, it is difficult even impossible to apply classic optimal control theory based on the accurate mathematic model. Therefore, based on the study of the human brain structure and intelligence characteristics, the idea of optimizing system by means of human-simulated intelligent control is proposed, guided by this idea, lots of fruitful research and develop work are performed (Peng et al., 1994; Zhao, 1996). By simulating the qualitative reasoning, fuzzy reasoning and knowledge-learning process of human brain, the theories and technologies of expert system, self-organizing and self-adaptive fuzzy control (Li, 1993), neural network adaptive control (Hu, 1993) were proposed, which were applied to solving the optimal control problems of complex process and achieved good performance.

10.4.4 Integrated system for monitoring, control and management Because of complexity, mutagenicity and uncertainty of industrial production processes, it is very difficult to implement overall automatic control. Applying synthetically modern management and decision technology, information science and technology, automation science and technology, system engineering technology and artificial intelligence technology, integrating expert group, data and information with computer technology, the computer information integrated system for monitoring, control and management can be constructed. The system integrates the information of three elements (that is, personnel, technology and management), information flow and material flow, can be used to elaborate overall advantages of enterprise and realize cooperative work, which has been an effective approach to develop the current potential, improve production technology indexes and improve the competitiveness for an enterprise. 10.4.4.1 System Structure and Working Principle

An integrated system for monitoring, control and management mainly includes: object layer, measuring and control layer, supervision and scheduling layer, information management and decision making layer and corresponding communication network. Its structure is shown in Fig. 10.11. Controlled object is the industrial production process. Various information reflecting production state is sent to the measurement and control layer by certain sensors and measurement instruments, production process are controlled directly by measurement and control layer.

10 Multi-objective Systematic Optimization of FKNME 

Measurement and control layer is also called digital direct control level (DDC), in general, it is composed of field computer control system (DCS), programmable logic controller (PLC), measurement, control network and so on. Its main tasks are: on-time measuring relational information and transmitting that to supervision and scheduling layer; receiving control orders from supervision and scheduling layer and controlling objects according to them. Supervision and scheduling layer, in general, is composed of a special computer system, short-range or remote communication network and its application software. Its main tasks are: providing accurate field production data for the information Fig. 10.11 Integrated system for monitoring, control and management management and decision-making layer on time; receiving, explaining and executing orders from the decision-making layer; according to the production plan, condition of raw materials, condition of production equipments and orders from decision-making layer, organizing production and executing optimal scheduling, so as to make full use of the production resource; condition monitoring and analysis, fault diagnosis, online adjusting and self-learning setting of control laws and control parameters, assistant decision-making and producing the optimal operation strategy; sending the control signal to the measurement and control layer. Information management and decision-making layer generally is composed of the computer management information system network and decision support system. Its tasks include: receiving data from the scheduling layer; based on these, predicting the development trend of important technology parameters, making optimal decisions so as to make production run in good state; sending orders to the scheduling layer; sharing related information of every links of production processes; managing the information about personnel, finance, marketing, equipment and production by means of a computer, doing statistical analysis on the related data, displaying and/or printing relational reports. To sum up, the connections among layers of the integrated system for monitoring, control and management are reflected in the information integration and exchanging. There are obvious forward information flow and feedback information flow between different layers. Meanwhile there are also relative independent information closed-loop circulation, which can run as a local information system. Hence, every layer of the system possesses good openness and expansibility.

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao 10.4.4.2 Example: nickel smelting workshop computer information integrated network system

Based on nickel smelting process intelligent decision support system, technology parameters online monitoring system and workshop computer management network, a nickel smelting workshop computer information integrated network system was developed, which gathers the functions of management, decision-making and technology parameters’ online monitoring in integral whole. By using it, the overall optimal decision-making of the production process in smelting workshop was achieved; meanwhile, the computerized management of production cost, production equipments and personnel matters were also realized. Therefore, it increased significantly the level of management and decision-making of the workshop, and laid a physical and technical foundation for developing the computer integrated management system of the whole factory. The structure of the smelting workshop computer information network system is shown in Fig. 10.12.The system adopts a two-level radial topological structure

Fig. 10.12 Topo of smelting workshop computer information network

10 Multi-objective Systematic Optimization of FKNME 

and a Novell net operation system, all application software are driven by menu and with prompt of Chinese character, so it possesses characteristics of simple structure, good compatibility and easy operation and maintenance. Workstation No.1 is a super workstation, it is also used as a mothball server. It is used to administer and maintain network. The Network operation system and application software are preinstalled in its hard disk, related data are copied termly from special server to it. Once the special server breaks down, super workstation can immediately be put into use as a server, so as to ensure the normal operation of network. Workstation No.2 adopts an industrial control computer, which connects with smelting workshop technology parameters monitoring network. Data of each monitoring spot are send to workstation No.2 through communication routes and intelligent communication card, and workstation No.2 sends the processed data to the network server, which provides data service for other workstations in the network. Other workstations adopt compatible microcomputer. The main software of system includes: a) Nickel smelting process energy saving and intelligent decision support system; b) Realtime displaying software of production technology parameters; c) Management software for production technology, quality, personnel matters, cost, equipment and so on. System functions include: a) Nickel smelting process energy saving and intelligent decision support system: taking energy saving as the goal, performing adaptive optimal decision-making for overall nickel production process of the“submerged arc furnace-converter-slag cleaning furnace”. b) Realtime displaying software of production technology parameters: using realtime measured data of technology parameters in the server disk, displaying time-varying curves of several key parameters such as transformer-oil temperature of the submerged arc furnace and slag cleaning furnace, working out the current and voltage of each electrode, the blast pressure and so on, and ringing the limit alarm for them in time. c) Management software for production and quality: managing routine production data and statistics reports, performing general production statistics analysis using computer. d) Cost management software: performing production cost statistics, calculation and analysis, outputting related cost report forms and analysis charts. e) Equipment management software: managing information of equipments in smelting workshop. f) Spare parts management software: managing information of spare parts.

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

g) Personnel matters management software: managing information of personnel matters.

References Chang Chunxin (1982) Conspectus of Modern Control Theory (in Chinese). China Machine Press, Beijing Computational Mathematics Group, Department of Mathematics, Nanjing University (1978) Optimization Methods (in Chinese). Science Press, Beijing Feng Chunbo (1986) Adaptive Control (in Chinese). Electronic Industry Press, Beijing Gao Rong (1991) General Optimization Design (in Chinese). China Coal Industry Publishing House, Beijing Gong Xifang (1979) Computational Methods of Optimization Control (in Chinese). Science Press, Beijing Hetsroni G (1997) Handbook of Multi-phase Flow and Heat Transfer (in Chinese). Metallurgical Industry Press, Beijing Hu Shouren (1993) Application Technology of Neural Network (in Chinese). National University of Defense Technology Press, Changsha Hu Yuda (1990) Practical Multi-targets Optimization (in Chinese). Shanghai Science Press, Shanghai Katsuhiko Ogata (1980) Modern Control Engineering (in Chinese). Science Press, Beijing Kunii D, Levenspied O (1969) Fluidization Engineering. Wiley, New York. Landau I D (1985) Adaptive Control: Model Reference Method. Translated by Wu Baifan. National Defence Industry Press, Beijing Li Youshan (1993) Fuzzy Control Theory and its Application in Process Control (in Chinese). National Defence Industry Press, Beijing Liu Honglin, Bao Hong (1999) Artificial Intelligent Optimization in Chemical Metallurgical Process (in Chinese). Metallurgical Industry Press, Beijing Liu Weixin (1994) Optimization of Mechanical Design (in Chinese). Tsinghua University Press, Beijing Mei Chi (1987) Principle of Transport Processes in Metallurgy (In Chinese). Metallurgical Industry Press, Beijing Mei Chi (2000) Handbook of Nonferrous Metallurgical Furnaces Design (in Chinese). Metallurgical Industry Press, Beijing Mei Chi, Peng Xiaoqi, Zhou Jiemin (1994) Fuzzy and adaptive control model for process in Nickel matte smelting furnace. Trans. of Nonferrous Metals Society of China; 4(3): 9~11 Mei Chi, Wang Qianpu, Peng Xiaoqi et al (1996) Simulation and optimization of nonferrous metallurgical furnaces (in Chinese). Transactions of Nonferrous Metals Society of China, 6(4): 19~23 Oeters F (1994) Metallurgy of Steelmaking. Germany: Verlag Stableisen Gmbh

10 Multi-objective Systematic Optimization of FKNME  Peng Xiaoqi (1998) The development and application of the intelligent decision technique for economizing electric energy and reducing consumption in nickel smelting and the integrated computer information network in smelting workshop (in Chinese), thesis(Ph.D). Central South University of Technology, Changsha Peng Xiaoqi, Mei Chi (1996) An intelligent decision support system in the operation process of electric furnace for cleaning slag. Journal of Central South University of Technology, 3(2) Peng Xiaoqi, Mei Chi, Zhou Jiemin (1994) An intelligent decision support system (IDSS) on the process of nickel matte smelter (in Chinese). Journal of Central South Institute of Mining and Metallurgy, 25(4): 526~529 Peng Xiaoqi, Mei Chi, Zhou Jiemin et al., (1994) An identification method of multivariable fuzzy control model and its application in the decision support system of smelting furnace (in Chinese). Control Theory & Application, 11(5): 582~587 Peng Xiaoqi, Mei Chi, Zhou Jiemin et al (1995) A fuzzy neural networks control model of multivariable systems and its application (in Chinese). Control Theory & Application, 12(9): 351~357 Qin Shoukang et al (1980) Optimization Control (in Chinese). National Defense Industry Press, Beijing Wang Yu, Dai Ruwei (1990) A method to build an expert system with artifical neural network (in Chinese). Chinese Journal of Computers, 13(5):391~396 Xiao Xingguo, Xie Yunguo (1997) Metallurgical Reaction Engineering foundation (in Chinese). Metallurgical Industry Press, Beijing Zhao Zhenyu (1996) Groundwork and Application of Fuzzy Theory and Neural Network (in Chinese). Tsinghua University Press, Beijing

 441 Simulation and Optimization on the Furnaces and Kilns for Nonferrous Metallurgical Engineering

Index

A

Circulating fluidized bed  236

Absorptivity  29

Critical fluidization velocity  236

Activation energy  39

Cumulatine probability function  33

Activation temperature  39

D

Anisotropic  12

Damkohler number  46

Ar(Archimedes number)  236

Damping function  22

Arrhenius law  39

Dense phase  236

Artificial intelligence model, AI model 101

Diffusion term  17

Average particle size  243

Dilute phase  236

B

Dilute phase model  244

Barreiros  52

Direct exchanging area  31

Bath smelter  2

Discrete transfer radiation model, DTRM 34

BET surface area  58

Dissipation rate  44

Beta function  45

Dissipation rate of turbulent kinetic energy 

Biot-Savart law  62

252

Black box model  7

Dissipation term  17

Blast smelting furnace  236

Double delta function  45

Boiler  102

Dynamic simulation  127

Boundary condition  137

E

Boussinesq  16

Eddy-break-up model, EBU model  45

Bubble assemblage model, BAM model 261

Effective radiation  31

Bubble clouds  259

Electric cleaning furnace  193

Bubble phase  259

Electric smelting furnace  175

Bubbling bed model  259

Electrical conduction equation  153

Bubbling fluidized bed  236

Electrode paste  176

C

Elkem-D  178

CAD (computer-aided design)  318

Elkem-S  177

CFD (Computational fluid dynamics)  6

Elkem-3X  178

Chaos  271

Elkem-T  178

  Index Emissivity  29

I

Emulsion phase  258 Enthalpy transport equation  28 Equivalent magnetic dipole method 68 ESCIMO (engulfment-stretching-

Integrated system for monitoring, control and management  330 Intelligent decision support system, IDSS  320

coherence-interdiff-usion-interaction

Intelligent fuzzy analysis  107

-moving observer model)  45

Isotropic  25

Enthalpy  28

J

Euler coordinates  244

Jet flow  12

Evans formula  168

K

k - ε model  16

Expert system  8 F Favre-averaging  13

Kolmogorov-Prandtl expression  17 Kronecker number  13 L

Fenimore mechanism  54 Fick’s law  39

Lagrange coordinates  253

Field model  50

Lagrangian method  244

Fixed beds  302

Laplace force  147

FKNME (furnace and kilns for nonferrous

Large eddy simulation, LES  70

metallurgical engineering)  1

Lillebuen formula  168

Flash (smelting) furnace  213

Logarithmic probability distribution  240

Flow field  11

Low - Reynolds - number k - ε models  21 M

Fluidized roaster  235 Frequency factor  39

Magnetic attenuation factor  67

Furnace and kilns  1

Magnetic dipole  68

Fuzzy neural network analysis  116

Magnetic field  69

Fuzzy simulation  104

Magnetic shielding factor  67

G

Magnus force  96

Gas-particle two-phase flows  235

Magnussen soot model  47

Gaseous combustion  42

Mathematic simulation  99

Generalized fluidization  90

Mathematical model  8

General Navier-Stokes equations  13

Mathematical model building  83

Gibb model  51

Matte  69

Grey box model  8

Mixed-is-burnt model  43

H

Mixture fraction  42

Heat conduction equation  133

Mechanism analysis  7

High temperature erosion  306

Monochromatic radiation heat flux  30

Hologram simulation  5

Monte Carlo method  28

Hybrid models  8

Moving bed  236

Index  Multi-field coupling  87

Rate constant  39

Multi-fluid model  245

Rate equations  267

Multi-purpose synthetic optimization  309

Raw coal (dry and ash free)  48

N

Reaction rate  51

Navier-Stokes equations  70

Reaction rate constant  268

Neural network model  119

Recirculation flow  12

NOx model  53

Relaxation time  96

Non-homoge nous turbulent flow  19

Renormalization group (RNG) k - ε

Normal distribution model  239 Non-slip model  245

model  25 Reynolds stress  16

Nukiyawa-Tanasawa distribution function  Reynolds stresses model  26 Robl formula  168

242 Nusselt number  22 O Oil combustion  52

Robust  16 Rosin-Rammler distribution  241 Runge-Kutta integration method  254 S

Online optimization system  327 Optical thickness  34

Scattering  35

Optimization  5

Schmidt number  308

Optimization methods  309

SER tubes  276

P

Sherwood number  260

Particle group trajectory model  245

Shrinking core model  268

PSIC method  48

Side ledge thickness  157

Particle source terms  247

Simple

Physical property  16

homogeneous

reaction

model 

268

Potential field  136

SIMPLE scheme  279

Prandtl number  19

Simulation  5

Pre-exponential factor  39

Simulation test  319

Probability density function  44

Single-phase fluid  246

Probability density function models (PDF

Slag  21 Small-slip model  245

models)  287 Pseudo-partial-premix-PDF model 287 Q Quality control functions  305 R Radiation coefficient  36

Soderberg electrode  176 Source term  56 Standard wall function  288 Suspension flow  250 T

Radiation model  28

Terminal velocity  236

Random cumulative  33

The Reynolds-averaging  13

Rate coefficient  51

Thermal system of the furnaces and kilns 3

  Index Three Āhighsā   Preface 

Two-equation model  18

Tower trays  216

Two Ālowsā Preface 

Tower type zinc distillation furnace  213

Two-phase simple bubble model  258

Transport equation  17

U

Turbulent diffusion  245

Unit output functions  300

Turbulent diffusion time  46

Unit output rate  300

Turbulent dynamic viscosity  202

V

Turbulent kinematic viscosity  93

Vertical shaft furnace  236

Turbulent models  15

Viscosity  13

Turbulent Prandtl number  19

Viscous stresses  13

Turbulent thermal diffusivity  202

Viscous stresses tensor  13

Turbulent diffusivity  41 Turbulent drift  247 Turbulent flow  11 Turbulent fluidized bed  236

W Weibull probability distribution function 241 Z

Turbulent kinetic energy  16

Zeldovich mechanism  55

Turbulent viscosity  93

Zone method  301