Engineering Physics and Mechanics: Analyses, Prediction and Applications (Engineering Tools, Techniques and Tables)

ENGINEERING TOOLS, TECHNIQUES AND TABLES SERIES

ENGINEERING PHYSICS AND MECHANICS: ANALYSIS, PREDICTION AND APPLICATIONS

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ENGINEERING TOOLS, TECHNIQUES AND TABLES SERIES Computational Methods in Applied Science and Engineering A.K. Haghi (Editor) 2010. ISBN: 978-1-60876-052-7 Engineering Physics and Mechanics: Analyses, Prediction and Applications Matias Sosa and Julián Franco (Editors) 2010. ISBN: 978-1-60876-227-9 Precision Gear Shaving Gianfranco Bianco and Stephen P. Radzevich 2010.ISBN: 978-1-60876-861-5 Hydraulic Engineering: Structural Applications, Numerical Modeling and Environmental Impacts Gerhard Hirsch and Bernd Kappel (Editors) 2010. ISBN: 978-1-60876- 825-7 Primer to Kalman Filtering: A Physicist Perspective Netzer Moriya 2010. ISBN: 978-1-61668-311-5

ENGINEERING TOOLS, TECHNIQUES AND TABLES SERIES

ENGINEERING PHYSICS AND MECHANICS: ANALYSIS, PREDICTION AND APPLICATIONS

MATIAS SOSA AND

JULIÁN FRANCO EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Engineering physics and mechanics : analyses, prediction, and applications / editors, Matias Sosa and Julián Franco. p. cm. ISBN 978-1-61324-550-7 (eBook) 1. Mechanical engineering. 2. Physics. I. Sosa, Matias. II. Franco, Julián. TJ146.E54 2009 621--dc22 2009038660

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Chapter 1

Solar Absorption Systems as the Foundation for the New Generation of Heat-Pumping, Refrigerating and Air Conditioning Technologies A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

1

Chapter 2

Causes of Failures and the New Prospects in the Field of Space Material Sciences A.I. Feonychev

147

Chapter 3

Modeling of Interaction Kinetics during Combustion Synthesis of Advanced Materials: Phase-Formation-Mechanism Maps B.B. Khina

237

Chapter 4

Identification and Control of Large Smart Structures Yeesock Kim, Reza Langari and Stefan Hurlebaus

295

Chapter 5

Transfer Processes in a Heat Generating Granular Bed Yu.S. Teplitskii and V.I. Kovenskii

361

Chapter 6

The Influence of ND Laser Irradiation Parameters on Dynamics of Metal Condensed Phase Propagating Near Target V.K. Goncharov, K.V. Kozadaev and M.V. Puzyrev

441

Chapter 7

Thermodynamic and Kinetic Study of Oil Shale Processing G.Y. Gerasimov, E.P. Volkov and E.V. Samuilov

473

Chapter 8

Radiation Induced Synthesis and Modification of Carbon Nanostructures G.Y. Gerasimov

493

Chapter 9

Monitoring a 22-Story Building under Severe Typhoons with Bayesian Spectral Density Approach Ka-Veng Yuen and Sin-Chi Kuok

509

vi

Contents

Chapter 10

Characteristics of Ohm Law for Metal at Low Temperature A.N. Volobuev and V.V. Galanin

535

Chapter 11

Calculating and Experimental Researches of Free-Flowing Substance Axisymmetric Movement as Quasi-Newton Liquid V.V. Lozovetsky, F.V. Pelevin and S.N. Leontiev

547

Chapter 12

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method Ming-Hung Hsu

559

Chapter 13

Multiclass Fuzzy Classifiers Based on Kernel Discriminant Analysis Ryota Hosokawa and Shigeo Abe

577

Index

599

PREFACE The study of engineering physics emphasizes the application of basic scientific principles to the design of equipment, which includes electronic and electro-mechanical systems, for use in measurements, communications, and data acquisition. Engineering mechanics is the basis of all the mechanical sciences - civil engineering, materials science and engineering, mechanical engineering and aeronautical and aerospace engineering. This new book gathers the latest research from around the globe in this field of study. The analysis of existing models of high-temperature synthesis (SHS) is presented with special emphasis on the kinetics of interaction in strongly non-isothermal conditions typical of SHS. A novel multiple model approach is also proposed in order to model and control nonlinear behavior of large structures equipped with nonlinear smart control devices. In addition, this book examines the description of the processes which take place during the interaction of neodymium laser radiation (moderate power density) with metal targets. Other chapters in this book examine the main features of oil shale transformation under thermal processing, recent progress in application of radiation techniques for the synthesis and modification of carbon nanostructures, a brief analysis calculating and theoretical models describing free-flowing substance movement, and a discussion of fuzzy classifier based on kernel discriminant analysis (KDA) for two-class and multiclass problems. As presented in Chapter 1, since 1992 at a number of conferences held under the aegis of UNO they have discussed the problem directly of humanity—the problem of global warming caused by the constantly increasing concentration of so-called greenhouse gases (GG) in the atmosphere [129, 130]. Burning fossil fuel as a source of releases of carbon dioxide, which is one of the main greenhouse gases, makes the greatest contribution to their continuous accumulation in the atmosphere. The greenhouse gases, the release of which is controlled by the UNO Convention, include methane; its source is also power engineering and decomposition of domestic and industrial waste. The UNO Convention (the climate convention) was adopted in 1992 in Rio de Janeiro at the UNO conference on the environmental protection and development and was devoted to adopting measures by the world community for smoothing the global warming caused by the increase of the GG concentration in the atmosphere. In December 1997 in Kyoto (Japan) at the third session of the conference of member-countries of the frame UNO Convention on climate change, the Kyoto Protocol was adopted which was ratified by participating countries (55 countries including Russia and Ukraine); it is these countries that provide 55% of global carbon dioxide releases [130].

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In Chapter 2, the methodical study of the crystal growth processes and electrophoretic separation of the biomixtures have been carried out under microgravity conditions. The mathematical simulation of technological processes was performed by dint of the NavieStokes equations, the equations for heat and mass transfer and the Maxwell equations for magnetic and electric fields with reasoned assumptions. Analysis of crystal growth by the Bridgeman-Stockbarger and moving heater methods has shown that these methods does not give the expected positive results due to particularities of the fluid flows and heat and mass transfer under microgravity conditions and zero gravity. New condition for the dopant concentration at the crystallization boundary is used under calculations and comparison of the calculation data and the results obtained in experiments on board spacecraft. Thermocapillary convection stability and the process of crystal growth by the floating zone method are studied with use of different control actions. A rotating magnetic field, additional fluid layer (encapsulation of crystallizing melt) and standing surface waves, generated by axial vibration are applied as the control action. Axial static magnetic field is additional applied in two last cases. It is shown that under optimum parameters of the external action on thermocapillary convection, dopant segregation in micro- and macro-scales can be significantly reduced. The new idea on eigenfrequency of convective cell is used for analysis of the calculation results. The analysis of special space experiment on continuous flow electrophoresis showed that the failures of experiments on biomixture separation with the help of this method are due to hydrodynamic instability of biocomponent jet by the action of vibrations and ponderomotive force in electric field. The considered modifications of the floating zone method, as well as the use of the standing surface waves generated by vibrating crystal in the Czochralski method can be used for crystal growth in terrestrial conditions. Combustion synthesis (CS), or self-propagating high-temperature synthesis (SHS) is a versatile and cost efficient method for producing refractory compounds (carbides, borides, intermetallics) and composite materials. During CS, interaction between condensed reactants accomplishes in a short time (~0.1-1 s) whereas the traditional furnace synthesis of the same compounds takes several hours for the same particle size and close final temperature. Uncommon, non-equilibrium interaction mechanisms were observed experimentally, e.g., the dissolution-crystallization route rather than the traditional solid-state diffusion-controlled (SSDC) growth of a continuous product layer separating the starting reactants. Despite extensive experimental and theoretical investigation, the interaction pathways during CS are not well understood yet. In Chapter 3, the analysis of existing models of SHS is presented with special emphasis on the kinetics of interaction in strongly non-isothermal conditions typical of SHS. It is shown that in the modeling works employing the most used SSDC kinetics of the product formation, the diffusion coefficients used for calculations exceeded the experimentally known values by up to 3 orders of magnitude in a wide range of temperature. New models are developed for two typical SHS-reactions, Ti+C TiC (CS of interstitial compound) and Ni+Al NiAl (CS of intermetallic compound), basing on the SSDC kinetics and independent data on diffusion in the product phase. For CS of TiC, all possible situations are analyzed. Elastic stresses in a spherical TiC layer growing on the Ti particle surface are calculated, and a criterion for transition to the non-equilibrium dissolution-precipitation route is obtained. For CS of NiAl, competition between the growth of solid NiAl and its dissolution in the liquid Al-base and solid or liquid Ni-base solutions is considered for non-isothermal

Preface

ix

conditions. The Ni-Al phase diagram is used for numerical modeling along with the temperature dependencies of phase densities. Simulation has revealed the limits of applicability of the traditional SSDC approach, which is based on the assumption of local equilibrium at phase boundaries. The criteria are determined for transition to non-equilibrium reaction routes, namely dissolution-precipitation with and then without a thin solid interlayer of NiAl between the parent phases. As a final result, phase-formation-mechanism maps for the Ti-C and Ni-Al systems are constructed in coordinates ―initial metal particle size-heating rate‖, which permit predicting a pattern of structure formation during interaction in the non-isothermal conditions typical of CS. The existence of uncommon interaction pathways, which were observed experimentally and debated in literature, is confirmed theoretically ex contrario. In Chapter 4, a novel multiple-model approach is proposed in order to model and control nonlinear behavior of large structures equipped with nonlinear smart control devices in a unified framework. First, a novel Nonlinear System Identification (hereinafter as ―NSI‖) algorithm, Multiinput, Multi-output (hereinafter as ―MIMO‖) AutoRegressive eXogenous (hereinafter as ―ARX‖) inputs-based Takagi-Sugeno (hereinafter as ―TS‖) fuzzy model, is developed to identify nonlinear behavior of large structures equipped with smart damper systems. It integrates a set of MIMO ARX models, clustering algorithms, and weighted least squares algorithm with a TS fuzzy model. Based on a set of input-output data that is generated from large structures equipped with MagnetoRheological (hereinafter as ―MR‖) dampers, premise parameters of the MIMO ARX-TS fuzzy model are determined by the clustering algorithms, while the consequent parameters are optimized by the weighted least squares algorithm. Second, a new Semiactive Nonlinear Fuzzy Control (hereinafter as ―SNFC‖) algorithm is proposed through integration of multiple Lyapunov-based state feedback gains, a Kalman filter, and a converting algorithm with TS fuzzy interpolation method: (1) the nonlinear MIMO ARX-TS fuzzy model is decomposed into a set of linear dynamic models that are operated in only a local linear operating region; (2) Then, based on the decomposed dynamic models, multiple Lyapunov-based state feedback controllers are formulated in terms of linear matrix inequalities (hereinafter as ―LMIs‖) such that the large structure-MR damper system is globally asymptotically stable and the performance on transient responses is also guaranteed; (3) finally, the state feedback controllers are integrated with a Kalman filter and a converting algorithm using a TS fuzzy interpolation method to construct semiactive output feedback controllers. To demonstrate the effectiveness of the proposed MIMO ARX-TS fuzzy model-based SNFC systems, it is applied to a 3-, an 8-, and a 20-story building structure employing MR dampers. It is demonstrated from the numerical simulations that the proposed MIMO ARXTS fuzzy model-based SNFC algorithm is effective to control responses of seismically excited large building structures equipped with MR dampers. As explained in Chapter 5, heat generating granular beds are practically an important type of disperse systems (beds of nuclear fuel microcells of atomic power stations, beds of solid fuel particles in layer burning, heat generating beds of biological origin, etc.). Heat, generated in solid particles, produces temperature fields of specific character in the system; it is influenced by a whole number of factors: heat release intensity, heat carrier filtration velocity, size of particles, etc.

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A full system of boundary conditions and the conjugation conditions for the temperatures of phases and pressure that take into account the preliminary heating of a heat carrier and the degree of its turbulence at the bed exit have been formulated. Using the analogy between the processes of convective heat and mass transfer, dependences for calculating the effective coefficients of the thermal conductivity of a granular bed have been obtained. The influence of heat release and gas compressibility on the granular bed resistance has been elucidated, and an engineering technique of its calculation has been worked out. Based on the analysis of the trends in the near-wall hydrodynamics of the bed, the dependence is obtained to calculate the coefficient of heat exchange of a granular bed with the environment that takes into account the near-wall thermal resistance; it has been verified in a wide range of Reynolds numbers. Within the framework of the two-temperature model of an infiltrated granular bed, basic laws governing steady heat and mass transfer over the space of a bed have been investigated for various types of heat generating. The criterion of ―quasi-homogeneity‖ was introduced. It allows one, on the basis of the operative conditions, to estimate the thermal state of the two-phase system. Mathematical simulation of the process of filtrational evaporative cooling of a heat releasing bed has been carried out. An engineering method of calculation of bed cooling has been developed making it possible to determine the position and size of the evaporation zone. Unsteady processes of the propagation of compression and expansion waves originating on a instantaneous increase (decrease) in the gas pressure at the granular bed inlet were investigated. Equations in a dimensionless form have been obtained to calculate a maximum and minimum temperature of a heat carrier, as well as the time of establishment of a new stationary state. In Chapter 6, erosion laser jet of metals with presence of condensed phase particles ejected from the target was investigated experimentally. Jets were generated during interaction of intensive laser irradiation with metal surfaces. Optimal intensities for precision and rapid laser processing were empirically determined for several metals. Mechanisms of drop-liquid phase formation were studied for the metal targets. Time delay between the start of bulk vaporization and the beginning of laser action was experimentally defined. Observed data allowed to reveal the fact of metal drops fragmentation near the target surface. The possibility of liquid phase parameters control with an implementation of electrical and electrical-magnetic fields was showed. Real-time way of nanoparticle separation during their formation and method of metal nanoparticle suspensions producing were described. Chapter 7 describes the main features of oil shale transformation under thermal processing. Thermodynamic investigation of oil shale gasification in oxygen was performed using the TETRAN computational software and associated database. In this way, adiabatic gasification temperatures were calculated as a function of oxygen excess. Optimal conditions of synthesis gas generation were determined. Kinetic model of oil shale thermal decomposition (pyrolysis) was constructed on the base of analysis of available experimental data. Model includes the kinetics of organic matter (kerogen) decomposition at high temperatures, the processes of heat and mass transfer inside of single oil shale particle, polydispersity of the particles, their fragmentation, and the secondary chemical reactions (cracking, condensation, and polymerization of the shale oil inside of the particles with coke deposition). The model was applied to engineering procedure that simulates the Galoter

Preface

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process (pyrolysis of oil shale in a horizontal rotary drum-type reactor in contact with finegrained solid heat carrier, namely, hot ash obtained from solid residue combustion). Chapter 8 gives a brief review of the recent progress in application of radiation techniques for synthesis and modification of carbon nanostructures. The review includes the examination of available experimental data as applied to radiation-induced formation of carbon nanostructures such as fullerenes, concentric-shell carbon clusters (onions), singleand multiwall carbon nanotubes, and graphene sheets. The considerable part of these experiments deals with modification of initial carbon materials under electron beam irradiation in an electron microscope, which is of paramount importance because it allows in situ observation of dynamic processes on an atomic scale. It is also discussed the modification of carbon nanostructures under irradiation: polymerization of fullerenes films, modification of carbon nanotubes structure, junction of crossing carbon nanotubes, etc. The research part of the work contains the theoretical estimations of carbon nanostructures stability under electron beam irradiation, which use the analytical approximation of the cross-section for the Coulomb scattering of relativistic electrons by carbon atom nuclei as well as results of classical molecular dynamics simulations. As explained in Chapter 9, typhoon is a frequently-occurred natural phenomenon. It is valuable to investigate the behaviour of the infrastructures in coastal cities under this severe aerodynamic condition. Two severe typhoons, namely Nuri and Hagupit, attacked the southern China coast in August and September 2008. Nuri and Hagupit passed by Macao from the northeast side and south side, respectively, generating significantly different aerodynamic condition to the infrastructures. The East Asia Hall, which is a 22-story 64.70 m reinforced concrete building in Macao, is investigated in this study. The floor layer is in Lshape with unequal spans of 51.90 m and 61.75 m and the height-to-width aspect ratio of the building is close to unity. Due to the particular geometry of this building, the structural response is sensitive to both the wind speed and the wind attacking angle. Its acceleration time histories were measured for the complete duration of these two typhoons. The Bayesian spectral density approach is applied to identify the modal parameters of the building and the excitation, such as the modal frequencies of the structure and the spectral intensity of the modal forces. Moreover, the associated uncertainties of these estimated parameters can be quantified by Bayesian inference to reflect the reliability of the identification results. It is important to distinguish whether the changes are due to statistical uncertainty or other factors. During the two typhoons, substantial changes appeared in the modal frequencies and damping ratios, but these changes recovered almost immediately after the typhoon dissipated. Aerodynamic effects such as vortex shedding will also be investigated. In order to develop a reliable structural health monitoring system, it is important to identify the factors, except for structural damage, that influence the modal parameters. The characteristics of Ohm law for a metal at low temperature are considered on the basis of research kinetics of charge carriers in the conductor, which can transit to a superconducting state. In work has shown that at decrease of absolute temperature to zero in the conductor some current can be conserved. According to quantum representations this current has identified with the current of superconductivity. The relation of the researched model with Ginzburg-Landau superconductivity theory is investigated in Chapter 10. Chapter 11 contains brief analysis of existing calculating and theoretical models describing free-flowing (granular) substance movement. There is given a survey of advantages and disadvantages of the analyzed models. In the paper, it is noted that the model

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suggested by V.N. Krymasov provides satisfactory results, which were proved by experimental data. Based on the model there are suggested the analytical expressions for the calculation of free-flowing substance movement and is justified the choice of numerical calculating scheme. Based on the data of the experiments the authors suggest the differential equation set. The authors justify the choice of boundary conditions describing two variants: one that takes into account the slipping on the boundary and the other – the so-called universal boundary conditions. The calculations made using the boundary conditions mentioned above are compared with the test results; the coincidence is satisfactory. The authors conclude that the suggested method may be used successfully for the calculation of hoppers of axisymmetric form where there is free-flowing substance movement. In Chapter 12, the natural frequencies of non-uniform beams are numerically obtained using the spline collocation method. The spline collocation method is an effective numerical approach for solving partial differential equations. The boundary conditions accompanied the spline collocation procedure to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalue problem. The beam model considers the taper ratio, , , inertia, boundary conditions, and other factors, all of which affect the dynamic behavior of non-uniform beams. In Chapter 13, the authors discuss a fuzzy classifier based on kernel discriminant analysis (KDA) for two-class and multiclass problems. For two-class problems, in the onedimensional feature space obtained by KDA the authors define, for each class, a onedimensional membership function and generate a classification rule. To improve classification performance of the fuzzy classifier, the authors tune the membership functions based on the same training algorithm as that of a linear support vector machine (SVM). A data sample is classified into the class with the maximum degree of membership. For multiclass problems, the authors define a membership function for each pair of classes and then tune the membership functions of each pair of classes by the same method as that for two-class problems. A data sample is classified as follows: calculate the membership degree of the sample for each class by taking the minimum value among the membership degrees associated with the class, and classify the sample into the class with the maximum degree of class membership. Through computer experiments, the authors show that the performance of the proposed classifier is comparable to that of SVMs and least squares (LS) SVMs, and show that they can easily analyze the behavior of the proposed classifier using the membership functions.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 1-146

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 1

SOLAR ABSORPTION SYSTEMS AS THE FOUNDATION FOR THE NEW GENERATION OF HEAT-PUMPING, REFRIGERATING AND AIR CONDITIONING TECHNOLOGIES A.V. Doroshenko1,a, Y.P. Kvurt 2 and L.P. Kholpanov2,b 1

2

Odessa state academy of refrigeration, Odessa, Ukraine, 65082, The Institute of Problems of Chemical Physics of RAS, Chernogolovka, Russia, 142432

Introduction Since 1992 at a number of conferences held under the aegis of UNO they have discussed the problem directly of humanity—the problem of global warming caused by the constantly increasing concentration of so-called greenhouse gases (GG) in the atmosphere [129, 130]. Burning fossil fuel as a source of releases of carbon dioxide, which is one of the main greenhouse gases, makes the greatest contribution to their continuous accumulation in the atmosphere. The greenhouse gases, the release of which is controlled by the UNO Convention, include methane; its source is also power engineering and decomposition of domestic and industrial waste. The UNO Convention (the climate convention) was adopted in 1992 in Rio de Janeiro at the UNO conference on the environmental protection and development and was devoted to adopting measures by the world community for smoothing the global warming caused by the increase of the GG concentration in the atmosphere. In December 1997 in Kyoto (Japan) at the third session of the conference of member-countries of the frame UNO Convention on climate change, the Kyoto Protocol was adopted which was ratified by participating countries (55 countries including Russia and Ukraine); it is these countries that provide 55% of global carbon dioxide releases [130]. Obligations to reduce the annual amount of GG releases are considered in the Kyoto Protocol from different viewpoints: participating countries undertook the obligation to reduce a b

E- mail address: [email protected] E- mail address: [email protected]

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

releases by the year 2000 by 6–8% (and refused from initial intentions to reduce releases by 15% because of their being unreal) and only 4 countries—Russia, Ukraine, Norway and New Zealand—can maintain releases at the level of 1990. It should be noted that it is much less than what is necessary for showing the rates of accumulating GG in the atmosphere [130]. Unfortunately, most of the countries have failed to reduce releases in recent years. Only Germany has a positive experience (due to the stopping of industry in the eastern zone), as well as Great Britain (coal was replaced by natural gas), France (atomic energy), in other industrially developed countries the releases continue to grow (for example, in Canada by 10%). Due to quite understandable reasons, releases were somewhat reduced in countries with a transitional economy (countries of the former USSR). The main measures to slow the global warming include: • • • •

increase of energy use efficiency; wide practical application of renewable energy sources; utilization of domestic and industrial waste (releases of methane); spreading of modern agricultural and forestry technologies.

The problem of global warming becomes decisive in perspective planning and developing traditional power engineering and hence in all energy-consuming branches of industry and agriculture without exception, changing and defining all the ideology of life support being made on the threshold of a new century. The main point in perspective planning power engineering will be determined by the use of energy-efficient technologies and the intensive development of alternative power engineering—the use of renewable energy sources. Refrigerating and air-conditioning machinery today consumes on average in industrially developed countries up to 30% of the total energy generated [20, 97-100]. Renewable energy resources (RER) will play an important role in the European energy structure in the near future. Till 2020 RER can become the only significant contribution to the supply of Europe with primary energy and will be able to provide over 50% of the world demand for energy till 2060 [70]. RERs are modern technologies which find all-round support from the public and have many advantages as compared to the usual energy sources used. Renewable energy resources make it possible to considerably reduce releases of CO2 and other pollutants connected with the activities of the energy sector of the economy. Besides, they increase the safety of various sources and decrease the dependence on imports. The development of an industrial base for deliveries to the potentially large market will help revive the regions of Europe where industry is falling into decay. The employment potential for renewable power engineering is almost 5 times lager than that for the fossil fuel. It provides employment at the local level and can play an important role in the regional development at the expense of introducing profitable and stable sources of income in rural areas. Spreading of renewable energy resources enables them to become a means of developing distant regions and communication between them. Renewable energy resources involve: •

Wind power engineering. The cost of electric power generated by wind power stations has sharply reduced in recent years, in many cases being less than 0.04 eku/kWh. This method will become more acceptable for the public, if problems of visual and noise affects of such stations, located near inhabited areas, are solved.

Solar Absorption Systems…

• •

•

•

• •

3

Wind power stations located near the seashore have commercial significance now; they have become more numerous in Denmark, and their development is planned in the United Kingdom and the Netherlands. It is wind power stations based on the shore that are preferable for the Ukraine, too, where wind potential is evidently insufficient on its main territory. 10% of the electric power for Europe can be generated by wind turbines which occupy no more of the Earth’s surface than the island of Crete [36]. Bioenergetics. The use of biomass makes a significant contribution to the supply of Australia and Denmark with energy and is very prospective for the Ukraine. Water power engineering. Water power stations make a significant contribution to the supply of Europe with electric power. Ecological effects and conflicts take place mostly in cases when large-scale systems are used. To avoid strong ecological influence, it is possible to design small water power stations. Though over 40% of a total potential of water power stations in Europe have already been used, there are many potential areas for placing small-sized water power station, especially in the Ukraine. There is also a potential for modernizing and restoring exiting systems. Solar power engineering. Passive and active solar heaters, solar thermal (hot water supply and heating) and photo-voltaic stations can make a considerable contribution to the energy structure of Europe. Photo-voltaic stations can provide 450000 MW power in the EEC, i.e. provide 16% of Europe’s demand for electric power. The covering of faces and roofs of buildings by photo-voltaic batteries create for the whole of Europe a potential, which is estimated only for roofs in 500 TW with existing technologies [18]. It is necessary to attract architects for designing buildings with the use of both passive and active solar heaters; it provides the conformed approach to the development of industrial infrastructure in order to stimulate the market. Mass production in Europe should result in considerable reduction of the price due to saving in sizes and increasing of the scales of manufacture. But as it was proved, the photocell cost can become competitive in reaching peak values of electric power cost during the nearest ten years [68]. The photocell production causes ecological problems and only the technologies using silicon are considered to be acceptable from the viewpoint of ecological consequences. It is important to widen the sphere of practical use of solar energy, and first and foremost, it concerns the creation of solar refrigerating and air-conditioning equipment. Wave power engineering. Energy potential of wave power stations for the EEC approximately amounts to 155 TW (which makes 6.7% of the current production of electric power in the European Community. Tide power engineering. The affect of tide power stations on the environment is the problem of great importance, and in many cases it exceeds the potential benefit. Geothermal power engineering. Geothermal power stations can be used only when there is no considerable influence on the sensitivity of ecosystems and where it is possible to form closed cycles.

The global warming problems, as well as problems peculiar to vapour compression refrigerating and air-conditioning equipment, caused by the necessity of developing ozonenondepleting working bodies, have aroused a great, increasingly growing interest to potentialities of open absorption systems which operate at extremely low temperature

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

differences and use solar energy as a heating source. Schematic solutions, configurations and designation of such systems are extremely various, just as the list of working bodies used in them (solid and liquid sorbents). The open cycle can be the basis of a new generation of refrigerating, heat pumping and air-conditioning systems which are wholly based on the use of renewable energy sources, such as solar energy, and provide a reduction of energy consumption simultaneously with ecological cleanliness of solutions used.

1. Development of Optimal Schematic Approaches and Selection of Working Parameters 1.1. General State of the Problem and Main Objectives of the Project The number of papers devoted to studying the capabilities of the open absorption cycle in conformity with problems of air-conditioning and refrigerating systems (pre-dehumidifying the air, evaporative cooling the media, solar regeneration of the sorbent) is extremely numerous and is continuously increasing. It is caused by the search for fundamentally new approaches in connection with quickly aggravating interrelated problems of power engineering and ecology. The wide practical use of renewable energy sources is referred to only as some known basic measures on slowing global warming. The practical use of solar energy is most prospective for European countries, the Ukraine and Russia in particular. It is connected both with comparative simplicity of equipment and operation of solar plants, and with a great amount of coming solar energy. The idea of solar cooling and air-conditioning has been known for practical use since the 1890s. Various versions of such systems were created in the USSR in the 1960s in Tashkent. A solar cooler using LiBr solution was manufactured in Brizbene, Australia, in 1958. Later on (1966) a solar house with the solar cooler was made in Queensland. In the USA about 500 solar air-conditioners were installed in 1976, and they operated just on solar energy during 75–80% of the year—the rest of the year on electric power or liquid fuel [90]. The following years showed the increasingly growing interest for the capabilities of solar systems. Some conferences of IIR/IIF (Jerusalem, 1982 [125]; "Advances in the Refrigeration Systems, Food Technologies and Cold Chain", Sofia, Bulgaria [27, 29], 1998; Symposium Nantes '98 "Hygiene, Quality and Security in the Cold Chain and Air-Conditioning", 1998, Nantes, France; "EuroSun 98"— the second ISES— Europe Solar Congress, 1998, Portoroz, Slovenia; 20th International Congress of Refrigeration IIR/IIF, Sydney, 1999 ) paid great attention to this problem. The program of solar heating and cooling, which has existed since 1977, up to now was one of the first joint projects of the International Energy Agency (IEA). The last decade was especially active in this field in Japan [90, 101, 102, 114] and the USA [90, 97-100]. One of the most important national programs on renewable sources of energy is the Japanese Solar Project which is funded by the Japanese Ministry of International Trade and Industry (MITI) by means of the organization of new energy development (NEDO). The American Department of Energy (DOE) supports solar cooling on a wide basis. Attention is paid to the development of collectors, closed and open absorption systems. In 1996, over 110 reports were made at the conference devoted to absorption heat pumps in Montreal, Canada, and only one of them dealt with the use of solar energy (open system using zeolite). The situation is somewhat worse in the EEC, but here too

Solar Absorption Systems…

5

the last years have been marked by growing interest for problems of solar cooling. An important conference devoted to solar energy systems, "EuroSun 96" was held in Freiburg, Germany, in 1996. Over 340 reports were submitted and only 5 of them were about solar cooling. In the work “Solar Energy for Building Air Conditioning” presented in Dresden, Germany, in 1994 50 participants, mainly from Germany, discussed 8 different papers balanced on the themes between open and closed systems. In the work “Solar-assisted Air Conditioning of Buildings using Low-grade Heat” presented in Freiburg, Germany, in 1995, more than 50 participants, mainly from Germany too, discussed 17 different papers: 5 researches were devoted to open systems and 5 to closed ones. However, 4 papers devoted to open systems were related to the commercial use, and only one to R&D, on closed systems— 2 papers are connected with industrial application, and 3 with systems being developed. "Munich Discussion Meeting on Solar-assisted Cooling with Absorption Type Chillers" was held in Munich, Germany, in 1995. International experts in solar cooling with the use of sorption systems discussed the fundamental item of the solar problem to find out the most promising trends. 38 participants from 10 countries submitted 13 papers among which there were 4 papers devoted to the development of solar collectors, 3 to open cooling systems, and 5 to closed systems [90]. Principal capabilities both of open and closed systems were manufactured in one-, two[52, 103] and three-stage version combinations of similar systems [58, 53] Open absorption systems operating at extremely low gradients of temperature and moisture content and at atmospheric pressure are more flexible in their operation, consume less power, and can ruin temperatures of the heating source from 60 to 100 °C [95, 96]. The idea of solar airconditioning both for comfort and technological use [97, 101] is rather prospective, in particular, due to the existence of certain correlation between the insulation and the required level of cooling (the complex of thermal-humid air parameters for AACS), it occurs at the peak of energy consumption during the day time, and therefore can be especially advantageous. One should note an active prospective market for solar-system air-conditioning that is easy to operate, has low energy consumption, and causes no harmful ecological consequences [53, 98]. A classical scheme for open systems is the so-called Pennington cycle [5, 9, 20, 98], in which the fresh air entering the system is dehumidified in the absorber and then is cooled in the process of direct evaporation and passes into the room. The air from the room is cooled by evaporation, provides fresh air cooling in the heat exchanger and then, after additional heating, provides the regeneration process in the desorber. There are lots of versions of this scheme simple enough and well-known since the beginning of the century. As a rule, such systems include as the main components [4, 9, 19, 20, 23, 26, 29, 39, 41-42, 47, 52-53, 55-56, 57, 63, 66, 73-74, 91, 92-93, 97-100, 101-102, 105, 106, 108, 114, 121, 125, 126, 128, 131]: the absorber (adsorber) where the air flow is dehumidified, the evaporative cooler of the direct and indirect evaporative type, and the desorber (regenerator) of the direct or indirect type, as well as systems of heat-exchangers the necessity of which is due to low temperature gradients [41-42, 52-53, 97-100]. In dehumidifying the air, its moisture content is reduced, and hence, the values of the wet-bulb temperature and dew point which provides the possibility for deep cooling in the evaporative cooler. For air-conditioning systems it means a possibility of providing comfortable thermal-humid parameters with the use of evaporative cooling methods only, without using vapour-compression coolers. In conformity with solar air-conditioning systems the main schemes can be of a ventilation mode, VM, and a

6

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

recirculation mode, RM, when a part of the air flow leaving the room is used to this or that degree for organizing the main air-conditioning process, or is simply mixed with the fresh air flow entering the room [20, 49]. Usually the amount of air recirculating in such a system is 10–20% [20, 23-24]. Solid sorbents (silica gel, zeolites, hydrides) [5, 9, 63, 66] and liquid sorbents on the basis of solutions LiBr-H2O, LiCl-H2O [4, 23, 26-27, 29, 30, 39, 41-42, 52-53, 63, 73-74, 89-90, 103, 126] are used with the number and variety of working media constantly increasing. To organize a continuos process in the case of using solid sorbents (solid sorption system), they use either switching adsorbents or drums rotating at a definite speed [9, 20, 66] whose sections are filled with the adsorbent, at constant and simultaneous pumping through different sectors of the drum the air flows being dehumidified and regenerated. The adsorber is characterized by small overall dimensions and good characteristics of the process, but has high resistance to the movement of heat carriers and requires much higher temperatures of regeneration. In this respect the use of liquid sorbents (liquid sorption systems) is more preferable, but the influence of working substances on the main microclimate characteristics and the construction materials stability is of importance. An interesting solution for the rotational film absorber (absorption heat-pumping system) has been used in the work [69], where a mathematical model of the absorption process was developed for such a rotating absorber. The solar regeneration of the absorbents in open systems can be direct [39, 101-102] and can occur in direct contact of the air flow and the absorbent film in the air solar collector, that is the absorbent regeneration here takes place under conditions of direct effect of solar radiation and the air flow. Here there is a danger of contaminating the absorbent and problematic character of providing heated air in amount needed for the regeneration [41-42]. Conclusions made in the work [42] with respect to advantages of systems with direct regeneration of the absorbent are not convincing and have not been confirmed in later researches. It should be noted that the comparative study made by these authors for systems with direct and indirect regeneration was based on the use of flat solar air collectors in both cases, which is not perspective for systems with indirect regeneration of the absorbent and is practically met nowhere else. The indirect regeneration provides the availability of the desorber with an external outstanding heat-exchanger or built-in heat-exchanger [52-53], to which the water heated in the heliosystem is supplied [73-74, 126]. In the absorber we can observe the opposite situation, cooling in required there. A cooling tower is most frequently introduced into the scheme for this purpose. The absorber with internal evaporative cooling is very appealing, however, its creation is connected with certain constructive difficulties [49, 97-100]. Along with apparatuses of direct evaporative cooling [39, 73-74] it is promising to use, as evaporative coolers, apparatuses of the indirect evaporative type [22-27, 29, 37, 52-53, 86, 94, 97-100, 126, 127], in particular, regenerative ones [24, 25, 66, 73-74]. In these apparatuses contact-free cooling of the main air-flow is achieved, that is cooling at the unchangeable moisture content, which is undoubtedly favorable for air-conditioning systems. The direct evaporative cooler [24, 37] can be used as an additional cooler after the indirect type cooler. Its use for the air-flow which has already been cooled and dry, does not caused any problem of an excess moisture in the room being conditioned. The most important matters defining the future of alternative solar systems is the creation of highly efficient (absorber, desorber, evaporative coolers, heat exchangers). As the number

Solar Absorption Systems…

7

of such apparatuses, included in the systems, is rather high, it will required considerable energy consumption for organizing the movement of heat carriers, that is the electrical power expenses for the operation of air fans and liquid pumps. From the thermodynamic viewpoint, allowing for small moving forces of processes characteristic of apparatuses; but it is, of course, connected with the increase of energy consumption and to some extent, it results in the depriciation of advantages of the principle used. Unfortunately, this problem is not considered in most of the works, which is evidently connected with today’s theoretical and experimental level of developments. It is perspective to use apparatuses of the film type providing separate gas and liquid flows in the multi-channel ordered packing bed of such apparatuses [22-32, 86, 97-100] for alternative solar systems. The cross-flow scheme of the flows interaction in apparatuses [24, 25, 49, 73-74, 127, 134] makes it possible to obtain additional advantages, as it is characterized by slight air resistance. We can say, that it is promising to use in such systems apparatuses with movable quasi-liqnefied packing bed which has comparatively high air resistance, but capable to operate in a stable regime in complicated, particularly, highly polluted media [24, 134]. It should be supported that the field of such apparatuses practical use is systems of considerable capacity. The possibility to use heliosystems with thermal solar collectors (SC) as external heating sources for providing solar energy regeneration [4, 23, 26-27, 29, 30, 39, 41-42, 52-53, 63, 73-74, 89-90, 103, 126] is of certain interest. These collectors can be nonevacuated flat collectors, evacuated flat or tube collectors, evacuated low-concentrating collectors, etc. They are rather expensive and today the development of solar energetics is under certain state support. Unfortunately, their efficiency decreases with the increase of the temperature achieved, and most of solar refrigeration technologies require the temperature level of about 100 °C. the situation is a little better for solar air-conditioning systems for which the temperature level of 60-100 °C can be sufficient [52-53, 97-100, 101-102]. It should be noted that the cheapest and most common type of the solar collector today – flat SC can provide only 50-65 °C and the operation on its basis even of the least fastidious in this sense, open solar system on the LiBr|H2O solution is rather problematic. Solar alternative systems require the creation of a compensation mechanism, connected with the problem of natural variations of solar activity. For such systems it is important to create efficient heat energy accumulators. It is also advantageous in this to developed various combined systems providing the possibility of joint using different sources of low-grade heat along with solar energy, gas and liquid boilers [95-96, 97-100]. The commercial situation with solar refrigerating and conditioning systems [90] is rather uncertain. It is explained not only by today’s stage of developing such new systems but the rushing change of priorities and contradictory estimations of the situation in this field. Solar alternative systems, as the available though rather limited experience of their practical use shows, [4, 101-102] can provide two-fold reduction of energy consumption [24, 101-102] as compared to vapour-compression coolers. They are undoubtedly deprived of ecological problems characteristic of traditional vapour-compression equipment (the global warming problem, ozone security) but they have serious disadvantages too – large overall dimensions, problematical character of solar regeneration only, hazards of corrosion effect on construction materials, the need for highly efficient energy accumulators. However, these drawbacks are characteristic of all alternative sources of energy without exception and the evaluation of prospects in this field must be based not on technical and economical comparison with

8

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

traditional approaches, well-known today, but proceeding from long-term perspective concerns where the importance of solar refrigerating and conditioning systems must be decisive allowing for the available and constantly aggravating energy-ecology problems.

Main Conclusions ‰

‰

‰

‰

‰

‰

it should be noted that the interest in capabilities of solar systems for air conditioning and media cooling is considerable and constantly growing. It is defined by the ecological cleanliness of these systems and low energy consumption. At the same time, these systems are at the stage of theoretical developments and experimental tests, and today we should speak about their prospective importance rather than a serious commercial aspect; along with various combined cycles the most prospective one is the open absorption cycle, based on the use of liquid and solid sorbents and the possibility of using solar energy as an external heating source; the preferable field of practical use of alternative systems based on the use of the open absorption cycle is solar air-conditioning, which is defined by a certain correlation between the insulation and the required level of cooling (the complex of heat-humidity air parameters for AACS), comparatively low as compared to refrigerating systems, temperatures of the sorbent regeneration and much wider prospective commercial demand for such systems [90], solar refrigerating systems require higher regeneration temperatures, they have a limited climatic zone of application (low moisture content of the outdoor air) [53]; solar air-conditioning systems—AACS—on the basis of the open absorption cycle using liquid sorbents (absorbents) have obvious advantages over the systems using solid sorbents: a lower required temperature level of the sorbent regeneration, the construction design simplicity, lower energy expenditures on the organization of the heat carriers movement; solutions of LiBr are referred to the most widely spread types of absorbents today; solar AACS with direct and indirect absorbent regeneration have their advantages but it is indirect regeneration that makes it possible to exclude the possibility of the sorbent pollution and to lessen the overall dimensions of the solar system on the whole; the main problems of solar AACS, requiring the urgent solution to enter the market in the future are as follows: 9 the necessity to improve the solar heat receiver (heliosystem with solar collectors), increasing its efficiency and attaining temperatures of 100 °C and more; 9 the use of new ideas in the field of the cooling part of the system—evaporative cooler, which is of a special significance for open systems and more important than in the case of closed systems; 9 the creation of compact heat-and-mass transfer equipment for open absorption systems, unified for all its main components and providing the minimization of energy expenditures on organizing the heat carriers movement and high compactness. Obviously, it is the most important task, the successful solution of which will influence the development of this trend in the near future; 9 the creation of combined systems for autonomous cooling (air conditioning) and heating, which are flexible and operative enough in control; 9 modeling operation processes in the main links of alternative systems and creating fundamentals of the alternative solar system design.

Solar Absorption Systems…

9

1.2. The Development of New Approaches The application of the open absorption cycle provides new possibilities for creating a prospective generation of engineering heat-and-cold supply systems—those of refrigeration, heat pumping and air conditioning. The cycle can operate at extremely small temperature differences, is ecologically clean and low power-intensive. Low-grade heat, natural gas or solar energy can serve as an external heating source. The solar energy source, in a practical sense, can be a heliosystem with flat solar collectors, i.e. the cheapest and most reliable type of heliosystem, developed by the authors [23, 26, 30] for hot water supply and including the necessary number of collectors and a tank-accumulator, depending on the required capacity [30, 118]. Versions of a schematic for an alternative system developed by the authors is given in figures 1.1–1.5 (in conformity with the problem of air-conditioning by AACS) on the basis of the open absorption cycle and solar regeneration of the absorbent. The scheme consists of two ) and refrigerating (denoted ). In main parts: dehumidifying air (denoted the dehumidifying part the heat needed for the absorbent regeneration is provided by the heliosystem with flat solar collectors 6 (7, 8, 9 are solar collectors, a tank-accumulator, an additional heating source, respectively), and cooling the absorber is provided by a fan coolingtower 5. The given schemes includes as the main components absorber 3 (the air dehumidifier), desorber 4, designed for solar regeneration of the absorbent; combined evaporative cooler 1-2 and a system of regenerative heat exchangers 10-12, 17, whose necessity is caused by low temperature heads available. Air flow 13 (fresh outdoor air) being dehumidified in absorber 3 decreases moisture content x g and dew point temperature t dp , which provides a considerable potential of cooling in the evaporative cooler. The system includes fan cooling tower 5, cooling the absorbent at the inlet to the absorber (figures 1.1-1.4) or (figure 1.5) an absorber with internal evaporative cooling development by the authors is used. As an evaporator is used the indirect evaporating cooling apparatus IEC [22, 23, 24, 26, 27, 29, 32, 94] developed by combined scheme in the form of the multi-channel packing bed with alternating «wet» (the auxiliary air flow and water recirculating through the apparatus) and «dry» channels (the main air flow cooled in the IEC at the unchangeable moisture content) (figures 1.1-1.5, figures 1.6-1.7 – position 1, IEC is a film cross-flow heat-and-mass transfer apparatus.). The amount of water evaporated water loop is replenish by feeding fresh water 15. As a results of evaporative cooling the water in “wet” channels, contact free cooling the main air flow in “dry” channels IEC is provided through a thin heat conducting wall which separates these channels. The wet bulb temperature at the inlet of IEC is a natural limit of evaporative cooling in a single-stage IEC. Figure 1.2 shows the schematic diagram of two-stage indirect evaporative cooler. The authors had studied theoretically and experimentally [22, 24, 32, 81] possibilities of combined coolers IEC. The number of such stages should not exceed three, as the further growth of their number results in a slight increase of the cooler efficiency at the considerable growth of energy consumption for running the process. It is undoubtedly useful to include into schemes the coolers of regenerative heat exchangers (HEX 12 in figure 1.1 and HEX 12 and 17 in figure 1.3), in this case, however, one should take into account the growth of attendant energy consumption. Figure 1.1 shows a combined evaporative cooler as a part of the IEC (the first stage of cooling) and the DEC (the direct evaporative cooler as the second stage of cooling) –

10

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

IEC/DEC. As it follows from previous studies of the authors [22, 23, 24, 32, 86] and American researches [37], such an evaporative cooler is promising for dry and hot climates. According to the authors’ data, its efficiency is slightly lower than that of the two-stage evaporative cooler in figure 1.2 but the latter requires much more energy consumption for organizing the process. The regenerative scheme providing a high efficiency of the process is of a special interest, but it is distinguished by high power intensity. In figure 1.1 it is a joint operation of the indirect evaporative cooler IEC (1) and the heat exchanger HEX 12, and in figure 1.2 it is given for the interpretation of the two-stage evaporative cooler scheme. If we allow, that the regenerative scheme of the indirect evaporative cooler can potentially provide cooling up to the dew point temperature of the air coming to the evaporative cooler, then the level of cooling can be rather high, taking into account its preliminary dehumidification in the absorber. The combined evaporative cooler CEC can operate both by the ventilation and recirculation schemes, but in the latter case the flow 13 at the inlet to the IEC is the air leaving the room being conditioned. The internal evaporative cooling of the absorber (figure 1.5) provides a high efficiency of the absorption process (3-6 times as much as that of the usual absorber [97, 100]), which makes it possible to considerably reduce the absorbent consumption and due to it to cut expenditures down needed for its regeneration and to increase the total C.O.P of the system by 30-35% [97, 100]. The main problem in designing such apparatuses is to separate liquid flows at the outlet from the absorber [97, 100]. It can be solved with the help of the indirect evaporative cooler design developed by the authors, in which packing bed components from closed channels thus providing the separate movement of air flows and the required tightness of channels (figure 1.8C). Of a special interest for cooling and air-conditioning systems is the scheme given in figures 1.3 and 1.4, in which cooling tower 5 is used as the second stage of the combined evaporative cooler. The air dried in absorber 3 and having a low dew point temperature (the limit of evaporative cooling in the regenerative indirect evaporative cooler IEC – HEX 12) is cooled, while the moisture content is unchangeable, in the IEC and enters the cooling is provided. This water can be used in ventilated heat exchangers-cooler (18) installed directly in air-conditioned rooms (20) or refrigerating chambers. In this case the refrigerating unit can be outside air-conditioned rooms and the building. Cooling tower 5 is also a film cross-flow device, and the refrigerating unit CEC includes, along with CTW, two regenerative heat exchangers for air flows leaving IEC and CTW (figure 1.7). Figures 1.8A-1.8B give the schematic description of the absorber (dehumidifier) and the desorber (solar regenerator of the absorbent) in some basic versions of the design. In figure 1.8A these apparatuses are shown as a version with placed outside (external) heat exchangers for cooling and heating the absorbent which corresponds to the schematic shown in figures 3.1– 3.4. In figure 1.8B apparatuses are made in combination with appropriate heat exchangers, which corresponds to schemes in figures 1.1–1.5 the packing bed of these apparatuses has the tube-plate construction, previously developed and investigated by the authors in conformity with evaporative condensers of refrigerating plants [24, 81], and in these developments the longitudinally corrugated sheet with regular roughness of the surface is also used. The design of all the components of the schemes (DEC, IEC, CTW, ABR, DBR) is unified. They are manufactured in the form of film counter- or cross-flow apparatuses [24, 81], in which a longitudinally (in the direction of the liquid film flow) corrugated sheet of the packing bed with regular roughness on its surface as an intensification method is used as a

Solar Absorption Systems…

11

part of the packing bed (figure 2.10). In this case the jet-film mode of liquid flowing (in the riffles of corrugation) and wet-dry mode of contacting gas and liquid flows are created. It provides the minimization of energy consumption and replenishment of the system with fresh water. The problems of such flows stability, relationship of heat-and-mass transfer surfaces, etc. have been studied by the authors theoretically and experimentally. The design of such units as the absorber, the desorber, and the cooling tower can also be executed in the form of apparatuses with a movable pseudoliquefied packing bed [24, 40, 134] developed and put into production by the authors in producing fan cooling towers, they proved to be good while operating in any contaminated media. This solution is preferable for large capacity systems.

Figure 1.1. Schematic of the alternative air conditioning system (AACS) with heliosystem as an external source of heating. Nomenclature: 1 – indirect evaporative cooler; 2 – direct evaporative cooler; 3 – absorber; 4 –desorber; 5 – cooling tower; 6 – heliosystem (7, 8 – solar collector, tank-accumulator); 9 – additional heating source; 10, 11, 12, 17 – regenerative heat-exchangers; 13 – fresh (outdoor) air; 14 – release into the atmosphere; 15 – water (replenishment of the system or fresh water); 16 – air in the room; 18 – fan-coils; 19 – rooms air; 20 – room; - - - - - - -dehumidifying part; -.-.-.-.-.-.-.-refrigerating part ( CEC).

12

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 1.2. Variant of the AACS schematic from figure 1.1. Evaporative cooler in the form of a twostage indirect-evaporative system (17 – additional regenerative heat-exchanger). Nomenclature is the same as in figure 1.1.

The heliosystem (figures 1.1-1.5) with flat solar collectors 7 is used as an external heating source during the absorbent regeneration. Such a system has been developed by authors (Research-Production firm “New Technologies”) and proved to be good for hot water supply. Figure 2.27 gives a general view of the developed solar collector SC. The heat absorbing panel of SC (figure 2.26 is made in the form of a pipe register. Pipes have float-type fins made of corrosion-resistant aluminium alloy. The SC includes hydraulic collectors (2, 3), the housing of shaped aluminium. The special construction of fasteners provides a simple and reliable method of installing glass 6. Two SC modifications with the area of the heat absorber

Solar Absorption Systems…

13

1.1 and 2.0 m2 are produced. One collector of SC-1.1 modification provides heating 80 litres of water up to 60-65 °C under the July conditions in Odessa, which is quite acceptable for the organization of the absorbent regeneration process. An additional heating source 9 is provided in the tank-accumulator 8 of the heliosystem (figures 1.1-1.5); it compensates natural variations of solar activity.

Figure 1.3. Schematic of the alternative air-conditioning system (AACS) with heliosystem (solar energy) as an external source of heating and feeding cold water to heat exchangers-coolers. Nomenclature is the same as in figure 1.1. Additional nomenclature: 17 – regenerative heat exchanger; 18 – ventilated heat exchanger-cooler; 19 – room air; 20 – conditioned room; - - - - - -unit of air cooling; -.-.-.-.-.-.-.-unit of evaporative cooling.

14

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 1.4. The schematic of the AACS evaporative unit on the basis of the two-stage combined system of IEC/CTW. Nomenclature is the same as in figure 1.1 and 1.3.

Solar Absorption Systems…

15

To meet the own needs of the system for electric power (pumps, fans) it is possible to use solar energy of photo-voltaic stations (direct transformation of solar energy into electricity). In this case the solution is completed with renewable ecologically clean sources of energy. It should be noted, that the experience gained by the authors in the development, production and use of heat-and-mass transfer apparatuses, heliosystems and photo-voltaic transformers of solar energy can greatly speed up process of bringing into production alternative system under consideration. As these elements were brought into production for other problems in the form of standard-size series of various unit capacity, there are no fundamental problems in putting AACS into production (as well as refrigerating and heat pumping systems) of any capacity and configuration. 6 8 7

7

9

7

15

13

4

1

IEC

Strong Solution

14

DBR

15 3

12

11

DEC ABR

10

2

DEC

HEX

13 15

16

14

HEX

13

Ambient air

14

HEX 13

Figure 1.5. Schematic of the alternative air conditioning system (AACS) with heliosystem as an external source of heating and absorber with internal evaporative cooling. Nomenclature is the same as in Figure 1.1.

16

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 1.6. Schematic description of the combined evaporative cooler CEC on the basis of IEC/DEC (A, B) and IEC/R (C). Notation: 1 – indirect evaporative cooler IEC; 2 – direct evaporative cooler DEC; 3 – HEX; 4 – ambient air; 5 – exhaust air; 6 – water tank; 7 – water (replenishment of the system or fresh water).

Solar Absorption Systems…

17

Figure 1.7. Schematic description of the combined evaporative cooler CEC on the basis of IEC|CTW. Nomenclature is the same as in figure 1.6. Additional notations: 12 ,13 – chilled water.

Figure 1.8. Schematic description of the absorber and the desorber (regenerator). A – apparatuses with placed outside heat exchangers, the cooler and the heater, respectively; B – apparatuses of the combined with heat-exchangers type; C – the absorber with internal evaporative cooling. Notations in figure 1.6, additional notations: 8 – the liquid distributor, 9 – the absorbent; 10 – the liquid pump; 11 – the recirculation water loop in the absorber; 14 – built-in heat-exchanger; 15 – absorber; 16 – desorber; 18 – the heat transfer medium (water) from the heliosystem; 19 – the dehumidified air flow.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

2. Simulation of Working Processes in the Alternative Air-Conditioning Systems 2.1. The Execution of the General Thermodynamic Analysis Thermodynamic efficiency of cycles is determined by their heat COP and degree of thermodynamic perfection θ

COP =

Q0 COP ,θ = , Q DBR (COP )K

(2.1)

where

Q0

–

refrigerating capacity of the cycle;

Q DBR

–

heat load of the regenerator (desorber);

–

heat coefficient of the Carnot cycle, drawn in the same temperature

(COP) K

range for heat supply and heat removal.

(COP )K

=

TH − TaA TH

⋅

TC TaA − TC

,

(2.2)

Refrigerating capacity and heat load of the regenerator are determined by the amount of moisture evaporated in the evaporator

( ΔG ) DBR

( ΔG )IEC L

and absorbed in the regenerator

L

Q 0 = ( ΔG L ) As

IEC

⋅ rIEC , Q D = ( ΔG L )

DBR

⋅ rDBR ,

( ΔG w ) IEC = ( x out − x in ) IEC ⋅ ( G g ) IEC , ( ΔG w ) DBR = ( x out − x in ) DBR ⋅ ( G g ) DBR

(2.3)

(2.4)

where

x out , x in

–

moisture content of the air at the inlet and outlet from the

corresponding device. As the moisture is not accumulated in the scheme under established working conditions

( x out − x in )IEC = (x out − x in ) ABR .

(2.5)

Solar Absorption Systems…

19

In the single-flow (by the air flow) scheme and atmospheric the following conditions are observed in apparatuses:

(G )

g IEC

( )

= Gg

DBR

= Gg ,

rIEC = rDBR = r. Then

COP =

Q0 Q DBR

=

(x out − x in )IEC = (x out − x in )IEC . (x out − x in ) DBR (x out − x in ) ABR

(2.6)

(2.7)

For an idealized cycle at zero under-recuperation at the ends of heat-and-mass transfer apparatuses

COPid =

x s (Tc ) − x min x aA − x min

,

(2.8)

where

x s (Tc ) – moisture, content of the saturated air at the outlet from the evaporator; x min

– minimum moisture content, corresponding to the partial pressure of water

x aA

vapour in the strong absorbent solution; – moisture content of the outdoor air, coming into the system.

Figure 2.1. COP as a function of

TH

for ϕ var.

20

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 2.2. θ as a function of

TH

for ϕ var.

Figures 2.1 – 2.2 shows the dependencies of COP and θ for an source temperature at different values of relative humidity of the outdoor air (it is accept that TC = 7 °C,

TaA = 30 °C). The relationships are draw for relative humidity values of air corresponding to wet bulb temperature (dew point) which is higher than the required temperature of cooling in the cycle. The case of lower relative humidity was not considered, as the required reduction of temperature is realized in the usual evaporative cooler under these conditions. As one can see from figure 2.1 the temperature of the heating source of 60 °C and above provides the acceptable COP values in the idealize cycle practically for all values of relative humidity of the outdoor air. In going to the real cycle, for the determination of moisture content values in corresponding points at the COP calculation it is necessary to introduce into calculations the value of heat efficiency η for apparatuses. For accepted in all devices the cross-flow scheme of movement η can be determined by F.Trefni equation with the accuracy sufficient for the analysis and engineering calculations [107, 124]:

η=

{ [

) ]} , ⋅ exp{− S ⋅ [1 + A ⋅ (1 − 2 ⋅ f ) ]}

(

1 − exp − S ⋅ 1 + A ⋅ 1 − 2 ⋅ f ϕ

(

)

1 + A ⋅ 1 − fϕ − A ⋅ fϕ

where

fϕ

– flow scheme coefficient for the accept scheme f ϕ =0.495;

S

– number of transfer units,

S= k w;

ϕ

(2.9)

Solar Absorption Systems…

A=

w max w min

21

;

w max , w min

– minimum and maximum (for the given apparatus) total heat capacity of mass flow rate.

In calculating the moisture content of the air flow at the inlet and outlet from the absorber, desorber (regenerator) and evaporative cooler it is necessary to previously determine the temperature of each flow at the inlet and outlet from the apparatus for the determination of COP of real cycles. For schematic solutions under consideration temperatures of each flow at the inlet into the apparatus T1′ and T2′ are usually known. The outlet temperatures of each flow T1′′ and

T2′′ can be determined by:

T1′′ = T1′ − E Φ ⋅ (T1′ − T2′ );

T2′′ = T2′ − E Z ⋅ (T1′ − T2′ ),

(2.10)

where

EΦ

EΦ = EZ

EZ =

–

efficiency of heating;

T1′ − T1′′ ; T1′ − T2′ –

efficiency of cooling;

T2′′ − T2′ . T1′ − T2′

Calculations of moisture content values for determining COP and θ of the real cycle with real heat and hydraulic losses in heat-and-mass transfer apparatus for a wide range of temperatures of a heating source and relative humidity of outdoor air will be carried out at further stages of the work.

2.2. Selection of Working Substances Taking into Account Thermophysical Properties To raise the efficiency of evaporative cooling process in AACS atmospheric air is predehumidified in the absorber. In case of high moisture content in the air at the system inlet its predehumidification is the only condition for the possible use of evaporative cooling to attain comfort air parameters. Aqueous solutions of calcium chloride, lithium chloride, lithium bromide as well as multicomponent solution on the basis of the above-mentioned substances are usually substances are usually used as absorbent in open system.

22

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The solution of CaCl2 is the cheapest, ecologically pure, well studied absorbent, still its application in AACS, as it will be shown below, is problematic. The solution of LiCl is widely used as an absorbent. Its main advantages are bactericidal effect and harmlessness for people. The air treated by this solution is highly sterilized – in dehumidified air the amount of microorganisms can be reduced to 97%. It is characterized by the capability to absorb harmful smells; the possibility to be regenerated by low-grade heat (low-temperature water from heat-and-power stations, industrial waste, solar energy); the possibility of significant reducing moisture content in the air being dehumidified; a wide range of working parameters [136]. The disadvantage of the lithium chloride solution is the corrosive effect on metals, which requires the use of special covers of surfaces or the introduction of inhibitors into the solution. Brass, aluminium, tin, etc. are not affected by the solution. Aqueous solution of lithium bromide (LiBr) [65, 136] has the best absorbing capability among the above-mentioned absorbents. Nevertheless, it is very agressive with respect to metals and other material and more expensive than the lithium chloride solution. The lithium bromide solution is most widely used as a working body in closed absorption systems. Data on the solution thermophysical properties in a wide range of parameters were obtained with the help of calculation equations given in papers [65, 136]. To achieve the required moving force in AACS the required concentration of LiBr should be 60–65%, i.e. the process line is quite near the crystallization line in the temperature range of 30–60 °C. In this connection, new working substances are being developed, which contain components increasing the solubility and decreasing the corrosive activity. LiNO3, ZnCl2, CaBr2, LiI and other components are used as additives. The system LiBr+LiNO3+H2O (4:1 LiBr:LiNO3 in moles) has been recently used, which, in the authors opinion, [65], decreases the corrosive activity and improves the cycle characteristic as compared to the solution of lithium bromide. The authors [65] have measured the density, the viscosity and other thermophysical properties of the solution in a wide range of parameters. The system LiBr+ZnCl2+CaBr2+H2O (1.0:1.0:0.13 LiBr:ZnCl2:CaBr2 in mass fraction respectively) is undoubtedly of interest as a working body, which equally with LiBr+LiNO3, decreases the corrosive activity and improves the cycle characteristics as compared to the solution of LiBr, as the authors show [64]. The said system, as well as LiBr+LiNO3 and LiBr, has a bactericidal effect and is harmless for people. Data on thermophysical properties of the solution in wide range of parameters are given in paper [64]. The saturated vapour pressure and the solubility of the five-component solution H2O+LiBr+LiI+LiNO3+LiCl were studied in paper [85]. It is difficult to make a conclusion concerning the efficiency of this solution, as the results of measurments presented in the graphical form for the temperature range, we are interested in, have a very small scale. Moreover, there are no data on thermophysical properties of this solution. This section gives the preliminary analysis of working substances on the basis of their thermophysical properties. Later on the solution will be designated: H2O+LiBr – LiBr; H2O+LiBr+LiNO3 – LiBr+; H2O+LiBr+ZnCl2+CaBr2 – LiBr++.

Solar Absorption Systems…

23

2.2.1. Vapour Pressure The vapour pressure

ps is the most important characteristic of working substances in

AACS. To raise the system's efficiency the relationship

ps (ζ ) should match the following

requirements: • • •

low pressure at the absorption temperature (~30 °C); strong temperature dependence of the vapour, and, hence, high vapour pressure at desorption temperatures (~60 °C in using solar energy as a heating source); steep form of the crystallization line.

The last factor providing high reliability of using heat-and-mass transfer equipment in the vicinity of the crystallization line, as the probability of salt deposits on surfaces of the equipment is reduced when the conditions of using are disturbed. As it was shown above, the use of the calcium chloride solution in open AACS is 1

1

problematic. It, for instance, the parameters of the surrounding air t aA = 30 oC, x g = 10 g/kg, (a typical situation for regions with moderate climate), the final moisture content at the outlet from the absorber is 1,6 kPa and

2

1

x g = 5 g/kg then partial pressure of the water vapour are equal to p g =

2

p g = 0,8 kPa, respectively. The surrounding temperature by the wet bubl is about

20 °C. With the reality of the heat-and-mass transfer processes in the cooling tower and heatexchanger between water-strong solution taken into account, the solution temperature at the inlet of the absorber is not less than 25 °C. The minimal partial pressure (on the line of crystallization) over the solution surface at this temperature is 0.93 kPa. Even in such an almost ideal situation (in reality the absorbent temperature will be higher) the moving force of the process becomes negative, therefore this solution won't be considered as an absorbent below. The vapour pressure of the LiBr++ solution at different temperatures and concentrations was calculated by Antoine formula [47, 65]:

log p s =

6

6

∑ A n ⋅ ζ + [1000/(T - 43.15)] ⋅ ∑ Bn ⋅ ζ , n

n =0

n

(2.11)

n=0

The vapour pressure of the LiBr+ solution at different temperatures and сoncentrations was calculated by Antoine formula [64-65] for temperature range 278.35 Т ≤ 335.95 К:

log p s =

4

4

∑ A n ⋅ ζ + [1000/(T - 43.15)] ⋅ ∑ Bn ⋅ ζ .

n =0

n

n

(2.12)

n=0

Figure (2.3) gives vapour pressures of aqueous solutions of LiCl(1), LiBr(2), LiBr+(3) and LiBr++(4) on two isotherms characteristic of open absorption cycle: 303.15 K (absorption)

24

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

and 333.15 K (desorption). At the same pressures on isotherms 303.15 K solutions LiBr and LiBr+ are father from the crystallization line than solution LiCl, therefore their use is preferable from the viewpoint of reliability of the equipment operation in AACS at the same moving forces of absorption processes. The steeper line of crystallization is also an advantage of solution LiBr and LiBr+. At high concentrations (over 45%) the LiBr solution has a lower pressure than the LiBr+ solution (the difference is about 0.1 kPa near the line of crystallization), i.e. it has a better absorption capacity. At the temperature 333.15 K and ζ = 47% (concentration on the line of crystallization at T = 303.15 K) the vapour pressure over the LiCl solution is 1.73 kPa. At this temperature and the similar concentration the vapour pressure over the LiBr solution is 1.34 kPa, and over the LiBr+ solution is 1.8 kPa. The fact that under the same conditions the lower desorption temperature is required for the LiBr+ solution is of great importance when low grade sources of energy are used as a heating source. Low corrosive activity is also an advantage of this solution.

Figure 2.3. Vapour pressure over the absorbent surface. _________ crystallization line.

The solution of LiBr++ is also of interest as a working body. As it follows from figure 2.3, it is characterized by: high solubility (79.2%) at T = 303.15 K, low pressure (p = 0.01 kPa on the crystallization line at T = 303.15 K), almost vertical position of the crystallization line. It one assumes that the pressure of the LiBr++ solution at T = 303,15 K (point B1) is the same as that of the LiBr+ solution on the crystallization line (point A1), then the pressure of LiBr++ (point B2) at desorbtion will be practically the same as that of the LiBr+ (point A2) solution, however, the line B1B2 is rather far from the crystallization line. Unfortunately, it is difficult to use almost vertical position of the crystallization line for the solution LiBr++ in AACS with solar regeneration, as a too high temperature is required for the desorbtion near the line of crystallization. This region is interesting for deep dehumidifying air. On the basis of analysis performed it is possible to make a conclusion about the perspectiveness of using solutions LiBr, LiBr+, LiBr++ as working bodies in suitable from

Solar Absorption Systems…

25

the viewpoint of reliability from the heat-and-mass transfer equipment. The working interval of this solution concentrations is approximately 70-75%.

2.2.2. Heat Conduction There are no experimental data on heat conduction of solution LiBr+ and LiBr++, therefore the heat conduction of these solutions was calculated by the formula [136]: h ⎛ ⎞ λ = λ 0 ⋅ ⎜⎜1 − ∑ β i ⋅ ζ i ⎟⎟ , ⎝ i =1 ⎠

(2.13)

where

λ0

–

is the heat conduction of water Wt/(m⋅K);

βi

–

are coefficients being determined by the data [136];

ζi

–

is the concentration of the i-th component, kg of substance per 1 kg of the solution.

The heat conduction of water is approximated by the expression [136]:

λ 0 = 0.5545 + 0.00246 ⋅ t + 0.00001184 ⋅ t

2

(2.14)

with the error of 0.01 Wt/(m⋅K) in the temperature interval from 0 to 100 °C. Coefficients

βi

obtained by mathematical processing the experimental data for solutions under consideration are: 1) for LiBr β = 0.16442; 2) for LiNO3 β = 0.19961; 3) for ZnCl2 β = 0.37108; 4) for CaBr2 β = 0.41701. Paper [65] gives the investigations of the LiBr+ solution in which the relationship between LiBr and LiNO3 was 4:1 in mole fractions. In this connection mole concentrations were converted into mass ones. In the mixture LiBr++ the relationship between components LiBr:ZnCl2:CaBr2 was 1.0:1.0:0.13 in mass fractions, so the concentration of each component was O.469×ζ, 0.469×ζ and 0.069×ζ.

2.2.3. Volumetric Heat Capacity The heat flow, which is transferred by the working substance at the temperature difference in the outlet and inlet of the apparatus equal to ΔT is defined by the well-known equation:

26

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

(

)

Q = ρ ⋅ C p ⋅ V ⋅ ΔT .

(2.15)

The heat capacity per unit volume or the product ( ρС p ) is a complex characterizing the heat carrier “transport capability”. The dependence of volumetric heat capacity on the concentration is shown in figure 2.4 for solutions LiBr, LiBr+ and LiBr++. In the range of working concentrations ρС p of the LiBr+ solution is slightly higher than that of the LiBr solution and equals about 3400 kJ/(m3⋅K) and greatly lower than that of the LiBr++ solution (volumetric heat capacity of the latter is equal to 5000 kJ/(m3⋅K) at ζ = 72%). Thus from the viewpoint of “transport capability” LiBr++ is also the best heat carrier.

Figure 2.4. Volumetric heat capacity of absorbents. The nomenclature is the same as in figure 2.3.

2.2.4. Viscosity Being a thermophysical property viscosity is of a special interest, as it defines to a great extent the mode of the working body motion in the heat-and-mass transfer equipment and affects the pressure losses value. Overviscosity does not permit organizing the turbulent flow at rational transport expenditures, and the transition to the laminar mode results in decreasing the heat transfer coefficient, hence, in increasing the surface of apparatuses at the same heat load. For heat exchangers of the plate type the critical Reynold’s number is equal to 200; the appropriate diameter of meshed-flow plates is 8 mm; the recommended flow rate is 0.25-0.8 m/s. Thus, the “critical” viscosity value is *

ν =

w ⋅ de Re

*

= 10

−5

m2/с.

(2.16)

Solar Absorption Systems…

27

Figure 2.5. Kinematics viscosity of absorbents. The nomenclature is the same as in figure 2.3.

Figure 2.5 shows the dependence of kinematic viscosity of LiBr, LiBr+ and LiBr++ solutions, solutions LiBr and LiBr+ have practically the same viscosity and are below the “critical” line. When the concentration is 70%, the viscosity of the LiBr++ solution is less than that of the solutions LiBr and LiBr+. When the concentration is 75% on the isotherm 303.15 K the viscosity of the LiBr++ solution exceeds the “critical” value. The “critical” value of concentration ζ* corresponding to the onset of the turbulent mode is 74%. Thus, from the viewpoint of viscosity the solution LiBr++ is most suitable, at the concentration of 70%, the working range of concentrations of this solution is from 70% to 74%.

2.2.5. Heat Transfer Factor At turbulent flow in plate heat exchangers heat transfer is described by criterion equation [61]

Nu = 0.135 ⋅ Re

0.73

⋅ Pr

0.43

⋅ (PrL /Prwl )

0.25

(2.17)

On the basis of (2.17) it is possible to form a complex consisting of thermophysical properties which characterizes the intensity of heat transfer. This complex is called a heat transfer factor and has the form: 0.57

Fα = 0.135 ⋅ λ

(

⋅ ρ ⋅ Cp

)

0.43

⋅ν

0.48

It is possible to determine the heat transfer coefficient with the help of

(2.18)

Fα

28

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov 0.73

w α = Fα ⋅ de Figure 2.6. shows the dependence of

(2.19)

Fα on the concentration for solutions LiBr, LiBr+ and

LiBr++. In the crystallization line area the solution LiBr has a larger value of the heat transfer factor than the solution LiBr+, but both solutions have less, than at a solution LiBr++ Fα at ζ = 70%. At the concentration of 75% the LiBr++ solution has a approximately the same

Fα as

the LiBr+ solution. On the basis of the analysis made it is possible to conclude about the perspectiveness to use the solution LiBr++ from the viewpoint of the highest intensity of the heat transfer process in the concentration interval of 70-75%.

Figure 2.6. Heat transfer factor.

2.2.6. Pressure Drop Factor The hydraulic resistance of “one run” of the plate apparatus can be calculated by similarity equation [61].

Eu L = 1350 ⋅ Re

−0.25

.

As in case of the heat transfer process, let us form complex

(2.20)

Fp characterizing the hydraulic

resistance of the apparatus

Fp = 1350 ⋅ ν

0.25

⋅ρ .

(2.21)

Solar Absorption Systems…

29

Hydraulic losses are connected with the pressure drop factor * in the following way: −0.25

Δp = Fp ⋅ d e The concentration dependence of complex

⋅w

1.75

.

(2.22)

Fp for solutions LiBr, LiBr+ and LIBr++ is given

in figure 2.7. Solutions LiBr and LiBr+ have practically the same values of concentration of 70% the LiBr++ solution has the same value of the line of crystallization, and at ζ = 75%

Fp . At the

Fp as the LiBr+ solution on

Fp of LiBr++ is by 30% larger. Thus, from the

viewpoint of the pressure drop the solution of LiBr++ at 70% has the same characteristics as LiBr and LiBr+ solutions, and at higher concentrations yields to them.

Figure 2.7. Pressure drop factor.

Conclusions: On the basis of the analysis made in this section it is possible to conclude that: • •

•

the solution CaCl2 is not suitable for use in AACS as an absorber due to too high vapour pressure values; among substances used as absorbents the most prospective from the viewpoint of thermophysical properties are aqueous solutions on the base of lithium bromide; in this case it is expedient to use additives reducing corrosive activity and increasing solubility (LiNO3, ZnCl2, CaBr2 etc); it is expedient to use solutions LiBr, LiBr+ and LiBr++ for solving air-conditioning problems;

30

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov •

the solution LiBr++ is the most preferable one from the viewpoint of heat-and-mass transfer characteristics, as well as the reliability of using (high solubility and relatively low corrosive activity). But at high concentrations it has a high temperature of regeneration. The working interval of concentrations is approximately 70–75%.

2.3. Modelling of the Working Processes in the Main Components of Alternative Air-Conditioning and Cooling Systems (AACS, ARFS) 2.3.1. General Principles of Designing Heat-and-Mass Transfer Apparatuses for Alternative System 2.3.1.1. Main Heat-and-Mass Transfer Apparatuses of Alternative System The low aerodynamic resistance during the transport of working substances (air and liquid flows) is the general requirement of heat-and-mass transfer apparatuses (HMA) for alternative systems, as the number of HMA and heat-exchangers (HEX) used in schemes is rather great. On the basis of the authors’ long-time experience in the development (theory and experiment), production and maintenance of HMA [24, 81], the film-type HMA was chosen as the main universal solution for all HMA of alternative systems. It provides separate movement of gas and liquid flows with low aerodynamic resistance. The cross-current scheme of contact between gas and liquid was chosen as the most suitable when it is necessary to put together numerous HMA and HEX in a single unit of apparatus. The crosscurrent scheme provides a smaller number of flow turns nodes and lower Δp values as compared to the counter-current scheme of contacting, as it is characterized by higher values of maximum permissible velocity of the gas flow movement in the channels of the HMA packing-bed [82, 83] (loss of stability, liquid drops removal by the gas flow from the working zone). The packing-bed of film cross-current HMA (figure 2.10) is formed by longitudinally corrugated (in the direction of the liquid film flow) thin-walled sheets of material, placed equidistant to each other. The surface regular roughness is used as the main method of intensifying processes of joint heat-and-mass transfer in HMA. Optimal values of roughness parameters are

k opt =

p = 8…14 , e

(2.23)

where p and e–are the spacing and the height of roughness fins, determined by the authors during theoretical and experimental studies of peculiarity of film flows of viscous liquid thin layers on surfaces of a complicated shape [10, 24, 27, 81, 82, 83]. Fins of roughness are evenly dispersed along the surface of longitudinally corrugated sheets of packing-bed (P and E are the spacing and the height of the main corrugation of pacing-bed sheets – figures 2.10, 2.20). Optimal values of the main corrugation parameters, densities of pacing-bed dimensions were determined by the authors. Within the framework of theoretical and experimental studies of the film two-phase flows stability problem the

Solar Absorption Systems…

31

maximum permissible values of gas flow movement velocities in [10, 83] were determined. They were used in calculations below (Section 2.3.2). The construction of all HMA, comprising the alternative systems under development, is unified (DBR, ABR, CTW, DEC). There is some difference in the construction of the IEC (figures 2.19 – 2.20), where the packing-bed is an alternation of «dry» and «wet» channels, designed for the movement of the main and auxiliary air flows, respectively. It is equally valid for the construction of DBR and ABR, shows in figure 1.8. The design scheme in figures 3.1 – 3.4 gives the simplified solution for these HMA with HEX 11 and 12 carried out. The variant of an absorber with internal evaporative cooling (figure 2.22), which was not discussed at this stage of the project, it of a special perspective interest. The problem in conformity with a film cross-flow heat-and-mass exchange apparatus, has been solved for the case when you take into account the real character of liquid film flowing along the longitudinally corrugated vertical surface forming the walls of «dry» and «wet» channels. In this case the film-jet flow with a characteristic distribution of “dry” and “wetted” sections of the surface (figure 2.20) is formed in “wet” channels of IEC and CTW. This circumstance must be taken into account in calculating the joint heat-and-mass transfer, and particularly in case of high heat conduction materials of the packing bed, as it occurs, for example, in IEC or in ABR with internal evaporative cooling (figure 2.22). These problems were studied by the authors in particular [24, 81, 82, 83], including the real character of the surface, the stability of film flowing at the interaction of laminar wave liquid film and the turbulent gas flow, the liquid film thickness, etc., in conformity with heat-and-mass transfer apparatuses included in alternative systems. 2.3.1.2. Heat Exchangers for Alternative Systems Because of low temperature heads AACSs characterized by rather a large number of heat exchangers construction should provide an optimal combination of heat efficiency, comfort and reliability of using at low capital investments and operating costs. The use of plate heat transfer to take place between two liquids. It is due to their following advantages: • • •

• • •

high operating reliability at moderate pressure (up to 2 MPa) and temperatures (up to 150 °C); high value of the heat transfer surface per a volume unit; high degree of the liquid flow turbulization between plates. Even viscous liquids can be pumped through winding channels under turbulent conditions at low Reynolds numbers (for most types of plates the critical Re number value does not exceed 200). It results in the intensification of heat transfer between liquid flows and reduces the formation of most varieties of sediments to the minimum; simplicity of disassembly and assembly during manual cleaning the heat exchanger; variety of materials used to manufacture the plates. this circumstance makes it possible to avoid corrosion of the plates while using aggressive absorbents; light pressure losses. Coefficients of pressure loses in plate heat exchangers are much greater than those in pipe heat exchangers at the same Reynolds numbers [120]. However, the velocities of the flow between plates are much lower, and the length of plates, necessary to attain specified values of heat transfer units number, is much

32

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov shorter than in pipes. Due to it pressure losses are usually lower under the same conditions than in flowing through pipes.

The construction of plate heat exchangers and geometry of plate surfaces are described in papers [61, 119, 120]. Tables 2.1 and 2.2 give the characteristics of most commonly types of plates. Table 2.1. Characteristics of band-flow plates [7] Characteristics of plates Sizes: L×B, mm Thickness of the wall, mm Heat transfer surface, m2 Mass, kg Appropriate diameter of the channel, m Cross-section area of the channel, m2 Plate spacing, mm Corrugation spacing along the flow, mm Number of corrugations on the plate Reduced lengh of the channel, m Cross-section area of the angular hole, m2 Diameter of the connecting union, mm

Р-5 (П-1) 800×225 1,2 0,15 1,4 0,005 0,0004 2 23 7 0,7 0,002 50

Р-11 (П-2) 1020×315 1,25 0,21 3,2 0,006 0,00075 3 22,5 7 0,8 0,003 50

Р-12 1170×420 1,3 0,4 5,6 0,008 0,0015 4 22,5 7 1,0 0,0045 76

Р-13 890×280 1,25 0,2 3,0 0,008 0,0008 3,5 23 7 0,8 0,002 50

Table 2.2. Characteristic of net-flow plates [7] Characteristics of plates Sizes: L×B, mm Thickness of the wall, mm Heat transfer surface, m2 Mass, kg Appropriate diameter of the channel, m Cross-section area of the channel, m2 Plate spacing, mm Characteristics of plates Corrugation spacing (mm): along the flow along the normal to corrugation Number of corrugations on the plate Reduced lengh of the channel, m Cross-section area of the angular hole, m2 Diameter of the connecting union, mm Slope of the corrugations to the vertical axis of symetry, °

ПР-0,2 650×650 1,2 0,2 3,6 0,0075 0,0016 3,8 ПР-0,2

ПР-0,3 1370×300 1,0 0,3 3,2 0,008 0,0011 4 ПР-0,3

ПР-0,5Е 1380×500 1,0 0,5 5,4 0,008 0,0018 4 ПР-0,5Е

ПР-0,5М 1380×550 1,0 0,5 5,6 0,0096 0,0024 5 ПР-0,5М

20,8 18 21 0,44 0,0082 100 60/30

20,8 18 50 1,12 0,0045 50 60

18 16 66 1,15 0,017 150 60

20,8 18 66 1,0 0,017 150 60

While designing AACS it is convenient to define heat carrier temperatures at the outlet of the heat-exchanger, assuming its efficiency with the further determination of the heat

Solar Absorption Systems…

33

exchange surface, and constructive parameters. The efficiency of the heat exchanger through which two heat carriers pass: heated (1) and cooled (2) is defined by the relation:

( ) ( = (G ⋅ C ) ⋅ (t

out

in

) ( ) ⋅ (t − t ) = ) /(G ⋅ C ) ⋅ (t − t )

E = G ⋅ C p ⋅ t1 − t1 / G ⋅ C p 1

p

2

out 2

in

− t2

min

in

in

2

1

p min

in 2

in 1

,

(2.24)

where

(G ⋅ C )

p min

is the least of

(G ⋅ C ) and (G ⋅ C )

p 2 values.

p 1

The preliminary thermal and hydraulic calculations were carried out for heat exchangers with plates given in tables 2.1 and 2.2 for aqueous solutions of LiBr+ZnCl2+CaBr2, LiBr+LiNO3 and LiBr when the absorbent concentration ξ = 55% and the mass flow rate of solutions G = 10000 kg/h for efficiency values from 0.7 to 0.9. The velocity of the heat carrier movement has been defined by the formula

ω=

G , 3600 ⋅ ρ ⋅ N ⋅ f

(2.25)

where N is the number of channels. Let us find the Reynolds number changing, it necessary, the velocity to attain the turbulent conditions. The values of the Nusselt-number are calculated using the equation of similarity in a plate apparatus [61, 119]

Nu = 0.135 ⋅ Re

0.73

⋅ Pr

0.43

⎛ Pr ⎞ ⎟ ⋅ ⎜⎜ ⎟ Pr ⎝ w1 ⎠

0.25

.

(2.26)

The relations known were used to find coefficients of heat transfer from both heat carriers and the coefficient of heat exchange. Then the heat flow was defined

(

Q = G ⋅ Cp ⋅ t

out

−t

in

)

(2.27)

and the total area of the apparatus too

F=

Q . K ⋅ Δt

(2.28)

34

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

On the basis of performed calculations the number of plates was found and the lay-out diagram of the apparatus was made. The one pass pressure loss for plates of Р-5 (П-1), Р-11 (П-2), Р-12 and Р-13 marks was calculated by the equation of similarity [61]

Eu = 1850 ⋅ Re

−0.25

,

(2.29)

.

(2.30)

and for plates ПР-0.2, ПР-0.3, ПР-0.5Е and ПР-0.5М

Eu = 1350 ⋅ Re

−0.25

The pressure loss in one pass was found by the relation 2

Δp = Eu ⋅ ρ ⋅ ω .

(2.31)

The technico-economic analysis and the selection of the optimal type of plates for the plate heat exchanger were made with the help of above-mentioned relations at the efficiency of Е = 0.8 for solutions LiBr+ZnCl2+CaBr2. The cited expenditures were taken as a criterion of optimality. The calculation of the cited expenditures was carried out for some tentative prices for electric power, equipment, etc. With other prices quantitative results will change, the method of calculation and the qualitative consideration will be the same. The cited expenditures can be calculated by the following expression [51]

Z = K ⋅ E n + C a + C r + τ ⋅ Ce ⋅ Wpump ,

(2.32)

where K is the cost of a heat transfer apparatus, hryv (К = 135⋅F, hryv.);

En

is the normative coefficient of the capital outlays efficiency, ( E n = 0.15);

Сa

I the part of the heat exchanger cost, allocated annually for its depreciation ( Сa =

0.125⋅K);

Сr

is the same for its current repair ( Сr =0.45⋅K);

τ is the number of the heat exchanger operation hours a ear (τ = 8000);

Сe

is the electric power cost ( Сe =0.15 hryv/(kWt⋅h));

Wpump

is the pump capacity depending on the pressure loss, mass flow rate and number

of passes in the heat exchange apparatus ( Wpump at n = 1).

W1 was calculated by the formula

= W1 ⋅ n , where W1

is the pump capacity

Solar Absorption Systems…

35

G ⋅ Δp , ρ ⋅ 3600 ⋅ η pump

(2.33)

W1 = Where:

ηpump is the coefficient of performance of the pump ( ηpump = 0.6).

Figure 2.8. Dependence of the referred expenditures on the heat exchanger area. Plate type: 1-ПР-0.2; 2-ПР-0.3; 3-ПР-0.5Е; 4-ПР-0.5М; 6-Р11; 7-Р12; 8-Р13.

Figure 2.9. Dependence of the area of the heat exchanger with ПР-0.3 plates on its efficiency; ________ solution LiBr+; - - - - - - LiBr++.

36

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The calculation results are given in figure 2.8. The representation of relation Z(F) in the form of continuous lines is to some extent tentative, as the heat exchange surface is a discrete value multiple of the area for one plate of the said type. Such a representation has been used to make the analysis visual and convenient. The use of ПР-0.2 plates results in expenditures indicated, which are beyond the limits of the figure. It follows from the figure that the use of ПР-0.5Е plates (curve 3) allows the minimization of the said expenditures while the surface of heat exchange is relatively small. Figure 2.9 gives the dependence of the heat exchanger surface on its efficiency for solutions of LiBr+ (weak and strong solution) and LiBr++. As it should be expected, according to ideas given in Section 2.2, at the same efficiency the solution of LiBr++ requires a smaller surface than LiBr+, though this difference is comparatively small (∼6% at Е = 0.8). For heat transfer to make place between two gas flows it is expedient to plate-fin heat exchangers. The thermal and hydraulic calculations for these heat exchangers have been carried out by the method described in paper [6].

Conclusions •

•

The thermal and hydraulic calculations have been made for heat exchangers being components of AACS. On the basis of the technico-economic analysis performed the type of heat transfer plates corresponding to the minimum of the said expenditures has been selected. It has been shown, that the use of the LiBr++ solution results in smaller heat exchange surface as compared to the rest of the absorbents discussed in the report. This conclusion agrees with the results of the analysis of working bodies’ thermophysical properties.

2.3.2. Peculiar Features of Wave Film Flows of a Viscous Liquid Along the Longitudinally Corrugated Vertical Surface of Heat-and-Mass Transfer Apparatuses 2.3.2.1. Wave Flow of a Viscous Liquid Thin Layer Along the Flat Vertical Surface Having Regular Roughness To calculate HMA it is necessary to know flow wave parameters as they are included in calculation formulas for the determination of mass transfer coefficients [10, 24, 77-79, 81]. Therefore the theoretical description of a liquid film flow along the surface with RR, when the interaction with a gas flow is taken into account, is timely. Packing beds of HMA under discussion (figure 2.10) consist of equally separated from each other flat or corrugated along the liquid flow sheets with counter flow RR made in the form of ordered bulges having specified configuration. The section of such a surface perpendicular to the plane, will be a flat line whose equation is a periodic function, moreover, a continuous one. It can be given in the form of a Fourier series [81]

f (z ) =

a0 ∞ ⎛ nπz nπz ⎞ + ∑ ⎜ a n ⋅ cos + b n ⋅ sin ⎟, 2 n =1 ⎝ e e ⎠

Solar Absorption Systems…

37

where e is the function half-period; e

1 a 0 = ⋅ ∫ f (z )dz ; e −e e

e

1 nπz 1 nπz a n = ⋅ ∫ f (z ) ⋅ cos dz ; b n = ⋅ ∫ f (z ) ⋅ sin dz ; (n=1,2,3,...). e −e e e −e e

Figure 2.10. Schematic of gas and liquid cross flows (film-jet liquid flow) in channels of the regular packing bed (RP) of the head-and-mass exchanging apparatus (HMA), which is formed by longitudinally corrugated sheets (in the direction of the liquid flow) with regular surface roughness (RR) located equidistantly. Notation: Ö gas; Ø liquid; P, E are the pitch and the height of the main corrugation of sheets; p, e – the pitch and the height of the regular roughness of the sheet surfaces; H, L, B are the height, the length and the width of the packing bed of the heat exchanging apparatus; a is the distance between sheets; δmax is the liquid film thickness.

38

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The movement of a non-compressed liquid in the general case is described by the NavierStocks equations and that of continuity, connecting components of velocity and pressure vectors. When a thin layer of liquid is flowing down the flat vertical surface with RR by gravity (figure 2.10), these equations will take the form [81]:

⎧ ∂w ⎛ ∂2wx ∂2wx ⎞ ∂w x ∂w x 1 ∂PL x ⎟, + wx ⋅ + wz ⋅ =− ⋅ + ν⋅⎜ + ⎪ 2 2 ⎜ ⎟ t x z x ∂ ∂ ∂ ρ ∂ x z ∂ ∂ ⎪ ⎝ ⎠ ⎪ 2 2 ⎛ ∂ wz ∂ wz ⎞ ∂w ∂w ⎪ ∂w z 1 ∂P ⎟ + g, + wx ⋅ z + wz ⋅ z = − ⋅ L + ν ⋅ ⎜ + ⎨ 2 ⎜ ∂x 2 ⎟ t ∂ x ∂ z z ∂ ρ ∂ z ∂ ⎪ ⎝ ⎠ ⎪ ∂w ∂w z ⎪ x + = 0, ∂z ⎪ ∂x ⎩ where t – time. Boundary condition: 1. Adhesion condition

w x = w z = 0 ; at х=f(z); Kinematics condition on the free surface

wx =

∂h ∂h + w z ⋅ , at х=h(z,t); ∂t ∂z

Dynamic condition of tangential stresses 1

12

1 − h z ⎛ ∂w z ∂w x ⎞ Pnτ = τ г ; − 4 ⋅ μ ⋅ +μ⋅ ⋅ + ⎟ = τ g , at х=h(z,t); 12 12 ⎜ ∂z ⎠ 1 + hz 1 + h z ⎝ ∂x hz

Dynamic condition of normal stresses equality 1

Pnn = Pnn ; 12

1

11

1 − h z ∂w z h z ⎛ ∂w z ∂w x ⎞ σ ⋅ hz , − PL − 2 ⋅ μ ⋅ ⋅ − 2 ⋅μ ⋅ ⋅ + ⎟ = − Pg + 12 12 ⎜ 12 3 2 ∂z ⎠ 1 + h z ∂z 1 + h z ⎝ ∂x 1 + hz

(

at х=h(z,t).

)

(2.34)

Solar Absorption Systems… The

film

HMT

the

ranges

ε = e δ N < 1, α = δ N p < 1

of

(α

2

input

parameters

)

39 changes

are

such

that

<< I , 1 < (α ⋅ Re fL ) ∼ α . In this case the 2

wave condition is defined only by stagnant waves, and running waves will be slight ripples spreading over the surface of stagnant waves. The procedure of making dimensionless is used:

x=

x − f ( z) , f ( z) = max f ( z) ⋅ f1 ( z) = e ⋅ f1 ( z), δN

f ( z) e = ⋅ f ( z) = ε ⋅ f1 ( z ) , δN δN 1 k=

Pg PL w wx ⋅ p p z P = , z = , wz = z , wx = , PL = , , 2 2 g e p w0 δN ⋅ w0 ρ ⋅ w0 ρ ⋅ w0

h=

τг ⋅ δ N t ⋅ w0 h−f h δ , δ= = = h − ε ⋅ f1 ( z) , t = , τ= , μ ⋅ w0 δN δN δN p

Re fL =

w 0 ⋅ δN ν

2

2

2

ρ ⋅ p ⋅ w0 , Fr = , We = . σ ⋅ δN g⋅p w0

It resulted in obtaining the following equations with an accuracy to terms of the

2

α order:

⎧ ∂PL ⎪ ∂x = 0, ⎪ 2 ∂w z ∂w z ∂PL 1 ∂ wz ⎪ 1 , + wz ⋅ =− + + ⋅ ⎨w x ⋅ 2 x z z Fr Re ∂ ∂ ∂ α ⋅ x ∂ f L ⎪ ⎪ ∂w ∂w z ⎪ x+ = 0. ∂z ⎩ ∂x w x = w z = 0, at x = ε ⋅ f1 ( z); w x = ∂w z = τg , ∂x

∂h ∂h + wz ⋅ , ∂t ∂z

at x = h (z ); − PL = − Pg +

1 ⋅ h ′′ , We z

(2.35)

at x = h (z, t ); at x = h (z, t ) . (2.36)

40

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

From the equation of continuity it follows:

wx = −

x

∫

∂w z ∂z

εf1 ( z )

dx .

Then we shall have 2 3 x ⎧ ∂w ∂w z ∂w z ∂ wz 1 1 ∂h 1 z dx + w z ⋅ , ⋅ ∫ = + ⋅ 3+ ⋅ ⎪− 2 x z x Fr We Re ∂ ∂ ∂ α ⋅ z x ∂ ∂ fL εf 1 ( z ) ⎪⎪ (2.37) ⎨ h ∂ ⎪ ⎪ ∂z ∫ w z dx = 0 ⎪⎩ εf1 (z )

w z = 0, at x = ε ⋅ f1 ( z); Let us substitute variables by formula

∂w z ∂x

= τ, at x = h( z) .

ξ = x − ε ⋅ f1 ( z) by averaging the firs equation by

layer thickness (i.e. integrating both of its parts by ξ from 0 to δ and dividing by δ), then we shall get: 3 δ δ δ 2 ⎧ ∂w z ∂w z ∂ wz δ δ ∂h 1 − ⋅ ξ + ⋅ ⋅ ξ = + ⋅ + ⋅ dξ, w d 2 w d ⎪ 2 3 ∫ ∫ z ξ=δ ∫ z ∂ ∂ α ⋅ z z Fr We Re ∂ ∂ξ z ⎪ L f (2.38) 0 0 0 ⎨ δ ⎪∂ ⎪ ∂z ∫ w z dξ = 0 ⎩ 0

w z = 0, at ξ = 0;

∂w z ∂ξ

= τ, at ξ = δ( z) .

According to [10, 81]: δ

∫ w z dξ =w ⋅ δ = 1.

(2.39)

0

conditions (2.38 – 2.39)

w z it is necessary to approximate by the following quadratic

function

wz =

− 6 + 3⋅ τ ⋅ δ 4 ⋅δ

3

2 2

⋅ξ +

6 − τ ⋅δ 2 ⋅δ

2

2

⋅ξ .

(2.40)

Solar Absorption Systems…

41

Substitution of (2.40) in the first equation of system (2.37) results in such an equation with respect to the dimensionless film thickness: 2

3

δ ⋅ δ′′′ +

2

4

2

48 − 2 ⋅ τ ⋅ δ − τ ⋅ δ ⎛ 3 ⋅ We ⎞ 3 6 − 3⋅ τ ⋅ δ ⋅ We ⋅ δ′ + ⎜ + ε ⋅ f1′′′⎟ ⋅ δ − ⋅ We = 0 . (2.41) 2 ⋅ α ⋅ RefL 40 ⎝ α ⋅ Re ⎠

Supposing that

δ ′ = δ 1 , δ ′′ = δ 1′ = δ 2 we get

⎧δ′ = δ1 , ⎪ ′ ⎪δ1 = δ 2 , ⎪ ⎨ 2 2 4 2 ⎪ ⎛ ⎞ ⎪δ′ = − 48 − 2 ⋅ τ ⋅ δ − τ ⋅ δ ⋅ We ⋅ δ − ⎜ 3 ⋅ We − ε ⋅ f ′′′⎟ + 6 − 3 ⋅ τ ⋅ δ ⋅ We. 3 3 1 1⎟ ⎜ α ⋅ Re ⎪⎩ 2 40 ⋅ δ ⎝ ⎠ 2 ⋅ α ⋅ RefL ⋅ δ fL The periodic solution of this system has been found by the Kutta-Merson method at boundary conditions

δ(0) = δ( p), δ 1 (0) − 0, δ 2 (0) = 0 .

if 0 ≤ z ≤ ( P − T) 2 ⎧0, ⎪ π ⋅ ( z − p 2) ⎪ , if ( P − T) 2 < z < ( P + T) 2 f ( z) = ⎨e ⋅ cos T ⎪ if ( P + T) 2 ≤ z ≤ P ⎪0, ⎩

(2.42)

From the obtained solution it follows that the availability of gas counter-flow with velocities up to 5-6 m/s does not practically affect the average film thickness, the flow is stable, the flooding of the apparatus will not take place. 2.3.2.2. The Flow of Viscous Liquid Thin Layer Along the Longitudinally-Corrugate Surface with Regular Roughness (RR) It is known that if the packing bed component is a sheet longitudinally corrugated along the flow (figure 2.10), the liquid is stored and flows along riffles of corrugations, rather than covers the whole sheet, i.e. we have a wet-dry condition of contacting which, under the conditions of packing bed sheets heat-conduction, results in the necessity to take into account real transfer surfaces [24]. Along the riffle the corrugated sheet with RR (figure 2.10) a stream of liquid flows by gravity. In the set up flows of liquid and gas the weight force is balanced by the tangential stress on the channel wall and free surface of the liquid (the surface wettability effect and the force of the surface tension are not taken into consideration): 2

2 ρg ⋅ w g ρL ⋅ w L ρ ⋅ g ⋅ Sf = Pw ⋅ λ L ⋅ + PM ⋅ λ g ⋅ . 2 2

(2.43)

42

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Taking into account that

w L = G L SL ,

w g = G g Sg , we shall get: 2

2

ρ ⋅ g ⋅ Sf = Pw ⋅ λ L ⋅ As

ρL ⋅ G L 2

2 ⋅ SL

+ PM ⋅ λ g ⋅

ρg ⋅ G g

.

2

2 ⋅ Sg

(2.44)

S L << Sg , then the total area between neighboring sheets by the length, equal to the

period, can be approximately taken as

Sg , i.e. Sg ≈ P × H . For the parabolic profile of

corrugation its equation for a half-period has the form:

)

(

2

(8 ⋅ E )

P = b , then we get

2

y = 8 ⋅ E P ⋅ x (as designated in

Figure 2.10), for the simplification we assume that

2

2

y = b ⋅ x . The maximum thickness of the liquid in the channel is δ max , then ⎛ ⎜ δ S L = 2 ⋅ ⎜ δ max ⋅ max ⋅ ⎜ b ⎜ ⎝

⎞ ⎟ δ max δ max 2 ⎟ = 4 ⋅δ ; P = 2 ⋅ ; ⋅ b x dx ⋅ ∫ м ⎟ 3 max b b 0 ⎟ ⎠

δ max b

δ max

Pw = 2 ⋅

b

∫

2

2

1 + 4 ⋅ b ⋅ x dx = −

0

]

[(

)

1 − ln 2 ⋅ b ⋅ δ max + 1 + 4 ⋅ b ⋅ δ max + 2⋅b

+ 2 ⋅ b ⋅ δ max ⋅ 1 + 4 ⋅ b ⋅ δ max . Let us introduce

∇ = 2 ⋅ b ⋅ δ max , then S = ∇

3

(6 ⋅ b ) 2

∇ 1 ⎡ ⎛ 2 2 ⋅ ln⎜ ∇ + 1 + ∇ ⎞⎟ + ∇ ⋅ 1 + ∇ ⎤ , Pм = − ; ⎥⎦ ⎠ b 2 ⋅ b ⎢⎣ ⎝ 9 5 2 2 2 ∇ = 5.51 ⋅ λ L ⋅ b ⋅ G L ⋅ ⎡ln ⎛⎜ ∇ + 1 + ∇ ⎞⎟ + ∇ ⋅ 1 + ∇ ⎤ + ⎢⎣ ⎝ ⎥⎦ ⎠ Pw = −

2

ρg ⎛ G g ⎞ ⎟ ⋅ ∇7 + 0.306 ⋅ λ g ⋅ b ⋅ ⋅ ⎜⎜ ρ ⎝ P ⋅ H ⎟⎠

.

(2.45)

Solar Absorption Systems…

For

the

laminar-wave

⎛ e 68 ⎞ ⎟ λ L = 0.012 ⋅ ⎜⎜ + ⎟ d Re ⎝ e ⎠

jet

flow

λL =

12.3 , Re

43

for

the

turbulent

one

0.25

,

λ g was experimentally found in paper [24]. Then (2.45) for

the laminar-wave condition of the film flow will take the form: 2

9 4 2 2 ∇ = 8.4719 ⋅ ν ⋅ b ⋅ G L ⋅ ⎡ln⎛⎜ ∇ + 1 + ∇ ⎞⎟ + ∇ ⋅ 1 + ∇ ⎤ + ⎢⎣ ⎝ ⎥⎦ ⎠ 2

ρg ⎛ G g ⎞ ⎟ ⋅ ∇7 + 0.306 ⋅ λ g ⋅ b ⋅ ⋅ ⎜⎜ ρ ⎝ P ⋅ H ⎟⎠

,

(2.46)

for the turbulent flow 9

−2

∇ = 5.56 ⋅ 10 ⋅ b

4.75

2 17 ⋅ ν ⎤ 2 ⎡ 1.5 ⋅ b ⋅ e ⋅ GL ⋅ ⎢ + ⎥ 3 GL ⎦ ⎣ ∇

0.25

2 2 ⋅ ⎡ln⎛⎜ ∇ + 1 + ∇ ⎟⎞ + ∇ ⋅ 1 + ∇ ⎤ ⎢⎣ ⎝ ⎥⎦ ⎠

1.25

+

(2.47)

2

+ 0.306 ⋅ λ g ⋅ b ⋅

ρg ⎛ Gg ⎞ ⎟ ⋅ ∇7 ⋅⎜ ρ ⎜⎝ P ⋅ H ⎟⎠

The solution of these equations with respect to ∇ enables us to calculate all the magnitudes of jet-film flow parameters, we are interested in, namely δ max , Pw , PM , FM , FH , α HMT = FM FH depending on the liquid and gas flow rates for any values of the main corrugation parameters of the surface with RR. 2.3.2.3. The Stability of Separated Two -Phase Flow in the Flat Channel of the HMTA with a Regular Packing Bed In film HMA-s for optimal progress of transfer processes it is necessary to provide the condition of active hydrodynamic phase interaction. However, the loads on the part of the liquid and gas should not be so large that intensive drop carrying-over and «flooding» of the apparatus could take place. The available results of theoretical and experimental investigations of maximum loads are contradictory. The authors have offered a mathematical model for determining the stability of the two-phase flow in the vertical channel, over whose surface the liquid film is flowing, when the gas flow is available, and the conditions of stationary regime of heat-and-mass transfer. The result is generalized for the case of channels the surface of which has RR [24, 81]. As it is shown in paper [81] for long waves the contribution to forces of fluctuating tangential stress which strive to change the wave shape, is mach less than that of normal stress. It makes possible in investigating the stability of the liquid film surface to regard liquid and gas as ideal media and to use the theory of potential flows. Equations of continuity and Koshi-Lagrange integrals for liquid and gas have the following form:

44

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov 2

2

∂ ϕi ∂x

+

2

∂ ϕ1 ∂y

= 0,

2

(2.48)

2 2 Pi 1 ⎡⎛ ∂ϕ i ⎞ ⎛ ∂ϕ i ⎞ ⎤ + ⋅ ⎢⎜ ⎟ ⎥ − g ⋅ x + = f (t) . ⎟ +⎜ ∂t 2 ⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦ ρi

∂ϕ i

Here the i=1 index refers to liquid, 2 – to gas, defined by equalities

w Li = ∂ϕi ∂x ;

(2.49)

ϕ i = ϕ i ( t , x, y) are potential of velocities,

w gi = ∂ϕi ∂y .

Boundary conditions are:

∂ϕ i

= 0, at y = 0;

∂y

∂ϕ 2 ∂y

∂ϕ1 ∂h ∂h = + wL ⋅ , ∂y ∂t ∂x ∂ϕ 2 ∂h ∂h = + wg ⋅ , ∂y ∂t ∂x

= 0, at y = r; at y = h (t, x ); y = h (t, x ).

at

The dynamic condition with forces of surface tension taken into account is:

P1 − P2 =

σ ⋅ h ′′xx

(1 + h ) 12

32

,

(2.50)

x

where y=h(t,x)

wL,

– wg –

is an equation of free liquid surface; are average, by the flow rate, velocities of liquid and gas, respectively.

Let the equation of the free film surface have the form

h( t , x) = δ N + α ⋅ e

i⋅k⋅( x − c⋅t )

,

с = с1 + i ⋅ c 2 (i is an imaginary unit).

Supposing that the length of the mach wave (perturbation wave) is large, i.e.

k ⋅ δ N << 1 , we get ( h ′x ) << 1 , and the boundary condition (2.50) will be written: 2

Solar Absorption Systems…

45

at y = h (t, x ) .

2

P1 − P2 = σ ⋅ k ⋅ h,

(2.51)

Velocity will have the form:

ϕ1 (t, x , y ) = w L ⋅ x + Ψ1 (y ) ⋅ e

i ⋅ k (x − c ⋅ t )

ϕ 2 (t, x , y ) = w g ⋅ x + Ψ2 (y ) ⋅ e

,

i ⋅ k (x − c ⋅ t )

.

Substituting these expression in equations (2.48) – (2.49) and using boundary conditions (2.50) – (2.51) we shall get:

ϕ1 (t, x , y ) = w L ⋅ x + i ⋅ a ⋅ (w L − c ) ⋅

h⋅k⋅y i ⋅ k (x − c⋅ t ) ⋅e , s ⋅ h ⋅ k ⋅ δN

) s–⋅ ⋅hh⋅ ⋅kk⋅ ⋅(z(z−−δy)) ⋅ e

(

ϕ 2 (t , x , y ) = w g ⋅ x + i ⋅ a ⋅ w g − c ⋅

i ⋅ k ⋅ (x − c ⋅ t )

.

N

The dispersion relationship is obtained by excluding pressure

Pi from (2.49, 2.51). With an

accuracy to terms of the second-order infinitesimal it has the form:

(

)

P ⋅ (w L − – ) + G L ⋅ w g − c = σ ⋅ k , 2

where

2

(2.52)

P = ρ1 ⋅ c ⋅ th( k ⋅ δ N ) , G L = ρ 2 ⋅ c ⋅ th (k ⋅ (z − δ N )) .

Separating real and imaginary parts in (2.52) we shall get:

[

]

[(

)

]

⎧⎪ P ⋅ (w − c )2 − c 2 + G ⋅ w − c 2 − c 2 = σ ⋅ k, g L 1 2 L 1 2 ⎨ ⎪⎩c1 = P ⋅ w L + G L ⋅ w g (P + G L ). Excluding

(

)

c1 from equations (2.53) we have:

(

)

⎛ P ⋅ GL ⎞ 2 ⋅ w L − w g − σ ⋅ k ⎟⎟ c 2 = ⎜⎜ ⎝ P + GL ⎠ Hence, if the velocity of the gas

w g satisfies the condition

(P + G ) . L

(2.53)

46

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

w g > w L + [(P + G L ) ⋅ σ ⋅ k (P ⋅ G L )] , 12

then

с 2 > 0 and the surface of phase separation is unstable, long-wave perturbations will

exponentially increase as time goes on. When

2

с 2 < 0 , the surface of the liquid will be stable.

= 0 ) the maximum velocity of the gas, after which instability

For the neutral curve ( с 2

begins, is determined by the formula

w g = w L + [(P + G L ) ⋅ σ ⋅ k (P ⋅ G L )]1 2 ∗

For the counter flow, when

GL << 1 , we get P

∗ wg

⎡σ ⋅ k ⎤ =⎢ ⋅ th (k ⋅ (r − δ N ))⎥ ⎣ ρ2 ⎦

Let us write the wave number in the form

∗ wg

12

− wL .

k = ξ δ N , (ξ = const ) then

⎡ σ⋅ξ ⎛ ⎛ r ⎞ ⎞⎤ =⎢ ⋅ th⎜ ξ ⋅ ⎜⎜ − 1⎟⎟ ⎟⎥ ⎜ ⎟ ⎢⎣ ρ 2 ⋅ δ N ⎠ ⎠⎥⎦ ⎝ ⎝ δN

12

− wL .

(2.54)

Constants ξ in equation (2.54) is selected on the basis of the generalized criterion equation, defining maximum loafs of the gas and liquid phase with the counter-flow available and taking into account the length and the diameter of the channel. The equation is given in paper [24], it is a generalization of a lot of experiments and has the form: ∗ wg

2 ν 0.38 0.113 ⎛ ρ ⎞ = 1.346 ⋅ ⋅ Re L ⋅ We ⋅ ⎜⎜ 2 ⎟⎟ d ⎝ ρ1 ⎠

f (H, d) =

0.513

⎛μ ⎞ ⋅ ⎜⎜ 1 ⎟⎟ ⎝ μ2 ⎠

0.955

⎛ d ⋅ ⎜⎜ ⎝ δN

1.628

⎞ ⎟ ⎟ ⎠

− 0.8 H ⋅ (0.38 ⋅ d − 0.015) + 0.07 ⋅ d d

⋅ f (H , d ) ,(2.55)

(2.56)

The relative root-mean-square deviation of experimental data from relationship (2.55) does not exceed 4%. According to [24] we specify ξ in the following form:

Solar Absorption Systems…

ξ = γ ⋅ We where

α1

⎛μ ⎞ ⋅ ⎜⎜ 1 ⎟⎟ ⎝ μ2 ⎠

47

b1

γ , α 1 , b 1 have been found by the least square method. It yields: γ = 0.01,

α1 = −0.5,

b1 = −0.12 .

(2.57)

Relationship (2.54) obtained corresponds to the flow in the vertical channel with smooth walls. For vertical channels with RR of the surface, the height of which is comparable with δ L , in flowing stationary waves of a large amplitude are formed on the film surface. When hydrodynamic interaction of phase is great, wave amplitudes tend to increase as time goes on, which leads to the decrease of

∗

w g . The correction for the surface RR is introduced in

formula (2.55) in the form of the power dependence on the relative height of the roughness bulges: ∗ w g, gRR

where

=

∗ wg

b ⎛ ⎛ e ⎞ 2 ⎞⎟ ⎜ , ⋅ 1 − α2 ⋅ ⎜ ⎟ ⎜ p⎠ ⎟ ⎝ ⎠ ⎝

(2.58)

∗

w g is defined by relationship (2.55) and a 2 , b 2 are to be determined. These

constants are found by generalizing experimental data from paper [24, 81]. It yields

a 2 = 0.88,

b 2 = 0.572 .

(2.59)

Relationships (2.55) – (2.58) are valid in the following range of varying parameters:

20 < Re L < 100 ; 0.3 m < H < 0.85 m ; 16 mm < d e < 24 mm ; 0 <

e < 0.2 . p

The deviation from experimental data does not exceed ±10.1 %. 2.3.2.4. Experimental Study of Film Flows in Heat-and-Mass Transfer Apparatuses In numerous experimental works devoted to the study of peculiar features of liquid film flowing along vertical surfaces they have mainly analyzed the flow through tubes or along the flat smooth surface. The number of works where they studied the flows along the flat surface with regular roughness (RR) is scarce, and only some aspects of flowing along the longitudinally corrugated surface have been considered conformably to the special features of the condensation process. But it is longitudinally corrugated sheets that found wide

48

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

application in designing film HMTA. The flow along such surfaces is characterized by some distinguishing features. Well-known methods of experimental measuring the liquid film thickness, divided into two groups. Methods of measuring local thickness

δ L , can be

δ L : the method of touching

or electrocontact method; photoshady method, method of light flux absorption; method of measuring electrical capacitance between the sprayed wall and the sensor; method of electrical resistance or local conductance; method of measuring radioactive traces in the liquid film, etc. Methods of measuring average thickness of the liquid film

δ L : the method of

instant power supply cut-off; method of consecutive weighing the dry and moistened part; neutron diagnostics method; modified method of electrical capacitance; method of electrical resistance or conductance. The electrical conductance method is most widely used in studying two-phase systems [10, 87, 88]. It makes possible to study a lot of parameters (wave characteristics, gas content nature of flowing); it can be used in channels of complicated configurations; it has a good reproducibility of results obtained; it can be used with the gas flow both available and not available; it is suitable for local measurements when there are many sensors, in this case the flow does not get perturbed. To obtained the information about the distribution of δ L on the channel perimeter and the value of the wetted surface the method of local electrical conductance (the ratio of the electrode size by the flow D to the distance between the electrodes is about 1) is used; to measure the average thickness

δ L the method of integral

electrical conductance (l/D >> 1) is used. The idea of the method is to measure the ohmic resistance of the liquid film by means of several electrodes located on the surface of sheets in such a way that the bulge should not be formed (flush with). The authors [10, 24, 81, 87] used tap water the natural is sufficient for making measurements. The alternating current is used to eliminate the polarization effect. The value basis of the film resistance

δ L , on the part being studied, was defined on the *

R f and that of the liquid column R f , which responds to

complete filling the experimental channel with the liquid. Proceeding from the Ohm’s law, *

*

δL = D′ ⋅ R f / R f , where D′ is the channel width taken from the conditions: D′ >> δ f *

( δ f is the limiting value of measured). Then

Ra = f ⋅

δ L ) and D ≤ L (L is the length of the channel part being *

R f and R f will be of approximately the same order, that is the slope

1 will be practically constant. Thus it is possible to replace ω

⎛ R* ⎛ R* ⎞ ⎜ f⎟ → ⎜ a ,f ⎜R ⎜ Rf ⎟ ⎝ ⎠f → ∞ ⎝ a ,f

⎞ ⎟ ⎟ ⎠f

Solar Absorption Systems…

49

The working frequently has been chosen as equal to 5 kHz, referring to the linear part of the relationship through the whole range of changes in

δ L . Two schemes of placing

electrodes on the sheets have been used. In the first one three electrodes (strap electrodes of aluminium foil 5 mm wide, 2 mm thick) are placed perpendicular to the film flow, which makes it possible to avoid leak flows. But in defining

δ L , especially in case of steam-line

flow around the fin of regular roughness, an error appears, which is connected with the change of equality of experimental parts lengths arising in defining

*

R f and R f . This

equality is guaranteed by the scheme of two electrodes located along the flow but this scheme is not suitable for studying flows along the longitudinally corrugated sheet. To check the method an experiment has been carried out in the channel with the clearance of varying height, wholly filled with a liquid. This is equivalent to plotting the calibration curve. The stand – a one-channel model (figure 2.11) provides the possibility to conduct research both with the gas flow net available and available. Here 1 is the liquid distributor, 2 is the fastening frame; 3 is a sheet of the pacing bed (backing) with the dimensions of the part being measured 0.25×0.5 m; 4 is the water rotameters block PC; 5 is electrodes; 6 is the liquid receiver; 7, 8, 9 are the filter, the pump, the tank; 10 is the heater; 11 is the receiver; 12 is the air rotameter PC; 13 is the air distributor; 14 is the observation panel; 15 is the contact thermometer. The width of the working channel is 15 mm. The observation panel is made of plexiglas’s. To distribute the liquid film uniformly and to smooth pulsations a band of thin capillary-porous material (phlizeline) is placed in the upper part of the sheet. The same bands 5 mm wide are placed along the edges of the sheet to prevent the film taking-off at low loads. The investigation has been conducted on vertical flat and longitudinally corrugated sheets (figure 2.11B). Flat sheets were made of plexiglas’s: 1 is smooth sheets (a sheet without regular roughness); 2 is a sheet with regular roughness in the form of horizontal bulges uniformly distributed along the surface of the sheet (the bulges have a rectangular form with rounded edges) – figure 2.11B.a. Height of bulges (of 1 mm) is accepted on the basis of the recommendations [24]; width at the basis of 1 mm. The quantity k= p/е was changed discretely in experiments (k = 2.5; 4.5; 6.5; 8.5; 11; 20; 36.5). Longitudinally corrugated sheets were formed from epoxy resin (to exclude the samples electrical conduction) in some versions: 1 – without regular roughness; 2 – with fins of roughness – figure 2.11B.b. The sheet main corrugation parameters are follows: P =10 mm, E = 3.8 mm, those of roughness are: k = 12.5 at e =1 mm. The values of

δ L were averaged by

the channel width. In papers [10, 24] it is shown by the “searching needle” method that the maximum increase of the surface at the expense of thickening at the side walls does not exceed 0.7%, therefore such averaging is grounded. The investigation has been carried out both without the gas flow and with it, in the latter case special attention was paid to the problems of stability. The former case corresponds to the absence of hydrodynamic phase interaction (it is characteristic of the cross flow scheme). Figures 2.12 – 2.13 give the main results. The transition from the laminar to the turbulent *

flow region ( Re L ) depends on the availability of regular roughness and the magnitude k. The availability of two-dimensional roughness accelerates the transition, reducing the

50 threshold value of

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov *

Re L . It follows from the stability theory – the surface roughness causes in

the laminar flow perturbations which are added to the perturbations available due to a certain degree of turbulence in the external flow. The latter accelerates the flow mode transformation.

Figure 2.11. А. Experimental stand (the single channel model) for studying film flows. Nomenclature: 1 is the liquid distributor; 2 is the fastening frame; 3 is the packing bed sheet; 4 is the water rotameters block; 5 is electrodes; 6 is the liquid collector; 7, 8, 9 are the filter, the pump, the tank; 10 is the heater; 11 is the receiver; 12 is the air rotameter; 13 is the air distributor; 14 is the vision panel; 15 is the contact thermometer. В. The types of investigated sheets: a is the flat sheet with regular roughness RR; b is the longitudinally corrugated sheet with RR.

Solar Absorption Systems… For a smooth sheet

51

*

Re L ≈ 1650 . On a smooth sheets and sheets with k > 20 one can

observe a slightly noticeable irregular wave flow. At с k < 20 a monotonous and periodical wave flow is formed, in this case the initial part length becomes less with the decrease of k. At the same time the film becomes thicker (k = 20 → 4) with subsequent thinning at k < 4. The minimum value of the growth of

*

Re L , corresponds to k = 8-14. The further reduction of k results in

Re L , thus the relationship Re L (k ) is described by a complicated curve. The *

*

type of the relationship is explained by the fact that at sufficiently low k fins of roughness are practically linked and conditions of film flows approach to those characteristic of the smooth sheet without regular roughness. The film thickening takes place which is connected with the growing influence of capillary forces on the liquid delay. It was shown by Japanese researchers [57, 82] who noted, that in this case the evaporative cooling efficiency could be even lower than that of the flow along the smooth sheet. According to k = p e three peculiar regions of liquid film flow can be distinguished: 1. k = 8-14. A stable regular wave mode of flow with the predomination of standing waves is notable; the initial part of the wave formation is practically missing: the splash formation at flowing around the fin is minimum: 2. k < 8. The flow laminarization takes place.; the values of

*

δ L and Re L increase; at

sufficiently low k the situation is practically the same as in flowing along the smooth sheet; 3. k > 14. The regular wave mode of flowing is changed; the initial part of the wave formation increases; the splash formation grows; at sufficiently high k the situation is the same as in flowing along the smooth sheet. Thus, the range of values k = 8-14, where there are optimal conditions for revealing roughness and maximum intensity of transfer processes in the liquid film is provided, is of special concern. Experimental results are given in the form: 2

δL = x 1 ⋅ The parameters

3⋅ νL 4⋅g

x

⋅ Re L 2 ⋅ k

x3

⋅ exp(x ⋅4 k ) ⋅ sin α

x5

(2.60)

x 1 … x 5 were found by minimizing the efficiency function n

(

)

2

ϕ = ∑ δL, ex − δ L, cal , i =1

i

i

(2.61)

52

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

with the help of the search method by a polyhedron under deformation (Neldor-Mid), where n is the number of experimental points. The following expression are obtained: for the laminar wave region 2

4

δ L = 0.83 ⋅10 ⋅ 3

3⋅ νL

0.251

⋅ Re L ⋅ k

− 0.655

(

)

−1

⋅ exp 0.22 ⋅10 ⋅ k ⋅ sin γ

4⋅g (the average error is 9%; Re L = 280 − 1260 ; k = 2.25-40)

− 0.48

(2.62)

for the turbulent region 2

3

δ L = 0.81 ⋅10 ⋅ 3

3⋅ νL

0.56

⋅ Re L ⋅ k

− 0.401

(

−2

)

⋅ exp 1 ⋅10 ⋅ k ⋅ sin γ

4⋅g (the average error is 4%; Re L = 720 − 2100 ; k = 2.25-40)

− 0.327

(2.63)

The boundary of flow modes existence is defined by the expression *

2

Re L = −3.36 ⋅10 ⋅ k

0.661

⋅ exp(− 0.039 ⋅ k ) + 1650

(2.64)

(the average error is 12.1%). The comparison of calculated (see 2.3.2.1 – 2.3.2.2) and experimental values of δ L shows that they are in good agreement. The gas flow influence on the quantity

δ L was not registered in the range ωg = 1.5 − 7.0 m/s.

Flow Along the Vertical Longitudinally Corrugated Sheet with Regular Roughness The transition from a flat to longitudinally corrugated sheet means the transition from frontally flowing film to jet-film flow in channel cavities with alternating wetted parts (figure 2.13). With the predominant abundance of corrugated sheets in film HMA-s such a flow is traditionally considered as the film one. The problem of correlation of heat-and-mass transfer surfaces appears to be important, especially in case of high heat conduction of the material the sheet is made of. While analyzing experimental data instead of the average

δ L notion, we

used the idea of the maximum film thickness in cavities of channels. In calculating

δ L , the

wetted perimeter and the wetted surface quantity we supposed that the tree surface of the jet was smooth with some curving at the boundaries. The latter increases the wetted surface quantity by 10% at an average (figure 2.13B). It can be seen that the introduction of regular roughness results in film thickening (figure 2.12). The retaining of optimal range k opt = 8 − 14 has been established for the flow along the longitudinally corrugated sheet

Solar Absorption Systems…

53

too. The regular wave mode of the flow with the predomination of standing waves is notable. The following expression for the flow along the longitudinally sheet have been obtained: 2

4

δ L , L = 0.198 ⋅10 ⋅

3

3⋅ νL 4⋅g

0.404

−0.591

0.698

− 0.327

⋅ Re L ⋅ k

2

3

δ L, T = 0.178 ⋅10 ⋅ 3

3⋅ νL 4⋅g

⋅ Re L ⋅ k

(

)

0.549

)

0.602

⎛ P ⎞ ⋅ exp 1.44 ⋅10 ⋅ k ⋅ ⎜ ⎟ ⎝ 2⋅E ⎠

(

−2

−2 ⎛ P ⎞ ⋅ exp 0.33 ⋅10 ⋅ k ⋅ ⎜ ⎟ ⎝ 2⋅E ⎠

⋅ cos α

−0.249

⋅ cos α

, (2.65)

− 0.219

(2.66)

Figure 2.12. Results of experimental studying film flows in heat exchanging apparatuses. А is the relationship δL (ReL, k) for a flat sheet with the surface regular roughness (ϑ is the smooth sheet); B is the influence of the longitudinal corrugation of the sheet: ▲ – the flat sheet with RR, Ì – the longitudinally corrugated sheet with RR, (_ _ _ ) is the smooth flat sheet (k = 8.5).

54

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov The design and experimental values of

δ L for the longitudinally corrugated sheet are in

good agreement. It should be noted that for the smooth longitudinally corrugated sheet the magnitude

*

Re L coincides with the magnitude generally accepted for the film flow along the

smooth flat surface (≈ 1600, figure 2.12). The shift of

*

Re L accounting for the surface the flat

sheet with regular roughness registered by us. The value

a HMT = FM FH has been obtained

by calculating experimental data. The control was exercised by the method of local electrical conduction in spot electrodes (d =0.5 mm, n is the number of electrodes equal to 15-30) were located around the channel perimeter in some of its sections by the height. The value of the wetted perimeter was registered by shorting a pair of extreme electrodes with the liquid film. On the average, the surface correlation a HMT = FM FH is equal to 0.3-0.5 and it was later used in calculating transport coefficient (figure 2.13A). The design and experimental values of a HMT are in good agreement. The influence of regular roughness fins on the value of the channel wetted perimeter was not taken into account in calculations (figure 2.13B).

Figure 2.13. Results of experimental studies of film flows in heat-and-mass transfer apparatuses. А is the value of relations for surfaces of heat-and-mass transfer in heat-and-mass transfer apparatuses with longitudinally corrugated sheets of the packing bed: 1 – without the surface regular roughness (RR), 2 – with the surface regular roughness (k = 12.5, P = 10); В is the influence of ReL on the value of the wetted surface of the longitudinally corrugated sheet.

Solar Absorption Systems…

55

The longitudinally corrugated sheet provides a film-jet mode of flowing with the alternation of dry and moistened parts (wet-dry mode of heat-exchange). In this case the former, under conditions of high heat-conduction of the packing bed material, served as finning of moistened parts, to some extent compensating the loss of totally transported heat in the system due to the reduction of FM . This problem is specially dealt with in the paper [10] which shows the attainment of considerable water saving for the replenishment of the system with a slight loss of refrigerating capacity in conformity with evaporative coolers. Unlike the solution [10] our sheet construction simplifies the distribution of the liquid film, providing the same effect. The gas flow influence was studied on the counter-flow and cross-flow schemes of contacting. For the counter-flow the absence of this influence was registered up to the values of w g ≈ 6.5 m/s, when the flooding phenomenon is established. For the cross-flow the visualization model was used. The high corrugation is characterized by the picture which corresponds on the whole to the case of one-phase liquid flow. The shift of the latter in the direction of the gas movement is not height at all. In reducing the corrugation height (E ≈ 2.2 mm) at high velocities ( w g > 10 м/с) the film is shifted to the peak of the longitudinal corrugation and partly dispersed; drops are gone with the gas flow. For E ≈ 1.5 mm this phenomenon appears even at w g ≈ 7 − 8 m/s. These results confirm high stability of the organized film-jet flow along the surface of the longitudinally corrugated sheet and justify the choice of just the cross flow scheme of gas and liquid contacts in HMTA.

2.3.3. Experimental Study of Working Characteristics of Cross-Flow Film HMTA 2.3.3.1. Experimental Equipment and Research Program The stand (figures 2.14.A and 2.15) consist of the main part and systems of preparing the flows. The main part of the stand is the working (4), water distributing (3) and water catching (5) (multi-sectional) chambers and two chambers for measuring air flow parameters. All chambers are made of thick-walled transparent plexiglass. The water distributor is lamellar (the upper ends of the packing bed sheets are protruded into the water distributing chamber, and layers of capillary-porous material are arranged between and over them). The upper layer of foamed plastic serves for the uniform distribution and filtration of the liquid. The watercatcher consists of 6 pockets, the last of which is located outside the module in the direction of the gas movement. It provides differentiated measuring of the liquid flow-rate and its longitudinal (for the gas) drift with the air flow. Frame pockets the liquid passes to the unit of measuring the local flow-rates (12) with temperature sensitive elements placed in its measuring chambers. The chamber’s dimension are 460×400× ×180 (width) mm. The chamber capacity is up to 2000 m3/h for the gas. The air was supplied by a centrifugal fan, installed lower in the flow which made it possible to take into account the heating in the fan and high stability and uniformity of the distribution. The flow rate control was achieved by even changing the voltage, supplied to the electric motor. The air electric heaters are located in front of the fan and they switched on the two-position temperature regulator (the

56

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

controlling accuracy is 0.3 °C). Flow meters (7, 8) of the collector type were used. To control the relative humidity the by-pass line with flow meter 8 is provided. The heat load was controlled by the one-phase regulator of the voltage; the liquid temperature – by the twoposition temperature controller – with the accuracy of 0.1 °C. The liquid flow meter is a block of rotameters of PC-5 and PC-7 types (11).

Figure 2.14. А. The experimental stand for investigating characteristics of cross-flow film heat-andmass transfer apparatuses: 1 – electrical heater; 2 – fan; 3 – water distributing chamber; 4 – working chamber; 5 – accumulating sections; 6 – separator of drop moisture; 7, 8 – air flow-meters; 9, 10 – air flow-rate controls; 11 – water rotameters; 12 – sectional liquid flow-rate meter; 13 – water tank; 14 – pump; 15 – filter; 16 water heater; 17 – liquid temperature control. B – main types of the studied sheets of the packing bed: a – scheme of the packing bed sheet; b – straight double riffle; c – “straight oblique riffle”; d – “double oblique rifle” (____ – arrangement of the main channel, _ _ _ _ – arrangement of the regular roughness fin).

Solar Absorption Systems…

57

Figure 2.15. Pictures of the experimental stand (A) for investigating working characteristics of crossflow film heat-and-mass transfer apparatuses and the working chamber (B) with a pack of beds “direct double riffle” installed in it. The nomenclature is as in figure 2.14.

Table 2.3. Geometric characteristics of investigated cross-flow apparatuses

Solar Absorption Systems…

59

The preference was mainly given to the study of conditions of evaporative water cooling (CTW), processes of direct (DEC) and indirect (IEC) evaporative cooling at the cross-flow scheme of contacting. The program was formed by the step planning method and consist of the following sections: the investigation of the working characteristics in a wide range of the layer density values; the determination of the optimal d e (equivalent diameter of the channel) magnitude; the study of the influence of regular roughness parameters (RR); the study of the influence of the module height and length in the gas flow movement (cross-flow); the study of the effect of the nonuniformity of distributing contacting flows on the working characteristics. On the whole, five types of packing beds with stacks of longitudinally corrugated sheets were studied (Table 2.3): the longitudinally corrugated sheet without regular roughness; the longitudinally corrugated sheet with regular roughness (RR) of the surface; in this case regular roughness fins are located horizontally (along the air flow movement) – the “flat double sheet” type – figure 2.14B.b; the same sheet with RR is placed at an angle to the running-on flow – the “straight oblique riffle” type – figure 2.14B.c; the sheet with obliqueangle double corrugation, when both channels themselves and fins of surface regular roughness are located at an angle to the vertical – the “double-oblique row” type – figure 2.14B.d; •

the sheet with horizontal main channels and the slope of RR fins towards the running-on flow (in the table is not shown).

The magnitude of the channels equivalent diameter ranged from 12 to 30 mm; the construction surface of the packing-bed in the layer volume unit was in the range of 170-200 m2/m3; the slope angle of the main packing-bed channel is 0-90°; the slope angle of the regular roughness fins to the horizontal is 0-30°; the roughness parameter value k = 10-14 ( k = p e = 12.5 in the majority of experiments). 2.3.3.2. Hydrodynamic Characteristics The investigation was carried out within the range of values

d e = 12.0 − 29.9 m

(calculated by the middle section of the channel) at k = 12.5 which is among the optimal values of k opt = 8 − 14 . We studied the influence of the surface regular roughness, the arrangement of regular roughness fins and the main channels with respect to the running-on air flow, the main corrugation parameters P and E, on the transfer intensity and energy consumption. The transition to the cross-sectional scheme provides the reduction of the Δp level as compared to the counter-flow and the possibility to increase the loads considerably. The introduction of regular roughness for the vertical arrangement of the main corrugation does not practically affect the value Δp in contrast to the counter-flow both for one- and two-phase flows. The hydrodynamic interaction of phases is missing in the whole range of gas and liquid flow-rates. The traditional phenomenon of flooding for the cross-flow scheme is not available at all, up to values w g ≈ 10 − 12 m/s and at large w g it is replaced by the

60

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

phenomenon of the longitudinal liquid drift which results in its unfavorable redistribution in the packing bed volume and removal from the layer. The influence on the main corrugation parameters P and E was studied. The increase of E (2-5.3 mm (P = idem)) results in the growth of Δp; at

*

E ≤ 2.2 mm the longitudinal drift of the liquid begins from w L >10.0

m/s. The decrease of p (14 →10 mm) results in the drop of Δp. In both cases the lines of the gas flowing become more flatted in channels of the regular packing bed, which reduces transfer intensity.

Figure 2.16. Results of experimental studies of working characteristics of cross-flow film heat-andmass transfer apparatuses. А is the influence of regular roughness on the processes intensity: □ is the corrugated packing bed without RR; Ì is the corrugated packing bed “direct double riffle”; ○ is the packing bed “direct oblique riffle”; В is the influence of the main corrugation parameters: ▲ – Е = 3 mm, ● – Е = 4 mm, ■ – Е = 2 mm. (Р = 10мм.)

Solar Absorption Systems…

61

Figure 2.17. Results of experimental studies of working characteristics of cross-flow film heat-andmass transfer apparatuses. А is the slope of the surface regular roughness (RR) to the run-on gas flow (packing bed sheet “direct oblique riffle”): 1 – wg = 3 m|sec.; 2 – wg = 4 m|sec.; 3 – wg = 5 m|sec. В is the influence of the slope of the main packing bed sheet (main channel) corrugation to the run-on gas flow: ________ vertical channel (“direct double riffle” – figure 2.14Bb); Ì is the packing bed sheet “oblique double riffle” – figure 2.14Bd.

The channel slope within the limits of 75-90° does not practically affect the amount of resistance, but the situation with vertically and horizontally arranged channel is different. The latter is characterized by the abrupt change of the longitudinal drift of the liquid already at w g = 2.5 m/s (the transformation of the cross-flow scheme into the parallel current one) and by the increase of the spray density influence on Δp. The vertical arrangement of the main channel is preferable: the film-jet liquid flow in its cavities is characterized by high stability in case of the cross-flow. 2.3.3.3. Heat-and-Mass Transfer in the System The maximum increase of the transfer intensity was obtained for the “straight oblique riffle” (figures 2.16 and 2.17), and one can notice the favorable liquid distribution over all the surface of the sheet in the module. The influence of the slope angle of regular roughness fins to the running-on flow is illustrated by the data in figure 2.17. The angle α = 15° is optimal,

62

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

which corresponds to results of visual observations of the liquid distribution. Unlike the counter-flow, where regular roughness fins are arranged horizontally, for the cross-flow scheme their optimal arrangement has been concretized for the first time. The absence of liquid on sheets in the lower part of the inlet section, noted in literature is practically not available, which provides the reduction of specific consumption of materials of the module by 15-20%. One can note the particular form of the relationship K h = f q L for α = 15°

( )

which shows the absence of the longitudinal redistribution of the liquid. Considerable influence of the main channel slope was not detected and its vertical arrangement is preferable (figure 2.17B). It should be noted that RP of the “oblique double riffle” (figure 2.14B.a) type which is very popular abroad not provide any advantages over the vertical arrangement of the channel, according to the results of the present investigation. The version with the horizontal arrangement of the channel presents quite a different picture. In the velocity range w g ≤ 3.5 m/s low energy consumption, but with the further increase of w g the longitudinal drift of the liquid sharply increases. With the decrease of value

q L the critical

*

w g increases and for q L =4 m3/(m2⋅h) amounts to 1.5-5.0 m/s. Such an arrangement

of the module is favorable under the conditions of thermal-moist air treatment (evaporative coolers IEC, DEC). It should be noted that with the growth of q L , K h is unusually decreasing, which is explained just by the phase interaction. The sheet with “double corrugation” should be recommended for the version with a vertical channel – for evaporative cooling of the liquid (CTW); with a horizontal channel – for thermal-moist treatment of the gas (IEC, DEC), the optimal arrangement of the regular roughness fins proves to be identical. The influence of the main corrugation parameters is shown in figure 2.16B. The increase of E (2-5 mm) results in the increase of K h , it is registered at two values of P (10 and 14 mm). The optimal range

K=

P = 1.4-2.0 is defined by the influence of this parameters 2⋅E

on the distribution of flow lines in the channel bent in the direction of the gas movement. This range of K values is recommended for both schemes of contacting. The intensifying affect of regular roughness fins under the conditions of the cross-flow scheme is revealed in the reduction of thermal resistance in both phase [10, 24, 81] as compared to the smooth (background) sheet, and the increase of K h = f w g is practically

( )

provided at the expense of reducing the thermal resistance of the phase

R g , which is

connected with the lack of hydrodynamic phase interaction. The intensification mechanism can be explained on the basis of conclusions from papers [124], experiments with the sublimation of naphthaline under the conditions of the cross-flow scheme of flowing in channels with corrugated walls (the channel from and the distance between walls in our investigation and the works [124] are practically identical). Curvilinearity of the air flow lines and the appearance of secondary flows overlaying the main one are noted. The separation of flow on the lee side of the sheet results in abrupt reducing the transfer velocity, which increases in the same abrupt manner in the range of repeated addition of the flow. And it can be noted that the transfer intensity on the concave surface is much greater, than on convex

Solar Absorption Systems…

63

one. It is important to show that in our case the places of appearing repeated flows are located just in the range of the film-jet flow on the convex walls of the sheet. The analysis of the specific energy consumption and the attainable degree of cooling confirms advantages of the sheet “straight oblique riffle” for the cross-flow scheme.

Figure 2.18. Schematic of the fan cross-flow-tower (А) and the scheme of gas liquid flows (B) in the channels of the regular packing bed (RP) of the heat-and- mass exchanging apparatus. Notations are given in figure 2.10.

entering air flow;

discharged flow;

water.

64

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The investigations were carried out in the range d e = 12.0-29.9 mm (k = 12.5) for the sheet of the “straight oblique riffle” type. With the increases of d e the transfer intensity decreases. Experimental data allow the concretization of the range d e. opt = 20-25 mm, i.e. it is displaced to the side of higher values of d e . The energy consumption for obtaining the specified value of the process efficiency E L is much lower under the conditions of the cross-flow scheme than that of the counter-flow. The problem of the relations between main overall dimensions of cross-flow HMTA (H is the height, L is the length in the direction of the air flow, B is the width) was only raised in literature. The optimal formula of the main dimensions relation according to the results of this investigation is: H = B; the magnitude L is limited by the admissible magnitude of pressure losses in the air flow (but the ban features). Besides, one should be guided by the optimal range of working loads while choosing values H and B. The film-jet contacting provides the reduction of water consumption for feeding the system by decreasing the evaporation surface. As unsprayed parts of the heat transfer sheet surface play the role of fins in wetted parts the decreases of refrigerating capacity of the system on the whole is insignificant. It was first discussed in paper [35, see also the patent of France N2409481, F28F25/02, 1979], where special grooves were made on sheets for the liquid to flow. On the average, for the load range under the investigation it was possible to lower the water consumption for the evaporative process (feeding the system) by 30-40% while the refrigerating capacity being reduced only by 1015% (IEC, DEC). The wet-dry principle of contacting, laid in the basis of developed HMTA, provides the possibility of using them in autonomous systems, and its combination with regular roughness of the surface makes it possible to control the relation between heat-and-mass transfer surface and the distribution of phase thermal resistances in the system. The vertical arrangement of the main channel is recommended for multi-channel HMTA with regular roughness. For the cross-flow – the “direct oblique riffle” with an inclined fin of regular roughness (α = 15°). All cases are characterized by the film-jet laminar-wave (or transitional) liquid flow at the turbulent gas flow. The wet-dry principle of containing makes it possible to decrease the amount of the liquid needed for feeding the system when the decrease of the refrigerating capacity is insignificant. The cross-flow scheme has some advantages over the counter-flow: the range of working loads is much wider; the energy consumption is reduced at higher efficiency of the process; the height of HMTA is lowered; it is possible to install a fan outside the wet air flow and, if necessary, to reverse it. The joint arrangement of some heat-and-mass transfer apparatuses into one unit of an alternative cooler is simplified to a considerable extent. The possibility to use the cross-flow scheme at high loads without notable redistribution (a horizontal drift) is decisive for the use of it in HMTA for alternative air-conditioning and refrigeration systems, especially in designing absorbers and desorbers.

2.3.4. Simulation of Working Processes Sin the Combined Evaporative Cooler (CEC) on the Basis of IEC/DEC (IEC/CTW) 2.3.4.1. Direct Evaporative Cooling (DEC/CTW) The cooling part of AACS is based on the use of the combined evaporative cooler (CEC) including indirect evaporative cooler (IEC) as the first stage and direct evaporative cooler

Solar Absorption Systems…

65

(DEC) or cooling tower (CTW) as the second one: IEC/DEC (figure 1.6, 2.19) or IEC/CTW (figure 1.7). All apparatuses are of the film-type crosscurrent with multi-channel packing-bed of the structure made of longitudinally corrugate sheets (in the direction of liquid film flowing) with regular roughness of the surface as the basic method of heat-and-mass transfer processes intensification (figure 2.10).

Figure 2.19. А. Evaporative unit of the ASAC. Schematic of the combined evaporative system on the IEC|DEC basis. Notations are given in the figure 2.10. В. Picture of the heat exchanging component – “wafer” – channels for the movement of the main air flow in IEC. Additional notations: air flow;

the auxiliary air flow;

the water flow.

the main

66

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Simulation of the processes in DEC/CTW – figure 2.18, CTW – was carried out in conformity with flat-and-parallel packing bed and cross-current scheme of phase contact and after it in conformity with more complicated situation with longitudinally corrugated sheets. The heat-and mass transfer taken place on the free water surface, contacting with air, if their temperatures are different and the partial pressure of water vapour at the interface as well as in the air space are also different. For the evaporation (condensation) of the amount of moisture and supplied heat from interface Fs :

g s = β p ⋅ ( p s − p) ⋅ Fs ,

(2.67)

Q s = g s ⋅ r = r ⋅ β p ⋅ ( p s − p) ⋅ Fs .

(2.68)

This amount of heat can be provided, if there are no other sources, only convective mode from the side of gas and liquid:

(

)

Q g = α g ⋅ t s − t g ⋅ Fs ,

(2.69)

Q L = α L ⋅ ( t L − t s ) ⋅ Fs .

(2.70)

At the phase of boundary the heat balance is maintained:

Q ∑ = Qg + Qs ,

(

)

α L ⋅ ( t L − t s ) ⋅ Fs = α g ⋅ t s − t g ⋅ Fs + r ⋅ β p ⋅ ( p s − p) ⋅ Fs .

(2.71)

The moving force of mass-transfer on the free surface of the liquid is the difference of partial pressure Δp = p s − p , at Δp > 0 the liquid evaporates, and hence, the air gets wet, when Δp < 0 the moisture condenses or, as it happens with liquid absorbents, condensation, absorption and dissolving of moisture in the solution takes place, hence, the air is dehumidified. Let us consider the laminar (laminar-and-wave) liquid film flowing on the vertical surface of the flat-and-parallel packing-bed in contact with the turbulent gas flow under the conditions of phase cross-current (it is this mode of flows movement that is characteristic of film heat- and –mass transfer apparatuses in real ranges of changing gas and liquid working loads). For elementary liquid volume Δv = Δx ⋅ Δz ⋅ δ , the amount of heat lost in the time unit

dQ L = − c L ⋅ G L ⋅ Δt ,

(2.72)

where

G L = ρ L ⋅ δ ⋅ Δz ⋅ v L

–

is the mass flow rate of the liquid ( v L is the average by

the flow rate velocity of the liquid).

Solar Absorption Systems…

67

On the other hand, the amount of heat removed by convection on the free surface ΔFs = Δx ⋅ Δz is equal to:

dQ L = α L ⋅ ( t L − t s ) ⋅ ΔFs

(2.73)

and the equation of the heat balance for liquid volume Δv has the form:

− c L ⋅ g L ⋅ Δt = α L ⋅ ( t L − t s ) ⋅ Δx where m

g L = ρL ⋅ δ ⋅ v L =

GL

2⋅n⋅L

–mass flow rate of the liquid referred to the unit of the packing-bed sheet width, kg/(m.s) m

GL

– mass flow rate of the liquid in a film apparatus, kg/s;

n L

– number of packing-bed sheets; – packing-bed size in the direction of the air movement, m.

Taking the relationship (2.71) into consideration and going to the limit obtain:

∂t L ∂x

(

Δx → 0 , well

)

= a 1 ⋅ t g − t L + b1 ⋅ ( p − p s ) ,

(2.74)

where

a1 =

αg cL ⋅ gL

,

b1 =

βp ⋅ r cL ⋅ gL

.

Let us consider the air movement in the flat-and-parallel pacing bed between two surface wetted by liquid film running off. We distinguish the elementary volume of air Δv = Δx ⋅ Δz ⋅ b («b» is the distance between sheets). Then from the element of area ΔFs = Δx ⋅ Δz the free surface of liquid film

Δg s = β p ⋅ ( p s − p) ⋅ ΔFs will evaporate. The increase of moisture content in the gas volume

Δg g = G g ⋅ Δx p , where

(2.75)

Δv g will be (2.76)

68

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

G g = ρ g ⋅ b ⋅ v g ⋅ Δx

–mass flow rate of the air in the channel cross-section

g

ΔFs = b ⋅ Δx ; x p = 0.622 ⋅ The material balance equation

p 0.622 = ⋅ p (as pg − p pg

p g >> p) .

(2.77)

Δg g = 2 ⋅ Δg s at Δz → 0 with equation (2.77) taken into

account has the form

β p ⋅ pg ∂p = 3.2154 ⋅ ⋅ ( p s − p) , ∂z gg

(2.78)

Where: m

g g = ρg ⋅ v g ⋅ b = –

Gg

n⋅H

mass flow rate of the air reduced to the unit of the channel height, kg/(m.s); m

Gg –

total mass flow rate of the air of the film apparatus, kg/s;

n H

number of sprayed channel of the packing-bed ; packing-bed height, m.

– –

As the result of heat-and-mass transfer for the air volume of

Δv g , it is enthalpy changes

ΔQ L = G g ⋅ Δh ,

(2.79)

Where:

h = c g + r0 ⋅ x p + c p ⋅ t g ⋅ x p

–

specific enthalpy of wet air;

r0 = 2500 kJ / kg

–

heat of vaporization at 0 °C.

Then equation (2.79) will take the form

(

)

dQ L = G g c v ⋅ Δt g + rt ⋅ Δx p , Where:

c v = cg + c p ⋅ x p ;

rt = r0 + c p ⋅ t g .

(2.80)

Solar Absorption Systems…

69

On the other hand, the amount of heat removed from the free surface of water ΔS L = Δx ⋅ Δz at convection and evaporation, is determined by the relationship

[ (

]

)

dQ g + dQ r = α g ⋅ t L − t g + r ⋅ β p ⋅ ( p s − p) ⋅ Δx ⋅ Δz .

(2.81)

Hence, the heat balance equation for the volume of the air under consideration is

(

dQ L = 2 ⋅ dQ g + dQ or going to the limit of

)

Δz → 0 :

∂t g ∂x p ⎞ ⎛ ⎟ = 2 ⋅ α L ⋅ t L − t g + 2 ⋅ r ⋅ β p ⋅ ( p s − p) gg ⋅ ⎜ c v ⋅ + rt ⋅ ∂z ∂z ⎠ ⎝

(

∂x p ∂z gg ⋅ c v ⋅ Where:

∂t g ∂z

=

)

(2.82)

0.622 ∂p 2 ⋅ β p ⋅ = ⋅ (ps − p ) , p g ∂z Gg

(

)

= 2 ⋅ α g ⋅ t L − t g + 2 ⋅ β p ⋅ Δr ⋅ ( p s − p) ,

(2.83)

Δr = r − rt .

As the numerical analysis shows for characteristic values defining the evaporation (condensation) of the water vapour

β p ⋅ Δ p ⋅ Δr α g ⋅ Δt g

<< 1 .

(2.84)

Hence, the gas temperature change can be found by the equation:

∂t g ∂z

(

)

= a2 ⋅ tL − tg , a2 =

2 ⋅ αg cv ⋅ gg

.

(2.85)

The mathematical model of evaporative cooling (condensation) processes in DEC/CTW at cross phase current is given by the system of equations:

70

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

(

)

⎧ dt L ⎪⎪ dx = a 1 ⋅ t g − t L + b1 ⋅ p g − p * ; ⎨ ⎪ dt g = a ⋅ t − t ; dp g = b ⋅ p * − p g 2 2 L g ⎪⎩ dz dz

(

)

(

(

)

(2.86)

)

and boundary conditions at

0

0

0

x = 0, t L = t L ; at z = 0, t g = t g , p g = p g .

(2.87)

The partial pressure value for saturated vapour on the free surface depends on the temperature of the liquid:

p* = p * ( t L ) .

(2.88)

Coefficients of heat-and-mass transfer for the flat-and-parallel packing-bed were found by relationships (2.33) 0.8

0. 4

Nu g = 0.023 ⋅ Re g ⋅ Pr , Sh = 0.95 ⋅ Nu g .

(2.89)

The boundary problem (2.86) – (2.87) is solved by the method of finite differences. Equations (2.86) are approximated by the following difference scheme:

[ ( (

] )

( )

⎧t i+1,k = 1 − a ⋅ Δx ⋅ t i ,k + a ⋅ Δx ⋅ t i ,k + b ⋅ Δx ⋅ p i ,k − b ⋅ Δx ⋅ p * t i ,k ; g g 1 1 1 1 L L ⎪ L ⎪⎪ i ,k +1 i ,k i ,k (2.90) ⎨ t g = 1 − a 2 ⋅ Δz ⋅ t g + a 2 ⋅ Δz ⋅ t L ; ⎪ ⎪ p i ,k +1 = 1 − b ⋅ Δ z ⋅ p i ,k + b ⋅ Δ z ⋅ p * t i , k . 2 2 g L ⎪⎩ g

( )

)

For boundary nodal points 0 .k

0

i.0

i.0

0

t L = t L , t g = t g , pg = pg .

(2.91)

Liquid flowing on the corrugated surfaces has some features caused by the fact that because of forces of surface tension the liquid is accumulated in cavities a curved surface and the movement of the liquid in such a packing-bed is of a regular-jet nature, and effect of gas flow doesn’t practically result in the removal of the liquid from the apparatus. On the basis of theoretical and experimental researches in jet flows on corrugated surfaces with regular roughness (P, E are corrugation parameters; p, e – regular roughness parameters) the authors obtained (section 2.3.2):

Solar Absorption Systems… 0.3277

δ 0 = 0.7986 ⋅ G Lv

71

,

(2.92)

Where:

G Lv –

the volume flow rate of the liquid in a jet, cm3/s (at 8 mm≤ P ≤ ≤12 mm, 3

mm≤ E ≤4 mm; 8 mm≤ p ≤10 mm, 0.6 mm≤ e ≤0.8 mm.) The value of free liquid surface ( 1 ), as well as perimeters of wetted (

0

) and dry (

2

)

sections of the corrugated surface are determined by the relations (figure 2.20): ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪⎩

1

δ0

= p⋅

2⋅E

⎛ 4 4 ⋅ E⎞ ; ⎜ Δ = ⋅ 2 ⋅ δ 0 ⋅ E , Δ1 = ⎟ p p ⎠ ⎝

(2.93)

2

0

p 2 2 = ⋅ ⎡ln⎜⎛ Δ + 1 + Δ + Δ 1 ⋅ 1 + Δ ⎟⎞ ⎤ ⎝ ⎠ ⎥⎦ ⎢ 16 ⋅ E ⎣ 2

2

=

p 2 2 ⋅ ⎡ln⎛⎜ Δ + 1 + Δ 1 ⎞⎟ + Δ 1 ⋅ 1 + Δ 1 ⎤ − 0 ⎠ 32 ⋅ E ⎣⎢ ⎝ 1 ⎦⎥ 2

Processes of the evaporative cooling (condensation) in DEC/CTW with corrugated components having regular roughness are again defined by the problem (2.86) – (2.88), in this case coefficients a 1 , a 2 , b 1 , b 2 in Equations (2.86) take into account dry and wetted sections of the surface and are calculated by formulas:

a1 =

a2 =

(

αL ⋅

2⋅ αL ⋅

1

+ kα ⋅

0

+K

cL ⋅ gL 1

+ kα ⋅

p ⋅ cv ⋅ gg

0

+ 2⋅K

, b1 =

),

βp ⋅ r ⋅

1

cL ⋅ gL

b 2 = 3.2154 ⋅

, β p ⋅ pg p ⋅ gg

,

(2.94)

⎛ 1 δ 1⎞ ⎟, + + kα = 1 ⎜ ⎝ αs λ L α L ⎠ Where:

k α – heat transfer coefficient; K

takes into account the efficiency of the fin BD of the corrugated surface.

Heat transfer coefficient are determined by the Kader formulas, usually used when the liquid (gas) is flowing through the channels with large bulges of the walls roughness [24, 81].

Figure 2.20. Schematic of gas and liquid flow (film-jet liquid flow, cross-current scheme) in channels of the regular packing bed (RP), which is formed by longitudinally corrugated sheets. Evaporative unit of AACS, the combined IEC|DEC cooler. Notation are given in figure 2.19. Additional notations: main air flow;

the auxiliary air flow;

the water flow.

the

Solar Absorption Systems…

73

2.3.4.2. Simulation of Working Processes in IEC The evaporative cooling process in the IEC is described by a system of equations (Figures 2.19 – 2.20):

⎧ dt w ⎪ dx = a 1 ⋅ ( t AX − t w ) + b 1 ⋅ ( p − p *) + c1 ⋅ ( t mn − t w ); ⎪ dp AX ⎪ dt AX = a 2 ⋅ ( t w − t AX ); = b 2 ⋅ ( p * − p); ⎨ dz dz ⎪ ⎪ dt mn ⎪ dz = c 2 ⋅ ( t w − t mn ). ⎩ at

0

0

0

(2.95)

0

x = 0, t w = t w ; at z = 0, t AX = t AX , p = p , t mn = t mn .

(2.96)

Here constants a 1 , a 2 , b 1 , b 2 , c1 , c 2 take into account dry and sprayed sections of corrugated surface with regular roughness (2.92), (2.93), and heat exchange coefficients are determined by Kader formulas [24, 81]. The solution of equations (2,95) was carried out by the method of finite difference set:

[

]

(

)

⎧t i+1,k = 1− (a + c ) ⋅ Δx ⋅ t i,r + a ⋅ t i,k − b ⋅ pi,k + b ⋅ pi,k* + c ⋅ t i,k ⋅ Δx, 1 1 1 1 1 1 w AX mn ⎪w ⎪ i,k+1 i,k i,k ⎪t AX = ⋅(1− a2 ⋅ Δz) ⋅ t AX + a2 ⋅ t w ⋅ Δz, (2.97) ⎨ ⎪pi,k+1 = c ⋅ (1− b ⋅ Δz) ⋅ pi,k + b ⋅ pi,k* ⋅ Δz, 2 2 2 ⎪ i,k ,k+1 ,k i i ⎪t ⎩ 0 = (1− c2 ⋅ Δz) ⋅ t mn + c2 ⋅ t w ⋅ Δz, Where: i, k

–

define nodal points on x and z coordinates.

For boundary nodal points they define parameters of all the channels in the IEC both for single – multistage scheme. System (2.86) – (2.95) is easily transformed for the use with DEC (for the IEC/DEC or IEC/CTW scheme). Discussed are the processes of regenerative indirect evaporative cooling of air (IEC/R) during which the whole air-flow is first cooled without contracting the liquid in «dry» channels and only then a portion of the flow (an auxiliary air flow) is passed to "wetted" channels of the packing bed, where the water film is cooled by evaporation in direct contact with the air flow. "Dry" and “wetted” channels alternate multichannel packing bed with heat conducting walls of high heat conductivity made of a thin-walled aluminium sheet. Thus,

74

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

unlike the IEC scheme, shown in figure 1.6, the flows are separated not at the inlet of the apparatus but at the outlet of it, and a portion of the flow returns to “wetted” channels of the apparatus packing bed. The limit of cooling in IEC/R is the dew point temperature of the air to be cooled. The combination of such a regenerative IEC with the heat-exchanger HEX 13 (figures 1.1 – 1.5) without DEC is one of its versions. It is this version of CEC that was used in AACS designs (Section 3.3). Mathematical models developed make it possible to calculate temperature and humidity parameters (fields) in “dry” and “wetted” channels of air evaporative coolers and to optimize the cooling process allowing for the minimization of the energy consumption for its development.

2.3.5. Simulation of Working Processes in the Air-Solution Contactor (PreDehumidification of Air by Liquid Absorbent and Solar Regeneration of Absorbent) The drying part of AACSs including the dried (absorber) and the regenerator (desorber) is schematically shown in figure 1.8A in the form of film cross flow heat-and-mass transfer apparatuses (A) and apparatuses of combined with heat exchangers types – the cooler and the heater (Figure 1.8B). In the latter case heat-and-mass transfer apparatuses – the absorber and the desorber – have the packing bed in the form of tube-plate surfaces, which was developed by the project authors in conformity with air condensers of refrigerating plants [24]. The construction in figure 1.8A does not practically differ in any way from the cooling tower CTW described above, or the direct evaporative cooler DEC. Figures 1.8C and 2.22 show the absorber with internal evaporative cooling. The packing bed design of such an absorber is similar to that of the apparatus for indirect evaporative cooling. In the inner space of heat exchanging components-wafers the air flow being dehumidified moves in contact with the absorbent film streaming down the internal surfaces; in the interwafer space there moves the air flow and the water film streaming down the outer surfaces which removes the absorption heat, when the evaporative cooling takes place, through a thin heat conducting wall of heatexchanging components (figure 2.22) The design of the air dehumidifier (absorber) and the regenerator of solution (desorber) is carried out on the basis of the developed mathematical model of absorption (desorption) processes. Aqueous solutions of salts are used as absorbents for dehumidifying the air. Aqueous solutions of LiCl and LiBr are most widely used for dehumidifying the air, as well as aqueous solutions of salts with additives (inhibitors) which reduce the corrosive effects of the given solutions on construction materials (see 2.2). Mathematical model of absorption (desorption) processes under the conditions of cross phase-contacts is presented by a system of equations

⎧ ∂t s ⎪⎪ ∂x = a 1 ⋅ t g − t s + b 1 ⋅ p g − p ** ; ⎨ ⎪ ∂t g = a ⋅ t − t ; ∂p g = b ⋅ p ** − p s g g 2 2 ⎪⎩ ∂z ∂z

(

(

and boundary conditions

)

)

(

)

(

)

(2.98)

Solar Absorption Systems… 0

0

ξ=ξ

0

pg = pg .

at x

= 0,

ts = ts ,

at z

= 0,

tg = tg ,

75

(initial concentration of the solution), 0

(2.99)

Figure 2.21. Dehumidifying unit of the AACS. Schematic of the absorber (A) and the scheme of gas and liquid flows (B) of the air flow being dried and the absorbent – in the channels of the regular packing bed (RP). Nomenclature is the same as in figure 2.19 In addition to the above entering air flow;

dried flow;

absorbent.

76

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 2.22. Dehumidifying unit of the ASAC. Schematic of the absorber (A) with internal evaporative cooling and the scheme of gas, water and absorbent flows (B) in channels of HMA packing bed. Nomenclature is the same as in figure 2.19: outlet of the auxiliary air flow;

inlet of air;

absorbent;

outlet of the dried air flow;

water.

Significant distinction between the problem under consideration (2.98), (2.99) and that of evaporative liquid cooling (2.86), (2.87) lies in the fact that problem (2.98), (2.99) is mainly non-linear, as coefficients a 1 , a 2 , b 1 , b 2 , as well as p s are function of the solution

Solar Absorption Systems…

77

temperature and concentration. Coefficients of heat transfer in this case are determined by formulas as before [24, 81]. The problem (2.98), (2.99) is solved by the finite differences method. In designing the desorber the same equations (2.98) are used, but the solution temperature at the inlet to the apparatus is selected by the iteration method so that the concentration of the solution should reach the needed value

0

ξ at the outlet from the

apparatus. The design programs for air-solution contactor (absorber – desorber) include equations approximating thermophysical properties of working liquids (solution) in the required ranges of working parameters (temperatures and concentrations) and thus the programs provide the possibility to calculate the working processes in the AACS irrelative of types of absorbents used.

2.3.6. Modeling the Processes in a Heliosystem with Flat Solar Collectors (An External Heating Source of AACS) and in a Tank-Accumulator In developed AACS schemes a heliosystem with flat SCs is used as an external heating source. The heliosystem usually includes a SC system, a thermally insulated tankaccumulator, a system of downcomimg and lifting pipelines. The circulation of the heat carrier is realized by the natural way due to density differences in downcoming and lifting parts of the heliosystem, or by the forced one with the help of a pump (figures 2.33, 2.25). This section describes the methods of thermal and hydraulic designs developed by the authors for both types of heliosystems. 2.3.6.1. Heliosystem with Natural Circulation of the Heat Carrier Natural circulation of the heat carrier is used in comparatively small heliosystem. Despite the simplicity of these systems design as compared to systems with forced circulation, methods of theoretical investigations are more complicated because of the necessity to conduct both thermal and hydraulic designs simultaneously, as the heat-carrier flow rate is the value to be determined. Some additional difficulties occur because of the nonstationarity of processes taking place in heliosystem components. The heliosystem design has been made on the basis of a mathematical model accounting for the most important characteristics of thermal and hydrodynamic processes occurring in the system on the whole and its every component, at the same time mathematical description provided for the conjugation of all heliosystem components. The model being developed is presented by nonstationary equations of energy, movement and continuity for all components of the plant with the relevant initial and boundary conditions which provide conjugation of these components. The design process can be conditionally divided into some stages: thermal design of the SC; thermal design of TA together with connecting pipelines; determination of the heat carrier flow rate at its natural circulation. A mathematical SC model has been developed (figures 2.26 – 2.27) which had some modifications and additions as compared to know solutions [3, 17, 21]: •

the offered model is based on the two-dimensional concept of heat processes allowing for the material heat capacity;

78

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov • •

the longitudinal heat conduction of the absorber plate has been taken into consideration in the direction of the liquid movement; the effect of cooling the absorber by the liquid coming from the TA was taken into account.

The transparent coating temperature

δ s ⋅ ρs ⋅ c s ⋅

∂Ts ∂τ

Ts = Ts ( τ, x, y) is found by the equation:

(

)

= U ps ⋅ Tp − Ts + U s0 ⋅ (Ts − T0 ) .

(2.100)

In calculating U (the coefficient of heat losses) we took into account the plate and glass radiation, free convection between the plate and the glass, the heat conduction through the thermal insulation, etc. according to the methods given in [17]. The plate temperature is defined by the equation

⎛ ∂ 2Tp ∂ 2Tp ⎞ ⎟ + J − U ⋅ T − T + U ⋅ T − T , (2.101) δp ⋅ ρ p ⋅ c p ⋅ = λ p ⋅ δp ⋅ ⎜ 2 + 2 i p o ps s p ⎜ ∂x ⎟ ∂τ ∂ y ⎠ ⎝ ∂Tp

where

Ui =

λi δi

(

(

))

(

)

.

The initial condition of equations (2.100) and (2.101) has the form at

τ = 0, Tp = Ts = T0 .

Boundary conditions for the plate range

0 ≤ x ≤ w 2 , 0 ≤ y ≤ L sc can be presented in

the following way:

w , 2

∂Tp

at

x=

at

x = 0, 2 ⋅ λ p ⋅ δ p ⋅

∂x

= 0, ∂Tp ∂x

(

)

= α L ⋅ π ⋅ d L ⋅ Tp − TL .

(2.102)

Here one can suppose that the heat flow is uniformly distributed around the pipe perimeter

at

y = 0,

∂Tp ∂y

= 0,

Solar Absorption Systems…

at

∂Tp

y = L sc ,

∂y

Temperature fields

79

= 0.

(2.103)

Ts = Ts ( τ, x, y) and Tp = Tp ( τ, x, y) are determined in the range: 0 ≤ τ ≤ τ s , ( τ s is the light day duration) 0≤ x≤

w , 0 ≤ y ≤ L sc . 2

The typical SC construction is defined by inequality

(2.104)

Lsc >> W 2 , therefore it seems

possible to average the temperature field by the coordinate x, i.e. to reduce the problem to a one-dimensional one, supposing w

w

2 2 2 2 Tp ( τ, y) = ⋅ ∫ Tp ( τ, x, y)dx , Ts ( τ, y) = ⋅ ∫ Ts ( τ, x, y )dx , w 0 w 0 In heliosystems with the natural circulation the laminar mode of the liquid flow is usually ound, for which the following relationship is valid:

α L = 4.36 ⋅

λL dL

.

Taking into account this relationship and the above boundary conditions one can write equations (2.100) and (2.101) in the form:

δ s ⋅ ρs ⋅ c s ⋅ δ p ⋅ ρp ⋅ c p ⋅

∂Ts ∂τ

∂Tp

(

)

))

∂ 2 Tp

(2.105)

λL ⋅ Tp − TL + ∂y w + U ps ⋅ Ts − Tp .

= λp ⋅δp ⋅

∂τ + J − U i ⋅ Tp − To

(

(

= U ps ⋅ Tp − Ts + U s0 ⋅ (Ts − T0 ) ,

2

− 13.7 ⋅

(

)

(

)

(2.106)

Averaging signs in these formulas are omitted. The system of equations (2.105) and (2.106) was solved under the following conditions: initial conditions at

τ = 0, Ts = Tp = T0 ,

boundary conditions

(2.107)

80

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

at y=0 and at

y = L sc ,

∂Tp ∂y

=0

(2.108)

The temperature distribution in the liquid flowing through SC pipes is defined by the differential equation of convection 2

π ⋅ d L ∂TL G ∂T ρL ⋅ c L ⋅ ⋅ + c L ⋅ L ⋅ L = 13.7 ⋅ λ L ⋅ Tp − TL . 4 ∂τ n ∂y

(

)

(2.109)

The thermal SC design is to solve equations (2.105), (2.106) and (2.109) at the following initial and boundary conditions: at

τ = 0, Ts = Tp = TL = T0 ; ∂Tp

= 0, TL = T2 (L 2 ) ; ∂y ∂Tp =0 . at y = L sc , ∂y at

y = 0,

(2.110)

The mathematical TA model (figure 2.26) takes into account the effect of mixing stratified liquid layers as a result of convection when the hot heat carrier is supplied. The energy conservation equation for the TA has the form:

ρL ⋅ c L ⋅ fT ⋅

∂TT ∂τ

The initial conditions is: at

+ cL ⋅ G L ⋅

∂z

2

= λ L ⋅ fT ⋅

∂ TT ∂z

2

+ U T 0 ⋅ PT′ ⋅ (T1 − T0 ) . (2.111)

τ = 0, TT = T0 .

The boundary conditions is: at

at

∂TT

z = 0, TT = T2 ( τ, L 2 ) ,

z = HT , λL ⋅

∂TT ∂z

The liquid temperature in the lifting pipelines

= U T 0 ⋅ (TT − T0 ) .

T2 = T2 ( τ, y) is defined by equation

(2.112)

Solar Absorption Systems… 2

ρL ⋅ cL ⋅

π ⋅d2 4

⋅ L2 ⋅

∂ T2 ∂τ

The initial conditions is: at

+ cL ⋅ G L ⋅ L2 ⋅

∂ T2 ∂y

= U 20 ⋅ π ⋅ d 2 ⋅ L 2 ⋅ (T2 − T0 ) .

y = 0, T2 = TL ( τ, L sc ) .

The liquid temperature in the downcomimg pipeline 2

ρL ⋅ c L ⋅

4

∂T1

⋅ L1 ⋅

∂τ

The initial condition is: at

(2.113)

τ = 0, T2 = T0 .

The boundary conditions is: at

π ⋅ d1

81

+ c L ⋅ G L ⋅ L1 ⋅

∂T1 ∂y

(2.114)

T1 = T1 ( τ, y) is defined by equation

= U10 ⋅ π ⋅ d 1 ⋅ L1 ⋅ (T1 − T0 ) .

(2.115)

τ = 0, T1 = T0 .

The boundary conditions is: at

y = 0, T1 = TT (τ, H T ) .

(2.116)

The determine the liquid flow rate around the close loop we use the Navier-Stokes equation for a one-dimensional liquid flow through the pipe [59] 2

1 ∂p ∂ω ∂ω ∂ω +ω⋅ = − ⋅ + ω ⋅ 2 + gξ , ∂τ ∂ξ ρ L ∂ξ ∂ζ or 2

⎛ ∂ω ∂ω ⎞ ∂p ∂ω + ω ⋅ ⎟ = − + μ ⋅ 2 + ρ ⋅ gξ . ρL ⋅ ⎜ ∂ξ ⎠ ∂ξ ⎝ ∂τ ∂ζ

(2.117)

Averaging equation (2.18) by the cross-section of the pipe we shall obtain

⎛ ∂ω ∂ω ⎞ ∂p ρL ⋅ ⎜ + ω ⋅ ⎟ = − + τ st + ρ ⋅ g ξ , ∂ξ ⎠ ∂ξ ⎝ ∂τ

(2.118)

Where:

ω = ω( τ , ξ), p = p( τ, ξ ) are the cross-section average velocity and pressure of the liquid;

τ st is the tangential stress on the pipe wall. Here it is taken into account that on the pipe axis ( ζ

= 0)

82

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

τ = μ⋅ and on the pipe wall ( ζ

∂ω = 0, ∂ζ ζ=0

=d 2) μ⋅

∂ω = τ st . ∂ζ ζ = d 2

The sign of averaging is further omitted. As the liquid motion as a result of natural convection takes place in the closed loop of the hydraulic system of the plant, we shall integrate equation (2.118) by this loop:

∫ρ

L

⋅

∂ω ∂ω ∂p ∂ω . (2.19) dξ + ∫ ρ L ⋅ ω ⋅ dξ = − ∫ dξ + ∫ τ st ( ξ )dξ + ∫ ρ L ⋅ g ξ ⋅ dξ ∂τ ∂ξ ∂ξ ∂τ

Let us analyse the values in the right part of equation (2.119). It is possible to show that [118]

∫ τ st (ξ)dξ = −ρ Where:

h i is

L

⋅ g ⋅ ∑ hi , i

specific head losses for overcoming all forces of resistance on i-th part of

the hydraulic loop. Let us consider the integral .

⎡

∫ ⎢⎣ρ

L

⋅ gξ −

∂p ⎤ dξ ∂ξ ⎥⎦

As it is know, the difference between the volume force and the pressure force can be taken as equal to

g ⋅ (ρ − ρ 0 ) at the natural convection of the liquid, thus

⎡

∫ ⎢⎣ρ where

L

⋅ gξ −

∂p ⎤ dξ = − ∫ g ⋅ (ρ L − ρ 0 ) ⋅ sin ϕ ξ dξ = − g ⋅ ∫ (ρ L − ρ 0 ) ⋅ sin ϕ ξ dξ , ∂ξ ⎥⎦

ϕ ξ is the slope angle of the loop dξ with respect to the horizontal.

Let us analyse the values in the left part of equation (2.119). As the liquid velocity at the natural circulation is low, the velocity component

∫ ρL ⋅ ω ⋅

2 ∂ω ∂ dξ =ρ ⋅ ∫ ⎛⎜ ω 2 ⎞⎟ dξ ⎠ ∂ξ ∂ξ ⎝

can be neglected. Let us represent the first component of the left part in the form:

Solar Absorption Systems…

∫ ρL ⋅

83

∂ω ∂ω dξ = ∑ ρ i ⋅ i ⋅ L i . ∂τ ∂τ i

As a result of it, equation (2.119) will have a form more convenient for practical calculations

∑i

∂ω i ∂τ

⋅L i = − g ⋅ ∫

1 ⋅ (ρ L − ρ 0 ) ⋅ sin ϕ ξ dξ − g ⋅ ∑ h i . ρ0 i

(2.120)

Writing down the equation of continuity for all heliosystem components (SC, TA, of the downcoming and lifting pipelines) and allowing for the relation

HТ fТ

<<

L1 f1

we can write

the first component in the form:

∑

∂ω i ∂τ

⋅ Li =

L L ⎞ ∂G L L ⎛ L sc ⋅⎜ + 1 + 2⎟⋅ . ρ L ⎝ n ⋅ f sc f1 f 2 ⎠ ∂τ

Introducing the volume expansion coefficient, we will write the first component in the right part of equation (2.120) in the form: 1 ⋅ (ρ L − ρ0 ) ⋅ sin ϕdξ = β ′ ⋅ ρ0 ∫

Lsc

∫0 [T ( y) − T ( L )] ⋅ sin ϕ sc dy − β ′ ⋅ ∫ [T ( y) − T ( L )]dz . L

2

T

2

1

1

(2.121)

Where:

ϕ sc is the slope angle of the SC; the temperature difference along the length of connecting pipelines is neglected. The last integral in the right part of equation (2.121) is the moving force for the free convection in the TA resulting in the formation of closed liquid currents inside the tank; therefore, the said integral can be neglected. Writing down the expression for hydraulic losses, neglecting local resistances as compared to the resistance along the length, and allowing for the fact that for the laminar mode of flowing, which, as a rule, takes place at natural circulation, the coefficient of resistance along the length is defined by the relation: λ ′ =

64 . The second component in the Re

right part of formula (2.121) can be written down in the form:

84

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

g ⋅ ∑ hi =

32 ⋅ ν L ρL

Here it is taken into account, that

⎛ L L1 L2 ⎞ sc ⎟. ⋅ G L ⋅⎜ + + 2 2⎟ ⎜ n ⋅ d2 ⋅ f f1 ⋅ d 1 f 2 ⋅ d 2 ⎠ ⎝ sc sc HТ 2

d Т ⋅ fТ

L1

<<

2

d 1 ⋅ f1

(2.122)

hence, the equation of the liquid

movement will have the form:

⎛ L sc L L ⎞ ∂G L ρ L gβ ′ ⎜ + 1 + 2⎟ = 2 ⎝ n ⋅ f sc f1 f 2 ⎠ ∂τ

Lsc

∫ [ T ( y) − T ( L )]dy − 0

L

2

2

(2.123)

⎛ L L1 L2 ⎞ sc ⎟. − 32 ⋅ ν L ⋅ G L ⋅ ⎜⎜ + + 2 2 2⎟ ⎝ n ⋅ d sc ⋅ f sc f1 ⋅ d 1 f 2 ⋅ d 2 ⎠ The initial condition is: at

τ = 0, G L = 0 .

Thus, the heliosystem with the natural circulation is defined by the following system of equations, initial and boundary conditions: For The Solar Collector

(

)

∂Ts ⎧ ⎪δ s ⋅ c s ⋅ ρ s ⋅ ∂τ = U ps ⋅ Tp − Ts + U s0 ⋅ (Ts − T0 ), ⎪ 2 ⎪ ∂Tp ∂ Tp λ = λ p ⋅ δ p ⋅ 2 − 13.7 ⋅ L ⋅ Tp − TL + ⎪δ p ⋅ c p ⋅ ρ p ⋅ w ∂τ ∂y ⎪ ⎪ + J − U i ⋅ Tp − T0 + U ps ⋅ Ts − Tp , ⎪ 2 ⎪ G ∂T π ⋅ d 1 ∂TL ⎪c L ⋅ ρ L ⋅ ⋅ + c L ⋅ L ⋅ L = 13.7 ⋅ λ L ⋅ Tp − TL , 4 n ∂y ∂τ ⎨ ⎪ ⎪ ⎪at τ = 0, Ts = Tp = TL = T0 ; ⎪ ∂Tp ⎪ ⎪at y = 0, ∂y = 0, TL = T2 (L 2 ); ⎪ ⎪ ∂Tp = 0. ⎪at y = L sc , ∂y ⎩

(

[

(

)]

(

)

)

(

)

(2.124)

Solar Absorption Systems…

85

For the Tank-Accumulator (TA) in Downcoming and Lifting Pipelines 2 ⎧ ∂ TT ∂TT ∂TT ⎪c L ⋅ ρ L ⋅ f T ⋅ = λ L ⋅ f T ⋅ 2 + U T 0 ⋅ PT′ ⋅ (T0 − TT ), + cL ⋅ G L ⋅ ∂z ∂τ ∂z ⎪ 2 ⎪ ⎪c ⋅ ρ ⋅ πd 2 ⋅ L ⋅ ∂T2 + c ⋅ G ⋅ L ⋅ ∂T2 = U ⋅ π ⋅ d ⋅ L ⋅ (T − T ), 2 L L 2 20 2 2 2 0 ⎪ L L 4 ∂y ∂τ ⎪ 2 ⎪ πd 1 ∂T ∂T ⋅ L1 ⋅ 1 + c L ⋅ G L ⋅ L1 ⋅ 1 = U10 ⋅ π ⋅ d 1 ⋅ L1 ⋅ (T1 − T0 ), ⎪⎪c L ⋅ ρ L ⋅ ∂τ ∂y 4 ⎨ ⎪ ⎪ ⎪at τ = 0, T1 = TT = T2 = T0 ; ⎪at y = 0, T = T (L ), T = T (H ); sc 1 L 2 T T ⎪ ⎪at z = 0, TT = T1 (L1 ); ⎪ ⎪at z = H, λ ⋅ ∂TT = U ⋅ (T − T ). L T0 0 T ⎪⎩ ∂z

(2.125)

The equation of the movement is L ⎧⎛ L ⎤ L L ⎞ ∂G L ρ L ⋅ g ⋅ β′ ⎡ sc ⎪⎜ sc + 1 + 2 ⎟ ⋅ ⎢ ∫ TL (y )dy − T2 (L 2 ) ⋅ L sc ⎥ − = ⋅ 2 ⎪⎜⎝ n ⋅ f sc f 1 f 2 ⎟⎠ ∂τ ⎢⎣ 0 ⎥⎦ ⎪ ⎪ ⎛ L L1 L 2 ⎞⎟ sc ⎨ − 32 ⋅ ν ⋅ G ⋅ ⎜ ; + + 2 2 2 L L ⎜ n ⋅d ⋅f ⎪ f1 ⋅ d 1 f 2 ⋅ d 2 ⎟⎠ sc sc ⎝ ⎪ ⎪ ⎪at τ = 0, G = 0. ⎩ L

(2.126)

The calculation was done by the method of finite differences using the implicit scheme of Krank-Nikolson which is absolutely stable. The heat design program for a heliosystem was developed which makes it possible to determine the heat carrier temperature in all components of the plant at any time. Figure 2.24 gives the comparison of our temperature calculation results for the TA of the helioplant with the results of approximate calculations [17]. The design was carried out for a helioplant with the natural circulation of the heat carrier having 12 SCs of 12 m2 total area and the volume of the TA 1 m3. The solar radiation intensity and the ambient temperature were taken from the Reference-book on Climate for Odessa in July. It follows from Figures that the account of factors, mentioned in the beginning of the section, in the model results in the temperature increase in the lower part of the TA, and, hence, at the inlet to the SC, which decreases the design capacity of the plant. The temperature in the upper part of the TA is lower than that in reference [17] because of which the average liquid temperature in the TA decreased. Such behavior agrees quite will with physical conceptions of the influence of changes occurring in the model on heliosystem characteristics.

86

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 2.23. Schematics of heliosystems. A – heliosystem with natural circulation of heat carrier, B – heliosystem with forced circulation of heat carrier. 1-SC, 2-lifting pipe-line, 3-tank-accumulator, 4duplicate source of heat, 5-hot water extraction for the consumer, 6-cold water replenishment, 7lowering pipe-line, 8-water pump.

Figure 2.24. Water temperature in the tank-accumulator. 1-upper part of tank-accumulator, 2-lower part of tank-accumulator. - - - - - results of approximated calculations; –––––––– calculation by developed mathematical model.

2.3.6.2. Heliosystem with Forced Circulation of the Heat Carrier Mathematical model of the heliosystem with forced circulation is simplified as compared to the above-mentioned one since the flow rate of the heat carrier is not the parameter to be defined. With the increase of the flow rate the COP of the heliosystem increases, but at the same time more energy is consumed for the movement of the heat carrier. In practice they usually select the flow rate proceeding from the relation G ⋅ c p Fsc ⋅ U = 2-4, which corresponds to the efficiency of SC equal to 86 – 90 % of the maximum possible one under given outer conditions [17]. In modeling heliosystems with forced circulation two approaches are usually used. One is based on large-scale averaging the characteristics of heliosystems

Solar Absorption Systems…

87

proceeding from accumulated experimental data. In this case the results are obtained rather easily, however because of low accuracy they can be used only for approximate evaluation of heat capacity of the heliosystem. The other approach, which is being developed by the authors of the project, is based on solving differential balance equations for heliosystems and makes it possible to predict accurately enough time dependencies of heat characteristics of heliosystems allowing for their construction peculiarities and varying climatic conditions. Both approaches are analysed below. Normative method of calculating heliosystems. COP of the heliosystem can be calculated by formula [17]

[ (

)]

⎧ ⎫ out in 0.5 ⋅ t − t − t 0 ⎪ ⎪ η = 0.8 ⋅ ⎨θ − 9 ⋅ U ⋅ ⎬, J ∑ i ⎪ ⎪ ⎩ ⎭ i

(2.127)

where 0.8 is a coefficient taking into account the dust content and shadowing; the summation is done by hours of the light day. The solar radiation intensity in the solar collector plan is founded by expression:

J = Js ⋅ ks + J D ⋅ k D . Coefficient

k s is defined by Tables [17], and k D is calculated by the relation 2

k D = cos β 2 , where β is the slope of the SC plane to the horizon (30° for the latitude of Odessa). The data for J s , J D and t 0 during every hour of the current month are given in the construction norms and rules CN&R 2.04.05-91 (“Heating, ventilation and air-conditioning”, State Construction Committee of the USSR, Moscow, 1997). The required area of the SC is defined by formula

Fsc =

Q′

η ⋅ ∑ Ji

,

(2.128)

i

where Q′ is the required 24-hour heat load for the warmest month when there is a duplicate source of heat in the heliosystem. The volume of tank-accumulator is defined by empirical formula

VTA = (0.06 − 0.08) ⋅ Fsc , m3/m2.

(2.129)

88

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 2.25. The heliosystem – the external heating source, providing solar regeneration of the absorbent in an alternative system; A - Two circuit heat supply system with two tanks- accumulators; B - Two circuit heat supply system with a combined tank- accumulator. Notation: 1 – solar collectors SC; 2 – pumps; 3 – expansion tank; 4 – tank-accumulator; 5 – tank for hot water; 6, 7 – heat exchangers; 8 – cold water replenishment; 9 – additional source of heat; 10 – water for the heat supply system.

The largest values of the volume in this formula refer to the IV climatic zone. To find a value of COP seasonal ηseas depending on Fsc and VTA one can use the nomograph given in [17]. The amount of heat energy generated by the heliosystem during a season (AprilSeptember) is equal to

Q p = ηseas ⋅ Q seas ,

Solar Absorption Systems… where

89

Qseas is the amount of heat falling on the SC of the heliosystem during a season.

Mathematical model of the heliosystem. The calculation procedure described above is of an approximate character and can be used for orientation evaluation of its efficiency. The mathematical model offered is to a large extent deprived of disadvantages, characteristic of the normative procedure, and makes it possible to obtain more realistic results. While deriving equations which describe processes of heat exchange in the heliosystem it was assumed: •

•

•

•

processes in the heliosystem are of a quasi-steady nature. It allows the use of characteristics, obtained under stationary conditions, in calculations. The legitimacy of this assumption is stipulated for a slow change of the solar radiation intensity and the temperature of surroundings; heat losses through pipe walls into surroundings were not taken into account. As the calculations and the experience in using heliosystems show, these losses are insignificant due to small surface of pipes and the heat insulation available; the temperature separation (stratification) of the heat carrier in TA was not taken into account. With the forced circulation the velocity of the heat carrier is greater than that with the natural one, which results in stirring the liquid. The assumption of complete stirring leads to under-estimated values of heat capacity, providing in such away a calculation store of generated heat energy. The heat balance equation for the heliosystem has the form

J ⋅ η ⋅ Fsc dτ = M TA ⋅ c pdt + k TA ⋅ FTA ⋅ ( t − t 0 )dτ

,

(2.130)

where t is the current temperature of the heat carrier in TA; M TA is the mass of the heat carrier in TA. Equation (2.130) can’t be solved with respect to the unknown temperature as ηdepends on t. Let us the known expression for COP of SC

[

out ⎧⎪ 0 .5 ⋅ ( t − t ) − t 0 η = 0.8 ⋅ ⎨θ − U ⋅ J ⎪⎩

]⎫⎪⎬ .

(2.131)

⎪⎭

From the heat balance equation of the heliosystem

(

J ⋅ η ⋅ Fsc = G ⋅ c p ⋅ t

out

−t

)

we shall find temperature tout introducing the idea of specific flow rate t

out

=t+

J⋅η . g ⋅ cp

(2.132)

g = G Fsc (2.133)

90

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 2.26. On modeling working processes in a flat solar collector. A is the absorber of SC (heat receiver), B is the cross section of the SC absorber. Nomenclature: 1 – the pipe register SC; 2 – collector pipes; 3 – glass (transparent insulation); 4 – heat insulation; 5 – the collector housing; 6 – the bottom; 7 – the clamping device (fastening the glass); 8 – the air gap.

From expressions (2.132) and (2.133) we obtained

t

out

⎛ g ⋅ с p − 0.4 ⋅ U ⎞ ⎛ ⎞ ⎟ + 0.8 ⋅ ⎜ J ⋅ θ + U ⋅ t 0 ⎟ . = t ⋅⎜ ⎜ g ⋅ с + 0.4 ⋅ U ⎟ ⎜ g ⋅ с + 0.4 ⋅ U ⎟ p p ⎝ ⎠ ⎝ ⎠

(2.134)

Solar Absorption Systems…

91

Thus, the problem of determining the time dependence of temperature in TA is reduced to solving the nonlinear differential equation of the first order

dt J ( n, τ) ⋅ η ⋅ Fsc k TA ⋅ FTA ⋅ [ t − t 0 ( n, τ)] = − dτ M TA ⋅ c p M TA ⋅ c p

(2.135)

allowing for expressions (2.131) and (2.134). In equation (2.34) n means the number of the month for which the calculation is done. The equation was solved by the method of RungeKutta with the time period of 1 hour. The calculations given with the time period twice as much showed a slight deviations of results.

Figure 2.27. The flat solar collector developed by the SPF "New Technologies" produced since 1991. A is a picture of the CK-1.1 modification collector. B is the general view of the collector. Notation: 1 – pipe register; 2, 3 – collector pipes; 4 – SC housing; 5 – fastening angle; 6 – glass; 7 – fastener; 8 – thermal insulation. C is the heat receiver in the form of a pipe register with fins.

92

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov In using the model the following factors were taken into account: •

•

•

the dependence of the heat loss coefficient on temperature. It is usually assumed that U = const. However, as the calculations show, with the increase of the absorbing plane temperature from 50 °C to 100 °C the heat loss coefficient increases by 30 %; the influence of the wind velocity on the heat loss coefficient. The increase of the wind velocity from 0 to 5 m/s increases the heat loss coefficient by 15 – 20 %, therefore the calculations were done allowing fore a real wind load for the area where AACS is located; the dependence of the penetration capacity of the transparent insulation from the angle of incidence of solar rays on the SC plane. It is necessary to take this factor into account, as the glass will reflect practically total radiation falling on its surface, if the said angle is over 60°, [17].

Using the described procedures the calculation was done of the heliosystem for hot water supply of a 50-flat house situated in Odessa. The total area of SC was 236 m2, the hot water temperature – 50 °C. The seasonal generation of heat calculated by an approximated method amounted to 310 GJ, and the heat obtained by means of the developed mathematical model was 275 GJ. Such a discrepancy is caused both by a greater physical correctness of the approach offered by the authors and the account of the above-mentioned factors.

3. Engineering Supplement (Techniko-Economical) Characteristics of a Pilot Plant 3.1. Modeling of Working Processes in Alternative Refrigerating and AirConditioning Systems (The Design Procedure) The main design scheme is given in figure 3.1, it shows the numbers of design points (parameters of flows). The simplified design of heat-and-mass transfer apparatuses is given – of the absorber and the desorber—with outstanding heat-exchangers HEX 11 and 12. The scheme is oriented to the AACS and includes a combined evaporative cooler IEC/R or IEC/DEC in the cooling part. In figure 3.2 this scheme is given with the use of cross-flow heat-and-mass transfer apparatuses (the absorber, desorber, cooling tower, indirect and direct evaporative coolers), which corresponds to the calculations carried out and executed developments of alternative air-conditioning systems. The cross-flow scheme corresponds to all other schematic developments of the AACS given in figure 3.3 – 3.4 and 3.23. The algorithm of alternative systems design is built in the following sequence: •

•

the calculation of the air dehumidification process in the absorber and further the process of cooling the air in the evaporative cooler IEC/DEC, the determination of heat-moisture parameters of the main air flow leaving the evaporative cooler (of the temperature and the relative humidity) and coming to the conditioned room; the determination of the AACS refrigerating capacity; the calculation of processes in the dehumidification part of the AACS (in the absorber and the desorber), the determination of the cooling water temperature and flow rate (the calculation of the process in the cooling tower) and the temperature and capacity of the heating source;

Solar Absorption Systems… •

93

the calculation of the heliosystem with the determination of the solar collectors type and the required surface of heat receivers.

1. We assume the type, concentration ξ (Section 2.2) and the temperature of the absorbent 8

t s (in the first approximation) at the inlet to the absorber. All geometrical characteristics of heat-and-mass transfer apparatuses are selected on the basis of the investigation carried out in the work (Section 2.3). The rate flow of the air entering the conditioned room is defined by the area and the height of this room, and the required multiplicity of air exchange in the room k (CN&R 2.04.05-91, “Heating, ventilation and air-conditioning”, State Construction Committee of the USSR, Moscow, 1997). Let us select the working value of air movement velocity in channels of crossflow heatand-mass transfer apparatuses (Section 2.3.3) which enables the calculation of the main overall dimensions of the packing bed of these apparatuses and their air resistanse magnitude. The design parameters of the outside air (temperature and moisture content) are defined for a chosen city or region allowing for the existing norms (construction norms and rules CN&R 2.04.05-91 “Heating, ventilation and air-conditioning”, State Construction Committee of the USSR, Moscow, 1997).

Figure 3.1. Schematic description of the alternative air-conditioning system AACS on the basis of the open absorption cycle with external (outstanding) heat exchangers HEX 11 (cooler) and HEX 12 (heater) as the design scheme. Nomenclature:

1 ¿ 13

are heat-and-mass transfer apparatus (in

figure 1.1); 1 – 22 – are flow parameters (numbers of design points).

94

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.2. Sketch of the AACS – the dehumidifying unit and the evaporative cooling unit (A), and the cooling tower (B) in the form of film cross-flow Heat-and-mass transfer apparatuses. Nomenclature is the same as in figure 3.1.

Solar Absorption Systems…

95

Figure 3.3. The design scheme of the alternative air-conditioning system AACS on the basis of the regenerative evaporative cooler IEC/R. Nomenclature is the same as in figure 3.1.

•

Solving the systems of equations (2.98) with boundary conditions (2.99) which describe heat-and-mass exchange in the absorber, we shall get concentration ξ and temperature t s of the absorbent at the outlet from the absorber. Aqueous solutions of LiBr, LiBr+ and LiBr++ were considered as working bodies. Thermophysical properties of the solutions were defined on the basis of the literature data and formulas given in Section 2.2. Then let us calculate the main parameters of the air 2

•

2

flow leaving the absorber – temperature t and moisture content x . Then let us calculate the evaporative cooling processes in the cooling part of the plant. In the main, the generative indirect evaporative cooler IEC/R is used here; it includes the indirect evaporative cooler IEC proper and the heat exchanger HEX 13 (figure 3.3). The air flow rates on the process line 2–3 and on the line 5–7 amount to G g and G g / 2 , respectively. The heat balance equation at

(

2

3

)

Gg ⋅ с ⋅ t − t =

Gg 2

(

7

5

)

(

7

⋅ c ⋅ t − t , at E HEX = t − t

5

) (t

2

5

)

−t .

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.4. The design scheme of the alternative system on the basis of the open absorption cycle and cooled water supply to heat exchangers-coolers. Nomenclature is the same as in figure 3.1

Let us define the air temperature values which are 5

3

2

t = 2.5 ⋅ t − 1.5 ⋅ t ,

(

7

3

2

(3.1)

)

t = 0.5 ⋅ t + t .

(3.2)

for EHEX = 0.8. All geometrical characteristics of IEC are chosen on the basis of the investigation carried out (Section 2.3). On solving the differential problem (2.95) – (2.96) by 3

the specified value of air parameters at the inlet to the IEC ( t and temperature of the auxiliary air flow at the outlet from the cooler: 5

() 3

t = IEC t ,

3

x ), we shall find the

(3.3)

Solar Absorption Systems…

97

when the temperature of the water recirculating through the IEC reaches the steady state (the main condition of stabilizing the process in the indirect evaporative cooler). Thus, the heat calculation of IEC/R is reduced to a repetitive procedure with the use of equation (3.1) and operator equation (3.3). We assume the initial value of the auxiliary air flow temperature (2.95) – (2.96) of defining the temperature of the auxiliary air flow

t

3, 0

and solve the problem

5

t under the condition of the

steady temperature of the water recirculating through the IEC t w . It also defined in the course of the repetitive procedure by the variation of t w at the inlet to the apparatus till the water temperature values at the inlet to and outlet from the apparatus become the same. As a result

t 3,1

5,1

( ),

= IEC t

5,1

2

3, 0

(3.4) 3, 0

3,1

from (3.1) follows t = 0.4 ⋅ t + 0.6 ⋅ t . Then, having chosen t = t as an initial value of the auxiliary air flow temperature, we continue the repetitive procedure till the 3, k

subsequent value of t differs little from the previous one (with the required accuracy of calculating this value). Finally, on solving the initial value problem Koshi (2.95) – (2.96) we define the temperature of the main air flow

4

t at the outlet from IEC. The moisture content of 2

the main air flow in this case does not change in cooling and is equal to x . Then, if it is necessary, we calculate the process in the direct evaporative cooler DEC [by solving the system of equations (2.86) – (2.87)], to which the main air flow is directed for additional cooling. 2. The calculation of processes in the dehumidifying part of the AACS ( in the absorber and the desorber). We perform the heat calculation of the desorber, for which we 11

define the absorbent temperature t s in the course of the repetitive procedure; this temperature provides the restoration of the initial concentration of the absorbent ξ at the specified parameters of the outside air. This repetitive process corresponds to the solution of the differential problem (2.98) – (2.99), describing the process of heatand-mass exchange in the desorber. Then, we define the temperature of the strong absorbent solution at the outlet from the desorber. •

•

The absorbent temperature values found make it possible to turn to the calculation of heat-exchangers HEX 10, HEX 11 and HEX 12. The calculation of heat carriers temperature are done by equation (2.24), Section 2.3.1.2. The area of the heat transferring surface is calculated by relations (2.25) – (2.88). On the basis of results obtained during the calculations the value of E HEX = 0.8 was chosen, which corresponds to rather a wide range of changing the temperature of working bodies for an acceptable surface of heat-exchange. Let us do the heat calculation of cooling tower CTW. Geometrical characteristics of the apparatus packing bed, density of the layer and the relationship between the flow rates of water and air are chosen on the basis of the investigation performed (Section

98

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov 2.3). By solving the system of equations (2.86) – (2.87), describing the processes of heat-and-mass exchange in the direct evaporative cooler, we obtain the water temperature at the outlet from the cooling tower

( )

16 ,1

15

t w = CTW t w

(3.5)

in the first approximation. From equation (2.24, HEX 11) we shall find a new value of the absorbent temperature t s at the inlet to the absorber. Then in the course of the repetitive procedure we define the final value of the absorbent temperature at the inlet to the absorber 8

t s with the required accuracy. 17

18

3. The heating source temperatures t w and t w (heat-exchanger HEX 12) were defined as a result of the heat calculation of AACS. At the known flow rate of the heat carrier (it was assumed to be equal to the absorbent flow rate) we found the heat load of the heliosystem Q and by formulas (2.128) and (2.129) calculated the area of SC (solar collector) and the volume of the tank-accumulator in the first approximation. The type of the SC is defined by the level of required temperatures and was chosen in accordance with the data from table 3.1. In calculating the COP of the heliosystem (the expression (2.127)) we used the values of the reduced optical COP θ and the reduced coefficient of heat losses U, also given in the table. For the flat SC with single glazing produced by RPF “New Technologies”, the data have been obtained as a result of full-scale tests [30, 118], for others the data have been taken from literature sources [17, 21, 90]. Depending on the required area of SC the type of the heliosystem was chosen by the mode of the heat carrier movement. For small heliosystems it is possible to use the scheme with natural circulation of the heat carrier. To define the time dependence of the temperature in the TA equations (2.124) – (2.126) have been solved. For heliosystems of higher capacity the scheme with forced circulation was used, and the problem was reduced to numerical solving differential equation (2.135). If the water temperature in the TA did not reach the required value, the correction of the total area (number) of SC was done, and the calculation was repeated. Table 3.1. Characteristics of the main types of SCs Type of solar collector (SC) Flat SC with single glazing without selective coating of the heat receiver surface The same with double glazing Flat SC with single glazing and selective coating of the heat receiver surface Evacuated glass tube SC

Working temperature, °C

θ

U, W/(m2×°C)

60-70

0.73

5.6

60-80

0.65

4.5

70-90

0.75

3.5

100-150

0.7

2.0

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3.2. Analysis of the Results Obtained On the basis of the developed procedure for calculating processes in alternative systems the calculations were carried out with respect to problems of air-conditioning and cooling the media. The main results of calculations for AACS are given in tables 3.2 – 3.3 in the form of tabulated data by the basic parameters for three types of absorbents LiBr+H2O, LiBr+LiNO3+H2O and LiBr+ZnCl2+CaBr2+H2O (further on LiBr, LiBr+ and LiBr++, respectively). The main part of calculations is done conformably to LiBr++. Calculations of AACS were done for a wide range of possible climatic conditions and operating parameters: •

1

1

x g = 7-25 g/kg of dry air (for the range of values x g = 7-13 g/kg of dry air calculations were carried out in conformity with a combined evaporative system CEC, Table 3.4; for the range of values

1

x g = 13-25 g/kg of dry air the calculation

results are given in table 3.6); 1

•

t g = 20-45 °C;

•

t w = 27-45 °C;

•

for each of three types of absorbents in changing % and

•

ξ LiBr = 45-50 %, ξ LiBr + = 45-60

ξ LiBr + + = 60-75 %;

the relationship of flow rates of gas and liquid for CTW is assumed, on the basis of the previous investigations, to be equal to unit [24, 81], for other apparatuses (IEC, ABR, DBR) the optimal value of this relationship has been defined by calculations and is given below.

We studied the influence of initial parameters of climatic condition: the moisture content and the temperature of outdoor air; the type and the concentration of the absorbent solution; the relationship of gas and liquid flow rates through the apparatus; temperatures of the heating source and the cooling medium; the required level of cooling on capabilities and working characteristics of AACS. The requirements of thermal-moist air parameters in a conditioned room are of decisive importance for air-conditioning system. Systems of air-conditioning should provide favorable microclimate in rooms. The main characteristics of a microclimate air are: the temperature in specific zones of rooms, humidity and hygienic state of inner air. Temperature and humidity, their maximum calculated values, fluctuations and changes during 24 hours and within a year are the most important factors influencing the people in the room. The velocity of air moving in the room is of some importance. The main task of AACS is to supply the conditioned room with the required amount of fresh air having the temperature and humidity which provide comfort heat and moisture parameters for residential and public premises. According to standards of the former USSR [Construction norms and rules CN&R 2.04.05-91 “Heating, ventilation and air-conditioning” the State Construction Committee of the USSR, Moscow, 1997] the zone of optimal heat and moisture parameters for a warm (summer) period of the year is limited by isotherms of 20 and 25 °C and by lines ϕ = 30% and 60 % (figure 3.7, the comfort parameters zone is marked with grey color).

Table 3.2. The calculation performances of AACS on the basis of the open absorption cycle (nomenclature – figures 3.1, 3.2)

Note: 1. The table does not show the cooling process 2–3 of air (mn) in HEX 13, this calculation is performed on the basis of efficiency value EHEX=0.8. The process 13–

8 of cooling the absorbent solution in the heat-exchanger HEX 11 is not shown. 2. Value basis of value EHEX=0.8 (HEX 12).

11

ts

defines the required value of the heating source temperature on the

Table 3.3. The calculation performances of AACS on the basis of the open absorption cycle (nomenclature – figures 3.1 and 3.2)

Table 3.3. Continued

Note: The dependence f(ξ) has been obtained with packaging density of the apparatus packing bed which is defined by the distance between sheets of the packing bed equal to b = 0.016 m and with the relationship of flows Gg/Gs=1.0; all other dependencies have been obtained for b = 0.01 m and the relationship of flows Gg/Gs = 2.0.

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103

The standard specifies an extended comfort zone with admissible parameters: isotherms of 18 and 28 °C, on the left side the curve ϕ = 30 %, on the right side the broken line with typical points (t = 27 °C, ϕ = 60 %; t = 26 °C, ϕ = 65 %; t = 25 °C, ϕ = 70 %; t = 24 °C, ϕ = 75 %) – figure 3.7. According to American Association of engineers on heating and ventilation ASHRAE 55 – 56 [ASHRAE 1989 Fundamentals Handbook (SI)] the comfort zone is limited for the summer period by the isotherms t = 10 and 24 °C and the lines ϕ = 30% and 70%, that is, the zone is even wider than it is shown above. In the basic calculations we oriented ourselves to the zone of admissible comfort air parameters by CN&R 2.04.05-91. Calculations of combined evaporative coolers CEC were done allowing for new parameters of pre-dehumidified air in the open absorption system. The efficiency of multistage evaporative coolers (multistage IEC) is equal for the three-stage scheme of IEC E db = 0.9-0.95 (is calculated with the orientation to the dew point temperature), but the flow rate of the main air flow with such a design of IEC. (figure 1.2 shows a two-stage scheme of IEC in the composition of AACS) is continuously (from stage to stage) decreasing which shows the growth of specific power consumption. For the combined scheme (IEC/DEC) the efficiency value, under comparable conditions, is lower and amounts to E db = 0.84 but the power consumption is much less, which completely corresponds to the authors’experimental data [22 – 27, 94] and makes it possible to consider such a scheme to be more preferable for being included in AACS. As it is seen from the analysis of processes in AACS for a wide variety of climatic conditions, it is quite enough to use IEC/R or a combined scheme IEC/DEC in the cooling part to enter the zone of comfort heat-moisture parameters, and only in rare cases, at high outdoor air temperatures one has to use the double-stage scheme IEC/IEC. Figures 3.5 – 3.6 show the characteristic calculation distribution of the basic gas and liquid flows parameters in the crossflow scheme of the interaction for the absorber, desorber, and indirect evaporative cooler for some typical calculation points. One can note a considerable separation of values for temperatures and moisture contents in the air flow leaving the desorber. The character of distributing the temperature of water recirculating through IEC at the outlet from the apparatus is of some interest, and the mean value of this temperature in the cycle is unchangeable and equal to the temperature of water at the inlet to IEC. Figures 3.7 – 3.10 illustrate on the H-X diagram for moist air the course of working processes in the alternative air-conditioning system for the ventilation scheme, when the whole air flow being cooled comes into the conditioned room (ventilation mode, VM). The diagrams show the main processes taking place in AACS: dehumidification of the air in the absorber, followed by the increase of its temperature: its cooling in IEC (shown is only the process of the main air flow cooling which proceeds with unchangeable moisture content); the process of changing the air flow state in the desorber (the absorbent regeneration process). 1

It is quite obvious, that even at the highest values of moisture content in air ( x g =25g/kg) the alternative system is capable to provide obtaining comfort parameters without using vapourcompression cooling. With the growth of moisture content the temperature of the heating source providing the absorbent regeneration increases too. The temperature increase of the water cooling the absorber results in the decrease of the air dehumidification process efficiency and the growth of the heating source temperature. In this case, of course, the energy expenditures on the organization of processes in cooling tower decrease too (the

104 growth of

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

t w corresponds for instance to the decrease of the air flow rate in the cooling

tower) and the possibility is evident for optimizing the process of evaporative cooling the water in the cooling tower which serves the absorber. It is also possible to use an auxiliary air flow leaving IEC for cooling the absorbent in the heat exchanger HEX 11. Of course, one should take into account the resistance, in the first place, aerodynamic drag, of all service lines in the scheme. The increase of the absorbent concentration intensifies the process of dehumidifying the air but causes the increase of the heating source temperature.

Figure 3.5. Characteristic distribution of flow parameters (calculation) with the cross-flow scheme of gas and liquid flow interaction ( ξ LiBr + + = 60%).

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105

Figure 3.6. Characteristic distribution of flow parameters (calculation) with the cross-flow scheme of gas and liquid flow interaction ( ξ LiBr + + =70%).

The processes taking place in IEC/R are shown in figure3.17 in more details, particularly, the figure shows the process in the auxiliary air flow. It also shows conventionally the state of water recirculating through the apparatus.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.7. The representation of working process in AACS on H,X-diagram for wet air. (influence of the moisture content of outside air). Nomenclature: K is the zone of comfort heat-humidity parameters; 1–2 is the dehumidification of the air in the absorber; 2–4 is the air cooling in IEC (assimilation process in the space being conditioned is not shown); 1–14 – changing the state of the air flow during the absorbent regeneration in the desorber.

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107

Figure 3.8. The representation of working processes in AACS on the H,X-diagram of wet air. A. The absorbent is LiBr + LiNO3 +H2O. The influence of the absorbent concentration on the working processes in the system. В. The absorbent is LiBr + ZnCl2 + CaBr2 + H2O. The influence of the outdoor air temperature 3.7.

t g = 25-40 °С, at x g = 15 g/kg and ξ s = 60%. Nomenclature is the same as in figure

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.9. Working processes in the alternative system. Influence of the moisture content of the outside sir. Nomenclature is the same as in figure 3.7.

Figure 3.10. Continued on next page.

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109

Figure 3.10. The representation of working processes in AACS on H,X-diagram of wet air. The absorbent is LiBr + ZnCl2 + CaBr2 + H2O. А is the influence of the cooling liquid temperature

tw ; В

is the influence of the absorbent concentration on the working processes in the system. Nomenclature is the same as in figure 3.7.

Figure 3.11. Dependence of COP on the temperature of ambient air.

110

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.12. Dependence of COP on the moisture content of ambient air. • data of the work [ 52 ] at 1

t g = 29.4 °C for the solution LiBr. The coefficient of performance (COP) for AACS was calculated by formula COP = Q IEC Q DBR , i.e. as the ratio of cooling capacity to heat obtained by the absorbent from the heliosystem. Figure 3.11 shows the dependence of COP on the surrounding air temperature at the constant moisture content. The character of this dependence is defined by a considerable reduction of Q DBR and comparatively week growth of

1

Q IEC with increasing t g . As it follows from figure 3.12, it is possible to note

monotonous decreasing COP with the increase of the moisture content of the surrounding air. Such a behavior agrees both with general physical considerations and the results of the work [52], in which they calculated COP of the idealized cycle as the ratio of changes of moisture content in the evaporative cooler and the absorber. We can make the following conclusions: •

1

1

for the whole discussed range of initial air parameters ( x g = 13-25 g/kg; t g = 20-45 °C) the pre-dehumidification of air makes it possible to reduce its moisture content to the value

1

x g < 13 g/kg, which, in its turn, gives the possibility to enter the zone of

comfort air parameters by means of evaporative cooling; •

when

1

x g < 13 g/kg of dry air, it is quite enough to use CEC in the composition of

IEC/R or IEC/DEC and there is no need for the dehumidifying part of AACS; when

Solar Absorption Systems…

111

1

x g > 13 g/kg it is impossible to provide entering the zone of comfort air parameters •

•

•

without the dehumidifying part of AACS only by means of evaporative cooling; the configuration of the cooling part of CEC in the composition of AACS, as it can be seen from the calculations carried out, can be the combination of IEC and regenerative heat exchanger HEX – 13, and only in case of a dipper dehumidification of air it is necessary to use CEC in the form IEC/DEC or the double-stage evaporative system IEC/DEC; the comparative estimation of working bodies-absorbents – LiBr and LiBr+ and LiBr++ showed their suitability for solving the problem of air-conditioning, and LiBr+ provides some reduction of the required temperature of the heating source at a lower extent of dehumidifying the air. Taking into account general considerations (see Section 2.2 of the given report) we may prefer LiBr++, which has, besides everything, lower corrosive activity; the analysis of parameters varying in calculations showed that the moisture content 1

1

of the air ( x g )was important rather than the outdoor air temperature value ( t g ); the

•

increase of the absorbent solution concentration greatly influences the results of the dehumidification process; the optimal value of the relationship between working flows of the gas and the liquid in the apparatus VR is equal to unit for the cooling tower, to 2.0 for the absorber and the desorber [for example, the change of G g G s from 1.0 to 2.0 results in a slight growth of the air moisture content at the outlet from the absorber (from 9.7 to 10.0 1

1

g/kg for initial parameters of the air flow being dried t g = 30 °C and x g = 13.5 g/kg) and the increase of the absorbent temperature at the inlet to the desorber from 50.5 °C to 53.9 °C which justifies, of course, the double-fold reduction of the absorbent flow-rate)]; for apparatuses of direct and indirect evaporative cooling – 10.0 which completely corresponds to experimental investigation conducted by the authors previously [22 – 27, 81, 94].

3.3. Description and Technico-Economic Characteristics of a Pilot Plant The general requirement to heat-and-mass transfer apparatuses (HMA and HEX) for alternative systems is their small overall dimensions [22-32, 97-100] due to high intensity of heat-and-mass exchange processes at low energy expenditure for the transportation of working substances (air and liquid flows), as HMA and heat exchangers HEX, used in systems, are numerous. The main universal development for all HMA of alternative systems has been realized through the film type apparatuses which provide a separate movement of gas and liquid flows in packing bed channels at low air resistance and the cross-flow scheme of contacting gas and liquid flows as the most acceptable one if it is necessary to arrange numerous HMA and HEX in a single block. The cross-flow scheme provides minimum turnings of flows and lower air resistance as compared to the counter-flow scheme, because it is characterized by higher values of extreme velocities of gas flow movement in packing bed channels HMA [24, 81].

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The packing bed of cross-flow HMA is universal for all apparatuses use the developed multi-channel packing bed of a regular structure, it is formed by longitudinally corrugated thin sheets equally distant from each other. Regular roughness of the surface is used as the main method of intensifying processes of joint heat-and-mass exchange in HMA. Optimal values of roughness parameters are k opt = p e = 8…14 , where p and e are the pitch and the height of roughness riffles. Roughness riffles are uniformly dispersed on the surface of the packing bed longitudinally corrugated sheets. Optimal values of the main corrugation parameters (P and E are the pitch and the height of the main corrugation of the packing bed sheets), of the packing bed layer density (the distance between sheets in a stack, the magnitude of the equivalent diameter of the apparatus packing bed channels) and the main overall dimensions of the packing bed have also been defined above (Section 2.3.2 and 2.3.3). Within the farmers of theoretical and experimental studies of the problem of two-phase film flows stability the extreme values of the gas flow velocities in HMA were defined. In developing heat-and-mass transfer apparatuses comprising alternative systems, on the basis of the researches carried out it was assumed: the value of the equivalent diameter of packing bed channels 25 mm, the pitch and the height of the main corrugation of the packing bed sheets P = 14 mm, E = 6 mm the magnitude of the parameter of the sheet surface regular roughness k = 12.5; the work range of the air flow velocity in the channels of the packing bed 5-7 m/s. the design of the packing beds for all HMA comprising the alternative systems under developing the unified (desorber DBR, absorber ABR, cooling tower CTW, direct evaporative cooler DEC).

Figure 3.13. Schematic description of the combined evaporative cooler CEC on the basis of IEC/R. Nomenclature: 1 – indirect evaporative cooler IEC; 3 – heat exchanger HEX; 4 – entering (outdoor) air; 5 – air flow discharged into the atmosphere; 6 – water tank; 8 – liquid distributors; 9 – fan; 10 – water pump; 11 – recirculation water loop in IEC; 14 – replenishment of the system with water.

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113

Figure 3.14. Schematic description of the combined evaporative cooler CEC on the basis of IEC|DEC. Nomenclature is the same as in figure 3.13 (2 – direct evaporative cooler DEC).

Figure 3.15. Continued on next page.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

Figure 3.15. Schematic description of the combined evaporative cooler CEC on the basis of IEC/CTW. Notation: 1 – indirect evaporative cooler IEC; 2 –cooling tower CTW; 3 – heat exchanger HEX; 4 – entering (outdoor) air; 5 – air flow discharged into the atmosphere; 6 – water tank; 7 – combined evaporative cooler СЕС; 8 – liquid distributors; 9 – fan; 10 – water pumps; 11 – recirculation water loop in IEC; 12, 13 – chilled water; 14 – replenishment of the system with water.

Figures 3.13 –3.15 shows the schematic description of combined evaporative coolers CEC: on the basis of IEC/R (figure 3.13); on the basis of joint arranging evaporative coolers of direct and indirect types IEC/DEC (figure 3.14), evaporative cooler and the cooling tower IEC/CTW (figure 3.15). All these coolers are based on the use of only the evaporation principle and are intended for the operation in dry and hot climate. The main component of these approaches is the device of indirect evaporative cooling IEC, the packing bed of which is the alternation of dry and wetted channels designed for the movement of the main and auxiliary air flows respectively. Longitudinally corrugated parts of the packing bed form closed components-wafers in the inner space of which moves the main air flow, cooled contactlessly at the unchangeable moisture content. The auxiliary air flow moves in the space between closed components (wafers) arranged vertically and uniformly. This flow is perpendicular to the main one and counter-flow with respect to the water film streaming down the outer surfaces of closed components. In this case evaporative cooling of water removes heat from the main air flow. The apparatus has a vessel for water 6, recirculating through its packing bed with the help of a pump 10. The water distributor 8 is a system of perforated tubes uniformly dispersed over closed components-wafers. Natural losses of water during its evaporation are compensated by automatic replenishment of the system with fresh water – 14. The division of air flows into the main and auxiliary flows from the one which enters the apparatus, occurs automatically and is defined by air resistances of the packing bed and air service lines. It can be regulated

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115

by a shutter of the main flow air-duct. The movement of air flows is provided by a single fan located at the entrance of air into the CEC.

Figure 3.16. General view of the alternative air-conditioning system (AACS) with the combined evaporative air cooler СЕС (1), located outside the building, and the system of water-air heat exchangers (2), located directly inside the rooms being served. All the nomenclatures shown in the CEC unit, are the same as in figure 3.15.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The auxiliary air flow first enters the space of the apparatus under the packing bed and then into the packing bed of the apparatus. In the CEC version, shown in figure 3.13, a regenerative variant of the cooler is represented. It comprises IEC and a heat exchanger, where the air to be cooled is precooled by the cold and wetted auxiliary air flow which is discharged into the surroundings. The cooler based on the joint arrangement of evaporative direct and indirect coolers IEC/DEC (figure 3.14) includes the DEC apparatus providing additional cooling of the main air flow which leaves the IEC. The cooling process takes place at the direct interaction of the air and the water film flowing down the surfaces of the packing bed sheets is directed on the isoenthalpy line. The water recirculating through the DEC apparatus takes the temperature equal to the wet bulb temperature of the air coming into the a DEC. The air is cooled and rewetted which provides an additional possibility to regulate it heat-moisture parameters. The design of the DEC apparatus is simpler than that of the indirect evaporative cooler. The packing bed is a set of longitudinally corrugated sheets equally distant from each other on both sides of which the water film streams. The cooler based on the joint arrangement of the indirect evaporative cooler and the cooling tower IEC/CTW (figure 3.15) includes a cooling tower 2, which provides evaporative cooling of water by the main air flow leaving the IEC. As the moisture content of the air is not changed while the air is being cooled in the IEC, its dew point temperature is reduced which provides deeper cooling of water in the cooling tower. Cold water (13) comes into the system of water-air heat exchanger. The cooling tower design is similar to the DEC apparatus. A separator for drop-like moisture made in the form of louvers is provided at the outlet from the cooling tower. The system includes two heat exchangers where the precooling of the air entering the CEC is provided by air flows leaving IEC and CTW. Figure 3.16 shows the general view of the air-conditioning system based on the IEC/CTW system. The CEC unit is located outside the system of water-air heat exchangers placed directly in rooms being served. The general analysis of principal capabilities of CEC coolers is given in Table 3.4 and on the wet air diagram, figure 3.17. Calculated air parameters of the table are given for different cities of the world for climatic conditions when x g < 13 g/kg (zone A). The diagram shows the working processes, occurring in the cooler, on the example of two most specific points with parameters t g = 40 °С, и t g = 45 °С,

x g = 8 g/kg

x g = 10 g/kg , and it can be seen that an additional stage of cooling DEC is

necessary only at high initial air temperatures ( t g > 40 °C). The diagram shows the processes of changing the main air flow state (3–4) and those of the auxiliary flow (3–5) in the indirect evaporative cooler IEC (the state of water recirculating through the apparatus and having constant temperature – figures 3.5-3.6 is shown by the arbitrary point 21 in the diagram; the dashed line indicates the direction of the process line in the “wet” part of cooler). Lines 2–3 and 5–7 are the processes taking place in the heat exchanger. It is evident, that at x g > 13 g/kg using only evaporative methods it is impossible to provide necessary comfort parameters of the air.

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117

Figure 3.17. Representation of working processes in the combined evaporative cooler CEC on the basis IEC/R (the scheme is the same as in figure 3.13) on the H,X-diagram of wet air for different cities of the world (for climate conditions

1

xg

< 13 g/kg – zone А). Notations of cities are the same as in table

3.4. Progressing of the processes in IEC/R is shown on the example of two characteristic points with calculation parameters t g = 40 °С,

x g = 8 g/kg and t g = 45 °С, x g = 10 g/kg, where: 2–3 and 5–7 are

processes occurring in the heat exchanger HEX 13; 3–4 and 3–5 are the processes of changing the state of the main and auxiliary air flows in IEC; 21 –the state (temperature) of the water recirculating through IEC (is shown conventionally); 4–6 is the additional air cooling in DEC for the point with the calculation parameters if air t g = 45 °С,

x g = 10 g/kg.

1

Table 3.4. Design air parameters for an AACS of the evaporative type (IEC/R) ( x g < 13 г/кг)

Table 3.4. Continued

Notes: design air parameters (summer air-conditioning) are adopted according to Construction Norms and Rules CN&R, 2.04.05-91 “Heating, ventilation and Airconditioning”, State Construction Committee of the USSR, Moscow, 1997, ASHRAE 1989 Fundamentals Handbook (SI); for cities with a design air temperature

over 40 °C a combined evaporative system IEC/DEC is used, and there is a column for the temperature

6

tg

(calculation scheme in figure 3.1) in the table.

Table 3.5. Technical-and-economical characteristics of a pilot plant of AACS (alternative air-conditioning system; figure 3.18)

Table 3.5. Continued

Note: The direct evaporative cooler DEC can be not available in the scheme AACS, which is defined by initial calculation parameters of the air flow; Geometrical characteristics of the “wet” and “dry” parts of the indirect evaporative cooler IEC are identical; The absorbent concentration (the operating range) and the temperature level of regeneration depend on the type of the absorbent and the required degree of dehumidifying the air in the absorber.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

The schematic description of alternative air conditioning systems AACS on the basis of the open absorption cycle and solar regeneration of the absorbent is given in figures 3.18 – 3.20. Figure 3.18 shows the simplified version of AACS for the absorber and desorber (figure 1.8A) with outstanding heat exchangers of cooling and heating the absorbent. The above mentioned main versions of evaporative coolers are included in alternative systems as an evaporative part. In this case it is a version of the cooler on the basis of IEC/R. The design of the absorber and desorber is similar to that of the cooling tower described above. The apparatuses are supplied with a separator of drop-like moisture in the form of louvers. The layout diagram shows the heat exchanger of preheating the air entering the desorber and the heat exchanger of flows of the hot strong and cold weak solutions of the absorbent. The diagram shows two fans providing the movement of all air flows.

Figure 3.18. Schematic description (layout diagram) of the alternative air-conditioning system AACS on the basis of the open absorption cycle. Nomenclature: 1 – indirect evaporative cooler IEC; 2 – cooling tower CTW; 3 – heat exchangers HEX; 4 – entering (outdoor) air; 5 – air flow discharged into the atmosphere; 6 – tank for the liquid; 7 – AACS; 8 – liquid distributors; 9 – fan; 10 – liquid pumps; 11 – recirculation water loop in IEC; 12, 13 – chilled water; 14 – replenishment the system with water; 15 – absorber; 16 – desorber; 17 – air flow to the room being conditioned; 18 – heat carrier (water) from the heliosystem. In addition to the above: absorbent; water (loop of cooling); water (loop of heating).

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Figure 3.19. Schematic description (layout diagram) of the alternative air conditioning system AACS on the basis of open absorption cycle. Heat exchangers of the heating loop (desorber) and the cooling one (absorber) are joint with the desorber and the absorber respectively. Nomenclature is the same as in figure 3.18.

In figure 3.19 the layout diagram of the AACS is shown in the variant of combining heat exchangers of heating and cooling with the desorber and the absorber, respectively (figure 1.8B). The packing bed of these apparatuses is a tube-plate structure, and longitudinally corrugated sheets are used as plates like in all other apparatuses. The author of the project developed such apparatuses before as evaporative condensers of refrigerating plants, and they proved to be good [24, 81]. Such a development makes it possible to avoid additional heat exchangers and to reduce the total air resistance to the movement of heat carriers. All the above-mentioned AACS are represented in the version of ventilation schemes (ventilation mode, VM), when all the cooled air flow comes into the room. The recirculation apparatus (recirculation mode, RM – figure 3.23) is more economic, when the amount of the fresh air entering the system can be about 20% [20, 24]. Various versions of similar schematic approaches have been discussed in Section 1.1 of the present report. As a rule, the air leaving the room is cooled in the direct evaporative cooler DEC and then can be used, for example, for precooling the air flow which has been dehumidified in the absorber and enters the cooler, as is shown in figure 3.33, or is directed for the recirculation in the room.

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Figure 3.20. Schematic description (layout diagram) of the alternative refrigeration system ASRF on the basis of the heliosystem as a heating source (solar regeneration of the absorbent) and the supply of chilled water to heat-exchangers-coolers (room fan-coils). Nomenclature is the same as in figure 3.18.

In figure 3.20 the layout diagram of the ARFS is shown in the variant of the evaporative cooler on the basis of IEC/CTW and the cold water supply to heat exchangers-coolers. This development makes it possible to avoid the system of branched cost of construction work and the plant on the whole. Table 3.5 gives the design technico-economical characteristics of a pilot plant AACS (the layout diagram is the same as in figure 3.18). All the calculating have been carried out for the flow rate of the air being dehumidified G g = 10000 kg/h when the correlation

VR ABR (VR DBR ) is equal to unit, in this case overall dimensions of these apparatuses are

the same (H×B×L = 650×650×1000 mm) and the velocity of the air flow in packing bed channels is w g = 5.8-6.8 m/s, which is quite acceptable from the view point of the flow stability. The required capacity of the pilot plant has been defined owing to the necessity of providing comfort air parameters in rooms, like offices, of the area 200 m2 situated in the city of Odessa, for the summer period of air-conditioning. In this case •

design air parameters in the city of Odessa in the summer period of air-conditioning amount to t = 28.6 °C; h = 62 kJ/kg; ϕ = 52% (“B” parameters, Construction Norms

Solar Absorption Systems…

•

•

•

125

and Rules CN&R, 2.04.05-91 “Heating, ventilation and air-conditioning”, State Construction Committee of the USSR, Moscow, 1997). optimal characteristics of the micro climate (comfort air parameters) for an office amount to t = 23-25 °C; ϕ = 40-60% (CN&R, 2.04.05-91). Let us choose from this range of parameters the values of t = 25 °C, ϕ = 50%. the required temperature of the air fed to room through the cooler AACS at the working temperature difference of 4 °C (the difference between the air temperature in the room and that of the air coming from the air cooler) will be t = 21 °C; h = 46 kJ/kg; x = 10.5 g/kg. These parameters, as it can be seen from results obtained by us, are quite possible to be provided by AACS. In this case, the air leaving the absorber will have the parameters t = 29 °C; x = 10.5 g/kg. The required content x = 10.5 g/kg is achieved in the single-stage IEC/R. the capacity of the forced-exhaust ventilation at the required multiplicity of the air exchange in the room k = 4-7 (CN&R, 2.04.05-91) will amount to G = k⋅F⋅Н = 5.5⋅200⋅4.5 = 4950 m3/h

(it was assumed in the design that k = 5.5, F = 200 m2 – the aria of an office space, H – the height of the space), •

the refrigerating capacity of the pilot plant is

Q = G ⋅ c ⋅ Δ ⋅ h = 28.6 kW in

ventilation mode (VM; the flow rate of air is taken equal to 5000 m3/h); in the recirculation mode (RM), when the amount of fresh air fed to the room is only 20%, Q = 14.3 kW (figure 3.23, the design scheme and the general view of the recirculation system AACS). Figures 3.21A and 3.21B shows the space layout diagrams of AACS in the form of separate units of cooling, and figures 3.22 and 3.24 shows the general view of alternative airconditioning systems including the very unit of cooling 1, heliosystem 3 located on the roof of the building and including a tank-heat accumulator 4 with a duplicate source of heating 5, a system of air ducts of force ventilation 6 and exhaust one 7, and for version of AACS with a cooling tower – a system of water-air heat exchangers being ventilated 2 which are located in rooms of the building. The general analysis of the principle capabilities of alternative air-conditioning systems is given in Table 3.6 and diagram H-X of the wet air, figure 3.25. The design parameters of the air in the table are given for different cities of the world for climatic conditions when xg > 13 g/kg (zone B). The absorbent LiBr++ is used at the concentration ξ = 70%. The course of processes in AACS is shown on the example of typical points with calculation parameters t g = 21.1 °C, x g = 11.99 g/kg (Bogota); t g = 45.6 °C, x g = 22.7 g/kg (Port-Said); t g = 32.2 °C,

x g = 24.96 g/kg (Duala); t g = 35 °C, x g = 18.47 g/kg (Tel Aviv). The diagram shows

only the processes of dehumidifying the air in the absorber and the following evaporative cooling of the air in IEC, it is seen in this case that the need for an additional stage of cooling (DEC and IEC) appears only at high initial temperatures of air ( t g > 40 °C) – Port-Said. The

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alternative solar system of air-conditioning is capable to provide the obtaining of comfort air parameters for any climatic conditions of the world and in the form of an evaporative cooler it is optimal to use a single-stage regenerative indirect evaporative cooler IEC/R. As compared to traditional vapour – compression systems of air – conditioning the alternative system AACS provides considerable redaction of energy consumption (30-60%), which is confirmed by not numerous data of using similar plants [24, 101, 102].

Figure 3.21. Layout diagrams of alternative air-conditioning systems AACS (A) and those of cooling ASRF (B) on the basis of the open absorption cycle. Nomenclature: IEC – indirect evaporative cooler; CTW – cooling tower; ABR – absorber; DBR – desorber; 12-13 – chilled water; 17 – air flow to the room being conditioned; 18 – heat carrier (water) from the heliosystem.

1

Table 3.6. Design air parameters for a pilot plant of AACS to be used in different cities of the world ( x g > 13 г/кг)

Table 3.6. Continued

Notes: design air parameters (summer air-conditioning) are adopted according to Construction Norms and Rules CN&R, 2.04.05-91 “Heating, ventilation and Airconditioning”, State Construction Committee of the USSR, Moscow, 1997, ASHRAE 1989 Fundamentals Handbook (SI)

Table 3.7. Calculation results of the main parameters for the alternative refrigerating system

Notes: a two-stage evaporative cooler with further cooling of water in the cooling tower is used in the cooling part of the plant.

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Figure 3.22. General view of AACS located outside the building with the system of air ducts in the room. Nomenclature: 1 – AACS; 3 – heliosystem; 4 – tank-heat accumulator; 5 – duplicating heater; 6 – air ducts for supplying air to rooms; 7 – exhaust ventilation the rest nomenclature is the same as in figures 3.18 and 3.21.

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Figure 3.23. Design scheme (А) of the alternative recirculation system (recirculation mode, RM) of air conditioning. Nomenclature: 1–24 – flow parameters (numbers of design points). General view of AACS (В). The nomenclature is the same as in figures 3.18 and 3.21.

The general analysis of the principle capabilities of alternative cooling systems (the layout diagram of the alternative cooling system AACS on the basis of the open absorbtion cycle with a heliosystem used as a heating source and chilled water supply to heat exchangers-coolers (room fan coils) is given in figure 3.20; the calculating scheme is given in figure 3.4) is given in Table 3.7 and diagram H-X of the wet air, figure 3.26. The design parameters of the air in the table are given for particular points with design parameters – t g = 25 °C, x g = 10 g/kg and t g = 28 °C, x g = 20 g/kg (the processes of air dehumidification in the absorber 1–2, of the 2-stage evaporative cooling in IEC – 2–4; and of water cooling in the cooling tower are shown; the temperature of the water cooled is given in the form of a point in the line ϕ = 100%). The temperature of the cooled water is 8.5-11 °C, in this case the required temperature of the heating source is in the range of t 17 = 95-122 °C. The use of two-stage cooler IEC/DEC greatly increases energy expenditures for the organization of the process, as from the second evaporative stage only ¼ of air which had been dehumidified is carried out. The possibility of operation of such a cooling system is limited by the range of comparatively low values of moisture content of the outdoor air x g < 10-15 g/kg.

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Figure 3.24. General view of AACS located outside the building with the system of water-air heatexchangers in the rooms. Nomenclature: 1 – AACS; 2 – heat exchangers; 3 – heliosystem; 4 – tank-heat accumulator; 5 – duplicating heater; the rest nomenclature is the same as in figures 3.18 and 3.21.

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Figure 3.25. Representation of working processes in AACS (the scheme on figure 3.18) on the H,Xdiagram of wet air for different cities of the world (for climate conditions of

1

x g > 13 g/kg – zone B).

Notations of cities are the same as in Table 3.6. Progressing of the processes in AACS is shown on the example of characteristic points with calculation parameters:

t g = 21 °С, x g = 11.99 g/kg (Bogota),

t g = 45.6 °С, x g = 22.7 g/kg (Port-Said), t g = 32.2 °С, x g = 24.96 g/kg (Douala); t g = 35 °С, x g = 18.47 g/kg (Tel Aviv). Shown are only processes of dehumidifying the air in the absorber and those of evaporative cooling in IEC.

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Figure 3.26. The picture of working processes in the alternative refrigerating system (the scheme is the same as in figure 3.20) on H, X diagram of wet air. The course of processes in the system is shown on the example of typical points with calculation parameters

xg=

t g = 25 °C, x g = 10 g/kg and t g = 28 °C,

12 g/kg (shown are the processes of dehumidifying the air in the absorber, double-stage

evaporative cooling in the IEC and cooling the water in the cooling tower; the temperature of water cooling is shown conventionally in the form of a point on the line ϕ = 100%).

Basic Conclusion The report presented includes the thermodynamic analysis of developed schematic approaches to alternative refrigerating and air-conditioning systems, the selection of working

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135

substances (absorbents), the developed mathematical models (software) of operating processes in all the main components of alternative systems (refrigerating and airconditioning systems), providing the interrelated calculation of such systems and the following design. A pilot plant AACS is developed on the basis of the research carried out and obtained results. ‰ It is preferable to use liquid sorbents (absorbents) in alternative solar systems of airconditioning AACS and of cooling ARFS. The analysis of characteristics of working bodies used as absorbents for solar alternative systems being developed showed low suitability of CaCl2+H2O because of high partial pressures of water vapours at absorption temperatures, the prospectiveness of solutions LiBr+H2O (LiBr), LiBr+LiNO2+H2O (LiBr+) and LiBr+ZnCl2+CaBr2+H2O (LiBr++), in this case LiBr+ has better absorption capacity, and LiBr++ has less corrosive activity and allows the decrease of the required temperature of the outside heating source during the desorption, which is important in principle while using low-grade sources of heat; The calculation analysis has been done for working characteristics in a wide range of initial parameters (temperature and moisture content of outdoor air, absorbent type and concentration, cooling water temperature, relation of gas and liquid flow rates in apparatuses of alternative systems) and of construction features of heat- and- mass transfer apparatuses which made it possible to reveal qualitative (predictable) characteristics of the system and to obtain quantitative information necessary for their further engineering calculation and designing. ‰ The general analysis of principal capabilities of solar air-conditioning systems AACS showed:

− when x g < 13 g/kg (zone A – dry and hot climate), providing comfort air parameters is quite possible when only evaporative methods of cooling are used, without predehumidifying the air flow, at x g >13g/k evaporative cooling alone cannot provide required comfort air parameters. The need for an additional stage of cooling DEC, appears only, at high initial air temperatures ( t g >40 °C).

− when x g > 13 g/kg (zone B) the alternative solar air-conditioning system is quite capable of providing the comfort air parameters for any climatic conditions of the world. And the use of a single-stage regenerative indirect-evaporative cooler IEC/R as an evaporative cooler is optimal, the need for an additional stage of cooling DEC or a double-stage system IEC/IEC appears only at high initial air temperatures ( t g > 40 °C);

− the required temperature of the heating source for providing solar regeneration of the absorbent amounts to a range of values t 17 = 55-125 °C. The possibility of using the simplest and cheapest flat solar collectors is limited for these systems by the range of

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov values of low moisture contents and temperatures of outdoor air. To solve the problem of solar air conditioning in a wide range of climatic conditions it is necessary to have solar collections with selective coating or evacuated glass pipe collectors. The general analysis of principal capabilities of solar cooling systems ARFS showed that it is possible to achieve the cooling water temperature of 8.5–12 °C, and the required temperature of the heating source is in the range of t 17 =95-125 °C. The possibility of functioning such a cooling system is limited by the range of rather low values of moisture content in outdoor air x g <10-15 g/kg. The use of the double-stage cooler IE/IE as a component of ARFS greatly increases total energy expenditures for the organization of the process, since only ¼ of air dehumidified in the absorber goes out of the second evaporative stage. Further development of solar refrigerating systems ARFS requires the use of highly efficient evacuated collectors and the improvement of the cooling part of the plant. 1

‰ The pilot plant. For the calculation condition of Odessa (summer period – July, t g = 30 °C,

1

x g = 13.3 g/kg, flow rate of the air being dehumidified G = 10000 kg/h, flow rate of

the air fed to the conditioned room is 5000 kg/h ventilation scheme of air-conditioning – ventilation mode, VM), to provide a continuous cycle (absorbent regeneration ) the solar collectors area of 150 m2 (LiBr++) is necessary, in this case the calculation temperature of regeneration is 65–70°C. As compared to traditional vapour – compression systems of air – conditioning the alternative system AACS provides considerable redaction of energy consumption (30–60%), which is confirmed by not numerous data of using similar plants [24, 101, 102].

Nomenclature t, T p h x cp ρ r D g ϕ σ λ ν μ ξ

– – – – – – – – – – – – – – –

temperature, °C, K; pressure (air), Pa; entalphy, J|kg; water content of air (kg water / kg dry air); specific heat, J|kg K; density, kg|m3; latent heat of phase transition (evaporation), J|kg; diffusion coefficient, m2|s; gravitational acceleration, m|s; relative humidity of air (%); surface tension, N|m; thermal conductivity, W|m K; kinematics viscosity, m2 |s; dynamic viscosity, kg|m s; absorbent concentration in solution (kg absorbent / kg solution);

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J τ δ δo A f (F) H, L, B d E, P

– – – – – – – – –

solar radiation intensity, kJ|m2 s; time, s. liquid film thickness, m ; maximum thickness of liquid film in cavities of corrugated surfaces, m; transfer aria, m2; surface cross-section area; surface aria, m2; height, length, width, m; diameter, m; the height and the pitch of the main corrugation of the packing bed sheets, m;

k opt

–

optimal values of roughness parameters;

e, p n

– –

W G Q w VR

– – – – –

U E COP θ α k β Nu Sh Pr Re We Le Fr Eu

– – – – – – – – – – – – – – –

the height and the pitch of roughness riffles, m; number of sheets in packing bed of heat-mass-transfer apparatus; number of tubes in SC set; distance between tubes in SC, m. mass flowrate, kg|s; amount of heat, kW; velocity, m|s; air to water flow ratio (DEC, CTW), air to solution flow ratio (ABR, DBR), main air flow to auxiliary air flow ratio (IEC); coefficient of heat losses, W|m2 K; efficiency; coefficient of performance; optical coefficient of performance; heat transfer coefficient, W|m2 K ; overall heat transfer coefficient, W|m2 K; mass transfer coefficient (air side or solution side), kg|m2 s; Nusselt number; Sherwood number Prandtl number; Reynolds number; Weber number; Lewis number; Frud number; Euler number.

Subscripts g, L w s wk, str mn, ax wb, db

– – – – – –

gas, liquid; water; solution; weak and strong solution; main and auxiliary air flow; wet bulb and dew point temperature of air;

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov aA, exA f H, M e * in, out id p wl 1, 2 i, k min max #

– – – – – – – – – – – – – –

ambient and exhaust air; film; heat-and-mass transfer; equivalent diameter of packing elements, m; interface, critical value; inlet, outlet parameters; ideal; fin of absorber plate SC; wall; lifting and lowering pipelines of a heliosystem; nodal points on x and z coordinates; minimum value; maximum value; difference.

Abbreviations AACS ARFS CEC DEC IEC R CTW ABR DBR HMA HEX FNC SC TA RR RP GG VM RM

– – – – – – – – – – – – – – – – – – –

alternative air conditioning systems; alternative refrigeration systems; combined evaporative cooler; direct evaporative cooler; indirect evaporative cooler; regenerative scheme; cooling tower; absorber; desorber; heat-mass-transfer apparatus; heat exchanger; fan-coil; solar collector; tank-accumulator; regular roughness; regular packing bed; greenhouse gases; ventilation mode; recirculation mode.

Other notation are given in the text of the report in places of their use.

References [1]

Antonopoulos, K.A., Rogdakis, E.D., 1996, Perfomans of solar-driven ammonialithium nitrate and ammonia-sodium thiocyanate absobtion systems operating as coolers or heat pumps in Athens, Appl. Therm. Eng., vol. 16, n. 2, p. 127-147.

Solar Absorption Systems… [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12]

[13] [14]

[15] [16]

[17]

[18] [19] [20]

[21]

139

ARI Air-Conditioning and Refrigeration Institute, Catalog of Publications. Beckman, Y.A., Klane, S.A., Daffi, J.A., 1982, The design of solar heat supply systems, Moscow, Energoizdat, pp. 72. Best, R., Pilatowsky, I., 1998, Solar assisted cooling with sorption systems: status of the research in Mexico and Latin America, Int. J. Refrig., vol. 21, no. 2: pp. 100 -115. Biel, S., 1998, Sorptive luftentfeuchtung und verdunstungskühlung, KI Luft- und Kältetechnick, no. 7., pp. 332-336. Case, V.M., London, A.L., 1962, Compact heat exchangers, Moscow-Leningrad, Gosenergoisdat, p. 160. Catalogue for plate heat exchangers, Edit. P.I.Bazhan, Moscow, Mashinostroenie, p. 59. Chant, E.E., Jeter, S.M., 1994, A steady-state simulation of an desiccant-enhanced cooling and dehumidification system, ASHRAE Trans., US, vol. 100, n. 2, p. 339-347. Chau, C.K., Worek, W.M., 1995, Interactive simulation tools for open-cycled desiccant cooling systems, ASHRAE Trans., US, vol. 101, n. 1, pp. 725-734. Kholpanov, L., Doroshenko, A., 1986, Peculiarities of liquid film flowing along the elements of the packing bed with regular roughness of the surface, Izvestiya vuzov, series – Khimiya i khimicheskaya Tekhnologiya (Chemistry and Chemical Technology), Vol. 29, issue 10, pp. 117-120. Claassen, N.S., 1995, Developments in multi-stage evaporative cooling, Afr. Refrig. Air Cond., ZA, vol. 11, n. 2, p. 55-67. Collares Pereira, M., 1995, CPC-type collectors and their potential for solar energy cooling applications, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 5. Colvin T.D., 1995, Office tower reduce operating costs with two stage eveporative cooling system, ASHRAE, vol. 37, n. 3, p. 23-24. Commission of the European DGXVII, 1994, The European renewable energy study, Office for Official Publications of the European Comminities, Luxembourg, vol. 1, p. 38. Costa, A. Et al., 1996, Potential of solar assisted cooling in Southern Europe, Proceedings of EuroSun 96, Freiburg, pp. 1201-1206. Czederna, A., Tillman, N.N., Herd, G.C., 1995, Polimers as advanced materials for desiccant applications. 3. Alkalisalts of PSSA and poliAMPSASS, ASHRAE Trans., US, vol. 101, n. 1, p. 697-712. Daffi, J.A., Beckman W.A., 1974, Solar Energy Thermal Processes, John Wiley & Sons, N.Y. – London – Sydney – Toronto; Daffi, J.A., Beckman, Y.A, 1977, Heat processes with the use of solar energy, Moscow, Mir, pp. 566. Das Potential der sonnenergie in der EU, Eurosolar, Bonn, 1994. Deng, S.M., Ma, W.B., 1999, Experimental studies on the characteristics of an absorber using LiBr/H2O solution working fluid, Int. J. Refrig., vol. 22, P. 293-301. Dhar, P.L., Kaushik, S.V., 1995, Analysis of a fild-installed hybryi solar desiccant cooling system, desiccantaugmented evaporativecooling cycles for Indian conditions, ASHRAE Trans., vol. 101, n. 1, p. 735-749. Donets, Ya.I, 1991, Mathematical modelling of flat solar collectors with liquid heat carriers. Heat pipes and heat pumps, Minsk, p. 131-137.

140

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

[22] Doroshenko, A., 1988, Indirect evaporative cooler for air-conditioning, Cholodilnaya Technika, no. 10: p. 28-33 (Russian). [23] Doroshenko, A., 1996, Alternative Air-Conditioning, International Conference of Research, Design and Conditioning Equipment in Eastern European Contries, September 10-13, Bucharest, Romania, IIF/IIR: p. 102-108. [24] Doroshenko, A., Compact heat-and-mass transfer equipment for refrigerating engineering (theory, calculation, engineering practice), Doctor dissertation, Odessa State Academy of Refrigeration. [25] Doroshenko, A., et al., 1985, Indirect-evaporative cooling as a perspective trend in airconditioning, Izvestiya Akademii Nauk Turkmenskoi SSR, series – Physico-technical, chemical and geological sciences, No 3, pp. 30-35. [26] Doroshenko, A., 1996, New Developments of Air-Conditioning, International Conference of Applications for Natural Refrigerants’ 96, September 3-6, Aarhus, Denmark, IIF/IIR: p. 339-345. [27] Doroshenko, A., Karev, V., Kirillov, V., Kontsov, M., 1998, Heat and Mass Transfer in Regenerative Indirect Evaporative Colling, Intern. Conference IIR/IIF of Advanes in the Refrigeration Systems, Food Technologies and Cold Chain- Sofia'98, September 23-26, Sofia, Bulgaria. [28] Doroshenko, A., Kirillov, V., 1988, Special features of liquid film flowing along surfaces roughness, Eng.-Phys.-Journal, vol. 54, no. 5: p. 739-745 (Russian). [29] Doroshenko, A., Kirillov, V., Kontsov, M., 1998, Alternative Refrigerating Systems on the basis of Open Absorption Cycle Using Solar Energy as a Heat Source, Intern. Conference IIR/IIF of Advanes in the Refrigeration Systems, Food Technologies and Cold Chain- Sofia'98, September 23-26, Sofia, Bulgaria. [30] Doroshenko, A., Kontsov, M., 1993, Mathematical modeling and optimization of a Solar Hot-Water System, 8-th Intern. Conference on Thermal Eugng and Thermogrammetry, Budapest, Hungary; 1995, Theoretical and Experimental investigations of solar Hot Water Systems; The optimization of contraction of flat-plat Solar Collectors in Hot Water supply Systems, 9-th Intern. Conference on Thermal Engn and Thermogrammetry, June 14–16 Budapest, Hungary; Doroshenko, A., Glikson., 1997, Non-Convenctional Energy Sources in To-day's Heat Supply systems, 10-th Intern. Conference on Thermal Engn and Thermogrammetry, June 14–16 Budapest, Hungary. [31] Doroshenko, A.V., Omelchenko, Yu.M., 1999, Alternative energetics: experience of application and real perspectives. Untraditional and renewable sources of energy, Ministry of transport of Ukraine, State Department of marine and river transport, Odessa, No.2, pp. 12-13. [32] Doroshenko, A., Yarmolovich, Y., 1987, Indirect Evaporative Cooling, Cholodilnaya Technika, no. 12: p. 23-27 (Russian). [33] El-Chalban, A.R., 1995, Simulation and performance analysis of a compound heat transformer – refrigeration system, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 2. [34] Engelhorn, H.R., 1998, Solar-cooling plant Jiangxi, Fachhochschule GiessenFriedberg.

Solar Absorption Systems…

141

[35] Erhard, A., Spindler, K., Hahne, E., 1995, Test and simulation of solar powered dry absorption cooling machine, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 4. [36] EUREC Agency, 1996, The future renewable energy – Prospects and directions, James & James Ltd, London. [37] Foster, R.E., Dijkastra, E., 1996, Evaporative Air-Conditioning Fundamentals: Environmental and Economic Benefits World Wide, International Conference of Applications for Natural Refrigerants’ 96, September 3-6, Aarhus, Denmark, IIF/IIR: p. 101-109. [38] Franke, U., 1998, Beispielsammlung für die Anwendung des SECOSorptionsgenerators, Fachbericht, ILK Dresden. [39] Gandhidasan, P., 1994, Performance analysis of an open liquid desiccant cooling system using solar energy for regeneration, Int. J. Refrig., vol. 17, no. 7: p. 475 - 480. [40] Grandov, A., Doroshenko, A., 1995, Cooling Tower with Fluidized Beds for Contaminated Environment, Int. J. Refrig., vol. 18, no. 8: p. 512-517. [41] Grossman, G., Devault, R.C., Creswick, F.A., 1995, Simulation and perfomans analysis of an ammonia-water absorption heat pump based on the generator-absorber heat exchange (GAX) cycle, ASHRAE Trans., vol. 1, n. 1, p. 1313-1323. [42] Grossman, G., Zaltash, A., Devault, R.G., 1995, Simulation and perfomance analysis of a four-effect water-lithium bromide absorbtion chiller, ASHRAE Trans., US, vol. 101, n. 1, p. 1302-1312; Haim, .I., Grossman, G., Shavit, A, 1992, Simulation and analysis of open cycle absorption system for solar cooling, Solar Energy, Vol. 49, n., 6, pp. 515534. [43] Guillemnot, J.J.G., Chalfen, J.B., Poyelle, F.G.R., 1995, Mass and heat transfer in consolidated adsorbing composites. Effect on the performance of a solid-adsorption heat pump, 19-th int. Congr. Refrig., The Hauge, NL, pp. 261-268. [44] Gusel E., Gurgor, A., 1995, Air conditioning using absorption and solar energy in an open system, Proc. ULIBTK’95, Ankara, TR., pp. 303-313. [45] Hahne, E., 1996, Solar heating and cooling, Proceedings of EuroSun 96, Freiburg, pp. 3-19. [46] Hamad, M., 1995, Experimental study of the performance of a solar collector cooled by heat pipes, Renewable Energy, GB., vol. 6, n. 1, pp. 11-15. [47] Hammand, M.A., Audi, M.S., 1992, Performance of solar LiBr-water absorption refrigeration system, Renewable Energy, 2(3), pp. 275-282. [48] Hanna, W.T., Saunders, J.H., Wilkson, W.H., Philips, D.B., ASHRAE Trans., US, vol. 101, n. 1, p. 1189-1198. [49] Hara, T., Oka, M., Nikai, I., 1996, Adsorption/desorption characteristics of adsorbent dehumidifier for air conditioners, International Conference of Applications for Natural Refrigerants’ 96, September 3-6, Aarhus, Denmark, IIF/IIR: p. 111-120. [50] Hartmann, K., 1998, Lithiumbromid / Wasser-Absorptions-kältemaschinen, DIE KÄLTE & Klimatechnick, no. 10, pp. 780-790. [51] Heat exchange apparatuses of refrigerating plants, 1986, Edit. G.G. Danilova, Leningrad, Mashinostroenie, p. 34. [52] Hellman, H.M., Grossman G., 1995, Simultation and analysis of an open-cycle dehumidifier-evaporator (DER) absorption chiller for low-grade heat utilization, Int. J. Refrig., vol. 18, no. 3: p. 177-189.

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A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

[53] Hellman, H.M., Pohl, J.P., Grossman, G., 1996, IIF/Proc. Aarhus Meat, IIR, FR., p. 121-132. [54] Henning, H. (Co-ordinator), Solar desiccant cooling system for an office building in Portugal, Contract EU Thermie No. REB/78/95/DE/PO. [55] Henning, H., Erpenbeck, T., 1996, Integration of solar assisted open cooling cycles into building climatisation systems, Proceedings of EuroSun 96, Freiburg, pp. 1248-1253. [56] Henning, H.-M., Häberle, A., Gerber, A., 1995, Building climatization with solar assisted open cycle solid sorption cooling systems – design rules and operation strategies, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 9. [57] Hilali, I., Okuyan, C., Aktacir, M.A., 1995, Investigation of absorption cooling systems using solar energy in Sanliurfa, Proc. ULIBTK’95, Ankara, TR., p. 323-332. [58] Höper, F., 1999, Optimierte anlagenschaltung zur solaren Kühlung mit absorptionskältetechnick, KI Luft- und Kältetechnick, no. 8., pp. 397-400. [59] Idelchik, I.E., 1976, Deference-book on hydraulic resistances, Moscow, Machinebuilding, pp. 560. [60] Intensification of heat exchange in evaporators of refrigerating machines, Editor A.A. Gogolin, Moscow, Light and food industry (Legkaya and pishchevaya promishlennost), 1982, pp. 224. [61] Intensification of heat exchange in evaporators of refrigerating machines, 1982,Edit A.A. Gogolin, Moscow, Light and food industries, p. 224. [62] International Institute of Refrigeratinf, Publications 1998. [63] Ismail, I.M., Mahmoud, K.G., 1994, Comparative study of different air-conditioning systems incorporative air washers, Int. J. Refrig., vol. 17, no. 6: p. 364-370. [64] Iyoki, S., Uemura, T., 1988, Physical and thermal properties of the water-lithium bromide-zinc chloride-calcium bromide system, Int. J. Refrig., vol. 12, Sept.: p. 272277. [65] Iyoki, S., Yamanaka, R., 1993, Physical and thermal properties of the water-lithium bromide-lithium nitrate system, Int. J. Refrig., vol. 16, no. 3: p. 191-200. [66] Jain, S., Dhar, P.L., Kaushik, S.C., 1995, Evaluation of solid-desiccant-based evaporative cooling cycle for typical hot humid climates, Int. J. Refrig., vol. 18, no. 5: p. 287 - 296. [67] Jin-Soon Kim, Huen Lee, Sun I1 Yu, 1999, Absorption of water vapour into lithium bromide-based solutions with additives using a simple standart pool absorber, Int. J. Refrig., vol. 22, pp. 188-193. [68] John Browne, 19/5/97, Addressing climate change, a speech by the Chief Executive of BP delivered to Stanford University, California. [69] Kang, Y.T., Christensen, R.N., 1995, Transient analysis and design model of a LiBrH2O absorber with rotating drums, ASHRAE Trans., US, vol. 101, n. 1, pp. 1163-1174. [70] Kassler, P., 1994, Energy for development, Shell Selected Paper, London. [71] Kauffeld, M., Aarhus, Indirekte Kälteanlagen, DK, KK/9. [72] Kaushik, S.C., Kaudinya, J.V., 1989, Open cycle absorption cooling – a review, Energy Conversion Management, 29(2), pp. 89-109. [73] Kessling, W., Laevemann, E., Peltzer, M., 1996, Efficient energy storage for desiccant cooling systems, Proceedings of EuroSun 96, Freiburg, pp. 1254-1260.

Solar Absorption Systems…

143

[74] Kessling, W., Laevemann, E., Peltzer, M., 1998, Energy storage in open cycle liguid desiccant cooling systems, Int. J. Refrig., vol. 21, no. 2, pp. 150-156. [75] Khairullah, A., Singh, R.P., 1991, Optimization of fixed and fluidized bed freezing processes, Int. J. Refrig., vol. 141, May, pp. 176-181. [76] Khan, A.Y., 1996, Parametric analysis of heat and mass transfer performance of a packed-type liquid desiccant absorber at part-load conditions, ASHRAE Trans., US, vol. 102, n. 1, p. 349-357. [77] Kholpanov, L., Ismailov, B., Vlasak, P., 2005, Modelling of multiphase flow containing bubbles, drops and solid particles, Engineering Mechanics, vol.12, n.6, p.111. [78] Kholpanov, L., 2008, Mathmodeling of nonlinear dynamics and mass transfer, Czasopismo techniczne. Mechanika, .z. 2-M, p. 95-112. [79] Kholpanov, L., Zaporozhets V., Zibert G., Kaschickii Yu., 1998, Mathematical modeling nonlinear termohydrogasdynamic processes, Moscow, Nauka, pp. 320. [80] Khrustalev, D., Faghri, A., 1996, Fluid flow effects in evaporative from liquid-vapour meniscus, Heat Transfer, US, vol. 118, n. 3, p. 725-730. [81] Kirillov, V., 1994, Hydrodynamics and heat-and-mass exchange in 2-phase flows of film apparatuses for refrigerating engineering. Doctor dissertation, Odessa, Odessa State Academy of Refrigeration, (Russian). [82] Kirillov, V., Doroshenko, A., 1988, Peculiarities film flowing along the surface with regular roughness, Eng.-Phys.-Journal, vol. 54, no. 5: p. 139-145 (Russian). [83] Kirillov, V., Doroshenko, A., 1996, Maximum permissible velocity of gas in heat-andmass transfer equipment, Eng.-Phys.-Journal, vol. 69, no. 2: p. 269-285 (Russian). [84] Koepel, E.A., Klein, S.A., Mitchell, J.W., 1995, Commercial absorption chiller models for evoluation of control strategies, ASHRAE Trans., US, vol. 101, n. 1., p. 1175-1184. [85] Koo, K.K., Lee, H. R., Jeohg, S. Y., atal, 1998, Solubility and vapor pressure characteristics of H2O/( LiBr + LiJ + LiNO3 + LiCl ) system for air- cooled absorbtion chillers. International Conference " Naturals working fluids, Oslo, Norway, IIF / IIR: p. 531-536. [86] Kvurt, Yu., Doroshenko, A., Shestopalov K., 2008, Research and modeling evaporating cooling equipment of an indirect type, 21-th International conference on Mathematical Methods in Engineering and Technology - MMET-21, 29 May - 1 June, Saratov, vol. 3, p. 71-72 (Russia). [87] Kvurt, Yu., Kholpanov, L., 2008, Hydrodynamics of film current in channels with screw roughness, 3-th International conference Heat and mass transfer and hydrodynamics in swirling flow, October 21-23, Moscow, p. 37-40. [88] Kvurt, Yu., Kholpanov, L., 2000, Method for measuring the consumption of solid inclusions in slurries, Proceedings of 13th Advance in Filtration and Separation Technology, March 14-17, Myrtle Beach, SC, vol., 14, p. 789-794. [89] Lamp, P. (Co-ordinator), Solar assisted absorption cooling machine with optimized utilization of solar energy (SACMO). Contract EU JOULE No. JOR 3CT 9500 2O. [90] Lamp, P., Ziegler, F., 1998, Review Paper, European research on solar-assisted air conditioning, Int. J. Refrig., vol. 21, no. 2, pp. 89-99. [91] Lauritano, A., Marano, D., 1995, Open-cycle solar absorption cooling and its applicability to residential buildings in Italy, Cond. Aria, JT., vol. 39, p. 495-502.

144

A.V. Doroshenko, Y.P. Kvurt and L.P. Kholpanov

[92] Lävermann, E., 1995, Desiccant cooling, Included in Potential of Solar Assisted Cooling in Southern Europe, Final Report, EU Contract RENA-CT94-0017. [93] Lävermann, E., Keвling, W., 1995, Energy storage in open cycle liquid disiccant cooling systems, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 10. [94] Lavrenchenko, G., Doroshenko, A., 1988, Development of Indirect Evaporative AirCoolers for Air-Conditioning Systems, Cholodilnaya Technika, no. 10: p. 28-33 (Russian). [95] Lazzarin, R., Longo, G.A., Gasparello, A., 1996, Theoretical analysis of an open-cycle heating and cooling systems, Int. J. Refrig., 19(4), pp. 239-246. [96] Lazzarin, R.M., Gasparella, A., Longo, G.A., 1999, Chemical dehumidification by liquid disiccants: theory and experiment, Int. J. Refrig., vol. 22, pp. 334-347. [97] Lowenstein, A, Novosel, D., 1995, The seasonal performance of a liquid-desiccant air conditioner, ASHRAE Trans., US, vol. 101, n. 1, pp. 679-685. [98] Lowenstein, A., 1993, Liquid desiccant air-conditioners: An attractive alternative to vapor-compression systems. Oak-Ridge nat. Lab/Proc. Non-fluorocarbon Refrig. AirCond. Technol. Workshop. Breckenridge, CO, US, 06.23-25, p. 133-150. [99] Lowenstein, A.I., Dean, M.N., 1992, The effect of regenerator performance on a liquiddesiccant air conditioner, ASHRAE Trans., US, vol. 98, n. 1, p. 704-711. [100] Lowenstein, A.J.,Gabruk, R.S., 1992, The effect of absorber dising on the performance of a liquid-desiccant air conditionaer, ASHRAE Trans., US, vol. 98, n. 1, p. 712-720. [101] Lu, S.-M., Yan, W.-J., 1995, Development and experimental validation of a full-scale solar desiccant enhanced radiative cooling system, Renewable Energy, vol. 6, n. 7, pp. 821-827; Yan, W., Wu, W., Shyu, R., 1994, Study on solar liquid desiccant cooling system, The 5th ASEAN Conference on Energy Technology, April 25-27, The TaraImpala Hotel, Bangkok, Thailand. [102] Lu, S.-M., Shyu, R.-J., Yan, W.-J., Chung, T.-W., 1995, Development and experimental validation of two novel solar desiccant-dehumidification-regeneration systems, Energy, vol. 20, n. 8, pp. 751-757. [103] Ma, W.B., Deng, S.M., 1996, Theoretical analysis of low-temperature hot source driven two-stage LiBr/H2O absorption refrigeration system, Int. J. Refrig., vol. 19, no. 2, pp. 141-146. [104] Matsuda, A., Munafata, T., Yoshimura, T., Fuchi, H., 1980, Measurement of vapor pressure of lithium-bromide-water solution, Kagaku Kagaku Ronbunshu, no. 5: p. 119 – 122. [105] Mendes, L.F., Collares Pereira, M., Ziegler, F., 1998, Supply of cooling and heating with solar assisted absorption heat pumps: an energetic approach, Int. J. Refrig., vol. 21, no. 2, pp. 116-125. [106] Merkle, Th., Hahne, E., 1996, Solar thermal energy concept for water heating, space heating and cooling of an industrial buildings, Proceedings of EuroSun 96, Freiburg, pp. 256-365. [107] Mikheev, M., 1968, Fundamentals of heat Transfer, MIR Publishers, Moscow, 376 p. [108] Mostafavi, M., Agnew, B., The effect of ambient temperature on the surface area of components of an air-cooled lithium bromide-water absorption unit, Appl. Eng., GB, vol. 16, n. 4, p. 313-319.

Solar Absorption Systems…

145

[109] Nahredorf, F., Blank, U., Iliev, N., Saumweber, M., Stojanoff, C.G., 1998, Development of a fixed bed absorption refrigerator with high power density for the use of low grade heat sources, Int. J. Refrig., vol. 21, no. 2, pp. 126-132. [110] Ney, A., 1995, Room air conditioning by means of adiabatic evaporative cooling, Tech. Bau, DE, n. 1, p. 35-40. [111] Nguen, M.M., Riffat, S.B., Whitman, D., 1996, Appl. Therm. Eng., GB, vol. 16, n. 4, p. 347-356. [112] O’gorman, T., 1995, Cooling from the sun, Air Cond., GB., vol. 98, n. 1167, p. 25-26. [113] Oertel, K., 1995, Design of stand-alone solar-hybrid cooling system for decentralized storage of food, Proceedings of 2nd Munich Discussion Meeting ‘solar Assisted Cooling with Sorption System’, München, Paper No. 12. [114] Ono, T., 1992, High efficiency chemical refrigeration system using solar heat, Japan’s Sunshine Project, 1991, Annual Summary of Solar Energy R&D Program. New Energy Development Organization. [115] Orlov, A., Kholpanov, L., 1997, On new aspects of using solar energetics in obtaining hydrogen, Preprint of ROAN, Institute of general physics, Moscow, (Russian). [116] Oppermann, G., 1996, Using heat from a depth of 1500 metres, IEA Heat Pump Cent. Newsl., US, GB, vol. 16, n. 4, p. 347-356. [117] Otternein, R.T., 1995, A theory for heat exchangers with liquid-desiccant-wetted surfaces, ASHRAE Trans., US, vol. 101, p. 317-325. [118] Palatnik, I.L., 1998, Development and investigation of heliosystems with flat solar collectors for heat and cold supply. Dissertation… Sci (Eng.), Odessa. [119] Rats, I.I., 1962, Construction, investigation and design of heat-exchange apparatuses, Moscow, p. 230. [120] Reference-book on heat exchangers, 1997, Edit. O. G. Martynenko et al., Moscow, Energoatomizdat, Vol. 2, p. 352. [121] Sönke Biel, 1998, Soptive Luftentfeuchtung Und Verdunstungs-kühlung, Ki Luft-und Kältetechnik, no. 7: p. 332-336. [122] Saman, N,F., Jostone, H.W., 1996, Maintaining temperature and humidity in nonhumidity-generating spaces, ASHRAE Trans., US., vol. 102, n. 2, p. 73-79. [123] Schweigler, C., Demmel, S., Riesch, P., Alefeld, G., 1996, A new absorption shiller to establish combined cold, heat and power generation utilizing low temperature heat, ASHRAE Transaction 102(1) and ASHRAE Technical Data Bulletin 12(1), Absorption/Sorption Heat pumps and Refrigeration Systems, pp. 81-90. [124] Sherwood, T.K., Pigford, C.R., 1982, Mass Transfer, Chimiya, Moscow, 697 p. [125] Solar Energy for Refrigeration and Air Conditioning, Proceedings of Meetings of Commissions E1 and E2, Jerusalem, 14-15 March 1982.Int. Inst. Refrigeration, Paris. [126] Steimle, F., Development in Air-Conditioning, International Conference of Research, Design and Conditioning Equipment in Eastern European Contries, September 10-13, Bucharest, Romania, IIF/IIR: p. 13-29. [127] Stoitchkov, N. J., Dimirov, G.J., 1998, Effectiveness of Crossflow Plate Heat Exchanger for Indirect Evaporative Cooling, Int. J. Refrig., vol. 21, no. 6: 463-471. [128] Stojanoff, C., Nahredorf, F., Blank, U., Iliev, N., Saumweber, M., 1995, Development of solar-driven absorption cooling system, Proceedings of 2nd Munich Discussion Meeting ‘Solar-Assisted Cooling with Sorption System’, München, Paper No. 3.

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[129] UNEP. Montreal Protocol on Substances That Deplete The Ozone Layer. Final Act: date – 11 September 1987, 6 p. [130] United Nations on Climate Change. General Convention Kyoto, 1997. [131] Wenjyh Yan., Weiyih Wu., Rueyjong Shyu., 1994, Study on a solar liquid desiccant cooling system, The 5th ASEAN Conference on Energy Technology, April 25-27, Bankok, Thailand, vol. I, pp. 304-311. [132] West, M.K., Lyer, S.V., 1995, Analysis of a field-installed hybrid solar desiccant cooling system, ASHRAE Trans., US, vol. 101, n. 1, pp. 686-696. [133] Westerlund, L., Dahl, J., 1994, Absorbers in the open absorption system, Applied Energy, n. 48, pp. 33-49. [134] Westerlund, L., Dahl, J., 1994, Use of an open absorption heat-pump for energy conservation in a public swimming-pool, Applied Energy, n. 49, pp. 275-300. [135] Yang, R., Wang, P.L., 1995, Experimental study of a glased solar collector/regenerator operated under a humid climate, INT. J. Sol. Energy, CH, vol. 16, n. 3, pp. 185-201. [136] Zayitsev, I.D., Aseev, G.G., Physical and chemical properties of binary and multicomponent solutions of inorganic substances Reference edit, Chemistry (Khimiya), Moscow, p. 416 (Russian), [137] Zech, S., 1995, Solar cooling system based on water/zeolite absorption, Luft Kaltetechn, DE., vol. 31, n. 12, pp. 559-561.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 147-235

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 2

CAUSES OF FAILURES AND THE NEW PROSPECTS IN THE FIELD OF SPACE MATERIAL SCIENCES A.I. Feonychev* Institute of Technical Acoustics, National Academy of Sciences, Belarus

Abstract The methodical study of the crystal growth processes and electrophoretic separation of the biomixtures have been carried out under microgravity conditions. The mathematical simulation of technological processes was performed by dint of the Navie-Stokes equations, the equations for heat and mass transfer and the Maxwell equations for magnetic and electric fields with reasoned assumptions. Analysis of crystal growth by the Bridgeman-Stockbarger and moving heater methods has shown that these methods does not give the expected positive results due to particularities of the fluid flows and heat and mass transfer under microgravity conditions and zero gravity. New condition for the dopant concentration at the crystallization boundary is used under calculations and comparison of the calculation data and the results obtained in experiments on board spacecraft. Thermocapillary convection stability and the process of crystal growth by the floating zone method are studied with use of different control actions. A rotating magnetic field, additional fluid layer (encapsulation of crystallizing melt) and standing surface waves, generated by axial vibration are applied as the control action. Axial static magnetic field is additional applied in two last cases. It is shown that under optimum parameters of the external action on thermocapillary convection, dopant segregation in micro- and macro-scales can be significantly reduced. The new idea on eigenfrequency of convective cell is used for analysis of the calculation results. The analysis of special space experiment on continuous flow electrophoresis showed that the failures of experiments on biomixture separation with the help of this method are due to hydrodynamic instability of biocomponent jet by the action of vibrations and ponderomotive force in electric field. The considered modifications of the floating zone method, as well as the use of the standing surface waves generated by vibrating crystal in the Czochralski method can be used for crystal growth in terrestrial conditions.

*

E-mail address: [email protected]

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Introduction Methodical investigations in the field of materials sciences under microgravity conditions began in 1973, with the flight of the orbital station Skylab. Crystal growth by the BridgmanStockbarger method and fractionation of biomixtures by continuous flow electrophoresis can be marked among multitudinous directions of investigation. These two directions are distinguished by both high flying goal to produce high-quality homogeneous materials which are unattainable on the ground and the immensity and the number of accomplished experiments. The history of studies in these directions is distinct. Contrary to expectations, a high nonhomogeneity of dopant distribution on radius had been produced in all crystals grown by the Bridgman-Stockbarger method [1-3]. In spite of this, such experiments have been continued for long time after the first failures. Moreover, experiments on crystal growth by the Bridgman-Stockbarger and other ampoule methods with their associated installations are present in program of scientific experiments on the Russian Segment of International Space Station. It should be mentioned that the quintessential information for following analysis of crystal growth by the Bridgman-Stockbarger method is present in the first publications [1, 2]. However, detailed analysis of these experiments showed some problems due to the specific conditions that took place during the performance of space experiments. Then problems were the following: residual accelerations, vibrations existing aboard spacecraft, the possibility of melt separation from ampoule wall, etc. It a long time took to obtain required information and accomplish methodical numerical investigation of both the standard method und different modifications which were proposed or realized in other experiments in space conditions. Eventually it has been demonstrated [4, 5] that the Bridgman-Stockbarger method with all possible modifications and any ampoule methods are unsuitable for crystal growth during the flight of orbital spacecraft. By contrast, experiments with fractionation of biomixtures by the continuous flow electrophoresis were quietly discontinued in USA and Russia after the announcement about progress. The published data on the results of these experiments are deficient for rigorous analysis. However, the available results of fluid flow observation in the course of the model experiment on the “Ruchey” facility during the flight of the “Mir” orbital station and some results of numerical investigations led to a better understanding of physical causes responsible for failures of full-scale experiments. Revealed causes point the way to further investigations. There is the reason to think that the goal being sought for these experiments at the beginning can be successfully implemented. Furthermore, there are two other methods for fractionation of biomixtures, for example zone electrophoresis, isotachophoresis and isoelectric focusing. Productivity of these methods is less than the former but they are more suitable for delicate biological investigations under microgravity conditions. The floating zone method is used in several experiments on crystal growth under microgravity conditions. These studies have met with only limited success. However, the numerical investigations of this technological process were show that extended free surface of a fluid creates prerequisite to effective control of fluid flow and of dopant transfer. Axial vibration, static and rotating magnetic fields and additional liquid layer on cylindrical surface of liquid column have been investigated as the control action [6-8]. The following investigations show that considerable reduction of dopant segregation can be obtained under optimal parameters of the control action. Moreover, the all advantages of this method as

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compared to ampoule methods, for example, the low level of dislocation, absence of pollution from ampoule wall, a small consumption of energy, the possibility to remove fusible impurities from free fluid surface and to carry out multiple recrystallization, etc., are retained. There is opportunity to use the studied technological techniques in terrestrial production. The second aim is to obtain new data for the behavior of fluid and gas media, transfer processes in a fluid and gas phases, including the influence of conditions on interfaces of fluid-gas, fluid-fluid under microgravity conditions. The principal results of investigations have been obtained by the mathematical simulation methods since physical modeling of these processes aboard spacecraft is unduly expensive and practically impossible with the help of modern hardware. If results of these investigations are not always rightly interpreted the core of technological process, they have stimulated the advancement of computing methods. It is supposed that obtained data will be used for the progress of terrestrial technology. The specialists in different fields of science took part in the discussion of scientific problems in the framework of annual Congresses of the International Astronautics Federation, COSPAR’s Symposiums and local Symposiums. Unfortunately, investigations in the field of microgravity have decreased recently. It is caused by aforementioned failures. The collections of articles devoted to problems of space material sciences [9-11], do not contain reasonable analysis of failed experiments and do not show the ways for solution of emersed problems. Such ways are marked in our part of Summary Report on the project INTAS-2000-0617. The determining role of convection and heat and mass transfer in the formation of dopant nonhomogeneities in growing crystal are presented in detail considered in monograph [12]. However, more attention needs to be given to particularity of these processes under microgravity conditions. It might be supposed that the presentational results of numerical investigation on hydrodynamics and heat and mass transfer in context of noted technological processes will provide interest in investigations under microgravity conditions and lead to the solution of problems set more than thirty years ago.

General Mathematical Formulation of Problems Numerical solution for all considered problems is based on a system of two-dimensional equations for processes of convective transport in viscous incompressible fluid. The distinction consists in different expression for the outside forces, additional equations for determination of these forces and peculiar boundary conditions. The finite difference approximation of differential equations and calculation scheme are different and conditioned by peculiar features of problem. These peculiarities are noted and given corresponding mathematical expressions in items of article. The governing equations are cast in dimensionless form using R, R2/ν, ν/R, ρυ2/R2, Δ T = Tmax – Tmin, c0 to scale the length, time, velocity, pressure, force, temperature, and dopant concentration, where ν is the melt’s kinematic viscosity, ρ is the melt’s density, c0 is initial concentration of dopant, R is radius of fluid, Tmax and Tmin are maximum and minimum temperature in fluid, respectively. The dimensionless equations that govern momentum, heat and solute transport in fluid are given in vectorial form as follows: ∂v/∂τ + (v ∇ ) v = − ∇ p + ∇ 2v + F

(1)

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∇v = 0

(2)

∂θ/∂τ + (v ∇ ) θ = − Pr -1 ∇ 2θ

(3)

∂C/∂τ + (v ∇ ) C = − Sc -1 ∇ 2C

(4)

where Pr = ν/χ is the Prandtl number, Sc = ν/D is the Schmidt number, χ is the melt’s thermal diffusivity, D is diffusion coefficient of dopant in melt. A concrete expression of an outside force F will be given below in the solution of specific problem. All the components of the velocity at the solid walls are fount to be zero (sticking condition). The axial symmetry of a flow means that the radial component of the velocity is given to zero under r = 0. For studied problems of crystal growth the lateral area of fluid (r = R) and end-wall (z = L) are dopant tight. The temperature on boundary of crystallization (z = 0) is kept constant (θ = 0). It should be noted the following for the dopant concentration condition at the boundary of crystallization (z = 0). This condition is typically written as: (∂C/∂z)s = - ScRecr (1 – k0)Cs

(5)

where Recr = vcrR/ν is the dimensionless crystal growth rate, k0 is the equilibrium coefficient of dopant distribution; the s index references to crystallization boundary. However, as our calculations showed, this condition can be using only for the cases of strong mixing melt. The new condition has been derived from dopant balance on the phase boundary for the cases of weak mixing of melt. (∂C/∂z)s = - ScRecr (1 – k0Cs)

(6)

This condition was in particular used in [4]. The implicit finite-difference scheme used for obtaining the solution of the Navier-Stokes equations in pressure-velocity variables has the third order of accuracy in space coordinates and possesses conservative properties. The scheme is constructed on a five-point pattern. The Neumann problem for the pressure increment is solved using a computational algorithm and velocity values at half-integer points. This provides the fulfillment of the difference analog of the incompressibility condition. The convergence and stability of the solution is provided at the following conditions for the space and time steps (hmin and Δτ, respectively): hmin ≤ 0.2 (Remax) -0.5 , Δ τ ≤ (Remax) -1

(7)

where Remax = vmaxR/ν is the (dimensionless) maximum velocity in fluid. In the calculations, the time step is corrected automatically for the fulfillment of the second condition (7). The pressure increment is determined by the conjugate-gradient method, and the temperature, velocity, and concentration are determined by the implicit Lanczos method. He difference scheme and a computational algorithm are verified by comparison with the available results of solution of the test problem on a fluid flow in a

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cavity with a moving cover, and experimental data. This allows one to obtain stable solutions for the Reynolds numbers up 104, that is for oscillatory convection.

I. Crystal Growth The development of microelectronics imposes more and more high demands on the quality of single crystals. They must have a perfect crystal structure and highly homogeneous composition on micro- and macroscales. The Czochralski method is not always convenient for growing promising crystals of the A3B5 group (AsGa, InP, GaSb, InSb) or especially pure crystals, when impurity from a crucible is impermissible. The Bridgman-Stockbarger method was commonly employed for the experiments under microgravity conditions owing to its simplicity. The floating zone method requires more complicated equipment compared to the Bridgman-Stockbarger method. This is especially true with regard to the means for creation of the heating zone with the requisite temperature profile. Besides, process of crystal growth happens at strong thermocapillary convection. Thus, it is necessary to solve problems of terrestrial technology. Because of the above-named reasons the floating zone method has been only occasionally realized in space experiments.

1. The Bridgman-Stockbarger Method The process of crystal growth is considered in the case when gravity vector and vibration are directed along the cylindrical ampoule axis. The following expression for F is introduced into the equation (1) F = (GrT + ∑ GrT vib i cos Ωi τ)θ n

(8)

The first summand in the right-hand side of (8) defines the Archimedean buoyancy. The second term describes the influence of vibration with a set of frequencies with number i = 1, 2, 3, … on nonisothermal fluid. Here n (0, nz, o) is the unit vector, GrT = g βT ΔT R3/ν2 is the Grashof number, Ω = ωR2/ν is the (dimensionless) circular frequency, τ is the dimensionless time, g is gravitational acceleration, gvib = aω2 is the amplitude of vibration acceleration, a is the vibration amplitude, βT is coefficient of thermal expansion. We assume that fluid fills the ampoule, and thus, we do not take into account thermocapillary effect at z = L. This effect, as well as changing height of fluid column in the process of crystal growth, was investigated earlier [13]. The velocity of crystallization front moving is usually small that one can consider this front as static, neglecting the change of the length of fluid. These two effects are essential for the final stage of crystal growth, when the crystallization boundary is close to the end z = L, but this stage is not considered in [13].The end surfaces z = 0, L are flat. Parabolic temperature distribution is set on the lateral surface of a fluid θ =

z/R . This function agrees well with the experimental data. Aspect ratio of L/R

fluid L/R was varied through a range of 2 – 8. In the figures presented below, L/R is equal to 3. If it required to use these data for any other length of melt, the Rayleigh number is

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multiplied by a factor equal to (3R/L)0.5. This is possible from the fact that the flow velocity of melt along the interface is defined by temperature gradient on the ampoule length. All results presented below have been obtained for dimensionless crystallization rate Recr = 0.2. However, calculations with Recr in the range from 0.05 to 1 showed that the dependence of dopant concentration from crystallization rate is linear between 0 and 0.3. Consequently, the application of the results on the other crystallization rates is not difficult. All experiments in space conditions have been carried out under Recr < 0.2.

1.1. Results of Calculations 1.1.1. Radial Segregation of Dopant Already in the first experiments on crystal growth by the Bridgman-Stockbarger method was discovered the two facts: the dislocation reduction in crystals and a high radial nonhomogeneity of the dopant distribution in the space-grown crystals. The former was explained by possible detachment of the melt from the ampoule wall immediately at the boundary of the crystallization by virtue of the growth angle. The method guaranteeing a detaching of melt from the ampoule wall in the immediate of crystallization boundary was proposed in [14]. For this purpose, it was proposed to coat the layer of additional material with the lower temperature than base material. It was also presumed that this will allow radial segregation of dopant to reduce. As is clear from the subsequent text, the second goal cannot be achieved. Numerical investigations convective heat and mass transport with simulation of all conditions for crystal growth by the Bridgman-Stokbarger method have been carried out at first for rectangular infinite-rectangular gap [15] and shortly for cylindrical ampoule [16]. These investigations showed that as the velocity of fluid along the crystallization boundary increases, radial nonhomogeneity of dopant distribution heightens initially, attains to maximum and then falls. This effect of maximum, as such, is of little interest and indicates the moment when transport of dopant along the boundary of crystallization overbalances maximally the removal away from this boundary. The conclusion of this maximum can be heuristically obtained. Thus, if the dependence of radial segregation of dopant from velocity of fluid flow is continuous function, that is evident, and is equal to zero in regime of diffusion transfer and under infinite mixing of melt, this function will peaks. Once the level of vibration accelerations existing aboard spacecraft has been measured, radial nonhomogeneity of dopant distribution is often associated with the action of these accelerations [17, 18]. The methodical numerical investigation has been carried out to define exactly the role of residual steady-state and vibration accelerations under crystal growth aboard spacecraft [4]. All calculations were performed for typical semiconductor materials, which were widely used in experiments under microgravity conditions aboard spacecraft. They are Ge(Ga), GaAs(Te), InSb(Te), Si(P), Si(B). The Rayleigh number is common for generalization of the results obtained under investigations in the field of gravitational convection. However this dimensionless number fails for the dopant transport during the crystal growth by the Bridgman-Stockbarger method under microgravity conditions. This is clearly shown in figure 1. The same situation is observed (figure 2) for time-averaged flow induced by vibration. A combined parameter Ra2 /Ω, is described the intensity of fluid flow by vibration. In this case the Rayleigh number (Ra

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= gvib βT ΔT R3/(χν)1/2) is evaluated with use of unit equivalent vibration frequency. The analogy between gravitational convection and time-averaged convection by the actionof vibration has been mathematically evidenced in [19] for nonisothermal fluid. The relative radial inhomogeneity of the dopant distribution on the crystallization boundary (ΔC s / C s ) is evaluated at the point in time after which this parameter reaches a steady-state value. Here ΔC s = Cs max – Cs min is difference of dopant concentration in melt at crystallization boundary,

C s is the mean dopant concentration on this boundary.

ΔCs/Cs

0,12

4 2

0,10 1

0,08

3

0,06 0,04 0,02 0,00 -2 10

-1

10

0

10

1

10

2

Ra

10

Figure 1. Relative radial nonhomogeneity as a function of the Rayleigh number under action of residual acceleration for the four semiconductor materials. (1) GaAs(Te); (2) Ge(Ga); (3) Si(P); (4) InSb(Te). Recr = 0.2.

The flow patterns (isolines of stream function) for gravitational convection and for timeaveraged flow being created by vibration are shown in figure 3. Dashed and solid lines in these and other examples correspond to counter-clockwise and clockwise flow of fluid, respectively. It should be pointed out that the flow patterns at diametrically opposite directions of gravity vector (figure 3a and figure 3b) are not quite identical. The flow patterns shown in figure 3d and figure 3c are practically alike and are distinguished by the direction of fluid flow. This means that the proportioned vibration can be used for the reduction of gravitational convection up to the reasonable level. The last two figures (3d and 3e) show that frequency set can be substituted by one equivalent frequency. A method of substitution is presented in [4]. The three dimensionless frequencies in figure 3d correspond to the following dimensional parameters for fluid with a radius R = 2cm and ΔT = 20 K: f1 = 0.8 Hz, g1 = 4.5·10-3 ms-2, 1.35 Hz, g1 = 4·10-2 ms-2, 2 Hz, g1 = 1.17·10-3 ms-2. The first two frequencies in this frequency spectrum were obtained from results of measurements made with the help of the SAMS (Space Acceleration Measurement System) during active physical exercises of an astronaut on the Mir orbital station. The third frequency is obtained as a result of summation of several peaks in the range of 2-9. The frequencies above 10 Hz were not taken into account since they do not contribute considerable to the transport processes in fluid.

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0,14

1

2

0,12

4

ΔCs/Cs

0,10 0,08

3

0,06 0,04 0,02 0,00 10

2

10

3

4

2

Ra /Ω

5

10

10

Figure 2. Relative radial nonhomogeneity as a function of vibration parameter Ra2 /Ω for the four semiconductor materials. (1) GaAs(Te); (2) Ge(Ga); (3) Si(P); (4) InSb(Te). Recr = 0.2.

z

z

z

3

3

3

3

2

2

2

2

1

1

1

ψ max= - 0.0212 1

z

0.016

-0.016

- 0.003716

0

0.0

0.5

(a)

1.0

r0

0.0

0.5

(b)

1.0

r0

0.0

0.5

(c)

Figure 3. Continued on next page.

1.0

r

0 0.0

1.0

(d)

r

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155

z

2

1

- 0.00366

0 0.0

1.0

r

(e) Figure 3. Flow pattern for gravitational convection (Ra = 2.79, vector of gravity is directed toward the interface boundary (a); Ra = - 2.79, vector of gravity is directed from the interface boundary (b) and for time-averaged flows by vibration. The single frequency is acted, Ra = 9.3·102, Ω = 7.94·102, material is GaAs(Te) (c); three frequencies typical for the Mir station are acted, Ra1 = 6.89·101, Ω1 = 1.49·104, Ra2 = 5.74·102, Ω2 = 2.51·104, Ra3 = 1.8·102, Ω3 = 3.72·104 (d); one frequency equivalent to case (d) is acted, Raeq = 9.3·102, Ωeq = Ω1 = 1.49·104; material is Ge(Ga) (d and e).

Remax 10

1

10

0

10

-1

10

-2

10

-3

------ g down -- -- g up

vmax

umax

10

-4

10

-2

10

-1

10

0

10

1

(a) Figure 4. Continued on next page.

10

2

Ra

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A.I. Feonychev

, 10

0

10

-1

10

-2

10

-3

10

-4

- - ---

10

1

10

2

10

2

3

R a /Ω

(b) Figure 4. The characteristic velocities as a function of the dynamic parameters, Ra (a) or Ra2 /Ω (b). 0 ,1 4

1 3

0 ,1 0 ΔCs/Cs

4

2

0 ,1 2

0 ,0 8 0 ,0 6 0 ,0 4 0 ,0 2 0 ,0 0 10

-2

10

-1

10

0

P e um ax

10

1

Figure 5. Relative radial nonhomogeneity as a function of the diffusion Peclet number for four semiconductor materials. (1) GaAs(Te); (2) Ge(Ga); (3) Si(P); (4) InSb(Te). Recr = 0.2.

As it is shown in figure 4a and figure 4b, the dependences of the characteristic velocities (maximal values of velocity components) on the dynamic parameters for gravitation and vibration convection are similar. This fact combined with similarity of figure 3a and figure 3c gives grounds to establish a unified functional correlation between ΔC s / C s and the Diffusion Peclet number for gravitational and vibration convection. The data presented in figure 3 and figure 4 show that the velocity fields coincide with the given value of the Peclet number. However a relationship between ΔC s / C s and Peumax is different for different materials (figure 5). This points to the fact that a strong influence of the diffusion coefficient (the Shmidt number) and equilibrium coefficient of dopant distribution (k0) on the concentration field. For each individual material the function of ΔC s / C s (Peumax) becomes

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universal in the field of very small velocity of fluid (“creeping” flow) when Peumax < 3 (for vibration under Peumax < 1). This is evident in figure 6. The versatility of the function of ΔC s / C s (Peumax) is based on the possibility for “creeping” flow to neglect the convective

ΔCs /Cs

term (v ∇ )v and to linearize the equation (1). The condition of Peumax < 1 was fulfilled in all experiments aboard spacecraft.

4

0,4

2 1 6 3

5

0,3 0,2 0,1 0,0 -1

10

0

10

1

10 Pe U

max

Figure 6. Relative radial inhomogeneity of the dopant distribution in melt at the crystallization boundary as a function the intensity of liquid flow (the Peclet number) for the crystal GaAs(Te) grown by various method: (1) Bridgman- Stockbarger, acceleration is directed away from the crystallization boundary; (2) the same with acceleration directed to the crystallization boundary; (3) the same with a rotating magnetic field; (4) floating zone; (5) Bridgman-Stockbarger under the action of vibration; (6) ampoule variant of zone melting. Pr = 0.0186, Sc = 32, k0 = 0.04, Recr = 0.2.

The admissible levels of residual gravitational and vibration accelerations as a function of frequency are presented in figure 7 for five typical semiconductor materials. These dependences have been obtained for an ampoule with R = 20 mm and ΔT = 20 K under the condition that the radial dopant inhomogeneity in a melt at the interface does not exceed 2 %. The plot ΔC s / C s (Ω) was obtained with using of the average interface steady-state concentration. The residual gravitational acceleration consists of the constant and slowly varying with the orbital frequency (forb) parts. The vibration acceleration showed as a function of the equivalent frequency (feq). The boundary between these accelerations (Ω = 102) is rather conventional. In real conditions of spacecraft flight, the vibration frequencies and the orbital frequency are located far from this boundary. This boundary can be determined by using the data for the eigenfrequency of the convective cell (feig) [20]. In [20] it has been shown that the eigenfrequency of the convective cell is a function of the Rayleigh number. Thus, the boundary between the parts of admissible accelerations can be defined by the following conditions: forb << feig (Ra) << minimum feq.

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A.I. Feonychev

ng 0

10

4 5 3 2 1

-1

10

-2

10

-3

10

-4

10

5 4 3

-5

10

-6

10

12345-

2

-7

10

1

InSb(Te) Ge(Ga) GaAs(Te) Si(P) Si(B)

Ω

-8

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

Figure 7. Permissible levels of steady-state acceleration and the dependence vibration accelerations from its frequency for five typical semiconductors.

The results presented in figure 7 show that the unit function for permissible accelerations feasible for all materials and all technological processes cannot exist. In numerous publications the dependence of admissible vibration accelerations from frequency is inclined at angle of 450 to the Ω-axis. This corresponds to a linear flow of fluid, for example, flow through a pipe. In our case, transport processes are two-dimensional, the presented dependences with an angle of 22.50 to the Ω-axis are considerably realistic. Contrary to existing opinion, vibrations aboard spacecraft, with exception of course extreme conditions, pose no hazard to crystal growth. The residual accelerations of gravity, which exist aboard orbital spacecraft, outreach the permissible limits for Ge and GaAs, are on the permissible level for InSb and reasonable only for Si. It is seen from figure 5, figure 6 and figure 8 (lower dotted curve) that a lowering of the residual gravitational accelerations is the most suitable method for the reduction of radial dopant segregation in the crystal growing by the Bridgman-Stockbarger method under microgravity conditions. To do this, the orbit of spacecraft must be significantly increased, that is unreal. The residual acceleration due to motion of spacecraft in rarefied atmosphere can be decreased at the expense of reverse acceleration set up by an electric engine by small thrust. This proposal is considered in [21] and can be used for the small spacecraft of the Foton, Eureca class. The main limitation of this method is to regulate engine thrust with the high precision. Another way to reduce the gravitational convection is to compensate for small convective flow by proportioned vibration. This way is readily apparent from an examination of the flow patterns, as shown in figure 3b and figure 3c. In the realization of this method we have to determine the direction of acceleration vector, to orientate the ampoule so that the acceleration is directed from the interface boundary, and to create the appropriate intensity of the axial vibration. The success of this method is based on the full identity of the flow

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patterns under the action of week gravitational convection and small vibration. The experiment described in [22] is also based on the fact that reverses of fluid flow, as shown in figure 3a and figure 3b. In this experiment the ampoule is rotated through 1800. The radial segregation of dopant in the HgCdTe crystal is decreased. The obtained effect is related to the transition process in the concentration field including dopant distribution at the interface. It should be mentioned that the residual acceleration does not need to be directed along the axis of the ampoule. However, a homogeneous crystal of large length can be obtained by this method only under the repeated turning of the ampoule. Quite a different result comes out, if the characteristics features of fluid flows are not taken into account. The detached solidification method [23, 24] belongs to this category. In this method, it was assumed that thermocapillary convection arising due to the artificially being created detachment of melt from the ampoule wall would compensate gravitational convection. A modified method, known as the detached Bridgman method has been realized in an experiment aboard the Foton-M1 spacecraft in 2002. Unfortunately, the experimental data have been lost during the capsule landing. Growing crystal vibration as a supplementary means for the action on convection was used for the modification. The analysis of these methods has been presented in [5]. Numerical investigations have shown [25] that two convective zones induced by the thermocapillary effect and residual gravity (or the combined action of residual gravity and vibration) interact without breaking apart. The intensity of convective flow close to the interface increases and, as a result, the radial dopant segregation rises. The application of static magnetic field is the simplest method for lowering of convective velocities. If is applied axial magnetic field the additional term (the Lorentz force) in the projection on the r-axis in (8) is written as: - Ha2 u

(9)

where Ha = BR(κ/ρν)1/2 is the Hartmann number, B is magnetic induction, κ is the electric conductivity. It should be noted that the Lorentz force is a volumetric force and hence does not act on a free fluid surface, whereas the thermocapillary effect, on the contrary, has a molecular nature and operates within infinitely small thicknesses. The results of calculations with the static magnetic field for two volumes of residual accelerations (Ra = 1.86 and 4.65) are presented in figure 8. The arrows with numbers indicate the quantity ΔC s / C s corresponding to the given Hartmann number. Two arrows with Ha = 0 note the initial points of calculations. A relatively small magnetic field (Ha = 5060) allows lowering of radial dopant segregation up to 1-1.5 % [7]. It should be remarked that the efficiency of the static magnetic field falls off exponentially with decay of convective motion in a fluid. The positive effect of can turn into its direct opposite if With the availability of even small detachment of the melt from the ampoule wall in microgravity conditions static magnetic field tends to increase of radial dopant segregation [5]. This negative effect could be more if the rotating magnetic field of small intensity is used [5]. The effects of thermocapillary convection and rotating magnetic field will be considered below in part 2.

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A.I. Feonychev Ra=4.65, Ha=0

0,20

Ha=3 10

0,15

ΔCs/Cs

15

0,10

25 50

0,05

40

Ra=1.86, Ha=0

Ha=60 10

0,00

Ha=50

40

-1

10

25 0

10

Peumax

Figure 8. Relative radial inhomogeneity of the dopant distribution in melt at the crystallization boundary as a function the intensity of liquid flow (the Peclet number) for the crystal GaAs(Te) grown by the Bridgman-Stockbarger method under the action of static magnetic field. The solid line was obtained under Ra = 4.65; the dot-and-dash line was obtained under Ra = 1.86. The arrows indicate the point on the curves with the corresponding value of the Hartmann number (the arrows with Ha = 0 mark the onset of calculation with magnetic field). The dotted line corresponds to calculation without magnetic field.

1.1.2. Longitudinal Segregation of Dopant As shown in 1.1.1, the radial dopant segregation can be practically lowered up to 1-2 % under crystal growth by the Bridgman-Stockbarger method or other ampoule methods under microgravity conditions. Dopant segregation along the length of crystal has not attracted adequate attention in contrast to the radial segregation. The first experiments in space [1-3] showed that dopant concentration along the length of growing crystal is monotonically varied at a considerable distance from the onset of process. The length of crystal with height longitudinal inhomogeneity is dissimilar for different materials. Analysis of dopant distribution along a growing crystal was carried out in [4] for the different conditions of process. The investigations were performed under the weightlessness conditions (diffusion regime of transport), as well as under a dynamic situation beyond the normal state aboard the spacecraft. It should be noted that if the calculations are executed under the boundary condition for dopant at the interface (5), since in the course of time this leads to unreasonably high values of Cs (curve 1 in figure 9). The new boundary condition (6) takes properly into account the specific conditions of week convection in melt (curve 2 in figure 9). The results of calculations are shown in figures 10 and 11 for two typical semiconductor materials. These two materials had been chosen, because they clearly demonstrate the different duration of the transition period from the beginning of crystal growth to the steadystate condition, when C s becomes constant. For a melt of indium antimonide, equilibrium coefficient of dopant distribution k0 is close to 0.5, while the coefficient of dopant diffusion in a fluid is very small, and hence the Schmidt number is large. As a result, the regime with practically uniform dopant distribution occurs very quickly. For InSb(Te) crystal with the

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growth rate Recr = 0.2, as an example, this regime begins when τ > ~ 3.5 (solid line in figure 10) or when z > ~ 0.7R. The melt of germanium with gallium impurity has a combination of two parameters, k0 and Sc, for which the mentioned transition period lasts much longer than for InSb(Te) (solid line in figure 11). The length of Ge(Ga) crystal with the inhomogeneous dopant distribution along the z-axis will be in the range ~ (35-40) R under Recr = 0.2.

Cs 1 2 ,5

In S b (T e ) R e c r = 0 .2

2 ,0 2

1 ,5

1 ,0

0

1

2

3

4

τ

5

Figure 9. Dopant concentration in melt at the crystallization boundary as a function of time under diffusion transfer of dopant. Comparison of old (5) and new (6 boundary conditions. InSt(Te). Recr = 0.2.

1 ,6

2 3

Cs

1

1 ,4 4 1 ,2 1 ,0 0

1

2

3

τ

4

Figure 10. The average concentration of Te in InSb(Te) crystal at crystallization boundary as a function of time. (1) diffusion regime, Recr = 0.2; (2) small convection, Ra = 23 or Ra/Ω2 = 2.67·104, Recr = 0.2; (3) more strong convection, Ra = 230 or Ra/Ω2 = 1.67·105, Recr= 0.2; (4) diffusion regime, Recr = 0.1.

The physical properties (primarily D and k0) of Ge(Ga) and GaAs with different dopants are closely and , as calculations wewre shown, the results obtained for Ge(Ga) can be used for the GaAs-crystals. The data presented in figures 10 and 11 show that, unlike the radial dopant segregation, the longitudinal one is fairly insensitive to residual gravity accelerations and vibrations. It can be said with confidence that under the conditions existing aboard spacecraft, the longitudinal dopant transport occurs in in a purely diffusion regime. Dopant distribution

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along the length of growing crystal is defined by physical parameters of melt, first and foremost coefficient diffusion of dopant D and equilibrium coefficient of dopant distribution k0. 1 2 3

2,5

4

s

3,0

C 2,0

1,5

1,0 0

2

4

6

τ

8

10

Figure 11. The average concentration of Ga in the Ge(Ga) crystal at crystallization boundary as a function of time. (1) diffusion regime; (2) small gravitational convection, Ra = 9 or vibration, Ra/Ω2 = 1.476·103; (3) gravitational convection, Ra = 18; (4) gravitational convection, Ra = 29.9. Pr = 0.018, Sc = 10, k0 = 0.023, Recr = 0.2.

CTe, at/cm

3

2

1.445x10

18

1

2

18

10 9 8

0

2

4

6

z, mm

8

10

Figure 12. Comparison of the calculations with the experimental data for the InSb(Te)- crystal experiment in the Skylab-III flight [1]; (2) calculation for diffusion regime, Sc = 52.3, Pr = 0.23, k0 = 0.5, Recr is varying.

This conclusion is confirmed by comparison of calculations and experimental data (figures 12 and 1). The time-varying rate of crystallization obtained by processing of the experimental data for the Ge(Ga) crystal [2] was used in calculation of dopant distribution along the z-axis of crystal (figure 13). During the M562 experiment the crystal growth rate has not been measured [1]. As the furnace for crystal growth was the same and temperatures were closely the same in these two experiments, the dimensionless function Recr(τ) obtained

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under processing of the experiment data [2] was used in calculation of the InSb(Te) crystal growing.

1

19

10

8

CGa, at/cm

3

2

6 4

2

10

18

0

5

10

z , mm

15

20

Figure 13. Comparison of the calculations with the experimental data for the Ge(Ga)- crystal grown during the Apollo-Soyuz mission [2]; (2) calculation for diffusion regime, Sc = 10, Pr = 0.018, k0 = 0.023, Sc = 10, Recr is varying.

The large radial inhomogeneity of gallium distribution is noted in [2]. The position of maximum and minimum values of dopant concentration on the interface is changed during experiment, especially in the end. This fact indicates that the value of acceleration vector and its direction are varied with time. However, this factor cannot adequately taken into account at present since vibration accelerations are not measured during the flight of these spacecraft and dynamics of these orbital spacecraft is impossible to determine. Therefore, the diffusion transport of dopant, that under g = 0, is assumed in all calculations. Taking into consideration above-mentioned assumption, it may be inferred that a comparison of the experimental data and the calculation results shows rather good agreement. The following conclusions obtained under numerical investigation were confirmed by comparison with experiments: • • •

•

the longitudinal transfer of dopant is slightly dependent on residual accelerations and vibration aboard spacecraft; the dopant transfer from the interface can be calculated in the condition of diffusion mechanism; this transport is defined by physical properties of melt with solute dopant, primarily diffusion coefficient of dopant in melt and equilibrium coefficient of dopant distribution; large length of growing crystal with inhomogeneous dopant distribution will exists under crystallization of many semiconductor materials under the condition of full weightlessness.

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The Bridgman-Stokbarger method will be repeatedly considered in the subsequent text as an antithesis to the floating zone method under analysis of the realistic control actions on crystal growth process. A rotating magnetic field was used in an experiment with growing of the CaTe:Cl, Cd0.9Zn0.1Te:Cl and CdTe0.9Se0.1:Cl crystals in the ampoule with use of the traveling heater method on- board the spacecraft “Foton” [26]. It was assumed that the longitudinal dopant segregation would be reduced, as shown in figure 11. In the other limiting case, for example on the ground, when the melt is strongly mixed by gravitational convection, Cs quickly reached to the steady-state value which is close to 1. However, on the other hand, a secondary flow generated by a rotating magnetic field with Ha2Reω from 3·103 to 7·105 [26] will be reasonably strong according to the data given in [27]. Under microgravity conditions, the growth of convective motion in a melt leads to an increase of radial dopant segregation, as this is evident from the data presented in figure 6 (curve 6). The data for radial dopant segregation are absent in [26]. In any case it is symptomatic, that this method is no longer used in space experiments.

2. Floating Zone Method The floating zone method is the second method of crystal growth, which is used in experiments under microgravity conditions. A high level of dopant microsegregation is characteristic for crystals obtained by this method both on the ground and in space conditions. Methodical investigations of fluid flows and heat and mass transfer under different external actions were carried out in order to understand the influence aboard spacecraft conditions on the quality of growing crystals,. To simulation of crystal growth by the floating zone method, the certain bounder conditions have been changed. On the lateral free fluid surface, the heat flux distribution is given by the exponential curve with a maximum at the middle of fluid zone and the kinematic condition for tangential velocity is connected with thermocapillary effect

[

∂θ 2 = A exp − B (z − L / 2 ) ∂r ∂v ⎛ ∂θ ⎞ = − Ma Pr −1 ⎜ ⎟ ∂r ⎝ ∂z ⎠

]

(10)

(11)

where A and B are constant, Ma = - (∂σ/∂T)ΔTR/(ρνχ) is the Marangoni number, σ is the surface tension. The number of experiments performed by the floating zone method is little. Among them it should be noted the experiments [28, 29], in which the attempt to reduce dopant microsegregation is made. An oxide film was applied to the exterior surface of initial crystal in several cylindrical zones. A melt was in contact with solid wall in these zones. It was assumed that, by doing so, temperature oscillations and dopant microsegregation would be decreased. Positive effect of this method appears to be insignificant. This, apparently, can be explained by the inertia of thermocapillary flow and periodic change of boundary condition

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for fluid flow, when flow turns from a shear flow nearby free fluid surface, which is hydrodynamically stable, to a less stable flow along solid oxide film, and vice verse. Static axial magnetic field is used in the space experiment [30] for purpose of lowering microsegregation of dopant in the Ge(Ga)-crystal. Numerical investigation of Ge(Ga)-crystal growing have been carried out for two regimes of laminar thermocapillary convection with the Hartmann number from 0 to 103. Investigation reveals that radial dopant segregation grows drastically with the increase of magnetic intensity (figure 14). In this figure the points on the curves indicates the parameter ΔCs/ C s and the corresponding Hartmann number shown by number. The calculations for oscillatory thermocapillary convection (Ma/Pr = 3.27·105) demonstrate (figure 15) that in this case the function of ΔCs/ C s (Ha) is nonmonotone, although the negative tendency for increase of ΔCs/ C s under elevation of magnetic intensity remains. In the range of the Hartmann number from 0 to 35, in which the experiment [30] is carried out, relative radial segregation of germanium in gallium increases by a factor of ~ 5. This effect is evident when considering flow patterns presented in figures 16a-16c. With elevation of magnetic intensity an area of convection circulation is steadily shifted to a free fluid surface (r = 1). In this area a mixing of melt increases, whereas motion of fluid close to the axis (r = 0) is damped. It is interesting to note that relative radial dopant segregation increases in the similar area of magnetic intensity when this crystal is grown by the Bridgman-Stockbarger method under terrestrial conditions (figure 17).

1,0 3

3

Ha = 10

0,8

1

2

o o

ΔCs/Cs

Ha = 10

5 10

2

5 10

o

2

o

0,6

2o

2 10 o

3

1.5 10 o

10

2

2

0,4

o

o

10

2

50

0,2 o

o

50

o

0,0 1 10

10

2

10

3

4

Peumax10

10

5

10

6

Figure 14. The dependence of relative radial segregation of gallium in germanium on the intensity of convection and static magnetic field. Ge(Ga). (1) Ma/Pr = 2.3·104, (2) Ma/Pr = 1.3·105, (3) calculation without magnetic field. The point on the curves show the value of ΔCs/ C s under the corresponding value of Ha. Sc = 10, Pr = 0.018, k0 = 0.023, Sc = 10, Recr = 0.2.

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A.I. Feonychev

0,4

ΔCs/Cs

0,3 0,2 0,1 0,0

0

20

40

60

80

100

120

Ha Figure 15. The dependence of relative dopant segregation gallium in germanium on the intensity of static magnetic field for oscillatory thermocapillary convection. Ma/Pr = 3.27·105, Sc = 10, Pr = 0.018, k0 = 0.023, Sc = 10, Recr = 0.2.

z 3

3

z

3

z

- 224.8 - 92 205.9

- 87

2

- 142.2

2

2

2.6

1

1

- 2.6

162.2

1

- 201.2

87 92

232.2

0

0

1

(a)

r 0

0

1

(b)

r0

0

1

r

(c)

Figure 16. Flow patterns (stream function fields) for oscillatory thermocapillary flow. Momentary structure of flow without magnetic field (a); steady-state flow with magnetic field at Ha = 35 (b) and Ha = 60 (c). Ma = 5.88·103, Pr = 0.018.

Causes of Failures and the New Prospects in the Field of Space Material Sciences

0,6

Ha = 300 240

1

ΔCs/Cs

167

200

0,5 0,4

150 100

0,3

90

2

Ha = 0

0,2 75

0,1

55

10

1

10

2

Peumax

10

3

Figure 17. The dependence of relative dopant radial segregation for the Ge(Ga) crystal growing by the Bridgman-Stockbarger method under terrestrial conditions from the intensity of gravitational convection (the Peclet number) and static magnetic field (1). The curve 2 shows the case without magnetic field. The arrows indicate the point on the curve 1 with the corresponding value of the Hartman number. Ra = 5.6·107, Recr= 0.2.

Thus, the steep rise of radial dopant segregation by the increase in magnetic intensity is far superior to the decrease of dopant microsegregation in all real condition of crystal growth by the floating zone method with use of static magnetic field under microgravity conditions. The calculations were carried out in the operating range of the Marangoni number (1.7·104 – 2.7·105) in order to study the influence of vibrations aboard the Mir orbital station on thermocapillary convection and crystal growth. It is found that under Ma/Pr > 1.2·105, i.e. in the range of experiments, oscillation amplitudes of all parameters in a fluid are not in excess of 0.1 % and reduce exponentially with increase in Ma/Pr. Parameters oscillate with frequency equal to at the lowest frequency of all existing frequencies (0.8 Hz). The influence of time-varying residual gravitational accelerations during the flight of spacecraft on its orbit is negligible small, because the oscillation amplitude is a value of the second order of smallness with respect to itself acceleration. It is obvious that, when temperature oscillations in a fluid induced by hydrodynamic instability of the fluid flow or the external action (vibration and so on), the quantity of heat supplied to the interface from a liquid phase varies with time, and consequently crystal growth rate of is a time-dependent quantity. When the temperature at the interface is constant, it is suggested that a variation of fluid undercooling due to the required variations of crystallization rate is a negligibly small. If to write the momentary heat balance and heat balance averaged over period of oscillations and to subtract one from other, the following relationship for the time-dependent rate of crystallization results:

Re cr (τ ) =

1 Kq

⎡⎛ ∂θ ⎞ ⎛ ∂θ ⎞ ⎤ (0 ) ⎢⎜ ⎟ − ⎜⎜ ⎟⎟ ⎥ + Re cr , ⎢⎣⎝ ∂z ⎠ s ⎝ ∂z ⎠ s ⎥⎦

(12)

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A.I. Feonychev

where

Kq =

ρqcrυ (0 ) is dimensionless heat complex, Re cr is the dimensionless rate of κΔT

crystallization averaged over period of oscillations which is equal to initial value,

⎛ ∂θ ⎞ ⎜ ⎟ is momentary temperature gradient in a fluid at the interface, ⎝ ∂z ⎠ s

⎛ ∂θ ⎞ ⎜⎜ ⎟⎟ is the same but averaged over period of oscillations. ⎝ ∂z ⎠ s To illustrate this technique, relative radial dopant segregation as a function of time is shown in figure 18. The calculation have been performed for crystallization of Ge(Ga)-melt under hydrodynamic instability of a fluid flow (regime of oscillatory thermocapillary convection). The determined parameter (Kq = 1.08) is overestimated. The process of crystal growth has been calculated with the constant rate of crystallization ( Re cr

(0 )

= 0.2) up to τ = 0.065 (this

time is shown by an arrow) and thereafter the crystallization rate is calculated according to the relationship (12). For all semiconductor materials this effect is slight, because heat of phase transition is small by comparison with heat trough phase boundary. For example, for Ge(Ga) under ΔT = 10K parameter Kq is approximately equal to 10. However under other materials, this effect can be noticeable.

0,090

ΔCs/Cs

0,085 0,080 0,075 0,070 0,065 0,06

0,07

τ

0,08

Figure 18. The effect of time-dependent rate of crystallization in the case of oscillatory thermocapillary convection. The arrow shows at point in time, when this effect is taken into account. Ma/Pr = 3.75·105, Recr(0) = 0.2, Kq = 1.08.

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2.1. Use of Rotating Magnetic Field A rotating magnetic field is applied in industry during casting of steel to intensify the stirring of the liquid metals in order to obtain a better structure of the alloy and decrease the porosity and liquation inhomogeneity of ingots for a long time. The approximate mathematical models have been used under calculations of magnetohydrodynamic flows as applied to this technology. This is due to the fact that fluid flows are in the area of high-level turbulence and correct solution of this problem is a complicated task of computational mathematics. Investigations of a rotating magnetic field as a means to control dopant transfer under crystal growth started recently. In so doing, different methods of single-crystal growth were considered: the Czochralski [31] and Bridgman-Stockbarger [32] methods and the traveling heater [33, 34] and floating zone [35] methods. These processes were discovered under the action of the earth’s gravitational acceleration in [31, 35], of a reduced gravitational acceleration in [33, 34]. The first experiment with use of a rotating magnetic field under crystal growth under microgravity conditions [26] was a poor attempt. The statement of this experiment in itself exemplifies an insufficient understanding of processes of hydromechanics and dopant transfer. In the ground preparation of this space experiment [33, 34] attention focuses exceptionally on the interface form by rotating magnetic field. The dopant transfer was quite not considered. The effects of rotating magnetic field were considered to fit crystal growth by the floating zone method in terrestrial conditions [35]. The secondary flows generated by rotating magnetic field were considered more detail in [7] individually and in combination with gravitational and thermocapillary flows. It was shown as a rotating magnetic field affects on dopant transfer under crystal growth by different methods. These investigations were accomplished with publication [27] in which the results of study of crystal growth by different methods on the ground and under microgravity conditions were described. Some results of these studies are presented below. 2.1.1. Statement of Problem Fluid flow in a cylindrical volume of radius R and length L is considered. An external rotating magnetic field is generated by coils located around the fluid cylinder and the axes are directed along the radius. The coil windings are energized by alternating current so that the pole pairs lie on one diameter. When alternating current with a cyclic frequency ω is supplied, the magnetic field rotates about the cylinder axis z. The magnetic induction components along the radius r and in azimuth angle φ are written, as Br = B sin (ωt), Bφ = B cos (ωt).

(13)

It is assumed that the magnetic induction module is constant with r and the z axis. The action of an externally imposed magnetic field on the fluid is considered in the inductionless approximation that is justified, because the Magnetic Reynolds number Rem = μ0λRVmax << 1, where Vmax is the characteristic (maximum) velocity of fluid. This condition is fulfilled for all regimes under investigation. Melts of semiconductors are high conductance and the convective electric currents are neglected small as compared to conduction current; Joule heating is negligible small. It was assumed that an azimuth component of velocity exists but is independent from an azimuth angle φ. This assumption is based on the fact that the velocity of rotation of the fluid

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is much more than the secondary flow velocity in a rotating fluid. The skin-layers close to the solid boundaries are thick. This follows immediately from the small frequency of magnetic field rotation which is less than several kilohertz. The electrical conductivity of fluid and crystal are close. Therefore electrovortex flow arising from the conduction jump at the interface is negligibly small. Since the characteristics frequencies for transport processes of momentum, heat and mass (these frequencies are proportional to νR-2, χR-2 and DR-2, respectively) significantly lower than the rotating magnetic field frequency, the transport equations can be averaged for the rotation period of magnetic field. It should be also mentioned that the eigenfrequencies of convective cells in a fluid (see [20]) are much less than the rotation frequency of fluid around the z axis. These assumptions provided a basis to simplify the Navier-Stokes, Maxwell equations and the Ohm law. The investigation of crystal growth by the floating zone method with use of the rotating magnetic field was carried out in the pseudo three-dimensional statement. In the projection of Eq. (1) on the r axis must be added the term −

u descriptive of the centrifugal force in a r2

rotating fluid. Upon averaging over the rotation frequency the Lorentz force projections in Eq. (1) on the axes r, z and φ can be written in the form Fr = −

Ra ∂θ 1 − Ha 2 u Pr ∂r 2

Fz = − Ha

Fφ =

2

v 2

⎛ w ⎞ 1 ⎟ Ha 2 Reω ⎜⎜1 − 2 Reω ⎟⎠ ⎝

(14)

(15)

(16)

where w is the velocity component on the azimuth coordinate, Reω = ωR2/ν is the dimensionless angular rotation velocity of the magnetic field. Mathematical formulation of the problem had been verified in [27] by comparison of calculations and the experimental data. 2.1.2. Results of Calculations At the expense of centrifugal force, a fluid moves from the z axis to a free fluid surface in the middle section (z = L/2) and returns back to the z axis close to the solid ends (z = 0, L). As a result the secondary flow (figure 19a) is closely similar to thermocapillary flow (figure 19b). Characteristic velocities of the secondary flow (maximum values of velocity components) as a function of the complex parameter Ha2Reω are given in figure 20. The parameter Ha2Reω is characterized the rotating magnetic field intensity and, as calculations were shown, is suitable for generalization of obtained results. Note first that the secondary flow intensity is very high and this flow can only be neglected if Ha2Reω < ~ 102. Secondly, the transition from laminar flow to oscillatory one (beyond this boundary the corves in figure 20 are shown by dots) occurs when Ha2Reω ~ 9.5·103, in this case wmax = 3·103.

Causes of Failures and the New Prospects in the Field of Space Material Sciences

3

z

3

171

z

- 93.36 - 92.7

2

2

1

1 93.36

0

92.7

0

1

0

r

0

1

(a)

r

(b)

Remax

Figure 19. Flow patterns (stream function fields) under thermocapillary convection, Ma/Pr = 3·104 (a) and of the secondary flow generated by a rotating magnetic field, Ha2Reω= 4·103 (b).

10

3

10

2

10

1

10

0

10

-1

10

-2

wmax vmax

umax

1

10

10

2

10

3

4

10

5

2

10

Ha Reω Figure 20. Chracteristic velocities of the secondary flow as a function of the rotating magnetic field intensity. Ge. Ha = 1.

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A.I. Feonychev

2.1.2.1. Flow Instability and Dopant Microsegregation The turbulent and oscillatory regimes of convection are one of the main reasons for the appearance of dopant striations in the crystals (dopant microsegregation). Therefore, in choosing the parameters of rotating magnetic field, one should know the position of the transition boundary to oscillatory regime of convection to avoid this zone.

z

z

3

3 - 37.86 - 112.47 - 6.8

2

2 6.75

- 6.75

6.8

1

1 112.47

37.86

0 0.0

0.5

1.0

0 0.0

r

0.5

(a)

1.0

r

(b)

-20

u (0.5; 0.02)

-30 -40 -50 -60 -70 1,6

1,8

2,0

2,2

2,4

τ

2,6

2,8

(c) Figure 21. Flow patterns (stream function fields) at the instant in time τ = 2.76 (а) and 2.392 (b) and oscillations of radial velocity nearby the crystallization boundary (r = 0.5, z = 0.02) Ma/Pr = 2.7·104, Ha2Reω= 9.5·103.

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173

The action of rotating magnetic field on thermocapillary convection causes ambiguous consequences. If thermocapillary convection is located at the area of laminar flow, the action of rotating magnetic field placed below the boundary of stability (Ha2Reω < 9.5·103) often transforms flow to oscillatory regime. In figures 21a and 21b structures of oscillatory flow were shown in anti-phase. The initial thermocapillary flow is in a laminar regime (Ma/Pr = 2.7·104) and rotating magnetic field (Ha2Reω= 9.5·103) was placed up to the stability boundary. The oscillations of radial component of velocity close to crystallization boundary (z = 0.02) are shown in figure 21c. Quite a different result is obtained when a rotating magnetic field affects on oscillatory thermocapillary convection. In this case, the magnetic field can lead to increase of flow stability and to disappearance of oscillations of parameters in a fluid. This effect is shown in figures 22a and 22b. Thermocapillary convection at Ma/Pr = Ma/Pr = 1.35·105 is oscillatory (figure 22a). Oscillations cease and flow becomes stationary when a specially chosen magnetic field is applied (figure 22b).

3

z

3

z

- 237.9

2

66.5

- 65.8

1

2

- 115.66

1

115.66

233.4

0

0

1 (a)

r

0

0

1

r

(b)

Figure 22. Momentary flow patterns (stream function fields) of oscillatory thermocapillary flow (a) and steady-state flow under the action of the rotating magnetic field on therocapillary convection (b). Pr = 0.023. Ma/Pr = 1.35·105, Ha2Reω= 4·103.

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A.I. Feonychev

z

z

z

3

3

3

- 1.1

-3

- 6.84

9.1

- 0.12

2

2

1

1

2

- 10.33

10.33

1 0.12

- 9.1

6.84 3

0 0.0

0.5

1.0

0 0.0

r

(a)

1.1

0.5

0 0.0

r

1.0

(b)

0.5

1.0

r

(c)

Figure 23. The structure of secondary flow induced by the rotating magnetic field under different value of Ha2Reω : 1·102 (a), 9.4·103 (b) и 1·104 (c).

8000

(Reω)*

7000

6000

turbulent flow

5000

laminar flow

4000

3000

2000 0

500

1000

1500

2000

2500

Ma Figure 24. The boundary of thermocapillary convection stability under the action magnetic field. Ha = 4, Reω is varying.

of a rotating

The interaction of thermocapillary convection with the secondary flow generated by a rotating magnetic field is a complicated phenomenon. The complexity of flow pattern for a combined convection is related to different direction of displacement for convective cells of the secondary and thermocapillary flows with increase of parameters Ma/Pr and Ha2Reω. If

Causes of Failures and the New Prospects in the Field of Space Material Sciences

175

the former is shifted to solid boundaries (figures 19b, 23a, 23b and 23c) the second is held against free fluid surface. Figure 23b and figure 23c fall in the points placed at the left and at the right from the boundary between the laminar and oscillatory regimes (see figure 20). The value of

∂u in a fluid at the solid boundaries determines the transition to oscillatory regime ∂z

of convection in closed volumes. As a result of interaction of two flows, the boundary curve between laminar and oscillatory regimes on the plane of Ma/Pr – Ha2 Reω must be nonmonotone. In particular case of Ge-melt (Pr = 0.018) with the given Hartmann number (Ha = 4) this curve is shown in figure 24. The effect of oscillation suppression has been obtained with use of a specially chosen rotating magnetic field in solution of the two-dimensional problem. In a real threedimensional flow, oscillations arise earlier than in a two-dimensional axially symmetric flow because of the appearance of a small azimuth velocity. However, it is apparent that a large azimuth velocity w (see Ris. 20) should eliminate a small azimuth inhomogeneity of velocity distribution and suppress oscillations in fluid. Therein lay the effect of flow symmetrization in rotating fluid. This conclusion is confirmed by experiment in earth’s conditions [35]. For the Bridgman-Stockbarger method, the rotating magnetic field does not promote an increase in the stability of thermal gravitational convection in all the regimes considered. If flow is oscillatory, then oscillations continue at any intensity of rotating magnetic field. The oscillations in the unstable regime of gravitational convection are retained under the imposition of rotating magnetic field. The boundary of thermal gravitational convection stability by the action of rotating magnetic field is shown in figure 25 for the two direction of gravity vector. The zone of laminar flows lies below the curves given in these figure. The vertical dotted lines and numbers nearby them show the transition boundaries and the Grashof numbers such that oscillatory regime begins for gravitational convection without magnetic field.

5

2

(Ha Reω)*

10

10

9.5x10

2

8

1 1.2x10

6

4

10

1

10

2

10

3

4

10

10

5

6

10

7

Gr

10

10

8

Figure 25. The stability boundary of thermal gravitational convection with the rotating magnetic field. (1) vector g is directed to the crystallization boundary; (2) vector g is directed from away the crystallization boundary.

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A.I. Feonychev

z

z 3

3

- 6.12

2

2

1

1

- 31.4

- 32.6

0 0.0

0.5

1.0

0 0.0

r

(a)

0.5

1.0

r

(b)

θ (0.5; 0.02)

0,0156 0,0154 0,0152 0,0150 0,22

0,24

0,26

τ

0,28

(c) Figure 26. Flow patterns (stream function fields) of steady-state gravitational convection without a rotating magnetic field (a) and of unstable gravitational convection with this magnetic field (b) and temperature oscillation in a melt near the crystallization boundary (r = 0.5, z = 0.02) the second case. Gr = 1·107, Ha = 0.158, Reω = 1.6·106.

In contrast to thermocapillary convection the structure of thermal gravitational convection (figure 26a) is all out of proportion to the structure of the secondary flow generated by rotating magnetic field (figure 19b). Negative effect can be explained by this incompatibility. The instantaneous structure of thermal gravitational convection with rotating magnetic field and temperature oscillations in a fluid close to the crystallization boundary (r = 0.5, z = 0.02) are shown figure 26b.

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177

2.1.2.2. Dopant Macrosegregation As it was shown in part 2.1.2.1, the dopant striations caused by unstable convection can substantially eliminated by choosing the optimum parameters of a rotating magnetic field. The radial impurity distribution at the crystallization boundary is highly sensitive to the ratio of the radial and axial velocity components of fluid flow near this boundary. The first calculations of convective heat and mass transfer with use of a rotating magnetic field [7] showed that there is a possibility to control fluid flow and impurity distribution. A subsequent numerical study of this problem led to the desired result [27]. The dependence of radial inhomogeneity of the dopant distribution in melt at the crystallization boundary from the parameter Ha2Reω, which determines the rotating magnetic field intensity, is given in figure 27. The calculations are carried out for Ge(Ga)-melt under the three values of the Marangoni number including the oscillatory regime of thermocaoillary convection (Ma = 2430). The Hartmann number in these calculations is equal to 0.5, 1 and 2; the dimensionless cyclic velocity of magnetic filed is varied over a wide range. The addition calculations are made for Ma = 2430 These three variants of calculations are shown in figure 27.

ΔCs/Cs

0,3

3 2

0,2

1 0,1

0,0 10

1

10

2

10

3

10

4

2

10

5

10

6

Ha Re ω Figure 27. Effect of the rotating magnetic field on the relative radial magrosegregation for Ge(Ga)-crystal. (1) Ma = 2430, Ha = 0.5, 1 and 2, Reω is variable; (2) Ma = 680, Ha = 2; (3) Ma = 130, Ha = 2. Recr = 2.

ΔCs/Cs

0,085

0,084

0,083

0,082 0,29

0,30

0,31

0,32

τ

Figure 28. Oscillations of the radial dopant segregation under the action of a rotating magnetic field on thermocapillary convection during Ge(Ga) crystal growth by the floating zone method. Ma/Pr = 1.7·105, Recr = 0.2, Ha = 1, Reω = 6·104.

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A.I. Feonychev

ΔCs

The small oscillations (~ 1.5 %), just as it shown in figure 28, should be eliminated by an increase of rotating magnetic field induction (the Hartmann number) or by the application of additional axial static magnetic field.

2

0,20

3

1

0,15 0,10

4

0,05 0,00 10

2

3

10

10

4

2

Ha Reω

(a) 0,20 1 0,15

ΔCs

2 3

0,10

4

0,05

0,00 10

2

10

3

10

4

2

Ha Reω

10

5

(b) Figure 29. The dependence of the radial inhomogeneity of the Sb-dopant in the Si(Sb) me at the interface from the rotating magnetic field intensity. Vector g is directed to crystallization boundary (a) and from this boundary (b). Recr = 0.2, k0 = 0.023, Sc = 5, Ha = 0.158. (1) Gr = 1.6·105, (2) Gr = 4·105, (3) Gr = 1·106 and (4) Gr = 1·107 for (a); (1) Gr = 3·104, (2) Gr = 1.6·105, (3) Gr = 1·106, (4) Gr = 1·107 for (b).

In spite of negative result obtained in studies of gravitational convection stability, the investigation of crystal growth by the Bridgman-Stokbarger with this modification is continued, during which some intriguing results are received. The dependences of the radial inhomogeneity of the Si-dopant distribution in the Si-melt at the interface from the rotating magnetic field intensity are presented in figure 29 for two orientations with respect to the

Causes of Failures and the New Prospects in the Field of Space Material Sciences

179

gravity vector. Gravity vector acting along the z axis is directed to crystallization boundary (figure 29a), or this vector has the opposite direction (figure 29b). The vertical dotted lines (for the curves 1, 2, 3 and 4 in figure 29a and for curves 1, 2 and 3 in figure 29b) show the value of the parameter Ha2Reω corresponding to the transition from laminar flow to oscillatory regime of gravitational convection. The curve denoted by 4 in figure 29b refers to oscillatory gravitational convection. Reducing of radial dopant segregation is obtained only for the second case in the regime of laminar convection (figure 29b). The most intriguing result is observed when vector g directs to the crystallization boundary (figure 29a). The jump of dopant radial inhomogeneity at the crystallization boundary takes place in laminar regime of fluid flow; in so doing the maximum of inhomogeneity is consistently moved to the growing intensity of rotating magnetic field with an increase of gravitational convection (the Grashof number). But this phenomenon requires an explanation and further studies, although practical application of this result is not clear.

ΔCs

0,040

0,038

0,036

0,9

1,0

1,1

1,2

1,3

τ

1,4

1,5

Figure 30. Oscillations of radial inhomogeneity of dopant distribution in a melt at crystallization boundary during crystal growth by the Bridgman-Stockbarger method on the ground in pulsating regime of convection under the combination of gravitational convection and a rotating magnetic field. Gr = 1·107, Pr = 0.023, Recr = 0.2, Ha = 0.158, Reω = 8·105.

Specific regime of fluid flow with the effect of the cyclic rapid variations of parameters in a fluid and, as result, the large width of dopant striations is shown in figure 30. The pulsating regimes of oscillatory flow is observed in some cases under the combination of thermocapillary and gravitational convection with a rotating magnetic field (figures 31, 32). The regimes of this type are possible under crystal growth by the floating zone method on the ground (figure 31). In this case, the wide dopant striations can be formed since the pulsating frequency is commonly small (figure 31). The arrow shows the point in time after which the crystallization rate is varied and the calculation is carried out with the help of the relationship (14). As one might expect, the effect of the crystallization rate change of this oscillatory regime is negligible small. This calculation was performed to analyze the experiment [35].

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A.I. Feonychev

Wmax

680

660

640

620 0,10

0,15

0,20

0,25

0,30

0,35

τ

Cs max - Cs min

Figure 31. Pulsating oscillations of the azimuthal velocity maximum at the combination of thermocapillary and thermal gravitational convection with a rotating magnetic field. Gr = 3.4·104, Pr = 0.023, Ma = 1.7·103, Ha = 0.882, Reω = 4.4·104.

0,170

0,165

0,160

0,20

0,25

0,30

τ

0,35

Figure 32. Radial difference of dopant concentration in a melt at the crystallization boundary in pulsating regime of fluid flow under the combination of thermocapillary and thermal gravitational convection with a rotating magnetic field. The arrow shows the point in time when the timedependence crystallization rate is taken into account. Gr = 3.4·104, Pr = 0.023, Ma = 1.7·103, Ha = 0.882, Reω = 4.4·104.

2.2. The Incapsulation Method (Two-Layer Liquid Systems) 2.2.1. Brief Historical Survey Two-layer liquid systems have aroused considerable scientific interest in both the fluid dynamics area and crystal growth area due various reasons. Firstly, these systems facilitate the understanding of nonlinear problems in theoretical hydromechanics, like flow stability, mechanism of transition to oscillatory regime of convection, spectral characteristics of

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181

oscillatory (turbulent) convection, etc. Due to this reason, beginning with the initial works of Pearson [36], Sterling and Scriven [37], the majority of early publications on two-layer liquid systems were devoted to the stability problem [36, 39]. Thermocapillary convection in twolayer systems was being considered in [40-42]. The Marangoni-Benard instability in a symmetrical three layer system has been investigated in the flat cuvette during the IML-2 mission on the Space Shuttle [43]. Limited studies have been reported with regard to the interaction of thermocapillary and thermal gravitational convection [44]. The flows in twolayer systems, which are of theoretical and practical interest, are associated with large velocities and large gradients of parameters in fluid. Specific effect of viscous braking of fluid flow in two-layer liquid system with the exterior free surface has been detected in [45]. It was shown that under certain relation between the thicknesses and physical properties of layers, first and foremost density and viscosity, fluid in the lower layer begins to move along the division boundary in the opposite direction to thermocapillary effect on this boundary. The studies examing this effect have been conducted later [45]. The problem of thermocapillary convection stability in two-layer liquid systems was considered in [46]. The second aspect of the usefulness of two-layer liquid system investigation is connected with the practical use of such systems for growing of single crystals. Most part of these investigations was directed to use in ampoule methods of crystal growth. One distinct advantage of this new technology is the absence of physical contact of the growing crystal with the ampoule wall which leads to a reduction in the dislocations in the grown crystal. This idea forms the basis for the soviet patent [14]. Impurity transfer in the two-layer liquid systems with the exterior solid boundary was not considered. Only starting with work [45] the problem of dopant transfer under crystallization of one of liquid layers has become the subject of study. The use of magnetic field is a useful method for gaining flow stability during a crystal growth process. Preliminary investigations of this problem [47] have demonstrated this possibility not the effect of the static magnetic field on dopant distribution in a two-layer system was not quite clear. In succeeding work [48] flows in a two-layer liquid system with the exterior free surface are classified by a marker of the predominant thermocapillary effect on one of two boundaries. It is also shown that an optimum static magnetic field can be effective means for reduction of both dopant microsegragation and dopant macrosegregation in crystals growing by the floating zone method in weightlessness. A patent for this method has been obtained [49]. The results of an investigation into problem of crystal growth in twolayer liquid systems not only by the floating zone but also by any ampoule method of crystal growth are presented in [50].

2.2.2. Statement of Problem The physical setup used in the investigation is two-dimensional and consists of a rectangular area of length L in the x direction, and width d = d1 + d2, the vertical extent of the two layers in the y direction. Such a configuration is used to study the action of an external static magnetic field in two directions, x and y. The external heat flux on the upper boundary (y = d) is prescribed as ∂θ/∂x = A exp [- B (y – L/2)2] for the floating zone and horizontal melting zone methods of crystal growth. This equation is similar to (12). For the BridgmanStockbarger method the temperature distribution on the upper boundary is given as

z/d . L/d

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The temperatures at the ends (x = 0 and x = L) are fixed at T0 and the lower boundary (y = 0) is heat-insulated. All boundaries besides the crystallization front (x = L) are assumed to have zero dopant flux. At the crystallization front, there is rejection or absorption of the dopant according to thermodynamic equilibrium on the phase boundary (∂C/∂x)s = - ScRecr (1 – k0)Cs . This equation is similar to (5) and is applied in calculations because it is assumed that melt in lower layer is good mixing. Specific conditions for the component of velocity along the x-axis (u) on the lower boundary y = 0 allows the flexibility of considering several configurations: the floating zone, vertical melting zone melting and Bridgman-Stockbarger methods with a condition of symmetry at y = 0, ∂u/∂y = 0 (pseudocylinder geometry) and the horizontal zone melting method of crystallization (boat geometry), if the lower boundary is treated as a solid and hence u = 0 at y = 0. The action of an external imposed magnetic field on the fluids is considered in the inductionless approximation that is justified above. The Lorentz force is described in the form -

1 2 1 Ha u or - Ha2v with according to the direction of applied magnetic field (on the y– 2 2

axis or the x-axis, respectively). In two-layer liquid system additional dimensionless parameters are formed from the ration of the corresponding physical constant and thickness of the second layer liquid to the first some: Kρ = ρ2/ρ1, Kν = ν2/ν1, Kλ = λ2/λ1, Kε = ε2/ε1, Kd = d2/d1. The system of equations (1) – (4) is remained but is added multipliers when these equations are described for the second upper layer. Kν is inserted in the right part of the equations (3) and (4). (Kρ)-1is multiplied by ∇ p. The term with the Lorentz force is multiplied by Kν· Kε-1. The conditions for the fluid free surface and the two-component interface have the following dimensionless forms: ∂u2 / ∂y = - Ma2 Kν Pr2-1 (∂θ2 /∂x), v2 = 0; y = 1

(17)

∂u1 / ∂y = - Ma12 Pr1-1 (∂θ1 /∂x) + Kν Kρ (∂u2 / ∂y), ∂θ1 /∂y = Kλ(∂θ2 /∂y),

(18)

θ1 = θ2 , u1 = u2 , v1 = v2.

(19)

2.2.3. Results of Investigation The first calculations of crystal growth in two-layer liquid systems showed that the possibility exists of dopant macrosegregation lowering. This success was determined by appropriate choosing of the parameters Kν, Kρ and Kλ and the Marangoni numbers (Ma2 and Ma12) [45]. The multiparameter problem demands to systematize a host of variants on some sign. The predominance of thermocapillary effect at either of two phase boundaries or the equilibrium of these effects is convenient sign. Calculations have been carried out for a pseudocylinder and boat geometry with a fixed ratio of liquid layer thicknesses, d2/d1 = 0.25, and length L = 3d. The results for these

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geometric variants are little different; because of this, the data for boat geometry are not given. The parameters describing the dopant flux on the phase boundary are invariant with Recr = 0.2, Sc = 10, and k0 = 0.087. 2.2.3.1. Floating Zone Method The common parameters for all the calculations of the floating zone method are shown in table 1. Table 1. Numerical simulation parameters for floating zone Regime Ma12 Ma2 Pr1 Pr2 Kν Kλ Kρ A B

1 2.8·102 1·103 0.1 10 80 0.5 0.8 0.8 0.2

1a 5.2·102 3.74·103 0.1 10 80 0.5 0.8 15 10

2 2.16·103 7·102 0.01 0.021 2 1.5 0.4 0.8 0.2

3 7.6·103 2.35·102 0.02 1 30 0.5 0.3 0.8 0.2

Based on the magnitude of the thermocapillary effect on the boundaries, y = 0.8 and y = 1.0 and on the observed flow pattern in the layers, the system can be systematized into three distinct regimes: Regimes 1 and 1a: effect of the fluid free surface dominates (surface tension effects); Regime 2: effects on both boundaries (free surface and interface) are balanced; Regime 3: effect of the interface dominates (interfacial effects). The zone of external heat flux on the fluid free surface in the 1a regime is narrower than in the other regimes. Typical thermo-fluid characteristics of these regimes are shown in figures 33-41. All the data are presented in steady-state conditions. For the regime 1 the flow pattern is presented as a stream function plot, the temperature field by an isotherm plot and the evolution of the radial dopant concentration in the melt along the crystallization boundary is plotted as a ratio ∆Cs/ C s , where ∆Cs = Cs

max

– Cs

min,

and C s is the average concentration of dopant. Ad

addition to these plots, distributions of the temperature and tangential velocity at the free surface (y = 1.0) and at the interface (y = 0.8) are presented. In the first regime the fluid motion in the upper layer dominates and drives the flow in the melt layer (figures33a, 33f). The free surface velocity profile (figure 33e) shows the flow being drawn from the warmer center area to the cooler sides and the interface velocity profile (figure 33f) shows the flow being drawn in the opposite direction. The surface Marangoni number, Ma2 = 1·103 is little more than the interfacial Marangoni number, Ma2 = 2.8·102, but the viscosity of upper layer is much more (Kν = 80). Hence is follows that dynamic effect of upper layer to the melt layer is strong. The relative dopant segregation approaches a value of ~ 0.042 after an initial transient period.

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A.I. Feonychev

y 1.0

- 13.7

13.7

0.8 0.6 0.4

- 20.11

20.11

0.2 0.0

0

1

2

3

2

3

x

(a)

y 1.0 0.8 0.6 0.4 0.2 0

1

(b)

θ (x, 1.0)

0,6 0,5 0,4 0,3 0,2 0,1 0,0 0,0

0,5

1,0

1,5

2,0

2,5

x

3,0

(c) 0,4

θ (x, 0.8)

0.0

0,3 0,2 0,1 0,0 0,0

0,5

1,0

1,5

2,0

x

2,5

(d) Figure 33. Continued on next page.

3,0

x

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300 200

u(x, 1.0)

100 0 -100 -200 -300

0,0

0,5

1,0

1,5

2,0

1,5

2,0

2,5

3,0

x

(e) 200

u (x, 0.8)

100

0

-100

-200

0,0

0,5

1,0

x

2,5

3,0

(f)

ΔCs/Cs

0,05

0,04

0,03

0,02

0,06

0,08

0,10

0,12

τ

0,14

(g) Figure 33. Regime 1(surface tension dominates at top free surface. Ha = 0 (no magnetic field). (a) stream function distribution; (b) isotherm distribution; (c) temperature distribution at the free surface (y = 1.0); (d) temperature distribution at the interface; (e) tangential velocity profile on the free surface(y = 1.0); (f) tangential velocity profile on the interface (y = 0.8); (g) time-dependence of dopant in fluid at the crystallization boundary (x = 3.0; 0≤ y ≤ d1).

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A.I. Feonychev

y 1.0

- 15.1

15.1

0.8 0.6

- 14.04

0.4

14.04

0.2 0.0

0

1

2

x

3

Figure 34. Regime 1(surface tension dominates at top free surface. Stream function distribution. Magnetic field is applied in lower layer along the x-axis, Ha1x = 11.

y 1.0

- 2.82

2.82

0.8 0.6

- 1.21

0.4

1.21

0.2 0.0

0

1

2

x

3

Figure 35. Regime 1(surface tension dominates at top free surface. Stream function distribution. Magnetic field is applied to the upper layer along the y-axis, Ha2y = 500.

The effect of the static magnetic field on the flow pattern is shown in figure 34 for the case when magnetic field is applied also to the lower layer in the x direction. Under change of the Hartmann number from 0 to 11 fluid flow along the crystallization front transforms, so that dopant distribution at the phase boundary (x = 3.0) becomes more uniform (see figure 42). If very strong magnetic field is applied to the upper layer along the y axis the structure of fluid flow transforms greatly (figure 35), resulting in an ambiguous affect of dopant distribution in growing crystal (figure 43).

y 1.0

40.92

- 40.92

0.8 0.6 0.4

87.13

- 87.13

0.2 0.0

0

1

2

3

x

Figure 36. Regime 1a (surface tension dominates at top free surface). Stream function distribution; Ha = 0 (no magnetic field).

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y 1.0 0.8 0.6 0.4 0.2 0.0

0

1

2

3

x

Figure 37. Regime 1a (surface tension dominates at top free surface). Stream function distribution. Magnetic field is applied to the lower layer along the x-axis, Ha1x = 20.

In the regime 1a the surface tension gradient at the free surface and the flow intensity are higher (figure 36) than in the regime 1 (figure 33a); this being so, the magnetic field induction in the regime 1a must be twice large to obtain the effect is similar to regime 1 (see figure 42).

y 1.0

58.2 - 46.4

- 86.3

0.8

- 58.2 46.4

86.3

0.6 0.4

- 360.68

360.68

0.2 0.0

0

1

2

3

x

(a) 0,0330

Relative radial nonhomogeneity

0,0328 0,0326 0,0324 0,0322 0,0320 0,0318 0,0316 1,10

1,15

1,20

Time, τ

(b) Figure 38. Regime 2 (surface and interfacial tension forces are balanced). Ha = 0. Momentary stream function distribution (a). Radial dopant segregation is a function of time (b).

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A.I. Feonychev

y 89.06

1.0 - 110.7

- 45.22

45.22 - 78.35

0.8

110.7

78.35

0.6

- 95.64

95.64

0.4

- 89.06

0.2 0.0

0

1

2

3

x

(a)

u (2.25, y)

2000 1500 1000 500 0 -500 -1000 0,0

0,2

0,4

0,6

0,8

y

1,0

(b) Figure 39. Regime 2 (surface and interfacial tension forces are balanced) Magnetic field is applied to the lower layer along the x-axis. (a) Momentary Stream function distribution; Ha1x = 50. (b) Tangential velocity distribution in the x = 2.25 section; Ha1x = 40.

In the second regime (figures 38 and 39) there is a balance between the surface and the interfacial tension forces. The tangential velocity gradients in the upper layer are large (figure 39b) and fluid flows in the both layers are unsteadiness that is clearly reflected in the dopant inhomogeneity evolution (figure 38b). When the magnetic field is applied to the lower layer in the x direction the convective circulation contours are displaced to the interface boundary (y = 0.8) and the tangential velocity at this boundary increases (figure 39b). If the magnetic field is applied in the y direction (figure 40), these contours are displaced to solid boundaries (x = 0 and x = 3.0). With a rise of the magnetic field induction the oscillation amplitudes of parameters in fluid are decreased but the oscillation spectra are enriched with high frequencies. Lastly in the third regime where interfacial tension dominates another motion source, the flow direction in the liquid layers is exactly opposite of that in the first case (figure 41). The liquid in the upper layer moves along the free liquid surface in opposition to surface tension gradient, from the cold areas (x = 0, x = 3.0) to the area with maximum temperature (x = 1.5).

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y - 86.1

86.1

1.0

91.2

- 91.2

0.8 0.6

64.05

- 64.05

0.4 0.2 0.0

0

1

2

3

x

Figure 40. Regime 2 (surface and interfacial tension forces are balanced). Momentary stream function distribution. Magnetic field is applied to the lower layer along the y-axis; Ha1y = 80.

y 1.0

3.6

- 56.2

- 3.6

56.2

0.8 0.6 0.4

131.6

- 131.6

0.2 0.0

0

1

2

3

x

Figure 41. Regime 3 (interfacial tension dominates at interface). Stream function distribution; Ha = 0 (no magnetic field).

0,4 2

3

0,3

ΔCs /Cs

1 1a

0,2 0,1 0,0

0

20

40

60

80

100

Ha1x

Figure 42. Relative radial dopant segregation is a function of Ha1x. Magnetic field is applied to the lower layer on the x-axis. 1 is regime 1; 1a is regime 1a; 2 is regime 2; 3 is regime 3.

The effect of the magnetic field on the dopant inhomogeneity for the four regimes discussed early is presented in figures 42-46 for two different orientation of the magnetic field

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A.I. Feonychev

applied individually to the lower and upper layer. Only for the 1 and 1a regimes there is a reduction in the dopant inhomogeneity on the crystallizing front. The area of the Hartmann number, more suited for using, is located from 0 to 11 when the magnetic field applied to the lower layer in the x direction (figure 42) and from 0 to 7 when the magnetic field in the lower layer acts to transverse direction (figure 43). It should be noted that the 1 regime provides an opportunity for reduction of dopant segregation without the magnetic field (figures 43 and 44 under Ha = 0). If the magnetic field is applied to the upper layer the most positive effect is observed when the magnetic field is directed on the y-axis and the Hartmann number reaches ~ 200 in the 1 regime and ~ 600 in the 1a regime (figure 46). The second and third regimes are of no interest for practical using. The patent [49] is based on using of the first regime with and without magnetic field.

0,20

ΔCs /Cs

3

0,15 2

0,10 0,05 1 0,00

0

20

40

60

80

100

120

Ha1y

Figure 43. Relative radial dopant segregation is a function of Ha1y. Magnetic field is applied to the lower layer on the y-axis. 1 is regime 1; 2 is regime 2; 3 is regime 3.

ΔCs /Cs

0,14

3

0,12 0,10 2

0,08 0,06

1 0,04

0

20

40

60

80

100

120

Ha 2x

Figure 44. Relative radial dopant segregation is a function of Ha2x. Magnetic field is applied to the upper layer on the x-axis. 1 is regime 1; 2 is regime 2; 3 is regime 3.

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3

ΔCs /Cs

0 ,1 5 2

0 ,1 0

0 ,0 5

0 ,0 0

1

0

20

40

60

80

100

120

H a 2y

Figure 45. Relative radial dopant segregation is a function of Ha2y. Magnetic field is applied to the upper layer on the y-axis. 1 is regime 1; 2 is regime 2; 3 is regime 3.

0,10

ΔCs /Cs

0,08 0,06

1

0,04 1a 0,02 0

200

400

600

Ha2y

800

Figure 46. Relative radial dopant segregation is a function of Ha2y. Magnetic field is applied to the upper layer on the y-axis. 1 is regime 1; 1a is regime 1a.

2.2.3.2. Ampoule Methods For the most part the investigations in the field of two-layer systems have been carried out for the configurations appropriated to ampoule methods of crystal growth. Until recently, there were attempts to realize crystal growth by ampoule method in space experiment. For example, the GaAs and GaAs (Te) crystals was obtained with the help of the fluid encapsulation melting zone technique during the experiment on the Space Shuttle [51]. These crystals are lower dislocation densities than those grown on ground. Reduction of dopant striations (dopant microsegregation) was also noted in all crystals [51]. However, the data about macrosegregation of Te in crystal are absent. The simulation of crystal growth by the encapsulated Bridgeman-Stockbarger and encapsulated melting zone methods with and without static magnetic field have been fulfilled to verify possibilities of these modified methods. The parameters for numerical simulation are shown in tables 2 and 3. ]. The parameters describing the dopant flux on the phase boundary are fixed with Recr = 0.2, Sc = 10, and k0 = 0.087.The magnetic field was applied only to the lower crystallizing liquid layer in the longitudinal and transversal directions. The flow

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A.I. Feonychev

patterns are shown in figures 47 and 50. The radial inhomogeneity of dopant in melt at the crystallization boundary (x = 3.0) in dependence of the magnetic field induction (the Hartmann number) is presented in figures 48 and 49 for the Bridgeman-Stockbarger method and in figures 51 and 52 for the ampoule melting zone method. Table 2. Numerical simulation parameters for the Bridgeman-Stockbarger method Regime Ma12 Pr1 Pr2 Kρ Kλ Kν

1 2·104 0.02 1.0 0.3 0.5 30

2 1·104 0.01 0.02 0.4 1.5 2

3 1·103 0.01 0.02 0.4 1.5 2

Y 1.0

0.5

- 883.3

0.0 0.0

1.0

3.0 X

2.0

ΔCs /Cs

Figure 47. Flow pattern (stream function distribution) for the two-layer Bridgman Stockbarger method. Regime 2. Ha = 0 (no magnetic field).

0,30

3

0,25

1

0,20

2

0,15 0,10 0,05 0

20

40

60

80

100

Ha1x

Figure 48. Relative radial inhomogeneity of dopant distribution in a melt at the crystallization boundary as a function of magnetic field applied to the lower layer along the x-axis. The two-layer Bridgman-Stockbarger method. (1) regime 1; (2) regime 2; 3 is regime 3.

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ΔCs /Cs

3 0,3 1 0,2

2

0,1

0

20

40

60

80

Ha1y

100

Figure 49. Relative radial inhomogeneity of dopant distribution in a melt at the crystallization boundary as a function of magnetic field applied to the lower layer along the y-axis. The two-layer Bridgman-Stockbarger method. (1) regime 1; (2) regime 2; 3 is regime 3.

Table 3. Numerical simulation parameters for the melting zone method Regime Ma12 Pr1 Pr2 Kρ Kλ Kν

1 2·103 0.02 1.0 0.3 0.5 30

2 3.2·103 0.01 0.02 0.4 1.5 2

Y 1.0 0.8 0.6 0.4

- 432

432

0.2 0.0 0.0

0.5

1.0

1.5

2.0 X

Figure 50. Stream function distribution for the two-layer melting zone method. Regime 2. Ha = 0 (no magnetic field).

194

A.I. Feonychev

2

ΔCs /Cs

0,50 1

0,25

0,00

0

20

40

60

80

100

Ha1x

Figure 51. Relative radial inhomogeneity of dopant distribution in a melt at the crystallization boundary as a function of magnetic field applied to the lower layer along the x-axis. The two-layer melting zone method. (1) regime 1; (2) regime 2.

0,3

ΔCs /Cs

1

0,2 2

0,1

0

20

40

60

Ha1y

80

100

Figure 52. Relative radial inhomogeneity of dopant distribution in a melt at the crystallization boundary as a function of magnetic field applied to the lower layer along the y-axis. The two-layer melting zone method. (1) regime 1; (2) regime 2.

So, in all the cases of incapsulation ampoule methods of crystal growth reduction of radial dopant segregation is not obtained. This result was expected as the additional (upper) layer decreases surface tension gradient at the interface boundary (y = 0.8) and so diminished flow velocity along the crystallization boundary. In response to the decrease of velocity the radial dopant segregation is raised (motion on the right branch of a curve in figure 53). These curves were obtained for the usual one-layer methods of crystal growth but the mechanism of dopant transfer in crystallizing fluid of two-layer systems is well described by these relationships.

Causes of Failures and the New Prospects in the Field of Space Material Sciences

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Figure 53. Relative radial inhomogeneity of dopant distribution in a melt at the crystallization boundary as a function of dimensionless velocity along crystallization boundary for different crystal growth methods: ampoule zone melting method (1), the Bridgman-Stockbarger method (2) and floating zone method (3).

2.3. Floating Zone Method with Surface Standing Waves The behavior of the free surface of a fluid column by axial vibration under microgravity conditions was repeatedly studied both theoretically [52-54] and experimentally [55, 56]. This problem was posed in connection with works on single crystal growth by the floating zone method onboard spacecraft. The analytical models of vibration-induced free surface oscillations [52-54] assumed that the standing wave on fluid free surface consists of one period and oscillates with the applied vibration frequency. In the models [52, 53] it assumed also that a resonance increase in the standing-wave amplitude happens when vibration frequency reaches the eigenfrequency of the fluid column. The experiments on the Cleveland‘s “drop tower” showed that expected resonance increase of the oscillation amplitude of free surface under the action of vibration directed along this boundary was not observed [56]. It should be noted that the experiments on sounding rockets (TEXUS project) and orbital spacecraft (Projects SL-1, SL-D1 and SL-D2) were performed in a narrow range of influence vibration parameters. Because of this, it is not practical to verify reliably analytical models [52-54]. In the framework of agreement between NASA and the Russian Space Agency the numerical investigation of fluid column deformation by the axial vibration had been carried out in 1995-1997 (project TM-7) [57]. The calculations were made under the action one frequency from a wide range of frequencies, under the simultaneous action of two different frequencies, and under the impact action on the fluid column. It was found that both one and several lengths of the standing wave could be placed on the free surface. The one-period standing wave placed on the free fluid surface L = R is shown in figure 54. The three-period standing wave on L = 2R is presented in figure 55.

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A.I. Feonychev

ξ

0,04 0,02 0,00 -0,02 -0,04 0,0

0,2

0,4

0,6

0,8

z

1,0

Figure 54. Surface standing wave with n = 1. L/R = 1.0, Kσ = 3.26·103, Reω = 10, Ωv = 10. 0,15

ξ

0,10 0,05 0,00 -0,05 -0,10 0,0

0,5

1,0

1,5

z

2,0

Figure 55. Surface standing wave with n = 3. L/R = 2.0, Kσ = 3.26·103, Reω = 30, Ωv = 30.

Figure 56. Oscillation of point at free surface z = L/4. L/R = 1.0, Kσ = 3.2·103, Ωv = 200, Ω = 50, Reω = 3.12, n = 1.

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Oscillations of the standing wave could be performed at the frequency equal to the applied vibration or smaller than the latter. In the last case (Fig. 56) the oscillation frequency of the standing wave is equal to the applied vibration frequency divided by natural number (Ω = Ωv/4) . Under the action of two frequencies, the low frequency was determined and the high frequency distorted the wave form, creating a “ripple”. Analytical relations for standing waves in zero gravity were obtained as the result of calculation data generalization [58]. The numerical investigations of interaction of the standing waves with convection flow showed [58, 59] that such waves may serve a means of controlling the processes of heat and mass transfer. The study of thermocapillary convection stability under the action of the standing surface wave of different configuration was carried out in [60]. The data obtained in this work provided the basis on which the use of the standing waves in technology of crystal growth was possible. In [61] it shown that considerable reduction the dopant segregation in crystals grown by the floating zone method can be possible under optimum parameters of the standing wave. Additional positive effect is achieved by application of static magnetic field.

2.3.1. Analytical Model of Surface Standing Waves The analytical model of surface standing waves has been constructed for reciprocating motion of the fluid column by axial vibration. Under motion of fluid column as a unit the deformation of fluid free surface is determined by the action of surface tension and inertia. The latter cannot be expressed in the terms of physical parameters. The numerical solution of this problem had been obtained with the help of the volume of fluid method (VOF- method) [62]. Waves observed in this case are rated in the class of capillary waves. The following conditions are used in creation of this model: •

the vibration rate must be no more than the phase velocity of standing wave. Phase velocity is inversely proportional to the length of wave, therefore if this condition is violated the number of standing wave periods on the length of free surface is increased as long as the following condition will be fulfilled Vσ ≥ Rev,

(20)

where Vσ =

2.732 (Kσ nR / L )1/ 2 2π

is dimensionless phase velocity of capillary wave, Rev = aωR/ν is dimensionless velocity of vibration, Kσ = σRρ-1ν -2 is dimensionless capillary constant, n is the number of the standing wave periods on the length of fluid column, a is amplitude of vibration, ω is cyclic frequency of vibration; •

the fluid column oscillates with the frequency which is equal to vibration frequency or less this frequency. This condition has a connection with the concept of the eigenfrequency of fluid column.

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The dimensionless eigenfrequency for fluid column depends from the length of fluid column and capillary constant Kσ Ωe = 2.732 Kσ1/2 (L/Rn)-3/2,

(21)

where Ωe = ωe R2/ν is the (dimensionless) cyclic eigenfrequency of surface standing wave. The relation (21) is obtained by the numerical experiment with the impact action on fluid column. After the initial period with random nonharmonic oscillations the free surface of fluid column is harmonically oscillated at the eigenfrequency with damping. The dependence of the capillary standing wave eigenfrequency from the length of fluid column is shown in figure 57 for four numbers n (n = 1, 2, 3 and 4). If vibration frequency Ωv (where Ωv = fvR2/ν) is more than the eigenfrequency Ωe the standing wave oscillates with frequency is equal to vibration frequency lowered by the factor of m (m is natural number). The number m is so chosen that the condition (22) asserts under a minimum value of m (m = 1, 2, 3,…) Ωv /m ≤ Ωe

(22)

For example, in the case of figure 55 Ωv = 200 and Ωe = 54.64, and hence minimum m is equal to 4. The standing wave is oscillated in calculation with the frequency Ω = 50. The result is obeyed the condition (22). The data of the numerical calculations showed (figures 54 and 55) that the standing wave is well described by a sinusoid. Let us write the equation for the standing wave in dimensionless form at any number of wave periods on the length L: ξ/R = sin(2πzn/L)cos Ωτ,

(23)

where ξ is deviation of the free surface from the unperturbed state, Ω = ωR2/ν is the dimensionless circular frequency of standing wave oscillation, ω is circular frequency of standing wave oscillation. Oscillations in the standing wave give a demonstrative example of the kinetic-to-potential conversion. The energy for sustaining a continuous oscillation process is supplied due to the applied vibration. Thus, the establishment of the relation between these energy-balance components will permit determination of the standing wave amplitude δ. The equation linking the standing-wave amplitude with the vibration amplitude has been obtained in [59] with use of the McRobert E-function. On the substitution of the McRobert E-function by the special function of the second kind elliptic integral and approximate estimates of disturbed surface layer of fluid, the sought-for equation is written as E [(1 – (4nδ)2)1/2 ; π/2] = 1 + 5(a/R) Rev 3/2 Kσ-1,

(24)

where E [ς; π/2] is the second kind elliptic integral, δ = b/R is dimensionless amplitude of standing wave, b is amplitude of standing wave. The obtained relation describes effect of proportional reduction of the standing-wave amplitude with the change of standing wave periods (δn+1 /δn = n/(n+1)). The relation (24) has restrictions on the side of small and high vibration frequencies. The dimensionless thickness of disturbed surface layer is proportional to the dimensionless vibration frequency Ωv-1/2.

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199

Hence it follows that the vibration frequencies must be more than the frequency with which the disturbing layer surpasses a radius of fluid column. On the side of large vibration frequencies this approximation is restricted to the use of the incompressible fluid model. Real vibrations existing onboard spacecraft or vibrations that can be created artificially for control of fluid flow are within these limits.

20

Ωe Kσ

-0.5

1

2

3

4

15 10 5 0 0

1

2

3

L/R

4

Figure 57. Eigenfrequency of the capillary standing waves as a function of the length of fluid column and the number of standing-wave periods on this length. (1) n = 1; (2) n = 2; (3) n = 3; (4) n = 4.

The dependence of the ratio of the standing-wave and vibration amplitudes on the vibration frequency is shown in figure 58 for two melts, Si and GaAs. Capillary constant of the GaAs-melt is less than Si by a factor of ~ 6. Because of this, the standing-wave amplitude for GaAs-melt is considerably more. It can obtain a unified curve if the ratio of amplitudes to multiply on Kσ0.5 (figure 59).

b/a

0,14 0,12 1

0,10 0,08

2

0,06 0,04 0,02 0,00

0

50

100

150

200

250

F

300

Figure 58. The ratio of the standing-wave and vibration amplitudes as a function of the dimensionless vibration frequency. L/R = 2, n = 1. (1) GaAs, Kσ = 3.9·106 ; (2) Si, Kσ = 2.3·107.

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A.I. Feonychev

250

Kσ

0.5

b/a

300

200 150 100 50 0 0

50

100

150

200

250

Fv 300

Figure 59. The universal ratio of the standing-wave and vibration amplitudes as a function of the dimensionless vibration frequency. L/R = 2, n = 1.

v, cm/s

The standing waves discussed above can be named as inertial-capillary waves. The Faraday’s waves (inertial-gravitational waves) are analogies of these waves in terrestrial conditions. If one of solid ends of fluid column (growing or melting crystal) vibrates the usual surface waves takes place. In this case the oscillation frequency of standing waves is given by vibrating crystal. The three possible situations for waves of this type are shown in figure 60 for Si-melt in terrestrial conditions. The waves with a length less than 1-2 mm could be considered as capillary waves even in terrestrial conditions. The dependence of the capillary standing wave amplitude on the vibration amplitude for five configuration standing wave is shown in figure 61. This function was obtained on the assumption that a mono-molecular surface film is not stretched by vibration. For these waves the relation between wave amplitudes under change of the number of wave periods on fluid column length is satisfied: δn+1/ δn = n/n+1.

140 120 100 80 60 3

40 20 0

2 0

1 10

20

30

40

50

λ , cm

Figure 60. Velocity of surface waves as a function of wave length. Si-melt. The curve 1 is capillary waves; the curve 2 is gravitational waves; the curve 3 is combined waves.

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δ

201

1 2

0,2

3 4 5

0,1

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

a/R

Figure 61. Amplitude of usual capillary standing wave as a function of vibration amplitude for 5 case of configurations of wave; Si-melt. The curves 1, 2, 3, 4, and 5 correspond to n = 1, 2, 3, 4, and 5.

2.3.2. Stability of Thermocapillary Convection under the Action of Surface Standing Waves The study of the thermocapillary convection stability in a fluid with an infinitely large free fluid surface by the linear method with use of infinitely small disturbances was begun almost 50 years ago. The case when the deformation of the free surface happens with the finite value of surface tension (capillary constant Kσ is finite) was estimated in [64]. In this and following works the deformation of free fluid surface was considered as the result of the velocity head action of moving fluid on free surface. Currently the thermocapillary convection stability came be regarded under the condition when the fluid free surface oscillates by the vibration action [58, 60, 65, 66]. It was shown [58] that the surface standing waves generated by vibration had much stronger effect on the fluid flow stability than convective fluxes in fluid. A flow of fluid in cylinder of length L and radius R with deforming free surface is considered. It is suggested that a standing wave with an amplitude b and a number of periods n acts on the free surface of this cylinder. The system of equations and the boundary conditions corresponding to the floating crystal growth method are described above. The changes in boundary conditions connected with oscillating free surface is as follows: •

on the fluid free surface oscillating by the standing-wave law r = f(z) = b sin (2πnz/L) cos Ωτ

∂θ = − A(cos α ) exp − B( z − L / 2 R ) 2 , u = b sin (2πnz/L) sin Ωτ, ∂r

[

]

(25)

where α is the angle between the normals to the points on the disturbed free surface and the surface r = R, u is the component of velocity on the r axis. The difference grid along the radius is constructed with respect to the oscillation amplitude (r + b). In the process of calculations, the boundary points on the oscillating free

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A.I. Feonychev

surface are determined by a definite law of motion. It is assumed that a point lies on the fluid boundary if its r coordinate exceeds half the grid pitch. The main calculations are carried out with L/R = 2, A = 0.8, B = 0.3 for fluids with Pr < 1 (0.018 and 0.023), characteristic of the majority of semiconductor materials, and the other calculations are carried out for Pr = 1. As previously, parameter Ma/Pr characterizes the intensity and structure of the initial thermocapillary flow. The ranges of change in the problem parameters being investigated were as follows: δ = 0.002-0.1, Ω = 4.4·102 - 7.8·104, n = 1-15, Ma/Pr = 3.2·102 – 8.6·104. The procedure of investigating the stability of a fluid flow is the following. At first, calculations are performed for the process of formation of a stationary flow without the surface standing wave action, and then the action of a standing wave with oscillations of definite amplitude and frequency are taken into account. The calculations have been carried out as long as a new stationary or oscillation regime with steady-state values of the amplitude and frequency of oscillations of the fluid parameters is established. A new regime called for from one hundred to several thousands of the oscillation periods for the standing wave is established. The number of these periods is increased with increase in Ω.

2.3.2.1. Results of Calculations The calculations showed that surface standing wave have exerted a strong influence on thermocapillary flow. This is due to the fact that the standing wave acts in the area where thermocapillary effect manifests itself at the most. In figure 62 it is shown as the surface standing wave has the influence on the initial laminar thermocapillary flow. The flow patterns are shown for the four phase moments. z

z

2.0

2.0

z 2.0 - 0.53

- 1.59

- 1.26

1.5

1.5

1.0

1.5

1.0

- 0.91

1.0

0.89

0.5

0.5

0.5 1.94

1.34

0.55

0.0 0.0

0.5

(a)

1.0

r

0.0 0.0

0.5

1.0

r

(b) Figure 62. Continued on next page.

0.0 0.0

0.5

(c)

1.0

r

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203

z 2.0

- 1.95

1.5

1.0

0.5 1.65

0.0 0.0

0.5

1.0

r

(d) Figure 62. Momentary flow patterns (the fields of stream function) of thermocapillary flow during the one cycle of standing wave oscillations. (a) φ = 0, (b) φ = π/2, φ = π, and (d) φ = 3π/2. Ma = 3.2·102, Pr = 1, Ω = 30, δ = 0.1 n = 3.

θ (0.5; 0.02)

In connection with this example, the effect of antisymmetry in flow structure at points in the antiphase times (location of maxima of stream function) and the nonlinear interaction of thermocapillary flow and vibration (the values of above-mentioned maxima are unequal). The oscillations of temperature in fluid at the point nearby the crystallization boundary (r = 0.5; z = 0.02) on the action of the three-period surface standing wave on laminar thermocapillary convection are shown in Fig. 63. Besides the standing wave frequency (Ω = 30) the twice frequency (2 Ω) is present in oscillation temperature spectrum.

0,017 0,016 0,015 0,014 1,1

1,2

1,3

1,4

τ 1,5

Figure 63. Temperature oscillations by surface standing wave on laminar thermocapillary convection. Ma/Pr = 3·102, Pr = 1, n = 3, Ω = 30, δ = 0.1.

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A.I. Feonychev

Such surface standing waves are of little interest for the flow stability problem and for optimization of crystal growth process. Comparative small amplitudes of surface standing waves are of prime interest for these problems. Clearly the standing-wave amplitude is most important parameter for studied problem. However two conditions play a part in investigation of standing wave effects, namely, mutual symmetry of thermocapillary flow and surface wave and, secondly, the standing wave frequency in reference to the eigenfrequency of a convective cell [20]. Thermocapillary flow for floating zone configuration is symmetric about the plane passed through the middle of fluid column length (z = L/2). The surface standing waves with different number of periods give diverse conditions of mutual symmetry, from full symmetry, when n = 2, to full antisymmetry (n = 1). In so doing, thermocapillary flow after the action the symmetric standing wave (n = 2) does not need to be symmetric. This is shown in figures 64a, 64b and 64c. The flows in figures 64b and 64c are opposite to the direction of fluid circulation and are obtained each other by rotation about the plane of thermocapillary flow symmetry (z = 1). In these flows, the absolute values of the stream function maximum are practically equal but have different sings. It should be note that the flow in figure 64b is characterized by oscillations of all parameters in a fluid with the frequencies are lower than the applied frequency Ω. The flow in figure 64c can be considered as conditionally stable. The positions of the convective cells remain unchanged and there occur a small jitter of the stream-lines and very weak oscillations of the velocity component at the fluid free surface with an applied frequency Ω. These peculiarities will be connected next with the dopant distribution in a fluid. As it has turned out that radial dopant segregation in fluid on the crystallization boundary is of an excellent indicator of flow pattern and subtle alterations of flow structure by the action of the surface standing waves.

z

z

z 2

2

2

- 69.9

- 45.9

1

1

1

- 187.12

0 0.0

0.5

(a)

1.0

r

0 0.0

21.5

187.3

93.6 0.5

(b)

1.0

r

0 0.0

0.5

1.0

r

(c)

Figure 64. Flow patterns (fields of stream function) of laminar thermocapillary convection in the absence of the standing-wave action (a) and by the action of surface standing waves (b and c). Ma/Pr = = 6.6·104, Ω = 2.5·103, n = 2; δ = 0.04 (b), δ = 0.043 (c).

Causes of Failures and the New Prospects in the Field of Space Material Sciences

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The case when initial thermocapillary flow and the configuration of surface standing wave are full antisymmetry (n = 1) is readily illustrated in figure 65. To δ = 0.0111 the flow is stable and thereafter falls dramatically to oscillatory regime. Under δ > 0.01115 oscillations are built up with increase of δ. This regime can be named as regime of developed oscillations or regime of weak turbulence.

ΔCs /Cs

1 0,04 0,03

2 3

0,02 0,01

0,0111

0,0112

0,0113

0,0114

δ

0,0115

Figure 65. Relative radial dopant segregation as a function of the standing-wave amplitude at n = 1. Ω = 2.5·103; Ma/Pr = 2.19·104 (1), 5.5·104 (2), and 6.6·104.

ΔCs /Cs

The transition from laminar flow to turbulent one happens less dramatically when n = 3 and n = 5 (figure 66). The sharp transition to developed oscillatory (turbulent) regime takes place in both case at δ ≈ 0.0076. There is zone of flow with small oscillations between a stable regime and turbulent flow. The zone of small oscillations is narrower at n = 3 since the mutual asymmetry of the standing wave and the initial thermocapillary flow is more evident. These cases are intermediate between the cases when n = 1 and n = 2.

3

0,04 0,03 1

2

0,02 0,01

0,004

0,006

δ

0,008

Figure 66. Relative radial dopant segregation as a function of the standing-wave amplitude at n = 5 (1), n = 3 (2), and in the absence of the standing wave (3). Ω = 2.5·103; Ma/Pr = 3.9·104.

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A.I. Feonychev

ΔCs /Cs

For the case of full symmetry of thermocapillary convection and the standing-wave configuration (n = 2), the boundary between the stable flow and small oscillating flow is not strong as at n = 1 and happens at δ = 0.003-0.004 (figure 66). The boundary between stable flow and flow with small oscillations is wide and depends on the parameter Ma/Pr. The regime of small oscillations is located between this boundary and the boundary of weakly turbulent flows. The transition to weakly turbulent flow stands out conspicuously at δ = 0.008 for all the numbers Ma/Pr. The intriguing peculiarity of transition zone between these boundaries is pronounced hump on the curves in figures 67-69. In the zone of hump (δ = 0.004- ~ 0.006) flow is conditionally stable and the flow pattern has analogy with flow at δ < 0.003-0.004. The reason of this phenomenon is not understood at present but it should be noted that the middle of this zone (δ ≈ 0.005) and the value δ such that flow with n = 1 falls dramatically to strongly oscillating regime (δ ≈ 0.011) corresponds to the common value of vibration amplitude. As mentioned above, the standing-wave amplitude is reduced by one half under change of n from 1 to 2.

0,04

0,02

0,00

0,004

0,006

0,008

δ

Figure 67. Relative radial dopant segregation as a function of amplitude δ for Ma/Pr = 3.9·104 (curves 1 and 3), Ma/Pr = 6.6·104 (curves 2 and 4). The standing wave has parameters: Ω = 2.5·103 and n = 2. The lines of 3 and 4 are related to the cases when the standing wave is not available.

It is amply clear that the inverse of symmetry and antisymmetry are most conspicuous in these phenomena. As for the conditionality of flow stability, it should be remarked that very small oscillations in a fluid are available. In addition to it, if a condition Ω > Ωe is fulfilled, besides an applied frequency Ω, overtones are obtained in spectra of all the parameters in a fluid. If a condition Ω >> Ωe takes place, infra-frequencies are increased. The degeneration of symmetry phenomenon is demonstrated in figure 68. The hump at n = 2 is transformed into a small peak at n = 4 and disappears at n = 6. To this point the cases discussed above concern to the condition when the frequency of the standing-wave oscillations is large than the eigenfrequency of the convective cell. For example, the eigenfrequency of convective cell Ωe at Ma/Pr = 3.9·104 is equal to 480 [20]. If a standing wave oscillates with a frequency Ω = 440, an approximate equality between Ω and Ωe (curve 3 in figure 69) takes place and the oscillations of the standing wave correspond to the integral period of the fluid rotation in the convective cell. In this case, the flow is stable in

Causes of Failures and the New Prospects in the Field of Space Material Sciences

207

ΔCs /Cs

a wide range of change in the standing-wave amplitude as long as δ = 0.008, which corresponds to the break of the flow stability and the transition to the regime of small turbulence. If the frequency Ω is much higher than Ωe (curve 1 in figure 69), a complex nonlinear interaction between thermocapillary flow and the surface standing wave takes place, with the result that different flow regimes are increased: from weakly oscillating flows (illustrated in figure 64b) to stable flows in the zone of hump (figure 64c). If Ω >> Ωe (on one order or more), the concrete value of Ω is inessential. Curve 2 in figure 69 corresponds to the intermediate case where the frequency of the applied disturbance (Ω = 103) is approximately two times higher than the eigenfrequency of the convective cell.

0,04

4 1

0,03

2 0,02

3

1

0,01

0,004

0,006

0,008

δ

ΔCs /Cs

Figure 68. Relative radial dopant segregation as a function of the standing-wave amplitude at n = 2 (1), 4 (2), 6 (3). The line 4 is related to the cases when the standing wave is not available. Ma/Pr = 3.9·104, Ω = 2.5·103.

0,05 0,04

3

0,03

2

1 0,02 0,01 0,004

0,006

0,008

δ

Figure 69. Relative radial dopant segregation as a function of amplitude δ for surface standing wave with n = 2. The standing-wave frequency Ω is equal 2.5·103 (1), Ω = 1·103 (2), and 4.4·102 (3). Ma/Pr = 3.9·104.

208

A.I. Feonychev The regime of small oscillations is characterized by a limited number of oscillation

frequencies, as rule two-three frequencies. In this case, for the parameter ΔC s / C s , a low frequency inherent in the usual oscillating regime of thermocapillary convection (F = 100200) is always present at the frequency spectrum. If the frequency of standing-wave oscillations Ω is far in excess of the eigenfrequency of convective cell Ωe, by the factor of 10 and more, the frequency Ω exists only in the oscillation spectrum of velocity, especially at the points close by free fluid surface, but is absent in the oscillation spectra of temperature and impurity concentration. The obtained data were generalized for Ma/Pr = 3.9·104 and is presented in figure 70. The dependence of the standing-wave amplitude δ on the number n at the boundaries between the three regimes of fluid flow is shown. The lower curve separates the field of stable flows (I) and flows with small oscillations (II). The upper curve separates the oscillatory (II) and turbulent (III) regimes. With increase of n at n > 6, the value of δ at the upper boundary remains practically unchanged and the value of δ at the lower boundary decreases slowly; by this means at n = 15, δ = 0.0022 at the lower boundary and δ = 0.0074 at the upper boundary. These values can be considered as asymptotes at any other large values of n. All the fluid flows increasing at n > 6 can be characterized as flows with a deformed free surface of ripple type. The lower boundary is determined when the oscillation amplitude of the parameter

ΔC s / C s reaches 0.1 %. In this case, the temperature oscillations at the point (r = 0.5, z = 0.02) are reached 1 %. Above the upper boundary, the oscillation amplitudes of all the parameters in a fluid run into several tens of percent and the spectrum represent a set of frequencies. Even through the data presented in figure 70 are obtained for the concrete values of Ma/Pr and Ω, they can be serve for orientation in the case where the indicated parameters take other values because, as was shown above, the characteristics of the fluid-flow regimes change insignificantly with change in these parameters.

δ 0,012 0,010 III

0,008

0.0074 (n=15)

II

0,006 0,004

I 0.0012 (n=15)

1

2

3

4

5

6

7

n

Figure 70. Boundaries between the regimes of fluid flow as a function of the number of the standingwave period. Region of I, II, and III fall to laminar, oscillatory and turbulent regimes, respectively. Arrows and numbers show the value δ at n = 15. Ma/Pr = 3.9·104, Ω = 2.5·103.

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209

z

z

2

2

- 134.3

- 147.1

- 29.3

98.7 - 28.6

87.7

1

1

30.3

163.3

0 0.0

0.5

32.2

- 92.7

- 80.4

1.0

149.5

0 0.0

r

0.5

(a)

1.0

r

(b)

θ (0.5; 0.02)

Figure 71. Momentary flow patterns (fields of stream function) of oscillatory thermocapillary convection without the standing-wave action (a) and by the action of surface standing waves (b). Ma/Pr = 1.08·105; Ω = 7.55·104, n = 2, δ = 0.0026.

0,026

δ = 0.0026

δ = 0.004

δ = 0.006

0,024

0,022 0,15

0,20

0,25

0,30

0,35

τ

Figure 72. Temperature at the point close to crystallization boundary (r = 0.5, z = 0.02) In the regime of oscillatory thermocapillary convection as a function of time in the process of the three-stage change in amplitude of the surface standing wave. The arrow shows at the time of the standing wave application. Ma/Pr = 9.56·104, Ω = 7.55·104, n = 2.

In the oscillatory regime of thermocapillary convection, flow pattern is asymmetric about a plane z = L (figure 71a). The extent of asymmetry is the difference of stream function

210

A.I. Feonychev

maxima in the upper and lower parts fluid column. This difference is equal to 18.3, i.e. 18.5 % in relation to the maximum of stream function (98.7). By contrast to this, the configuration of surface standing wave with n = 2 is symmetric about a plane z = L. Because of this, the flow pattern in oscillatory thermocapillary convection becomes more symmetric toward this plane by the action of surface standing wave with n = 2 and δ = 0.0026 (figure 71b). In this figure it is shown that the difference above mentioned is equal to 6, i.e. 6.5 %. As illustrated in figure 72, the oscillation amplitude of temperature at the point nearby the crystallization boundary (r = 0.5, z = 0.02) is stepwise reduced with the step-by-step increase of the standing-wave amplitude. Eventually the temperature oscillations are reduced by the factor of 5.3 at δ = 0.006. This effect is obtained in the two-dimensional statement of problem. Because the surface standing waves is symmetric also about the z-axis, the effect symmetrization will take place in the real three-dimensional case, as happened with the rotating magnetic field. 2.3.3. Use Surface Standing Wave and Axial Static Magnetic Field under Crystal Growth by the Floating Zone Method

ΔCs /Cs

Once the study of flow stability was accomplished, a possibility to use the surface standing waves for crystal growth by the floating zone method became apparent. It is suggested that the surface standing waves will be combined with the static axial magnetic field. In this case the Lorentz force has only the projection onto the radius of the liquid cylinder and is described by equation (9). As an example the process of crystallization of silicon with dopant of phosphorus, Si(P) is considered. The following dimensionless parameters are used in calculations: Pr = 0.023, Sc = 5, k0 = 0.35, Recr = 0.1 and L/R = 2. The distribution of the heat flux on the cylindrical liquid surface is given by (12) which describe well the experimentally obtained distribution of temperature on the surface of the liquid zone [28, 29]. 0,06

0,05

Ha = 10

Ha = 20

Ha = 30

0,04

0,03

0,4

0,6

0,8

1,0

τ 1,2

Figure 73. Relative radial dopant segregation with the action of standing wave on laminar thermocapillary convection as a function of the time in the process of magnetic field optimization. Ma/Pr = 3.26·104, n = 2, Ω = 2.5·103, δ = 0.005. The horizontal line notes the value of the absence of surface standing wave.

ΔC s / C s

in

Causes of Failures and the New Prospects in the Field of Space Material Sciences

211

It has been shown above that the transport processes in a fluid under the action of the surface standing waves depend little on the frequency of standing wave oscillations if Ω >> Ωe. This allows one to carry out calculations at lower values of Ω than is required by the considered parameters of the surface standing waves, which will considerably accelerate the process of computations due to a greater step in time Δτ. In most part of the examples presented below, the determination of optimum parameters of standing wave and magnetic field is shown for the most clearness in the form of continuous process. The figures 73-76 show the process of static magnetic field optimization for the cases when initial thermocapillary convection takes place in the area of laminar regime of fluid flows. The surface standing wave operates from the beginning of crystal growth. The axial static magnetic field is switched on at the time instant τ = 0.341 and the calculation is continued up to the state of steady-state oscillations of ΔC s / C s (figure 73). Thereafter a new value for the magnetic field induction is assign, and the calculation is being continued by the same procedure. Under the Hartmann number Ha = 20 the time-average value of parameter

ΔC s / C s is equal to 0.0234 with the oscillation amplitude < 0.13 %. Both these parameters are increased with further increasing Ha. The time-average value ΔC s / C s is raised to 0.0294; oscillation amplitude is doubled, up to 0.26 %. The initial value of ΔC s / C s without the surface standing wave is equal to 0.0391 (showed by the straight stroke-dotted line) and the oscillations are unavailable.

ΔCs /Cs

0,05 0,04 0,03 Ha = 5 0,02

Ha = 10 Ha = 15

Ha = 20

0,01

0,1

0,2

0,3

0,4

0,5

0,6

τ

0,7

Figure 74. Relative radial dopant segregation with the action of standing wave on laminar thermocapillary convection as a function of time in the process of magnetic field optimization. Ma/Pr = 6.01·104, n = 2, Ω = 2.5·103, δ = 0.003. The arrow shows at the time of the magnetic field switchingon. The horizontal line notes the value of

ΔC s / C s in the absence of surface standing wave.

The calculation at Ma/Pr = 6.01·104 had been performed with the four-stage rise of magnetic field (figure 74). Here two values of the Hartmann number can be observed as optimum, Ha = 10 and Ha = 15. In the first case, the time-average value of ΔC s / C s is equal

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A.I. Feonychev

to 0.0097 under the oscillation amplitude ~ 0.85 %. In the second case, the time-average value of ΔC s / C s is just below (0.00865) but the oscillation amplitude is large (~3 %). It seems reasonable to say that the case with Ha = 10 is preferential. The initial value of

ΔC s / C s without the surface standing wave is equal to 0.0306 (horizontal line in figure 74) and the oscillations were nil.

ΔCs /Cs

0,04

0,03

Ha = 5 Ha = 10 Ha = 14.14 Ha = 20

Ha = 30

0,02

0,01 0,4

0,6

0,8

τ

Figure 75. Relative radial dopant segregation with the action of standing wave on laminar thermocapillary convection as a function of the time in the process of magnetic field optimization. Ma/Pr = 7.2·104, n = 2, Ω = 2.5·103, δ = 0.0034. The arrow shows at the time of magnetic field

ΔCs /Cs

application. The horizontal line notes the value of

ΔC s / C s in the absence of surface standing wave.

0,04 0,03 Ha = 5

Ha = 10

Ha = 15

20

Ha = 30

0,02 0,01

0,2

0,4

0,6

0,8

1,0

τ

Figure 76. Relative radial dopant segregation with the action of standing wave on laminar thermocapillary convection as a function of the time in the process of magnetic field optimization. Ma/Pr = 7.2·104, n = 4, Ω = 2.5·103, δ = 0.0032. The arrow shows at the time of magnetic field application. The horizontal line notes the value of

ΔC s / C s in the absence of surface standing wave.

The two following calculations (figures 75 and 76) are performed at Ma/Pr = 7.2·104, the same value of the standing wave frequency (Ω = 2.5·103) but with the different number of

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standing wave periods (n = 2 and n = 4) and slightly different amplitudes of standing waves (δ = 0.0034 and 0.0032, respectively). The five-stage change of magnetic field takes place to determine the optimal intensity of a magnetic field (the Hartmann number) in these cases. Without a surface standing wave and a magnetic field ΔC s / C s = 0.0295 (straight strokedotted lines in figures 75 and 76). In the presence of a surface standing wave without a magnetic field, the time-average value of ΔC s / C s is equal to ~ 0.0125 (figure 75) and to ~ 0.014 (figure 76). The amplitudes of fluctuations are ~ 40 % and ~ 13 %, respectively. The two values of the Hartmann number can be compared and used for a practical application: Ha = 14.4 (15 for the case n = 4) and Ha = 20. For a standing wave with n = 2 at Ha = 14.4, the time-average value of ΔC s / C s is 0.0084 and the amplitude of fluctuations is ~ 2.2 % (figure 75). In the case of n = 4 at Ha = 15, these values are equal to 0.0084 and ~ 1 %. At Ha = 20 the time-average values of ΔC s / C s are equal to 0.0102 with the amplitude of fluctuations ~ 1.2 % for n = 2 and these parameters are 0.098 and ~ 1.6 % for n = 4. Finally, at Ha = 30 the value of ΔC s / C s increases up to 0.037 for n = 2 and up to 0.04 for n = 4. In summation, the standing wave with n = 4, δ = 0.0032 and the static magnetic field with an induction corresponding the Hartmann number Ha = 15 can give the best result.

ΔCs /Cs

0,015

δ = 0.0026 δ = 0.004

δ = 0.006

0,014 0,013 0,012 0,15

0,20

0,25

0,30

τ

0,35

Figure 77. Relative radial dopant segregation in the regime of oscillatory thermocapillary convection as a function of time during the process of the three-stage change in amplitude of the surface standing wave. The arrow shows at the time of standing wave application. Ma/Pr = 9.56·104, n = 2, Ω = 7.55·104.

As it was remarked above, a surface standing wave symmetrizes fluid flow in the regime of oscillatory thermocapillary convection. At Ma/Pr = 9.56·104, the change in the amplitude of the ΔC s / C s oscillations in the case of a stepwise increase of the standing wave amplitude δ from 0.0026 to 0.006 is shown in figure 77. In the absence of surface standing wave, the time-average value of ΔC s / C s is 0.013 and the amplitude of oscillations is 7.27 %. At a standing wave with δ = 0.006 the time-average value of ΔC s / C s decreases slightly, on ~ 4

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%, but the amplitude of oscillations decreases to 1.47 %, i.e. almost fivefold. The dimensionless frequency of the initial oscillations F does not change here and is equal to 121.6. Note that at the amplitude of a surface standing wave δ = 0.0026, the amplitude of oscillations ΔC s / C s is even higher (8 %) than without standing wave. Figure 78 shows the changes of the parameter ΔC s / C s changes after successive switching-on of the surface standing wave and of the magnetic field. Before the switching-on of the surface standing wave the fluid flow corresponds to the beginning of oscillating thermocapillary convection (Ma/Pr = 9.6·104). In this regime of flow, the time-average value of ΔC s / C s is 0.0126 at an oscillation amplitude of ~ 7.6%. After the start of the action of the surface standing wave with the parameters n = 2, δ = 0.006, and Ω = 7.8·104 a transition regime begins, at the end of which the time-average value of ΔC s / C s decreases to 0.01074 and the amplitude of oscillations increases up to about 9%. Thereafter the static magnetic field (Ha = 5) is switched on, as a result of the action of which the oscillations damp out rapidly (the residual dopant microsegregation is ~ 0.1%) and the time-average value of

ΔC s / C s is established at a level of about 0.01055. In this example, the reduction of the dopant microsegregation is slight, about 16% of the initial level, but the principal effect in the practical elimination of the dopant microsegregation. 1

ΔCs /Cs

0,014 2

0,012

0,010 0,1

0,2

0,3

0,4

τ

Figure 78. Relative radial dopant segregation in the regime of oscillatory thermocapillary convection (Ma/Pr = 9.5·104) as a function of time under the sequential switching-on of standing wave (n = 2, Ω = 7.8·104, δ = 0.006) and static magnetic field (Ha = 5). The 1 and 2 arrows show at the times of the surface standing wave and magnetic field switching-on, respectively.

The process of determination of the optimum parameters of a surface standing wave and a magnetic field is presented in figure 79. The initial regime of fluid flow corresponds to a developed oscillatory thermocapillary convection close to turbulent regime (Ma/Pr = 1.2·105). In this regime of fluid flow, the time-average value of ΔC s / C s is 0.01087 and the amplitude of oscillations is about 21.5%. The amplitude of a surface standing wave with n = 2 and Ω = 2.5·103 is first set at the level of 0.0032 and then of 0.005. In the latter case, the time-

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average value of ΔC s / C s somewhat grows up to 0.01125 at an amplitude of oscillations equal to about 19.5 %. After this a magnetic field with a stepwise increase in the Hartmann number (5, 10, and 15) is switched on. In the optimum regime (δ = 0.005 and Ha = 10), the time-average value and the amplitude of oscillations of ΔC s / C s are equal to 0.0101 and ~ 1%, respectively. At Ha = 15 the oscillations of the parameter ΔC s / C s decrease to 0.4%,

ΔCs /Cs

but its time-average value increases sharply up to about 0.0385.

0,04 Ha = 15 Ha = 10

δ = 0.003 δ = 0.005 1

0,03

2 Ha = 5

0,02 0,01

0,1

0,2

0,3

0,4

0,5

0,6

0,7

τ

θ (0.5, 0.02)

Figure 79. Relative radial dopant segregation in the regime of oscillatory thermocapillary convection as a function of the time in the process of the standing wave and magnetic field optimization. The 1 and 2 arrows show at the times of the surface standing wave and magnetic field switching-on respectively. Ma/Pr = 1.2·105, n = 2, Ω = 2.5·103.

0,04

δ = 0.003

δ = 0.005

1

2

0,03

Ha = 5 Ha = 10

Ha = 15

0,3

0,6

0,02

0,01

0,1

0,2

0,4

0,5

0,7

τ

Figure 80. Temperature at the point (r = 0.5, z = 0.02) in the regime of oscillatory thermocapillary convection as a function of the time in the process of the standing wave and magnetic field optimization. The 1 and 2 arrows show at the times of the surface standing wave and magnetic field switching-on, respectively. Ma/Pr = 1.2·105, n = 2, Ω = 2.5·103.

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The determination of the optimal parameters of surface standing waves and magnetic field had been carried out with the purpose of the most demonstrativeness by continuous process with the transition stages between different regimes. In the real process of crystal growth these transition stages should be held prior to the onset of crystal growth. In this case, the period of the development of the dopant concentration profile in melt takes a time not longer than τ = 0.1. The length of the crystal in the regime of establishment of the concentration field is determined from Recrτ, i.e. under Recr = 0.1 this length is not longer than 0.01 of the crystal radius. The temperature at the point close to the crystallization boundary (r = 0.5, z = 0.002) is present in figure 80 as a function of the time in the process discussed above. This figure 80 shows that the possibility exists of temperature measurement during optimization process. The investigation carried out shows that the use of surface standing waves as well as the combination of surface standing wave with a static axial magnetic field allows one to obtain a considerable reduction of the micro- and macrosegregation of dopant in crystals grown by the floating zone method under microgravity conditions. Obtained results were used in the patent on new method and equipment for crystal growth [66]. An especially great decrease in the dopant macrosegregation is obtained when the initial condition is characterized by the small intensity of thermocapillary convection in a laminar regime of fluid flow. A positive result can be obtained when the optimal parameters of the surface standing wave (the number of the periods of a standing wave over the liquid zone length and the amplitude) and of the magnetic field are selected exactly. This is important to underline, since otherwise there is a danger of occurrence in the zone of conventional stability of fluid flow (the “hump” zone for n = 2), or of slip into the zone of a developed turbulent regime. The overshooting of the optimal magnetic field intensity may lead to a catastrophic increase in the dopant macrosegregation, as this is true in the normal floating zone method of crystal growth. Optimal parameters of a standing wave and magnetic field can be chosen for specific material and conditions of crystal growth by the method of mathematical simulation.

II. Electrophoretical Separation of Biomixtures It is well known that electrophoresis is one of the most efficient methods of analyzing, purifying and fractionating biomixtures (nucleic acids, proteins, cells, etc.). The principal obstacle to using electrophoretic biomixture separation on Earth is intense convective mixing of the components within the electrophoretic cell. To avoid this, various support media are employed or the linear dimensions of the cell are reduced. However, the presence of support media and the decrease in size sharply reduce the output of the cell. Moreover, the biocomponents may sometimes combine with the support media to form complexes which are difficult to separate and interfere with the identification and use of the mixture components. Fundamentally new possibilities are opened up to using electrophoretic methods under microgravity conditions, when the gravitational convection is much less intense. Since 1971 during flights of the “Apollo”, “Skylab”, Apollo-Soyuz” and “Sayut” spacecraft a series of demonstration experiments on the electrophoresis of biomixtures (blood proteins, cells, DNA, etc.) and model media (mixtures of latex beads, ions) had been carried out [67]. In these experiments various methods of liquid electrophoresis were employed: zone electrophoresis, isotachophoresis, isoelectric focusing, and continuous flow electrophoresis (free-flow

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electrophoresis). After these previous experiments, the main attention had been focused on the continuous flow electrophoresis as the most productive. Pilot experiments with this method had been spread by the MacDonnell-Douglas Co. (USA) on the Space Shuttles and the “Energy” NPO (USSR) on the “Salyut” and “Mir” Space Stations. These experiments closed down without analysis of causes for failures. It seems plausible, on the base of new data and mathematical models, such the analysis can be carried out.

2.1. Statement of the Problem and Method of Numerical Solution As a typical object of electrophoresis let us consider a protein molecule. The protein molecule, which consists of a certain set of amino acids, is believed [68] to contain a large number of ionogen groups (carboxyl, phenol, etc.), which, when the protein dissolves in water, dissociate with the release or absorption of a proton. As a result, the molecule itself acquires a charge. Part of the charge is neutralized by the ions of opposite sign absorbed by the molecule, the rest of the charge determining a diffusely scattered layer of ions which is mobile and depends on the external conditions (number and composition of the ions in the solvent-buffer, the presence of other protein molecules, an external electric field, etc.). This mobile part of the charge determines the electrokinetic potential (ζ-potential) of the molecule. When an external electric field is applied, the molecule moves towards one of the poles, its velocity being determined by the ζ-potential, the size and shape of the molecule, the viscosity of the fluid, and the strength of the electric field E. Each species of molecule (i) can be /

characterized by its mobility mi , which is equal to the velocity of the molecule at an electric /

field strength E equal to 1 V/cm. Thus, the electrophoretic velocity of the molecule u i =

mi/ E. The ζ-potential and hence the mobility mi/ depend on the medium ion concentration. For simplicity, it is deal with the hydrogen ion concentration cH in the solvent-buffer as characterized by the pH value pH = - log cH. Moving in the electric field, the biomixture molecules progressively separate. As a result, the zone containing molecules of the same species is forming. This is the basis of the zone method of electrophoretic separation. As a certain pH value specific for each molecular species the effective charge (or ζpotential of the molecule) will be equal to zero and, accordingly, the mobility of the molecule will also be equal to zero. This is the basis of isoelectric focusing method of separating macromolecules in an electric field. The pH value at which the mobility of the molecule is equal to zero is called the isoelectric point. When a pH gradient is created in the solventbuffer in the direction of the action of electric field, each species of molecule gradually forms a fairly thin layer near its isoelectric point. The width of this layer is determined by the pH gradient, the dependence of the mobility of the molecule on the pH near the isoelectric point and the diffusion coefficient. If continuous flows of buffer and biomixture are created in the electrophoretic cell, and the electric field is directed at right angles to the hydrodynamic flow, then as a result of the electrophoretic velocity each species of molecule will be differently displaced in the direction of the electric field at the cell outlet the biomixture will be separated into its components. This is the principle of continuous flow electrophoresis (free-flow electrophoresis).

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The electrophoretic cell may be conceived as a parallelepiped, one side of which (thickness) is usually much smaller than the other two (height L and width H). Two opposite faces, formed by the thickness and one of the other sides (H or L, according to the method of biomixture separation), constitute the electrodes (anode and cathode). The buffer fully occupies the volume of the cell. The gas cavities are impermissible. The solvent-buffer is a weak electrolyte based on water and dissolved acids, salts and bases. The fluid velocities are low (the Reynolds number generally less than 10), as are the temperature differences (several degrees or even fractions of a degree). Simple mathematical models of the electrophoretic separation of biomixture were used to obtain a qualitative analytic description of the processes in [69-72]. The use of more complex mathematical models [73] for the numerical calculations encounters difficulties associated with the duration of the individual experiments (for example, isoelectric focusing) and solving equations of parabolic type with a small parameter multiplying the highest derivatives in order to determination the spatial distribution of the biomixture components. These problems had been solved by the collaboration with the research workers from the M.V. Keldysh Institute of Applied Mathematics, the USSR Academy of Sciences [74-76]. The mathematical model of electrophoresis is based on the Navier-Stokes equations in the Boussinesq approximation and the heat transfer equation with allowance for heat conduction and convection in the presence of volume heat release due the passage of the electric current. The mass transfer equation is written for each component of the biomixture (the number of components is N) and, if necessary, for the component of the solvent-buffer (the number of these components is M). To these equations there must be added the Maxwell equations for the electric field. In writing the system of equations, note should be taken on the following physically justified assumptions: the biomixture concentration is low and the interaction of the components is not taken into account; the fluid is a weak conductor; the medium as a whole is electric neutral; magnetic effects are neglected. Given these assumptions, in the dimensionless variables vorticity ή, stream function ψ, temperature θ, concentration of the i-th component of the solution Ci and electric potential Φ in rectangle, [0 < x < H] × [0 < z < L] the system of equations has the form: N ∂Ci ∂η ∂η ∂ψ ∂η ∂ψ ∂ 2η ∂ 2η ∂θ + − = 2 + 2 + GrT + ∑ Gri , ∂τ ∂x ∂z ∂z ∂x ∂x ∂z ∂x i =1 ∂x

(26)

∂ 2ψ ∂ 2ψ + = −η , ∂x 2 ∂z 2

(27)

⎡⎛ ∂Φ ⎞ 2 ⎛ ∂Φ ⎞ 2 ⎤ 1 ⎛ ∂ 2θ ∂ 2θ ⎞ ∂θ ∂θ ∂ψ ∂θ ∂ψ = ⎜⎜ 2 + 2 ⎟⎟ + Rκ ⎢⎜ − + ⎟ ⎥, ⎟ +⎜ ∂z ⎠ ∂τ ∂x ∂z ∂z ∂x Pr ⎝ ∂x ⎢⎣⎝ ∂x ⎠ ⎝ ∂z ⎠ ⎥⎦

(28)

∂C i ∂ ⎡ ⎛ ∂ψ ⎞ ⎤ ∂ ⎡ ⎛ ∂ψ ⎞ ⎤ 1 ⎛ ∂ 2 Ci ∂ 2 C i ⎞ ⎜ ⎟, + + + − = + u C v C ⎜ ⎟ ⎜ ⎟ i i i i ∂τ ∂x ⎢⎣⎝ ∂z ⎠ ⎥⎦ ∂z ⎢⎣⎝ ∂x ⎠ ⎥⎦ Sci ⎜⎝ ∂x 2 ∂z 2 ⎟⎠

(29)

Causes of Failures and the New Prospects in the Field of Space Material Sciences

∂ ⎛ ∂Φ ⎞ ∂ ⎛ ∂Φ ⎞ ⎟ = 0, ⎟ + ⎜κ ⎜κ ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ where

u=

219

(30)

∂ψ ∂ψ ∂Φ ∂v ∂u ∂Φ ,v= − ,η= − , ui = − a i , vi = − a i , ∂z ∂x ∂x ∂z ∂x ∂z N

κ = 1 + K ∑ (κ i − 1)Ci ,

i = 1, 2, …., N.

i =1

Here, GrT = gβTT0H3/ν2 is the Grashof number for thermal convection, Gri = gβiTc0H3/ν2 is the Grashof number for concentration convection, R = κ0 (Δφ)2/(νρcpT0) is the heat release coefficient, Pr = ν/χ is the Prandtl number, Sci = ν/Di is the Schmidt number, K = c0/ρ is the maximum mass fraction of biomixture in the solution, θ = T/T0 is the dimensionless temperature, Ci = ci/c0 is the dimensionless concentration of the i-th component, Φ = φ/Δφ is the dimensionless electric potential, βT = - (1/ρ)∂ρ/∂T and βi = - (1/ρ)∂ρ/∂ci are the coefficients of thermal and concentration expansion of the fluid, respectively, ρ is the density, ν is the kinematic viscosity coefficient, χ is the thermal diffusivity, Di is the diffusion coefficient for the i-th component of biomixture, κ and κi are the conductivities of the substance and the i-th component of the solution, κ0 is the conductivity of the buffer, cp is the specific heat of the substance at constant pressure, u and v are the velocity components along the x and z axes, respectively, ui and vi are the electrophoretic velocity components, ai is the electrophoretic mobility, L and H are the length and the width of the domain, respectively, T0 is the initial temperature, c0 is the maximum initial value of the mixture concentration in the calculation domain, and Δφ is the potential difference across the electrodes. The system of equations (26) – (30) must be supplemented by the boundary conditions. For the temperature were given the two conditions

∂θ = 0, z = 0, z = L/H; 0 ≤ x ≤ 1 and θ = 1, x = 0, 1; 0 ≤ z ≤ L/H, ∂n where n is the normal to the boundary of the domain. The electrical potential takes the value Φ = 0 and Φ =1 on the electrodes; the condition

∂Φ =0 ∂n on the other boundaries.

∂Ci =0 ∂n on all the boundaries for the zone electrophoresis and the isoelectric focusing; for the continuous flow electrophoresis, the condition for Ci is specific and will be given below.

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A.I. Feonychev The condition for stream function is given in the conventional form

ψ = f1 ,

∂ψ = f2 , ∂n

where the functions f 1 and f 2 describe different cases on the boundaries (sticking, electroosmotic slipping, supply and removal of fluid). It is a feature of the system of equations (26) – (30) considered above that in the equation for the concentration (29) the coefficient 1/Sci is of exceptionally small number (10-4 – 10-6), i.e. the system (26) – (30) contain an equation of parabolic type (29) with a small parameter as a coefficient of the higher derivatives. The numerical solution of equations of this kind is known to involve certain difficulties. For solving the equation (29) with the above mentioned property a locally one-dimensional difference scheme with artificial dispersion has been proposed [76]. The idea is to add antidispersion terms to the usual scheme (with approximation of the convective terms by central differences). This considerably reduces the dispersion properties of the scheme and makes it possible to carry out the calculations on a grid with space increments of the order of

1 / Sci .

For solving the Navier-Stokes equations (26), (27) a conservative difference scheme was used. The equation for the temperature (28) was solved in accordance with an implicit conservative scheme using the scalar sweep method. A steady-state solution of the elliptic equation for the potential (30) was obtained by iteration.

2.2. Results of Numerical Calculations The mathematical model and the method of numerically solving the system of equations (26)-(30) were used to calculate the process of electrophoretic separation of biomixture by three different techniques. The numerical solutions were obtained using the scheme with artificial dispersion.

2.2.1. Zone Electrophoresis In this case, the electric field was directed along the z coordinate. The buffer solution has a constant given pH, the electrophoretic mobilities of the biomixture components being equal to a1 = 6, a2 = 3. These component were taken to be electrically nonconducting (κ1 = κ2 = 0). The solution of biomixture was diluted (K = 5·10-2). The other parameters were equal to: R = 1.22, Pr = 10, Sc1 = Sc2 = 104, L = 6. The location of the biomixture is shown in figure 80 for the initial instant (τ = 0) by the solid curve at 5 < z < 5.4. When the electric field is switched on, the components of the mixture begin to move at different speeds and separate along the z coordinate. Setting the residual acceleration of mass forces at g = 10-4g0 , where g0 is the acceleration gravity on Earth, gravitational convection can be neglected (Grt ~ 1, Gri ~ 10-2). In the absence of electro-osmotic motion of the fluid relative to the walls we have the case of onedimensional motion of the biomixture component, i.e. in this case the equation (29) takes the form

Causes of Failures and the New Prospects in the Field of Space Material Sciences

∂Ci ∂Ci 1 ∂ 2 Ci = + vi Sci ∂z 2 ∂z ∂τ

221

(31)

Carrying out the charge of variables y = z – vi τ, we obtain the equation

∂f i 1 ∂2 fi , Ci (( z , τ ) = f i (z − viτ , τ ) = ∂τ Sci ∂y

(32)

The exact analytical solution of equation (32) is well known.

Figure 81. The distribution of the concentration of the two components at τ = 2.4. The solid lines are appropriated to regime without the electro-osmotic motion. The dotted and dot-and-dash lines correspond to the localizations of components at the x = 0.5 and the wall, respectively, with electroosmotic flow along walls.. The number 1 and 2 are concerned to the relevant number of component.

The distribution of the concentration along the z direction of the two components is shown in figure 81 at the time τ = 2.4. The solid curves with the number 1 and 2 have been obtained for the case when the electro-osmotic slipping along the walls is absent. The curve with τ = 0 shows the initial state of biomixture. The exact analytical and numerical solutions of the equation (32) coincided. If electro-osmotic slipping of the fluid along the walls in a direction opposite to that of the electrophoretic motion of the biomixture components is taken into account, the component concentration distribution acquires a two-dimensional character. The electroosmotic slipping velocity of the fluid along the walls was given to 0.25. In figure 81 the concentration distribution on the z direction is present by the dotted curves for x = 0.5 and by the dot-and-dash curves at the wall for the same components. The electro-osmotic slipping leads to an acceleration of the displacement of the mixture components along the z-axis and to considerable noninformity in the distribution of components over the width of the cell. This complicates the sampling of the mixture components on completion of the separation process.

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As calculations showed [74], under terrestrial conditions (g = g0, GrT ~ 104, Gri ~ 102) gravitational convection leads to intensive mixing within the cell and the mixture cannot be separated into its components. In conclusion it should be noted that the selected potential difference Δφ and maximal initial value of the mixture concentration c0 the electric field was almost constant over the cell. Accordingly, in considering the other two types of electrophoresis the equation for the potential (30) was not solved.

2.2.2. Continuous Flow Electrophoresis Under calculations of the biomixture separation the action of gravitational convection was neglected and the electric field was taken to be constant. In this case the Navier-Stokes equations (26), (27) and the equation for the concentration of biomixture (29) are independent. The calculations scheme is as follows: for given boundary conditions the steadystate solution of the Navier-Stokes equation is found, then for a known flow the steady-state solution of the equations for the concentration of the two components (i = 1 and i = 2) is determined.

Figure 82. The trajectories of motion for two components of biomixture.

The calculations were carried out for a cell of height L = 1.5. As in zone electrophoresis, the buffer solution has a constant pH. In this case the electric field was directed transversely on the cell, along the x coordinate and the electrophoretic velocities of the components are directed at right angles to hydrodynamic flow: u1 = 0.3, u2 = 0.6, v1 = v2 = 0. Two variants of the hydrodynamic flow were considered (Fig. 82). First: supply and removal of buffer are performed uniformly in x with v = -3, supply of biomixture over a width x2 - x1 = 1/120 along the axis of the cell (x = 0.5) is also uniform. In this case the function Ci, 0 = 0 (0 ≤ x < x1 , x2 < x ≤ 1), Ci, 0 = 0.5 (x1 ≤ x ≤ x2), x1 = 0.5 – 1/120, x2 = 0.5 + 1/120. The second variant: the

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hydrodynamics of supply and removal are the same, but the condition of ideal fluid slipping is imposed at the walls. The concentration distribution for the two components at the outlet of the cell is presented in figure 83. The solid and the dotted curves are appropriated for the first and second hydrodynamic regime, respectively. The dot-and-dash curve represents the concentration distribution for another Schmidt number (Sc1 = 2·103) and only for the first component of biomixture. The data presented in figure 83 are clearly demonstrated that the flow hydrodynamics and, all the more, the diffusion coefficient have an important influence on the sharp and location of the concentration maximum at the cell outlet. The comparison of the numerical calculations with the exact analytical solution from [68] showed that discordance runs into 5% for the first variant of calculation and 30% for the second case.

Figure 83. The distribution of the concentration of the two components (noted as 1 and 2) of biomixture at the outlet of the cell. The solid and dotted lines are related in the first and second regimes, respectively. The dot-and-dash line is relative to the first regime of the component of the number 1 with Sci = 2·103.

In [77] the flow in the electrophoretic cell was considered in the pseudothree-dimensional approximation. The velocity distribution on the y coordinate was given by the parabolic curve w(y) = Wmax [1 - 4(y/h)2]. A similar method for solution of the three-dimensional problems was be used earlier under analysis of space experiment with thermocapillary convection [78]. It was assumed that the physical parameters and electrophoretic mobility of biomixture components are linear function concentration and are expressed by following relationships: N

κ = 1 + ∑ ci i =1

N 1 ∂κ 1 ∂μ , μ = 1 + ∑ ci , κ 0 ∂ci μ 0 ∂ci i =1

N

ε = 1 + ∑ ci i =1

μ 1 ∂ε , ai = ai ,0 0 μ ε 0 ∂ci

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Figure 84. Isolines of concentration for single-fraction jet in section z/H = 6. The solid lines correspond to αi = 0.35 (a) and αi = - 0.2 (b) The dotted lines describe the case when concentration effect is absent, i.e. ci → 0.

The isolines Ci = constant for single-component sample jet at cell outlet are shown in figure 84 for the two values of the parameter αi. The complex parameter αi was given by a function of the i-th component concentration

⎛ 1 ∂κ 1 ∂μ ⎞ ⎟⎟Ci + α i = ⎜⎜1 + ∂ c ∂ c κ μ i i ⎠ 0 0 ⎝ As indicated in figure 84, the concentration effect is quite considerable for reasonably large concentration of biomixture. If the solution is much diluted (the dotted curves for Ci → 0 when the parameter αi at supply in the electrophoretic cell, αi, 0 is equal to zero) the isolines Ci = const for the (a) and (b) cases are identical, i.e. the concentration effect is absent.

2.2.3. Isoelectric Focusing The calculations of isoelectric focusing have being carried out for a cell of height L = 9 with a buffer solution whose pH varies linearly from 4 to 10 over the height of the electrophoretic column. The dependence of the acidity index pH from the height of cell is shown in figure 85 by the dot-and-dash line. The dependence of the electrophoretic velocities of the five components on the pH over the height of the column is shown in the same figure. The location of the isoelectric points on the height is defined by condition vi = 0. The distribution curves for all five components are shown in figure 86 at time τ = 102. The initial position of the biomixture is represented by the dot-and-dash line in this figure. Three components (i = 1 – 3) have reached their isoelectric points, while the other two (i = 4 and 5) are still in motion. Calculations show that the components i = 4 and i = 5 reach their isoelectric points at time τ = 4·102 and 6·102, respectively, when their profiles differ only slightly from those reproduced in figure 86. Accordingly, instead of carrying out isoelectric focusing up to the moment at which all the components reach their isoelectric points, it can be limited by a shorter focusing time interval and sample the components after most of them

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have stopped moving. This moment and the component sampling point can be determined by calculation.

Figure 85.The change of electrophoretic mobilities with height of electrophoretic cell components of biomixture.

for five

Figure 86. The distribution of concentration of five components under isoelectric focusing of biomixture. Solid lines correspond to τ = 100, dotted lines correspond to τ = 400 for 4th component and τ = 600 for 5th component. The dot-and-dash line shows the initial position of biomixture.

The component zone width is determined by the diffusion coefficient and the rate of variation of the component mobility: the smaller the diffusion coefficient and the greater the variation of the velocity near the isoelectric point, the narrower the zone. Thus, at the isoelectric point the zone width for the component i = 3 is approximately three times narrower than that for the component i = 5 (figure 86). At their isoelectric points the mobility gradients of these components differ approximately by a factor of 10 (figure 85).

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2.2.4. Hydrodynamic Instability of Biocomponent Jet Towards the end of 1997 a work on the TM-7 project [57] have been completed and it became apparent that the jets of biocomponents during the continuous flow electrophoresis experiments aboard spacecraft can be in unstable state under the action of vibrations existing on board. A liquid column during the process of crystal growth by the floating zone method is insensitive to these vibrations, because the capillary constant Kσ is large number for semiconductors (106 – 108). The situation for jet of an aqueous solution of biocomponent flowing in aqueous solution of buffer can be much different, since Kσ is considerably less. It should be noted that the vibration vector can have the arbitrary direction in space. Moreover, in electrical field a ponderomotive force takes into account if permittivities of buffer and biomixture (biocomponents) are different. In the general case the ponderomotive force must be added on the Navier-Stokes equations and written in addition the equation of free electrical charge density [79] F = ρe E -

1 2 E ∇ε , 8π

∇ (εE) = ρe ,

(33) (34)

where ρe is free charge density, ε is permittivity. The second term on the right of (33) gives the force due to the different values of permittivity in the biocomponent jet and the buffer solution. This force acts on the normal to the boundary of biocomponent jet. By this means in the general case, under the action of vibration at unrestricted angles to jet and of the ponderomotive force the combination of the Kelvin-Helmholz and Rayleigh-Taylor instabilities can take place. Even though the biocomponent jet is not collapsed, biocomponent outlet from the electrophoretic cell in a well-defined area will not be guaranteed. Considering this circumstance we have upheld (in 1988 or 1989) a suggestion of French colleagues to carry joint experiment with the aim to study the complex of hydrodynamic processes during continuous flow electrophoresis. The French installation “Selecte” have been designed for scientific investigations of continuous flow electrophoresis. The experiment was presumed to carry out on board the “Mir” orbital station. Unfortunately, instead of scientific investigation the modernization of the electrophoretic installation “Ruchey“ have been given preference (change of mechanical pump on peristaltic one, new electric system, etc.). The model experiment on the “Ruchey“installation was fulfilled by cosmonaut A. A. Serebrov aboard the “Mir” orbital station during February 14-15, 1990. This experiment showed clearly that the hydrodynamic instability of biocomponent jet takes place. The smooth jet from the solution of hemoglobin was injected in the travelling flow of tris-glycine buffer. Once the jet having reached the other side of the electrophoretic cell, a voltage was applied to the electrodes aligner transversely on the flow. Immediately afterwards a considerably deformation of jet was observed. The picture of initial stage of jet deformation is presented in figure 87.This picture is draw copy of the electrophoretic cell photograph “in clear gap” (the jet is shaded). The photograph have been fulfilled at the moment t = 3 s after switching on electrical current.

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The jet deformation in electric field is shown in figures 88a and 88b at the moment t = 3 s and 6 s after switching on electric current, respectively [77]. It should be noted that groundbased experiments were carried out in the electrophoretic cells of large thickness (on the y direction). In this case, the hydrodynamic instability development is damped by the viscous interaction of fluid with the two x-z planes.

Figure 87. The picture of solution jet in traveling flow of the tris-glycin buffer solution at the moments t = 3 s after switching on electric current was obtained during experiment aboard the “Mir” orbital station in 1990.

Figure 88. The pictures of hemoglobin solution jet in traveling flow of the tris-glycin buffer solution at the moments t = 3 s (a) and t = 6 s (b) after switching on electric current were obtained by the numerical calculation [77].

The second experiment was that the jet from coloured water was injected in traveling flow of clean water. In spite of the fact that electric current was absent and thermal gravitational convection was impossible, but the jet disintegrated very rapidly and paint extends over all the volume of the cell. Molecular diffusion process cannot proceed so fast. In

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this case, the surface tension between two aqueous media is extremely small and the negligible perturbation, for example vibration, can destroy a jet. The video-recording of this experiment was accompanying by a very strong noises from operating installation. One cannot to make out the words of A. A. Serebrov owing to noises. The poor quality of experiment video-recording interferes with using of this recording for publication.

Conclusion The analysis of experiments on space material sciences has shown that the initial premises at the beginning studies were incorrect and appreciably reflected the degree of incomprehension of the hydromechanical and heat and mass transfer processes in microgravity condition. It has taken several decennial events for undertaking the studies in the conditions, existing on board spacecraft, to contribute certain clarity in particularity of these processes applied to technology of crystal growth and to electrophoretic separaton of the biomixtures. Mathematical simulation of the process of crystal growth by method BridgmanStockbarger and of the ampoule method of the moving heater has shown that the uniform distribution of dopant on radius is possible at the reduction of the residual accelerations of gravity force on the 2-3 orders or when the static magnetic field is used. The longitudinal homogeneity of the dopant distribution for majority semiconductor materials is reached on too much length of the growing crystal and herewith connecting process does not depend on level of the remaining speedups of power to gravity. Furthermore, the transition process for steady-state concentration field of dopant in melt occurs similarly under full zero gravity. The undertaken modifications of these methods not only do not perfect the situation, but also, opposite, caused to increase of dopant segregation in crystals. By this means it is reasonably safe to confirm that all ampoule methods of crystal growth in microgravity conditions have a no outlook for the further studies. The floating zone method, which attracted vastly smaller attention, turned out to be more perspective for studies and improvements. Three different modifications of this method, namely, use of rotating magnetic field, encapsulation i.e. use of additional fluid layer, a creation of the standing surface waves by axial vibration, give essential reduction to dopant segregation in micro- and macro-scales. The additional positive effect is gotten if in the two last cases to use the static magnetic field. The numerical calculations have been demonstrated, as it is possible to achieve the maximum positive effect at the optimum parameters of the external action on crystallizing melt. These results have been obtained on a basis of the numerical studies of thermocapillary and thermal gravitational convection stability, the conditions of mutual symmetry for fluid flow and external influences. Amongst these modifications, a use of the standing surface waves in combination with axial static magnetic field is most interesting and perspective. In the numerical calculations it was discovered that standing surface waves, generated by vibration directed along free cylindrical surface of the liquid, possess unknown earlier properties. The generalization of numerical study results has allowed the connexion between the wave period number and its oscillation frequency with the parameters of applied vibration, geometric sizes and physical properties of the fluid column. By the end of 90-th years the results were received, which did not leave doubts in that that floating zone method with modifications and pertaining with him scientific problems must be

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priority in the field of space material sciences. The method and the facility for the experimental verification of standing surface waves characteristics in microgravity conditions were prepared. The experiment under name "Ivolga" was included in program of the scientific experiments on Russian segment of the International space station in 1998. The model experiment executed in 1990-th year on the orbital station "Mir" has graphically shown that the failure of biomixture separation by the continuous flow electrophoresis in microgravity conditions have been risen from hydrodynamic instability of biocomponent jet. The hydrodynamic instability of this jet could be conditioned by the action of vibration and (or) ponderomotive force in electric field. Under random orientation of the vibration vector in space, the Rayleigh-Taylor and Kelvin-Helmholz instabilities can be arised. This scientific problem could be timely solved because by which time the programs for the numerical investigation of fluid flows and processes of the heat and mass transfer in fluid volumes with free moving boundary have been developed. Unfortunately, this was not made. Both process of crystal growth by the floating zone method with use of the standing surface waves and continuous flow electrophoresis posed the new scientific problem: to study the behaviour of the gas-liquid and liquid-liquid boundaries by the action of different perturbations in condition of zero gravity. This problem was considered in purely demonstration aspect earlier. At present, the study of this problem is not only scientific but also practical interest. Results of these studies can be used in space technology not only, but also in terrestrial industrial processes. The structure of fluid flow due to the standing surface wave action on thermocapillary flow in the floating zone method evidences that this method can be used under crystal growth in terrestrial conditions. As shown in experiments on model fluids [81], low-frequency vibrations of crystal are shaped the specific pictures of standing waves on free surface of fluid for geometry of the Czochralski method. The problem of the standing surface wave use under crystal growth by the Czochralski method was not considered in other scientific publications. It is probable that in this case investigation with the aim of parameter standing wave optimization will allow an perfection of the Czochralski method. Thereby, currently new directions of investigations are defined in the field of space material sciences which should lead to the solution of problems set 40 years ago.

References [1] Witt, A. F., Gatos, H. C., Lichtensteiger, M. et al. (1975). Crystal growth and steadystate segregation under zero gravity: InSb. J. Electrochem. Soc., vol. 122, No 2, 276-283. [2] Witt, A. F., Gatos, H. C., Lichtensteiger, M., and German, C .I. (1978). Crystal growth and segregation under zero gravity: Ge. J. Electrochem. Soc., vol. 125, No 11, 18321840. [3] Ivanov, L. I., Zemskov, V. S., et al. (1978). Melting, crystallization and formation of phases in microgravity conditions. The “Multipurpose Furnace” experiment on the “Soyus-Apollon” programme, 197p. Moscow, USSR: Nauka. (in Russian). [4] Feonychev, A. I., and Dolgikh, G. A. (2001). Effects of Constant and Variable Accelerations on Crystal Grown onboard Spacecraft by the Method of Directional

230

[5]

[6]

[7]

[8]

[9] [10] [11]

[12]

[13]

[14] [15]

[16]

[17]

A.I. Feonychev Crystallization. J. Cosmic Research, vol. 39, No 4, 365-373 (translated from Kosmicheskie Issledovaniya. (2001), vol. 39, No 4, 390-399). Feonychev, A. I., Dolgikh, G. A. (2004). Effects of Local Detachment of a Melt from Ampoule Walls, Crystal Vibration, and Magnetic Field at Crystal Growth under Zero Gravity. J. Cosmic Research, vol. 42, No 2, 117-128 (translated from Kosmicheskie Issledovaniya. (2004). Vol. 42. No 2, 123-135). Feonychev, A. I., Dolgikh, G. A., and Kalachinskaya, I. S. Convective Heat and Mass Trasfer the Production of Materials in Microgravity. (1992). In Walter, H. (Ed.). Hydrodynamics and Heat/Mass Transfer in Microgravity, pp. 47-52. Switzerland: Gordon and Breach Sci. Publ. Feonychev, A. I., Dolgikh, G. A. (1995). Effects of magnetic field on crystal growth process under action of gravity and capillary force. Ninth European Symp. “GravityDependent Phenomena in Physical Sciences”. 2-5 May, Berlin, Germany. Abstracts, p. 246. Feonychev, A. I., and Kalachinskaya, I .S. (2001). The Impact of Variable Accelerations on Crystal Growth onboard Spacecraft by the Floating Zone Method. J. Cosmic Research. (2001), vol. 39, No 4, 374-379 (translated from Kosmicheskie Issledovaniya, vol. 39, No 4, 400-406). Material Sciences in Space. (1986). Feuerbacher, B., Hamacher, H., Naumann, R. (Eds.). Berlin, Heidelberg, Germany: Springer-Verlag. Fluid Sciences and Materials Science in Space. A. European Perspective. (1987). Walter, H. U. (Ed.). Berlin, Heidelberg, Germany: Springer-Verlag. Materials and Fluids Under Low Gravity. (1996). Ratke, L., Walter, H., Feuerbacher, B. (Eds.). Proc. of the IXth European Symposium on Gravity-Dependent Phenomena in Physical Sciences. (Lecture notes in physics; vol. 464). Berlin, Germany, 2-5 May 1995. Berlin, Heidelberg, Germany: Springer-Verlag. Muller, G. Convection and Inhomogeneities in Crystal Growth from the Melt. Crystals, Growth, Properties, and Applications. Vol. 12. (1988). Berlin, Heidelberg, Germany: Springer-Verlag. Dolgikh, G. A., and Feonychev, A. I. (1982). Numerical investigation of processes of heat and mass exchange under directional crystallization in weightlessness conditions. In Problems of mechanics and heat exchange in space engineering. O. M. Belozerkovskiy (Ed.), 224-232 . Moscow, USSR: Mashinostroenie. Feonychev, A. I., Medov, V. M., and Savichev, S. S. Inventor’s Certificate No 1306169 with piority dated by March 07. 1985. Polezhaev, V. I., and Vedyushkin, A. I. (1980). Hydrodynamic Effects of the Concentration Stratification in Closed Volumes. Fluid Dynamics, vol. 15, No 3. (Translated from Izv. Akad. Nauk USSR, Mekh. Zhidk. Gaza. (1980), No 3, 11-18). Feonychev, A. I. (1981). Numerical Study of the Heat and Mass Exchange Processes during the Crystallization under the Action of Slight Accelerations of Mass Forces. Fluid Dynamics, vol.16, No 1, 155-160. (Translated from Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. (1981), No 1, 186-192). Alexander, J. I. D. (1990). Low-Gravity Experiment Sensitivity to Residual Accelerations: A Review, Microgravity Sci. Technol., vol. 97, p. 52.

Causes of Failures and the New Prospects in the Field of Space Material Sciences

231

[18] Alexander, J. I. D., Amiroudine, S., Ouazzani J., and Rosenberger, F. Analysis of the Low Gravity Tolerance of the Bridgman-Stockbarger Crystal Growth. II Transition and Periodic Accelerations. (1991), J. Cryst. Growth, vol. 113, 21. [19] Gershuni, G. Z., and Zhukhovitskii, E. M. (1979). About free thermal convection in vibration field in weightlessness conditions. Doklady Acad. Nauk USSR, vol. 249, No 3, 580-585 (in Russian). [20] Feonychev, A. I. (2004). Application of the eigenfrequency of a convective cell to determination of the frequency characteristics of oscillatory thermal gravitational and thermocapillary convection in closed volumes. J. of Engineering Physics and Thermophysics, vol. 77, No 5, 990-998. (Translated from Inzhenerno-Fizicheskii Zhurnal. (2004), vol. 77, No 5, 105-112). [21] Feonychev, A. I., Popov, G. A., and Kim, V. 2000. Use of electric propulsion for compensation of atmospheric drag acceleration during experiments on space material sciences. Proc. 3rd Intern. Conf. on Space Propulsion. Cannes, 10-13 October, 2000. ESA Sp-465, 869-872. [22] Watring, D. A., Gillies, D. C., Lechotsky, S. L. et al. (1996). Convective Influence on Radial Segregation during Unidirectional Solidification of the Binary Alloy HgCdTe. Proc. of SPIE. Vol. 2809. Space Processing of Materials. Denver, Co. USA. 4-5 August 1996, 57-68. [23] Wilcox, W. R., and Regel, L. L. (1995). Detached Solidification. Microgravity Sci. Technol., VIII/1, 56-61. [24] Regel, L L., Popov, D. I., and Wilcox, W. R. 1995. Modeling of Detached Solidification. 46th International Astronautical Congress. October 2-6, 1995. Oslo. Norway. IAF-95J.3.08. [25] Feonychev, A. I., and Dolgikh, G. A. (1994). Influence of vibration on heat and mass transfer in microgravity conditions. Microgravity Quart., vol. 4, No 4, 233-240. [26] Salk, M., Eiche, C., Fiederle, M., Joerger, W., Benz, K. W., Senchenkov, A. S. & Egorov, A. V. (1995). The Influence of Different Material Transport Regimes on the Radiation Detector Properties of CaTe:Cl, Cd0.9Zn0.1Te:Cl, and CdTe0.9Se0.1:Cl. Ninth European Symposium “Gravity-Dependent Phenomena in Physical Sciences” , Berlin, Germany, 2 – 5 May, 1995. Abstracts, 253-254. [27] Feonychev, A. I., and Bondareva, N. V. (2004). Effect of a rotating magnetic field on convection stability and crystal growth in zero gravity and on the ground. J. Phys. and Thermophys., vol. 77, No 4, 731-742 (translated from Inzhenerno-Fizicheskii Zhurnal (2004), vol. 77, No 4, 50-61). [28] Croll, A., Muller, W., and Nitsche, R. (1986). Floating-zone growth of surface-coated silicon under microgravity. J. Crystal Growth, vol. 79, 65-70. [29] Croll, A., Muller-Sebert, W., Benz, K. W., and Nitsche, R. (1991). Natural and thermocapillary convection in partially confined silicon melt zones. Microgravity Sci. Technol., III/4, 204-215. [30] Barmin, I. W., Gel’gat, Yu. M., Senchenkov, A. S., and Smirnova, I. G. (1988). Influence of a magnetic field on crystallization of semiconducting materials by the method of crucible-less zone melting under weighrlessness conditions. Magn. Gidrodyin, No 4, 110-116 (in Russian). [31] Devdariani, M. T., Pelevin, O. V., Prostomolotov, A. I., Rubinraut, A. M., and Verezub, N. A. (1992). Computer analysis of combined rotating and static asial magnetic fields

232

[32]

[33]

[34]

[35]

[36] [37] [38]

[39]

[40] [41] [42]

[43]

[44] [45]

A.I. Feonychev influence on flow and heat transfer in Cz-silicon crystal growth. Institute for Problem in Mechanics, the USSR Academy of Sciences. Preprint No 516. Moscow, 32p. Gelfgat, Yu. M., Sorkin, M. Z., Priede, J., and Mozgirs, O. (1992). On the possibility of heat and mass transfer and interface shape control by electromagnetic effect on melt during unidirectional solidification. In Walter, H. (Ed.). Hydrodynamics and Heat/Mass Transfer in Microgravity, pp. 429-434. Switzerland: Gordon and Breach Sci. Publ. Senchenkov, A. S., Friazinov, I. V., and Zabelina, M. P. Mathematical modeling of convection during crystal growth by the THM. (1992). In Walter, H. (Ed.). Hydrodynamics and Heat/Mass Transfer in Microgravity, pp. 455-459. Switzerland: Gordon and Breach Sci. Publ. Marchenko, M. P., Senchenkov, A. S., and Fryazinov, I. V. (1992). Mathematical simulation of the process of crystal growth from the solution-melt by the moving-heater method. Math. Model, vol. 4, No 5, 67-79. Dold, P., Croll, A., Lichtensteiger, M., Kaiser, Th., and Benz, K. W. (1995). Floating zone growth of silicon in magnetic fields: IV. Rotating magnetic fields. J. Cryst. Growth, vol. 231, No 1-2, 95-106. Pearson, J. R. A. (1958). On convection cells induced by surface tension. J. Fluid Mech., vol. 4, pt. 5, 489-500. Sterling, C. V., and Scriven, L. E. (1959). Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AIChE J., vol. 5, No 4, 514-523. Nepomnyashchy, A. A., and Simanovski, I. B. (1983). Thermocapillary convection in two-layer system. Izv. Akad. Nauk of USSR, series Mekh. Zhidk. I Gasa, No 4, 158-163 (in Russia). Gershuni, G. Z., Zhuchovitski, E. M., Nepomnyashchy, A. A., and Simanovski, I. B. (1984). Development of disturbances and transition to turbulence at the convection in two-layer systems. In Laminar – Turbulent Transition. IUTAM Symp. Novosibirsk, USSR: Springer-Verlag. 1985, 749-754. Doi, T., and Koster, J. N. (1993). Thermocapillary convection in two immiscible liquid layer with free surface. Physics of Fluid A: Fluid Dynamics, vol. 5, 1914-1927. Koster, J. N. (1994). Multilayer fluid dynamics of immiscible liquids. Second Microgravity Fluid Physics Conference. NASA Conference Publication .3276, 65-71. Crespo del Arco, E., Extremet, G. P., and Sani, R. L. (1994). Thermocapillary convection in a two-layer fluid system with flat interface. Adv. Space Res., vol. 11, No 7, 129-132. Georis, P, and Legros, J. C. (1996). Pure Thermocapillary Convection in a Multilayer System: First Results from the IML-2 Mission. In L. Ratke, H. Walter, and B. Feuerbacher (Eds.), Materials and Fluids Under Low Gravity, pp. 299-311. Berlin: springer-Verlag. Ramachandran, N. (1993). Thermal Buoyancy and Marangoni Convection in a Two Fluid Layered System. J. Thermophysics and Heat Transfer, vol. 7, No 2, 352-360. Feonychev, A. I., Dolgikh, G. A., and Kalachinskaya, I. S. (1992). Convective heat and mass transfer in the production of materials in microgravity. In H. Walter (Ed.), Hydromechanics and Heat/Mass Transfer in Microgravity, pp. 47-52. Amsterdam: Gordon and Beach Sci. Publ.

Causes of Failures and the New Prospects in the Field of Space Material Sciences

233

[46] Feonychev, A. I. (1995). Comparative analysis of thermocapillary convection in oneand two-layer systems. Problem of oscillatory convection. Adv. Space Res, vol. 16, No 7, 59-65. [47] Feonychev, A. I., and Pokhilko, V. I. (1996). Use of magnetic fields under crystal growth in space and on Earth. Crystallization on two-layer liquid systems in weightlessness with and without an external magnetic field. In The 20th Intern. Symp. On Space Technol. and Sci. May 1996. Gify, Japan. Abstract 96-d-22. [48] Feonychev, A. I., Ramachandran, N., Pokhilko, V. I., and Mazuruk, K. (1997). Flow and crystallization in two-layer liquid systems with and without a magnetic field. In Material Research in Low Gravity, vol. 3123, pp. 272-282. Proc. of SPIE Conference. San Diego California. 28-29 July 1997. Washington: The Society of Photo-Optic. Inst. Eng. [49] Feonychev, A. I. (2004). Method of single crystal growth by the floating zone method. Patent No 10885. Republic of Byelorussia. 2004.12.27. [50] Feonychev, A. I. Crystal growth in two-layer liquid systems in microgravity conditions. J. Eng. Phys. and Thermophys. (in press). [51] Lopez, C. R., Mileham, J. R., and Abbaschian, R. (1999). Microgravity growth of GaAs single crystal by the liquid encapsulated melt zone (LEMZ) technique. J. Crysr. Growth, vol. 200, 1-12. [52] Meseguer, J., and Perales, J. M. (1992). Non-steady phenomena in the vibration of viscous cylindrical long bridges. Microgravity Sci. Technol., vol. 5, 69-72. [53] Bauer, H. F. (1990). Axial response and transient behavior of a cylindrical liquid column in zero-gravity. Z. Flugwiss. Weltraumforshung, Bd. 14, 174-182. [54] Schulkes, R. S. M. (1991). Liquid bridge oscillations: analytical and numerical results. Microgravity Sci. Technol., No IV/2, 71-72. [55] Martinez, I., Perales, J. M., and Meseguer, J. (1996). Response of a liquid bridge to an acceleration varying sinusoidally with time. In L. Ratke, H. Walter, and Feuerbacher (Eds.), Materials and Fluids under Low Gravity, pp. 271-279. Berlin: Springer. [56] Kamotani, Y., and Ostrach, S. (1991). Effects of g-jitter liquid free surfaces in microgravity. Microgravity Sci. Technol., No IV/2, 144-145. [57] Feonychev, A. I., Kalachinskaya, I. S., and Pokhilko, V. I. (1996). Deformation of fluid column by action of axial vibration and some aspects of high-rate thermocapillary convection. In Proc Third Microgravity Fluid Physics Conf., Cleveland, Ohio, USA, June 13-15, 1996. NASA Conf. Publ. No 3338, pp. 493-498. [58] Feonychev, A. I., and Kalachinskaya, I. S. (2001). Effect of variable accelerations on crystal growth by the floating-zone method onboard spacecraft. J. Cosmic Research., vol. 39, No 4, 374-379. (Translated from Kosmicheskie Issledovaniya. (2001), vol. 39, No 4, 400-406). [59] Feonychev, A. I. (2004). Inertial-capillary surface waves and their influence on crystal growth in zero gravity. J. Eng. Phys. and Thermophys., vol. 77, No 2, 348-359. (Translated from Inzhenerno-Fizicheskii Zhurnal. (2004), vol. 77, No 2, 83-92). [60] Feonychev, A. I. (2007). Stability of thermocapillary convection and regimes of a fluid flow acted upon by a standing surface wave. J. Eng. Phys. and Thermophys., vol. 80, No 5, 961-969. (Translated from Inzhenerno-Fizicheskii Zhurnal. (2007), vol. 80, No 5, 108-115). [61] Feonychev, A. I. (2007).Use of standing surface waves and of a stationary magnetic field in growing crystals by the floating-zone method in weightlessness. J. Eng. Phys.

234

[62] [63] [64]

[65]

[66] [67] [68] [69] [70] [71] [72] [73] [74]

[75]

[76]

[77]

[78]

A.I. Feonychev and Thermophys., vol. 80, No 5, 970-976. (Translated from Inzhenerno-Fizicheskii Zhurnal. (2007), vol. 80, No 5, 116-121). Hirt, C. W., Nicols, B. D. (1981). Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comp. Phys., vol. 39, 201-205. Davis, S. H., and Homsy, G. M. Energy stability theory for free-surface problem: buoyanvy-thermocapillary layers. (1980). J. Fluid Mech., vol. 98, pt. 3, 527-553. Kawaji, M., Liang, R., Nasr-Esfahany, M., Simic-Stefani, S., and Yoda, Sh. (2004). The effect of small vibrations on Marangoni convection and the free surface of a liquidbridge. 55th Intern. Asronautical Congress. Vancouver, Canada. Paper IAC-04-J.08, 10p. Duan, L., Kang, Q., and Hu, W. R. (2004). Experimental study on surface deformation and surface wave of fluid convection with free surface. 55th Intern. Asronautical Congress. Vancouver, Canada. Abstract IAC-04-J.08, 1p. Feonychev, A. I. (2004). Method and equipment for single crystal growth by the floating zone method. Patent No 10056. Republic of Byelorussia. 2004.12.27. Hennig, K., and Wirt, T. (1980). Continuous flow electrophoresis in space. In Space Technology. (Russian translation). Moscow, USSR: Mir, pp. 275-283. Dukhin, S. S., and Deryagin, B. V. (1976). Electrophoresis. (in Rissian). Moscow, SSSR: Nauka, 328p. Ivory, C. F. (1980). Continuous flow electrophoresis. The crescent phenomena revisited. Pt. 1. Isothermal effects. J. Chromatogr. Vol. 195, No1, 165-179. Giannovario, J. A., Griffin, R. N., and Gray, E. L. (1978). A mathematical model of free-flow electrophoresis. J. Chromatogr. Vol. 153, No 2, 329-352. Mikkers, F. E. P., Everaerts, F. M., and Verheggen, Th. P. E. M. (1979). Concentration distributions in free zone electrophoresis. J. Chromatogr. Vol. 169, No 1, 1-10. Zhukov, M. Yu., and Yudovich, V. I. (1982). Mathematical model of isotachophoresis. Doklady Akad. Nauk SSSR. Vol. 267, No 2, 334-338. Babskii, V. G., Zhukov, M. Yu., and Yudovich, V. I. (1983). Mathematical theory of Electrophoresis. (in Russian). Kiev, SSSR: Navukova Dumka. 202 p. Mazhorova, O. S., Popov, Yu. P., Pokhilko, V. I., and Feonychev, A. I. (1987). Mathematical simulation of fluidic electrophoresis. SSSR Academy of Sciences. Preprint No 219. (in Russian). Moscow. 32 p. Mazhorova, O. S., Popov, Yu. P., Pokhilko, V. I., and Feonychev, A. I. (1988). Numerical investigation of liquid electrophoresis without support media. J. Fluid Dynamics. Vol. 23, No 3, 333-339. (Translated from Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. (1988), No 3, 14-20). Mazhorova, O. S., Popov, Yu. P., and Pokhilko, V. I. (1986). Difference scheme with artificial dispersion for an equation. USSR Academy of Sciences. Preprint No 142. (in Russian). Moscow. 33 p. Aksenov, A. A., Golovinkin, A. V., Meshkov, M. A., Gudzovsky, A. V., and Serebrov, A. (1991). Numerical and experimental investigation of increased concentration sample separation by continuous flow electrophoresis in space. In Proc. of AIAA/IKI Microgravity Science Symposium. Moscow, USSR. May 13-17, 1991, 216-220. Ermakov, S. V., and Feonychev, A. I. Thermocapillary convection in cavities wth one and two curvilinear free surfaces. (1987). Fluid Dynamics, vol. 22, No 3. (Translated from Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. (1987), No 3, 9-15).

Causes of Failures and the New Prospects in the Field of Space Material Sciences

235

[79] Landau, L. D., Lifshiz, E. M. (1982). Theoretical physics. Vo. VIII. Electrodynamics of continuous media. Moscow, USSR: Nauka, 620 p. (In Russian). [80] Myaldun, A. Z. (1995). The action of low-frequency vibrations on heat and mass transfer under crystal growth from liquid phase. Ph.D Thesis. The General Physics Institute. Academy of Sciences of the USSR. Moscow (in Russian).

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 237-293

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 3

MODELING OF INTERACTION KINETICS DURING COMBUSTION SYNTHESIS OF ADVANCED MATERIALS: PHASE-FORMATION-MECHANISM MAPS B.B. Khina* Physico-Technical Institute, National Academy of Sciences of Belarus.

Abstract Combustion synthesis (CS), or self-propagating high-temperature synthesis (SHS) is a versatile and cost efficient method for producing refractory compounds (carbides, borides, intermetallics) and composite materials. During CS, interaction between condensed reactants accomplishes in a short time (~0.1-1 s) whereas the traditional furnace synthesis of the same compounds takes several hours for the same particle size and close final temperature. Uncommon, non-equilibrium interaction mechanisms were observed experimentally, e.g., the dissolution-crystallization route rather than the traditional solid-state diffusion-controlled (SSDC) growth of a continuous product layer separating the starting reactants. Despite extensive experimental and theoretical investigation, the interaction pathways during CS are not well understood yet. In this Chapter, the analysis of existing models of SHS is presented with special emphasis on the kinetics of interaction in strongly non-isothermal conditions typical of SHS. It is shown that in the modeling works employing the most used SSDC kinetics of the product formation, the diffusion coefficients used for calculations exceeded the experimentally known values by up to 3 orders of magnitude in a wide range of temperature. New models are developed for two typical SHS-reactions, Ti+C→TiC (CS of interstitial compound) and Ni+Al→NiAl (CS of intermetallic compound), basing on the SSDC kinetics and independent data on diffusion in the product phase. For CS of TiC, all possible situations are analyzed. Elastic stresses in a spherical TiC layer growing on the Ti particle surface are calculated, and a criterion for transition to the non-equilibrium dissolution-precipitation route is obtained. For CS of NiAl, competition between the growth of solid NiAl and its dissolution in the liquid Al-base and solid or liquid Ni-base solutions is considered for non-isothermal *

E-mail address: [email protected], [email protected]. 10 Kuprevich St., Minsk 220141, Belarus.

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B.B. Khina conditions. The Ni-Al phase diagram is used for numerical modeling along with the temperature dependencies of phase densities. Simulation has revealed the limits of applicability of the traditional SSDC approach, which is based on the assumption of local equilibrium at phase boundaries. The criteria are determined for transition to non-equilibrium reaction routes, namely dissolution-precipitation with and then without a thin solid interlayer of NiAl between the parent phases. As a final result, phase-formation-mechanism maps for the Ti-C and Ni-Al systems are constructed in coordinates “initial metal particle size-heating rate”, which permit predicting a pattern of structure formation during interaction in the non-isothermal conditions typical of CS. The existence of uncommon interaction pathways, which were observed experimentally and debated in literature, is confirmed theoretically ex contrario.

1. Introduction: Advances and Challenges in Modeling Combustion Synthesis 1.1. Approaches to Modeling Non-isothermal Interaction Kinetics during CS Combustion synthesis (CS), or self-propagating high-temperature synthesis (SHS), also known as solid flame, is recognized as a versatile, cost and energy efficient method for producing refractory compounds (carbides, borides, nitrides, intermetallics, complex oxides etc.) and advanced composite materials possessing fine-grain structure and superior properties. The advantages of CS include short processing time, low energy consumption, high product purity due to volatilization of impurities, and unique structure and properties of the final products. Besides, CS can be combined with pressing, extrusion, casting and other processes to produce near-net-shape articles [1-8]. Despite vast literature available in this area, CS is still a subject of extensive experimental and theoretical investigation. Combustion synthesis can be carried out in the wave propagation mode, or “true SHS”, and in the thermal explosion (TE) mode. In the former case, a compact reactive powder mixture is ignited at one end to initiate an exothermic reaction which propagates through the specimen as a combustion wave leaving behind a hot final product [1-8]. In the latter case, a pellet is heated up at a prescribed rate (typically 1-100 K/s) until at a certain temperature called the ignition point, Tign, an exothermal reaction becomes self-sustaining and the temperature rises to its final value, TCS, almost uniformly throughout the sample. Typically, the value of Tign is close to the melting point of a lower-melting reactant or to the eutectic temperature. The unique features of the obtained products are ascribed to extreme conditions inherent in CS, which may bring about unusual reaction routes: (i) high temperature, up to 3500 °C, (ii) a high rate of self-heating, up to 106 K/s, (iii) steep temperature gradient in SHS waves, up to 105 K/cm, (iv) rapid cooling after synthesis, up to 100 K/s, and (v) fast accomplishment of conversion, from about 1 s to the maximum of 10 s [1-4]. It should be noted that traditional furnace synthesis of refractory compounds requires a much longer time, ~1-10 h, for the same initial composition, particle size and close final temperature. It has been demonstrated experimentally [9-16] that in many systems phase and structure formation during CS proceeds via uncommon interaction mechanisms from the point of view of the classical Physical Metallurgy [17,18].

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Modeling and simulation traditionally play in important part in the development of CS and CS-related technologies (see reviews [1-3,19-23] and references cited therein). An adequate mathematical model is supposed to describe both heat transfer in a heterogeneous reactive medium and the interaction kinetics, which is responsible for heat release during CS. In modeling CS, a quasi-homogeneous, or continual model [24,25], which is based on classical combustion theory, is widely used. Heat transfer, which is considered on the volume-averaged basis, and the reaction rate in a sample are described as follows: ρcp ∂T/∂t = ∇(λ∇T) + Q ∂η/∂t

(1)

∂η/∂t = (1−η)n exp(–mη) k exp(−E/RT),

(2)

where T is temperature, ρ is density, cp is heat capacity, λ is thermal conductivity, Q is the heat release of exothermal reactions, η is the degree of chemical conversion (from 0 in the unreacted state to 1 for complete conversion), n (the reaction order) k (preexponential factor) and m are formal parameters and E is the activation energy. This approach permitted modeling dynamic regimes of SHS, e.g., oscillating [24] and spin combustion [26,27]. It was also used for studying the effect of intrinsic stochasticity of heterogeneous reactions, which can be attributed to a difference in the surface morphology, impurity content and hence reactivity of solid reactant particles, on the dynamic behavior of a solid flame for a one-stage [28] and multi-stage reaction [29] employing the cellular automata method. It should be outlined that this model is not linked to any process-specific phase formation mechanism and hence is referred to as a formal one. When applying this approach to modeling CS in a particular system, the value of the most important model parameter, viz. activation energy E, is supposed to correspond to the apparent activation energy of the CS as a whole. The latter is determined from experimental graphs “the combustion wave velocity vs. temperature” plotted in the Arrhenius form, and in its physical meaning corresponds to a real rate-limiting stage of phase formation during CS, which may be different in different temperature ranges. For example, below the melting temperature, Tm, of a metallic reactant E always refers to solid-state diffusion in the product while at T>Tm it can refer to processes in the melt (diffusion or crystallization) [30]. This method for choosing the E value was used when studying numerically the conditions of arresting a high-temperature state of substances in the SHS wave by fast cooling for the cases of a one-stage [31] and two-stage exothermal reaction [32]. In recent papers [33,34], this formal model [see Eqs. (1) and (2)] was employed for studying the SHS of a NiTi shape memory alloy. The activation energy used in calculations was E=113 kJ/kg, which is equivalent to 12.05 kJ/mol (because the molar mass of NiTi is 106.6 g/mol). This is an extraordinary low value for a reaction in a condensed system and can correspond only to diffusion in a transient melt formed in the CS wave. However, according to reference data [35], the activation energy for diffusion in some pure liquid metals is the following: Li, E=12 kJ/mol; Sn, E=11.2 kJ/mol; Zn, E=21.3 kJ/mol; Cu, E=40.7 kJ/mol; Fe, E=51.2 kJ/mol. Thus the value of E used for calculations in [33,34] is close to that for diffusion in low-melting metals such as Li or Sn, and is by the factor of 4 lower than for iron whose melting point, Tm, lies between Tm of Ni and Ti (the activation energy for diffusion in liquid metals is known to be proportional to Tm [36]). All the more, this E value is

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incomparably lower than a typical activation energy for diffusion in intermetallic compounds. Hence in this case the most important parameter of the formal model, E, appears to be physically meaningless. Recently, new features of SHS were observed experimentally [37-40]. First, microscopic high-speed video recording [37,38] and photographing [39] demonstrated microheterogeneous nature of SHS which revealed itself in the roughness of the combustion wave front, chaotic oscillations of the local flame propagation rate and new dynamic behaviors such as relay-race, scintillation and quasi-homogeneous patterns. Second, the formation of nonequilibrium structure and composition of SHS products was examined experimentally and interpreted qualitatively in terms of relationships between characteristic times of reaction tr, structuring ts and cooling tc [40]. These features were attributed to two main factors: inhomogeneous heat transfer in the charge mixture and a specific reaction mechanism [39]. These results gave rise to new, heterogeneous models [41-44] involving heat transfer on the particle-to-particle basis [41-43] and percolation phenomena in a system of chaotically distributed reactive and inert particles [44]. However, in these models the traditional formal kinetics for a thermal reaction [Eq. (2)] was employed. Thus, an urgent and still unresolved problem in CS is an adequate description of fast interaction kinetics in a unit reaction cell containing particles of dissimilar reactants whose composition corresponds to the average composition of a charge mixture. The most widely used kinetic model, which is connected to a particular phase forming mechanism, is a “solid-state diffusion-controlled growth” concept first applied to CS in [45] for planar symmetry and in [46] for spherical symmetry of an elementary diffusion couple. As in a charge mixture there are contacts of dissimilar particles, a layer of an intermediate or final solid product forms upon heating thus separating the initial reactants. The growth rate of the reaction product and associated heat release necessary for sustaining combustion is controlled by solid-state diffusion through this layer. Then, the diffusion-type Stefan problem is formulated instead of Eq. (2). However, as demonstrated below in more detail, in most cases modeling was performed not with real diffusion data, which are known for many refractory compounds, but using either dimensionless coefficients varied in a certain range or fitting parameters chosen to match the calculated and measured results of the SHS temperature profile and velocity. Besides, in most of the CS-systems fast interaction begins after fusion of a lower-melting-point reactant [1-3,25] but within this approach melting does not alter the phase layer sequence in an elementary diffusion couple [45,46]. A number of experimental results obtained by the combustion-wave arresting technique in metal-nonmetal (Ti-C [10,11], (Ti+Ni+Mo)-C [12], Mo-Si [13]) and metal-metal (Ni-Al [14,16]) systems gave rise to an qualitative notion of a non-traditional phase formation route. It involves dissolution of a higher-melting-point reactant (metal or non-metal) in the melt of a lower-melting-point reactant and crystallization of a final product from the saturated liquid. Besides, there is much controversy over the presence of an intermediate solid phase in the dissolution-precipitation route. In [14] it is concluded that during SHS in the Ni-Al system, solid Ni dissolves in liquid Al through an interlayer separating it from nickel, which agrees with the phase diagram. In this case, the rate-limiting stage is solid-state diffusion across this layer. But in [16] for the same system it is found that above 854 °C a solid interlayer between nickel and molten Al is absent; then the overall interaction during CS is controlled by either diffusion in the melt or crystallization kinetics.

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Such a situation is considered in recent models [47-52], where a solid reactant (nickel [47-49] or carbon [50-52]) dissolves directly in the liquid based on a lower-melting component (Al and Ti, respectively) and product grains (NiAl and TiC, correspondingly) precipitate from the melt; the rate-limiting stage is liquid-phase diffusion [47-49] or crystallization kinetics [50-52]. However, within these approaches the fundamental problem of the existence of a thin solid-phase interlayer at the solid/liquid interface is not discussed nor a criterion is obtained for transition between the solid-state diffusion-controlled mechanism and the dissolution-precipitation route with or without a thin interlayer. Hence, the applicability limits of the existing modeling approaches have not been clearly determined so far. The role of high heating rates, which are intrinsic in CS, in most of the models is not accounted for in an explicit form. Thus, adequate description of the interaction kinetics in condensed heterogeneous systems in non-isothermal conditions of CS is an urgent problem in this area of science and technology, and the absence of a comprehensive model hinders the development of new CSbased processes and novel advanced materials. Hereinafter the situation where a reaction between condensed reactants proceeds through a solid layer, i.e. solid reactant (C for the Ti-C system or Ni for the Ni-Al system)/solid final or intermediate product (TiC or one of intermetallics of the Ni-Al system, respectively)/liquid (Ti or Al melt), will be provisionally called “solid-solid-liquid mechanism” since the interaction occurs at both solid/solid and solid/liquid interface. This term will be used both for the “solid-state diffusion-controlled growth” pattern where the product layer is growing and for dissolution-precipitation route where the interlayer remains very thin. As the diffusion coefficient in a melt is much higher than in solids, the rate-limiting stage in this mechanism is diffusion across the solid interlayer. The second route, viz. dissolution-precipitation without an interlayer, can be referred to as “solid-liquid mechanism” since the interaction of condensed reactants (solid C or Ni with molten Ti or Al, respectively) occurs at the solid/liquid interface while the product (TiC or NiAl) crystallizes from the melt. However, up to now the solid-liquid mechanism has not been validated theoretically, nor the applicability limits of the solid-solid-liquid mechanism based on solid-state diffusion kinetics have ever been determined with respect to strongly non-isothermal conditions typical of CS. Thus, the main goal of this Chapter is to develop a system of relatively simple estimates and evaluate the applicability limits of the “solid-solid-liquid mechanism” approach to modeling CS and determine criteria for a change of interaction routes basing the calculations on experimental data to a maximum possible extent [53,54]. Below, a brief discussion of the diffusion concept of CS is presented. Then, calculations for particulars systems, viz. Ti-C and Ni-Al, are performed using available experimental data on both the diffusion coefficients in the growing phase and thermal characteristics of CS. For the Ti-C system, different situations are considered that can arise during CS within the frame of the above concept and, wherever possible, a quantitative and/or qualitative comparison between the outcome of calculations and experimental results is drawn. Emphasis is made on the structural characteristics of the CS product, titanium carbide, that emerge from this approach. The conditions for a change of the geometry of a unit reaction cell in the SHS wave due to melting of a metallic reactant (titanium) are analyzed and a micromechanistic criterion for the changeover of interaction pathways is derived. For the Ni-Al system, calculations within the frame of the diffusioncontrolled growth kinetics are performed taking into account both the growth of the product phase, NiAl, and its dissolution in the parent phases due to variation of solubility limits with

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temperature according to the equilibrium phase diagram. Finally, the “solid-liquid mechanism” concept for CS is justified and phase-formation mechanism maps for these two systems are plotted.

1.2. Brief Review of Diffusion-Based Kinetic Models of CS The interaction kinetics controlled by solid-state diffusion was used for numerical [45,46,55-62] and analytical [63] study of CS for the case of planar diffusion couples (alternating laminae of reactants) [45,56,57,59,63] and spherical symmetry (growth of a product layer on the surface of a spherical reactant particle) [46,58,60-62]. Inherent in this concept are two basic assumptions: (i) the phase composition of the diffusion zone between parent phases corresponds to the isothermal cross-section of an equilibrium phase diagram, i.e. the nucleation of product phases occurs instantaneously over all contact surfaces and (ii) the interfacial concentrations are equal to equilibrium values. This results in the parabolic law of phase layer growth [64-66]. It should be noted that in many diffusion experiments the phase layer sequence deviates from equilibrium: the absence of certain phases was observed in solid-state thin-film interdiffusion [67,68] and in the interaction of a solid and a liquid metal (e.g., Al) [69,70]. These phenomena were ascribed to a reaction barrier at the interface of contacting phases [71] without considering the nucleation rate of a new phase. The effect of a nucleation barrier was examined theoretically using the thermodynamics of nucleation [72,73] and the kinetic mechanism of phase formation in the diffusion zone [74], and it was shown that in the field of a steep concentration gradient the formation of an intermediate phase is suppressed [72-74]. This effect has never been considered in the diffusion models of CS. As in the theory of diffusion-controlled interaction in solids the nucleation kinetics is not included and it is assumed that critical nuclei of missing phases continuously form and dissolve [65,66], this qualitative concept is sometimes used in interpreting the results of CS [14]. It will be fair to say that deviation of phase-boundary concentrations from equilibrium due a reaction barrier was examined qualitatively for SHS [57] in the case of planar geometry. This effect is noticeable only in the low-temperature part of the SHS wave, and at high temperatures a strong barrier can only slightly decrease the combustion velocity [57]. Also, the influence of such barrier on self-ignition in the Ni-Al system at low heating rates, dT/dt<60 K/min, was studied quantitatively using experimental data on both thin-film interdiffusion in the NiAl3 compound [75], which is the first phase to form in Ni-Al diffusion couples, and bulk diffusion data [76]. Similar calculations were performed using an experimental temperature profile of SHS to determine the NiAl3 layer thickness formed below the melting point of aluminum Tm(Al)=660 °C [60]. At a thick NiAl3 layer (low heating rates) the reaction barrier is of little significance, but it can slow down the interaction for thin layers (higher heating rates) [76,60]: e.g., at dT/dt > 35 K/min the formation of the primary product can be suppressed [60]. But, as noted in [76,60], these results refer not to the SHS itself but only to a preliminary stage because fast interaction begins at T>Tm(Al), the combustion temperature reaches 1400 °C and the final product is NiAl [60]. It should be outlined that in many works using the diffusion model of CS the calculations were performed with dimensionless (relative) parameters varied in a certain range. A known or estimated value of the activation energy for diffusion in one of the phases was used only as

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a scaling factor and thus the results obtained revealed only qualitative characteristics of the process [45,46,55,59]. Besides, many of the modeling attempts [45,46] did not account for a change in the spatial configuration of reacting particles due to melting and spreading of a metallic reactant. The effect of melting was reduced to a change of interfacial concentrations and the ratio of diffusion coefficients in contacting phases [55]. In more recent papers [60,61], the parameter values (the activation energy E and preexponent D0) used for calculating the diffusion coefficient in a growing phase were presented. However, those were not the real values measured in independent works on solidstate diffusion but merely fitting parameters calculated from the characteristics of CS. For example, the formation of NiAl above 640 °C was modeled using D0=4.8×10–2 cm2/s and E=171 kJ/mol [60]. As noted in [60], this E value was the experimentally determined activation energy for the CS process as a whole. Then the diffusion coefficient in NiAl at T=1273 K is D = D0exp(−E/RT) = 4.6×10−9 cm2/s. Let’s compare it with experimental data on reaction diffusion in the Ni-Al system. For NiAl, D=(2.5–3.6)×10−10 cm2/s at T=1273 K [77]. The parameters for interdiffusion in this phase are E=230 kJ/mol and D0=1.5 cm2/s [78], hence at T=1273 K D=5.4×10−10 cm2/s. Thus, the diffusion coefficient used in modeling SHS exceeds the experimental value by an order of magnitude. SHS wave in the Ti-Al system with the Ti-to-Al molar ratio of 1:3 in the charge mixture was modeled using E=200 kJ/mol and D0=4.39 cm2/s for phase TiAl3 [61]. This E value was obtained from experiments on combustion synthesis using Arrhenius plots, and D0 was chosen to match the calculated and measured results of the propagation speed. Again, these values refer to the SHS wave as a whole but not to interdiffusion in TiAl3. However, experimental data on SHS of TiAl3 for the same starting composition, which were analyzed using the classical combustion model [see Eqs (1),(2)], gave a substantially higher activation energy: E=483 kJ/mol [79]. If solid-state diffusion in the TiAl3 layer is really the rate-limiting stage of the process, then the values of apparent activation energy ought to agree (within an experimental error) regardless of the particular form of a model. Since diffusion coefficients for many refractory compounds are known in literature, we can verify the validity of the diffusion-based kinetic model of SHS employing a somewhat opposite method: estimating the product layer growth and heat release using the experimental characteristics of SHS and independent diffusion data. For this purpose, two classical CSsystems are chosen, viz. Ti-C and Ni-Al.

2. Modeling Diffusion-Controlled Formation of TiC in the Conditions of CS CS in the Ti-C system was a subject of extensive theoretical and experimental studies [46,10,11,55,62]. It is a suitable candidate for investigation in this Chapter the following reasons: (i) the Ti-C phase diagram [80] (see figure 1) contains only one binary compound TiC whose melting temperature Tm(TiC)=3423 K exceeds the experimental SHS temperature TCS=3083 K [81] and (ii) numerous diffusion data for titanium carbide are available in literature [82-84]. We consider the case of spherical symmetry which better fits a typical configuration of reacting particles in CS. With respect to the phase diagram, here the solid-

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solid-liquid mechanism [situation C(solid)/TiC(solid)/ Ti(liquid)] is quasi-equilibrium and the solid-liquid mechanism [situation C(solid)/Ti(liquid)] is truly non-equilibrium. Let’s consider solid-state diffusion-controlled formation of the product, titanium carbide, during heating of the Ti+C charge mixture in the SHS wave. Typical particle radii are 5 to 100 µm for Ti, about 0.1 µm for carbon black and 1 to 30 µm for milled graphite [10,11,62,81]. Two scenarios with a different geometry of a unit reaction cell are examined: (1) a solid Ti particle surrounded by carbon particles in a stoichiometric mass ratio at temperatures below the Ti melting point, Tm(Ti)=1940 K [figure 2 (a and d)], and (2) a solid carbon particle surrounded by liquid titanium at T>Tm(Ti) [figure 2 (c and e)]. A condition for the change of the reaction cell geometry due to titanium melting [figure 2 (b)] is analyzed later.

Figure 1. The equilibrium Ti-C phase diagram [80].

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Figure 2. Schematic of an elementary reaction cell in the SHS wave in the Ti-C system (a and c) and corresponding concentration profiles for solid-state diffusion (d and e) [53]: (a and d) growth of the TiC layer on the surface of a titanium particle at T
2.1. Scenario 1: Growth of a TiC Case on the Titanium Particle Surface In scenario 1, at T
∂c C ∂t

=

( c021 − c10 )

D[T( t )] ∂ ⎛ 2 ∂c C ⎞ ⎜r ⎟, ⎜ ∂r ⎟⎠ r 2 ∂r ⎝

dR 1 dt

= − D[T ( t )]

∂c C ∂r

R1 ( t )

(3)

,

(4)

where D is the chemical diffusion coefficient in TiC, which is usually associated with the diffusion coefficient of carbon in the carbide layer, cC is the mass concentration of carbon, R1(t) is the current position of the TiC/Ti interface, R2 is the outer radius of the Ti particle, and c01, c021 and c023 are the interface concentrations [figure 2 (d)] according to the equilibrium phase diagram. The boundary (at r=R2) and initial conditions to Eq. (3) are

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B.B. Khina cC(t, R2) = c023, cC(t, r=0) = c01, cC[t, R1(t)] = c021, R1(t=0) = R2.

(5)

In [55,56] the Stefan-type boundary condition, Eq. (4), was posed at both Ti/TiC and TiC/C interfaces. We should outline that in interstitial compounds such as nitrides, carbides and many borides, the partial diffusion coefficient of nonmetal species exceeds that of metal atoms by several orders of magnitude, which is due to the interstitial diffusion mechanism. Hence, the growth of titanium carbide occurs at the Ti/TiC interface and is controlled by the diffusion of C atoms across the TiC layer. But at the C/TiC interface the growth of TiC at the expense of graphite, which requires the supply of Ti atoms, cannot occur. Thus, the first-kind boundary condition, cC(t, R2) = c023 [see Eq. (5)] is used for the C/TiC interface, which actually denotes an ideal “diffusion contact” of carbon particles with the outer surface of the growing TiC layer due to fast surface diffusion of the C atoms from the C/TiC contact spots.

2.2. Scenario 2: Growth of a TiC Layer on the Surface of Solid Carbon Particles The physical background for scenario 2 [figure 2 (c)] is the following. Spreading of molten titanium towards solid carbon in the SHS wave was observed experimentally [85,86]. Since it is accompanied with chemical interaction, for a sufficiently small C particle size the spreading velocity is not the rate-limiting stage [86,87]. Hence we consider that at T≥Tm(Ti) the carbon particles are completely enveloped with liquid titanium, and a thin TiC layer forms at the Ti/C interface separating the parent phases. Table 1. Diffusion data for titanium carbide Species No.

1 2 3 4 5 6 7 8 9 10 11

C

12 13 14 Ti

E, D[Tm(Ti)], D(TCS), D0, Refs. ΔT, K cm2/s kJ/mol cm2/s cm2/s −2 –8 –6 235.6 2073-2973 2.3×10 5×10 5.1×10 [83,84,88,94] [82,88,89,91] 6.98 398.7 1723-2973 1.3×10–10 1.23×10–6 (TiC0.97) 10 438.9 1873-2573 1.5×10–11 3.7×10–7 [88,89] (TiC0.9) 45.44 447.3 1723-2553 4.1×10–11 1.2×10–6 [89,91] (TiC0.887) 114 460.2 2018-2353 4.6×10–11 1.8×10–6 [89,92] (TiC0.67) 0.1 259.4 1553-1773 1.0×10–8 4.0×10–6 [83,84] –2 269.9 1673-1973 3.5×10–9 1.7×10–6 [83,95] 6.5×10 –2 307.1 1983-2573 2.3×10–10 2.6×10–7 [83,84] (TiC0.9) 4.2×10 0.48 328.42 1473-2023 6.9×10–10 1.3×10–6 [96] (TiC1.0) 99.48 328.42 1473-2023 1.4×10–7 2.7×10–4 [96] (TiC0.5) 77.8 338.9 1473-1673 5.8×10–8 1.4×10–4 [82] [93] 220 405.8 2200-2600 2.6×10–9 2.9×10–5 (TiCx, x=0.86-0.91) –9 –5 [93] 370 410.0 2200-2600 3.4×10 4.1×10 (TiCx, x=0.86-0.91) 1.31×103 347.3 1173-1473 5.8×10–7 1.7×10–3 [83] [83,90] 4.36×104 736.4 2193-2488 6.5×10–16 1.5×10–8 (TiCx, x=0.67-0.97)

Note

D(TCS) > DC(m)(TCS) D(TCS) > DC(m)(TCS) D(TCS) ∼ DC(m)(TCS) D(TCS) ∼ DC(m)(TCS) D(TCS) >> DC(m)(TCS) DTi << DC

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The product growth occurs at the Ti(melt)/TiC interface, i.e. outwards, due to diffusion of carbon atoms across the TiC layer [figure 2 (e)]. Since the diffusion coefficient of carbon in the melt is at least an order of magnitude higher than in the carbide (table 1), it is reasonable to presume that the titanium melt is saturated with carbon (otherwise the TiC layer will be dissolving). Then the boundary condition at r=R0 to diffusion equation (3) and initial conditions to Eqs (3),(4) look as cC(t, R0) = c023, cC(t, r>R1) = c01, R1(t=0) = R0,

(6)

where R0=const is the initial radius of the carbon particle.

2.3. Diffusion Data for TiC The parameters for calculating the diffusion coefficient in TiC in the Arrhenius form D = D0 exp[−E/RT(t)]

(7)

are listed in table 1, wherein the experimental data available in literature [88-96] for different temperature intervals ΔT are collected. Since the extrapolation of D to the whole temperature range of SHS may bring about overestimated values, the diffusion coefficients in TiC calculated at T=Tm(Ti) and TCS must be compared with the diffusion coefficient in molten titanium. Because of the absence of experimental data, the diffusion coefficient of C in molten Ti is estimated by a simple Stokes-Einstein (or Sutherland-Einstein) formula, which was used for assessing the diffusion parameters of C, N, O and H in liquid metals (Fe, Co, Ni, etc.) [97,98] Di(m) = kBT/(nπaiμ),

(8)

where numerical factor n=4 for substantially differing atomic radii of the melt components and n=6 for close radii, ai is the atomic radius of i-th diffusing species in the melt, μ=ζρm is the dynamic viscosity, ζ is the kinematic viscosity and ρm is the liquid-phase density. For carbon atoms, the covalent radius is aC=0.077 nm [99]. The density of molten Ti is ρm=4.11 g/cm3 [100]. A typical value of the kinematic viscosity for such liquid metals as Al, Fe, Co, Ni et al. near the melting point is (0.5-1)×10−2 cm2/s [99]. For liquid titanium saturated with carbon, ζ=0.94×10−2 cm2/s at T=Tm(Ti) [100], then DC(m)(Tm) ≈ (4.8−7.2)×10−5 cm2/s. For higher temperatures, the value ζ=1.03×10−2 cm2/s at T=2220 K is known [100]; using it at T=TCS gives DC(m)(TCS) ≈ (6.9−10.4)×10−5 cm2/s. It should be noted that since Eq. (8) doesn’t account for chemical interaction in the melt, which may be substantial for the Ti-C system, these DC(m) values are upper estimates. Then the values of diffusion coefficients in TiC, which are close to or higher than the upper estimate of DC(m)(TCS), are excluded from consideration (lines 10 to 14 in table 1).

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2.4. Temperature of the Reaction Cell in the SHS Wave Self-heating from ambient temperature, T0, to TCS during the combustion synthesis is due to the adiabatic heat release of chemical reactions which are almost accomplished when maximal temperature is reached, and in the so-called after-burn zone (at T≈TCS) only coalescence and sintering of the product particles occur with minor heat release [1-3]. Hence calculations of the product layer thickness and relevant heat release should be done in the time interval [0, tCS] corresponding to the attainment of TCS. To perform calculations, we have to know the time dependence of temperature in the reaction cell, T(t). We consider a steady-state combustion regime. For a low-temperature portion of the SHS wave, T0
(9)

where v is the combustion velocity, x is a coordinate along the SHS-sample and κ is the thermal diffusivity. To determine T(t) for the reaction cell, a coordinate x0 is chosen for which T(x0, t=0) = T0′ = T0+0.01Tm, where T0=298 K. Then the heating time, tm, from T0′ to Tm is tm = −(κ/v2) ln[0.01Tm/(Tm−T0)].

(10)

For a stoichiometric Ti-C mixture (20 wt.% C), κ≈0.04 cm2/s [102] and v=6 cm/s [81,11].

Figure 3. Temperature profile of the SHS wave in the Ti-C system [53]: 1, analytical solution for a steady-state SHS wave [Eq. (9)] for T≤Tm(Ti); 2, cubic-spline approximation of experimental curve [81] in the range Tm(Ti)≤T≤TCS.

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For higher temperatures, Tm≤T≤TCS, we use the spline-approximation of the experimental temperature profile of a steady-state SHS wave in the Ti-C system, which was registered by a micro-thermocouple technique [81] (figure 3). It should be noted that the low-temperature tail [at T
2.5. Adiabatic Heat Release in the Reaction Cell Having the heating law of the reaction cell, we can calculate the heat release due to diffusion-controlled phase layer growth in non-isothermal conditions and thus the maximal temperature attained, and then compare it with experimental TCS. In adiabatic conditions, a heat balance equation for the formation of stoichiometric TiC1.0 is written as: Tad

−ΔH0298(TiC1.0)mTiC(t) = mTiC(t)

∫ c p (TiC)dT +

298

⎫ ⎧ Tad ⎪ ⎪ mC(t) ∫ c p (C)dT + mTi(t) ⎨ ∫ c p (Ti )dT + I[Tad − Tm (Ti )]ΔH m (Ti )⎬ , ⎪⎭ ⎪⎩298 298 Tad

(11)

where Tad is the adiabatic combustion temperature, cp(i) is heat capacity, mi(t) is a current mass of i-th substance, ΔH0298(TiC1.0) = −3.077 kJ/g is the standard enthalpy of TiC1.0 [103], ΔHm(Ti) = 0.305 kJ/g is the heat of fusion of Ti [103] and I[Tad−Tm(Ti)] is the Heaviside unitstep function. The masses of all the substances are determined using a solution of the Stefan problem for particular geometry of the reaction cell, and then Tad is calculated from Eq. (11).

2.6. Modeling of TiC Layer Growth on the Titanium Particle Surface 2.6.1. Analytical Solution to Scenario 1 Problem (3)-(5),(7) is non-linear and in a general case can be solved only numerically. However, for a similar linear problem (with D=const) an asymptotic solution for the growth of a spherical phase layer, which is valid for a small layer thickness h=R2–R1 << R2, is known [18,104,105]. To apply it, we linearize Eqs (4) and (5) using substitution t

τ(t) =

∫ D[T(θ)]dθ . 0

(12)

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B.B. Khina

Here t varies from 0 to tCS, where tCS is time at which the temperature of the reaction cell reaches its maximal value TCS. Then, according to [18,104], the asymptotic solution of Eqs (3)-(5) with respect to the product layer thickness, h, looks as h(τ) = R2 − R1(τ) = βτ1/2 + β1τ/R2 + β2τ3/2/(2R22),

(13)

where β2 = 2ββ1 − β12[24q/β+4β(5+6q) + 2β3(1+q)] [32+60q+β2(18+20q)+β4(1+q)]−1, β1 = β2/(3+β2/2), q = (c023−c021)/(c021−c01),

(14)

and β is a solution of transcendental equation π1/2(β/2) exp(β/2)2 erf(β/2) = q,

(15)

which arises in a similar Stefan problem for a semi-infinite sample. For scenario 1 [figure 2 (a)], mTiC(τ) = (4/3)π [R23−R13(τ)]ρTiC, mC(τ) = mC0 − 0.2mTiC(τ), mTi(τ) = (4/3)πR13(τ)ρTi, where ρi is the density of i-th substance. For a stoichiometric composition, the initial C-to-Ti mass ratio is mC0/mTi0 = 0.25, where mTi0 = (4/3)πR23ρTi. Then, ignoring the temperature dependence of heat capacities in Eq. (11), the maximal adiabatic heating of the reaction cell, ΔTad = Tad−298, is estimated as ΔTad=

− ΔH 0298 (TiC1.0 )ρ TiC ρ TiC [c p (TiC) − 0.2c p (C)] + ρ Ti [c p (Ti ) R 13 ( τ) + 0.25c p (C) R 32 ] /[ R 32 − R 13 ( τ)]

.

(16)

2.6.2. Results of Calculations for Scenario 1 In the temperature range T0′≤T≤Tm(Ti), equilibrium interfacial concentrations c01= 0.00138 and c021=0.11 corresponding to the Ti-TiC eutectic temperature Teu=1918 K [80] were used. For higher temperatures, Tm≤T≤TCS, it was assumed that molten titanium remains inside the spherical TiC shell, and the interfacial compositions were taken for an intermediate temperature T=2673 K: c01=0.065, c021=0.14 [80]; at the C/TiC interface c023=0.2 (maximal solubility of C in the carbide). Calculations have shown that varying the c021 and c01 values along the solidus and liquidus lines of the Ti-C phase diagram in the range T=Tm to TCS has a negligible effect on the TiC layer thickness and associated heat release. For the temperature range T0′≤T≤Tm the calculations were performed with all the diffusion data listed in table 1 [figure 4 (a and b)]. For the whole temperature range, T0′≤T≤TCS, only the data giving D(TCS)< DC(m) (lines 1 to 9 in table 1) were used [figure 4 (c and d)]. A maximal TiC layer thickness attained by the time of reaching the titanium melting temperature is small, h(Tm)=0.068 µm [figure 4 (a)], and corresponds to diffusion data No.14 in table 1. The relevant adiabatic heating is insignificant, ΔTad=57 K for R2=10 µm [figure 4 (b)], and decreases with increasing the particle radius. For the temperature range T0′≤T≤TCS, the maximal TiC layer thickness corresponds to the set of diffusion parameters No. 1 (see in table 1), and this value is small: h(TCS)≈1.6 µm [figure 4 (c)]. The corresponding adiabatic

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251

heating is only ΔTad=1064 K for the Ti particle radius of 10 µm and sharply drops with increasing R2 [figure 4 (d)]. Thus, heat release due to product growth is insufficient to sustain the SHS wave (i.e. to reach TCS=3083 K).

Figure 4. Thickness of the TiC layer formed on the surface of a titanium particle by the time of attainment of Tm(Ti) (a) and TCS (c), and relevant adiabatic heating (b and d) [53]. Numbers at curves correspond to diffusion data sets in Table 1.

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B.B. Khina

The obtained result, viz. a small thickness of TiC grown in the temperature range below Tm(Ti), qualitatively agrees with experimental data [10,11]: in rapidly cooled samples almost no interaction was observed in the so-called “preheating zone” of the SHS wave. However, at the attainment of T=Tm(Ti) the melting of titanium can bring about the rupture of the primary TiC shell and the spreading of the metallic melt. It should be noted that in [62] the diffusion-controlled TiC formation was assessed using 6 different sets of the diffusion data, but only an isothermal situation below the titanium melting point was examined. Besides, the TiC layer growth was considered on the surface of a carbon particle whereas, as mentioned above, the initial TiC film at T
2.7. Rupture of the Primary TiC Shell The density of solid β-Ti at T=Tm is ρs=4.18 g/cm3 while for molten titanium at the same temperature ρm=4.11 g/cm3 [100]. The conditions for the rupture of the TiC case because of the dilatation of the titanium core during melting can be determined from a continuity equation written for spherical symmetry [106]: grad div Ur = 0.

(17)

The boundary conditions at r=R1 (expansion due to melting implying that liquid titanium is an incompressible fluid) and r=R2 are written as Ur(r=R1) = R1[(ρs/ρm)1/3−1], σrr(r=R2) = p0.

(18)

Here Ur is the radial displacement, σrr is the radial stress and p0=0.1 MPa is the outer pressure. As the plasticity of TiC is low, we consider only elastic deformation. A solution to Eq. (17) is: Ur(r) = Ar + B/r2, urr = ∂Ur/∂r = A − 2B/r3, uθθ(r) = Ur/r = A + B/r3,

(19)

where urr and uθθ are the radial and shear strain, correspondingly, and A and B are constants which are determined from boundary conditions (18). Hooke’s law for spherical symmetry looks as σrr =

Y Y [(1–ν)urr + 2νuθθ], σθθ = (uθθ + νurr), (1 + ν)(1 − 2 ν) (1 + ν)(1 − 2 ν)

(20)

where σrr and σθθ are the radial and shear stress, correspondingly, Y is the elastic modulus and ν is the Poisson’s ratio [106]. Then the solution for σθθ is obtained from Eqs (18)-(20):

Y ⎡⎛ ρ s ⎢⎜ σθθ(r) = 1 − 2ν ⎢⎜⎝ ρ m ⎣

⎞ ⎟⎟ ⎠

1/ 3

⎤ 1+ f γf − f r r , − 1⎥ − p0 1 + γf ⎥ 1 + γf ⎦

Modeling of Interaction Kinetics during Combustion Synthesis…

f=

R 32 2 R 13

,

fr =

R 32 2r 3

γ=

,

1+ ν . 1 − 2ν

253

(21)

Rupture of the primary TiC shell occurs when the maximal shear stress in the spherical layer (at r=R2) exceeds the ultimate tensile stress σuts. Then from Eq. (21) we obtain a critical thickness, hcr = R2−R1, of the TiC layer:

h cr

ϕ −1 , = R2 ϕ

⎡ ψ − 2σ uts + p 0 ⎤ ϕ=⎢ ⎥ ⎣ γ ( σ uts + p 0 ) ⎦

3Y ⎡⎛ ρ s ⎢⎜ ψ= 1 − 2ν ⎢⎜⎝ ρ m ⎣

⎞ ⎟⎟ ⎠

1/ 3

1/ 3

,

⎤ − 1⎥ . ⎥ ⎦

(22)

The TiC case can burst at h ≤ hcr. This is an upper estimate because we don’t take into account partial dissolution of TiC in molten titanium due to the eutectic reaction at 1645 °C. To calculate the hcr value, we have to determine the mechanical properties of TiC at the melting temperature of titanium. The temperature dependencies of the elastic modulus, Y, and shear modulus, G, for TiC are known in the following form [89]: Y(T) = Y0 − bYT exp(−T0/T), G(T) = G0 − bGT exp(−T0/T),

(23)

where T0=320 K, Y0=461 GPa, bY=0.0702 GPa/K, G0=197 GPa and bG=0.0299 GPa/K. Then at Tm(Ti)=1940 K we have Y=346 GPa and G=148 GPa, thus the Poisson’s ratio is ν = Y/(2G)−1 = 0.17. As for σuts values for TiC at elevated temperatures, there are only disembodied data, e.g., σuts(T=1073 K) ≈ 380 MPa, σuts(T=1273 K) ≈ 280 MPa [83]. However, available are data on the bending strength, σb, of titanium carbide over a wide temperature range because it is a typical test for brittle refractory compounds; σb has a maximum of approximately 500 MPa around T=2000 K [89, p.233]. Then, using an estimate σuts ~ σb/2 = 250 MPa, from Eq. (22) we obtain hcr≈0.6R2. Since the calculated value h[T=Tm(Ti)] is very small, for any initial size of Ti particles used in SHS (R2=5 to 100 µm) melting of the titanium core will inevitably bring about the rupture of the primary TiC shell and spreading of the melt. This changes the geometry of a unit reaction cell as shown in figure 2 (a-c).

2.8. Growth of a TiC Layer on the Surface of a Solid Carbon Particle 2.8.1. Analytical Solution to Scenario 2 For scenario 2 [figure 2 (c and e)], an asymptotic solution to Eqs (4),(5)-(7) with respect to the TiC layer thickness, h, can be obtained similarly to Eq. (13) [18,104,105]:

254

B.B. Khina h(τ) = R1(τ)− R0 = βτ1/2 − β1τ/R0 − β2τ3/2/(2R02).

(24)

Here coefficients β, β1 and β2 are defined, as previously, by Eqs (14),(15) and τ is determined according to Eq. (12) where integration is performed over the time range 0 ≤ t ≤ ΔtCS, which corresponds to the temperature range Tm≤T≤TCS (figure 3). To calculate adiabatic heating, we turn to Eq. (11). For the reaction cell shown in figure 2 (c), mTiC(τ) = (4/3)π (R13(τ)−R03)ρTiC, mC(τ) = mC0 − 0.2mTiC(τ), mC0 = (4/3)π R03ρC and mTi(τ) = 4mC0 − 0.8mTiC(τ). Then, ignoring the temperature dependence of heat capacities and neglecting the melting enthalpy of titanium (because ΔHm(Ti) << |ΔH°298(TiC1.0)| [103]), the adiabatic heating of the reaction cell is estimated as ΔTad=

− ΔH 0298 (TiC1.0 )ρ TiC ρ TiC [c p (TiC) − 0.2c p (C) − 0.8c p (Ti m )] + ρ C [c p (C) + 4c p (Ti m )] /[ R 13 ( τ) / R 30 − 1]

, (25)

were subscript “m” denotes melt. For calculations, the values of heat capacities (according to [103]) were taken at T=TCS. Eq. (25) refers to incomplete conversion of carbon into titanium carbide, i.e. when 0<ηTiC<1, where the degree of conversion is expressed as ηTiC = 1 − mC(τ)/mC0 = 0.2[R13(τ)/R03 − 1]ρTiC /ρC.

(26)

For complete conversion ηTiC=1, the maximal adiabatic heating is ΔTad(max) = −ΔH0298(TiC1.0)/cp(TiC) = 3095 K, and the adiabatic SHS temperature Tad(max) = 298 + ΔTad(max) = 3393 K. It is somewhat higher than the value Tad=3210 K calculated taking into account the temperature dependence of heat capacities [3,107]. Thus, Eq. (25) gives an upper estimate for ΔTad.

2.8.2. Results of Calculations for Scenario 2 Numerical results are presented in figure 5. The TiC layer thickness, which can form in the SHS wave with the temperature profile shown in figure 3, was calculated using Eq. (24) not accounting for the exhaustion of reactants [figure 5 (a)]. The maximal value is h≈1.5 µm for a sufficiently large carbon particle size, R0=12.5 µm, at the 1st set of diffusion data in table 1. Adiabatic heating [figure 5 (b)] was calculated taking into account the degree of conversion of carbon into carbide [see Eqs (25),(26)]. A plateau with ΔTad=ΔTad(max) for small R0 values corresponds to complete conversion (ηTiC=1). Thus, from figure 5 (b) it is seen that the diffusion-controlled growth mechanism can provide sufficient adiabatic heating to sustain the SHS process, which results from almost complete conversion, only for small-sized carbon particles: R0<3 µm. This contradicts numerous experimental works where SHS of TiC was performed with coarse-grained graphite: 7 µm [62,108,109], 20 µm [62,108], and up to 63 µm [110] in diameter. The results obtained regarding the above concept suggest that fast and complete conversion of reactants into the final product providing the required heat release can be achieved via a different route (without diffusion control of the product formation). For further

Modeling of Interaction Kinetics during Combustion Synthesis…

255

analysis of the diffusion model it makes sense to estimate a structural parameter of the product, viz. porosity. To do this, it is necessary to evaluate the displacement of the C/TiC interface.

Figure 5. Formation of the TiC layer on the surface of a carbon particle in the SHS wave after spreading of molten titanium (Tm(Ti)≤T≤TCS): (a) product layer thickness and (b) corresponding adiabatic heating vs. carbon particle radius [53]. Numbers at curves correspond to diffusion data sets in Table 1.

2.8.3. Displacement of the C/TiC Interface in the “Emptying-Core” Mechanism Above we have calculated the thickness of a spherical product layer formed due to interstitial diffusion of C atoms through titanium carbide, i.e. outward growth of TiC on the surface of carbon particles. Hence the TiC particles formed after complete conversion of the reactants will be hollow. This pattern of diffusion-controlled product formation is sometimes called “the emptying-core mechanism” [108]. Let’s estimate the displacement of the C/TiC interface, i.e. inward growth of the product layer due to diffusion of Ti atoms across TiC. From figure 5 (a) it is seen that at R0≥5 µm the effect of curvature is minor: raising R0 from 5 to 12.5 µm increases the TiC thickness by less than 10%. Thus the diffusion problem can be considered for a semi-infinite rod. The diffusion equations are written for both C and Ti atoms

∂c i ∂t

= D i [T( t )]

∂ 2ci ∂r 2

, i ≡ C,Ti.

(27)

The Stefan-type boundary conditions to Eq. (27) are formulated at interfaces Ti(melt)/TiC (r=R1) and C/TiC (r=R0) taking into account that here R0=R0(t) and cC + cTi =1

( co21 − c1o )

∂c C dR 1 = − D C [T ( t )] dt ∂r

, R1 ( t )

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B.B. Khina

(1 − co23 )

∂c dR 0 = D Ti [T ( t )] C dt ∂r

.

(28)

R 0 (t)

The initial conditions are R0(t=0) = R1(t=0) = R00.

(29)

Here DC and DTi are the partial diffusion coefficients of C and Ti atoms in TiC (see table 1), r is the linear coordinate and R00 is the initial position of the C/Ti(melt) interface at which a thin TiC layer originates at t=0. Using substitution t

τi(t) =

∫ D i [T(θ)]dθ ,

i ≡ C,Ti,

(30)

0

the non-isothermal problem (27)-(29) is reduced to an isothermal (linear) case which has an analytical solution [111] for the displacement of phase boundaries Ti/TiC (h) and C/TiC (δ): h(τC) = R1(τC) − R00 = βC τC1/2, δ(τTi) = R0(τTi) − R00 = βTi τTi1/2.

(31)

The coefficients βC and βTi are determined from transcendental equations: π1/2(βC/2)exp(βC/2)2{erf(βC/2) + erf[βTi(τTi/τC)1/2/2]} = (c023−c021)/(c021−c01) π1/2(βTi/2)exp(βTi/2)2{erf(βTi/2) + erf[βC(τC/τTi)1/2/2]} = (c023−c021)/(1−c023).

(32)

The calculated displacement of the C/TiCx interface during interaction in the SHS wave (at Tm≤T≤TCS) is negligibly small: δ=4.7 nm << h, i.e. only about 10 lattice periods of TiC. This is due to the smallness of the partial diffusivity of Ti atoms, DTi << DC (table 1). That is, the inward growth of the TiC layer is insignificant, and we can reasonably assume R0=const.

2.8.4. Product Porosity in the “Emptying-Core” Mechanism Now let’s make an upper-level estimate (for TiC1.0) of the apparent density of titanium carbide particles formed through the above described route implying that the starting charge composition also corresponds to the TiC1.0 stoichiometry (20 wt.% C). Then the mass of TiC formed per single carbon particle is mTiC = (4/3)π(R13−R03)ρTiC = (4/3)πR03ρC/0.2, and its volume is VTiC = (4/3)πR03(1+5ρC/ρTiC). Thus the apparent density of hollow TiC particles is ρeff = [1/ρTiC+1/(5ρC)]−1 ≈ 3.3 g/cm3 (where ρTiC=4.91 g/cm3 for TiC1.0 [82,83,89] and ρC≈2 g/cm3 for graphite), which is only 67% of the density of TiC1.0. That is, closed porosity of titanium carbide produced by SHS will be εcl = 1−ρeff/ρTiC ≈ 0.33 and the pore size must be about the initial diameter of carbon particles.

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257

However, microstructure investigations [10,85,109,110,112] have revealed that assynthesized TiC samples possess mainly open porosity, εop≈0.5−0.6. This is ascribed to the presence of green porosity, ε0=0.3−0.65 [87], melt spreading and gas release [85,112]. The fraction of closed porosity is usually small: about 1% for SHS in vacuum [110]. Besides, dense titanium carbide articles (with relative density above 95%) can be readily produced by SHS under high pressure [107,109] or by short-term compaction of a hot pellet immediately after the completion of SHS [1,2 and literature cited therein]. If as-synthesized TiC particles were hollow, which follows from the above described route, the attainment of such relative density would require prolonged pressure sintering at a high temperature. Thus the structural characteristic of the SHS product emerging from the diffusion model disagrees with experimental observations, which therefore supports an idea of a dissolutionprecipitation route capable of producing dense TiC particles, which includes dissolution of carbon in molten titanium and subsequent crystallization of the product grains. It should be noted that a conclusion in favor of the diffusion-controlled growth of hollow TiC shells on the surface of carbon particles having a size of 2R0=7 µm (initial porosity of a sample was ε0=0.2) and 20 µm (ε0=0.4) was made in [108] basing solely on the porosity measurements and microstructures of as-synthesized specimens. Let’s analyze these experimental data. For all of the samples the initial temperature, T0, was 293 K, and the total porosity measured after SHS was almost the same, εt(m)=0.46−0.5. As shown above, SHS of TiC via this mechanism is possible for small-sized carbon particles, R0≈3.5 µm, but closed porosity will be εcl=0.33 which greatly exceeds the measured value εcl(m)=0.06−0.08 [108]. Besides, total porosity of an as-synthesized sample for the formation of TiC1.0 is estimated as εt = 1 − (1−ε0)[ρTiC(0.8/ρTi+0.2/ρC)]−1

(33)

implying that the specimen volume doesn’t change during SHS, which is true for strongly compacted green pellets (as in [108]). Here ρTi=4.51 g/cm3 [99] is the density of initial α-Ti particles. Then for samples with 7 µm diameter carbon particles the formation of dense TiC grains (ρTiC=4.91 g/cm3) yields the total final porosity εt=0.41, which is close to experimental data. But if hollow TiC particles are formed via the diffusion mechanism, then, substituting into Eq. (33) ρeff=3.3 g/cm3 instead of ρTiC we obtain εt′≈0.13. In this case εt′ signifies the fraction of pores between the hollow particles. But this value is less than 0.154±0.005 (the Scher-Zallen criterion), which is required by the percolation theory [113,114] for the existence of open porosity. Thus, the sample will contain only closed pores (inside the TiC particles and between them) whereas in [108] high open porosity was observed: εop(m) = εt(m)−εcl(m) ≈ 0.4. For larger carbon particles (2R0=20 µm), as demonstrated above, SHS via the diffusion mechanism is impossible (for T0≈298 K) because of low heat release per unit reaction cell [figure 5 (b)]. If dense TiC particles are formed, from Eq. (33) for ε0=0.4 we have εt=0.56 which is close to experimental porosity εt(m)=0.46−0.5. For the formation of hollow particles (εcl=0.33), Eq. (33) gives εt′=0.34, and then the total porosity will be εt = εt′ + εcl = 0.67, which substantially exceeds the measured value. Maximal closed porosity in the experiments was εcl(m)=0.22 for samples with the particle diameter 17 μm (Ti) and 20 μm (carbon) [108]. Since SHS was performed under isostatic gas pressure, the origin of closed pores should be

258

B.B. Khina

ascribed to partial sintering of dense TiC grains (presumably precipitated from melt) around voids formed on the sites of outflown titanium particles. This is supported by the fact that the closed porosity was noticed to increase with raising the gas pressure (from εcl(m)=0.12 at 1 bar to 0.22 at 70 bar) while the total porosity remained almost the same, εt(m)=0.47−0.5 [108].

2.9. Analysis of the “Shrinking-Core” Mechanism in the Ti-C System Let’s discuss a dissolution-precipitation route of the TiC formation, which can produce 100% dense particles. According to the idea first proposed in [115] and used for studying SHS in the Ni-Al [60] and Nb-C [116] systems, as soon as a metallic melt spreads and engulfs solid particles, a thin film of an intermediate phase (here TiC) forms around them instantaneously. In this interaction pattern, the phase layer sequence in the reaction cell corresponds to the equilibrium phase diagram (figure 1). Then the product particles (TiC) precipitate from the saturated melt due to diffusion of carbon atoms across this film. The film thickness remains constant: it is believed that its growth rate at the C/TiC interface is equal to the dissolution rate at the melt/TiC interface. Thus, the TiC film shrinks to the center of the carbon particle as the latter dissolves. This pattern is sometimes called “the shrinking-core mechanism”. It corresponds to the solid-solid-liquid mechanism which, for the Ti-C system, is truly quasi-equilibrium. However, the film thickness has not been previously estimated using realistic diffusion data. The concentration profile of carbon in the reaction cell is similar to that shown in figure 2 (a) but with R0=R0(t); final TiC particles precipitate from the melt in domain [R1(t), R2]. In a general case, the displacement of the melt/TiC and C/TiC interfaces is determined by Eqs (27)-(29) with the only difference that Eq. (27) should be written in spherical symmetry. But since the TiC film thickness is small and DC>>DTi, outdiffusion of carbon atoms through the film is not the rate-limiting stage. Thus the process is controlled by indiffusion of Ti atoms across the TiC layer, and radial shrinking of the film is described as 0

(1 − c 23 )

dR 0 dt

∂c

C = D Ti [T ( t )] ∂r

,

R0(t=0) = R00,

R 0 (t)

h0 = R1(t)−R0(t) = const,

(34)

where h0 is the layer thickness. For a thin film, a steady-state concentration profile can be used to determine the concentration gradient at r=R0(t) in Eq. (34): cC(r) = c023R0/r + (1 − R0/r)(R1c021 − R0c023)/h0, R0(t) ≤ r ≤ R1(t).

(35)

Using τTi defined by Eq. (30) and introducing z = R0/R00, from Eqs (34),(35) we obtain: 0

0

c 23 − c 21 dz . = − 0 0 2 zR 00 / h 0 + 1 dτ Ti (1 − c 23 )(R 0 ) z

(36)

Modeling of Interaction Kinetics during Combustion Synthesis…

259

By the attainment of the maximal SHS temperature TCS, which corresponds to time t=tCS, the carbon particle completely dissolves, i.e. R0(tCS)=0. Then, integrating Eq. (36) from 0 to tCS, we receive a non-linear equation linking the initial radius of the carbon particle R00 with the thickness of the TiC film

[

h 0 R 00

− h 0 ln

(

R 00

)]

/ h 0 + 1 = τ Ti ( t SHS )

0

0

c 23 − c 21 0

1 − c 23

.

(37)

The results of the numerical solution of Eq. (37) are shown in figure 6. It is seen that the thickness of the TiC film for the initial radius of carbon particles R00=0.5 μm is close to the crystal lattice period: h0=0.5 nm ~ aTiC=0.4327 nm [82], and still decreases with increasing R00. Hence the aforesaid quasi-equilibrium solid-solid-liquid mechanism loses its physical meaning: a minimal thickness of a crystalline phase must be about the size of a critical nucleus which is typically of the order of 10 lattice periods.

Figure 6. Calculated thickness of a titanium carbide film vs. carbon particle radius for diffusioncontrolled dissolution-precipitation (“solid-solid-liquid”) mechanism [53].

2.10. Phase-Formation-Mechanism Map for Non-isothermal Interaction in the Ti-C System The above-presented consistent analysis of the solid-solid-liquid (diffusion-controlled) mechanism, which was performed using available experimental data on both solid-state diffusion in the product phase and characteristics of the SHS wave, has demonstrated that this widely used concept is actually not applicable to modeling SHS of titanium carbide. This is because the physical meaning of the results obtained within this approach (e.g., the product structure and density) disagrees with experimental data.

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It is shown that formal calculation of the product-layer thickness and associated heat release for small particles of a nonmetallic reactant (for scenario 2) can bring about numerical data supporting the diffusion model. Thus the comparison of theoretical and experimental results should be performed using structural characteristics of SHS products, e.g., porosity. Therefore, only the solid-liquid mechanism, which for the Ti-C system is truly nonequilibrium, can operate during SHS to produce dense product particles. It involves direct contact of solid carbon with molten titanium without the presence of a continuous TiC interlayer. The product formation occurs via dissolution of carbon in liquid titanium at the C/Ti(melt) interface and precipitation of TiC grains. Because of fast diffusion in hightemperature melts (DC(m)~10−5−10−4 cm2/s), diffusion in liquid is not the rate-limiting stage and the phase-forming process responsible for major heat release is crystallization of the product particles. Because of the presence of the solid/liquid interface and strong chemical interaction between the C and Ti atoms, crystallization of the TiC particles will occur via heterogeneous nucleation at the C/Ti(melt) interface rather than through homogeneous nucleation in the melt. Besides, the activation energy for the former is generally lower than for the latter. The nucleated TiC grains must detach from the carbon particle surface, otherwise a thin TiC layer separating the parent phases will form and the situation will reduce to the solid-solid-liquid (quasi-equilibrium) pattern considered above. This process continues until complete consumption of solid carbon is achieved. Further, in the after-burn zone of the SHS wave, growth and coalescence of the TiC particles in the metallic melt can occur. In this case, the final size of product particles will depend on the conditions of crystallization and subsequent coalescence/sintering but not on the size of initial reactants. An important factor is the melt lifetime which depends on the Ti-to-C ratio in the charge, structure of a green pellet determining the melt spreading conditions and heat exchange with the environment. This solid-liquid mechanism qualitatively agrees with the results obtained in experiments on arresting SHS wave in the binary Ti-C [11] and multi-component Ti-C-Ni-Mo [12] systems where in rapidly quenched samples the formation of small uniform-sized TiC particles was observed in the molten metal around a graphite particle, which were apparently detaching from the surface of carbon particles [11]. This mechanism may also be valid for other interstitial compounds such as carbides, borides etc. for which the SHS temperature exceeds the melting point of a metallic reactant but is below the melting temperatures of the non-metal and product, and for certain intermetallics. In this situation, the critical thickness, hcr, of the layer of a primary product formed on the surface of a metal particle before the attainment of the melting temperature, which is determined by Eq. (22), acquires a precise physical meaning. This is a criterion for the changeover from the solid-solid-liquid (diffusion-controlled) mechanism (a “slow” route of product formation) to the solid-liquid mechanism (a “fast” rout). As stated above, in the wave propagation mode this thickness is small (h << hcr) due to high heating rate, and hence the changeover will always take place. But this criterion is important in the thermal explosion (TE) mode when a heating rate to the melting temperature of a lower-melting reactant may be small, down to ~1−10 K/min [60,76], and hence a sufficiently thick case of the primary product can be formed on a metal-particle surface to prevent the liquid core from spreading. Basing on the above, a diagram of interaction routes can be constructed for nonisothermal heterogeneous interaction in the Ti-C system. Consider linear heating with rate vT,

Modeling of Interaction Kinetics during Combustion Synthesis…

261

which corresponds to CS in the TE mode. Then the time tm corresponding to the attainment of melting temperature of titanium is determined as tm = [Tm(Ti)−T0]/vT.

(38)

The thickness of a primary TiC case (see scenario 1) corresponding to the changeover of interaction pathways, is written as hTiC[τ(tm)] = hcr,

(39)

where τ is determined by formula (12) and hcr is calculated according to Eq. (22). Solving Eq. (39) together with Eqs. (12)-(15) and (38) permits obtaining the aforesaid criterion for the changeover of interaction mechanisms at different heating rates vT and initial particle sizes of the metallic reactant (titanium) R00. A typical radius of titanium particles used in CS is 1-50 μm; in calculations the R00 value was varied within 0.5-150 μm. For numerical calculations, we use three data sets, viz. Nos. 1, 11 and 14 from table 1 for the following considerations. As seen in figure 4 (a), sets No. 1 and 11 give a reasonable thickness of the TiC layer grown on the titanium particle surface during heating in the SHS wave while data set No. 14 yields a maximal (probably overestimated) value. The results are presented in figure 7 as a map of phase formation mechanisms in coordinates “heating rate” and “initial radius of a titanium particle” where lines 1-3 refer to different sets of the diffusion parameters in the titanium carbide.

Figure 7. Diagram of phase formation mechanisms for synthesis of titanium carbide in non-isothermal conditions: domain I is the solid-state diffusion-controlled TiC growth, or “slow” route typical of furnace synthesis; domain II is the non-equilibrium crystallization mechanism, or “fast” route typical of CS. Calculated using different data sets for diffusion in TiC (see Table 1): data set No. 1 (line 1), No.11 (line 2) and No. 14 (line 3).

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Parametric domain I, which lies below the line corresponding to the attainment of the critical thickness, hcr, of the primary refractory product (TiC), refers to the diffusioncontrolled growth mechanism. This is a slow, quasi-equilibrium route typical of diffusion annealing in weakly non-isothermal conditions. Here, a sufficiently thick shell of a primary refractory product (in our system, TiC) is grown of the metal particle surface during heating to Tm so that after melting the metallic reactant remains inside the shell (hTiC>hcr) and further interaction (at T>Tm) proceeds slowly since the rate-limiting stage is still the solid-state diffusion across this spherical layer. Thus, the models of CS employing this approach [5559,61,62] are valid in this range of parameters (the heating rate and metal particle size). From figure 7 it is seen that for small-sized metal particles (R00≤0.5 μm) the quasi-equilibrium diffusion-controlled growth of the refractory product can proceed at high heating rates typical of the SHS wave, vT~104-105 K/s. This agrees qualitatively with certain results observed during SHS in mixtures of small (nanosized) particles and in mechanically activated SHS [117]. Domain II, in its physical meaning, corresponds to a fast route typical of CS where the non-equilibrium dissolution-precipitation mechanism operates to provide fast completion of the reaction. Here, the refractory product layer formed during heating from T0 to the melting point of the metallic reactant is thin (hTiC
3. Interaction Patterns in the CS of Nickel Monoaluminide The CS of nickel monoaluminide has gained much attention in literature [14,16,60] because this compound possesses a unique combination of strength properties and resistance to gas corrosion at high temperatures, and is used in a variety of applications as a structural material [118,119] and protective coating [120]. Besides, of substantial interest is CS in multilayer thin-film Ni-Al system where the layer thickness, h, varies from ~1 μm to ~10 nm and in stacked foils with h~10-100 μm [121]. The former process is used for joining of metallic glasses [122-124], welding of a pure NiAl layer to high-strength superalloys [125], welding/soldering of microscopic objects such as electronic components, and similar applications [126-128] while the latter can be used for near-net-shape manufacturing of NiAl articles [129]. Earlier [130], SHS in laminated Ni/Al foils was used for experimental modeling of the reaction mechanism in Ni-Al powder mixtures. In both cases, CS in this system is a subject of extensive experimental and theoretical investigation. However, an intricate physical mechanism of phase and structure formation during CS of NiAl is not well understood yet. It has been demonstrated experimentally that during SHS in powder mixtures [14,16] and in lamellar Ni-Al systems [129] with h~10-100 μm the dissolution-precipitation (DP) mechanism takes place: at heating above the Al melting

Modeling of Interaction Kinetics during Combustion Synthesis…

263

temperature solid nickel dissolves in liquid aluminum and NiAl grains crystallize from the supersaturated melt. This interaction route may have non-equilibrium nature since the Ni-Al phase diagram [131,132] contains four compound phases: NiAl3 (melts at 1127 K), Ni2Al3 existing at T<1406 K, NiAl whose congruent melting point is 1911 K and Ni3Al (exists at T<1668 K), and Ni-base solid solution(figure 8). It is important to outline that in literature there is a controversy concerning the presence of solid interlayers of equilibrium phases between liquid Al and solid Ni during the formation of NiAl via this route. According to [14], dissolution of solid nickel in Al-base melt occurs “through a continuous forming and dissolving of a reaction diffusion layer on the surface on the Ni particle”, which corresponds to the equilibrium Ni-Al phase diagram. This standpoint is based on the classical theory of reaction diffusion in condensed phases [17,18,133] and implies local quasiequilibrium at phase boundaries. But in [16] it is concluded that above the melting point of NiAl3 “the system consists of only solid nickel and liquid aluminum solution”, i.e. a solid interlayer is absent and hence the formation mechanism of NiAl is substantially non-equilibrium. A criterion for transition between these two possible interaction routes is not known.

Figure 8. The Al-Ni phase diagram [131,132] and the solubility limits (solid lines) used in modeling. [Reproduced with kind permission from Springer Science+Business Media: International Journal of Self-Propagating High-Temperature Synthesis, Modeling heterogeneous interaction during SHS in the Ni-Al system: a phase-formation-mechanism map, Vol. 16, 2007, pp. 51-61, B.B.Khina and B.Formanek, Figure 1, Copyright Allerton Press, Inc., 2007].

Besides, there is a difference in interaction patterns between SHS in thin (h<1 μm) [122125] and thick (h~10-100 μm) [129] Ni-Al films: in particular, outflow of liquid aluminum through the butt end of stacked thick foils was observed experimentally [129], and the reasons for this behavior were not revealed.

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In modeling phase formation during CS in a particulate Ni-Al system in the TE mode [134] and during non-isothermal annealing at a prescribed temperature regime [135] using the diffusion-controlled growth kinetics, the choice of parameter values, viz. the activation energy E and preexponential factor D0 for diffusion, was not substantiated. The diffusion model was used for studying SHS in a layered Ni-Al system [136,137,138], but the values of macroscopic parameters, D0 and E, were obtained by fitting the calculated and experimental results of the SHS wave velocity and temperature profile instead of using relevant diffusion data for the solid product phase, NiAl, which are known in literature [139-149]. Hence these parameters refer not to diffusion in a particular phase but characterize the process as a whole, and their intrinsic physical meaning is vague. In modeling CS of NiAl in the TE mode it has been assumed that a solid nickel particle is separated from the Al-base melt by a thin NiAl interlayer of a constant thickness and diffusion across this film is a rate-limiting stage while the product grains precipitate from the melt [60]. This concept was first put forward in [115] and will be hereinafter called “the quasi-equilibrium dissolution-precipitation (QEDP)” route. However, the values of E and D0 used in numerical simulation [60] were the fitting parameters but not real diffusion characteristics of NiAl. It should be noted that an important role of Ni melting in the SHS process, which was observed experimentally [130], is not considered in the above models. Basically, structure, composition and properties of the CS product depend to a large extent on the interaction pathway. Current situation in this area necessitates a new insight into the formation mechanism of NiAl in non-isothermal conditions. In particular, it seems important to determine criteria for changeover of reaction routes at high temperatures and a range of parameters where a quasi-equilibrium interlayer of a solid product can exist at the solid/melt interface. In connection with the above, the objective of this part of the Chapter is to develop a model for phase formation in the Ni-Al system in substantially non-isothermal conditions. The model is based on the classical solid-state diffusion-controlled growth kinetics taking into account the independently measured values of diffusion parameters in NiAl, the Ni-Al phase diagram and other properties of phases involved in interaction. The ultimate goal of simulation [150,151] is to determine the ranges of parameters, e.g., the heating rates and initial metal layer thickness, where this quasi-equilibrium approach is valid. Thereby, criteria for transition to uncommon reaction routes, e.g., the dissolution-precipitation mechanism with or without a thin solid interlayer, will be found. Basing of the results of numerical calculations, a map of phase-forming mechanisms in non-isothermal conditions in this system will be constructed. As is known experimentally [39], the system geometry substantially influences heat transfer during SHS, which results in non-uniform interaction throughout the specimen. In a particulate system, interparticle contacts yield additional thermal resistance as compared with stacked foils, but in an “elementary” volume containing both of the reactants the interaction mechanism will be the same. Thus, without sacrificing the generality of approach, it makes sense to limit our consideration to the TE mode of CS where the effect of heat transfer inhomogeneity is eliminated.

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3.1. Model Formulation for NiAl Formation in Non-Isothermal Conditions 3.1.1. Physical Background First, let’s calculate the adiabatic temperature, Tad, of reaction Ni + Al → NiAl implying that the reactants are initially at the room temperature, T0=298 K. As the heat release of the reaction may be insufficient to melt all the product, the heat balance condition is written using the method proposed in [152]:

ΔH 0298 ( NiAl) +

Tm ( NiAl)

∫ c p ( NiAl, s)dT +

298

I[Tad − Tm ( NiAl)]f m ΔH m ( NiAl) +

Tad

∫

c p ( NiAl, m)dT Tm ( NiAl)

= 0,

(40)

where ΔH0298(NiAl) = –118.4 kJ/mol and ΔHm(NiAl) = 62.8 kJ/mol are the standard formation enthalpy and melting heat of NiAl, correspondingly [153], cp(NiAl,s) and cp(NiAl,m) are the specific heat of solid (s) and molten (m) NiAl, cp(NiAl,s) = 41.8 + 13.8×10−3T J/(mol⋅K) [153], Tm(NiAl)=1911 K is the NiAl melting point, fm is the mass fraction of melt, 0≤fm≤1, I is the asymmetric Heaviside step function: I(x) = 0 at x<0, I(x) = 1 at x≥0. Upon solving Eq. (40) numerically we obtain Tad=1911 K, i.e. the adiabatic combustion temperature coincides with the NiAl melting point, and the fraction of liquid phase at Tad is fm=0.42, that is solid NiAl constitutes 58% of the reaction product. In thin-film combustion in the Ni-Al system, the TCS value may be substantially below Tad because of heat removal into the substrate, thus the formation of only solid final product, NiAl, during CS is possible. As is known [14,16,129,130], fast interaction during SHS in the given system begins after heating above the aluminum melting temperature, Tm(Al)=933 K. According to the Gibbs phase rule, in a binary system the contact of a solid phase layer (here pure nickel or Nibase solid solution) with a two-phase region (here solid NiAl particles dispersed in Al-base melt at T
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ignored, and the Al concentrations at phase boundaries 1/2 and 2/3 correspond to equilibrium values. For constructing the model, the following basic assumption is made. It is considered that during the growth of NiAl compound in the whole temperature range T0 ≤ T ≤ Tad=Tm(NiAl), the interlayers of other equilibrium phases of the Al-Ni system (figure 8) [131,132], viz. NiAl3 (Tm=1127 K), Ni2Al3 (Tm=1406 K) and Ni3Al (Tm=1668 K), are absent, and metastable equilibria at interfaces “Al-base melt/solid NiAl” and “NiAl/Ni(s)” are described by the corresponding equilibrium solubility-limit lines, viz. lines GFE/LK and HIJ/ABC, respectively (figure 8). The latter presumption is only for the sake of simplicity, in order to avoid cumbersome calculation of metastable phase equilibria. Since in intermetallic compounds adjacent to NiAl on the phase diagram, Ni2Al3 and Ni3Al, the diffusion coefficient is substantially lower than in Ni-base solid solution and a fortiori lower than in liquid Al (phase 1), in this situation the phase 2 layer growth will occur faster than in the presence of all the equilibrium phase interlayers. Thus, within this model we will obtain an upper estimate for the NiAl layer grown due to solid-state diffusion across this phase and therefore an upper estimate for the conditions where this phase formation mechanism is possible. This is consistent with the idea of the ex contrario method.

Figure 9. Schematic of the diffusion-controlled growth of the NiAl (phase 2) layer in the diffusion couple Al (phase 1)/Ni (phase 3) during CS. [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics].

It should be outlined that the above assumption concerning the contact of phases NiAl/Ni refers only to temperatures below the melting point of phase Ni3Al: at T > Tm(Ni3Al)=1668 K the phase boundary 2/3 is equilibrium and described by the corresponding solidus and liquidus lines, JD and CD (figure 8). Similarly, regarding the Al(m)/NiAl interface, this assumption is valid only at T < Tm(Ni2Al3)=1406 K: above this point the concentrations at the phase boundary 1/2 correspond to the equilibrium solidus and liquidus lines ED and KD, respectively (figure 8). As the interdiffusion coefficient in phase NiAl, which determines its growth, increases with temperature according to the Arrhenius law, in non-isothermal

Modeling of Interaction Kinetics during Combustion Synthesis…

267

conditions corresponding to CS the growth of the NiAl layer will occur mainly at high temperatures, hence the effect of the above assumption on the phase 2 layer thickness will be minor. This is clearly demonstrated below by numerical simulation. For modeling, the aforementioned solubility-limit lines as well as the equilibrium liquidus and solidus lines are approximated by splines (solid lines ABCD, HIJD, LKD and GFED in figure 8). In a narrow (60 K) temperature range Tm(Ni3Al)=1668 K < T < Tm(Ni)=1728 K there are 4 phases in equilibrium in this system: Al(m)/NiAl(s)/Ni(m)/Ni(s), while at T < Tm(Ni3Al) and T > Tm(Ni) there are three phases. So, to simplify calculations, it is assumed that Ni-base solid solution melts at 1668 K (a horizontal solid line in figure 8). At a constant temperature, the phase 2 layer thickness, h2, changes according to the parabolic law h2 ≅ (Kpt)1/2 where Kp is the parabolic growth-rate constant. In weakly nonisothermal conditions, the growth kinetics of solid NiAl is determined by the diffusion-type Stefan problem [17,18,133] where a number of parameters, e.g., diffusion coefficients, equilibrium concentrations at phase boundaries and phase densities, depend on temperature. According to the Al-Ni diagram (figure 8), the solubility of Ni in phase 1 (Al-base melt) increases with temperature, the same refers to solubility of Al in phase 3 (Ni-base solid solution or melt). As the diffusion coefficient in a melt exceeds that in intermetallics by orders of magnitude, the characteristic time of concentration leveling along the 0x axis in phases 1 and 3 (figure 9) is small. Since solid [at T
dh 2 ( t , T) =

∂h 2 ∂t

dt +

∂h 2 ∂T

dT ,

(41)

or, for the time interval [t0, ti] that corresponds to heating in temperature range [T0, Ti], the expression for h2 can be written in the integral form:

Δh 2 = h 2 ( t i , Ti ) − h 2 ( t 0 , T0 ) =

ti

∫

t0

∂h 2 ∂t

dt +

Ti

∫

T0

∂h 2 ∂T

dT .

(42)

In formulas (41) and (42), ∂h2/∂t is the growth rate of phase 2 due to solid-state diffusion while ∂h2/∂T is the rate of change of h2 due to variation of equilibrium concentrations c01(T), c021(T), c023(T) and c03(T) according to the Al-Ni phase diagram (figure 8) [131,132]. Therefore, the model for phase 2 formation should be composed of two parts: (i) a quasiisothermal (diffusion-controlled growth of NiAl) and (ii) non-isothermal submodel (dissolution of NiAl in neighboring phases due to inclination of the solubility lines).

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It should be noted that due to competition of two physical processes, the following tentative situations are possible depending on the heating rate and initial metal layer thickness: (1) the dissolution rate of phase 2 becomes comparable with its growth rate, i.e. the phase 2 thickness reaches a certain maximum and starts decreasing. This means that the rest of the product (solid NiAl grains) will be crystallizing from the melt, i.e. the quasiequilibrium dissolution-precipitation mechanism (with the presence of the NiAl interlayer between phases 1 and 3) starts operating; (2) the NiAl film dissolves completely, h2=0. In the latter situation, two more cases are possible: (a) the NiAl layer disappears at a relatively low temperature, TTm(Ni). In this case two melts, viz. the Al-base (phase 1) and Ni-base (phase 3), merge into one and it will be impossible that a solid NiAl layer forms to separate them once again. This will denote the onset of a nonequilibrium mechanism of NiAl formation through crystallization of the supersaturated melt. The conditions for the changeover of the product-phase formation mechanisms will be determined below by numerical simulation within the frame of the above-stated physical concept. As the concept is based on the fundamental postulate of quasi-equilibrium at phase boundaries, i.e. phases 1 and 3 should always be separated by the phase 2 interlayer, thereby the non-equilibrium interaction route during CS in the Ni-Al system, which was a subject of discussion in literature [14,16,115], will be proved theoretically ex contrario.

3.1.2. Quasi-Isothermal Submodel The growth of a planar NiAl layer due to solid-state diffusion at a certain temperature T (figure 9) is described by a one-dimensional diffusion-type Stefan problem. As is known experimentally [14,16,129], noticeable interaction during CS in the Ni-Al system begins after aluminum melting. So we consider the temperature range from the eutectic point Teu(AlNiAl3) = 913 K to the adiabatic temperature Tad = Tm(NiAl1.0) = 1911 K. The diffusion coefficients in phases 1 (Al-base melt) and 3 (Ni-base solid solution or melt) substantially exceed that in intermetallic NiAl (the estimates are given below), hence it is reasonable to consider that the concentration of Al in phases 1 and 3 is constant along the 0x axis and corresponds to the equilibrium value at a given temperature T (figure 9); the latter is especially true above the nickel melting point. The diffusion-type Stefan problem for the growth of phase 2 includes the equation for diffusion mass transfer in phase 2

∂c 2 ∂t

= D 2 (T )

∂ 2c 2 ∂x 2

,

and mass balance conditions: (i) at phase boundary 1/2 [Al(m)/ NiAl(s)]

(43)

Modeling of Interaction Kinetics during Combustion Synthesis… 0

0

[ρ1 (T)c1 (T) − ρ 2 c 21 (T)]

dX 12 dt

= ρ 2 D 2 (T )

∂c 2 ∂x

269

,

(44)

X 21 ( t ) + 0

and (ii) at interface 2/3 [NiAl(s)/Ni(s,m)] 0

0

[ρ 2 c 23 (T) − ρ 3 (T)c 3 (T)]

dX 23 dt

= − ρ 2 D 2 (T )

∂c 2 ∂x

.

(45)

X 23 ( t ) − 0

Here subscripts 1, 2 and 3 denote phases Al(m), NiAl(s) and Ni(s or m), correspondingly, X21(t) and X23(t) are the current coordinates of 1/2 and 2/3 phase boundaries, c2 is the aluminum concentration in phase 2, c01(T) and c03(T) are the compositions of phases 1 and 3 in equilibrium with phase 2 at current temperature, c021(T) and c023(T) are the equilibrium concentrations of Al in phase 2 at the 2/1 and 2/3 interfaces, ρi is the density of i-th phase, i=1-3, D2(T) is the interdiffusion coefficient in intermetallic NiAl. Hereinafter mass concentration of Al is used. Expressions (44) and (45) are written taking into account the difference in the density of contacting phases. Initial conditions to Eqs.(43)-(45) look as h1(t=0) = h01, h3(t=0) = h03, h2(t=0) = 0,

(46)

where h01 and h03 are the starting thicknesses of phases 1 [Al(m)] and 3 [Ni(s)] at T0 = Teu(AlNiAl3) = 913 K. As information on the temperature dependence of the density of solid NiAl is not available in literature, in Eqs. (43)-(45) it is assumed constant: ρ2=6.02 g/cm3 [156]. For phases 1 (Al-base melt) and 3 (Ni-base solid or liquid solution) the density is expressed as ρ1 = c1 (T)ρ Al( m ) (T) + [1 − c1 (T)]ρ Ni( m ) (T), ρ 3 = c 3 (T)ρ Al(s,m ) (T) + [1 − c 3 (T)]ρ Ni (s,m) (T ) . 0

0

0

0

(47)

The interdiffusion coefficient in NiAl (phase 2) is defined in the Arrhenius form D2(T) = D0exp[–E/(RT)],

(48)

where E is the activation energy and D0 the preexponent. The choice of parameter values for modeling is substantiated below. It should be noted that within the frame of the Stefan problem a situation may arise where the layer of one of the parent phases, 1 or 3, is spent completely. Then, on the external surface of phase 1 or 3 the second-kind boundary condition, J = D2∂c2/∂x = 0, is to be posed instead of Eq. (44) or (45), correspondingly. This formulation has been widely employed in modeling SHS using the solid-state diffusion-controlled growth kinetics [56,59]. However, in this case it will be impossible to detect the changeover of interaction mechanisms because after complete disappearance of any of the parent phases the diffusion-controlled growth of NiAl will still continue. So, hereinafter this trivial situation is not considered deliberately. The results presented below demonstrate that if the model were supplemented with such

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conditions, certain phenomena revealed by computer simulation would have remained unnoticed. For a thin planar NiAl (phase 2) film, a steady-state (i.e. linear) profile of Al concentration is typically assumed ∂c2/∂x = (c023(T)–c021(T))/h2, where h2(t)=X23(t)–X21(t) is the NiAl layer thickness. Then from Eqs. (43)-(45) we obtain for given temperature T:

h2

dh 2 dt

= ρ 2 D 2 (T)[c 021 (T) − c 023 (T)](k −231 + k −211 ) ,

k 23 = ρ 2 c 023 (T) − ρ 3 (T)c 30 (T), k 21 = ρ1 (T)c10 (T) − ρ 2 c 021 (T) .

(49)

At T=const Eq. (49) has a simple analytical solution for a time interval [ti, ti+1]: −1 1 / 2 h 2 ( t i +1 ) = {h 22 ( t i ) + 2ρ 2 D 2 (T)Δt[c 021 (T) − c 023 (T)](k −231 + k 21 )} ,

(50)

where Δt = ti+1 – ti is a time increment. Then from Eqs. (44), (45) and (50) the thickness of phases 1 (h1) and 3 (h3) is obtained: h1(ti+1) = h1(ti) – h2(ti+1)k23(k21+k23)–1, h3(ti+1) = h3(ti) – h2(ti+1)k21(k21+k23)–1.

(51)

3.1.3. Non-isothermal Submodel To determine the dissolution rate of solid NiAl in adjacent phases, it is necessary to write down mass balance equations for Al at interfaces NiAl/Al(m) (2/1) and NiAl/Ni(s,m) (2/3) and the total mass balance equation in the system bearing in mind that phases 1 and 3 do not contact each other directly but are separated by the NiAl interlayer. As stated above, the interaction kinetics at phase boundaries is not the rate-limiting stage and diffusion rates in metallic melts (Al and Ni base) and Ni-base solid solution are substantially higher than in solid NiAl. Thus, during heating the equilibrium concentrations corresponding to a current temperature are established quickly throughout the whole thickness of phases 1 and 3 (figure 9). This assumption is valid for a small change of temperature, i.e. calculation over the whole temperature range [T0, Tad] should be performed with a sufficiently small step ΔT = Tj+1−Tj. Consider the temperature dependence of the phase 2 layer thickness ∂h2/∂T ≡ dh2/dT:

dh 2 dT

=

dΔX 21 dT

+

dΔX 23 dT

,

(52)

where ΔX21 and ΔX23 are the displacements of phase boundaries 2/1 and 2/3, correspondingly. Let’s write the total mass balance at interfaces 2/1 and 2/3 during partial dissolution of NiAl in phase 1 (ΔX21) and phase 3 (ΔX23) taking into account that ρ2=const while the densities of phases 1 and 3 vary with temperature:

Modeling of Interaction Kinetics during Combustion Synthesis…

dΔX 21

ρ2

+

dT

d (ρ1h 1 ) dT

= 0 , ρ2

dΔX 23 dT

+

d(ρ 3 h 3 )

271

=0 .

dT

(53)

For small ΔT and therefore small ΔX21 and ΔX23 values, the Al mass balance equations at interfaces 2/1 and 2/3 are formulated taking into account the temperature variation of equilibrium concentrations c01, c023, c021 and c03:

ρ2

d (c 021ΔX 21 ) dT

+

d (ρ1c10 h 1 ) dT

= 0 , ρ2

d(c 023 ΔX 23 ) dT

+

d(ρ 3 c 30 h 3 ) dT

=0 .

(54)

Then from Eqs. (53), (54) the following relationships for ΔX21 and ΔX23 are obtained:

dΔX 21 dT dΔX 23 dT

ρ 2 (c 021

− c10 ) + ΔX 21ρ 2

ρ 2 (c 023 − c 30 ) + ΔX 23ρ 2

dc 021 dT dc 023 dT

+ ρ1h 1

+ ρ3h 3

dc10 dT

=0,

dc 30 dT

=0.

(55)

Hence for temperature range [Tj, Tj+1], i.e. for a regular step ΔT, from Eqs. (52) and (53) we have h2(Tj+1) = h2(Tj) + ΔX21(Tj+1) + ΔX23(Tj+1), h1(Tj+1) = (h1(Tj)ρ1(Tj) – ΔX21(Tj+1)ρ2)/ρ1(Tj+1), h3(Tj+1) = (h3(Tj)ρ3(Tj) – ΔX23(Tj+1)ρ2)/ρ3(Tj+1).

(56)

System of equations (55) can be rewritten in a more convenient form

dΔX 2i dT

+

ΔX 2i ⎛ dc i0 dc 02i ⎞ ρ i (T1 )h i (T1 ) dc i0 ⎟= ⎜ , i=1,3, − 0 0 ⎟ ρ (c 0 − c 0 ) dT ⎜ dT dT c i − c 2i ⎝ 2 i 2i ⎠

(57)

which admits an analytical solution. Having initial conditions ΔX21(T0)=0, ΔX23(T0)=0 where T0 = Teu(Al-NiAl3) = 913 K, the solution for temperature range [Tj, Tj+1] is

1 ΔX 2 i ( T ) = μ i (T )

Tj+1

z

∫ Q i (ξ)Fi (ξ)dξ , Fi (z) = exp ∫ Pi (ω)dω , i≡1,3,

Tj

Tj

where Fi is the integrating factor and parameters Pi and Qi are defined as

(58)

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B.B. Khina

ρ i (T j )h i (T j ) dc 0 ⎛ dc 0 dc 0 ⎞ i i 2i ⎟ ⎜ , Qi = , i≡1,3. Pi = 0 0 − 0 0 ⎟ ⎜ dT dT dT c i − c 2i ⎝ ( c c ) ρ − ⎠ 2 i 2i 1

(59)

On each step over temperature the values of ΔX21 and ΔX23 are found from Eqs. (58), (59) where the integrals can be evaluated numerically, and then the current thickness of phases 1-3 is determined by formulas (56). As mentioned above, at sufficiently fast heating during CS a situation is possible where at relatively low temperatures, T
h 1 (T j+1 ) =

h 3 (T j+1 ) =

h 1 (T j )ρ1 (T j )[c 10 (T j ) − c 30 (T j+1 )] + h 3 (T j )ρ 3 (T j )[c 30 (T j ) − c 30 (T j+1 )] ρ1 (T j+1 )[c10 (T j+1 ) − c 30 (T j+1 )] h 3 (T j )ρ 3 (T j )[c10 (T j+1 ) − c 30 (T j )] + h 1 (T j )ρ1 (T j )[c10 (T j+1 ) − c10 (T j )] ρ 3 (T j+1 )[c10 (T j+1 ) − c 30 (T j+1 )]

,

. (60)

3.2. Sequence of Calculations In this Chapter, we study linear heating of a Ni-Al bilayer, which may constitute a unit element of a multilayer Ni-Al film, at rate vT=dT/dt=const in temperature range [T0=913 K, Tad=1911 K]. This corresponds to the TE mode of CS. Then, introducing small steps over temperature ΔT and time Δt=ΔT/vT, the heating process is divided into a large number of identical stages: (i) isothermal soaking at temperature Tj for time Δt, and (ii) instantaneous heating to temperature Tj+1 = Tj + ΔT. In the isothermal stage, the thickness of phases 1-3 is calculated by Eqs. (50), (51). During the heating stage, the displacement of phase boundaries due to variation of the solubility limits is calculated by Eqs. (58), (59) and (56) or Eqs. (60). The calculation is performed until complete consumption of one of the parent phases, 1 or 3, is achieved. As the ultimate goal of modeling is to determine the parametric domain where the growth of phase 2 can occur via the solid-state diffusion mechanism, for each initial thickness of the Al layer, h0Al, the heating rate vT is varied to reveal the occurrence (or nonoccurrence) of two physical situations: (1) the onset of shrinkage of phase 2, and (2) complete disappearance of the NiAl film at T>Tm(Ni). Since in SHS the heating rate reaches 105–max 106 K/s [1-3] in simulation the vT value was varied within 5×100–5×106 K/s.

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The initial mass ratio of pure Al to Ni corresponds to stoichiometry NiAl1.0. So, in modeling only the initial thickness of the aluminum layer at room temperature (298 K), h0Al, was specified. Then the thickness of the initial Ni layer is calculated by a simple formula h0Ni = h0Al(ρ0Al/ρ0Ni)x0/(1–x0), where ρ0i is the density of pure i-th metal (i≡Al,Ni) at 298 K, ρ0Al=2.71 g/cm3 and ρ0Ni=8.96 g/cm3 [157], and x0=0.314 is the mass content of Al in NiAl1.0. As noted above, interaction during CS in the given system begins at T0=Teu(Al-NiAl3)=913 K. Then the thickness of phase 1 (Al-base melt), h01, and phase 3 (Ni-base solid solution), h03, at T0 is determined, similarly to Eq. (60), from the condition of mass balance of both Al and Ni: 0 h1

=

0 h Al

ρ 0Al

x 0 − c 30 (T0 )

ρ1 (T0 ) x [c 0 (T ) − c 0 (T )] 0 1 0 3 0

, (61)

0 0 h 3 = h Al

ρ 0Al ρ 3 (T0 )

c10 (T0 ) − x 0 , x 0 [c10 (T0 ) − c 30 (T0 )]

where the densities of phases 1 and 3, ρ1(T0) and ρ3(T0), are defined by formulas (47). For numerical modeling it is necessary to know (i) the temperature dependence of density of Al and Ni in order to evaluate the densities of phases 1 and 3 by Eq. (47), and (ii) diffusion parameters E and D0 for NiAl to calculate the interdiffusion coefficient D2(T) by formula (48).

3.3. Parameter Values for Modeling Interaction in the Ni-Al System 3.3.1. Densities of Phases Formula (47) includes the density of liquid aluminum ρAl(m)(T) and that of solid and liquid nickel ρNi(s,m)(T). The densities of liquid metals (in g/cm3) are determined as [97] ρAl(m)(T) = 2.38 – ΛAl(T – Tm(Al)), ΛAl = (2.4-4.0)×10–4, ρNi(m)(T) = 7.90 – ΛNi(T – Tm(Ni)), ΛNi = (8.7-12.5)×10–4.

(62)

For calculating the density of liquid phases 1 and 3, minimal values ΛAl=2.4×10–4 and ΛNi=8.7×10–4 were taken. At that, according to formulas (47) and (62), the density of molten NiAl1.0 (31.4 wt.% Al) at Tm(NiAl1.0)=1911 K is 5.95 g/cm3, i.e. less than the density of solid nickel aluminide156 ρ2=6.02 g/cm3 (at higher ΛAl and ΛNi values the density of liquid NiAl1.0 appears to be higher than that of solid NiAl1.0). To calculate the density of solid phase 3 [Ni-base solid solution at T
274

B.B. Khina ρNi(s)(T) = ρNi(s)[Tm(Ni)] exp{–βNi[T–Tm(Ni)]},

(63)

where ρNi(s)[Tm(Ni)]=8.10 g/cm3 is the density of solid Ni at its melting point [158]. For Al dissolved in phase 3, the value at a room temperature ρAl(s)=2.71 g/cm3 [157] was used.

3.3.2. Interdiffusion Parameters in NiAl Diffusion in the Ni-Al system and, particularly, in intermetallic compound NiAl has been extensively studied [139-149]. The interdiffusion coefficient in NiAl, D2, was measured experimentally during annealing of diffusion couples at T=650-1300 °C [139,140], aluminizing of nickel at T=870-1000 °C [141], and reaction diffusion in thin films at low temperatures T=250-440 °C [142]. At T=1173-1473 K, the activation energy for interdiffusion in NiAl is E=200 kJ/mol for Al concentration xAl≤40 at.% and E≈250 kJ/mol for xAl=43-50 at.% [143]. The partial diffusivity of Ni atoms measured experimentally at T=1273-1623 K [144], T=1050-1550 K [145], and T=1273 and 1523 K [146] appears to depend strongly on the composition of NiAl: in stoichiometric NiAl1.0 it is considerably lower than near the upper and lower solubility limits. The concentration dependence of the interdiffusion coefficient in NiAl was measured at T=1223-1773 K [147] but the obtained values appeared to be substantially smaller than those reported in other works [139,140,141,143,148,149]. Here for numerical modeling we use interdiffusion parameters in NiAl obtained for xAl=38 at.% in the temperature range T=1200-1600 K [148] and for xAl=54 at.% at T=11231323 K [149]. The values of preexponent D0 and activation energy E retrieved from the Arrhenius plots log(D2) vs. T−1 presented in the cited papers (figure 10 in [148] and figure 13 in [149]) are listed in table 2 as data sets No. 1 and No. 2. Since the interdiffusion coefficient for non-stoichiometric NiAl is higher than for NiAl1.0, simulation with these parameters will give an upper estimate for the phase 2 layer growth via the diffusion-controlled kinetics, which goes in line with the ex contrario method. The calculated D2 values extrapolated to the whole temperature range of CS, T=913-1911 K, are shown in figure 10; lines 1 and 2 corresponding to these diffusion data sets intersect at T=1667 K giving D2≈10−7 cm2/s. Table 2. Interdiffusion data for NiAl used for modeling phase layer growth in the Ni-Al system in non-isothermal conditions Data set number

D0, cm2/s

E, kJ/mol

xAl, at.%

Refs.

Note

1 2 3 4 5

1.0 4.75×10−4 1.0×10–3 1.0×10–3 2.18×10−2

222.0 115.9 100 92.05 137

38 54 -

[148] [149] [134] [135] [137,138]

diffusion experiment, T=1200-1600 K diffusion experiment, T=1123-1323 K modeling CS in the TE mode modeling non-isothermal annealing modeling thin-film CS (fitting parameters)

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Figure 10. Temperature dependence of the interdiffusion coefficient in NiAl (D2) according to experimental data [148,149] extrapolated to the temperature range of SHS (solid lines), and according to the parameters D0 and E used in modeling SHS by other authors (dashed and dash-dot lines). Numbers at lines correspond to the diffusion data set numbers in Table 2. [Reproduced with kind permission from Springer Science+Business Media: International Journal of Self-Propagating HighTemperature Synthesis, Modeling heterogeneous interaction during SHS in the Ni-Al system: a phaseformation-mechanism map, vol. 16, 2007, pp. 51-61, B.B.Khina and B.Formanek, Figure 9, Copyright Allerton Press, Inc., 2007].

At the NiAl melting point, Tm(NiAl)=Tad=1911 K, D2=8.6×10−7 cm2/s for data set No. 1 and 3.2×10−7 cm2/s for data set No. 2 (table 2). Let us compare the obtained values with diffusion coefficients in the melt at the same temperature. For close atomic radii ai (0.143 nm for liquid Al and 0.124 nm for molten nickel [99]), a plausible estimate is given by the Sutherland-Einstein formula (8) with n=4. For liquid Ni, ρm=7.90 g/cm3 at its melting point Tm=1911 K [97] and the kinematic viscosity is ζ=0.4×10−2 cm2/s at T=1923 K [99]. Then for diffusion of Ni atoms in the Ni-base melt at 1911 K formula (8) gives DNi(m)=5.4×10−5 cm2/s, while for diffusion of Al atoms DAl(m)=4.6×10−5 cm2/s. This is two orders of magnitude higher than the interdiffusion coefficient in solid NiAl at the same temperature. Therefore, data sets No. 1 and 2 for diffusion in NiAl (see table 2) give reasonable values of D2 in the whole temperature range of CS. The parameters E and D0 used in modeling SHS in a particulate Ni-Al system in the TE mode [134] and in simulation of non-isothermal annealing [135] are presented in table 2 as data sets No. 3 and 4, respectively. However, the choice of these values in [134,135] was not justified. In modeling SHS in multilayer Ni-Al thin films [137,138], the values of E and D0 for diffusion in NiAl were obtained by fitting experimental and numerical results of the combustion temperature and velocity (data set No. 5 in table 2). The corresponding diffusion coefficients are presented in figure 10 for the sake of comparison (lines 3 through 5). It is seen that in the temperature range of CS data sets Nos. 3-5 give a substantially overrated

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diffusion coefficient in NiAl than the upper estimates of the corresponding experimental parameter: D2 exceeds the values given by data set No. 1 (line 1 in figure 10) by 1 to 3 orders of magnitude and those calculated using data set No. 2 (line 2) by one order of magnitude. As noted previously, at T=const the phase layer growth via diffusion mechanism follows the parabolic law h2 ≅ [Kp(T)t]1/2, where Kp is connected with the interdiffusion coefficient D2 in the growing phase [see Eqs. (49), (50)]. In non-isothermal conditions T=T(t), this dependence can be written in the integral form assuming that equilibrium concentrations at phase boundaries are constant: 1/ 2

⎫ ⎧t h ⎪ ⎪ h 2 ≅ ⎨ ∫ D 2 [T( t )]dt ⎬ ⎪⎩ 0 ⎪⎭

,

(64)

where th~100−10−2 s (depending on the system) is the characteristic time of reactants heating in the SHS wave from T0 to Tad. The heat release, which is responsible for self-heating during CS, is proportional to the product phase-layer thickness h2, and hence the adiabatic temperature rise, ΔTad=Tad−T0, is estimated as 1/ 2

t ⎫ − ΔH 0298 ( NiAl)S ⎧⎪ h ⎪ ΔT ad ≅ ⎨ ∫ D 2 [T( t )]dt ⎬ Vc p ( NiAl) ⎪ 0 ⎪⎭ ⎩

,

(65)

where S in the total Ni/Al interface area and V is the specimen volume; here the temperature dependence of heat capacity, cp(NiAl), is neglected. From Eq. (65) it is seen that raising the interdiffusion coefficient by the factor of only 3 increases ΔTad by the factor of 1.73. Therefore, to attain a fair agreement between the calculated and experimental temperature of CS, diffusion coefficient in the growing phase has to be substantially overestimated, otherwise within the diffusion-controlled-growth kinetic model the combustion will be impossible. This is a possible reason of why in many modeling attempts employing this kinetic approach the diffusion parameters E and D0 were either obtained by data fitting [137,138] or chosen without any justification [134,135] instead of using experimental data on solid-state interdiffusion in the product phase.

3.4. Numerical Modeling of NiAl Formation in Non-isothermal Conditions 3.4.1. Evolution of the Phase Layers Figure 11 displays an example of numerical modeling of phase 2 growth during linear heating of a thin-film Al-Ni diffusion couple at a small initial thickness of the pure aluminum layer, h0Al=0.1 μm, using diffusion data set No. 1 for NiAl (table 2). In the range of heating rates from 50 to 7.5×105 K/s the conversion of initial metals into the product, NiAl, occurs by the quasi-equilibrium solid-state diffusion mechanism. With increasing vT the temperature Tfin at which the process accomplishes, i.e. one of the parent phase layers (1 or 3) is

Modeling of Interaction Kinetics during Combustion Synthesis…

277

completely consumed, rises, but it is still below the melting temperature of phase 3. At that, the Al layer is consumed earlier than Ni (lines 1-4 in figure 11), but at the heating rate of 7.5×105 K/s phase 1 and 3 end simultaneously (lines 5 in figure 11); obviously, this corresponds to a maximum thickness of phase 2. At higher heating rates, phase 1 again ends earlier than phase 3, and the NiAl layer thickness, which is attained by the time of disappearance of the Al layer, decreases. For vT=5×106 K/s, temperature Tfin exceeds the Ni melting point (lines 6 in figure 11). In this case, the jags observed on curves h2(T) and h3(T) and the inflection point on line h2(T) are connected with a stepwise increase of the solubility limits due to the melting of Ni (see figure 8). However, at a somewhat higher heating rate vT=8.5×106 K/s (lines 7 in figure 11) the phase layers 1 and 3 yet another time end simultaneously, and the thickness of NiAl has a second maximum (lines 7 in figure 11). Only at vT=1×107 K/s, which is substantially above a typical heating rate in the SHS wave (105-106 K/s), the phase 2 layer thickness starts decreasing; this is not shown in figure 11 merely to avoid encumbering.

Figure 11. Change of the layer thickness of phases 1 (Al-base melt), 2 (NiAl) and 3 (Ni-base solid or liquid solution) vs. temperature for initial aluminum layer thickness h0Al=0.1 μm at linear heating with different rates: vT=50 K/s (line 1), 5×102 K/s (2), 5×103 K/s (3), 7.5×104 K/s (4), 7.5×105 K/s (5), 5×106 K/s (6), and 8.5×106 (7). Calculations are performed with diffusion data set No. 1 for phase NiAl (Table 2). [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics].

The observed behaviors are connected with two reasons [see Eqs. (44), (45) and (55)]: (i) different displacement rates of interfaces 1/2 and 2/3, which is due to the asymmetry of solubility-limit lines in the Ni-Al phase diagram (figure 8), i.e. different temperature dependencies of boundary concentrations c 10 (T) , c 021 (T ) , c 023 (T ) and c 30 (T ) , and (ii) competition between the diffusion-controlled growth of NiAl and its dissolution in neighboring

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B.B. Khina

phases 1 and 3. It should be noted that the first maximum in the phase 2 layer thickness (lines 5 in figure 11), which is observed at TfinTm(phase 3) (lines 7 in figure 11) denotes the situation when the dissolution rate of NiAl in neighboring melts becomes comparable with its growth rate, i.e. the dissolution-precipitation mechanism of product formation starts operating, parallel with the diffusion-controlled growth of the continuous NiAl layer. Numerical simulation has shown that for any initial Al layer thickness, h0Al, there always ( cr ,1)

, at which this phenomenon is observed. exists a certain critical heating rate, v T Hence, for a small thickness of the initial metal layers (h0Al≤0.1 μm) the diffusion rate in solid NiAl can provide almost complete conversion of Al and Ni into NiAl via the solid-state diffusion-controlled growth mechanism even at high heating rates that exceed the values typical of an SHS wave. This outcome of numerical modeling demonstrates that the idea expressed qualitatively in [159] about some unusual “shear”, or “martensitic” mechanism of atomic mass transfer across a continuous NiAl layer, which is allegedly responsible for fast growth of the reaction product during SHS in a thin-film Ni/Al system, is physically meaningless (in the sense of “Occam’s razor”): the reaction can well be accomplished in a very short time via the known solid-state diffusion-controlled route. Martensitic structure of the final product observed experimentally [159] for a certain composition of NiAl, viz. 63.8 at.% Ni which is close to that corresponding to diffusion data set No. 1 (table 2), could most probably be formed during fast cooling of hot NiAl in the so-called after-burn zone of the SHS wave due to heat removal into the substrate. This agrees with earlier results of the same authors [160]: the SHS wave temperature in a thin-film Ni-Al system with h0Al=30-100 nm exceeded the melting point of Al and fast cooling after the completion of high-temperature interaction was observed. At a larger initial thickness of pure Al, increasing the heating rate results in a changeover of interaction mechanisms, which occurs at T>Tm(Ni). Simulation for h0Al=10 μm was performed using different diffusion data sets for NiAl (table 2): No. 1 [figure 12(a)] and No. 2 [figure 12(b)]. At relatively small values of vT, below 500 K/s, the system behavior [lines 1 and 2 in figure 12(a)] is similar to that shown in figure 11 for heating rates at which the maximal temperature did not exceed Tm(Ni). The above described situation when the phase 2 layer thickness attains its first maximum [at T
Modeling of Interaction Kinetics during Combustion Synthesis…

279

temperature rise there exists a thin solid NiAl layer separating two liquid phases, 1 and 3. For a given heating rate, the phase 2 thickness is either almost constant or slightly decreases with rising temperature, while the thickness of Al-base melt (phase 1) decreases and that of phase 3 increases, i.e. boundaries 2/1 and 2/3 move in the same direction with almost same velocity [lines 4 and 5 in figure 12(a)]. The process continues until phase 1 disappears.

Figure 12. Change of the layer thickness of phases 1 (Al-base melt), 2 (NiAl) and 3 (Ni-base solid or liquid solution) vs. temperature for initial aluminum layer thickness h0Al=10 μm at linear heating with different heating rates: (a) diffusion data set No. 1 for phase NiAl (Table 2): vT=10 K/s (line 1), 100 K/s (2), 1×103 K/s (3), 2×103 K/s (4), and 3×104 K/s (5) [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics]; (b) diffusion data set No. 2 (Table 2): vT=50 K/s (line 1), 175 K/s (2), 400 K/s (3), 1×103 K/s (4), and 5×103 K/s (5) [Reproduced with kind permission from Springer Science+Business Media: International Journal of Self-Propagating High-Temperature Synthesis, Modeling heterogeneous interaction during SHS in the Ni-Al system: a phase-formation-mechanism map, vol. 16, 2007, pp. 5161, B.B.Khina and B.Formanek, Figure 10, Copyright Allerton Press, Inc., 2007].

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B.B. Khina

This phenomenon is conditioned by the fact that at a high temperature the thin NiAl layer is permeable for diffusion of Al and Ni atoms and, as a result, the film displaces as a whole. The NiAl thickness is small: h2≈0.5 μm at vT=1×104 K/s and ~0.1 μm at vT=(2.5-4.0)×104 K/s. In this situation, the reaction product can form only by precipitation of NiAl grains from the Nibase melt (phase 3) during displacement of the thin continuous NiAl film. Thus, it is unambiguously demonstrated that the QEDP mechanism (via diffusion across a thin equilibrium film separating the parent phases) of CS, which was used in modeling on a merely conceptual basis [115,60] can really take place in a certain range of heating rates and initial metal layer thicknesses. In this regime, the direction of displacement of the thin solidproduct interlayer (in this particular system, towards point x=0 in figure 9) depends solely on functions c10 (T) , c 021 (T ) , c 023 (T ) and c 30 (T ) [see Eqs. (55)], i.e. on the shape of liquidus and solidus lines of the phase diagram (figure 8). Earlier it has been proved [53] that this mechanism cannot operate during SHS in the TiC system because of a low value of the diffusion coefficient in titanium carbide: the thickness of the TiC interlayer between liquid Ti and solid C will be below the characteristic size of a critical nucleus. In the Ni-Al system this interaction pattern is possible: at vT=5×104 K/s the minimal NiAl film thickness is h2min≈60-30 nm, i.e. (160-80)aNiAl where aNiAl=0.373 nm is the NiAl lattice period [35], which exceeds the typical critical nucleus size hcr~10aNiAl. However, at this high heating rate the NiAl interlayer exists only in a limited temperature range after phase 3 melts. Note that in the given system this route cannot take place below the Ni melting temperature and in this model the elastic stresses that can arise in the phase 2 interlayer and lead to its rupture are not accounted for. A further increase in the heating rate (above 4.25×104 K/s) brings about a situation where, at the attainment of a certain temperature Td, the phase 2 layer dissolves completely while phase 1 (Al-base melt) still exists. This transition occurs on a narrow temperature range, ΔTd≈1 K, which substantially exceeds the step over temperature (ΔT≤0.2 K) used in modeling, i.e. the calculation accuracy was sufficient. The value of Td almost does not change with increasing vT and is about 1860 K. Apparently, this is connected with the shape of liquidus and solidus lines on the Ni-Al phase diagram (figure 8): at T>1840 K they quickly converge to the NiAl melting point. In this situation, liquid phases 1 and 3 inevitably merge together and hence the only possible product formation mechanism is crystallization of NiAl grains from the melt. As mentioned above, this is a substantially non-equilibrium route. For any initial thickness of the Al layer h0Al there must exist a certain critical heating rate,

v (Tcr , 2) , when the thin NiAl interlayer disappears at T>Tm(Ni), i.e. a transition the non( cr , 2)

equilibrium reaction pathway occurs: e.g., v T

≈4.25×104 K/s for h0Al=10 μm and

v (Tcr , 2) =350 K/s at h0Al=100 μm. Note that the uncommon interaction routes, viz. QEDP via a thin NiAl film and nonequilibrium crystallization of NiAl grains, take place after phase 3 melts, i.e. when the functions c10 (T) , c 021 (T ) , c 023 (T ) and c 30 (T ) used in simulation correspond to the real phase diagram (figure 8). This confirms the appropriateness of the assumptions made in this model.

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281

Simulation using diffusion data set No. 2 (table 2) for the same h0Al value gives similar results [cf. figures 6(a) and (b)]. The main difference is that the heating rates corresponding to ( cr ,1)

the onset of decrease of the NiAl thickness, v T

, and to complete dissolution of phase 2,

v (Tcr , 2) , displace towards lower values. This is due to a smaller D2 values at T>1667 K (figure 10). Besides, the above described case, when at T
3.4.2. Estimation of Critical Heating Rates ( cr ,1)

( cr , 2)

To correctly determine critical heating rates v T and v T for any thickness of the initial Al layer, the following graphs have been plotted using the results calculated for two diffusion data sets (table 2): the dependence on heating rate of the final thickness of phase 2, h2fin, which is attained by the instant of complete consumption of any of the parent phases (figure 13), and the corresponding temperature value, Tfin, vs. vT (figure 14). The first maximum in lines h2fin(vT) denotes the situation when phase layers 1 and 2 are exhausted simultaneously [figure 13(a)] at T
maximum in all the curves in figure 13(a) corresponds to v T , and in figure 14(a) it is seen as the maximum point. For data set No. 2 this maximum is dome-shaped [figure 13(b)] and in lines Tfin(vT) it corresponds to the transition point to the horizontal branch [figure 14(b)]. These behaviors are determined by different values of D2 (see figure 10). The endpoint of a line corresponds to disappearance of phase 2 interlayer at T>Tm(Ni), which denotes the onset of the non-equilibrium crystallization mechanism for a given h0Al value ( cr , 2)

, for each of the diffusion [figure 14(a) and (b)]; this is the second critical heating rate, v T data sets. It should be noted that all the curves in figure 14(a) and figure 14(b) are similar, and ( cr ,1)

( cr , 2)

the values of Tfin corresponding to critical heating rates v T and v T are the same for any h0Al at a given set of diffusion parameters. Besides, a minimum in curves Tfin(vT) seen in figure 14(a) does not correspond to a definite physical process and is only a result of competition between growth and dissolution of the phase 2 interlayer; for data set No. 2 it is not observed [figure 14(b)]. ( cr ,1)

( cr , 2)

and v T can be found Thus, for any h0Al value the critical heating rates v T unambiguously. However, transition to the QEDP mechanism, where the NiAl interlayer with almost constant thickness displaces as a whole between two melts, occurs gradually in a relatively narrow range of heating rates. Hence there isn’t a strict criterion for the onset of this ( cr ,3)

interaction route. So, the third critical heating rate v T ( cr ,1)

[obviously, v T

( cr ,3)

< vT

( cr , 2)

< vT

corresponding to this transition

] was determined basing of the behavior of lines h2(T)

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B.B. Khina

(like those shown in figures 5 and 6), which were calculated by varying vT for given h0Al; for diffusion data set No. 1 it does not coincide with a minimum in curves for Tfin [figure 14(a)].

Figure 13. Final thickness of phase 2 (NiAl), h2fin, attained by the time of disappearance of any of the parent phases vs. heating rate for different initial thickness of the Al layer: h0Al=100 μm (line 1), 50 μm (2), 20 μm (3), 10 μm (4), and 5 μm (5); (a) calculated using diffusion data set No. 1 for NiAl (see Table 2) [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics]; (b) calculated using diffusion data set No. 2 for NiAl (see Table 2) [Reproduced with kind permission from Springer Science+Business Media: International Journal of Self-Propagating High-Temperature Synthesis, Modeling heterogeneous interaction during SHS in the Ni-Al system: a phase-formation-mechanism map, vol. 16, 2007, pp. 51-61, B.B.Khina and B.Formanek, Figure 11, Copyright Allerton Press, Inc., 2007].

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Figure 14. Final temperature, Tfin, at which the growth of phase 2 (NiAl) terminates vs. heating rate for different initial thickness of the Al layer, h0Al: (a) data set No. 1 for interdiffusion in NiAl (Table 2): h0Al=100 μm (line 1), 50 μm (2), 20 μm (3), 10 μm (4), 5 μm (5), 1 μm (6), and 0.5 μm (7) [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics]; (b) data set No. 2 for interdiffusion in NiAl (Table 2): h0Al=100 μm (line 1), 20 μm (2), 5 μm (3), 1 μm (4), and 0.5 μm (5) [Reproduced with kind permission from Springer Science+Business Media: International Journal of Self-Propagating High-Temperature Synthesis, Modeling heterogeneous interaction during SHS in the Ni-Al system: a phase-formationmechanism map, vol. 16, 2007, pp. 51-61, B.B.Khina and B.Formanek, Figure 12, Copyright Allerton Press, Inc., 2007].

3.5. Phase-Formation-Mechanism Map for Non-isothermal Interaction in the Ni-Al System As a final result of numerical simulation, a diagram of phase and structure formation mechanisms for the synthesis of NiAl in non-isothermal conditions is constructed (figure 15).

284

B.B. Khina ( cr ,1)

( cr , 2)

Lines 1 (1') and 2 (2') correspond to critical heating rates v T and v T , respectively, calculated using diffusion data set No. 1 (lines 1 and 2) and No. 2 (lines 1' and 2').

Figure 15. Diagram of phase formation mechanisms for synthesis of intermetallic compound NiAl in non-isothermal conditions: domain I (below line 1 or 1') is the solid-state diffusion-controlled NiAl layer growth, or “slow” route typical of furnace synthesis; domain II [between lines 1 (or 1') and 3 (or 3')] combines diffusion-controlled growth of the displacing NiAl layer and precipitation of NiAl grains from melt; domain III [between lines 3 (or 3') and 2 (or 2')] is the quasi-equilibrium dissolutionprecipitation route where the thickness of the intermediate NiAl film is almost constant; domain IV (above line 2 or 2') is the non-equilibrium crystallization mechanism, or “fast” route typical of CS. Diffusion parameters for NiAl (Table 2): data set No. 1 (lines 1, 2 and 3) and No. 2 (lines 1', 2' and 3'). Mark z corresponds to phase formation pattern observed in [16] and mark „ refers to the experimental conditions of [14] for SHS in the Ni-Al system. [Reprinted with permission from: B.B.Khina, Journal of Applied Physics, Vol. 101, No.6, 063510 (11 pp.), 2007. Copyright 2007, American Institute of Physics].

In parametric domain I, growth of a solid NiAl layer at the expense of liquid Al and solid Ni proceeds via interdiffusion through this layer, which corresponds to a “slow” reaction route typical of the traditional furnace synthesis, where the heating rate is low, below 10 K/s. Here, widely used models employing solid-state diffusion-controlled kinetics [56-63] can give an adequate description of the product formation, but calculations should be performed with real diffusion parameters. For a reactant particle size of 20-100 μm, which is typical of CS in particulate systems, this interaction pattern cannot provide fast conversion of initial metals into the product in the course of CS in both thermal explosion and wave propagation mode. Note that this mechanism can take place in thin-film Ni-Al systems (at h0Al<0.3 μm) even at high heating rates characteristic of the SHS wave, vT ~ 105-106 K/s.

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In domain IV, non-equilibrium crystallization of NiAl grains from the melt occurs (without intermediacy of a continuous solid NiAl interlayer). This brings about fast accomplishment of interaction, which is typical of CS. In this parametric region, models [47,48,49,50,51,52] are valid; the rate-limiting stage is either diffusion in the liquid phase [47,48,49] or crystallization kinetics [51,52]. As seen from figure 15, at h0Al>5 μm conversion of reactants into the final product in the SHS wave where the heating rate attains 105-106 K/s can occur only via this “fast” route. In experiments on SHS in multilayer Ni-Al foils in the TE mode [16], the half-thickness of Al layers was 56 μm and the measured heating rate at temperatures above 1300 °C was 104 K/s. It corresponds to domain IV were NiAl grains can form via non-equilibrium crystallization. Very fast formation of the product at T> 1300 °C was ascribed to precipitation of solid NiAl from a supersaturated liquid without an intermediate solid interlayer [16], which agrees with the outcome of our numerical modeling. If a large amount of melt is formed that cannot react completely to form the solid product (NiAl) during a short heating time in the SHS wave, its outflow can occur, which was observed experimentally in a thick-foil Ni-Al system [129]. ( cr ,3)

Line 3 or 3' refers to the 3rd critical heating rate, v T . Region II lying between lines 1 and 3 (or between 1' and 3') corresponds to a combined mechanism when the NiAl layer, whose thickness is still increasing with time due to solid-state interdiffusion, starts displacing between the Al-base and Ni-base melts, and a certain fraction of the final product is formed via melt crystallization. This domain is rather narrow, especially for diffusion data set No. 1 (see table 2), and this interaction pattern can be considered as a transient one. In domain III located between lines 3 and 2 for diffusion data set No. 1 or between lines 3' and 2' for set No. 2, the quasi-equilibrium dissolution-precipitation mechanism takes place: a thin continuous NiAl film separating two liquid phases, whose thickness remains almost constant during heating, displaces as a whole while the final product, viz. NiAl grains, can form only via precipitation from the melt. The kinetics of product crystallization can be controlled by solid-state diffusion across the film since the diffusion coefficient in NiAl is substantially lower than in the melt. This situation is close to that considered in [60]. However, as mentioned above, model calculations in [60] were performed using fitting parameters instead of real diffusion data for phase NiAl. Besides, that model cannot give information about the product grain size, which is important for practical applications. Thus, to correctly describe this interaction route, the approach developed in this Chapter should be combined with a relevant crystallization model. In studying SHS of NiAl in a Ni-Al powder mixture [14], the initial particle radius of Al was 50-75 μm and the measured heating rate in the SHS wave was vT≈2700 K/s. This is close to the border between domains III and IV (figure 15). Since the temperature and heating rate during SHS is non-uniform in the cross-section normal to the combustion wave propagation because of heat losses to the environment, which is especially true for small-sized samples, interaction at the Ni/Al interfaces in the specimen during heating in the SHS wave can occur both via QEDP in the presence of a thin NiAl film and via non-equilibrium crystallization of NiAl grains. Hence, the results of our modeling qualitatively agree with results of microstructural studies [14], where the product (NiAl) grains were observed to precipitate from the saturated Al-Ni liquid.

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It should also be noted that if we use a different set of E and D0 values for diffusion in the NiAl layer or take into account the concentration dependence of interdiffusion coefficient D2 (e.g., according to [143-147]) and consider metastable phase equilibria in the Ni-Al system below the Ni3Al melting temperature, this will result in displacement of lines in figure 15 (probably towards lower heating rates because the parameters listed in table 2 give upper estimates of D2 values), but the general physical picture revealed in this Chapter most likely will not change. For complete modeling of CS in the Ni-Al thin-film system, the above kinetic approach should be combined with a heat transfer model such as developed in [161], where the heat release rate will be proportional to ∂h2/∂t times the reaction enthalpy.

4. Conclusion In this Chapter, the validity of the quasi-equilibrium concept of solid-state diffusioncontrolled product-layer growth is examined for the formation of interstitial compounds such as carbides (on the example of TiC) and substitutional phases such as intermetallics (taking NiAl as an example) in strongly non-isothermal conditions, which are intrinsic in CS. In both systems, real (i.e. experimentally known) parameter values, viz. the activation energy E and preexponential factor D0, for diffusion in the product phase are used along with relevant equilibrium binary phase diagrams. For titanium carbide, spherical symmetry of a unit reaction cell is considered, which is typical of CS in particulate systems, and all the possible scenarios are examined. For nickel aluminide, the case of planar symmetry is studied, which corresponds to CS in multilayer thin films. As a final result, phase-formation mechanism maps for non-isothermal heterogeneous interaction in the above binary systems are constructed (figures 7 and 15), which permit classifying the experimentally known reaction pathways and determining the applicability limits of existing kinetic models of CS. Besides, the effect of high heating rates, which are inherent in CS, on the interaction pattern is revealed. It is demonstrated that at high heating rates typical of CS and the characteristic reactant sizes of above 5 μm for the Ti-C system and 20 μm for the Ni-Al system, which are typical of SHS in powder charges, complete conversion of reactants into a solid product can occur via non-equilibrium crystallization of the refractory product grains from a saturated melt without the presence of a thin solid interlayer of an equilibrium phase. This is connected with a low rate of solid-state diffusion across the product phase. Thereby the existence of nonequilibrium patterns of phase formation, which were observed experimentally and debated in literature, is proved theoretically ex contrario for certain parametric domains. In these situations (domain II in figure 7 and domain IV in figure 15), theoretical approaches [47,48,49,50,51,52] can be used for modeling CS. At a large characteristic size of reactants and relatively low heating rates, product formation can occur via the equilibrium solid-state diffusion-controlled route (domain I in figures 7 and 15). However, for the particle sizes typical of CS in a powder charge mixture, this reaction pathway refers not to CS but to traditional furnace synthesis where the heating rate is low. Besides, this interaction pattern can take place during CS, i.e. at high heating rates, only for a small size of reactants: R00≤0.5 μm for the TiC system and h0Al<0.3 μm in the Ni-Al system. In this parametric domain, equilibrium diffusion-based models such as [5559,61,62] are valid.

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A quasi-equilibrium dissolution-precipitation route via the intermediacy of a continuous solid-product interlayer (the “shrinking-core” mechanism in the spherical symmetry) appears impossible in the Ti-C system. This is connected with a substantial difference in partial diffusion coefficients of the Ti and C atoms in the titanium carbide layer (DTi << DC), whose diffusion fluxes are responsible for the displacement of different phase boundaries; typically in metal carbides MCx DM << DC in a wide temperature range. But in the formation of substitutional compound, e.g., NiAl, whose growth is determined by the interdiffusion coefficient (i.e. can occur at both Ni/NiAl and NiAl/Al interfaces), which typically exceeds the diffusion coefficient in refractory carbides, this interaction route is possible in a certain parametric range (see domain III in figure 15). This pattern was considered in [60] implying solid-state diffusion across the intermediate NiAl layer as a rate-limiting stage of CS and using fitting parameters for calculations instead of real diffusion data for NiAl. In the formation of an interstitial compound (here TiC) in non-isothermal conditions, a criterion for the changeover from the equilibrium solid-state diffusion controlled mechanism to the non-equilibrium dissolution-precipitation route is the rupture of a primary refractoryproduct shell formed on the surface of a solid metal particle at melting of the latter. Hence a diagram similar to figure 7 can be plotted for non-isothermal interaction in regard to CS in some other systems, such as metal-carbon, where the value of TCS exceeds the melting temperature of the metallic reactant but is below that of a non-metal, and probably for some metal-gas (e.g., nitrogen) systems. In the formation of a substitutional compound (here intermetallic NiAl) in non-isothermal conditions typical of CS, the criteria for transitions from the equilibrium (solid-state diffusion controlled) interaction pattern to the quasi-equilibrium (with the presence of a thin layer of solid product) mechanism and then to the non-equilibrium route (crystallization of product grains from a saturated melt) are determined by competition between the diffusion-limited growth of the product layer and its dissolution in the parent phases. Therefore, phaseformation diagrams similar to that shown in figure 15 can be constructed for other binary systems containing substitutional compounds (intermetallics). Finally, it should be noted that the development of novel advanced materials using CS and CS-related technologies and, in future prospect, creating controllable CS processes necessitates elaboration of complex models for combustion synthesis in binary and multicomponent systems. Such models are supposed to include, along with heat transfer equation, adequate description of the mechanism and kinetics of heterogeneous interaction, which, as shown above, may substantially differ in different ranges of a heating rate, temperature and characteristic size of reactants. In this aspect, the results obtained in this Chapter can be used as a basis for further advances in this promising and fascinating cross-disciplinary area.

References [1] [2]

[3] [4]

Merzhanov, A. G. Russ. Chem. Bull. 1997, 46, 1-27. Merzhanov, A. G. In Combustion and Plasma Synthesis of High-Temperature Materials; Munir, Z. A.; Holt, J. B.; Eds.; VCH Publishers: New York, NY, 1990; pp 153. Munir, Z. A.; Anselmi-Tamburini, U. Mater. Sci. Rep. 1989, 3, 277-365. Bockowski, M. Physica B 1999, 265, 1-5.

288 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

B.B. Khina Varma, A.; Lebrat, J. P. Chem. Eng. Sci. 1992, 47, 2179-2194. Merzhanov, A. G. Ceramics Int. 1995, 21, 271-279. Merzhanov, A. G. J. Mater. Proc. Technol. 1996, 56, 222-241. Merzhanov, A. G.; Mukasyan, A. S. Solid-Flame Combustion; Torus Press: Moscow, 2007. Shteinberg, A. S.; Knyazik, V. A. Pure Appl. Chem. 1992, 64, 965-976. Rogachev, A. S.; Mukasyan, A. S.; Merzhanov, A. G. Dokl. Phys. Chem. 1987, 297, 1240-1243. Merzhanov, A. G.; Rogachev, A. S.; Mukasyan, A. S.; Khusid, B. M. Comb. Expl. Shock Waves 1990, 26, 92-102. LaSalvia, J. C.; Kim, D. K.; Lipsett, R. A.; Meyers, M. A. Metall. Mater. Trans. A 1995, 26, 3001-3009; LaSalvia, J. C.; Meyers, M. A. ibid. 1995, 26, 3011-3019. Sung-Won, J.; Gi-Wook, L.; Jong-Tae, M.; Yong-Seog, K. Acta Mater. 1996, 44, 43174326. Fan, Q.; Chai, H.; Jin, Z. Intermetallics 2001, 9, 609-619. Fan, Q.; Chai, H.; Jin, Z. Intermetallics 2002, 10, 541-554. Zhu, P.; Li, J. C. M.; Liu, C. T. Mater. Sci. Eng. A 2002, 329-331, 57-68. Christian, J. W. The Theory of Transformations in Metals and Alloys; Pergamon Press: Oxford, UK, 1975. Lyubov, B. Ya. Kinetic Theory of Phase Transformations; National Bureau of Standards: Gaithersburg, MD, 1978. Khusid, B. M.; Khina, B. B.; Rabinovich, O. S. Comp. Chem. Eng. 1994, 18, 11831195. Moore, J. J., Feng, H. J. Progr. Mater. Sci. 1995, 39, 275-316. Makino, A. Progr. Energy Comb. Sci. 2001, 27, 1-74. Rogachev, A. S.; Baras, F. Int. J. SHS 2007, 16, 141-153. Mukasyan, A. S.; Rogachev, A. S. Progr. Energy Comb. Sci. 2008, 34, 377-416. Shkadinskii, K. G.; Khaikin, B. I.; Merzhanov, A. G. Comb. Expl. Shock Waves 1971, 7, 15-22. Aldushin, A. P.; Martem'yanova, T. M.; Merzhanov, A. G.; Khaikin, B. I.; Shkadinskii, K. G. Comb. Expl. Shock Waves 1972, 8, 159-164. Ivleva, T. P.; Merzhanov, A. G.; Shkadinskii, K. G. Sov. Phys. Dokl. 1978, 239, 255256. Ivleva, T. P.; Merzhanov, A. G. Phys. Rev. E 2001, 64, 036218 (4 pp). Astapchik, A. S.; Podvoisky, E. P.; Chebotko, I. S.; Khusid, B. M.; Merzhanov, A. G.; Khina, B. B. Phys. Rev. E 1993, 47, 319-326. Khusid, B. M.; Khina, B. B.; Podvoisky, E. P.; Chebotko, I. S. Comb. Flame 1994, 99, 371-378. Kirdyashkin, A. I.; Maksimov, Yu. M.; Nekrasov, E. A. Comb. Expl. Shock Waves 1981, 17, 377-379. Khusid, B. M.; Khina, B. B.; Bashtovaya, E. A. Comb. Expl. Shock Waves 1991, 27, 708-714. Khusid, B. M.; Khina, B. B.; Bashtovaya, E. A.; Quang, V. Z. Comb. Expl. Shock Waves 1992, 28, 389-395. Hu, G.; Zhang, L. Comp. Mater. Sci. 2008, 42, 558-563. Zhang, L.; Wang, Z. Exper. Thermal Fluid Sci. 2008, 32, 1255-1263.

Modeling of Interaction Kinetics during Combustion Synthesis…

289

[35] Smithells Metals Reference Book; Brandes, E. A.; Brook, G. B.; Eds.; ButterworthHeinemann: Oxford, UK, 1992. [36] Wilson, J. R. Metall. Review 1965, 40, 381-590. [37] Hwang, S.; Mukasyan, A. S.; Varma, A. Comb. Flame 1998, 115, 354-363. [38] Varma, A.; Mukasyan, A. S.; Hwang, S. Chem. Eng. Sci. 2001, 55, 1459-1466. [39] Kochetov, N. A.; Rogachev, A. S.; Merzhanov, A. G. Dokl. Phys. Chem. 2003, 389, 80-82. [40] Merzhanov, A. G.; Kovalev, D. Yu.; Shkiro, V. M.; Ponomarev, V. I. Dokl. Phys. Chem. 2004, 394, 34-38. [41] Merzhanov, A. G. Dokl. Phys. Chem. 1997, 353, 135-138. [42] Rogachev, A. S.; Merzhanov, A. G. Dokl. Phys. Chem. 1999, 365, 127-130. [43] Beck, J. M.; Volpert, V. A. Physica D 2003, 182, 86-102. [44] Rabinovich, O. S.; Grinchuk, P. S.; Khina, B. B.; Belyaev, A.V. Int. J. SHS 2003, 11, 257-270. [45] Aldushin, A. P.; Khaikin, B. I. Comb. Expl. Shock Waves 1974, 10, 273-283. [46] Aldushin, A. P.; Kasparyan, S. G.; Shkadinskii, K. G. In Combustion and Explosion; Stesik, L. N.; Ed.; Nauka: Moscow, 1977, pp 207-212. [47] Gennari, S.; Maglia, F.; Anselmi-Tamburini, U.; Spinolo, G. J. Phys. Chem. B 2003, 107, 732-738. [48] Gennari, S.; Maglia, F.; Anselmi-Tamburini, U.; Spinolo, G. J. Phys. Chem. B 2004, 108, 19550-19556. [49] Gennari, S.; Anselmi-Tamburini, U.; Maglia, F.; Spinolo, G.; Munir, Z. A. Acta Mater. 2006, 54, 2343-2351. [50] Locci, A. M.; Cincotti, A.; Delogu, F.; Orru, R.; Cao, G. Chem. Eng. Sci. 2004, 59, 5121-5128. [51] Locci, A. M.; Cincotti, A.; Delogu, F.; Orru, R.; Cao, G. J. Mater. Res. 2005, 20, 12571268. [52] Locci, A. M.; Cincotti, A.; Delogu, F.; Orru, R.; Cao, G. J. Mater. Res. 2005, 20, 12691277. [53] Khina, B. B.; Formanek, B.; Solpan, I. Physica B 2005, 355, 14-31. [54] Khina, B. B. Int. J. SHS 2005, 14, 21-31. [55] Nekrasov, E. A.; Smolyakov, V. K.; Maksimov, Yu. M. Comb. Expl. Shock Waves 1981, 17, 305-311. [56] Nekrasov, E. A.; Smolyakov, V. K.; Maksimov, Yu. M. Comb. Expl. Shock Waves 1981, 17, 513-520. [57] Smolyakov, V. A.; Nekrasov, E. A.; Maksimov, Yu. M. Comb. Expl. Shock Waves 1982, 18, 312-315. [58] Maksimov, Yu. M.; Smolyakov, V. A.; Nekrasov, E. A. Comb. Expl. Shock Waves 1984, 20, 479-486. [59] Nekrasov, E. A.; Tkachenko, V. N.; Zakirov, A. E. Comb. Sci. Technol. 1993, 91, 207223. [60] Biswas, A.; Roy, S. K.; Gurumurthy, K. R.; Prabhu, N.; Banerjee, S. Acta Mater. 2002, 50, 757-773. [61] Oliveira, A. A. M.; Kaviany, M. Int. J. Heat Mass Transfer 1999, 42, 1075-1095. [62] Vrel, D.; Lihrmann, J. M.; Tobaly, P. J. Mater. Synth. Proc. 1994, 2, 179-187. [63] Armstrong, R. Comb. Sci. Technol. 1990, 71, 155-174.

290 [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96]

B.B. Khina Kidson, G. V. J. Nucl. Mater. 1961, 3, 21-29. Gusak, A. M.; Gurov, K. P. Phys. Metal. Metallogr. 1982, 53, 842-847. Williams, D. S.; Rapp, R. A.; Hirth, J. P. Thin Solid Films 1986 142, 47-64. Baglin, J. E. E.; Poat, J. M. In Thin Films: Interdiffusion and Reactions; Poate, J. M.; Tu, K. N.; Mayer, J. W.; Eds.; J. Wiley and sons: New York, NY, 1978, chapter 9. d'Heurle, F. M.; Gas, P. J. Mater. Res. 1986, 1, 205-221. Dybkov, V. I. J. Mater. Sci. 1993, 28, 6371-6380; idem, ibid, 2000, 35, 1729-1736. Bouche, K.; Barbier, F.; Coulet, A. Mater. Sci. Eng. A 1998, 249, 167-175. Gosele, U.; Tu, K. N. J. Appl. Phys. 1982, 53, 3252-3260. Desre, P. J.; Yavary, A. R. Phys. Rev. Lett. 1990, 64, 1533-1536. Gusak, A. M.; Gurov, K. P. Solid State Phenom. 1992, 23-24, 117-122. Khusid, B. M.; Khina, B. B. Phys. Rev. B 1991, 44, 10778-10793. Atzmon, M. Metall. Trans. A 1992, 23, 49-53. Farber, L.; Klinger, L.; Gotman, I. Mater. Sci. Eng. A 1998, 254, 155-165. Janssen, M. M. P.; Riek, G. D. Trans. Met. Soc. AIME 1967, 239, 1372-1385. Janssen, M. M. P. Metall. Trans. 1973, 4, 1623-1633. Song, H. Y.; Wang, X. J. Mater. Sci. 1996, 31, 3281-3288. Elliot, R. P. Constitution of Binary Alloys: First Supplement; McGrow-Hill: New York, NY, 1965. Zenin, A. A.; Korolev, Yu. M.; Popov, V. T.; Tyurkin, Yu. V. Sov. Phys. Dokl. 1986, 31, 239-242. Samsonov, G. V.; Vinitskii, I. M. Handbook of Refractory Compounds; IFI/Plenum: New York, NY, 1980. Samsonov, G. V.; Upadhaya, G. S.; Neshpor, V.S. Physical Metallurgy of Carbides; Naukova Dumka: Kiev, 1974. Dergunova, V. S.; Levitskiy, Yu. V.; Shurshakov, A. N.; Kravetskiy, G. A. Interaction of Carbon with Refractory Metals; Metallurgiya: Moscow, 1974. Shkiro, V. M.; Borovinskaya, I. P. Comb. Expl. Shock Waves 1976, 12, 828-831. Nekrasov, E. A.; Maksimov, Yu. M.; Ziatdinov, M. Kh.; Shteinberg, A. S. Comb. Expl. Shock Waves 1978, 14, 575-581. Kirdyashkin, A. I.; Maksimov, Yu. M.; Merzhanov, A. G. Comb. Expl. Shock Waves 1981, 17, 591-595. Kiparisov, S. S.; Levinskiy, Yu. V.; Petrov, A. P. Titanium Carbide: Synthesis, Properties, Applications; Metallurgiya: Moscow, 1987. Andrievskii, R. A.; Spivak, I. I. Strength of Refractory Compounds and Materials on Their Basis: a Handbook; Metallurgiya: Chelyabinsk, 1989. Sarian, S. J. Appl. Phys. 1969, 40, 3515-3520. Sarian, S. J. Appl. Phys. 1968, 39, 3305-3310. Sarian, S. J. Appl. Phys. 1968, 39, 5036-5041. Kohlstedt, D. L.; Williams, W. S.; Woodhouse, J. B. J. Appl. Phys. 1970, 41, 44764484. Adelsberg, L. M.; Cadoff, L. H. Trans. Metall. Soc. AIME 1967, 239, 933-935. Ushakov, B. F.; Zagryazkin, V. N.; Panov, A. S. Inorg. Mater. USSR 1972, 8, 19211925. Van Loo, F. J. J.; Wakelkamp, W.; Bastin, G. F.; Metsalaar, R. Solid State Ionics 1989, 32-33, 824-832.

Modeling of Interaction Kinetics during Combustion Synthesis…

291

[97] Iida, T.; Guthrie, R. I. L. The Physical Properties of Liquid Metals; Clarendon Press: Oxford, UK, 1988. [98] Lepinskih, B. M.; Kaybichev, A. V.; Savel'ev, Yu. A. Diffusion of Elements in Liquid Metals of the Iron Group; Nauka: Moscow, 1974. [99] Properties of Elements: a Handbook, vol. 1: Physical Properties; Samsonov, G. V.; Ed.; Metallurgiya: Moscow, 1976. [100] Kostikov, V. I.; Varenkov, A. N. Interaction of Metallic Melts with Carbon Materials; Metallurgiya: Moscow, 1981. [101] Frank-Kamenetskiy, D. A. Diffusion and Heat Transfer in Chemical Kinetics; Plenum Press: New York, NY, 1969. [102] Tables of Physical Parameters; Kikoin, I. K.; Ed.; Atomizdat: Moscow, 1976. [103] Kubaschewski, O.; Alcock, C. B. Metallurgical Thermochemistry; Pergamon Press: Oxford, UK, 1979. [104] Lyubov, B. Ya. Theory of Crystallization in Large Volumes; Nauka: Moscow, 1975. [105] Kartashov, E. M.; Lyubov, B. Ya. Heat Transfer-Soviet Research 1976, 8 (2), 1-39. [106] Landau, L. D.; Lifshiz, E. M. Theory of Elasticity; Pergamon Press: Oxford, UK, 1986. [107] Holt, J. B.; Munir, Z. A. J. Mater. Sci. 1986, 21, 251-259. [108] Dubois, S.; Karnatak, N.; Beaufort, M. F.; Vrel, D. J. Mater. Synth. Proc. 2001, 9, 253257. [109] Adachi, S.; Wada, T.; Mahara, T.; Miyamoto, Y.; Koizumi, M.; Yamada, O. J. Amer. Ceram. Soc. 1989, 72, 805-809. [110] Efimov, Yu. E.; Zaripov, N. G.; Bloshenko, V. N.; Bokii, V. A.; Kashtanova, A. A.; Belinskaya, U. L. Comb. Expl. Shock Waves 1992, 28, 496-499. [111] Eremeev, V. S. Diffusion and Stresses; Energoatomizdat: Moscow, 1984. [112] Shcherbakov, V. A.; Shteinberg, A. S. Dokl. Phys. Chem. 1995, 345, 275-279. [113] Scher, H.; Zallen, R. J. Chem. Phys. 1970, 53, 3759-3761. [114] Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor & Francis: London, UK, 1994. [115] Aleksandrov, V. V.; Korchagin, M. A. Comb. Expl. Shock Waves 1987, 23, 557-564. [116] He, C.; Blanchetiere, C.; Stangle, G. C. J. Mater. Res. 1998, 13, 2269-2280. [117] Sytschev, A. E.; Merzhanov, A. G. Russ. Chem. Reviews 2004, 73, 147-159. [118] Deevi, S. C.; Sikka, V. K.; Liu, C. T. Prog. Mater. Sci. 1997, 42, 177-192. [119] Stoloff, N. S.; Liu, C. T.; Deevi, S. C. Intermetallics 2000, 8, 1313-1320. [120] Kolomytsev, P. T. High-temperature Protective Coatings for Nickel-base Alloys; Metallurgiya: Moscow, 1991. [121] Rogachev, A. S. Russ. Chem. Reviews 2008, 77, 21-37. [122] Swiston, A. J.; Hufnagel, T. C.; Weihs, T. P. Scripta Mater. 2003, 48, 1575-1580. [123] Wang, J.; Besnoin, E.; Knio, O. M.; Weihs, T. P. Acta Mater. 2004, 52, 5265-5274. [124] Trenkle, J. C.; Wang, J.; Weihs, T. P.; Hufnagel, T. C. Appl. Phys. Lett. 2005, 87, 153108 (3 pp). [125] Pascal, C.; Marin-Ayral, R. M.; Tedenac, J. C. J. Alloys Compounds 2002, 337, 221225. [126] Besnoin, E.; Cerutti, S.: Knio, O .M.; Weihs, T. P. J. Appl. Phys. 2002 92, 5474-5481. [127] Duckham, A.; Spey, S. J.; Wang, J.; Weihs, T. P. J. Appl. Phys. 2004, 96, 2336-2342. [128] Wang, J.; Besnoin, E.; Knio, O. M.; Weihs, T. P. J. Appl. Phys. 2005, 97, 114307 (7 pp).

292

B.B. Khina

[129] Shteinberg, A. S.; Shcherbakov, V. A.; Munir, Z. A. Comb. Sci. Technol. 2001, 169, 124. [130] Anselmi-Tamburini, U.; Munir, Z. A. J. Appl. Phys. 1989, 66, 5039-5045. [131] Binary Alloy Phase Diagrams; Massalski, T. B.; Okamoto, H.; Subramanian, P. R.; Kacprzak, L.; Eds.; AMS: Materials Park, OH, 1990, p 183. [132] Nash, P.; Singleton, M. F.; Murray, J. L. In Phase Diagrams of Binary Nickel Alloys; Nash, P.; Ed.; ASM: Metals Park, OH, 1991, p 3. [133] Gurov, K. P.; Kartashkin, B. A.; Ugaste, U. E. Interdiffusion in Multiphase Metallic Systems; Nauka: Moscow, 1981. [134] Lapshin, O. V.; Ovcharenko, V. E. Comb. Explos. Shock Waves 1996, 32, 299-305. [135] Kovalev, O. B.; Neronov, V. A. Comb. Explos. Shock Waves 2004, 40, 172-179. [136] Mann, A. B.; Gavens, A. J.; Reiss, M. E.; Van Heerden, D.; Bao, G.; Weihs, T. P. J. Appl. Phys. 1997, 82, 1178-1188. [137] Jayaraman, S.; Knio, O. M.; Mann, A. B.; Weihs, T. P. J. Appl. Phys. 1999, 86, 800809. [138] Jayaraman, S.; Mann, A. B.; Reiss, M.;. Weihs, T. P; Knio, O. M. Comb. Flame 2001, 124, 178-194. [139] Janssen, M. M. P.; Riek, G. D. Trans. Metall. Soc. AIME 1967, 239, 1372-1385. [140] Janssen, M. M. P. Metall. Trans. 1973, 4, 1623-1633. [141] Hickl A. J.; Heckel, R. W. Metall. Trans. A 1975, 6, 431-440. [142] Garcia, V. H.; Mors, P. M.; Scherer, C. Acta Mater. 2000, 48, 1201-1206. [143] Watanabe, M.; Horita, Z.; Nemoto, M. Defect Diffus. Forum 1997, 143-147, 345-350. [144] Nakajima, H.; Sprengel, W.; Nonaka, K. Intermetallics 1996, 4, S17-S28. [145] Herzig, C.; Divinski, S. V.; Frank, St.; Przeorski, T. Defect Diffus. Forum 2001, 194199, 317-336. [146] Herzig C.; Divinski, S. Intermetallics 2004, 12, 993-1003. [147] Nakamura, Ryusuke; Takasawa, Koichi; Yamazaki, Yoshihiro; Iijima, Yoshiaki. Intermetallics 2002, 10, 195-204. [148] Helander, T.; Agren, J. Acta Mater. 1998, 47, 1141-1152. [149] Wei, H.; Sun, X.; Zheng, Q.; Guan, H.; Hu, Z. Acta Mater. 2004, 52, 2645-2651. [150] Khina, B. B. J. Appl. Phys. 2007, 101, 063510 (11 pp). [151] Khina, B. B.; Formanek, B. Int. J. SHS 2007, 16, 51-61. [152] Novikov, N. P.; Borovinskaya, I. P.; Merzhanov, A. G. In Combustion Processes in Chemical Technology and Metallurgy; Merzhanov, A. G.; Ed.; Institute of Chemical Physics: Chernogolovka, 1975, pp 174-188. [153] Barin, I.; Knacke, I.; Kubaschevski, O. Thermochemical Properties of Inorganic Substances: Supplement; Springer-Verlag: Berlin, 1977, p 490. [154] Pelekh, A. E.; Mukasyan, A. S.; Varma, A. Ind. Eng. Chem. Res. 1999, 38, 793-798. [155] Kharatyan, S. L.; Chatilyan, H. A.; Mukasyan, A. S.; Simonetti, D. A.; Varma, A. AIChE J. 2005, 51, 261-270. [156] Sinelnikova, V. S.; Podergin, V. A.; Rechkin, V. N. Aluminides: a Handbook; Naukova Dumka: Kiev, 1965. [157] Properties of Elements: a Handbook; Drits, M. E.; Ed.; Metallurgiya: Moscow, 1985. [158] Zinoviev, V. E. Thermophysical Properties of Metals at High Temperatures: a Handbook; Metallurgiya: Moscow, 1989. [159] Myagkov, V. G.; Bykova, L. E. Dokl. Phys. 2004, 49, 289-291.

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[160] Myagkov, V. G.; Zhigalov, V. S.; Bykova, L. E.; Mal’tsev, V. K. Tech. Phys. 1998, 43, 1189-1192. [161] Rabinovich, O. S.; Grinchuk, P. S.; Andreev, M. A.; Khina, B. B. Physica B 2007, 392, 272-280.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 295-359

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 4

IDENTIFICATION AND CONTROL OF LARGE SMART STRUCTURES Yeesock Kim1,a, Reza Langari2,b and Stefan Hurlebaus1,c 1

Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3367, U.S.A. 2 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3135, U.S.A.

Abstract In this book, a novel multiple-model approach is proposed in order to model and control nonlinear behavior of large structures equipped with nonlinear smart control devices in a unified framework. First, a novel Nonlinear System Identification (hereinafter as “NSI”) algorithm, Multiinput, Multi-output (hereinafter as “MIMO”) AutoRegressive eXogenous (hereinafter as “ARX”) inputs-based Takagi-Sugeno (hereinafter as “TS”) fuzzy model, is developed to identify nonlinear behavior of large structures equipped with smart damper systems. It integrates a set of MIMO ARX models, clustering algorithms, and weighted least squares algorithm with a TS fuzzy model. Based on a set of input-output data that is generated from large structures equipped with MagnetoRheological (hereinafter as “MR”) dampers, premise parameters of the MIMO ARX-TS fuzzy model are determined by the clustering algorithms, while the consequent parameters are optimized by the weighted least squares algorithm. Second, a new Semiactive Nonlinear Fuzzy Control (hereinafter as “SNFC”) algorithm is proposed through integration of multiple Lyapunov-based state feedback gains, a Kalman filter, and a converting algorithm with TS fuzzy interpolation method: (1) the nonlinear MIMO ARX-TS fuzzy model is decomposed into a set of linear dynamic models that are operated in only a local linear operating region; (2) Then, based on the decomposed dynamic models, multiple Lyapunov-based state feedback controllers are formulated in terms of linear matrix inequalities (hereinafter as “LMIs”) such that the large structure-MR damper system is globally asymptotically stable and the performance on transient responses is also guaranteed; (3) finally, the state feedback controllers are integrated with a Kalman filter and a converting a

E-mail address: [email protected] E-mail address: [email protected] c E-mail address: [email protected] b

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Yeesock Kim, Reza Langari and Stefan Hurlebaus algorithm using a TS fuzzy interpolation method to construct semiactive output feedback controllers. To demonstrate the effectiveness of the proposed MIMO ARX-TS fuzzy model-based SNFC systems, it is applied to a 3-, an 8-, and a 20-story building structure employing MR dampers. It is demonstrated from the numerical simulations that the proposed MIMO ARXTS fuzzy model-based SNFC algorithm is effective to control responses of seismically excited large building structures equipped with MR dampers.

1. Introduction 1.1. Nonlinear System Identification (NSI) of Large Structure-Damper Systems One of the most difficult but important tasks in control system design for building structures subjected to natural hazards is the development of an accurate explicit mathematical model of the building system to be controlled because precise mathematical information related to the building structure is used for calculation of control forces. However, the development of a mathematical model for a nonlinear building system is still a challenging problem. One example of a nonlinear building structure occurs when highly nonlinear hysteretic actuators/dampers are applied to building systems for efficient energy dissipation. In this case, the building structure integrated with the nonlinear dampers behaves nonlinearly although the building structure itself is usually assumed to remain linear (Ramallo et al. 2004). The development of an appropriate nonlinear model of the integrated structuredamper system considering the interaction effect between the structural system and the nonlinear damper plays a key role in control system design because the building structure integrated with a nonlinear damper is intrinsically nonlinear. In what follows, it is stated that a solution is available by means of NSI based on fuzzy logic theory. Since Zadeh’s paper (Zadeh 1965), fuzzy logic theory has been applied to many NSI problems (Langari 1999). In recent years, there have been a number of studies that use the TS fuzzy model, which provides an effective representation of nonlinear systems with the aid of fuzzy sets, fuzzy rules and a set of local linear models (Wang and Langari 1996). Research related to fuzzy logic-based system identification for large building structures first started from an ad-hoc approach based on many trials and errors of experienced investigators. However, this approach becomes unpractical when the number of design variables is large. To compensate for the drawbacks of the ad-hoc approach, later research focused on using intelligent learning algorithms, e.g., genetic algorithms and neural networks. Jiang and Adeli (2005) developed a fuzzy wavelet neural network model for identifying nonlinear behavior of high-rise building structures. In their work, the Multi-input-Single-output (hereinafter as “MISO”) fuzzy wavelet neural network model was trained by a hybrid Levenberg-Marquardt least-squares algorithm. However, only a few papers have been published on MIMO fuzzy NSI algorithms for use with a building structure equipped with a nonlinear damper (Kim et al. 2006; Kim and Langari 2007).

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1.2. Semiactive Nonlinear Fuzzy Control of Large Structures Fuzzy logic has attracted great attention to control system design (Langari 1993; Langari 1999; Yen and Langari 1999; Lei and Langari 2000; Hong and Langari 2000). A number of design methodologies for fuzzy logic controllers have been successively applied to a variety of large-scale civil structures, including: trial-and-error-based methodologies (Abe 1996; Subramanian et al. 1996; Battaini et al. 1998; Symans and Kelly 1999; Loh et al. 2003; Battaini et al. 2004); a self-organizing approach (Al-Dawod et al. 2004; Samali et al. 2004); training using linear quadratic Gaussian (hereinafter as “LQG”) data (Al-Dawod et al. 2001); neural networks-based learning (Faravelli and Yao 1996; Tani et al. 1998; Faravelli and Rossi 2002; Schurter and Roschke 2001; Faravelli et al. 2002); adaptive fuzzy (Zhou et al. 2003); genetic algorithms-based training (Ahlawat and Ramaswany 2002a, 2002b, 2004; Yan and Zhou 2006; Kim and Roschke 2006); fuzzy sliding mode (Wang and Lee 2002; Kim et al. 2004; Alli and Yakut 2005); etc. However, no systematic design framework has been conducted to design the SNFC systems for a building structure equipped with a nonlinear semiactive device. From a practical point of view, research related to a systematic semiactive control system design framework is still required for vibration control of large-scale civil infrastructures subjected to destructive environmental forces, e.g., earthquakes or strong winds. Active nonlinear fuzzy control (hereinafter as “ANFC”) system design can be carried out in a systematic way via the so-called parallel distributed compensation (hereinafter “PDC”) approach, which employs multiple optimum linear controllers (Hong and Langari 2000; Joh et al. 1997). The linear controllers correspond to the local linear models with automatic scheduling performed through fuzzy rules. Tanaka and Sano (1994) proposed a theorem on the stability analysis of an ANFC system using the Lyapunov direct method. This theorem states sufficient conditions for an ANFC system to be globally asymptotically stable by finding a common symmetric positive definite matrix such that a set of simultaneous Lyapunov inequalities are satisfied. However, no systematic design framework has been investigated to design SNFC systems for building structures equipped with nonlinear semiactive devices based on LMIs for reduction of response to earthquakes and strong wind events.

1.3. Outline of Book The first focus of this book is to propose a new NSI procedure for robust identification of large-scale building structures equipped with MR dampers subjected to destructive environmental forces such as earthquakes and winds. The new identification algorithm, multiple MIMO ARX-TS fuzzy model, integrates multiple MIMO ARX models with a TS fuzzy model. The premise part of the MIMO ARX-TS fuzzy model is determined through clustering algorithms and the consequent part is optimized using weighted least squares estimation. The second focus of this book is to develop a new SNFC algorithm for vibration control of large-scale building structures employing MR dampers: (1) multiple state feedback controllers in terms of LMIs are derived such that global asymptotical stability is guaranteed and the performance on the transient response is also satisfied. These state feedback gains are

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augmented with a Kalman observer to construct output feedback control systems; (2) then, an ANFC is developed through integration of the multiple output feedback controllers with fuzzy logic; (3) finally, the ANFC is integrated with a converting algorithm such as a clipped algorithm and an inverse MR damper model to develop a MISO SNFC. To demonstrate the effectiveness of the proposed MISO SNFC algorithm, a 3-story building structure employing a MR damper system is investigated. The third focus of this book is to extend capacities of the MISO SNFC system into MIMO SNFC system. Practical point of view, it might difficult to apply a centralized control system, which is a single control unit operates all the actuators and sensors, to large-scale building structures due to high cost of installation and maintenance as well as vulnerability to small damage of structural systems. In such cases, a solution can be found in decentralized control techniques. In this book, a MIMO SNFC control system is proposed that combines multiple MISO SNFC systems with decentralized control strategies such as a fully decentralized and a supervisory control. To demonstrate the effectiveness of the proposed approaches, these decentralized MIMO SNFC system design methodologies are applied to an eight story building structure and a full scale Los Angeles 20 story building structure. This is organized as follows: In Section 2, a variety of analytical models for a MR damper are introduced, including a Bingham, a polynomial, a Bouc-Wen, and a modified Bouc-Wen model. In addition to the forward MR damper models, the corresponding inverse MR damper models are addressed. The associated equations of motion are described in detail. In Section 3, a nonlinear multiple MIMO ARX-TS fuzzy model, which integrates multiple MIMO ARX models with TS fuzzy logic, is introduced. In Section 4, based on the nonlinear multiple MIMO ARX-TS fuzzy models, multiple optimum linear controllers are formulated in terms of LMIs such that global asymptotical stability is guaranteed and the performance on transient response is also satisfied. The stability issue is formulated via the Lyapunov direct method and the transient response performance is achieved by the pole-assignment algorithm. Then, such multiple state feedback controllers are integrated with a Kalman estimator to construct multiple output feedback controllers. Finally, a SNFC system is developed though integration of the output feedback controllers with converting algorithms and MR dampers. In Section 5, a MIMO SNFC system is proposed that combines multiple MISO SNFC systems using decentralized control strategies. They include a fully decentralized control and a supervisory control. In Section 6, three building structural systems are investigated to demonstrate the performance of the proposed MIMO ARX-TS fuzzy model-based SNFC algorithm. The equations of motion of building structures equipped with MR dampers are derived. Finally, concluding remarks and recommendation on future works are given in Section 8.

2. Magnetorheological (MR) Damper 2.1. Introduction In recent years, smart structures have been adopted from many engineering fields because the performance of structural systems can be improved without either significantly increasing the structure mass or requiring high cost of control power. They may be called intelligent structures, adaptive structures, active structures, adaptronics, structronics, etc. These

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terminologies refer to a smart structure which is an integration of actuators, sensors, control units, and signal processing units with a structural system. The materials that are usually used to make a smart structure are: piezoelectrics, shape memory alloys, electrostrictive/magnetostrictive materials, polymer gels, and magnetorheological/ electrorheological fluids (Hurlebaus and Gaul 2006). Semiactive devices have been applied to large scale civil structures. Semiactive control strategies combine favorable features of both active and passive control systems. Semiactive control systems include devices such as variable-orifice dampers, variable-stiffness devices, variable-friction dampers, controllable-fluid dampers, shape memory alloy actuators, and piezoelectrics (Hurlebaus and Gaul 2006). In particular, one of the controllable-fluid dampers, MR damper developed by Lord Corporation has attracted attention in recent years because it has many attractive characteristics. In general, a MR damper consists of a hydraulic cylinder, magnetic coils, and MR fluids that consist of micron-sized magnetically polarizable particles floating within oil-type fluids as shown in Fig. 2.1. The MR damper is operated as a passive damper; however, when a magnetic field is applied to the MR fluids, the MR fluids are changed into a semi-active device in a few milliseconds. Its characteristics are summarized: 1) a MR damper is operated with low power sources, e.g., SD-1000 MR damper can generate a force up to 3000 N using a small battery with capacity less than 10 W; 2) it has high yield strength level, e.g., its maximum yield strength is beyond 80 kPa; 3) the performance is stable in a broad temperature range, e.g., MR fluids operates at the temperature between -40 oC and 150 oC; 4) the response time is a few milliseconds; 5) the performance is not sensitive to contamination during manufacturing the MR damper. Moreover, the operating point of the MR damper, which is a current-controlled device, can be changed by a permanent magnet.

Figure 2.1. A schematic of the prototype 20-ton large-scale MR damper (Yang 2002).

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To fully use the best features of the MR damper, a mathematical model that portrays nonlinear behavior of the MR damper has to be developed first. However, this is challenging because the MR damper is a highly nonlinear hysteretic device. Therefore, research related to response control of building structures using MR dampers first started from development of a model that can describe the nonlinear behavior of MR damper (Spencer et al. 1997). In this section, several models for the MR damper are introduced. In Section 2.2, several forward models for a MR damper are introduced. It includes a Bingham, a polynomial, a Bouc-Wen, and a modified Bouc-Wen model. The associated inverse models are provided in Section 2.3. Finally, in Section 2.4, concluding remarks are made.

2.2. Forward Models of a MR Damper Ideally, a forward model of a MR damper has three inputs and a single output. The inputs include the piston displacement, the piston velocity, and the voltage applied to magnetic field of the MR damper; while the associated output is a control force signal. Many investigators have suggested several types of models that can effectively describe the relationship between the inputs and the output signals of the MR damper. In what follows, four different models that are widely recognized are introduced. They include a Bingham, a polynomial, a BoucWen and a modified Bouc-Wen model.

2.2.1. Bingham Model In general, the simplest model for a damper would be a viscous dashpot model

f = cx,

(2.1)

where the damper force f is linearly related to the applied velocity x . However, this cannot be used for the MR damper modeling because the relationship between MR damper forces and piston velocities is highly nonlinear. Furthermore, a MR damper has two more design parameters: the piston displacement and the applied voltage. Stanway et al. (1985, 1987) suggested a viscoplastic model, which is called Bingham model, by adding a Coulomb friction element into the viscous damper model for the highly nonlinear hysteretic behavior of an electrorheological (ER) damper as shown in Fig. 2.2 which is a schematic of Bingham model of a controllable fluid device. Such a Bingham model can be also applied to a MR damper (Spencer et al. 1997)

f MR

F = f C sgn ( x ) v + cx + f 0 ,

(2.2)

where f C is a Coulomb friction coefficient, x is the piston velocity, v is the applied voltage, c is the damping coefficient, and f 0 is an offset value to adjust a nonzero force value due to an accumulator. When a MR damper is designed, an accumulator can be

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incorporated into the MR damper in order to adjust expansion or contraction of MR fluids due to changed temperature.

Figure 2.2. Bingham model of an ER/MR damper (Stanway et al. 1985, 1987).

The reason that the Bingham model can be used to describe the behavior of a MR damper is that a MR damper has approximately two operation stages, i.e., pre-yielding and postyielding regions. Note that it is simple and easy for this Bingham model to be incorporated with a control system for analysis and design purpose; however, the piston displacement is not considered in this model, i.e., the effects of the MR damper stiffness is ignored. In addition, the performance is degraded when the magnitude of the piston velocity is small. The problem that the performance of the Bingham model is degraded at low velocity range can be solved by a polynomial model.

2.2.2. Polynomial Model Choi et al. (2001) developed a polynomial model such that it portrays nonlinear behavior of a MR damper. In this model, the hysteretic loop of the MR damper is divided into two parts, i.e., the upper loop and the lower loop. Fig. 2.3 shows a schematic of the polynomial model after the hysteretic loop is started: the solid line represents the upper hysteresis loop and the dotted line is the lower hysteresis loop.

Figure 2.3. A schematic of a polynomial model for a MR damper.

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The upper and lower parts of the hysteresis loop can be modeled via polynomials with the power of the piston velocity n

Upper f MR = ∑ aiUpper xi for the upper hysteresis loop,

(2.3)

i =0 n

Lower f MR = ∑ aiLower xi for the lower hysteresis loop,

(2.4)

i =0

Upper

where f MR

Lower

and f MR

are the level of the MR damper force that is represented by the Upper

upper part of the hysteresis loop and the lower hysteresis part, respectively; ai Lower i

a

and

are determined such that they match with experimental data; x is the piston velocity;

and n is the order of the polynomial that is selected based on trial and error approach. Upper

Since the coefficients ai

Lower

and ai

depend on input current I , they need to be

expressed in terms of the input current. Although the relationship between the current and the coefficients is nonlinear, they can be related linearly without loss of the performance

aiUpper = biUpper + ciUpper I ,

i = 0,1,..., n,

(2.5)

aiLower = biLower + ciLower I ,

i = 0,1,..., n.

(2.6)

Substitution of Eq. (2.5) and Eq. (2.6) into Eq. (2.3) and Eq. (2.4) yields n

Upper f MR = ∑ ( biUpper + ciUpper I ) xi ,

(2.7)

i =0 n

Lower f MR = ∑ ( biLower + ciLower I ) xi .

(2.8)

i =0

This polynomial model is as simple as the Bingham model. Moreover, it is also easy to derive an inverse model to implement a semiactive control system. Furthermore, the performance at low velocity range is improved comparing with the Bingham model. However, the effect of the piston displacement is still not considered in this polynomial model, i.e., the impact of the MR damper stiffness is ignored. However, the displacement parameter can be incorporated into the MR damper model by introducing a Bouc-Wen model.

2.2.3. Bouc-Wen Model One of the most popular mathematical models for modeling a MR damper is the BoucWen model (Wen 1976) depicted in Fig. 2.4.

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Figure 2.4. Bouc-Wen model of the MR Damper (Spencer et al., 1997).

The hysteretic behavior of the Bouc-Wen model for a MR damper is governed by the following equations (Spencer et al. 1997; Tse and Chang 2004)

f MR

F = c0 x + k0 ( x − x0 ) + α zBW ,

zBW = −γ x zBW zBW

where zBW and

n −1

n

− β x zBW + Ax,

(2.9) (2.10)

α = α a + α bu + α cu 2 ,

(2.11)

c0 = c0a + c0b u,

(2.12)

u = −η (u − v),

(2.13)

α , called evolutionary variables, describe the hysteretic behavior of the MR

damper; c0 is the viscous damping; k0 is the stiffness; x0 is the initial displacement, which is caused by an accumulator, of the spring that is corresponding to the stiffness k0 ; γ, β and A are adjustable shape parameters of the hysteresis loops; and v and u are input and output voltages of a first-order filter, respectively. Although this model describes the hysteretic behavior of the MR damper, it is still difficult for the Bouc-Wen model to capture the response behavior at the small piston velocities (Spencer et al. 1997). Such a problem can be solved by introducing additional stiffness and damping elements into the Bouc-Wen model, which is named a modified BoucWen model.

2.2.4. Modified Bouc-Wen Model To improve the performance at small magnitude of velocities, Spencer et al. (1997) proposed a modified version of the Bouc-Wen model, as shown in Fig. 2.5. The MR damper

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force f MR (t ) predicted by the modified Bouc-Wen model is governed by the following differential equations according to Spencer et al. (1997)

f MR

F = c1 y + k ( x − x0 ),

zBW = −γ x − y zBW zBW

y=

where zBW and

n −1

(2.14) n

− β ( x − y ) zBW + A( x − y ),

1 {α zBW + c0 x + k0 ( x − y)} , (c0 + c1 )

(2.15)

(2.16)

α = α a + α bu,

(2.17)

c1 = c1a + c1bu ,

(2.18)

c0 = c0a + c0b u,

(2.19)

u = −η (u − v),

(2.20)

α , called evolutionary variables, describe the hysteretic behavior of the MR

damper; c0 and c1 are viscous damping at high and low velocities, respectively; k0 and k1 control the stiffness at large velocities and the accumulator stiffness, respectively; the x0 is the initial displacement of spring with stiffness k1; γ, β and A are adjustable shape parameters of the hysteresis loops; and v and u are input and output voltages of a first-order filter, respectively. Note that the modified Bouc-Wen model is one of the most effective models to describe the nonlinear behavior of a MR damper; however, it is not easy to derive the inverse model for control system design purpose. In the following sections, the corresponding inverse models are introduced.

Figure 2.5. Modified Bouc-Wen model of the MR damper (Spencer et al. 1997).

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2.3. Inverse Models of a MR Damper A MR damper force cannot be directly controlled, but applied voltage only can be directly controlled, i.e., control force signals that are provided by a control algorithm should be converted into voltage or current signals to operate the MR damper. For the signal transformation, there might exist two ways: 1) use of an inverse model of the MR damper 2) use of an algorithm to be able to convert control forces into voltage or current signals. In this section, several types of inverse models for a MR damper are introduced. They include a Bingham, a polynomial, a Bouc-Wen, and a modified Bouc-Wen model.

2.3.1. Inverse Bingham Model An inverse Bingham model for a MR damper can be easily derived from Eq. (2.2) by solving for the voltage v

v=

f MR − cx + f 0 . f c sgn ( x )

(2.21)

2.3.2. Inverse Polynomial Model As one of the simplest ways of converting the control force level into a current signal, the inverse polynomial model is determined from Eq. (2.7) and Eq. (2.8) by solving for the current I (Choi et al. 2001)

⎧ fSNC − ∑ n biUpper xi i=0 ⎪ for the upper hysteresis loop, n ⎪⎪ ∑ i =0 ciUpper xi I =⎨ n Lower i x ⎪ fSNC − ∑ i = 0 bi for the lower hysteresis loop, ⎪ n Lower i x ⎪⎩ ∑ i =0 ci

(2.22)

where fSNC is a desirable control force that is generated by a semiactive nonlinear controller (SNC) in this research although it can be any type of control force.

2.3.3. Inverse Bouc-Wen Model Tse and Chang (2004) have derived an inverse Bouc-Wen model for a MR damper assuming that the evolutionary variable zBW can be approximated as its ultimate hysteretic strength and the MR damper is always operated in the postyielding region. The differential equations of the inverse Bouc-Wen model are given by

f MR ≅ (c0a + c0a u ) x + k0 x + (α a + α bu + α cu 2 ) zu ,

(2.23)

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⎛ A ⎞ z ≅ zu = sgn( x) ⎜ ⎟ ⎝γ +β ⎠

,

(2.24)

(α c zu ) u 2 + (α b zu + c0b x)u + (α a zu + c0a x + k0 x − f MR ) = 0,

(2.25)

u v=u+ .

(2.26)

η

2.3.4. Inverse Modified Bouc-Wen Model Tsang et al. (2006) derived an inverse dynamics of a modified Bouc-Wen model for a MR damper to synthesis a control system, assuming that the evolutionary variable zBW can be approximated as its ultimate hysteretic strength; a MR damper is operated within postyielding region (Spencer 1986); and stiffness of the MR damper can be neglected. The differential equation for the inverse modified Bouc-Wen model is given by

I (t ) = −

⎤ 1 ⎡ f MR − fSNC + e − p2 I (t −Δt ) ⎥ , ln ⎢ p2 ⎣ p1 ⎦

(2.27)

where p1 and p2 are related to the MR fluid stress; they are found by ad-hoc approach using experimental results. More detailed description is given in Tsang et al. (2006).

2.4. Concluding Remarks In this section, four forward and the associated inverse models for a MR damper are presented, including a Bingham, a polynomial, a Bouc-Wen, and a modified Bouc-Wen model. The Bingham model can be quickly and easily applied to the control system of building structures; however, it is difficult to accurately capture the hysteretic loop of the MR damper, in particular, in the range of low velocities. Derivation of the polynomial model-based forward and inverse MR damper models can be easily carried out; however, the piston displacement is not considered as an input parameter, which is the same as the Bingham model, in the polynomial models; it means that the effects of the stiffness of the MR fluid are ignored. However, the impact of the MR damper stiffness can be incorporated with a Bouc-Wen model. Highly nonlinear hysteretic loop of a MR damper can be described with the Bouc-Wen model; however, it is still not effective to capture the nonlinear behavior of the MR damper at low velocities. Such a drawback of the Bouc-Wen model can be overcome by modifying the Bouc-Wen model, i.e., the modified Bouc-Wen model has good performance at both high velocity and low velocity ranges. However, it is difficult to derive the inverse dynamics for the Bouc-Wen and the modified Bouc-Wen models for the purpose of control system design.

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3. Nonlinear System Identification (NSI) 3.1. Introduction One of the most difficult but important tasks in control system design for seismically excited building structures is the development of an accurate explicit mathematical model of the building system to be controlled because precise mathematical information related to the building structure is used for calculation of control forces. However, the development of a mathematical model for a nonlinear building system as a dynamic system is still a challenging problem. One example of a nonlinear building structure occurs when MR dampers, which are highly nonlinear hysteretic devices, are applied to the building systems for efficient energy dissipation. In this case, the integrated building-MR damper system behaves nonlinearly although the building structure itself is usually assumed to remain linear (Ramallo et al. 2004). The development of an appropriate nonlinear model of the integrated system that includes the interaction between the structural system and the nonlinear MR damper plays a key role in control system design because the building structure equipped with a nonlinear MR damper is intrinsically nonlinear. A solution can be found in NSI based on TS fuzzy model. In this section, a framework for NSI is presented through a family of local linear MIMO ARX input-based TS fuzzy model. In particular, the building-MR damper system, which is modeled as a TS fuzzy model, can be used to design a PDC-based nonlinear fuzzy controller because the PDC-based nonlinear fuzzy controller shares linguistic information with the building-damper system. The goal is to achieve optimal estimation of a set of nonlinear MIMO data from building structures equipped with MR dampers. In the following sections, the ARX model and fuzzy models are reviewed briefly. Next, a hybrid multiple ARX-TS fuzzy model is introduced and then its parameters are optimized via clustering and least squares algorithms.

3.2. MIMO ARX Model 3.2.1. Single-Input-Single-Output (SISO) ARX Model An input-output relationship of a linear time-invariant dynamic system can be described via a linear difference equation that is often called an ARX model in which AR represents the autoregressive output and X represents the exogenous input

y ( kT ) + a1 y ( kT − 1) + a2 y ( kT − 2) +

+ an y ( kT − n)

= b0 u ( kT − 1) + b1 u ( kT − 2) + b2 u ( kT − 3) +

+ bm u ( kT − m),

(3.1)

where y ( kT ) is an output signal, u ( kT ) is an input signal, n is the number of delay steps in the output signals, and m is the number of delay steps in the input signals. In addition, T is

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the sample period, 1 T is the sample rate, and k is the integer value. However, the y ( kT ) and u ( kT ) are written simply as y ( k ) and u ( k )

y ( k ) + a1 y (k − 1) + a2 y (k − 2) +

+ an y ( k − n)

= b0 u (k − 1) + b1 u ( k − 2) + b2 u ( k − 3) +

(3.2)

+ bmu ( k − m).

This difference equation can be considered as an equation to determine the current output in terms of previous inputs and outputs

y ( k ) = − a1 y ( k − 1) − a2 y ( k − 2) −

− an y ( k − n)

+b0 u ( k − 1) + b1 u ( k − 2) + b2 u (k − 3) +

(3.3)

+ bmu ( k − m),

or n

m

i =1

i =1

y (k ) = −∑ ai y ( k − i ) + ∑ bi u ( k − i ) .

(3.4)

Eq. (3.4) can be represented in more compact form by defining the following vectors

θ = [ a1 ,..., an , b1 ,..., bm ] , T

(3.5)

H ( k ) = [ − y (k − 1), ... , − y ( k − n), u (k − 1), ... , u ( k − m) ] . T

(3.6)

Using Eq. (3.5) and Eq. (3.6), Eq. (3.4) can be rewritten as T

y (k ) = H (k )θ.

(3.7)

Eq. (3.7) can be thought as a linear regression expression because y(k ) depends on the parameters in θ as T yˆ ( k θ ) = H ( k )θ,

(3.8)

where the vector H ( k ) is called a regression vector. Also, the Single-input, Single-output (hereinafter as “SISO”) ARX model can be generalized as a MIMO ARX model. In what follows, the SISO ARX model is extended into a MIMO ARX model.

3.2.2. MIMO ARX Model A SISO ARX model that represents a relationship between single input and single output of a linear time-invariant (hereinafter as “LTI”) dynamic system can be easily generalized into a MIMO ARX model

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y (k ) = −a1 y (k − 1) − a 2 y (k − 2) −

309

− a n y (k − n)

+b 0u (k − 1) + b1 u(k − 2) + b 2u( k − 3) +

(3.9)

+ b mu ( k − m),

or n

m

i =1

i =1

y ( k ) = − ∑ ai y ( k − i ) + ∑ b i u ( k − i ) ,

(3.10)

y ( k ) = ⎡⎣ y1 ( k ) , y2 ( k ) ,

, y p ( k ) ⎤⎦ ,

(3.11)

u ( k ) = ⎡⎣u1 ( k ) , u2 ( k ) ,

, uq ( k ) ⎤⎦ ,

(3.12)

where

⎡ ai1,1 ⎢ 2,1 ⎢ ai ai = ⎢ ⎢ p −1,1 ⎢ ai ⎢ a p ,1 ⎣ i

T

T

ai1,2 ai2,2

ai1,n −1 ai2,n −1

p −1,2 i p ,2 i

p −1, n −1 i p , n −1 i

a a

a a

ai1,n ⎤ ⎥ ai2,n ⎥ ⎥, p −1, n ⎥ ai ⎥ p ,n ⎥ ai ⎦

(3.13)

bi1,m ⎤ ⎥ bi2,m ⎥ ⎥. ⎥ biq −1,m ⎥ biq ,m ⎥⎦

(3.14)

and

⎡ bi1,1 bi1,2 ⎢ 2,1 bi2,2 ⎢ bi bi = ⎢ ⎢ q −1,1 biq −1,2 ⎢bi ⎢ b q ,1 biq ,2 ⎣ i

bi1, m −1 bi2,m −1 biq −1,m −1 biq ,m −1

Eq. (3.10) can be represented in more compact form by defining the following matrices

θ = [a1 ,..., a n , b1 ,..., b m ] , T

(3.15)

H ( k ) = [ − y ( k − 1), ... , − y ( k − n), u( k − 1), ... , u( k − m) ] . T

(3.16)

Using Eq. (3.15) and Eq. (3.16), Eq. (3.10) can be rewritten as

y (t ) = H T (k )θ.

(3.17)

yˆ (k θ) = H T (k )θ.

(3.18)

This can be also a regression equation

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Unfortunately, the ARX model has the limited range of operating regions because it is a LTI system model; however, a solution on this drawback can be found in multiple ARX models that have a variety of operating regions. In the following sections, fuzzy models are first introduced; then, it is integrated with the multiple ARX models.

3.3. Fuzzy Model 3.3.1. Membership Functions (MFs) and Fuzzy Sets Membership functions (hereinafter as “MFs”) and fuzzy sets are the cornerstone of a fuzzy logic-based system that is appropriate for modeling complex nonlinear systems with uncertain parameters. There exist always a variety of uncertainties in engineering problems; e.g., “the structural damage is very large” and “the performance of a MR damper is sensitive to high temperature.” However, questions would arise: “How much damage would be thought as very large quantity?” or “Which degree of temperature is high?” In reality, it is impossible to model variables with these uncertainties in a conventional way; however, MFs can be used for modeling such variables as an element of a fuzzy set. Fig. 3.1 shows several types of MFs that is generally selected by judgment of engineers depending on given problems.

Figure 3.1. Type of fuzzy MFs.

Fuzzy sets are constructed from the MFs. For example, if the structural damage is categorized into three stages, e.g., small, medium, and large damage, a fuzzy set can be constructed as shown in Fig. 3.2. This fuzzy set is used for constructing a premise part of an IF-THEN rule, i.e., IF STATEMENT.

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Figure 3.2. A fuzzy set representing structural damage.

3.3.2. Fuzzy Rules A fuzzy rule base has a family of fuzzy IF-THEN rules; e.g., “if a building structure has large damage, a controller is operated such that an alarm is rung twice”, “if the structural damage is medium, the controller is operated such that the alarm is rung once”, and “if there is no damage in the building structure, the controller is not operated.” The set of IF-THEN rules is blended into an integrated system through fuzzy reasoning methods.

3.3.3. Fuzzy Reasoning Fuzzy reasoning is a mechanism to perform the fuzzy inference system that derives conclusions from a family of IF-THEN rules, i.e., fuzzy reasoning is a methodology to organize a set of the IF-THEN rules. In what follows, two types of fuzzy models that have been extensively applied to a variety of engineering fields are introduced in order to compare different type of reasoning mechanisms: Mamdani and TS fuzzy models. 3.3.3.1. Mamdani Fuzzy Model Mamdani fuzzy model uses fuzzy sets in both IF and THEN STATEMENTS; the IF STATEMENT and THEN STATEMENT are called a premise part and a consequent part, respectively. A typical fuzzy rule of a Mamdani fuzzy model has the form 2 R j : If z1FZ is p1, j and zFZ is p2, j and

Then, y = q,

i and zFZ is pi , j

(3.19)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

i ⎤⎦ is a premise vector that can be either a set where Rj is the jth fuzzy rule; z FZ = ⎡⎣ z1FZ ,… , zFZ of system inputs or outputs. Consider a simple Mamdani fuzzy model with: 1) two rules, 2) two input variables in the premise part, and 3) one output variable in the consequent part in order to describe the Mamdani’s reasoning mechanism 2 R 1 :If z1FZ is p11 and zFZ is p21 Then y is q1 , 2 R 2 :If z1FZ is p12 and zFZ is p22 Then y is q2 .

(3.20)

The Mamdani’s reasoning procedure has four main steps: calculating weights, weighing consequent parameters, aggregation, and defuzzification. Step 1: Computation of weight The first step is to calculate weighing values of each rule for the input values 2 Weight of R1 : w1 = μ p11 ( z1FZ ) ∧ μ p21 ( zFZ )

2 Weight of R 2 : w2 = μ p12 ( z1FZ ) ∧ μ p22 ( zFZ ),

( ) is a MF for the premise parameter

where μ pij zFZ i

(3.21)

pij ; ∧ represents min-operation (Yen

and Langari 1999). These weighting values are applied to consequent parameters to derive conclusions of each fuzzy rule. Step 2: Apply the weights of rules to consequent parameters By applying the weights of rules to the MFs about consequent parameters, the following conclusions of each rule are derived 2 Conclusion of R1 : μq∗1 ( z1FZ , zFZ ) = w1 ∧ μq1 ( y ) ,

2 Conclusion of R 2 : μq∗2 ( z1FZ , zFZ ) = w2 ∧ μq2 ( y ) .

(3.22)

These conclusions of each fuzzy rule are derived as a conclusion via aggregation process. Step 3: Aggregate the conclusions obtained in Step 2 The conclusions that are derived in Step 2 should be blended as an integrated conclusion using union of fuzzy sets

μ q ( y ) = μ q∗ ( y ) ∨ μ q∗ ( y ) , 1

2

(3.23)

where ∨ represents max-operation (Yen and Langari 1999). However, values obtained from Eq. (3.23) are not definite values, but fuzzy values. Thus, these fuzzy set values are converted into definite values with a defuzzification process.

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Step 4: Defuzzification Defuzzification is a way of extracting definite values from fuzzy sets. A commonly used method, centroid of area method is used here although there are many defuzzification methodologies, e.g., bisector of area, mean of maximum, smallest of maximum, largest of maximum, etc.

yfinal

μ ( y ) ydy ∫ = . ∫ μ ( y ) dy q

y

(3.24)

q

y

However, the computational cost of Mamdani fuzzy model is high because the defuzzification procedure is time-consuming. In addition, the model is not appropriate for rigorous mathematical analysis. Those drawbacks have led to development of other types of fuzzy models that are computationally efficient and mathematically meaningful. In what follows, one of such efficient models, TS fuzzy model, is introduced. 3.3.3.2. TS Fuzzy Model Takagi and Sugeno (1985) developed a systematic methodology for a fuzzy reasoning using linear functions in the consequent part. Because the TS-fuzzy model uses linear functions in the consequent part, the defuzzification procedure is not required. Therefore, the reasoning mechanism of the TS fuzzy model is simpler than the Mamdani fuzzy model, i.e., the lower computational cost is required for the TS fuzzy model than the Mamdani fuzzy model due to much smaller number of fuzzy rules of the TS fuzzy model. Furthermore, the TS fuzzy model provides a systematic framework for rigorous mathematical analysis. A typical fuzzy rule for the TS fuzzy model has the form 2 R j : If z1FZ is p1, j and zFZ is p2, j and

Then,

i and zFZ is pi , j

(3.25)

i y = f ( z1FZ ,..., zFZ ),

where Rj is the jth fuzzy rule; z FZ = ⎡⎣ zFZ ,… , zFZ ⎤⎦ is a premise vector that can be either a set 1

(

i

of system inputs or outputs; y = f zFZ ,..., zFZ

(

i In general, y = f z1FZ ,..., zFZ

)

1

i

) is a linear function in the consequent part.

is a polynomial function in terms of the premise vector

i ⎤⎦ although it can be any type of function. z FZ = ⎡⎣ z1FZ ,… , zFZ

Differently with the Mamdani’s reasoning method with relatively complicated procedure, TS fuzzy model-based reasoning is a simple process to compute weighted mean values

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Yeesock Kim, Reza Langari and Stefan Hurlebaus Nr

∑w y j

j =1 Nr

yfinal =

∑w j =1

j

,

(3.26)

j

where n

i w j = ∏ μi , j ( zFZ ).

(3.27)

i =1

However, this type of polynomial may not be appropriate for modeling dynamic systems. Thus, ARX models are integrated with the TS fuzzy model such that the TS fuzzy model can describe behavior of a dynamic system.

3.4. A Set of MIMO ARX TS Fuzzy Models A nonlinear dynamic system that has multiple operation regions can be described through multiple linear dynamic models. In this section, multiple ARX models whose operating regions are blended with a fuzzy interpolation method are introduced (Johansen 1994; Abonyi et al. 2000; Abonyi 2003). A MIMO dynamic system can be described by the following multivariable nonlinear model z = f (t , z, u), (3.28) where t is the time variable; z is a state vector; u is an input vector; and f represents a multivariable nonlinear dynamic system. This MIMO nonlinear dynamic model can be described by a family of MIMO ARX-TS fuzzy models 2 R j : If z1FZ is p1, j and zFZ is p2, j and

i and z FZ is pi , j

Then,

(3.29)

y ( k ) = a1, j y ( k − 1) + a 2, j y ( k − 2 ) + + b1, j u ( k − 1) + b 2, j u ( k − 2 ) +

+ an, j y ( k − n ) + b m, j u ( k − m ) ,

or 2 R j : If z1FZ is p1, j and zFZ is p2, j and

i and zFZ is pi , j

(3.30)

Then, y (k ) =

n

∑a i =1

i, j

y (k − i) +

m

∑b i =1

i, j

u ( k − i ),

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i ⎤⎦ is a premise vector that can be either a set of where Rj is the jth fuzzy rule; z = ⎡⎣ z1FZ ,..., zFZ dynamic system inputs or outputs

z FZ ∈ { y1 (k − 1),..., y1 (k − n),..., y p (k − n), u1 (k − 1),...u1 (k − m),..., uq (k − m)} , (3.31) y ( k ) = ⎡⎣ y1 ( k ) , y2 ( k ) ,

, y p ( k ) ⎤⎦ ,

(3.32)

u ( k ) = ⎡⎣u1 ( k ) , u2 ( k ) ,

, uq ( k ) ⎤⎦ ,

(3.33)

ai , j

⎡ ai1,1, j ai1,2 ,j ⎢ 2,1 2,2 ai , j ⎢ ai , j =⎢ ⎢ p −1,1 aip, −j 1,2 ⎢ ai , j ⎢ a p ,1 aip, ,2j ⎣ i, j

T

T

ai1,, nj −1 ai2,, jn −1 aip, −j 1,n −1 aip, ,jn −1

ai1,, nj ⎤ ⎥ ai2,, jn ⎥ ⎥, ⎥ aip, −j 1,n ⎥ aip, ,jn ⎥⎦

(3.34)

bi1,, jm ⎤ ⎥ bi2,, jm ⎥ ⎥, ⎥ biq, −j 1, m ⎥ biq, ,jm ⎥⎦

(3.35)

and

bi, j

⎡ bi1,1 bi1,2 ,j ,j ⎢ 2,1 2,2 bi , j ⎢ bi , j =⎢ ⎢ q −1,1 biq, −j 1,2 ⎢bi , j ⎢ b q ,1 biq, ,2j ⎣ i, j

bi1,, mj −1 bi2,, jm −1 biq, −j 1, m −1 biq, ,jm−1

where n is the number of delay steps in the output signals; m is the number of delay steps in the input signals, p is the number of output signals, and q is the number of input signals. Note that the number of the fuzzy rules corresponds to the number of local MIMO ARX models, i.e., m MIMO ARX linear dynamic models represent m fuzzy rules that describe behavior of a nonlinear dynamic system. However, a question would arise on how to blend the multiple MIMO ARX dynamic models as an integrated system model, i.e., how to make a bridge for communication among each MIMO ARX models. One of solutions is found in fuzzy logicbased interpolation. The multiple MIMO ARX local models at the specific operating point i zFZ can be blended

n

Nr

m

Nr

i yˆ (k ) = ∑∑ w j ( zFZ ) ai, j y(k − i) + ∑∑ w j ( zFZi ) bi, ju(k − i), i =1 j =1

( )

i =1 j =1

where 0 ≤ w j zFZ ≤ 1 is the normalized true value of the jth rule, i

(3.36)

316

Yeesock Kim, Reza Langari and Stefan Hurlebaus n

i w j ( zFZ )=

∏μ

i, j

i ( zFZ )

∑∏ μ

i FZ

i =1

Nr

.

n

j =1 i =1

i, j

(3.37)

(z )

Once the multiple MIMO ARX-TS fuzzy model is set up, the premise parameters pi , j and the consequent parameters a i, j and b i, j are determined such that the multiple MIMO ARXTS fuzzy model describes behavior of a nonlinear dynamic system. In this book, the premise parameters are determined through clustering techniques and the consequent part is optimized using a weighted least squares estimation algorithms. In what follows, clustering algorithms are introduced.

3.5. Clustering Algorithms For efficient determination of the premise part, (i.e., the small number of MFs but reasonable pattern recognition), grouping of data with similar patterns would be desirable. In this book, subtractive and fuzzy C-means clustering algorithms are used to extract information on center of groups from a large data set.

3.5.1. Fuzzy C-means Clustering As a generalized version of the K-means clustering algorithm (Jang et al. 1997), fuzzy Cmeans clustering algorithm has been widely applied to a variety of engineering problems (Wang and Langari 1996). This algorithm generates fuzzy sets in an automatic way and does not require any previous knowledge about the data structure of a given problem. The fuzzy C-means clustering algorithm is formulated as a constraint optimization problem c

n

J ( U, σ 1 ,..., σ c ) = ∑∑ ( μi , j ) σ i − σ j

Minimize σ k −σ i ≠ 0

i =1 j =1

(3.38)

subject to c

∑μ i =1

where

m

i, j

= 1,

j = 1, 2,..., n,

μi , j is membership for σ j in the ith cluster whose values in between 0 and 1; σ i is

the cluster center of each group i ;

σ j is ith data point; m > 1 is a design parameter; and

U = ⎡⎣ μi , j ⎤⎦ is the partition matrix with c × n dimension.

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Using Lagrange multipliers, the constrained optimization can be transformed into an unconstrained optimization problem c

n

J (U, σ 1 ,..., σ c , λ1 ,..., λn ) = ∑∑ ( μi , j )

m

i =1 j =1

⎛ c ⎞ σ i − σ j + ∑ λ j ⎜ ∑ μi , j − 1⎟. (3.39) j =1 ⎝ i =1 ⎠ n

Differentiation of the augmented objective function leads to the necessary conditions n

σi =

∑(μ ) j =1 n

m

i, j

∑(μ ) j =1

σj m

,

(3.40)

i, j

where

⎛ σi −σ j μi , j = ∑ ⎜ k =1 ⎜ σ k − σ j ⎝ c

⎞ ⎟ ⎟ ⎠

−2 ( m −1)

.

(3.41)

This fuzzy C-means clustering procedure is a simple iterative algorithm: first, generate the initial MF matrix U = ⎡⎣ μi , j ⎤⎦ using a random number generator such that the constraint c

condition

∑μ i =1

i, j

= 1 is satisfied; second, calculate a cluster center using Eq. (3.40); third,

compute the cost function value using Eq. (3.38) and stop if the iteration stopping criteria is satisfied; and fourth, calculate a new MF matrix U = ⎡⎣ μi , j ⎤⎦ using Eq. (3.41) and then go to the second step. However, the fuzzy C-means clustering algorithm is sensitive to initialization of the MF matrix U = ⎡⎣ μi , j ⎤⎦ that might fall into local minima. In such a case, uncountable trial-anderrors would be required. Thus, a different clustering algorithm that is not sensitive to initial values is introduced in the following section: subtractive clustering.

3.5.2. Subtractive Clustering Differently with the fuzzy C-means clustering algorithm, subtractive clustering is not sensitive to initial values because it considers all the data points as a candidate for a cluster center. In addition, it is independent of the data dimension. In the subtractive clustering, a data point with the highest density neighborhood is selected as a cluster center. The density measure at data point σ i is given by (Liu et al. 2003)

⎛ σ −σ 2 ⎞ i j ⎟, Di = ∑ exp ⎜ 2 ⎜ Ra 2 ) ⎟ j =1 ( ⎝ ⎠ n

(3.42)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

σ i is the ith data point, n is the total number of data points, and Ra , which is chosen

by the user, represents range of data neighborhood to be considered or the degree to which a cluster center contributes to the density measure. After a data point is selected as the first cluster center with the highest potential, the selected cluster center and its neighborhood data points are subtracted in the following selection procedure

⎛ σ −σ 2 ⎞ i cj ⎟, Di = Di − Dc j ∑ exp ⎜⎜ 2 ⎟ j =1 ⎜ ( Rb 2 ) ⎟ ⎝ ⎠ n

(3.43)

where Dc j is the jth density measure, σ c j is the jth cluster center, and Rb is a parameter used to avoid closely spaced centers that is specified by Rb = η Ra , where η is a positive constant greater than 1. After subtraction procedure, the next density measure is calculated and the cluster center is selected. This procedure is repeated until a sufficient number of cluster centers are found in the input space. The cluster centers that have been calculated via either fuzzy C-means or subtractive clustering algorithms are used to construct the linguistic part, i.e., premise part, of a TS fuzzy model, e.g., the cluster center information is used as a center value of a Gaussian or triangular MFs. Once the premise part of the TS fuzzy model is determined by either fuzzy C-means or subtractive clustering algorithms, weighted least squares algorithm is applied to find optimum solutions of the consequent parameters of the TS fuzzy model.

3.6. Weighted Least Squares Once the premise part is determined, the consequent parameters can be optimized with weighted least squares algorithms. A least squares algorithm can be formulated as a quadratic optimization problem that minimizes difference between true values and estimated values

1 Min J = e(k )T e(k ), 2

(3.44)

where e(k ) = yˆ (k ) − y (k ) ; i.e., the error e(k ) is difference between the estimation model

yˆ (k ) and true values y (k ). A linear estimation model for used with the linear least squares algorithm is

yˆ (k ) = H(k )θ j .

(3.45)

On the other hand, the true model can be thought as a contaminated estimation model

y(k ) = H(k )θ j + e(k ).

(3.46)

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319

From Eq. (3.46), the error dynamics is given by

e(k ) = y(k ) − H(k )θ j .

(3.47)

Substituting Eq. (3.47) into Eq. (3.44) leads to the following objective function

J = J (θ j ) =

1 ⎡⎣ y (k )T y ( k ) − 2y ( k )T H ( k )θ j + θ j T H ( k )T H ( k )θ j ⎤⎦ . 2

(3.48)

In this problem, the objective is to find θ j such that the objective function, J is minimized. For minimization of the quadratic function of Eq. (3.48), the necessary condition can be derived

∇ θ j J = H ( k ) T H ( k )θ j − H ( k ) T y ( k ) = 0,

(3.49)

where ∇ θ j J is a Jacobian matrix, the 1st partial derivative of J about θ j . For solution of this equation, the following analytical least squares estimator is available (Wang and Langari 1996) −1

θ j = ⎡⎣ H ( k )T H ( k ) ⎤⎦ H ( k )T y ( k ).

(3.50)

This linear least squares estimation can be easily extended into a weighted least squares estimator by adding the weight matrix −1

θ j = ⎡⎣ H ( k ) T w j H ( k ) ⎤⎦ H ( k ) T w j y ( k ),

(3.51)

H (k ) = [y (k − 1)T ,..., y (k − n)T , u(k − 1)T ,..., u( k − m)T ],

(3.52)

θ j = [a1, j ,..., a n , j , b1, j ,..., b m , j ].

(3.53)

where and

In summary, the proposed nonlinear MIMO ARX-TS fuzzy modeling approach is: 1) nonlinear behavior of a building-MR damper system is represented by a family of multiple MIMO ARX LTI models that are integrated into a nonlinear time-varying model via fuzzy rules; 2) the premise part of the multiple MIMO ARX-TS fuzzy model are partitioned to subdivide the input space into several operating regions using either subtractive or fuzzy Cmeans clustering techniques; 3) the consequent parameters are optimized by a family of linear weighted least squares. Note that the nonlinear multiple MIMO ARX-TS fuzzy model will be used for design of a PDC-based SNFC system. The control system design is presented in the following section.

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4. Semiactive Nonlinear Control System 4.1. Introduction In recent years, attention has attracted to systematic applications of semiactive linear control (hereinafter as “SLC”) algorithms for vibration control of building structures subjected to natural hazards, e.g., earthquakes and strong winds. In particular, many design methodologies for the SLC have been developed for use with low-, mid-, and high-rise building structures employing MR dampers. In addition, semiactive nonlinear control (SNC), mainly SNFC, has been successively applied to the building-MR damper systems due to the effectiveness and robustness of the SNFC systems. However, the SNFC systems have been designed by trial and error approaches that use either experienced investigators or high cost computation, i.e., as a model-free controller, they are trained using a set of input-output data. Although useful for the performance purpose, the ad-hoc approach may not provide design guidelines in a systematic way. Furthermore, it is difficult for the ad-hoc approach-based SNFC systems to guarantee stability because stability conditions cannot be formulated using the ad-hoc design approach. Unfortunately, no systematic study has been conducted to design a SNFC system for structural vibration control of building structures such that stability conditions are guaranteed. Therefore, a new research is recommended to develop a systematic design methodology for the SNFC system of large scale building structures employing MR dampers. An ANFC system design has been carried out in a systematic way via the so called PDC approach, which employs multiple linear controllers (Hong and Langari 2000; Joh et al. 1997). The linear controllers correspond to the local linear structural models with automatic scheduling performed through fuzzy rule base. The PDC-based ANFC system provides sufficient conditions to be globally asymptotically stable by finding a common symmetric positive definite matrix such that a set of simultaneous Lyapunov inequalities are satisfied. However, no systematic design framework has been investigated to design a SNFC for building structures equipped with nonlinear MR dampers for mitigation of response to earthquake events. Section 4.2 describes the design framework. It includes TS fuzzy model, PDC, and LMIs. In Section 4.3, a LMI-based systematic design framework for the ANFC system design is described. Section 4.4 describes an optimal estimator and converting algorithms to derive a SNFC system. The proposed SNFC system is demonstrated in Section 4.5 with an illustrating example. Finally, concluding remarks are given in the last section.

4.2. Design Framework TS fuzzy model, PDC, and LMIs are the backbone for the proposed SNFC system. In this section, TS fuzzy model is briefly represented in terms of state space equations first. Its detail is described in Section 4. Then, fundamentals of the PDC and LMIs are discussed.

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4.2.1. TS Fuzzy Model As stated previously, Takagi and Sugeno (1985) suggested an effective way for modeling complex nonlinear dynamic systems by introducing linear equations in consequent parts of a fuzzy model, which is called TS fuzzy model. It has led to reduction of computational cost because it does not need any defuzzification procedure. However, a more important value of the TS fuzzy model is that it provides a framework for a rigorous mathematical analysis, i.e., many modern linear system theories can be applied to nonlinear system models in terms of IF-THEN rules. A typical fuzzy rule for a TS fuzzy model is of the form 2 R j : If z1FZ is p1, j and zFZ is p2, j and

i and zFZ is pi , j

(4.1)

Then, i y = f ( z1FZ ,..., zFZ ),

(

)

where the equation of the consequent part, y = f zFZ ,..., zFZ , can be any type of linear 1

i

equation. However, it is advantageous to represent the consequent part in terms of state space equations in order to apply modern control theories to the fuzzy control system design; a typical rule of the TS fuzzy model that is expressed in terms of state space equations in the consequent part is of the form n R j :If z1FZ is p1, j and ...and zFZ is pn , j

(4.2)

⎧x = A j x + B j u Then ⎨ , ⎩y = C j x + D j u

j = 1, 2,..., r ,

where r is the number of fuzzy rules, pi , j are fuzzy sets centered at the ith operating point, i zFZ is are premise variables that can be either input or output values, x is the state vector, u

is the input vector, y is the output vector, and A j , B j , C j , D j are system matrices. Note that Eq. (4.2) represents the j

th

local linear dynamic system of a nonlinear

dynamic system, i.e., a linear dynamic system model that is operated in only a limited region. Therefore, all the local dynamic system should be integrated into a global nonlinear dynamic system by blending operating regions of each local dynamic system. Such a blending job is performed through interpolation of all the local dynamic system models. Note that the local dynamic system involves only linear combinations of input and output vectors; however, the integrated dynamic system is truly nonlinear. The blended TS fuzzy model is of the following form

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Yeesock Kim, Reza Langari and Stefan Hurlebaus Nr

x=

∑ w ( z ) ⎡⎣ A x + B u ⎤⎦ j =1

j

i FZ

Nr

n

( ) ∏μ i

i =1

pi , j

j

∑w (z ) j =1

where w j zFZ =

j

j

,

(4.3)

i FZ

i i i ( zFZ ) and μ pi , j ( z FZ ) is the grades of membership of zFZ in pi , j .

To control the responses of the blended TS fuzzy model, an effective control law associated with Eq. (4.3), i.e., u should be designed. In this book, multiple optimal linear controllers associated with each local dynamic system are designed and then blended through a fuzzy interpolation method.

4.2.2. Parallel Distributed Compensation (PDC) Development of systematic design procedures for a nonlinear feedback control is still challenging due to its complexity. Thus, a curious question would be arising: “Can linear system theories be applied to the nonlinear control system design?” One of reasonable answers can be found in the PDC. In the PDC approach, linear system theories can be applied to state feedback controllers associated with each local dynamic system; in particular, the PDC approach is appropriate for the ANFC system because the local dynamic system in the consequent part of the fuzzy rules is described by a linear state space equation. The control rule j of an ANFC system is of the form n R j :If z1FZ is p1, j and...and zFZ is pn , j

(4.4)

Then u = K j x. The state feedback controller in the consequent part of the j

th

IF-THEN rule is a local linear

controller associated with a local dynamic system to be controlled. All the local state feedback controllers are integrated into a global nonlinear controller using fuzzy sets Nr

u=

∑ w ( z ) ⎡⎣K x ⎤⎦ j =1

j

i FZ

Nr

j

∑w (z ) j =1

j

.

(4.5)

i FZ

Notice that the blended state feedback controller is truly nonlinear. By substituting Eq. (4.5) into Eq. (4.3), the final closed loop control system is derived

Identification and Control of Large Smart Structures Nr

x=

Nr

∑∑ w ( z )w ( z ) ⎡⎣ A j

q =1

j

i FZ

Nr

Nr

j

q =1

q

i FZ

j

+ B j K q ⎤⎦ x

∑∑ w ( z )w ( z ) j

i FZ

323

.

(4.6)

i FZ

q

To implement the ANFC system Eq. (4.6), the next step is to compute the multiple state feedback gains, K q , q = 1,..., r such that the nonlinear dynamic system to be controlled is globally asymptotically stable while the performance on transient responses is also satisfied. Next, they are integrated with a Kalman filter to convert the state feedback mode into the output feedback mode and then are integrated with a converting algorithm to convert the active mode into semi-active one. Although there are many methodologies to design the state feedback gains K q ,

q = 1,..., r , LMI-based control formulations are carried out here because the LMI technique is appropriate to the formulation of multiple objectives and constraints. Basic backgrounds on LMIs is discussed and then stability issues are described.

4.2.3. Linear Matrix Inequalities (LMIs) In recent years, LMI techniques have attracted significant attention because a great variety of engineering problems can be re-formulated as convex or pseudo-convex optimization problems in terms of LMIs. Such problems include control system design, system identification, structural design, etc. In particular, many control problems can be recast in terms of LMIs because design objectives and constraint conditions can be formulated in numerically tractable manner. An LMI has the form of (Boyd et al. 1994) m

Q ( x ) Q 0 + ∑ xi Q i < 0,

(4.7)

i =1

Qi = QTi ∈ℜn×n , i = 0,..., m

where

are

given

as

the

symmetric

matrices

and

xi ∈ℜ , i = 1,..., m are the design variable to be solved. The inequality symbol of < 0 m

represents negative definite, i.e., the largest eigenvalue of Q ( x ) is negative. Finding a solution of the LMI Eq. (4.7) is a convex optimization problem because the LMI Eq. (4.7) is a convex constraint on x . One of the main advantages of the LMI formulation is that design objectives and design constraints that can arise in control system design can be combined in numerically tractable manner. Note that multiple LMIs that can arise in control system design can be considered as a single LMI, i.e.,

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⎧ Q1 ( x ) < 0 ⎫ ⎪ ⎪ . ⎪ ⎪ ⎨ ⎬ is equal to Q ( x ) := diag {Q1 ( x ) ,..., Q n ( x )} < 0. . ⎪ ⎪ ⎪Qn ( x ) < 0 ⎪ ⎩ ⎭

(4.8)

A typical example in control engineering using LMIs is the Lyapunov inequality (Khalil 2002),

A T P + PA < 0, n×n

where A ∈ℜ is the given system matrix of a dynamic system, P = P positive definite matrix as a design variable:

Find P = P T such that A T P + PA < 0.

(4.9) T

is a symmetric

(4.10)

In this book, from the Lyapunov inequality equation, a convex optimization problem to find a set of feasible solutions is formulated.

4.3. Formulations for LMI-based Control System Design This section presents an LMI-based systematic design approach for an ANFC system. This approach considers global asymptotical stability as well as transient response characteristics in a unified framework.

4.3.1. Stability Conditions In general, it is difficult to include a stability condition for the design of a typical fuzzy logic-based controller, e.g., Mamdani fuzzy model-based controller (Casciati 1997). The reason is that a Mamdani fuzzy system does not provide a rigorous mathematical framework for stability analysis. However, such a drawback of the Mamdani fuzzy system can be solved by a TS fuzzy model. Advanced system theories such as Lyapunov theorems can be applied to the TS fuzzy model to address stability conditions because the consequent part of the TS fuzzy model is expressed in terms of linear functions, e.g., state space equations. Consider the following LTI dynamic system

x = Ax,

(4.11)

where x is the state vector and A is a system matrix. The stability of this LTI dynamic system can be checked via eigenvalue analysis. The asymptotic stability can be also investigated using Lyapunov theorem:

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Theorem 1 (Khalil 2002). The equilibrium point of the LTI dynamic system is asymptotically stable if there exist a symmetric positive definite matrix P and a positive definite matrix QLy such that

A T P + PA = −Q Ly .

(4.12)

Based on this Lyapunov equation, Tanaka and Sugeno (1992) suggested a stability condition for a nonlinear TS fuzzy model. Theorem 2 (Tanaka and Sugeno 1992). The equilibrium point at the origin of the continuous nonlinear TS fuzzy model is globally asymptotically stable if there exist a common symmetric positive definite matrix P such that

A Tj P + PA j < 0 for all j = 1, 2,..., r ,

(4.13)

where r is the number of the fuzzy rules. This stability condition for an open loop system can be extended into a stability condition for a closed loop dynamic system. Theorem 3 (Wang et al. 1995). The equilibrium point at the origin of the continuous closed loop TS fuzzy model is globally asymptotically stable if there exists a common symmetric positive definite matrix P such that

( A j + B j K q )T P + P ( A j + B j K q ) < 0 for all j , q = 1, 2,..., r.

(4.14)

2

Note, to determine the common symmetric positive definite matrix P , r LMI formulations should be solved; however, the computational cost can be reduced by approximately a half by grouping terms. Corollary 1 (Wang et al. 1995). The equilibrium point at the origin of the continuous closed loop TS fuzzy model is globally asymptotically stable if there exists a common symmetric positive definite matrix P such that

( A j + B j K j )T P + P ( A j + B j K j ) < 0 for all j = 1, 2,..., r ,

(4.15)

G Tjq P + PG jq < 0 for all j, q = 1, 2,..., r ,

(4.16)

where

G jq =

(A j + B j K q ) + (A q + B qK j ) 2

, j < q ≤ r.

(4.17)

Using Eq. (4.15) to Eq. (4.17), the number of LMIs to be solved is reduced from r 2 of Eq. (4.14) to r ( r + 1) 2 . Note that from Eq. (4.15) to Eq. (4.17), control gains are not

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automatically obtained; the equations are only used to check if the designed control systems are stable, i.e., the system matrices A j and B j are given, the control matrix K j should be designed first. Then, they are checked to determine whether the controllers K 'j s, j = 1,2,..., r stabilize the closed loop systems. This procedure generally requires many trial-and-errors; therefore, it is desirable to integrate the stability conditions with the control system design procedure. In other words, it is necessary to formulate K 'j s and P as design matrix variables at the same time.

4.3.2. LMI Formulation of Stabilizing Control In this subsection, the control system design procedure is integrated with the stability condition checking process (Farinwata et al. 2000). Substitution of Eq. (4.17) into Eq. (4.15) and Eq. (4.16) yields Eq. (4.18) and Eq. (4.19), respectively

( A j + B j K j )T P + P ( A j + B j K j ) < 0, j = 1, 2,..., r , ( A j + B j K q )T P + ( A q + B q K j )T P + P ( A j + B j K q ) + P ( A q + B q K j ) < 0, j < q ≤ r = 1, 2,..., r.

(4.18)

(4.19)

By pre-and post-multiplying Eq. (4.18) and Eq. (4.19) by P −1 > 0 , Eq. (4.20) and Eq. (4.21) are obtained

P −1 ( A j + B j K j ) T PP −1 + P −1P ( A j + B j K j )P −1 < 0, j < q ≤ r = 1, 2,..., r , (4.20) P −1 ( A j + B j K q )T PP −1 + P −1 ( A q + B q K j )T PP −1 + P −1P ( A j + B j K q ) P −1 + P −1P ( A q + B q K j )P −1 < 0, j < q ≤ r = 1, 2,..., r.

(4.21)

By defining a new matrix variable Q = P −1

QA Tj + A j Q + QK Tj B Tj + B j K j Q < 0, j = 1, 2,..., r , QA Tj + A j Q + QA Tq + A q Q + QK qT B Tj + B j K q Q + QK Tj B qT + B q K j Q < 0, j < q ≤ r = 1, 2,..., r.

(4.22)

(4.23)

However, Eq. (4.22) and Eq. (4.23) are not LMIs because there exist nonlinear matrix terms

QK j T and K j Q. Thus, these coupled nonlinear terms are transformed into LMIs by defining a new matrix K j Q = M j

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327

QATj + A j Q + M Tj B Tj + B j M j < 0, j = 1, 2,..., r ,

(4.24)

QA Tj + A j Q + QA Tq + A q Q + M Tq Β Tj + B j M q + Μ Tj B Tq + B q M j < 0, j < q ≤ r = 1, 2,..., r.

(4.25)

Eq. (4.24) and Eq. (4.25) are used to design stabilizing feedback control gains. However, the stabilizing control formulations do not directly address transient response characteristics. The performance-based design can be achieved through integration of a pole-placement algorithm with the stabilizing controllers.

4.3.3. LMI Formulation of Pole-Assignment Control In large-scale civil structures, the performance on transient response is an important issue; however, the stability LMI formulation does not directly address that issue. Therefore, in this section, the pole-assignment concept is recast by LMI formulation. The formulation of the pole-placement in terms of LMI is motivated by Chilali and Gahinet (1996). A D-stable region is defined by first. Definition 1: Let D be a subset of left half plane in the complex plane that represents behavior of a dynamic system x = Ax . If all the poles (or eigenvalues) of the dynamic system are located in the subset region D, the dynamic system (or the system matrix A ) is Dstable. Definition 2: LMI Stability Region The closed loop poles (or eigenvalues) of a dynamic system are located in the LMI stability region

D = {s ∈ C : f D ( s ) := α + β s + β T s < 0} if and only if there exist a symmetric positive definite matrix

(4.26)

α = [α kl ] ∈ ℜm×m and a matrix

β = [β kl ] ∈ ℜm×m . The characteristic function f D ( s ) is a m × m Hermitian matrix. Based on Chilali and Gahinet (1996) theorem, Hong and Langari (2000) applied for a circular LMI region D that is convex and symmetric with respect to real axis to the poleassignment control formulation. Let consider the following circular LMI region that has a center at ( − qc , 0 ) and radius rc > 0

D = {x + jy ∈ C : ( x + qc )2 + y 2 < rc 2 }. The associated characteristic function f D ( s ) is given by

(4.27)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

⎛ − rc f D (s) = ⎜ ⎝ s + qc

s + qc ⎞ ⎟. − rc ⎠

(4.28)

Fig. 4.1 shows a schematic of the circular LMI region. This circular LMI region can be related to a LMI stability region described in terms of an m × m block matrix. Theorem 4 (Chilali and Gahinet 1996). The system dynamics of x = Ax is D-stable if and only if there exists a symmetric matrix Q such that

M D ( A, X) := α ⊗ Q + β ⊗ ( AQ) + β T ⊗ ( AQ) T = [α kl Q + β kl AQ + β lk QA T ]1≤ k ,l ≤ m < 0,

Q > 0.

(4.29)

It should be noted that M D ( A, Q) and f D ( s ) have close relationship, i.e., replacing

(1, s, s )

(

)

of f D ( s ) by Q , AQ , QA T of M D ( A, Q) yields

⎛ −rc Q qcQ + QA T ⎞ ⎜ ⎟ < 0, Q > 0. − rc Q ⎠ ⎝ qc Q + AQ From Eq. (4.30), a LMI for the pole-placement controller is derived.

Figure 4.1. Circular region (D) for pole location (Hong and Langari 2000).

(4.30)

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329

Theorem 5 (Hong and Langari 2000). The continuous closed loop TS fuzzy control system is D-stable if and only if there exists a positive symmetric matrix Q such that

⎛ − rc Q qc Q + Q( A j + B j K q )T ⎞ ⎜⎜ ⎟⎟ < 0. − rcQ ⎝ qcQ + ( A j + B j K q )Q ⎠

(4.31)

Remark: the inequality Eq. (4.31) is not a LMI because the matrices Q and K j are coupled. This nonlinear matrix inequality can be transformed into a LMI by defining a new matrix variable M j = K j Q . Corollary 2: The continuous closed loop TS fuzzy control system is D-stable if and only if there exists a positive symmetric matrix Q and M j such that

⎛ − rcQ ⎜⎜ ⎝ qc Q + A j Q + B j M j

qc Q + QA Tj + M Tj B Tj ⎞ ⎟⎟ < 0. − rcQ ⎠

(4.32)

This LMI (4.32) directly addresses the performance on transient response of the dynamic system. In summary, three LMIs Eq. (4.24), Eq. (4.25), and Eq. (4.32) are solved simultaneously to obtain Q and M j . Then the common symmetric positive definite matrix P and state feedback control gains K j are determined

P = Q -1 K j = M j Q −1 = M j P, j = 1, 2,..., r.

(4.33)

These state feedback control gains are integrated with a state estimator to construct output feedback controllers.

4.4. Output Feedback-Based SNFC In this section, two more design units, which are a state estimator and clipped algorithms, are introduced to convert the full state feedback-based active nonlinear control system into output feedback-based semiactive nonlinear control system.

4.4.1. State Estimator From practical point of view, it is not always available to measure all the states. Therefore, state estimators are designed to implement the full state feedback control systems as output feedback control systems.

330

Yeesock Kim, Reza Langari and Stefan Hurlebaus Consider the following state space equation

x = A j x + B j u,

(4.34)

y = C j x + D j u.

(4.35)

By adding and subtracting a term L j y into Eq. (4.34),

x = A j x + B j u + L j y − L j y.

(4.36)

Substitution of Eq. (4.35) into Eq. (4.36) yields

x = ( A j + L j C j )x + (B j + L j D j )u − L j y.

(4.37)

Consider a negative feedback controller

u = −K j x.

(4.38)

By substituting Eq. (4.38) into Eq. (4.37),

x = ( A j + LC j )x − (B j + LD j )K j x − L j y.

(4.39)

Then a continuous time state observer model of a dynamic system is derived

xˆ = ( A j + LC j )xˆ − (B j + LD j )K j xˆ − L j y ,

(4.40)

u = −K j xˆ .

(4.41)

Figure 4.2. Mechanism of combined controller and estimator.

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In this book, the optimal observer gains L j are obtained by Kalman filter estimation procedure (Crassidis and Junkins 2004) Fig. 4.2 is a schematic of a combined control law and observer system.

4.4.2. Clipped Algorithms Once the output feedback-based ANFC is designed, a converting algorithm and a MR damper are integrated with the output ANFC to develop a SNFC system. In general, a MR damper cannot be directly controlled by a control algorithm. The reason is that a controller generates force signals, while a MR damper requires voltage (or current) signals to be operated. Therefore, a unit that converts from a control force signal to a voltage (or current) signal should be integrated with the ANFC system to construct a SNFC system. Such a unit would be either to use an inverse MR damper model or to implement a converting algorithm. Candidates for the inverse MR damper models may include a Bingham, a polynomial, a Bouc-Wen, and a modified Bouc-Wen model whose detailed description is presented in Section 2. Another good candidate for the conversion is a clipped algorithm

v = Vmax H ({ f ANFC − f m } f m ) ,

(4.42)

where v is the voltage level, Vmax is the maximum voltage level, H is a Heaviside step function, f m is a measured MR damper force, and f ANFC is a control force signal generated by an ANFC. However, this clipped algorithm generates only either a maximum or a zero value. Therefore, this algorithm can be modified such that it takes any value between 0 and the maximum values (Yoshida and Dyke 2004)

v = Va H ({ f ANFC − f m } f m ) ,

(4.43)

where

⎧ μ ⋅ f ANFC Va = ⎨ ⎩ Vmax

for f ANFC ≤ f max, for f ANFC > f max,

(4.44)

where μ is a value relating the MR damper force to the voltage. Fig. 4.3 and Fig. 4.4 show the graphical representation for the modified clipped algorithm. Recall the ANFC Nr

f ANFC =

n

∑∏ μ j =1 i =1 Nr

i, j

i ( zFZ ) ⎡⎣K j xˆ ⎤⎦

n

∑∏ μ j =1 i =1

i, j

i FZ

(z )

.

(4.45)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

Figure 4.3. The function of the modified clipped algorithm.

Figure 4.4. Operation regions of the modified clipped algorithm.

By substitution of Eq. (4.45) into Eq. (4.42), the final voltage equation can be written as

⎛ ⎧ Nr n ⎫ ⎞ i ⎜ ⎪ ∑∏ μi , j ( zFZ ) ⎡⎣K j xˆ ⎤⎦ ⎪ ⎟ ⎪ j =1 i =1 ⎪ ⎜ − fm ⎬ fm ⎟ . v = Va H ⎨ Nr n ⎜ ⎟ i ⎪ ⎟ ⎜⎜ ⎪ ∑∏ μi , j ( zFZ ) ⎪⎭ ⎟⎠ j =1 i =1 ⎝ ⎪⎩

(4.46)

This SNFC system is applied to a 3-story building structure equipped with a MR damper to demonstrate its performance and is also used as a component of a MIMO control system that is explained in the following section.

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5. Supervisory Semiactive Nonlinear Control 5.1. Introduction In recent years, advanced control technologies, which include passive, active, and semiactive control systems, have been applied to large-scale civil structures for mitigation of natural hazards such as earthquakes and strong winds. However, these control systems have been implemented as the so-called centralized controllers. In the centralized control system, there exist only a single central control unit to operate many actuators and sensors. One of the severe problems for the centralized control technologies is that the overall control system of the large-scale civil structures will be broken down if the main central control unit malfunctions for some reasons during an earthquake event, e.g., shut down of power sources, broken sensors and wires. A solution to solve this problem is to use the so-called decentralized control concept. In general, a decentralized control is to divide the large-scale civil engineering structure into a number of sub-structures by first and then to implement several sub-controllers that are associated with each sub-structure, i.e., each sub-structure is controlled by a sub-controller independently (Hashemian and Ryaciotaki-Roussalis 1995). This decentralized control system increases fail-safe reliability of the overall control system. Thus, the decentralized control systems have attracted attention for use with large-scale building structures (Lynch and Law 2000; Rofooei and Monajemi-nezhad 2006). However, they have been mostly implemented based on linear control theories. Concurrently with the linear control-based decentralized control techniques, nonlinear decentralized controllers have been also applied to the large-scale civil structural systems, in particular, neuro, fuzzy, and neuro-fuzzy control systems because they are easy to handle with nonlinearity and are inherently robust with respect to uncertainties (Xu et al. 2003; Park et al. 2005). However, their applications are limited to active control system implementations. Later, as a breakthrough, Reigles and Symans (2006) suggested a supervisory nonlinear fuzzy control system for use with a baseisolated building structure employing controllable fluid viscous dampers. They designed two decentralized fuzzy controllers for a far- and a near-field earthquake disturbance and the control gains of the decentralized sub-controllers are adapted according to the command of a supervisor fuzzy logic system. However, their systems have been designed by a trial-anderror approach that uses either experienced investigators or high cost computation, i.e., as a model-free controller, they are trained using a set of input-output data. Although useful for the performance purpose, the ad-hoc approach may not provide design guidelines in a systematic way. Unfortunately, no systematic study has been conducted to design a decentralized fuzzy control system for structural vibration control of building structures equipped with nonlinear semiactive control devices. Therefore, research is needed to develop a systematic design methodology for the Decentralized SNFC (hereinafter as “DSNFC”) system of large scale building structures employing MR dampers.

5.2. DSNFC System A MIMO SNFC system is developed as a diagonal or a block-diagonal controller that consists of a set of MISO controllers. In what follows, fundamentals on decentralized control

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

techniques are discussed. Then, the decentralized control concept is applied to a MIMO SNFC system implementation.

5.2.1. Concept of Decentralized Control In a decentralized control system, it is assumed that the building structure to be controlled is close to diagonal, i.e., the building structure is a collection of a number of independent substructures. Fig. 5.1 shows a schematic of the decentralized control implementation for the large-scale building structure. As shown in Fig. 5.1, each sub-controller does not use all the state or output feedback information from the structural system, i.e., each local controller, which is independently operated, uses local feedback information.

Figure 5.1. Decentralized diagonal control concept.

The decentralized diagonal controller is given by

⎡ k1 ⎢0 ⎢ K = diag {ki } = ⎢ 0 ⎢ ⎢0 ⎢⎣ 0

0 k2 0 0 0

0 0 0 0

0 0 0 kN 0

⎤ ⎥ ⎥ ⎥, ⎥ ⎥ k N −1 ⎥⎦ 0 0 0 0

(5.1)

In Fig. 5.1, control signals ui are generated by the local sub-controllers ki , yi are output signals, ri are reference signals, and w is an external disturbance, i.e., earthquake acceleration record. A procedure to design the decentralized control system consists of four steps.

Identification and Control of Large Smart Structures

335

Step 1: Selection of locations to be controlled within the given building structure. Step 2: Development of a mathematical model for each sub-structure related to the locations to be controlled. Step 3: Design of each local sub-controller ki that is associated with each sub-structure. Step 4: Implementation of the independent sub-controllers into the given building. In this book, each local sub-controller is independently designed as a SNFC system. In the following section, the MISO SNFC system is generalized into a MIMO SNFC system via the decentralized control concept.

5.2.2. DSNFC Based on Eq. (5.1), a decentralized MIMO SNFC system is implemented as shown in Fig. 5.2. As shown in Fig. 5.2, a SNFC system is used as a sub-controller of the decentralized SNFC (hereinafter as “DSNFC”). In other words, each sub-controller is designed such that globally asymptotically stable is guaranteed and the performance on transient responses is also satisfied, i.e., the sub-controller is developed as the state feedback controller through solving Eq. (5.24), Eq. (5.25), and Eq. (5.32) simultaneously. Then, the state feedback-based sub-controllers are integrated with Eq. (5.40) and Eq. (5.42) to construct the output feedbackbased semiactive controller.

Figure 5.2. A schematic of a DSNFC system.

In Fig. 5.2, ki ( xˆ ) is the output feedback gain associated with the sub-structure; H ( ki ) is the semiactive converter that is implemented via either a clipped algorithm or an inverse MR damper model; MR # i is the MR damper; ui is the control force that is applied to the sub-structure; yi is the output; ri is the reference; w is the external disturbance, i.e., earthquake excitation signals; si is the sub-structure. Each DSNFC system is designed based on acceleration and drift feedback information. In this DSNFC system, any information

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between the local control units is not communicated. However, global performance of the closed loop system can be improved by adding the higher level of a controller, so-called supervisor controller (Lei and Langari 2000). It might be called a Supervisory SNFC (hereinafter as “SSNFC”).

5.2.3. SSNFC System In the DSNFC system, any information among the local sub-controllers is not communicated. However, the performance of the DSNFC system can be improved by adding the higher level of a controller, so-called a coordinate controller, into the DSNFC system. The supervisor controller is first developed as a state feedback controller through solving Eq. (5.24), Eq. (5.25), and Eq. (5.32) simultaneously such that globally asymptotically stable is guaranteed and the performance on transient responses is also satisfied. Then, the state feedback-based supervisor controller is integrated with Eq. (5.40) to construct the output feedback-based supervisor controller. The supervisor controller adapts the magnitude of control gains of the sub-controllers according to velocity feedback information. Fig. 5.3 shows a schematic of the SSNFC system configuration. To demonstrate the effective of the proposed DSNFC and SSNFC systems, an 8-story building structure is investigated.

Figure 5.3. A schematic of a SSNFC system.

6. Building Structures equipped with MR Dampers 6.1. Introduction In this section, three building structures employing MR dampers are presented. The goal is to create integrated building-MR damper system models that are used for the performance evaluation of the MIMO ARX-TS fuzzy model-based SNFC system design framework for building structures equipped with MR dampers.

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In Section 6.2, the equations of motion of a 3-story building structure are derived first. Then, differential equations of a MR damper are integrated with the equations of motion of the 3-story building structure. In Section 6.3, the equations of motion of an 8-story buildingMR damper are presented. In Section 6.4, properties of the 20-story building structure are described. Finally, in Section 6.5, concluding remarks are made.

6.2. A 3-Story Building Structure Equipped with a MR Damper In this section, the equations of motion of a 3-story building structure employing a MR damper are derived in terms of state space equations for use with the performance evaluation of a MISO SNFC system.

6.2.1. A 3-Story Building Structure Consider a deflected 3-story building frame shown in Fig. 6.1. The building structure is modeled as a lumped mass-spring system, i.e., the mass of each floor is lumped; the stiffness and the damping of columns are modeled as a spring and a dashpot element, respectively. In this model, each floor is assumed to be axially rigid and the vertical deformation and rotation of each column are assumed to be negligible.

Figure 6.1. Deflected 3-story building structure.

In Fig. 6.1, wg denotes the displacement of the ground that is induced by earthquaketype ground accelerations; yi are absolute displacements; mi are the mass of the i th floor;

ki are the stiffness of the i th floor columns; ci are the damping of the i th floor columns; and ui are control forces acting each floor.

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To derive the associated differential equations of motion, an isolated free body diagram (hereinafter as “FBD”) is depicted in Fig. 6.2 and then Newton’s second law is applied to each mass. Summation of all the forces acting on each mass to the horizontal direction leads to

∑F = m y

= − k1 ( y1 − wg ) + k2 ( y2 − y1 ) − c1 ( y1 − wg ) + c2 ( y2 − y1 ) + u1 , (6.1)

∑F = m y

= − k2 ( y2 − y1 ) + k3 ( y3 − y2 ) − c2 ( y2 − y1 ) + c3 ( y3 − y2 ) + u2 , (6.2)

1 1

2

2

∑F = m y

3 3

= − k3 ( y3 − y2 ) − c3 ( y3 − y2 ) + u3 .

(6.3)

Figure 6.2. An isolated FBD of a three story building structure.

With the definition of the relative displacements between the ground and each mass

x1 = y1 − wg ,

(6.4)

x2 = y2 − wg ,

(6.5)

x3 = y3 − wg ,

(6.6)

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and substituting Eq. (6.4) to Eq. (6.6) into Eq. (6.1) to Eq. (6.3), the equations of motion become

m1 x1 + ( c1 + c2 ) x1 − c2 x2 + ( k1 + k2 ) x1 − k2 x2 = u1 − m1wg ,

(6.7)

m2 x2 − c2 x1 + ( c2 + c3 ) x2 − c3 x3 − k2 x1 + ( k2 + k3 ) x2 − k3 x3 = u2 − m2 wg ,

(6.8)

m3 x3 − c3 x2 + c3 x3 − k3 x2 − k3 x3 = u3 − m3 wg .

(6.9)

Eq. (6.7) to Eq. (6.9) can be expressed in more compact form by defining new matrix variables

Mx + Cx + Kx = ΓU − MΛwg ,

(6.10)

where the system matrices are given by

⎡ m1 M = ⎢⎢ 0 ⎢⎣ 0

0 m2 0

0⎤ 0 ⎥⎥ m3 ⎥⎦

(6.11)

is the mass matrix,

⎡ c1 + c2 C = ⎢⎢ −c2 ⎢⎣ 0

−c2 c2 + c3 −c3

0 ⎤ −c3 ⎥⎥ c3 ⎥⎦

(6.12)

⎡ k1 + k2 K = ⎢⎢ −k2 ⎢⎣ 0

− k2 k2 + k3 − k3

0 ⎤ − k3 ⎥⎥ k3 ⎥⎦

(6.13)

is the damping matrix,

is the stiffness matrix,

⎡ u1 ⎤ U = ⎢⎢u2 ⎥⎥ ⎢⎣u3 ⎥⎦

(6.14)

⎡1 0 0 ⎤ Γ = ⎢0 1 0 ⎥ ⎢ ⎥ ⎢⎣0 0 1 ⎥⎦

(6.15)

is the control input vector,

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is the control input location matrix,

⎡1⎤ Λ = ⎢⎢1⎥⎥ ⎢⎣1⎥⎦

(6.16)

is the disturbance signal location matrix, the vector x is the displacement relative to the ground, and wg is the disturbance acceleration, i.e., earthquake. Properties of the 3-story building structure are taken from a scaled model (Dyke et al. 1996) of a prototype building structure that was developed by Chung et al. (1989). The mass of each floor m1 = m2 = m3 = 98.3 kg; the stiffness of each story k1 = 516,000 N/m, k2 = 684000 N/m, and k3 = 684,000 N/m; and the damping coefficients of each floor c1 = 125 Ns/m, c2 = 50 Ns/m and c3 = 50 Ns/m. It is advantageous to convert the second order differential Eq. (6.10) into the 1st order differential equation such that it is expressed in state space equations with the state space vector z = [ x

x ] . Eq. (6.10) can be expressed in the following form T

z = Az + Bu − Ewg y = Cz + Du + n,

(6.17)

where the system matrices are given by

I ⎤ ⎡ 0 A=⎢ −1 −1 ⎥ ⎣ − M K −M C ⎦

(6.18)

⎡ 0 ⎤ B = ⎢ −1 ⎥ ⎣M F ⎦

(6.19)

is the state matrix,

is the input matrix,

⎡ I C = ⎢⎢ 0 ⎢⎣ −M −1K

0 ⎤ I ⎥⎥ −M −1C ⎥⎦

(6.20)

is the output matrix, ⎡ 0 ⎤ D = ⎢⎢ 0 ⎥⎥ ⎢⎣ M − 1 F ⎥⎦

(6.21)

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is the feedthrough matrix,

⎡0⎤ E=⎢ ⎥ ⎣F ⎦

(6.22)

is the disturbance location matrix,

u = [u1 u2

u3 ]

T

(6.23)

is the control vector, and ⎡−1 F = ⎢⎢ 0 ⎣⎢ 0

1 −1 0

0 ⎤ 1 ⎥⎥ − 1 ⎦⎥

(6.24)

is the location matrix that a Chevron brace is located within a building structure. The vector n represents noise that is generated by a zero-mean Gaussian white noise generator. A Chevron brace is used to connect a MR damper into the 3-story building structure; the control force and disturbance are not acting only on each floor. Thus, the control force and disturbance matrices are transformed (Hart and Wong 2000) using a matrix F that represents control and disturbance locations that is derived from configuration of a Chevron brace. In the following section, this building structure is integrated with a MR damper.

6.2.2. An Integrated 3-Story Building-MR Damper System An integrated building-MR damper system is presented through integration of a MR damper with a building structure. Then, the integrated system is not linear anymore although it is assumed that the structural system itself is linear. Fig. 6.3 and Fig. 6.4 show a configuration on how a MR damper is integrated with a building structure. The associated equation of motion is given by

Mx + Cx + Kx = ΓfMR ( t , x1 , x1 , v1 ) − MΛwg ,

Figure 6.3. Integrated building structure-MR damper system.

(6.25)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

Figure 6.4. A schematic of a building-MR damper system.

where v is the voltage level to be applied. This second order differential equations can be converted into state space

z = Az + BfMR ( t , z1 , z4 , v ) − Ewg y = Cz + DfMR ( t , z1 , z4 , v ) + n,

(6.26)

where v is the voltage to be applied, z1 and z4 are the displacement and the velocity at the 1st floor level of the 3-story building structure, respectively. In this building structure, a SD-1000 MR damper (Spencer et al. 1997) is applied whose parameters are given in Appendix A. Fig. 6.5 compares the seismic response of the structure controlled by the proposed MIMO ARXTS fuzzy model-based SNFC system and uncontrolled structure: the dotted line represents the uncontrolled responses, while the solid line represents the controlled responses. It is demonstrated from Fig. 6.5 that the proposed MIMO ARX-TS fuzzy model-based SNFC system is effective to the seismic response control of the 3-story building structure equipped with a MR damper system. To evaluate the effectiveness of the proposed MIMO ARX-TS fuzzy model-based SNFC strategy with the larger scale building structure, an 8-story building structure is investigated in the following section.

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Figure 6.5. Response Time Histories of the 3-story Building subjected to 150% El-Centro Earthquake.

6.3. An 8-Story Building Structure Equipped with MR Dampers In order to demonstrate the effectiveness of the MIMO SNFC system with a larger scale example, an 8-story building structure is investigated here. The reason to choose this example is that it has been used as a benchmark problem by a number of researchers (Yang 1982; Yang et al. 1987; Soong 1990; Spencer et al. 1994). Note that the equations of motion of the 8-story building model are not derived here because its derivation can be easily extended from the equations of motion of the 3-story building model. The equation of motion of the 8story building structure is

Mx + Cx + Kx = ΓU − MΛwg ,

(6.27)

where the system matrices are given by

⎡ m1 ⎢0 ⎢ ⎢0 ⎢ 0 M=⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣ 0

0 m2 0 0 0 0 0 0

0 0 m3 0 0 0 0 0

0 0 0 m4 0 0 0 0

0 0 0 0 m5 0 0 0

0 0 0 0 0 m6 0 0

0 0 0 0 0 0 m7 0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ m8 ⎥⎦

(6.28)

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

is the mass matrix,

⎡c1 + c2 ⎢ −c 2 ⎢ ⎢ 0 ⎢ 0 C=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎣⎢ 0

−c2 c2 + c3 −c3 0 0 0 0 0

0 −c3 c3 + c4 −c4

0 0 −c4 c4 + c5 −c5 0 0 0

0 0 0 0

0 0 0 −c5 c5 + c6 −c6

0 0 0 0 −c6 c6 + c7 −c7

0 0

0 0 0 0 0 −c7 c7 + c8 −c8

0

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 0 ⎥ (6.29) 0 ⎥ ⎥ 0 ⎥ −c8 ⎥ ⎥ c8 ⎦⎥

is the damping matrix, ⎡ k1 + k2 ⎢ −k 2 ⎢ ⎢ 0 ⎢ 0 K=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢⎣ 0

− k2 k2 + k3 − k3 0 0

−k3 k3 + k4 − k4 0

0

0

−k5 0

0 0

0 0

0 0

0

0 0

0 0

0 0

0 0

− k4 k4 + k5

0

0 0

0 0 0

− k5 k5 + k6 − k6 0 0

− k6 k6 + k 7

− k7

− k7 0

k7 + k8 −k8

0 ⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ − k8 ⎥ ⎥ k8 ⎥⎦

(6.30)

is the stiffness matrix,

U = [u1 u2

u3

u4

u5

u6

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

u7

u8 ]

T

(6.31)

is the control input vector,

⎡1 ⎢0 ⎢ ⎢0 ⎢ 0 Γ=⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

0 1 0 0 0 0 0 0

is the control input location matrix, and

0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎥⎦

(6.32)

Identification and Control of Large Smart Structures

Λ = [1 1 1 1 1 1 1 1]

T

345 (6.33)

is the disturbance location matrix. The mass of each floor m1 = m2 = m3 = m4 = m5 = m6 = m7 = m8 = 345,600 kg; the stiffness of each story k1 = k2 = k3 = k4 = k5 = k6 = k7 = k8 = 340,400 kN/m; and the damping coefficient of each floor c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 = 2,937,000 Ns/m (Yang 1982). In this 8-story building structure, two MR dampers are installed on the 5th and 8th floor levels using Chevron braces. Fig. 6.6 shows a configuration on how the MR dampers are integrated with a building structure.

Figure 6.6. A building structure equipped with multiple MR dampers.

The equation of motion of the integrated building-MR damper system is given by

Mx + Cx + Kx = Γf MR ( t , x5 , x5 , v1 , x8 , x8 , v2 ) − MΛwg ,

(6.34)

where x5 and x5 are the displacement and the velocity at the 5th floor level relative to the 4th floor level of the 8-story building structure, respectively; x8 and x8 are the displacement and the velocity at the 8th floor level relative to the 7th floor level, respectively; v1 and v2 are the voltage levels to be applied to the MR damper installed on the 5th and the 8th floors of the structure, respectively; and n is noise. The second order differential equations can be converted into state space

z = Az + Bf MR ( t , z5 , z13 , v1 , z8 , z16 , v2 ) − Ewg (6.35)

y = Cz + Df MR ( t , z5 , z13 , v1 , z8 , z16 , v2 ) + n,

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where z5 and z13 are the displacement and the velocity at the 5th floor level relative to the 4th floor level of the 8-story building structure, respectively, and z8 and z16 are the displacement and the velocity at the 8th floor level relative to the 7th floor, respectively. Once the integrated building-MR damper system is constructed, a set of input-output data for NSI can be generated; the integrated systems are also used for the performance evaluation of SNFC systems as nonlinear dynamic models. However, since the 8-story building structure is much larger than the 3-story building structure, MR dampers with much larger capacity are needed. In the 8-story building structure, two 1000 kN MR dampers are used whose optimum parameters are given in Appendix D. Fig. 6.7 compares the performances of the structures controlled by the CSNFC, DSNFC, and SSNFC systems, while the uncontrolled system responses are used as the baseline. Note, in the CSNFC system, the two 1000 kN MR dampers are installed on the same floor (i.e., top floor). As seen in Fig. 6.7, all of the control systems are effective to the seismic response control of the 8-story building structure; in particular, SSNFC system provides the best solution.

Figure 6.7. Comparison of time history responses at the top floor of an eight story building structure equipped with two MR dampers controlled by CSNFC, DSNFC, and SSNFC systems.

6.4. A 20-Story Building Structure In this section, the effectiveness of the proposed MIMO ARX-TS fuzzy model-based SNFC algorithm are further studied here for nonlinear control of seismically excited high-rise building structures. A benchmark full-scale building structure is selected as a target model that meets seismic code for Los Angeles, California region designed by Brandow & Johnston Associates for the SAC Phase 2 Steel Project (Spencer et al. 1999; Ohtori et al. 2004). The reason to choose the building structure as a target model is that it has been chosen as a benchmark building structure to compare the performance of structural control algorithms by many other researchers (Spencer et al. 1999; Lynch and Law 2002). To evaluate the effectiveness of the proposed control system with respect to earthquake, four real-recorded earthquake signals which are El

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Centro, Kobe, Northridge, and Hachinohe are used. In what follows, a finite element model for the Los Angeles 20 story building structure is discussed first.

6.4.1.Properties of 20-Story Building Model As a full-scale 20 story building structure in Fig. 6.8 is a moment-resisting frame (hereinafter as “MRF”), the dimension is 30.48 m (100 ft) by 36.58 m (120 ft) in plane and 80.77 m (265 ft) in height. It has five bays in the north-south (hereinafter as “N-S”) direction while six bays in the east-west (hereinafter as “E-W”) direction. The dimension of the bay is 6.10 m (20 ft) on center in both N-S and E-W directions. The floor-to-floor height measured from center of beam to center of beam is 3.96 m (13 ft).

Figure 6.8. Los Angeles 20-story building structure (Spencer et al. 1999).

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5.32 ×105 kg (36.4 kips-sec 2 ft); the second floor is 5.65 ×105 kg (38.7 kips-sec 2 ft); the third floor The seismic mass of the structure is: the first floor is

to the 20th floor is 5.51×10 kg (37.7 kips-sec ft); and the roof level is 5.83 ×10 kg 5

2

5

(39.9 kips-sec 2 ft). The total seismic mass of the entire structure is 1.16 ×107 kg (794 kips-sec 2 ft). However, it is modeled using a plane frame element that contains two nodes in which each node has three degrees-of-freedom (DOFs), i.e., it is an in-plane finite element model of N-S MRF. Therefore, the seismic mass of the structure is modified: the first floor is

2.66 ×105 kg (18.2 kips-sec 2 ft); the second floor is 2.83 ×105 kg

(19.4 kips-sec2 ft); the third floor to the 20th floor is 2.76 ×105 kg (18.9 kips-sec2 ft); 2 and the roof level is 2.92 ×10 kg (20.0 kips-sec ft).

5

Each node of the plane frame element includes horizontal, vertical, and rotational DOFs. The total number of nodes and elements is 180 and 284, respectively. The total DOFs is 540 DOFs before boundary conditions and subsequent model reduction are applied. The boundary constrained DOFs at the horizontal direction are nodes of 1, 2, 3, 4, 5, 6, 13 and 18 and the vertical constrained DOFs are 1, 2, 3, 4, 5 and 6 (see Fig. 6.9). Then, the total DOFs are reduced to 526. However, more DOFs can be reduced because the floor slab in each horizontal plane is assumed to be rigid, i.e., each floor has the same horizontal displacements. Using a Ritz transformation (Craig 1981), the total DOFs are reduced to 418. However, this finite element model with 418 DOFs is too large to analyze/design a control system design. Therefore, the 418 DOF analysis model is reduced to a 106 DOF model using Guyan reduction (Craig 1981) of all the rotational and almost all vertical DOFs. Based on modal damping, the damping matrix is defined using the reduced 106 DOF model: the maximum value of a critical damping is 10 % and the damping in the first mode is assumed to be 2 %. The first ten eigen-frequencies of the benchmark building are: 0.29, 0.83, 1.43, 2.01, 2.64, 3.08, 3.30, 3.53, 3.99 and 4.74 Hz. The associated mode shapes are given in Figure 6.10.

Figure 6.9. First three mode shapes of the LA 20-story building structure (Spencer et al. 1999).

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Figure 6.10. Node numbers of an in-plane FEM for the LA 20-story building (Spencer et al. 1999).

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Fig. 6.11 compares the performances of the LQG-based controlled structure, the SNFCbased controlled one, and uncontrolled one: the dotted line represents the uncontrolled responses, the dashed dotted line represents the LQG-based controlled response, and the solid line represents the SNFC-based controlled responses. As seen in Fig. 6.11, the responses of the 20-story building structure are dramatically reduced via the SNFC-based control system.

Figure 6.11. Time history displacement responses of an uncontrolled, a LQG control, and a SNFC systems of a Los Angeles 20-story building excited by El Centro earthquake.

7. Conclusions and Future Studies 7.1. Summary of Concluding Remarks This book presents a multiple model approach for MIMO NSI of seismically excited building-MR damper systems. The proposed framework is developed through integration of multiple MIMO ARX inputs-based TS fuzzy model with weighted least squares and data clustering algorithms. This method does not require decoupling of identification procedures for subcomponents because it identifies a building structure and a MR damper as an integrated system. In other words, the MIMO nonlinear behavior of the building structure employing a MR damper are represented by a family of local linear MIMO ARX input models whose operating regions are integrated via a TS fuzzy interpolation method. The premise parts of the multiple MIMO ARX inputs-based TS fuzzy model are partitioned to subdivide the input space into several operating regions using clustering techniques. The consequent part is optimized by a set of linear weighted least squares algorithm.

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Second, this book also presents LMIs-based systematic design methodology for nonlinear control of building structures equipped with a MR damper. This approach considers the stability performance as well as transient characteristics in a unified framework. First, multiple Lyapunov-based controllers are formulated in terms of LMIs such that global asymptotical stability is guaranteed and the performance on transient response is also satisfied. Such Lyapunov-based state feedback controllers are converted into output feedback regulators with a Kalman estimator. Then, these Lyapunov-based controllers and a Kalman observer are integrated as an output feedback-based active nonlinear control system. Finally, the active nonlinear controller is integrated with a converting algorithm, e.g., either an inverse MR damper model or a clipped algorithm to convert the active system into a semiactive system. Third, this book also presents hierarchical SNFC techniques for a building structure equipped with multiple MR dampers. Based on acceleration and drift information, a set of sub-controllers using the SNFC algorithm are designed for sub-structures at the specific floor levels within the building structure for the lower level control systems. At the higher level, a supervisor controller, a velocity feedback-based ANFC is built up to supervise the performance of the sub-controllers at the lower level. Then, the nonlinear sub-controllers at the lower level are integrated with the supervisory nonlinear controller. Both higher and lower level nonlinear controllers are formulated in terms of LMIs such that global asymptotical stability is guaranteed and the performance on transient responses is also satisfied. Then, multiple Kalman estimators that are associated with the coordinator controller and subcontrollers are designed to construct output feedback regulators. Finally, the output feedbackbased active sub-controllers are converted into semiactive sub-controllers using converting algorithms. To demonstrate the effectiveness of the proposed MIMO ARX-TS fuzzy model-based SNFC system, 3-, 8-, and 20-story building structures employing MR dampers are studied. It is demonstrated from comparison of the uncontrolled and semiactive controlled responses that the proposed MIMO ARX-TS fuzzy model-based SNFC system design framework is effective in controlling vibration of a seismically excited building equipped with a MR damper.

7.2. Future Research The NSI and SNFC frameworks addressed in this book have been demonstrated theoretically and numerically. Although useful for design guidelines, further research is recommended to do experimental studies to demonstrate the effectiveness of the proposed methodologies. It is recommended that the proposed identification and control methods be applied to wind-excited high-rise building structures equipped with MR dampers. It is also recommended that the proposed methods be applied to large-scale bridge structures employing MR dampers. Further research is recommended to apply the proposed methods to structural systems equipped with a variety of semiactive devices such as an electrorheological damper and a piezoelectric friction damper.

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Appendix A Table A.1. Parameters of a Bouc-Wen model for SD-1000 MR damper (Spencer et al. 1997) Parameter c0

k0

Value 50 Nscm-1 25 Ncm-1

x0

3.8 cm

N

2 880 Ncm-1 100 cm-2 100 cm-2 120

α β

γ A

Appendix B Table A.2. Optimum coefficients of the polynomial model for MRF 132-LD damper (Choi et al. 2001) Parameters Upper 0

Value

b

-371.8

b1Upper

6.205

b2Upper

0.03728

b3Upper

-3.487e-4

b4Upper

-2.767e-6

b5Upper

6.924e-9

b6Upper

5.604e-11

c0Upper

-659.4

c1Upper

8.955

c2Upper

0.1062

c3Upper

-1.584e-4

c4Upper

-5.908e-6

c5Upper

1.137e-9

Upper 6

c

1.087e-10

b0Lower

-235.8

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Table A2. Continued Parameters Lower 1

Value

b

5.391

b2Lower

-0.02774

b3Lower

-3.788e-4

Lower 4

b

2.449e-4

b5Lower

8.804e-9

b6Lower

-5.374e-11

c0Lower

693.7

Lower 1

c

7.034

c2Lower

-0.1020

c3Lower

6.729e-5

c4Lower

4.967e-6

c5Lower

-4.924e-9

c6Lower

-8.196e11

Appendix C Table A.3. Parameters of a modified Bouc-Wen model for SD-1000 MR damper model (Spencer et al. 1997) Parameter

Value

c0 a

21.0 Nscm-1

c0b

3.50 Nscm-1V-1

k0

46.9 Ncm-1

c1a

283 Nscm-1

c1b

2.95 Nscm-1V-1

k1

5.00 Ncm-1

x0

14.3 cm

αa

140 Ncm-1

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Yeesock Kim, Reza Langari and Stefan Hurlebaus Table A3. Continued Parameter

Value

αb

695 Ncm-1V-1

γ

363 cm-2

β

363 cm-2

A

301

N

2

η

190 s-1

Appendix D Table A.4. Parameters for 1000 kN MR damper model (Jung et al. 2003) Parameter

Value

c0 a

110.0 Nscm-1

c0b

114.3 Nscm-1V-1

k0

0.002 Ncm-1

c1a

8359.2 Nscm-1

c1b

7482.9 Nscm-1V-1

k1

0.0097 Ncm-1

x0

0 cm

αa

46.2 Ncm-1

αb

41.2 N cm-1V-1

γ

164.0 cm-2

β

164.0 cm-2

A

1107.2

N

2

η

100 s-1

Identification and Control of Large Smart Structures

355

References Abe, M. (1996). “Rule-Based Control Algorithm for Active Tuned Mass Dampers.” J. Eng. Mech., 122(8), 705-713. Abonyi, J. (2003). Fuzzy Model Identification for Control, Birkhauser, Boston. Abonyi, J., Babuska, R., Verbruggen, H.B. and Szeifert, F. (2000). “Incorporating Prior Knowledge in Fuzzy Model Identification.” Int. J. of Sys. Science, 31(5), 657-667. Ahlawat, A.S and Ramaswamy, A. (2002a). “Multi-objective Optimal Design of FLC Driven Hybrid Mass Damper for Seismically Excited Structures.” Earth. Eng. Struct. Dyna., 31(7), 1459-1479. Ahlawat, A.S and Ramaswamy, A. (2002b). “Multiobjective Optimal FLC Driven Hybrid Mass Damper System for Torsionally Coupled, Seismically Excited Structures.” Earth. Eng. Struct. Dyna., 31(12), 2121-2139. Ahlawat, A.S and Ramaswamy, A. (2004). “Multiobjective Optimal Fuzzy Logic Control System for Response Control of Wind-Excited Tall Buildings.” J. Eng. Mech., 130(4), 524-530. Al-Dawod, M., Samali, B., Naghdy, F., and Kwok, K.C.S. (2001). “Active Control of Along Wind Response of Tall Building Using a Fuzzy Controller.” Eng. Struct., 23(11), 15121522. Al-Dawod, M., Samali, B., Naghdy, F., Kwok, K.C.S. and Naghdy, F. (2004). “Fuzzy Controller for Seismically Excited Nonlinear Buildings.” J. Eng. Mech., 130(4), 407-415. Alli H. and Yakut, O. (2005). “Fuzzy Sliding-Mode Control of Structures.” Eng. Struct., 27(2), 277-284.

Astrom, K.J. and Eykhoff, P. (1971). “System Identification-A Survey.” Automatica, 7(2), 123-162. Battaini, M., Casciati, F. and Faravelli, L. (1998). “Fuzzy Control of Structural Vibration. An Active Mass System Driven by a Fuzzy Controller.” Earth. Eng. Struct. Dyna., 27(11), 1267-1276. Battaini, M., Casciati, F. and Faravelli, L. (2004). “Controlling Wind Response through a Fuzzy Controller.” J. Eng. Mech. 130(4), 486-491. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia. Casciati, F. (1997). “Checking the Stability of a Fuzzy Controller for Nonlinear Structures.” Microcomputers in Civil Eng., 12(3), 205-215. Chilali, M. and Gahinet, P. (1996). “H∞ Design with Pole Placement Constraints: An LMI Approach.” IEEE Trans. Automatic Control.” 41(3), 358-367. Choi, K.M, Cho, S.W., Jung, H.J., and Lee, I.W. (2004). “Semi-active Fuzzy Control for Seismic Response Reduction using Magnetorheological Dampers.” Earth. Eng. Struct. Dyna., 33(6), 723-736. Choi, S.B., Lee, S.K., and Park, Y.P. (2001). “A Hysteresis Model for the Field-dependent Damping Force of a Magnetorheological Damper.” J. Sound and Vibration, 245(2), 375-383. Chung, L.L., Lin, R.C., Soong, T.T., and Reinhorn, A.M. (1989). “Experiments on Active Control for MDOF Seismic Structures.” J. Eng. Mech., 115(8), 1609-1627.

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Yeesock Kim, Reza Langari and Stefan Hurlebaus

Craig, R.R. (1981). Structural Dynamics: An Introduction to Computer Methods, John Wiley & Sons, New York. Crassidis, J.L. and Junkins, J.L. (2004). Optimal Estimation of Dynamic Systems, Chapman & Hall, Washington, DC. Dyke, S.J., Spencer, B.F. Jr., Sain, M.K., and Carlson, J.D. (1996). “Modeling and Control of Magnetorheological Dampers for Seismic Response Reduction.” Smart Mater. and Struct., 5(5), 565-575. Dyke, S.J., Spencer, B.F. Jr., Sain, M.K., and Carlson, J.D. (1998). “An Experimental Study of MR Dampers for Seismic Protection.” Smart Mater. and Struct., 7(5), 693-703. Faravelli, L. and Rossi, R. (2002). “Adaptive Fuzzy Control: Theory versus Implementation.” J. Struct. Control, 9(1), 59-73. Faravelli, L. and Yao, T. (1996). “Use of Adaptive Networks in Fuzzy Control of Civil Structures.” Microcomputers in Civil Eng., 11(1), 67-76. Farinwata, S.S, Filev, D., and Langari, R. (2000). Fuzzy Control-Synthesis and Analysis, John Wiley & Sons, New York. Hart, G.C. and Wong, K. (2000). Structural Dynamics for Structural Engineers, John Wiley & Sons, New York. Hashemian, H. and Ryaciotaki-Roussalis, H.A. (1995). “Decentralized Approach to Control Civil Structures.” Proc., the 1995 American Control Conf., Seattle, WA, USA, 4, 29362937. Hong, S.K. and Langari, R. (2000). “An LMI-based H∞ Fuzzy Control System Design with TS Framework.” Information Sciences, 123(3), 163-179. Hurlebaus, S. and Gaul, L. (2006). “Smart Structure Dynamics”, Mechanical Systems and Signal Processing, 20(2), 255-281. Jang, J.S.R, Sun, C.T., and Mizutani, E. (1997). Neuro-Fuzzy and Soft Computing-A Computational Approach to Learning and Machine Intelligence, Prentice Hall, Upper Saddle River, NJ. Jansen, L.M. and Dyke, S.J. (2000). “Semiactive Control Strategies for MR Dampers: Comparative Study.” J. Eng. Mech., 126(8), 795-803. Jiang, X. and Adeli, H. (2005). “Dynamic Wavelet Neural Network for Nonlinear Identification of Highrise Buildings.” Computer-Aided Civil and Infrastructure Eng., 20(5), 316-330. Jin, G., Sain, M.K., and Spencer, B.F. Jr. (2005). “Nonlinear Blackbox Modeling of MRDampers for Civil Structural Control.” IEEE Trans. on Control Syst. Technology, 13(3), 345-355. Joh, J, Langari, R., Jeung, F.T., and Chung, W.J. (1997). “A New Design Method for Continuous Takagi-Sugeno Fuzzy Controller with Pole Placement Constraints: An LMI Approach.” Proc., 1997 IEEE Int. Conf. on Syst., Man, and Cybernetics, Orlando, FL, 2969-2974. Johansen, T.A. (1994). “Fuzzy Model Based Control: Stability, Robustness, and Performance Issues.” IEEE Trans. Fuzzy Sys. 2(3), 221-234. Jung, H.J, Spencer, B.F. Jr., and Lee, I.W. (2003). “Control of Seismically Excited CableStayed Bridge Employing Magnetorheological Fluid Dampers.” J. Struct. Eng., 129(7), 873-883. Khalil, H. K. (2002). Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ.

Identification and Control of Large Smart Structures

357

Kim, Y., Langari, R. & Hurlebaus, S. (2007), Nonlinear System Identification of a BuildingMR Damper System, Journal of Intelligent Material Systems and Structures, (submitted). Kim, Y., Langari, R. & Hurlebaus, S. (2007), Semiactive Nonlinear Control of a Building Structure equipped with a Magnetorheological Damper System, Mechanical Systems and Signal Processing, (submitted). Kim, S.B, Yun, C.B., and Spencer, B.F. Jr. (2004). “Vibration Control of Wind-Excited Tall Buildings Using Sliding Mode Fuzzy Control.” J. Eng. Mech. 130(4), 505-510. Kim, Y. and Langari, R. (2007). “Nonlinear Identification and Control of a Building” New York City, NY, CD-Rom. Kim, Y., Langari, R., and Roschke, P.N. (2006). “Control of a Nonlinear Building Employing Magnetorheological Dampers.” Proc., 4th World Conf. on Struct. Control Monit., San Diego, CA. Kim, H.S. and Roschke, P.N. (2006). “Design of Fuzzy Logic Controller for Smart Base Isolation System Using Genetic Algorithm.” Eng. Struct., 28(1), 84-96. Kuehn, J.L. and Stalford, H.L. (2000). “Stability of a Lyapunov Controller for a Semi-active Structural Control System with Nonlinear Actuator Dynamics.” J. Mathematical Analysis and Applications, 251(2), 940-957. Langari, R. (1993). “Synthesis of Nonlinear Control Strategies via Fuzzy Logic.” Proc., the 1993 American Control Conf., San Francisco, CA, 1855-1859. Langari, R. (1999). “Past, Present and Future of Fuzzy Control: A Case for Application of Fuzzy Logic in Hierarchical Control.” Proc., Annual Conf. of the North American Fuzzy Information Processing Society, New York, NY, 760-765. Lei, S. and Langari, R. (2000). “Hierarchical Fuzzy Logic Control of a Double Inverted Pendulum.” Proc., IEEE Int. Conf. on Fuzzy Syst., San Antonio, TX, 1074-1077. Liu, W.Y., Xiao, C.J., Wang, B.W., Shi, Y., and Fang, S.F. (2003). “Study on Combining Subtractive Clustering with Fuzzy C-Means Clustering.” Proc., 2nd Int. Conf. on Machine Learning and Cybernetics, Xi’an, China, 2659-2662. Loh, C.H., Wu, L.Y., and Lin, P.Y. (2003). “Displacement Control of Isolated Structures with Semi-active Control Devices.” J. Struct. Control, 10(2), 77-100. Lynch, J.P. and Law, K.H. (2000). “A Market-based Control Solution for Semi-Active Structural Control.” Computing in Civil and Building Eng., 1, 588-595. Lynch, J.P. and Law, K.H. (2002). “Decentralized Control Techniques for Large-scale Civil Structural Systems.” Proc., the 20th Int. Modal Analysis Conf., Los Angeles, CA, USA, 1074-1077. Ohtori, Y., Christenson, R.E., Spencer, B.F. Jr., and Dyke, S.J. (2004). “Benchmark Control Problems for Seismically Excited Nonlinear Buildings.” J. Eng. Mech., 130(4), 366-385. Park, K.S., Koh, H.M., Ok, S.Y., and Seo, C.W. (2005). “Fuzzy Supervisory Control of Earthquake-excited Cable-stayed Bridges.” Eng. Struct., 27(7), 1086-1100. Ramallo, J.C., Johnson, E.A., and Spencer, B.F. Jr. (2002). “Smart Base Isolation Systems.” J. Eng. Mech., 128(10), 1088-1099. Ramallo, J.C., Yoshioka, H., and Spencer, B.F. Jr. (2004). “A Two-Step Identification Technique for Semiactive Control Systems.” Struct. Control Health Monit., 11(4), 273289. Reigles, D. G. and Symans, M.D. (2006). “Supervisory Fuzzy Control of a Base-isolated Benchmark Building Utilizing a Neuro-fuzzy Model of Controllable Fluid Viscous Dampers.” Struct. Control Health Monit., 13(2-3), 724-747.

358

Yeesock Kim, Reza Langari and Stefan Hurlebaus

Rofooei, F.R. and Monajemi-nezhad, S. (2006). “Decentralized Control of Tall Buildings.” The Struct. Design of Tall and Special Buildings, 15(2), 153-170. Samali, B., Al-Dawod, M., Kwok, K.C.S., and Naghdy, F. (2004). “Active Control of Cross Wind Response of 76-Story Tall Building Using a Fuzzy Controller.” J. Eng. Mech., 130(4), 492-498. Schurter, K.C and Roschke, P.N. (2001). “Neuro-Fuzzy Control of Structures Using Magnetorheological Dampers.” Proc., the 2001 American Control Conf., Arlington, VA, 1097-1102. Soong, T.T. (1990). Active Structural Control: Theory and Practice, Addison-Wesley Pub., New York. Soong, T.T. and Grigoriu, M. (1993). Random Vibration of Mechanical and Structural Systems, Prentice Hall, Upper Saddle River, NJ. Spencer, B.F. Jr. (1986). Reliability of Randomly Excited Hysteretic Structures, Springer, New York. Spencer, B.F. Jr., Christenson, R.E., and Dyke, S.J. (1999). “Next Generation Benchmark Control Problem for Seismically Excited Buildings.” Proc., the Second World Conf. on Struct. Control, Kyoto, Japan, 1351-1360. Spencer, B.F. Jr., Dyke, S.J., Sain, M.K., and Carlson, J.D. (1997). “Phenomenological Model for Magnetorheological Dampers.” J. Eng. Mech., 123(3), 230-238. Spencer, B.F. Jr., Suhardjo, J., and Sain, M.K. (1994). “Frequency Domain Optimal Control Strategies for a Seismic Protection.” J. Eng. Mech., 120(1), 135-158. Stanway, R., Sproston, J.L., and Stevens, N.G. (1985). “Non-linear Identification of an Electro-rheological Vibration Damper.” IFAC Identification and System Parameter Estimation, 7, 195-200. Stanway, R., Sproston, J.L., and Stevens, N.G. (1987). “Non-linear Modeling of an Electrorheological Vibration Damper.” J. Electrostatics, 20(2), 167-184. Subramaniam, R.S, Reinhorn, A.M., Riley, M.A., and Nagarajaiah, S. (1996). “Hybrid Control of Structures Using Fuzzy Logic.” Microcomputers in Civil Eng. 11(1), 1-17. Symans, M. and Kelly, S.W. (1999). “Fuzzy Logic Control of Bridge Structures using Intelligent Semi-active Seismic Isolation Systems.” Earth. Eng. Struct. Dyna., 28(1), 3760. Takagi, T. and Sugeno, M. (1985). “Fuzzy Identification of Systems and Its Applications to Modeling and Control.” IEEE Transactions on Syst., Man, and Cybernetics, 15(1), 116132. Tanaka, K. and Sano, M. (1994). “A Robust Stabilization Problem of Fuzzy Control Systems and its Application to Backing up Control of a Truck-trailer.” IEEE Trans. Fuzzy Syst., 2(2), 119-134. Tanaka, K. and Sugeno, M. (1992). “Stability Analysis and Design of Fuzzy Control Systems.” Fuzzy Sets and Syst., 45(2), 135-156. Tani, A., Kawamura, H., and Ryu, S. (1998). “Intelligent Fuzzy Optimal Control of Building Structures.” Eng. Struct. 20(3), 184-192. Tsang, H.H., Su, R.K.L., and Chandler, A.M. (2006). “Simplified Inverse Dynamics Models for MR Fluid Dampers.” Eng. Struct. 28(3), 327-341. Tse, T. and Chang, C.C. (2004). “Shear-Mode Rotary Magnetorheological Damper for SmallScale Structural Control Experiments.” J. Struct. Eng., 130(6), 904-911.

Identification and Control of Large Smart Structures

359

Wang, A.P and Lee, C.D. (2002). “Fuzzy Sliding Mode Control for a Building Structure based on Genetic Algorithms.” Earth. Eng. Struct. Dyna., 31(4), 881-895. Wang, H.O., Tanaka, K., and Griffin, M. (1995). “An Analytical Framework of Fuzzy Modeling and Control of Nonlinear Systems: Stability and Design Issues.” Proc., the 1995 American Control Conf., Seattle, WA, Proc., 2272-2276. Wang, L. and Langari, R. (1996). “Complex Systems Modeling via Fuzzy Logic.” IEEE Trans. Syst. Man. Cybernetics-Part B: Cybernetics, 26(1), 100-106. Wen, Y.K (1976). “Method for Random Vibration of Hysteretic Systems.” J. Eng. Mech., 102(2), 249-263. Xu, B., Wu, Z.S., and Yokoyama, K. (2003). “Neural Networks for Decentralized Control of Cable-stayed Bridge.” J. Bridge Eng., 8(4), 229-236. Yan, G. and Zhou, L.L. (2006). “Integrated Fuzzy Logic and Genetic Algorithms for MultiObjective Control of Structures Using MR Dampers.” J. Sound and Vibration, 296(1-2), 368-382. Yang, G., Spencer, B.F. Jr., Carlson, J.D., and Sain, M.K. (2002). “Large-scale MR Fluid Dampers: Modeling and Dynamic Performance Considerations.” Eng. Struct., 24(3), 309323. Yang, J.N. (1982). “Control of Tall Building under Earthquake Excitation.” J. Eng. Mech. Div., 108(EM5), 833-849. Yang, J.N., Akbrapour, A., and Ghaemmaghami, P. (1987). “New Optimal Control Algorithms for Structural Control.” J. Eng. Mech., 113(9), 1369-1386. Yen, J. and Langari, R. (1999). Fuzzy Logic-Intelligence, Control, and Information, Prentice Hall, Upper Saddle River, NJ. Yoshida, O. and Dyke, S.J. (2004). “Seismic Control of a Nonlinear Benchmark Building Using Smart Dampers.” J. Eng. Mech., 130(4), 386-392. Zadeh, L.A. (1965). “Fuzzy Sets.” Information and Control, 8(3), 338 – 353. Zhou, L., Chang, C.C., and Wang, L.X. (2003). “Adaptive Fuzzy Control for Nonlinear Building-Magnetorheological Damper System.” J. of Struct. Eng., 129(7), 905-913.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 361-439

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 5

TRANSFER PROCESSES IN A HEAT GENERATING GRANULAR BED Yu.S. Teplitskii and V.I. Kovenskii A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus

Abstract Heat generating granular beds are practically an important type of disperse systems (beds of nuclear fuel microcells of atomic power stations, beds of solid fuel particles in layer burning, heat generating beds of biological origin, etc.). Heat, generated in solid particles, produces temperature fields of specific character in the system; it is influenced by a whole number of factors: heat release intensity, heat carrier filtration velocity, size of particles, etc. A full system of boundary conditions and the conjugation conditions for the temperatures of phases and pressure that take into account the preliminary heating of a heat carrier and the degree of its turbulence at the bed exit have been formulated. Using the analogy between the processes of convective heat and mass transfer, dependences for calculating the effective coefficients of the thermal conductivity of a granular bed have been obtained. The influence of heat release and gas compressibility on the granular bed resistance has been elucidated, and an engineering technique of its calculation has been worked out. Based on the analysis of the trends in the near-wall hydrodynamics of the bed, the dependence is obtained to calculate the coefficient of heat exchange of a granular bed with the environment that takes into account the near-wall thermal resistance; it has been verified in a wide range of Reynolds numbers. Within the framework of the two-temperature model of an infiltrated granular bed, basic laws governing steady heat and mass transfer over the space of a bed have been investigated for various types of heat generating. The criterion of “quasi-homogeneity” was introduced. It allows one, on the basis of the operative conditions, to estimate the thermal state of the two-phase system. Mathematical simulation of the process of filtrational evaporative cooling of a heat releasing bed has been carried out. An engineering method of calculation of bed cooling has been developed making it possible to determine the position and size of the evaporation zone. Unsteady processes of the propagation of compression and expansion waves originating on a instantaneous increase (decrease) in the gas pressure at the granular bed inlet were investigated. Equations in a dimensionless form have been obtained to calculate a maximum

362

Yu.S. Teplitskii and V.I. Kovenskii and minimum temperature of a heat carrier, as well as the time of establishment of a new stationary state.

Notation B – radiation attenuation index: B = BH d ; *

Bi = K t R λf0 – Biot number; cf – heat carrier specific heat capacity at constant pressure, J/(kg⋅K) сv – gas specific heat capacity at constant volume, J/(kg⋅K); cliq , cv – liquid and vapor specific heat capacities at constant pressure, J/(kg·K); (division 6); cs – particle specific heat capacity, J ⁄ (kg⋅K); d – particle diameter, m;

( H μ ) ; D = (( p

D = patm d 3 ρf

2 f

0

0

− patm ) H ) d 3 ρ f0 μf02 ;

D0I =

3 p0 − patm d 3 ρ v0 p0 − patm d ρliq III ; ; D = 0 H μ v02 H μliq2 0

DI =

3 patm d 3 ρ f patm d 3 ρ v patm d ρliq II III ; = (division 6); = ; D D 2 H μf2 H μ v2 H μliq

I Er

D

(1 − ε ) = 150

DErIII = 150

ε

2

(1 − ε ) ε

Re0I + 1.75

3

3

(1 − ε )

2

Re0III + 1.75

ε

3

( Re ) ;

(1 − ε ) ε

3

I 2 0

( Re )

III 2 0

;

D – effective diffusion coefficient, m2 ⁄ sec (division 2);

Df0 – heat carrier molecular diffusion coefficient, m2 ⁄ sec; H – granular bed length, m; i – water-steam mixture enthalpy, J/kg; iliq – enthalpy of liquid saturation, J/kg; iv – enthalpy of vapor saturation, J/kg; J f = ρ f u – heat carrier mass flux, kg ⁄ (m2·sec);

J f0 = J f2 d 3 ( ε 2 H μf02 ) ; J f = J f2 d 3 ( ε 2 H μf2 ) ;

⎛ d 0.345R δ t K , K = 1 ⎜ + + ⎜ 10 ( λf0 + 0.0061cf ρf u∞ d ) ( λ r ) λt ⎝ ∞

⎞ 2 ⎟ –heat-transfer coefficients, W/(m ⋅K); ⎟ ⎠

K t = λt δ t , W/(m2⋅K); K 0 – permeability, m2; k , k* – filtration thermal boundary layer and thermal sublayer thickness, m;

Transfer Processes in a Heat Generating Granular Bed L – specific heat of vaporization, J/kg; l0 – heat carrier film near the heat-transfer surface thickness, m n – number of elementary layers in the pile;

αwd α d α Id α III d III I ; Nusselt numbers: Nu = 0 , Nu w = 0 , Nu = 0 , Nu = λv0 λf λf λliq p – pressure, kg/(m⋅sec2); p′ =

p ; p′ = ( p − patm ) patm

( p0 − patm ) (division 7);

p0 – inlet pressure, Pa; Δp = p0 − patm ;

P=

p0 − patm  patm H ; P= ; cf ρ f 0T0 cf ρf 0T0 d

Peclet numbers:

Pe = cf J f d ( 6α H (1 − ε ) ) , Pe∞ = cf ρf u∞ H λf0 ;

PefI = Pe I = PesI = Pef =

Pef =

cliq J f H

; Pef = III

ελ

I f

cliq J f d

6α (1 − ε ) H I

cliq J f H

(1 − ε ) λ

I s

cv J f H

ελfIII

; Pe

; Pes = III

III

=

;

cv J f d ; 6α (1 − ε ) H III

cv J f H ; (1 − ε ) λsIII

cJ H cf ρf0 v* d cs ρs v* d x s ; Pe = ; Pe = f fx ; λ 6 (1 − ε ) α H 6 (1 − ε ) α H cf J f H

ελfx

; Pef =

cf ρ f 0 Hv*

ελfx

(division 7);

cs ρs v* H cf J f H Pes = ; Pes = (division 7); (1 − ε ) λsx (1 − ε ) λsx Pr = cf μf λf0 – Prandtl number; Pr I , Pr III – Prandtl numbers for liquid and vapor;

q – emission incident flow, W/m2; q = 

q ρf0 q ; q = ; cf J f T0 J f patm

Q – heat release power, W ⁄ m3; Q = Q (1 − ε ) H

( cf J f T0 ) ;   III  Q (1 − ε ) H Q (1 − ε ) H Q (1 − ε ) H QI = ;Q = ; QL = ; Jf L cliq J f (Tsat ( patm ) − T0 ) cv J f (Tsat ( patm ) − T0 )

363

364

Yu.S. Teplitskii and V.I. Kovenskii

QH =

Wd – quasi-homogeneity criterion; cf J f T0

r – radial coordinate (across the heat carrier flow), m; r ′ = r R ;

rt – reflection coefficient of an elementary layer of the pile; Rb – reflection coefficient of an infinitely thick pile; R* – gas constant, m2/(sec2·K); R – granular bed radius, m; Reynolds numbers:

ρf0 v* H Re0 = J f d μf0 ; Re = J f d μf ; Re = (division 7); Re∞ = u∞ d ν f ; μf0 Recr = ucr d ν f ; Resd = usd d ρf0 μf0 ; Re0I =

Jf d

μliq0

; Re = I

Jf d

μliq

; Re = II

Jfd

μf

; Re0 = III

Jf d

μ v0

; Re

III

=

Jfd

μv

;

S – bed cross-section, m2; Sin – specific surface of particles (interphase surface per unit volume of bed) in the case

of a packing of spheres Sin = 6 (1 − ε ) d , 1 / m; Stanton numbers: St 0 =

α0 cf J f

, St H =

αH cf J f

;

t – time, sec; t ′ = t t ; *

T – temperature, K; Tf, Ts – heat carrier and particles temperature, K; Т0 – inlet heat carrier temperature, K; T* – temperature at the outer surface of the near-wall zone, K

Tw – tube outer surface temperature, K T0′ – heat carrier temperature for x → −0 , К; Tsat – saturation temperature, K;

T – cross-section-mean temperature at x = const , К; Tu – turbulence degree; u – heat carrier superficial velocity, m/sec; u∞ – heat carrier superficial velocity in the bed core, m/sec; u ′ = u u ∞ ;

ucr – critical velocity, m/sec; v – heat carrier local velocity ( v = u

ε ) , m ⁄ sec; v′ = v v* ;

vw – wave velocity, m ⁄ sec; vsd =

γ patm – sound velocity at atmospheric pressure and temperature T0 , m ⁄ sec; ρ f0

Transfer Processes in a Heat Generating Granular Bed

365

v0 – heat carrier inlet velocity, m ⁄ sec; W – mean power of heat release, W/m3; х – longitudinal coordinate, m; xa – active section length, m;

x – mass flow rate vapor content; y = R − r , m; α – interphase heat exchange coefficient, W ⁄ (m2·K);

α 0 , α H – heat carrier – particle skeleton heat-transfer coefficients for x = 0, H , W/(m2·К);

α =

1 – heat-transfer coefficient, W/(m2·K); 1 1 − K Kt

α w – near-wall heat transfer coefficient, W/(m2·K); α1 and α 2 – coefficients in (3.5), 1/sec and 1/m, respectively; δ – filtration boundary layer thickness, m; δ* –filtration viscous sublayer thickness, m; δ t – tube wall thickness, m; ε – porosity; ε ∞ – bed core porosity; ε b – bed emissivity; ε t – absorption coefficient of an elementary layer of the pile; ε s – particle emissivity; γ = cf cv – adiabatic exponent; κ – bed absorption coefficient, 1/m;

Κ –apparent mass factor ( Κ = 0.5 ) ;

λ – thermal conductivity, W/(m·K); λfo – heat carrier molecular thermal conductivity, W/m К; λliq0 , λv0 – liquid and vapor molecular thermal conductivities, W/(m·K);

λf , λs – heat carrier and particle skeleton thermal conductivities, W/(m·К); λs – particle material heat conductivity, W ⁄ (m⋅K);

λr – bed radiative conductivity, W/(m⋅К); μf – heat carrier dynamic viscosity, kg/(m·sec); μliq , μv , μf – liquid, vapor, and a vapor-liquid mixture dynamic viscosity, kg/(m·sec);

366

Yu.S. Teplitskii and V.I. Kovenskii

μf 0 – heat carrier dynamic viscosity at atmospheric pressure and temperature T0, kg/(m·sec);

ν f – heat carrier kinematic viscosity, m2/sec; θf =

Tf − T0 T − T0 , θs = s –heat carrier and particles dimensionless temperatures; T0 T0

θf = (Tf − T0 ) (Tsat ( patm ) − T0 ) ; θs = (Ts − T0 ) (Tsat ( patm ) − T0 ) (division 6); θsat = (Tsat − T0 ) (Tsat ( patm ) − T0 ) (division 6); ρ f , ρ s – heat carrier and particle density, kg/m3; ρf′ = ρf ρf0 ; ρ liq , ρ v , ρ f – liquid, vapor, and a vapor-liquid mixture density, kg/m3; (division 6) ρf 0 – heat carrier density at atmospheric pressure and temperature T0 , kg/m3; ρ v 0 – vapor density at atmospheric pressure and T=373 K, kg/m3; ρ0 – heat carrier density at p = p0 Tf = T0 , kg/m3; σ – Stefan–Boltzmann constant, W⁄(m2·K4); σ b –scattering index, 1 ⁄ m;

τ t – transmission coefficient of an elementary layer of the pile; τ x – optical thickness; ξ = x H ; ξ1 = h1 H ; ξ 2 = h2 H ; Superscripts 0 – molecular; 0 – initial stationary state; I, II, III – numbers of zones; Mod – modified; r – across the heat carrier flow; х – along the heat carrier flow.

Subscripts atm – atmospheric; b – bed; с – conductive; conv – convective; с-с – conductive-convective; с-r – conductive–radiative; cr – critical;

Transfer Processes in a Heat Generating Granular Bed

367

eff – effective; f – heat carrier; H – at the outlet; liq – liquid; max – maximum; opt – optimal; r – radiative; s – solid particles; sat – saturation; sd – sound; st – stabilized; t – elementary (thin) layer of the pile (division 2); t – tube wall; v – at constant volume; v – vapor (division 6); w – wall; w – wave (division 7); 0 – at the inlet; ∞ – bed core.

Introduction In engineering, there are a large number of thermal processes that proceed in granular beds, where temperature drops between particles and heat carrier are to be taken into account, thus rejecting the representation of the bed as a homogeneous heat-conducting medium. In the first place, these are various unsteady-state processes of heating or cooling of the bed by a heat carrier flow, when the temperatures of the phases have insufficient time to equilibrate. Another example concerns the case of the occurrence of temperature drops in bed when a heat flux move opposite to the heat carrier (transpiration cooling of the surfaces of flying vehicles, blades of high-temperature heat carrier turbines, etc.). Finally, mention should be made of the processes of filtration cooling of a heat-generating granular bed that can be encountered in nuclear power plants (cooling of fuel microelements) under both standard and emergency conditions.

1. Basic Equations, Boundary and Conjugation Conditions for Problems of Heat Transfer in Granular Beds on the Basis of a Two-Temperature Model In order to describe the hydrodynamics and heat transfer inside a heat-generating granular bed under steady state conditions, the following system of equations is used:

368

Yu.S. Teplitskii and V.I. Kovenskii

cf J f

0=

6 (1 − ε ) α ∂Tf ∂T ⎞ 1 ∂ ⎛ ∂ ⎛ r ∂Tf ⎞ = ⎜ ελfx f ⎟ + (Ts − Tf ) , ⎜ rελf ⎟+ d ∂x ∂x ⎝ ∂x ⎠ r ∂r ⎝ ∂r ⎠

(1.1)

6 (1 − ε ) α ∂ ⎛ 1 ∂ ⎛ x ∂Ts ⎞ r ∂Ts ⎞ (Tf − Ts ) + Q (1 − ε ) , (1.2) ⎜ (1 − ε ) λs ⎟+ ⎜ r (1 − ε ) λs ⎟+ ∂x ⎝ ∂x ⎠ r ∂r ⎝ ∂r ⎠ d

(1 − ε ) μf u − 1.75 (1 − ε ) ρf u 2 , ∂v ∂p ρf v = − − 150 d ∂x ∂x ε3 d2 ε3

(1.3)

p = ρ f R*Tf .

(1.4)

2

Equation (1.4) is used for a gas heat carrier. The force of heat carrier friction against particles is presented in (1.3) on the basis of the Ergun equation [1]. The gas is assumed to be perfect. Radial heat carrier velocity is assumed small.

Boundary Conditions 1. Boundary condition at x = 0 (heat carrier). We will consider an elementary volume of the active bed dV = Sdx near the interface x = 0 (figure 1). The balance of the heat carrier heat fluxes is

( q1 − q0 ) S = α (Ts − Tf ) dSin .

(1.5)

With allowance for dSin = Sin dV = Sin Sdx , we obtain

q1 − q0 = α (Ts − Tf ) Sin dx .

(1.6)

As dx → 0 , Eq. (1.6) yields the unknown boundary condition

q1 = q0 . Since q0 = cf J f T0′ and q1 = cf J f Tf ( 0 ) − ελf

x

∂Tf ∂x

(1.7)

, boundary condition (1.7) at x = 0 x =0

takes the form

cf J f (Tf − T0′ ) = ελfx

∂Tf . ∂x

(1.8)

Transfer Processes in a Heat Generating Granular Bed

369

Equation (1.8) is the first Danckwerts condition [2] widely used in modeling the processes of transfer in granular beds which are considered as homogeneous media. In this connection, the following conclusion can be drawn, which is important in the context of this chapter: in formulating boundary conditions for the two-temperature model, the phases at the boundaries of the bed can be considered as isolated from each other because of the absence of the interface surface there. 2. Boundary condition at x = 0 (particles). With allowance for the transfer of the heat that enters the system together with the heat carrier (preliminary heating), the boundary condition at x = 0 has the form

λsx (1 − ε )

∂Ts = α 0 ( Ts − T0 ) . ∂x

(1.9)

Figure 1. Toward the derivation of the boundary condition for the heat carrier at x = 0.

Using the relation

λsx (1 − ε )

∂Ts ∂x

= cf J f (T0′ − T0 ) , we rearrange Eq. (1.8) as x =0

cf J f (Tf − T0 ) = ελfx

∂Tf ∂T + (1 − ε ) λsx s , ∂x ∂x

(1.10)

Condition (1.10) can be called the generalized Danckwerts condition. 3. Boundary condition at x = H (heat carrier). With allowance for the independence of the phases, we may use the second Danckwerts condition

∂Tf =0 ∂x

(1.11)

which states that the entire heat flux transferred by the heat carrier is equal to the convective one.

370

Yu.S. Teplitskii and V.I. Kovenskii

4. Boundary condition at x = H (particles). In the absence of a heat flux from the outside, we have

(1 − ε ) λsx A difference of the coefficient [3, 4],

(α

0

dTs = α H (Tf − Ts ) . dx

(1.12)

α H in (1.12) from α 0 , which is usually used in this condition

= 0.5cf J f Re0−0.5 Pr −0.6 [5]) , is caused by the following two circumstances. The

first circumstance is associated with using the quantity Tf ( H ) instead of the third-kind classical boundary conditions needed [6] quantity T∞ being the heat carrier temperature far from the bed outlet. In view of this temperature, (1.12) will be:

(1 − ε ) λsx

∂Ts = α H* (T∞ − Ts ) . ∂x

(1.13)

(1 − ε ) λsx

∂Ts = cf J f (Tf − T∞ ) ∂x

(1.14)

Using the relation

and substituting into (1.13) the quantity T∞ determined from (1.14) yields condition (1.12) with the coefficient

αH =

α H* α H*

1+

(1.15)

cf J f

The second circumstance resulting in a difference of

α H from α 0 is connected with using

α H* instead of α H0 in (1.13). This is attributed to the fact that the heat carrier flow issuing from the bed has a turbulence intensity Tu ≠ 0 , whereas the undisturbed flow at Tu = 0 . Using the relationship

(

α H0 is the case of heat transfer with

α H* and α H0 for air ( Pr = 0.72 ) [7]

α H* = α H0 1 + 0.09 ( Re x= H Tu ) a final formula is obtained for calculation of

0.2

)

,

(1.16)

αH

α H = ψα H0 ,

(1.17)

Transfer Processes in a Heat Generating Granular Bed

371

where the parameter ψ is determined as follows:

1 + 0.09 ( Re x=H Tu )

ψ=

1 + 0.09 ( Re x=H Tu ) 0

1+ αH At Re x=H << 1

0.2

ψ = 1.64 ( Re x=H )

0.5

(1.18)

0.2

cf J f (for

α H0 = 0.5cf J f ( Re x=H )

−0.5

Pr −0.6 [5]); at

Re x=H >> 1 ψ = 0.09 ( Re x=H Tu ) . 0.2

5. Boundary condition at r = 0.

∂Tf ∂Ts = = 0 (symmetry condition). ∂r ∂r

(1.19)

6. Boundary condition at r = R.

ελfr

∂T λ ∂Tf + (1 − ε ) λsr s = t (Tw − T ∂r ∂r δ t

),

(1.20)

where T = ε Tf + (1 − ε ) Ts

∂Ts = 0. ∂r

(1.21)

Condition (1.21) means that the wall contact heat conduction of particles is neglected. With regard to the known experimental fact [8] ε → 1 at r → R (division 3) and relation (1.21), condition (1.20) will be

λfr

∂Tf λt = (Tw − Tf ) . ∂r δ t

(1.22)

Conjugation Conditions These conditions are used in those cases when at some x = h1 the bed porosity

ε and

(or) particle diameter d change spasmodically. 1. Conjugation condition for pressure. To obtain the sought-after condition, we use Eq. (1.3). With allowance for J f = const , we have

372

Yu.S. Teplitskii and V.I. Kovenskii

∂ ⎛ Jf v + ⎜ ∂x ⎝ ε

2 ⎛ 1 − ε ) μf u ( (1 − ε ) ρf u 2 ⎞⎟ . ⎞ + p ⎟ = − ⎜ 150 1.75 ⎜ d2 d ⎟⎠ ε3 ε3 ⎠ ⎝

(1.23)

We will integrate Eq. (1.23) within the limits h1 − Δx , h1 + Δx : h1 +Δx

∂ ⎛ Jf v + ⎜ ∫ ∂x ⎝ ε h1 −Δx

2 h1 +Δx ⎛ 1 − ε ) μf u ( (1 − ε ) ρf u 2 ⎟⎞ dx . (1.24) ⎞ p ⎟ dx = − ∫ ⎜ 150 + 1.75 ⎜ d ⎟⎠ ε3 d2 ε3 ⎠ h1 −Δx ⎝

We note that as Δx → 0 , the integral on the right-hand side of Eq. (1.24) also tends to zero, since it contains the expression for the interface surfaces

ΔSi 6 (1 − ε i ) = Δx ( i = 1, 2 ) and S di

Eq. (1.24) yields the conjugation condition for pressure:

⎛v v ⎞ Δp = p2 − p1 = J f ⎜ 1 − 2 ⎟ . ⎝ ε1 ε 2 ⎠

(1.25)

Relation (1.25) represents the conservation condition for the total flux of heat carrier impulse. 2. Conjugation conditions for the heat carrier temperatures and their derivatives. We will use Eq. (1.1) representing the heat carrier enthalpy as I f = cf Tf = cvTf +

p

ρf

⎞ 6 (1 − ε ) α ∂ ⎛ ⎛ p⎞ x ∂Tf (Ts − Tf ) . ⎜⎜ J f ⎜ cvTf + ⎟ − ελf ⎟= ∂x ⎝ ⎝ ∂x ⎟⎠ ρf ⎠ d

(1.26)

Having performed, as before, the integration of Eq. (1.26), for Δx → 0 we obtain

⎛ p p ⎞ ∂T ∂T J f cv (Tf1 − Tf2 ) + J f ⎜ 1 − 2 ⎟ = ε 1λf1x f1 − ε 2λf2x f2 . ∂x ∂x ⎝ ρ f1 ρ f2 ⎠

(1.27)

As is seen, relation (1.27) represents the conservation condition for the total heat flux of the heat carrier, which can be written

J f cf (Tf1 − Tf2 ) = ε1λf1x

∂Tf1 ∂T − ε 2 λf2x f2 . ∂x ∂x

(1.28)

Subjected to cf − cv = R , a comparison of Eqs. (1.27) and (1.28) yields the sought-after *

condition for the temperature jump:

Transfer Processes in a Heat Generating Granular Bed

⎛ p p ⎞ 1 ΔTf = Tf2 − Tf1 = ⎜ 2 − 1 ⎟ * . ⎝ ρ f2 ρ f1 ⎠ R

373

(1.29)

We note that condition (1.29) is also determined from the equation of state of the perfect gas (1.4). Substituting ΔTf into Eq. (1.28) yields the condition for the jump of the derivatives:

ε 2λf2x

Jc ⎛ p p ⎞ ∂Tf2 ∂T − ε1λf1x f1 = f *f ⎜ 2 − 1 ⎟ R ⎝ ρ f2 ρ f1 ⎠ ∂x ∂x

(1.30)

3. Conjugation conditions for the temperatures of particles and their derivatives. It is evident that these are the conventional fourth-kind boundary conditions [6]:

Ts1 = Ts2 ;

(1 − ε1 ) λs1x

∂Ts1 ∂T = (1 − ε 2 ) λs2x s2 . ∂x ∂x

(1.31)

When equations (1.1)-(1.4) under the corresponding boundary conditions for particular calculations are used, it is of importance to have reliable information on the quantities

λfx ,

λfr , λsx , λsr , α , α 0 that are the parameters of the model.

2. The Thermal Conductivity of a Granular Bed Thermal conductivities of phases entering (1.2), (1.3) are determined using the relations for calculation of conductive-convective components of effective thermal conductivities of a granular bed [9]: r x λc-c = λc + 0.1cf ρ f ud , λc-c = λc + 0.5cf ρ f ud

(2.1)

These quantities are the parameters of the one-temperature (quasi-homogeneous) model

cf J f

∂T 1 ∂ ⎛ r ∂T ⎞ ∂ ⎛ x ∂T ⎞ = ⎜ rλ ⎟ + ⎜λ ⎟ + Q (1 − ε ) , ∂x r ∂r ⎝ ∂r ⎠ ∂x ⎝ ∂x ⎠

(2.2)

into which the two-temperature model (equations (1.2), (1.3)) is reduced, when the temperature difference of phases

(Tf

 Ts = T ) can be neglected. This circumstance

provides a basis for deriving two connection equations

λ r = ελfr + (1 − ε ) λsr , λ x = ελfx + (1 − ε ) λsx

(2.3)

374

Yu.S. Teplitskii and V.I. Kovenskii

It is difficult to calculate the radiative thermal conductivity because of the absence of simple commonly accepted recommendations similar to (2.1). The same is true for the thermalconductivity coefficients of the phases

λfr , λfx , λsr and λsx .

Coefficients of Conductive-Convective Thermal Conductivity of the Phases Two connection equations (2.3) are insufficient to find four thermal conductivities

r λf,c-c ,

x r x λf,c-c , λs,c , λs,c . Therefore, we must use additional model representations of the mechanism

of transition in the two-phase system. We believe that the approach based on the analogy of convective heat and mass transfer and adopted in [9] is the most substantiated. The existing dependences were used for determination of the diffusion coefficients of the heat carrier in a granular bed [9]:

D r = 0.3Df0 + 0.1ud , D x = 0.3Df0 + 0.5ud

(2.4)

In accordance with (2.4), it was taken that r λf,conv = 0.1cf ρ f

u

ε

x = 0.5cf ρ f d , λf,conv

u

ε

d

(2.5)

Equations (2.3) and (2.1) yielded

λs,cr = λs,cx = λc (1 − ε ) .

(2.6)

We note that thereafter Ae′rov et al. [9] illegitimately used expressions (2.5) as

r λf,c-c and

x λf,c-c . In view of the molecular thermal conductivity of the heat carrier and Eqs. (2.1)–(2.3),

the thermal-conductivity coefficients of the phases at low temperatures can be determined more correctly: r λf,c-c = λf0 + 0.1cf ρ f

u

ε

d , λs,cr = λs,cx = λs,c =

λc − ελf0 , x u λf,c-c = λf0 + 0.5cf ρ f d 1− ε ε

(2.7)

To calculate the conductive component of the unblown (for u = 0) bed, we have obtained, by processing the experimental data [10], the following compact formula:

λc ⎛ λs ⎞ =⎜ ⎟ λf0 ⎝ λf0 ⎠

⎛ λs ⎞ ⎟ 0⎟ ⎝ λf ⎠

−0.06

(1−ε )⎜⎜

.

(2.8)

Transfer Processes in a Heat Generating Granular Bed

375

Dependence (2.8) represents a more simple and accurate (mean error ~4% for ε = 0.35–0.48) approximation compared to the formula

(1 − ε ) (1 − λf0 λs ) λc = 1+ 0.63( λ λ ) λf0 λf0 λs + 0.28ε 0 f

(2.9)

−0.18

s

proposed in [10] and much cited in the literature (mean approximation error ~15%).

Coefficient of Radiative Thermal Conductivity of the Bed Skeleton A pile model [11, 12], allowing for the cooperative effects, was used as a tool of assessment of this parameter. The model was developed for calculation of the transfer of radiation in a homogeneous dispersive medium of optically large particles with their different concentration (from a packed bed to very rarefied systems 1 − ε = 5 ⋅ 10 ). Within the model’s framework, the system of opaque gray spherical particles (uniformly distributed in space) with a diffusely reflecting surface is represented as a set (pile) of parallel plates (elementary layers in terms of the model) characterized by the coefficients of reflection rt, transmission τt, and absorption εt. The above coefficients are determined using the auxiliary model [11] in which their calculation is reduced to solution of the problem on radiative transfer in a closed system formed by isothermal gray and black surfaces. When rt, τt, and εt are known, the reflection, transmission, and absorption coefficients of a pile of k elementary layers are calculated from the following recurrence formulas: −3

rn = rn −1 +

τn =

τ n −12 rt 1 − rn −1rt

τ n −1τ t 1 − rn −1rt

( r1 = rt ),

( τ 1 = τ t ),

ε n = 1 − rn − τ n ,

(2.10)

(2.11)

(2.12)

which allow for the multiple reflection of radiation between the elementary layers. The emissivity factor and the reflection coefficient of the surface of an infinitely thick pile are determined as

Rb = lim rn , ε b = 1 − Rb n →∞

since τ n → 0 for n → ∞ .

(2.13)

376

Yu.S. Teplitskii and V.I. Kovenskii

For description of the transfer of radiation in a packed bed one widely uses a continuous approximation [13] in which the disperse system is considered as a homogeneous scattering, absorbing, and radiating medium. In this case the bed is characterized by the indices of absorption κ , scattering σ b , and attenuation κ + σ b and by the optical thickness

τ x = (κ + σ b ) x , and the transfer equation is used to determine the radiant flux and the temperature profile. The appreciable optical thickness of the bed ( τ x >> 1 ) makes it possible to consider the propagation of radiation as its diffusion [13] and to characterize the transport properties of the medium by the radiative thermal conductivity

λr =

16 σ Ts3 . 3 κ +σb

As is seen from (2.14), assessment of the radiative thermal conductivity requires

(2.14)

κ and σ b

values which can be calculated from certain model representations of a disperse system (with the pile model in this case). Comparing the expressions for the reflectivity and transmissivity of the plane-parallel bed of a closely packed gray granular medium, which have been established within the framework of the pile model and in the diffusion approximation of the transfer equation [14], we can obtain the following formulas:

κd = −

(where a =

1 − Rb 2 Rb 1 + a2 −1 1 + a2 −1 ln ln , σ bd = − a 2 (1 + Rb ) a (1 − Rb2 )

(2.15)

2 Rbτ t ) making it possible to calculate the indices of absorption and scattering (1-Rb2 ) rt

and hence the radiative thermal conductivity (2.14). The results of these calculations can be approximated by the formula

λr =

16 ( 0.35 + 0.52ε s0.85 )σ Ts3d = 4 ( 0.47 + 0.7ε s0.85 ) σ Ts3d = (1.87 + 2.77ε s0.85 ) σ Ts3d . (2.16) 3

The mean approximation error amounts to 0.28% with a maximum error of 0.84%. A great many formulas for calculation of the radiative thermal conductivity that are analogous to (2.16) and have been obtained from different models of radiation transfer in the bed are presented in the literature. To compare different dependences we represent (2.16), following [15], in the form

λr = 4 χσ Ts3 d ,

(2.17)

where the factor χ reflects the model representations used in deriving the formula. Different expressions for this parameter are given in table 1.

Transfer Processes in a Heat Generating Granular Bed

377

Figure 2 plots the model parameters χ as functions of the emissivity factors of particles εs; the dependences correspond to the formulas from table 1. As is seen, the curve corresponding to (2.16) lies in the zone of "accumulation" of the dependences taken from the given literature. The dependence (12) from [16] is the closest to (2.16). It is significant that formula (2.16) is not empirical, and the pile model underlying it allows for the effect of collective interaction of all particles, characteristic of concentrated systems. Therefore, (2.16) can serve as a certain reference point in the great number of the given recommendations (table 1).

Figure 2. Model parameter χ vs. particles emissivity εs. The curve number corresponds to the formula number in table 1.

Table 1. Formulas for the Model Parameter χ Parameter χ

No.

Literature source

1

0.47 + 0.7ε s0.85

2

0.0305ε s

[17]

3

0.05ε s2

[18]

4

8 ε 16 = 9 1 − ε 27 ε =0.4

[19]

5

8 ε2 6.4 = 9 1 − ε 27 ε =0.4

[20]

6

2ε s

[21]

(2.16)

378

Yu.S. Teplitskii and V.I. Kovenskii Table 1. Continued Literature source

Parameter χ

No. 7

εs ε 2 = εs 1 − ε 3 ε =0.4

8

1 − (1 − ε ) + (1 − ε ) 1− ε 2/3

[22] 4/3

ε s = 1.325ε s

[23] ε = 0.4

9

εs

[24]

10

2ε s 2 − 0.264ε s

[25]

11

εs 2 − εs

12

13

[26, 27]

1

ε 1− εs 1+ 1 − ε 2ε s

=

1 2 1− εs 1+ 3 2ε s

[16] ε = 0.4

2 B + ε s (1 − B ) 0.2 + 0.9ε s = ( 2 − ε s )(1 − B ) 0.9 ( 2 − ε s ) ε =0.4

[28]

B = 0.1

14

εs + B 1− B

=

10 ( ε s + 0.1) ε = 0.4 9

[29]

B = 0.1

Coefficients of Effective Thermal Conductivity of the Bed Since

λr refers to the skeleton of particles, λ r and λ x with account for (2.1) can be

represented as

where

λ r = λc-r + 0.1cf ρf ud , λ x = λc-r + 0.5cf ρ f ud ,

(2.18)

λc-r = λc + (1 − ε ) λr .

(2.19)

The validity of formula (2.19) proposed for calculation of the conductive-radiative thermal conductivity of an unblown dense bed was checked by comparing to the experimental data presented in [15, 30–32]. The packed-bed parameters corresponding to experimental

Transfer Processes in a Heat Generating Granular Bed

379

conditions are given in table 2. We processed ~430 experimental points. A comparison of the experimental data to the calculations from formula (2.19) is presented in figure 3. The mean deviation of the calculated values from those experimental amounted to ~15% with a maximum deviation of 58%.

Figure 3. Comparison of experimental data to the calculation from formula (2.19). Marker number corresponds to the line number in table 2.

λc-r , W ⁄ (m·K).

Thermal-Conductivity Coefficients of the Heat Carrier and the Bed Skeleton With account for λr and (2.7), these quantities are determined as follows: r λfr = λf,c-c = λf0 + 0.1cf ρ f

u

ε

x d , λfx = λf,c-c = λf0 + 0.5cf ρ f

λsr = λsx = λs,c-r =

u

ε

d,

λc − ελf0 + λr 1− ε

(2.20)

Table 3 gives systematized results of determination of the effective thermal-conductivity coefficient of a blown granular bed. Table 2. Granular Bed Characteristics Used in the Experiments Literature source

Solid–gas

1

[30]

Alumina– air

0.36

2.77

2

[30]

Alumina– air

0.38

6.64

3

[30]

Alumina– air

0.37

0.96

No.

ε

d, mm

εs

λs , W ⁄ (m·K)

0.83 − 4 ⋅ 10−4 ×

3.6 − 3 ⋅10−3 ×

× (T − 300 )

× (T − 300 )

0.83 − 4 ⋅ 10−4 ×

3.6 − 3 ⋅10−3 ×

× (T − 300 )

× (T − 300 )

0.83 − 4 ⋅ 10−4 ×

3.6 − 3 ⋅10−3 ×

× (T − 300 )

× (T − 300 )

Tmin - Tmax ,

K 350 - 1250 350 - 1250 350 - 1250

380

Yu.S. Teplitskii and V.I. Kovenskii Table 2. Continued Tmin - Tmax ,

No.

Literature source

Solid–gas

ε

d, mm

εs

4

[30]

Aluminiu m–air

0.37

3.24

0.043 + 3.4 ×

220 − 0.036 ×

× (T − 300 ) ⋅ 10−5

× (T − 300 )

5

[30]

Aluminiu m–air

0.38

6.33

0.043 + 3.4 ×

220 − 0.036 ×

× (T − 300 ) ⋅ 10−5

× (T − 300 )

6

[30]

Glass–air

0.37

2.85

0.95 − 6 ⋅ 10−4 ×

1.2 + 6 ⋅ 10−4 ×

× (T − 300 )

× (T − 300 )

7

[30]

Glass–air

0.37

6.0

0.95 − 6 ⋅ 10−4 ×

1.2 + 6 ⋅ 10−4 ×

× (T − 300 )

× (T − 300 )

8

[30]

Glass–air

0.39

13.5

0.95 − 6 ⋅ 10−4 ×

1.2 + 6 ⋅ 10−4 ×

× (T − 300 )

× (T − 300 )

9

[31]

Alumina– air

0.4

6.0

0.83 − 4 ⋅ 10−4 ×

3.6 − 3 ⋅10−3 ×

× (T − 300 )

× (T − 300 )

10

[31]

Steel–air

0.4

11.0

0.8

11

[31]

0.4

0.18

0.54

2.5

440 - 930

12

[31]

0.4

2.6

0.54

2.5

600 - 1110

13

[31]

0.4

3.6

0.54

2.5

540 - 1060

14

[31]

0.4

5.0

0.54

2.5

600 - 1130

15

[15]

Alumina– air

0.48

0.37

0.83 − 4 ⋅ 10−4 ×

3.6 − 3 ⋅10−3 ×

× (T − 300 )

× (T − 300 )

16

[15]

Sand–air

0.43

0.45

0.63

× (T − 550 ) ⋅ 10−3

17

[15]

Sand–air

0.43

0.65

0.63

× (T − 550 ) ⋅ 10−3

650 - 1200

18

[32]

Aluminiu m–air

0.4

50.0

0.1

167

300

19

[32]

Copper– helium

0.4

50.0

0.1

387

300

Cement clinker– air Cement clinker– air Cement clinker– air Cement clinker– air

λs , W ⁄ (m·K)

45 − 2 ⋅10−2 × × (T − 473)

4.65 − 3.4 ×

4.65 − 3.4 ×

K 340 - 660 320 - 670 350 - 600 330 - 600 360 - 590 310 - 950 413 - 1113

700 - 1250 620 - 1150

Transfer Processes in a Heat Generating Granular Bed

381

Table 3. Effective Thermal-Conductivity Coefficients of a Granular Bed Constitutive formula One-temperature model ⎛ λs ⎞ 0⎟ ⎟ ⎝ λf ⎠

−0.06

(1−ε )⎜⎜

⎛ λs ⎞ λc = λ ⎜ 0 ⎟ ⎝ λf ⎠ λc-r = λc + λr (1 − ε ) 0 f

Coefficient

conductive thermal conductivity of an unblown bed

conductive-radiative thermal conductivity of an unblown bed conductive-convective bed thermal r λc-c = λc + 0.1cf ρf ud conductivity across the flow conductive-convective bed thermal x λc-c = λc + 0.5cf ρf ud conductivity along the flow effective bed thermal conductivity across r r λ = λc-r + 0.1cf ρf ud = λc-c + λr 1 − ε the flow effective bed thermal conductivity along x x λ = λc-r + 0.5cf ρf ud = λc-c + λr 1 − ε the flow Two-temperature model

( (

r λf,c-c = λf0 + 0.1cf ρ f x λf,c-c = λf0 + 0.5cf ρf

λs,c

(λ =

c

− ελf0 )

u

ε u

ε

d = λfr d = λfx

(1 − ε )

16 0.35 + 0.52ε s0.85 )σ Ts3d ( 3 λs,c-r = λs,c + λr = λsr = λsx

λr =

) )

conductive-convective heat carrier thermal conductivity across the flow conductive-convective heat carrier thermal conductivity along the flow bed skeleton conductive thermal conductivity bed skeleton radiative thermal conductivity

bed skeleton conductive-radiative thermal conductivity Connection of the one- and two-temperature models

r λ r = ελf,c-c + (1 − ε ) λs,c-r x λ x = ελf,c-c + (1 − ε ) λs,c-r

effective bed thermal conductivity across the flow effective bed thermal conductivity along the flow

3. Heat Transfer in an Infiltrated Granular Bed at Moderate Reynolds Numbers As is known heat transfer in the infiltrated granular bed has a number of special features compared to the one-phase medium. The main of them are the anisotropy of the heat conduction coefficient, its dependence on the particle diameter and the rate of filtration,

382

Yu.S. Teplitskii and V.I. Kovenskii

substantial difference between the structural and transport characteristics in the bed core and the region adjacent to the macrosurface bounding the bed. The joint effect of these factors imparts a specific character to the processes of heat and mass transfer, which is inherent only in a granular bed.

Distinctive Features of Filtration in a Granular Bed Near the Wall Heat carrier flow in a granular bed near the wall is known to possess a number of distinctive features [8, 9]. Of them, the main one is a pronounced increase in the local velocity. A combination of properties of wall flow, which is related to the existence of the increased-velocity zone, is called the bypassing effect or “The wall effect” in the literature. In most cases this phenomenon impairs the operation of granular-bed apparatuses, since motion with an off-design velocity occurs in the wall region and an additional thermal resistance is created. The character of change in the local velocity is determined, in many respects, by the ordering action of the wall on the structure of adjacent particle layers. This results in a special distribution of the local velocity in the wall region, which is also related to the method of charging a dispersed material, particle shape, dimension of the apparatus, and others. In all cases we have a velocity maximum near the wall on a linear scale of the order of the particle diameter. The influence of the above factors on wall flow has mainly been investigated in the regimes of linear filtration [8]. We seek to establish a relationship between the wall velocity of the heat carrier and the local porosity distribution and to develop a procedure for calculating the profiles of this velocity in different regimes of flow (including nonlinear filtration).

Local-Porosity Distribution Near the Wall Random packings near the wall are known [8, 9] to be characterized by increased porosity values related to its ordering influence. In [33], experimental data obtained were generalized for particles of a spherical shape

⎛

π

⎝

3.5

ε = ε ∞ + ⎜1 − ε ∞ −

sin

πy ⎞

⎛ y⎞ exp ⎜ − ⎟ ⎟ d ⎠ ⎝ d⎠

(3.1)

and an arbitrary shape

⎛

ε = ε ∞ + ⎜1 − ε ∞ − ⎝

πy⎞⎞ y⎞ ⎛ ⎛ sin ⎜ 0.8 ⎟ ⎟ exp ⎜ −2.3 ⎟ 5 d ⎠⎠ d⎠ ⎝ ⎝

π

(3.2)

Based on (3.1) and (3.2), we propose a simple dependence

⎛ ⎝

ε = ε ∞ + (1 − ε ∞ ) c o s ⎜ 2

π y⎞

y⎞ ⎛ ⎟ e x p ⎜ − 1 .5 ⎟ , d ⎠ d ⎠ ⎝

(3.3)

Transfer Processes in a Heat Generating Granular Bed

383

for a bed of spherical particles; it satisfactorily describes experimental data (figure 4) and represents a smooth function convenient for subsequent analysis.

Models of Flow in the Granular Bed To describe flow in the granular bed one uses filtration theory based on Darcy’s law which in the simplest case has the form

−

∂p = αρ f u . ∂x

(3.4)

The coefficient α is given by the expression [8]

α = α1 + α 2u ,

(3.5)

which determines different filtration regimes. When the flow velocities are small, we have α ≈ α1 = const and (3.4) becomes the equation of linear filtration −

∂p = α 1ρ fu . ∂x

(3.6)

Under these conditions, use is also made of a very popular Brinkman model [34] which represents a generalization of Eq. (3.6):

−

∂p 1 ∂ ⎛ ∂u ⎞ . = α 1ρ f u − μ f ⎜r ⎟ ∂x r ∂r ⎝ ∂r ⎠

(3.7)

Figure 4. Distribution of the local porosity of a bed of spherical particles near the wall: 1 – d = 0.72 mm, 2 – 0.82, 3 – 1.3, 4 – 1.7, 5 – 2.2, 6 – 2.6, 7 – eq. (3.3), (1–5 – [33] and 6 – [9]).

384

Yu.S. Teplitskii and V.I. Kovenskii When the velocities are large, we have

α ≈ α 2u and (3.4) changes to the equation of

nonlinear filtration

−

∂p = α 2 ρf u 2 . ∂x

(3.8)

In practice, (3.4) is frequently used in the form of the Ergun empirical formula [1]

(1 − ε ) μf u + 1.75 1 − ε ρf u 2 , ∂p − = 150 ε3 d2 ε3 d ∂x 2

(3.9)

well describing the pressure drop in granular beds in a wide range of experimental conditions. We note that, with account for (3.9), the Brinkman equation (3.7) can be represented in the form

(1 − ε ) μf u − μ 1 ∂ ⎛ r ∂u ⎞ ∂p − = 150 f ⎜ ⎟ d2 r ∂r ⎝ ∂r ⎠ ∂x ε3 2

(3.10)

Model of Flow in the Wall Zone To formulate such a model we must primarily consider the question of the value of the heat carrier velocity on a solid wall. The experimental data available in the literature do not provide an unambiguous answer to this question and only a comparative qualitative analysis of different filtration-flow models is possible at the moment. The simplest assumption that u ( 0 ) = 0 is due to the influence of viscosity. However,

with allowance for the fact that distances comparable to the particle diameter are eliminated from macroscopic consideration of Eqs. (3.4) and (3.6) – (3.9), the slip

condition u ( 0 ) = uw ≠ 0 [8] is also acceptable on the wall. The latter is consistent with the fact that no sticking condition is set for the averaged motion on the particles within the bed. Clearly, in this case exterior walls are not fundamentally different from interior ones, i.e., particles. Taking account of the comments made, we analyze the possibility of using Eqs. (3.4) and (3.6)–( 3.9) for description of wall flow. The Brinkman equation (3.7) (or (3.10)) contains a viscous term and enables us to set the sticking or slip condition. A substantial drawback is that one can use it only under linear-filtration conditions. Equation (3.4) and its particular cases (3.6), (3.8), and (3.9) yield u ( 0 ) → ∞ with account for

ε ( 0 ) = 1 for a finite,

nonzero, pressure gradient. For regularization of the problem we can use the method of averaging

ε ( y ) [8]. Another method of regularization is allowance for the heat carrier

viscosity in these equations, which enables us to set the required boundary conditions. In the case of linear filtration this is implemented using the Brinkman equation (3.7) or (3.10). In the

Transfer Processes in a Heat Generating Granular Bed

385

general case such a regularization of the problem can be carried out based on the equation [35]

−

∂p 1 ∂ ⎛ ∂u ⎞ = αρ f u − μf ⎜r ⎟ , ∂x r ∂r ⎝ ∂r ⎠

(3.11)

which, with account for (3.9), will have the form

(1 − ε ) μf u + 1.75 (1 − ε ) ρf u 2 − μ 1 ∂ ⎛ r ∂u ⎞ . ∂p − = 150 f ⎜ ⎟ ∂x ε3 d2 ε3 d r ∂r ⎝ ∂r ⎠ 2

(3.12)

As is seen, (3.12) represents a generalization of the Brinkman equation (3.10) to the case of nonlinear filtration by superposition of the Ergun equation (3.9) and the equation for viscous flow in the channel. For small velocities (linear filtration), it is reduced to the Brinkman equation (3.10), whereas for large velocities (nonlinear filtration) (3.12) takes the form

−

(1 − ε ) ρf u 2 − μ 1 ∂ ⎛ r ∂u ⎞ . ∂p = 1.75 f ⎜ ⎟ d r ∂r ⎝ ∂r ⎠ ∂x ε3

(3.13)

The structure of (3.13) yields that the influence of the viscous term must be pronounced in a narrow wall (boundary) layer where we have large velocity gradients because of the deceleration of the liquid. The main role of this term is to ensure regularization of flow for y → 0 . In the remaining part of the layer, (3.13) is equivalent to Eq. (3.8).

Calculation of Flow in the Wall Region Calculations of u u∞ in the case of linear filtration were performed from the Darcy equation (3.6) and the Brinkman equation (3.10) with the sticking condition on the wall

u ( 0 ) = 0 . The value of the pressure gradient was calculated from the formula

(1 − ε ∞ ) μf u∞ . ∂p − = 150 d2 ε ∞3 ∂x 2

(3.14)

In accordance with this, the velocity profile, according to the Darcy equation, was determined by the expression

u (1 − ε ∞ ) ε3 = . 2 u∞ ε ∞3 (1 − ε ) 2

(3.15)

386

Yu.S. Teplitskii and V.I. Kovenskii

The velocity distribution in the case of the Brinkman equation (3.10) was found by numerical solution of the boundary-value problem

(1 − ε ∞ ) 150 ε ∞3

2

(1 − ε ) − 150

2

ε3

u u∞

2 u ⎛ d ⎞ 1 ∂ ⎛ ∂ ⎛ u ⎞⎞ +⎜ ⎟ ⎜ r′ ⎜ ⎟⎟ = 0 , u∞ ⎝ R ⎠ r ′ ∂r ′ ⎝⎜ ∂r ′ ⎝ u∞ ⎠ ⎠⎟

=0, r ′=1

(3.16)

∂u = 0. ∂r ′ r′=0

Figure 5 gives results of calculating the corresponding u u∞ values. It is seen that the solutions of (3.15) and (3.16) are coincident everywhere, in practice, except for the narrow wall region of thickness of the order d 4 . In the general case flow in the wall region was calculated within the framework of the Ergun equation (3.9) and the generalized Brinkman equation (3.12). The pressure gradient was prescribed in this case by a formula analogous to (3.14):

(1 − ε ∞ ) μf u∞ + 1.75 1 − ε ∞ ρf u∞2 . ∂p = 150 d2 d ε ∞3 ε ∞3 ∂x 2

−

(3.17)

The velocity profile determined by the Ergun equation was calculated from a formula yielded by (3.9): 2 ⎛ ⎛ (1 − ε ∞ ) Re + 1.75 1 − ε ∞ Re2 ⎞⎟ ⎞⎟ . u 150 (1 − ε ) ⎜ 7ε 3 150 = −1 + 1 + ⎜ ∞ ∞ 3 ⎟⎟ u∞ 3.5 Re∞ ⎜ ε ∞3 ε ∞3 1502 (1 − ε ) ⎜⎝ ⎠⎠ ⎝

(3.18)

Figure 5. Calculation of wall flow under linear-filtration conditions: 1 – from the Darcy equation (3.6); 2 – from the Brinkman equation (3.10).

Transfer Processes in a Heat Generating Granular Bed

387

The velocity distributions described by the generalized Brinkman equation (3.12) were found numerically by solution of the problem

(1 − ε ∞ ) 150 ε ∞3

2

+ 1.75

1− ε∞

ε ∞3

Re∞

(1 − ε ) − 150 ε3

2

⎛ u ⎞ ⎛ u ⎞ 1− ε ⎜ ⎟ − 1.75 3 Re∞ ⎜ ⎟ + ε ⎝ u∞ ⎠ ⎝ u∞ ⎠

2 ⎛d ⎞ 1 ∂ ⎛ ′ ∂ ⎛ u +⎜ ⎟ ⎜r ⎜ ⎝ R ⎠ r ′ ∂r ′ ⎜⎝ ∂r ′ ⎝ u∞

u u∞

=0, r ′=1

⎞⎞ ⎟ ⎟⎟ = 0 , ⎠⎠

(3.19)

∂u = 0. ∂r ′ r′=0

Figure 6 shows the wall-velocity profiles calculated from (3.18) and (3.19) for different

Re∞ numbers. As is seen, these solutions tend to converge with growth in Re∞ . As has been noted above, for large Re∞ numbers, the solutions of (3.18) and (3.19) are coincident everywhere, except for the narrow wall zone of thickness (0.1–0.15)d. Figure 7 compares the solutions of the Brinkman equation (3.16) and the generalized Brinkman equation (3.19). A trend for decreasing relative-velocity variations with growth in Re ∞ is observed. It is noteworthy that this fact has long been known in the literature and it is common practice to use it for explanation of the significant difference in the “effective” and “true” coefficients of interphase heat and mass exchange in granular beds [10].

Comparison with Experimental Data Figures 6 and 7 compare the calculated and experimental [36] values of the local heat carrier velocity in the wall region that has been obtained using a laser Doppler velocimeter. Despite the limited number of experimental points, we observe their good agreement; this confirms the correctness of the procedure proposed for calculation of the distributions of the heat carrier velocities that is based on the use of the generalized Brinkman equation (3.12).

Figure 6. Calculation of wall flow for different Reynolds numbers: a –

Re∞

= 50, b – 100; 1 – from

the solution of the Ergun equation (3.18); 2 – from the solution of the generalized Brinkman equation (3.19); points – experimental data [36].

388

Yu.S. Teplitskii and V.I. Kovenskii

Figure 7. Wall-velocity distribution under sticking conditions on the wall: a –

Re∞

= 10, b – 50 and c

– 100; 1 – from the solution of the Brinkman equation (3.16); 2 – from the solution of the generalized Brinkman equation (3.19); points – experimental data [36].

The experimental u u∞ values and those calculated from the Brinkman equation (3.16) and the generalized Brinkman equation (3.19) are compared in figure 7. As is seen, the generalized Brinkman equation, reflecting the evolution of the relative-velocity distribution with change in the Re0 number, enables us to describe experimental data much better than the classical Brinkman equation.

Transfer Processes in a Heat Generating Granular Bed

389

Heat Transfer Model Special features of heat transfer in the wall region, which greatly affect heat transfer, were accounted for in different ways: ((a) By introducing near the wall an effective heat carrier interlayer with thickness l0 and thermal conductivity λeff dependent on the rate of filtration [37, 38]. An analysis of experimental data on heat exchange of the bed with the surface within the framework of this two-layer model allowed one to obtain the following expressions for the interlayer parameters [38]: l0 = 0.1d , (3.20)

λeff = Aλf0 + 0.0061cf ρf u∞ d ,

(3.21)

where A = 1.6 (heat-conducting particles) and A = 1 (non-heat-conducting particles). The near-wall coefficient of heat transfer follows from (3.20) and (3.21) as

αw =

λeff l0

=

10 ( Aλf0 + 0.0061cf ρ f u∞ d ) d

;

(3.22)

b) By using a two-layer model in which the near-wall zone was presented in the form of an infinitely thin layer with a finite thermal resistance [9, 39, 40]. With such an approach, the heat conduction equation was solved at the boundary condition of the III kind

−λ r

∂T ∂r

= α w (T* − T ) .

(3.23)

r=R

α w was found from the comparison of model and experimental values of the heat-transfer coefficient α . It should be noted that the

The value of the near-wall coefficient of heat transfer

structure of (3.23) logically disagrees with an assumption on an infinitely thin near-wall layer where two different temperatures T* and T are possible. (c) The Mukhin−Smirnova model [41] it was assumed that considerable thermal resistance is concentrated near the wall in the region of a sharp temperature gradient where molecular heat transfer is realized. Assuming the thicknesses of the hydrodynamic and thermal boundary layers being equal, to calculate them we used the expression

(

δ = k = Recr ν f K 0

)

u∞ d ,

(3.24)

where Recr played the role of an adjusting empirical parameter. As is seen from the above analysis, the main feature of the mentioned models of heat transfer is their empirical character that is expressed by use of the effective parameters l0 ,

390

Yu.S. Teplitskii and V.I. Kovenskii

α w and Recr which are introduced a priori and are determined by the experimental data on the values of the heat-transfer coefficient. This indicates that being obtained in such a way they have a limited application region and cannot be justifiably used in calculations of heat transfer under new conditions. For definiteness, we consider heat transfer in a tube with a granular bed of the boundary condition of the first kind on the outer surface of the tube. In one-temperature approximation and with account of thermal anisotropy of the bed the heat conduction equation has the form of the equation of the 2nd order of the elliptic type

cf ρ f u ( r )

∂T 1 ∂ ⎛ r ∂T = ⎜ rλ ( r ) ∂x r ∂r ⎝ ∂r

∂T ⎞ ∂ ⎛ x ⎟ + ⎜λ (r) ∂x ⎠ ∂x ⎝

⎞ ⎟, ⎠

(3.25)

the boundary conditions are

x = 0, cf ρf u∞T0 = cf ρf u ( r ) T − λ x ( r )

∂T , ∂x

(3.26)

x = H,

∂T = 0, ∂x

(3.27)

r = 0,

∂T = 0, ∂r

(3.28)

r = R, − λ r ( r )

∂T λt = (T − Tw ) . ∂r δ t

(3.29)

Equation (3.25) and condition (3.26) are written without regard for the zone of hydrodynamic stabilization of heat carrier flow, which, as is known, amounts to several particle diameters [10]. This allows one to consider the quantities u ( r ) ,

λ r ( r ) and λ x ( r ) as functions only

of radius r. Despite a rather standard form of Eq. (3.25) with conditions (3.26)-( 3.29), calculation of temperature fields with the help of them is rather difficult due to the existing problem of correct determination of the functions u ( r ) ,

λ r ( r ) and λ x ( r ) .

The results of calculation of the functions u ( r ) (figure 6) allows one to introduce and identify, as shown in figure 8, the concepts of a filtration hydrodynamic boundary layer and a

viscous sublayer. As is seen, a special character of the function u ( r ) inherent only in the granular bed is realized in the filtration boundary layer. The quantities

δ and δ* calculated

by the technique shown in figure 8, are presented in figure 9. According to classification of [42], three flow models were considered: laminar 5 < Re ∞ < 80 , transient

Transfer Processes in a Heat Generating Granular Bed

391

80 < Re∞ < 120 , and turbulent Re∞ > 120 . The following approximation relation for δ * is obtained for a laminar region:

δ* d

= 0.12 Re∞−0.08 .

(3.30)

= 0.34 Re∞−0.32 .

(3.31)

= 0.33Re∞−0.31 .

(3.32)

We have for the transient region

δ* d and for the turbulent region

δ* d

Figure 8. Definition of filtration hydrodynamic boundary layer and a viscous sublayer

( Re∞ = 1) .

1 – calculation by (3.18), 2 – (3.19).

The relation

δ ≅ 1.78 δ* which holds for the three regions, was obtained for calculation of

(3.33)

δ.

392

Yu.S. Teplitskii and V.I. Kovenskii

Figure 9. Values of the filtration boundary layer and viscous sublayer. 1 – relation (3.30); 2 – (3.31); 3 – (3.32).

Calculation of Thicknesses of the Filtration Thermal Boundary Layer and Thermal Sublayer We use the known relation between

δ * and k* [7] k*

δ*

=

3

1 Pr

(3.34)

With account for (3.34) from (3.30)-(3.32) we obtain a laminar region

k* = 0.12 Re∞−0.08 Pr −0.33 , d

(3.35)

k* = 0.34 Re∞−0.32 Pr −0.33 , d

(3.36)

k* = 0.33Re∞−0.31 Pr −0.33 . d

(3.37)

a transient region

a turbulent region

Transfer Processes in a Heat Generating Granular Bed

393

The values of the thermal filtration layer were calculated by the relation k k* = 1.78 which is similar to that found earlier for

δ δ * (3.33).

Determination of Efficient Thermal Conductivities At a distance from the heat transfer surface (in the bed core) these quantities are calculated by the formulas (2.1), (2.2) and

(λ ) (λ ) x

∞

r

∞

Re∞ ≤ 10, ⎧1, =⎨ 0.32 ⎩0.66 Re∞ , Re ∞ > 10.

(3.38)

We note that relation (3.38) was obtained as a result of processing of experimental data by

(λ ) (λ ) x

r

∞

given in [9].

∞

Adaptation of relations (2.1) and (3.38) for calculation of thermal conductivities near the wall can be made with account for the existence of the thermal boundary layer and thermal sublayer. The expression for

λ r ( r ) was formulated in the form

⎧λeff , R − k* < r ≤ R, ⎪ λ r − λf0 ⎪ r λ ( r ) = ⎨λeff + ( R − k* − r ) , R − k < r < R − k* , k k − * ⎪ ⎪λ + 0.1c ρ u ( r ) d , 0 ≤ r < R − k . f f ⎩ c To calculate

(3.39)

λ x ( r ) we used the relation similar to (3.38) Re∞ ≤ 10, λ x ( r ) ⎧1, =⎨ r 0.32 λ ( r ) ⎩0.66 Re∞ , Re∞ > 10.

(3.40)

The value of the effective thermal conductivity in the thermal sublayer is given by relation (3.21) which is similar to (2.1).

Analysis of the Theoretical Model We write the system of equations (3.25)-(3.29) in the dimensionless form

Pe∞ u ′ ( r ′ )

where

2 r ∂θ ⎛ H ⎞ 1 ∂ ⎛ λ ( r ′ ) ∂θ ⎞ ∂ ′ =⎜ ⎟ ⎜r ⎟+ ∂ξ ⎝ R ⎠ r ′ ∂r ′ ⎝ λf0 ∂r ′ ⎠ ∂ξ

θ = (T − Tw ) (T0 − Tw ) .

⎛ λ x ( r ′ ) ∂θ ⎞ ⎜ ⎟, 0 ⎝ λf ∂ξ ⎠

(3.41)

394

Yu.S. Teplitskii and V.I. Kovenskii The boundary conditions are

ξ = 0,

⎛ T0 Tw = u′ ( r ′ ) ⎜ θ + T0 − Tw T0 − Tw ⎝

⎞ 1 λ x ( r ′ ) ∂θ , ⎟− 0 ⎠ Pe ∞ λf ∂ξ

(3.42)

ξ = 1,

∂θ = 0, ∂ξ

(3.43)

r ′ = 0,

∂θ = 0, ∂r ′

(3.44)

r ′ = 1, −

∂θ Bi θ . = ∂r ′ 1 + 0.0061Re∞ Pr

(3.45)

For comparison we considered the parabolic heat conduction equation (without regard for longitudinal thermal conductivity) with the corresponding boundary conditions 2 r ∂θ ⎛ H ⎞ 1 ∂ ⎛ λ ( r ′ ) ∂θ ⎞ Pe∞ u ′ ( r ′ ) =⎜ ⎟ ⎜ r′ ⎟, ∂ξ ⎝ R ⎠ r ′ ∂r ′ ⎝ λf0 ∂r ′ ⎠

(3.46)

ξ = 0, θ = 1 ,

(3.47)

r ′ = 0,

r ′ = 1, −

∂θ = 0, ∂r ′

∂θ Bi θ . = ∂r ′ 1 + 0.0061Re∞ Pr

(3.48)

(3.49)

The heat-transfer coefficient was calculated by the formula

K = −λ r

∂T ∂r

r=R

1 . T − Tw

(3.50)

Figure 10 shows the temperature fields calculated by (3.41)-( 3.45) and (3.46)-( 3.49) for the following parameters of the granular bed: T0 = 373 K, Tw = 273 K, H = 0.025 m,

R = 0.005 m, δ t = 0.002 m, d = 0.001 m, λs = 0.1 W/(m⋅K), cf = 1015 J/(kg⋅K),

A = 1 , λt = 62 W/(m⋅K), λf0 = 0.027 W/(m⋅K), μf 0 = 1.8 ⋅10−5 kg/(m⋅sec). As is seen, rather large temperature gradients due to the presence of the thermal sublayer are observed in

Transfer Processes in a Heat Generating Granular Bed

395

the near-wall region. At small Re ∞ , the difference between the solutions of the elliptic and parabolic heat conduction equations are rather appreciable (figure 10a) and decrease with an increase of Re∞ (figure 10b).

ξ

Figure 10. Temperature distribution in different cross sections of the tube with a granular bed: a –

Re ∞ = 1 0, b – 500. 1 – ξ = 0 , 2 – 0.25, 3 – 0.5, 4 – 1. Solid lines – solution (3.41)-( 3.45), dashed lines – (3.46)-( 3.49).

396

Yu.S. Teplitskii and V.I. Kovenskii

ξ

Figure 11. Variation of the local heat-transfer coefficient along the length of the tube with a granular bed:

Re∞ = 500 . Solid lines – solution (3.41)-( 3.45), dashed lines – (3.46)-( 3.49).

Figure 11 shows the calculated variations of the heat-transfer coefficient along the tube length. On generalization of the obtained values of α st in the case of stabilized heat transfer an important conclusion was drawn: the calculated values of

α st are approximated by the

formula

Nu st =

1 0.1λ

λeff

0 f

λ0 R + 0.345 rf ( λ )∞ d

with a rms error not exceeding 5% [ λeff and

(λ ) r

∞

,

(3.51)

are given, respectively, by (3.21), (2.1)

and (2.8)]. This formula was obtained in [38] as a result of analytical solution of the system of equations (3.46)-(3.49) at u ( r ) = u∞ = const with the use of the concepts on the existence

of a heat carrier film with the parameters (3.20), (3.21) near the wall. This fact can serve as a basis for using a simple two-layer model (3.20)-(3.22) in calculations. In figure 12, the calculation by (3.51) is compared with the experimental data available in the literature [40; 43, figure V.24]. As is seen, in all cases, the calculated data agree well with those obtained experimentally. Due to a great dependence of

(λ ) r

∞

on Re ∞ , the

contribution of the bed core to the total resistance [the second term in the denominator of (3.51)] to heat transfer decreases with an increase of velocity.

Transfer Processes in a Heat Generating Granular Bed

397

Figure 12. Heat exchange between the granular bed and the tube wall. Comparison with experimental data: a – glass spheres-air [43], 1 – D d = 5 , 2 – 7, 3 – 6-9, 4 – 7-10, 5 – 10, 6 – 10-14, 7 – 10-14, 8 – 40-50; b – glass spheres-water [40], 1 – D d = 6 , 2 – 16, 3 – 58. (D – granular bed diameter). Solid and dashed lines – formula (3.51)

An important parameter which determined the intensity of the heat transfer process is the value of the initial thermal section xst . For calculation of it, a simple approximation relation

398

Yu.S. Teplitskii and V.I. Kovenskii 0.4

xst R ⎛R⎞ = Re∞−0.3 ⎜ ⎟ , 1 ≤ Re∞ ≤ 2000, 1.5 ≤ ≤ 29, H d ⎝d⎠

(3.52)

is obtained. This relation indicates a decrease of xst with an increase of heat carrier velocity. This unusual relation that qualitatively differs from those similar for one-phase media [7] can be explained by a strong dependence of

λeff , λ r and λ x on the filtration velocity which

leads to enhancement of heat transfer and decrease of the inlet section with an increase of filtration velocity.

4. Thermomechanics of a Heat Generating Granular Bed As is known, the influence of thermal processes on the granular bed hydrodynamics can be significant, which under certain conditions leads to a radical change in the flow conditions [8, 44, 45]. From the practical point of view, of great importance is the investigation of the influence of the heat release on the resistance of a granular bed, largely determining the efficiency of a particular technical device.

Formulation of the Problem To describe the stationary longitudinal heat transfer and the pressure distribution inside the heat generating bed, we use the system of equations (1.1)-(1.4)

dTf dT ⎞ 6 (1 − ε ) α d ⎛ = ⎜ ελfx f ⎟ + (Ts − Tf ) , dx dx ⎝ dx ⎠ d

(4.1)

6 (1 − ε ) α d ⎛ x dTs ⎞ (Tf − Ts ) + Q (1 − ε ) , ⎜ (1 − ε ) λs ⎟+ dx ⎝ dx ⎠ d

(4.2)

cf J f

0=

1 − ε ) μf u 1 − ε ) ρf u 2 ( ( dv dp − 1.75 ρ f v = − − 150 , dx dx ε3 d2 ε3 d

(4.3)

p = ρ f R*Tf .

(4.4)

2

Equations (4.1)–( 4.4) are considered at the following boundary conditions:

x =0;

cf J f (Tf − T0 ) = ελfx

dT dTf + (1 − ε ) λsx s , dx dx

(4.5)

Transfer Processes in a Heat Generating Granular Bed

(1 − ε ) λsx

dTs = α 0 (Ts − T0 ) . dx p = patm ,

x=H ,

(1 − ε ) λsx where

399 (4.6)

dTf = 0. dx

(4.7)

dTs = α H (Tf − Ts ) , dx

(4.8)

α H is given by (1.16).

Determination of the Model Parameters In analyzing system (4.1)–(4.8) the correct choice of the coefficients

α , α 0 , λfx , λsx is

essential. The interphase heat-transfer coefficient is calculated by the formulas [46]

⎧ ⎛ Re ⎞ 2 / 3 1/ 3 Re > 200 ⎪0.4 ⎜ ⎟ Pr , ε αd ⎪ ⎝ ε ⎠ . Nu = 0 = ⎨ 1.3 λf ⎪ Re Re ⎛ ⎞ 1/ 3 −2 ⎪1.6 ⋅10 ⎜⎝ ε ⎟⎠ Pr , ε ≤ 200 ⎩

(4.9)

Gas-solids heat-transfer coefficient α0 defining the preheating intensity is found from the dependence [5]

St 0 =

α0 cf J f

= 0.5 Re0−0.5 Pr −0.6 .

(4.10)

The heat-transfer coefficient αH is found from (1.17), (1.18); thermal conductivity of phases

λfx and λsx are in part II; ρ f , μf and λf0 – from (5.4). Reduction to Dimensionless Form Let us write system (4.1)–( 4.8) in the dimensionless form:

dθ f d ⎛ 1 dθ f ⎞ 1 = ⎜ ⎟ + (θs − θ f ) , d ξ d ξ ⎝ Pef d ξ ⎠ Pe

(4.11)

400

Yu.S. Teplitskii and V.I. Kovenskii

0=

d J f ρf′ dξ

d ⎛ 1 dθ s ⎜ dξ ⎝ Pes d ξ

 ⎞ 1 ⎟ + (θ f − θs ) + Q , ⎠ Pe

(4.12)

⎛ 1 ⎞ (1 − ε ) Re− 1.75 1 − ε Re2 , dp′ − 150 ⎜ ′ ⎟ = −D dξ ε3 ε3 ⎝ ρf ⎠ 2

ρ f′ =

(4.13)

p′ . θf + 1

(4.14)

The boundary conditions are

ξ = 0,

θf =

1 dθ f 1 dθ s + ; Pef dξ Pes d ξ

(4.15)

1 dθ s = St 0 θs ; Pes dξ p′ = 1 ,

ξ = 1,

(4.16)

dθ f =0; dξ

(4.17)

1 dθs = St H (θ f − θs ) . Pes d ξ For

μf = μf0 (θ f + 1)

0.75

(4.18)

(approximation of data [47] for air) Eq. (4.13) in view of (4.14) can

be given in a form more convenient for analysis p′

ε 2 Resd2 d ⎛ θf + 1 ⎞ dp′ H p′ = − − ⎜ ′ ⎟ 2 dξ ⎝ p ⎠ γ Re0 dξ d

2 1.75 ⎛ (1 − ε ) (θf + 1) + 1.75 1 − ε θ + 1 ⎞⎟ . (4.19) ⎜150 ( f )⎟ ⎜ ε Re0 ε ⎝ ⎠

Consider special cases allowing integration of (4.19).

Resistance of the Granular Bed in the Isothermal Case At

θ f = 0 (4.19) will have the form ε 2 Resd2 d ⎛1⎞ dp′ H = − − p′ p′ ⎜ ′⎟ 2 γ Re0 dξ ⎝ p ⎠ dξ d

2 ⎛ 1− ε ) 1 ( 1− ε + 1.75 ⎜ 150 ⎜ ε ε Re0 ⎝

⎞ ⎟. ⎟ ⎠

(4.20)

Transfer Processes in a Heat Generating Granular Bed

401

Let us integrate (4.20) from ξ to 1:

p (ξ ) − patm ⎞ ⎛ ln ⎜ 1 + ⎟= patm ⎝ ⎠ 2 2 1− ε ) 1 ε 2 Resd2 ⎛ ⎛ p (ξ ) − patm ⎞ ⎞ H ⎛ ( 1− ε ⎞ ⎜ ⎟ 1 1 150 1.75 =− − + − + ⎜ ⎟ (1 − ξ ) . ⎜ ⎟ ⎜ ⎟ ⎟ ε ε 2γ Re02 ⎜ ⎝ patm d Re 0 ⎠ ⎝ ⎠

⎝

⎠

⎛

For Resd Re 0 >> 1 in (4.21) neglect of ln ⎜ 1 + 2

(4.21)

2

⎝

p (ξ ) − patm ⎞ ⎟ is admissible. Then for the patm ⎠

calculation of the bed resistance we obtain a simple dependence

p (ξ ) − patm ( Δp )E ≅ 1+ 2 (1 − ξ ) − 1 , patm patm

where

( Δp ) E patm

(4.22)

(1 − ε ) 1 + 1.75 1 − ε ⎞⎟ is a dimensionless writing of γ Re2 H ⎛ = 2 0 2 ⎜ 150 ε ε ⎟⎠ Resd ε d ⎜⎝ Re0 2

ρ f = ρf0

the Ergun formula for

( Δp )E H

(1 − ε ) = 150 ε3

2

J f μf0 1 − ε J f2 1.75 + d 2 ρf0 ε 3 d ρf0

(4.23)

As is seen, even in the isothermal case, in the bed a nonlinear pressure profile is formed. At small values of

( Δp )E patm

(4.22) will describe the pressure profile in the bed

p (ξ ) − patm ( Δp )E ≈ (1 − ξ ) . patm patm

(4.24)

The formulae for calculating of the total bed pressure drop follows from (4.22) at ξ = 0

( Δ p ) = ( Δ p )E

M od

p a tm

p a tm

≅

H 1+ 2 p a tm

⎛ (1 − ε ⎜150 ⎜ ε3 ⎝

) μ f0 u 2

d

2

+ 1 .7 5

1 − ε ρ f0 u 2 d ε3

⎞ ⎟ − 1 (4.25) ⎟ ⎠

Figure 13 presents the results of the calculation of the bed pressure drop Δp patm by (4.21) at ξ = 0 and (4.25), as well as by the Ergun formula (4.23) and by the same formula at

402

Yu.S. Teplitskii and V.I. Kovenskii

ρf = ρf ( p0 ) . It is seen that the exact solution of (4.21) practically coincides with the approximate solution of (4.25). Calculations by the Ergun formula for various densities of the gas give curves markedly differing from one another and from the solutions of (4.21) and (4.25). As is seen, the calculation of the bed resistance by the Ergun formula, where the influence of pressure on the gas density is neglected, can lead to great errors at

( Δp )E patm

> 0.1 .

This conclusion holds as well for other analogous formulas obtained from the modified Darcy equation (3.4) in using them to calculate the resistance of a granular bed when the bed pressure drop becomes comparable to the output pressure. The generalization of the results obtained is given in figure 14.

Figure 13. Bed pressure drop (isothermal case): a – H ⁄ d = 100; b – 1000; 1 – calculation by the Ergun formula (4.23); 2 – calculation by (4.21) and (4.25); 3 – calculation by the Ergun formula (4.23) with substitution of

ρf0

by

ρ 0 . d = 10−4 m. Jf, kg ⁄ (m2·sec).

Figure 14. Isothermal bed pressure drop versus the complex

(γ Re

2 0

Resd2 ε 2 ) ( H d ) :

1 –

calculation by the Ergun formula (4.23); 2 – calculation by (4.21) and (4.25); H ⁄ d = 100 — solid lines; H ⁄ d = 1000 — dashed lines. d = 10−4 m.

Transfer Processes in a Heat Generating Granular Bed

403

Resistance of the Granular Bed in the "Isobaric" Case At p′ ≈ 1 (this is justified to some extent for thin beds) and small J f Eq. (4.19) assumes the form 2 1.75 ⎞ 1 − ε ) (θ f + 1) ( ε 2 Resd2 dp′ H ⎛ dθ f 1−ε =− − + + ⎜ 150 1.75 θ 1 ⎟ . (4.26) ( ) f ⎟ dξ γ Re02 dξ d ⎜⎝ ε Re0 ε ⎠

Following [8], we neglect the terms containing

 and (4.12) for sufficiently large Q , we obtain

λf and λs in (4.1) and (4.2). Adding (4.11)

 dθ f ≈Q. dξ

(4.27)

θ f ( 0 ) = 0 following from (4.15) has the  form θ f = Qξ . The integration of (4.26) with respect to ξ from 0 to 1 in view of the solution The solution of (4.27) with the boundary condition

of (4.27) yields an expression:

Δp =

((

 J f2 ⎛  H 1 − ε 1 ⎛ 54.5 (1 − ε ) ⎜ 1+ Q ⎜Q + ρf0 ⎝ d ε Q⎝ Re0

)

2.75

)

((

)

 2 ⎞⎞ − 1 + 0.87 1 + Q − 1 ⎟ ⎟ . (4.28) ⎟ ⎠⎠

)

Resistance of the Granular Bed (General Case) The bed resistance is influenced by the thermal processes and the pressure dependence of the gas density. Therefore, to determine ∆p, one has to solve the complete system of equations (4.11)–( 4.18). Figures 15 and 16 show the calculated phase temperature and pressure profiles for various values of Jf, Q, and H. As is seen, with increasing Q the pressure profile in the bed becomes essentially nonlinear. Figure 17 shows the dependences ∆p ⁄ patm obtained as a result of the numerical solution and by Eqs. (4.25) and (4.28). The given curves make it possible to determine the value of the flow rate Jf at which ∆p in the heat generating bed begins to coincide with the pressure drop value in the isothermal bed (curves 4). It should also be noted that at J f < Jf the calculations by (4.28) practically coincide with the numerical solution. As is seen from figure 17, at

J f < Jf ∆p can be calculated by (4.28), and at J f ≥ Jf by (4.25). To determine Jf , we have obtained a simple relation

404

Yu.S. Teplitskii and V.I. Kovenskii

Q (1 − ε ) H Jf = 41.7 cf T0

⎛H⎞ ⎜ ⎟ ⎝d ⎠

−0.2

(4.29)

Note that the extreme character of the dependence ∆p(Jf) points to the existence of two Jf values corresponding to two stationary regimes of filtration at ∆p > ∆pmin. In [45], it has been shown that the regime with a lower flow rate is unstable. The system either slowly goes from this state to the stable stationary regime or is heated with no limit.

Figure 15. Temperature and pressure profiles in the granular bed: a, b, c – H ⁄ d = 100; d, e, f – 1000; a, d – Jf = 5·10−4 kg ⁄ (m2·sec); b, e – 5·10−2; c, f – 5; 1 – Ts; 2 – Tf; 3 – (p − patm) ⁄ patm; Q = 5·103 W ⁄ m3, d = 10−4 m. p, atm, T, K..

Transfer Processes in a Heat Generating Granular Bed

405

Figure 16. Temperature and pressure profiles in the granular bed: a, b, c – H ⁄ d = 100; d, e, f – 1000; a, d – Jf = 0.5 kg ⁄ (m2·sec); b, e – 50; 1 – Ts; 2 – Tf; 3 – (p − patm) ⁄ patm. Q = 5·107 W ⁄ m3, d = 10−4 m. p, atm, T, K.

406

Yu.S. Teplitskii and V.I. Kovenskii

Figure 17. Bed pressure drop (general case): a – H ⁄ d = 100; b – 1000; 1 – Q = 5·103 W ⁄ m3; 2 – 5·105; 3 – 5·107; 4 – calculation by (4.25); dashed lines — calculation by (4.28), solid lines — numerical calculation. d = 10−4 m. Jf, kg ⁄ (m2·sec).

5. Steady Heat Transfer Processes in a Heat Generating Granular Bed at Different Heat Release Distributions Steady processes of heat carrier filtration through a granular bed are considered for four types of distribution of heat sources (figure 18) having practical importance. a) Q = const Taking into account the simplicity of describing heat transfer processes within the framework of one-temperature model (2.2), it is first reasonable to analyze it. The equation of stationary heat conduction and the corresponding boundary conditions (figure 18a) in the onedimensional case are of the form

cf J f

x =0,

x=H , The solution of (5.1)-(5.3) at

∂T ∂ ⎛ x ∂T ⎞ = ⎜λ ⎟ + Q (1 − ε ) , ∂x ∂x ⎝ ∂x ⎠ cf J f (T − T0 ) = λ x

∂T , ∂x

∂T = 0. ∂x

λ x = const is of the form

⎛ 1 exp ( Pe x (ξ − 1) ) ⎞ ⎟ θ = Q⎜ x +ξ − ⎜ Pe ⎟ Pe x ⎝ ⎠

(5.1)

(5.2)

(5.3)

Transfer Processes in a Heat Generating Granular Bed where

θ=

407

T − T0 . It is shown in figure 19. T0

Within the framework of the two-temperature model the fields of phase temperature and pressure were calculated using system (4.11)-(4.18). Since the quantities

ρ f , μf and λf0

depend on temperature and pressure, we have approximated data on the thermophysical properties of air in the range of temperatures from 300 to 1400 K and pressures from 1 to 20 atm [47]:

ρ f = 0.00352 p Tf , μf = 2.64 ⋅ 10−7 T 0.74 ,. λf0 = 0.00021 T 0.84

(5.4)

Some results are plotted in figures 19 and 20. For the bed-mean relative temperature difference of phase we have

Figure 18. Distributions of heat sources used in calculations.

To calculate a pressure drop within the entire bed, the relation is obtained

Δp

( Δp )E

Mod

⎛H ⎞ = 1 + 0.93Q 0.8 Re0−0.04 ⎜ ⎟ ⎝d ⎠

Δp ( Δp )E = patm patm

Mod

or

−0.2

−0.2 ⎛ ⎞ 0.8 −0.04 ⎛ H ⎞ 1 0.93 Q Re + ⎜⎜ 0 ⎜ ⎟ ⎟⎟ , ⎝d ⎠ ⎠ ⎝

(5.5)

408

Yu.S. Teplitskii and V.I. Kovenskii

where

( Δp )E

Mod

is calculated by (4.25). Formula (5.5) holds true for the conditions:

 H 0.4 ×10 −3 ≤ Q ≤ 31 , 5.4 < Re0 < 54500 , 15 < < 100 . d It should be noted that calculations by formula (5.5) agree well with the results on the ascending section of curve

Δp ( J f ) (figure 17). This formula generalizes the calculation patm

results on the pressure drop in the region of increased Q and J f , and supplements relations (4.25) and (4.28). As is seen from figure 20, at small flow rates of heat carrier (small pressure drops) calculations by Ergun’s formula (4.24) practically coincide with the values of p calculated by (4.22) that allows for the influence of gas compressibility alone. As the flow rate of heat

carrier is increased, p (ξ ) determined by (4.22) approaches the numerical solution. Just this fact was taken into consideration when ( Δp ) E

Mod

was used in (5.5).

The bed-mean relative temperature difference of phases Θ is related by

Θ=

Ts − Tf 0.83 = 0.35 ( QH ) 3.2 × 10−5 < QH < 1.3) . ( T0

Figure 19. Temperatures vs. dimensionless coordinate (constant heat release), 1 – Ts , 2 –

Tu = 0.15 , а, b, c – Q = 10

7

W/m3, d –

Q = 10

9

, T, K.

(5.6)

Tf , 3 – T ,

Transfer Processes in a Heat Generating Granular Bed

Figure 20. Pressure drop vs. dimensionless coordinate (constant heat release), a, b, c – d –

Q = 109 ,

409

Q = 107

W/m3,

1 – calculation by Ergan’s formula (4.24), 2 – calculation by (4.22), 3 – numerical

calculation. Regimes correspond to those in figure 19.p, atm.

Figure 21. Dependence of Ts, Tf, and

p − patm

on the dimensionless coordinate ξ (Q = 5×107 W/m3): a

– H = 0.01 m, d = 0.001 m; b – 0.1 and 0.001; c – 0.1 and 0.003. p, Pa; Т, К, Jf, kg/(m2sec).

410

Yu.S. Teplitskii and V.I. Kovenskii

Figure 22. Dependence of Ts and Tf on the coordinate x (Q = 5×103 W/m3; d = 10−3 m, Jf = 1 kg/(m2⋅sec)): 1, 2, 3, and 4 – H = 0.01 m, 0.0193, 0.0373, 0.072. x, m, T, K.

⎧const , 0 ≤ x ≤ h h<x≤H ⎩0,

b) Q = ⎨

The calculations of Ts, Tf and p − patm by the two-temperature model at h = H / 2 (figure 18b) are shown in figure 21. Figure 22 shows the behavior of temperatures Ts and Tf at different heights of a granular bed. As is seen, the values of the temperatures of phases for beds with different H are strictly coordinated and form a single set. The pressure drop in a granular bed

 −1.85 D0 = ( 0.022ε −5.7 Re1.55 Re 02.3 ) Q 0.42 . 0 + 0.0066ε c) Q =

(5.7)

qB x⎞ ⎛ exp ⎜ − B ⎟ d (1 − ε ) d⎠ ⎝

This case(figure 18с) corresponds to the radiate heat flux q to the bed when x = 0 . In the case of semi-transparent particles a considerable amount of incident radiation penetrates deep into the bed, and absorption is volumetric in character. A rate of radiation attenuation depends on the properties of particles. In this case, the function Q can be presented as

x⎞ ⎛ Q ( x ) = A exp ⎜ − B ⎟ . d⎠ ⎝ The coefficient A is found from the condition

(5.8)

Transfer Processes in a Heat Generating Granular Bed

411

∞

x⎞ ⎛ A (1 − ε ) ∫ exp ⎜ − B ⎟ dx = q . d⎠ ⎝ 0

(5.9)

With regard to (5.9), for Q ( x ) we have

Q=

qB x⎞ ⎛ exp ⎜ − B ⎟ . d (1 − ε ) d⎠ ⎝

(5.10)

The case of such heat release is called the direct-current helioreceiver. The solution to system (5.1)-(5.3) is of the form ⎛ B* B* ⎞ ⎞ 1 1 ⎞⎛ ⎛ − * ⎟ ⎜ exp ( − B*ξ ) + x exp ( − B* − Pe x (1 − ξ ) ) − ⎜1 + x ⎟ ⎟ * x B ⎠⎝ Pe ⎝ B + Pe ⎝ Pe ⎠ ⎠

θ (ξ ) = qB* ⎜

(5.11)

Figure 23 shows the calculations by (5.11) and the dimensionless coordinate dependences of Ts , Tf obtained from the numerical calculations by two-temperature model (4.11)-(4.18). The quantity ( J f ) min was estimated from the balance condition

( J f )min ≅

q

cf (Tcr − T0 )

(5.12)

when Tcr was taken equal to 800 К. From the comparison of the phase temperatures Ts and

Tf with the temperature T calculated by the one-temperature model (5.1)-(5.3) – formula (5.11) – it is well seen that the functions Tf

(ξ )

and T (ξ ) are close in form. At the same

time, Ts (ξ ) as a rule is essentially different from T (ξ ) both in value and in form of the function. This fact obviously follows from the fact that the equations and the boundary conditions for Tf and T are close and those for Ts .are much different The active zone length xa was determined for

(θs − θf ) ≥ 0.01 . To calculate the zone

length the following relation is obtained

xa d = 4 B −0.85 Re00.04 , H H where 0.1 < B < 0.33 , 13.6 < Re 0 < 54500 , 15 <

H < 100 d

(5.13)

412

Yu.S. Teplitskii and V.I. Kovenskii

H opt d

= 4 B −0.85 Re00.04 ,

(5.14)

An important characteristic of a granular bed is its resistance that depends on many factors and, including, on a heat release power. The calculation data shown in figure 24 are generalized to obtain the following dependence:

Δp

( Δp ) E

Mod

= 3.35q 0.35 B 0.04 Re0−0.085 ,

(5.15)

where 0.03 < q < 1.33 , 0.1 < B < 0.33 , 13.6 < Re 0 < 54500 and ( Δp )E

Mod

– (4.25).

As seen from figure 24, as in the case of Q = const the functions p (ξ ) are close to those calculated by (4.24) (small flow rates of heat carrier) and agree with (4.22) at increased Jf. A specific power spent for heat carrier pumping is

N=

ΔpJ f

ρ0

=

Δp , ρ f0 ⎛ Δp ⎞ + 1⎟ ⎜ p ⎝ atm ⎠ Jf

(5.16)

where Δp is determined by (5.15). Note that in (5.16) the bed length is calculated by (5.14). An important characteristic responsible for the operation efficiency of a helioreceiver as a heat exchange device is its efficiency determined as

η = 1−

Δp N = 1− q −1 . q Δ p + p ( atm )

(5.17)

Figure 27a shows the calculation results by (5.17) that allow choosing an optimal particle diameter in different regimes. The bed-mean (at H ≅ H opt ) relative temperature difference of phases Θ is related by

Θ=

Ts − Tf 0.83 = 0.75 ( QH ) ( 0.003 ≤ QH ≤ 0.067 ) . T0

(5.18)

An optimal length of a direct-current helioreceiver determined at xa H ≈ 1 follows from (5.13):

Transfer Processes in a Heat Generating Granular Bed

413

Figure 23. Temperatures vs. dimensionless coordinate (direct-current helioreceiver ), 1 – Ts , 2 – Tf , 3 – T , Tu = 0.15 , а, b, c – q = 105 W/m2, d, e, f – 107 . T, K.

d) Q =

⎛ ( H − x) ⎞ qB exp ⎜ − B ⎟ d (1 − ε ) d ⎠ ⎝

This case (figure 18d) corresponds to the radiate heat flux to the bed for x = H . By convention, call it counter-current helioreceiver. The solution of one-temperature model (5.1)-(5.3) with Q (1 − ε ) =

⎛ ( H − x ) ⎞ is of the form qB exp ⎜ − B ⎟ d d ⎠ ⎝

414

Yu.S. Teplitskii and V.I. Kovenskii ⎛

θ (ξ ) = q ⎜ 1 − ⎝

⎞ B* ⎞ ⎛ B* B* exp ( B*ξ ) + x − 1 − x exp ( B* − Pe x (1 − ξ ) ) ⎟ exp ( − B* ) . (5.19) x ⎟⎜ B + Pe ⎠ ⎝ Pe Pe ⎠ *

p-patm

(p-patm)⋅103

(p-patm)⋅103

p-patm

(p-patm)⋅103

p-patm

Figure 24. Pressure drop vs. dimensionless coordinate (direct-current helioreceiver): а, b, c – 7

q = 105

,

W/m2; d, e, f – 10 . 1 – calculation by Ergan’s formula (4.24), 2 – calculation by (4.22), 3 – numerical calculation. Regimes correspond to those in figure 23.p, atm.

Transfer Processes in a Heat Generating Granular Bed

415

Figure 25 plots the calculations by (5.19) and the dimensionless coordinate dependences of Ts , Tf obtained from the numerical calculation results. To calculate the active zone and working length of the device, when the active zone occupies the entire bed volume the relations similar to (5.13) and (5.14) are obtained

xa d = 5.25 B −0.85 Re 0.014 . 0 H H H opt d

(5.20)

= 5.25B −0.85 Re00.014 .

where 0.1 < B < 0.33 , 13.6 < Re 0 < 54500 , 15 <

(5.21)

H < 100 . d

Having generalized the pressure drop calculations (figure 26), as before for the directcurrent helioreceiver, we obtain

Δp

( Δp )E

Mod

= 1.7 q

0.08

B

−0.05

Re

−0.02 0

⎛H ⎞ ⎜ ⎟ ⎝d ⎠

−0.09

.

(5.22)

where 0.03 < q < 1.33 , 0.1 < B < 0.33 , 13.6 < Re 0 < 54500 , 15 ≤ H d ≤ 100 . The specific power and the efficiency are determined by formulas (5.16) and (5.17), respectively, wherein Δp is calculated by (5.22).

η for the counter-current helioreceiver are shown in figure 27 b. Since in calculation of the bed resistance ((5.15) or ( 5.22)), involved in η , the heat exchanger optimal length ((5.14) or (5.21)) was used, the dependence of η on B is very The calculations of

essential. The bed-mean (at H ≅ H opt ) relative difference of temperatures of phases Θ is determined by the relation similar to (5.5) and (5.18),

Θ = 0.65 ( QH )

0.83

( 0.003 ≤ QH ≤ 0.067 ) .

(5.23)

Quasi-Homogeneity Criterion By comparing (5.5), (5.18) and (5.23), it can be concluded that the criterion QH plays an essential role in estimating the extent of difference of temperatures of phases in this or that heat transfer regime. With this in view, it is possible to formulate a rather universal criterion

416

Yu.S. Teplitskii and V.I. Kovenskii

QH =

Wd , cf J f T0

(5.24)

where W is the mean power of heat release per unit volume of the bed. In this case, W = Q (1 − ε ) is the heat releasing bed with Q = const ; W=

q is the direct-current and counter-current helioreceiver. H

Since QH determines a relative difference of temperatures of phases, it can be called the criterion for quasi-homogeneity of a two-phase system that permits determining the thermal state of the granular bed (extent of difference of temperatures of phases) under the given conditions.

Figure 25. Temperatures vs. dimensionless coordinate (counter-current helioreceiver), 1 – Ts , 2 – Tf , 3–

T , Tu = 0.15 , а, b, c – q = 105 W/m2, d, e, f – 10

T, K.

7

. Regimes correspond to those in figure 23.

Transfer Processes in a Heat Generating Granular Bed

417

It is obvious that the estimate of QH allows one to judge about a possible use of a simple one-temperature model. If Θ = 0.01 is taken as the upper boundary, then from (5.5) it follows that (5.1)-(5.3) can be used describing heat transfer in a heat-releasing granular bed at QH < 0.014 . As seen from (5.18) and (5.23), the one-temperature model can be adopted at

QH < 0.005 in the case of helioreceivers. (p-patm)⋅103

(p-patm)⋅103

(p-patm)⋅103

p-patm

p-patm

p-patm

Figure 26. Pressure drop vs. dimensionless coordinate (counter-current helioreceiver), а, b, c –

q = 105

W/m2, d, e, f –

107 . 1 – calculation by Ergan’s formula (4.24), 2 – calculation by (4.22), 3 –

numerical calculation. Regimes correspond to those in figure 23.p, atm.

418

Yu.S. Teplitskii and V.I. Kovenskii

Figure 27. Efficiency of the helioreceiver vs. particle diameter. а – direct-current helioreceiver, b – counter-current helioreceiver. 1 – lines – 0.1.

q = 105

Jf = 1

кг/(м2⋅с), 2 – 3, 3 – 5. Solid lines –

B = 0.33 ,

dashed

W/m2.

6. Filtrational Cooling of a Heat Generating Granular Bed in the Presence of a First-Order Phase Transition As is known [48], liquid evaporative cooling of heat generating granular bed possesses a number of qualitatively new properties as compared to similar processes, the heat carrier in which is a gas. The major features of such a process are: the high heat-transfer intensity in phase conversion of a heat carrier inside a granular bed, a substantial increase in the efficiency of cooling due to the heat of vaporization, and a low flow rate of a liquid coolant. Despite the fact that the major principles of this method of cooling have been known for a long time [49], up to now there has not been a single consistent method of calculating heat exchangers with bulk heat generation in the presence of phase transition.

Transfer Processes in a Heat Generating Granular Bed

419

Basic Assumptions A simplified scheme of evaporative cooling of a heat-releasing bed is presented in figure 28. The disperse bed consists of spherical particles of diameter d. A liquid of flow rate Jf and temperature T0 is supplied to the bed inlet. In the general case, there are three zones: I, the zone of liquid motion, II, the zone of evaporation, and III, the zone of vapor motion. In formulating the model, the following assumptions were made: 1) there are two interpenetrating continua (a heat carrier — solid particles); 2) the process is stationary, the position of the evaporation zone does not change in time; 3) the power of heat release is constant; 4) vapor is considered as a perfect gas; 5) the slip of the heat-carrier phases in the evaporation zone is neglected; 6) the values of

dTf dTs and in the beginning and at the end of the evaporation zone dx dx

⎛ ⎞ dT are assumed to be rather small ⎜ dTf = s ≈ 0⎟ ; ⎜ dx x = h ;h ⎟ dx x =h1 ;h2 1 2 ⎝ ⎠ 7) the heat-carrier temperature in the evaporation zone is equal to the saturation temperature and does not differ noticeably from the temperature of the particles; 8) the pressure drop is calculated from the Ergun equation (3.9).

Figure 28. Schematic diagram of evaporative cooling of a heat generating granular bed.

420

Yu.S. Teplitskii and V.I. Kovenskii

Equations of the Evaporative-Cooling Model Subject to the above-made assumptions, the equations describing the process of heat transfer in the system have the following form: in zone I ( 0 ≤ x ≤ h1 ) , liquid

cliq J f

I dTfI d ⎛ I dTfI ⎞ 6 (1 − ε ) α = ⎜ λf ε + TsI − TfI ) , ( ⎟ dx dx ⎝ dx ⎠ d

(6.1)

d ⎛ I dTsI ⎞ 6 (1 − ε ) α 0 = ⎜ λs (1 − ε ) + Tf I − TsI ) + Q (1 − ε ) , ( ⎟ dx ⎝ dx ⎠ d

(6.2)

2 1 − ε ) μliquliq 1 − ε ) ρ liquliq ( ( dp I ; − = 150 + 1.75 ε3 ε3 dx d2 d

(6.3)

I

2

in zone II ( h1 < x ≤ h2 ) , two-phase mixture II di d ⎛ II dTf II ⎞ 6 (1 − ε ) α Jf = ⎜ λf ε + TsII − TfII ) , ( ⎟ dx dx ⎝ dx ⎠ d

0=

II d ⎛ II dTsII ⎞ 6 (1 − ε ) α 1 − + TfII − TsII ) + Q (1 − ε ) , λ ε ) ( s ( ⎜ ⎟ dx ⎝ dx ⎠ d

dp II (1 − ε ) μf uf + 1.75 (1 − ε ) ρf uf2 ; − = 150 dx ε3 d2 ε3 d

(6.4)

(6.5)

2

(6.6)

in zone III ( h2 < x ≤ H ) , vapor

0=

III dTfIII d ⎛ III dTfIII ⎞ 6 (1 − ε ) α cv J f = ⎜ λf ε TsIII − TfIII ) , ( ⎟+ dx dx ⎝ dx ⎠ d

(6.7)

d ⎛ III dTsIII ⎞ 6 (1 − ε ) α 1 − + λ ε ( ) s dx ⎜⎝ dx ⎟⎠ d

(6.8)

III

(T

f

III

− TsIII ) + Q (1 − ε ) ,

Transfer Processes in a Heat Generating Granular Bed

421

1 − ε ) μ v uv 1 − ε ) ρ v uv2 ( ( dp III . − = 150 + 1.75 dx ε3 d2 ε3 d

(6.9)

p III = ρ v R*TfIII .

(6.10)

2

Boundary Conditions. With allowance for assumptions 6) and 7) and using the Danckwerts conditions for the heat carrier at x = 0; H we have

cliq J f (Tf I − T0 ) = λfIε

x = 0,

λsI (1 − ε ) x = h1 ,

x = h2 ,

dTsI = α 0 (TsI − T0 ) ; dx

p I = p II , TfI = TfII = Tsat ,

TsI = TsII ,

III

II

III

x = H , p III = patm ,

(6.12)

dTfII dTfIII = = 0, dx dx

dTsII dTsIII Ts = Ts , = = 0 ; x = 1 dx dx II

(6.11)

dTfI dTfII = =0, dx dx

dTsI dTsII = = 0 , x = 0 ; dx dx

p = p , Tf = Tf = Tsat , II

dT I dTfI + λsI (1 − ε ) s dx dx

III

(6.13)

dTfIII =0, dx

λs (1 − ε )

dTsIII = α H (TfIII − TsIII ) . dx

(6.14)

Integral Relations These are the heat-balance equations derived on the basis of the heat-conduction equations (6.1), (6.2), (6.4), (6.5), (6.7), and (6.8).

422

Yu.S. Teplitskii and V.I. Kovenskii

Zone I. We combine Eqs. (6.1) and (6.2) and integrate the resulting equation over x from 0 to h1:

(

)

cliq J f Tsat ( p ( h1 ) ) − TfI ( 0 ) = = λfI ε

dTf I ( h1 ) dT I ( h ) dT I ( 0 ) dT I ( 0 ) + λsI (1 − ε ) s 1 − λfIε f − λsI (1 − ε ) s + Q (1 − ε ) h1 . (6.15) dx dx dx dx

Using the boundary conditions (6.11) and (6.12), we obtain the sought-for heat-balance equation from Eq. (6.15)

(

)

cliq J f Tsat ( p ( h1 ) ) − T0 = Q (1 − ε ) h1 .

(6.16)

which yields the following equation for calculating the magnitude of zone I

(

h1 = cliq J f Tsat ( p ( h1 ) ) − T0

)

Q (1 − ε ) .

(6.17)

Zone II. We combine Eqs. (6.4) and (6.5) and integrate over x from h1 to h2 subject to conditions (6.12) and (6.13):

J f ( iv − iliq ) = Q (1 − ε )( h 2 − h1 ) .

(6.18)

di dx =L dx dx

(6.19)

Integrating the equation [7]

over x from h1 to h2, subject to the conditions and , we write

iv − iliq = L .

(6.20)

For the magnitude of the evaporation zone, Eqs. (6.18) and (6.20) yield

Δh = h2 − h1 =

Jf L . Q (1 − ε )

(6.21)

Zone III. An analogous operation of integration of Eqs. (6.7) and (6.8), using boundary conditions (6.13) and (6.14), gives

(

)

cv J f TfIII ( H ) − Tsat ( p ( h2 ) ) = Q (1 − ε )( H − h2 ) .

(6.22)

Transfer Processes in a Heat Generating Granular Bed

423

The heat-balance equation for the whole bed will be obtained as a result of summation of Eqs. (6.16), (6.18), and (6.22):

(

)

(

)

Q (1 − ε ) H = J f ⎡cliq Tsat ( p ( h1 ) ) − T0 + iv − iliq + cv TfIII ( H ) − Tsat ( p ( h2 ) ) ⎤ . (6.23) ⎣ ⎦ The physical meaning of Eq. (6.23) is that the heat released from the disperse bed is spent as follows: 1) to heat liquid from T0 to Tsat(p(h1)) in zone I; 2) to raise the enthalpy of the heat carrier from iliq to iv (liquid evaporation) in zone II; 3) to heat the vapor from Tsat(p(h2)) up to

TfIII ( H ) in zone III.

We should note that formulas (6.17) and (6.21) determine the dependence of the dimensions of zones I and II on the basic parameters of the process: J f , Q, L, T0 and. cliq . Dimensionless form

dθ fi d ⎛ 1 dθ fi ⎞ 1 i i = ⎜ i ⎟ + i (θs − θ f ) , dξ dξ ⎝ Pef dξ ⎠ Pe d 0= dξ

−D

i

d ( pi ) ′ dξ

(6.24)

i ⎛ 1 dθsi ⎞ 1 i i ⎜ i ⎟ + i (θ f − θ s ) + Q , ⎝ Pes dξ ⎠ Pe

(1 − ε ) = 150 ε3

2

Rei + 1.75

(1 − ε ) ε3

(6.25)

( Re )

i 2

;

ρf′ = ( p III )′ (θ fIII + 1) (for zone III), where p′ = p patm , i=I – zone I ( 0 ≤ ξ ≤ ξ1 ) ; i=III – zone III II

(6.27)

(ξ2 < ξ ≤ 1) .

( )≈T

Zone II (ξ1 < ξ ≤ ξ2). With allowance for Tf = Tsat p

(6.26)

II

s

II

, to describe zone II we

use the equations

dx = QL , dξ 0.25 d ( p II )′ patm ) ( dθ fII 1 = 1.405 . dξ Tsat ( patm ) − T0 ⎛ II ′ ⎞0.75 dξ ⎜( p ) ⎟ ⎝ ⎠

(6.28)

(6.29)

424

Yu.S. Teplitskii and V.I. Kovenskii

−D

II

d ( p II ) ′ dξ

(1 − ε ) = 150 ε

3

2

Re II + 1.75

(1 − ε ) ε

3

( Re )

II 2

;

(6.30)

Equation (6.28) is based on the condition that the entire heat released in zone II is spent to vaporize the liquid. Equation (6.29) was obtained by approximating the data of [47] on the saturation temperature for water:

Tsat ( p ) = 5.62 p 0.25 + 273 .

(6.31)

The boundary conditions are

ξ = 0,

θfI =

1 dθ fI 1 dθsI + , PefI dξ PesI d ξ

1 dθsI = St 0 θsI ; I Pes d ξ

ξ = ξ1 ,

( p )′ = ( p )′ , θ I

II

I f

= θ fII = θ sat ,

dθsI dθsII dθ fI dθ fII = = 0, θsI = θsII , = = 0, x = 0 ; dξ dξ dξ dξ

ξ = ξ2 ,

( p )′ = ( p )′ , θ II

III

II f

(

(6.33)

= θ fIII = θ sat ,

dθ sII dθ sIII dθ fII dθ fIII = = 0, θ sII = θ sIII , = = 0, x = 1 ; dξ dξ dξ dξ

ξ = 1,

(6.32)

1 dθsIII = St H (θ fIII − θsIII ) (6.35) III Pes dξ

dθ fIII =0 dξ

p )′ = 1, III

(6.34)

Parameters of the Evaporative-Cooling Model The effective thermal conductivity coefficients are (division 2)

λ

I f

λ = 1 + 0.5 0 liq

Re I

ε

Pr I ,

(6.36)

Transfer Processes in a Heat Generating Granular Bed

λfIII λv0 = 1 + 0.5

Re III

ε

425

Pr III ,

(6.37)

λsI =

λcI − ελliq0 + λrI , 1− ε

(6.38)

λsIII =

λcIII − ελv0 + λrIII . 1− ε

(6.39)

We note that the radiative component

λri is calculated by (2.15), λci – by (2.7) (i =I, III). The

coefficients of interphase heat exchange are calculated by (4.14). The thermophysical characteristics of a two-phase heat carrier in zone II [48] are

ρf =

ρ v ρ liq , μ = μliq (1 − x ) + μ v x. ρ v + x ρ liq f

(6.40)

Calculations by the developed model were performed for the system "water–steam" for which, based on reference values [47], the following approximating dependences were used:

μliq = 0.01 (TfI ) The values of

−0.76

, λliq0 = 0.5 ( TfI )

0.06

;

(6.41)

ρ v , μ v and λv0 were calculated by (5.4).

Numerical Simulation of the Process of Filtrational Evaporative Cooling The formulated boundary-value problem (6.24) – (6.30) with boundary conditions (6.32) – (6.35) was solved numerically by the method of collocations [50]. Calculation procedure for the case of existence of zone I alone For this purpose, Eqs. (6.24)–( 6.26) with conditions (6.32) and (6.35) are used. Calculation procedure for the case of existence of zones I and II [ 0]

First, using Tsat ( patm ) and Eq. (6.17), the value of h1

was estimated; Δh was

calculated from Eq. (6.21), and the conditions of the absence of zone III

( h2 ≥ H )

was

checked. Then, within the framework of the iterative process, the following procedures were performed successively: a) the problem for zone II was solved with the boundary conditions

x = 0; ξ = 1, θ fII = 1,

( p )′ = 0 ; II

ξ = ξ1[k ] ,

426

Yu.S. Teplitskii and V.I. Kovenskii b) the problem for zone I was solved with the boundary conditions (6.32) at

ξ = ξ1[k ] , ( p I )′ = ( p II )′ ; [ k +1]

c) the value of h1

dθ dθ = =0; dξ dξ I f

ξ = 0,

I s

( ( [ ] )) , where

was confirmed with the aid of Eq. (6.17) at Tsat p h1

( )

k

p h1[k ] is the pressure at the interface of zones I and II calculated in the course of the kth iteration ( k ≥ 1) ; [k ]

d) the value obtained was compared with h1 . The condition of completion of the iteration process is

1−

h1[k ] −3 [ k +1] ≤ 10 . h1

(6.42)

Calculation procedure for the general case of the existence of all three zones [ 0]

Preliminarily, the magnitude of zone I h1

at Tsat ( patm ) was estimated and the

condition h2 < H was checked. Thereafter, within the framework of an analogous iterative process the following procedures were carried out: a) the problem for zone III was solved with the boundary conditions

dθsIII = 0; dξ

θ fIII = θsat ( p (ξ 2 ) ) ,

ξ = 1,

dθ fIII 1 dθ sIII = 0, = St H (θ fIII − θsIII ) ; dξ PesIII d ξ

( p )′ = 0, III

b) the problem for zone II was solved with the boundary conditions

( p )′ = ( p )′ , θ II

ξ = ξ 2[k ] ,

III

II f

ξ = ξ 2[k ] , x = 1 ,

= θ fIII = θ sat ( p (ξ 2 ) ) ;

c) the problem for zone I was solved with the boundary conditions (6.32) at

ξ = ξ1[k ] , ( p I )′ = ( p II )′ ; [ k +1]

d) the value of h1

[k ]

e) the quantities h1

dθ fI dθ sI = =0. dξ dξ

( ( [ ] )) ;

was refined by Eq. (6.17) at Tsat p h1 [ k +1]

and h1

k

were compared;

f) the computation was considered completed if condition (6.42) is satisfied. In the case where

ξ = 0,

Transfer Processes in a Heat Generating Granular Bed

h1 =

cliq J f (Tsat ( h1 ) − T0 ) Q (1 − ε )

≥H,

427

(6.43)

which at Tsat ( h1 ) = Tsat ( patm ) yields

 QI ≤ 1

(6.44)

only zone I exists. For the pressure drop the following formula is obtained:

D0I ⎛d ⎞ = 1.15 − 1.9 ⎜ ⎟ I DE ⎝H ⎠

0.25

 ⎛ d ⎞ Q I ⎜ 2 ⋅10−3 ≤ ≤ 6 ⋅10−3 ⎟ , H ⎝ ⎠

(6.45)

which approximates the calculated data with a standard error of about 4%. The results of calculations of the temperatures of phases and of the pressure drop are shown in figures 29 and 30; figure 29 presents the temperature and pressure profiles for the case of existence of zones I and II, which, with account for Eqs. (6.17) and (6.21), corresponds to the condition

Jf cliq (Tsat ( h1 ) − T0 ) + L ≥ H . Q (1 − ε )

(

)

(

(6.46)

)

Relation (6.46) in dimensionless form, subject to Tsat p ( h1 ) ≈ Tsat ( p0 ) (see figures 29 and 30), takes the form

θsat ( p0 )  QI

1 +  ≥1 QL

(6.47)

(p-patm)⋅103

Figure 29. Temperature and pressure profiles in a granular bed in the presence of zones I and II (H = 0.5 m, d = 0.001 m, Jf = 1 kg/(m2sec), T0 = 293 K): a – Q = 1.8·106 W/m3; b – 4.3·106; solid lines, heat carrier; dashed lines, particles. p, atm, T, K.

428

Yu.S. Teplitskii and V.I. Kovenskii

Figure 30. Temperature and pressure profiles in a granular bed in the presence of all three zones at Q = 4.7·106 W/m3. The remaining parameters and symbols are same as in figure 29. p, atm, T, K.

To determine the magnitude of pressure at the inlet into the bed, the following relation was obtained:

D0III = DEIII

(d

H)

−0.45

⎛d ⎞ 0.008 − 3 ⎜ ⎟ ⎝H⎠

 QL7.5 −0.38

 ⎛ −3 d ⎞ −3 ⎜ 10 ≤ ≤ 3 ⋅10 , 0.18 ≤ QL ≤ 0.58 ⎟ , (6.48) H ⎝ ⎠

 QL6.4

which describes the calculated values with a standard error of about 6%. Figure 30 presents the temperature and pressure profiles for the general case where there are three zones in the heat-releasing bed, i.e.,

h1 + Δh < H ,

(

(6.49)

)

Subject to (6.17), (6.21), and Tsat p ( h1 ) ≈ Tsat ( p0 ) , this condition has the form

θsat ( p0 )  QI

1 +  < 1. QL

(6.50)

The value of p0 in this case was calculated by a formula similar to Eq. (6.48):

D0III = DEIII

(d

H)

0.25

 QL2.2

⎛d ⎞ 0.5 + 0.22 ⎜ ⎟ ⎝H⎠

0.25



 QL2.2

(1.6 ≤ Q

L

)

≤ 5.2 .

(6.51)

Transfer Processes in a Heat Generating Granular Bed

429

The error in using Eq. (6.51) does not exceed 10%. It should be noted that, just as in the case of existence of zone I alone, so in the presence of several zones in zones I and III (figures 29 and 30), virtually linear profiles of Tf and Ts are realized for the given specific conditions. Here, the temperatures of water and particles differ insignificantly. Evidently, such changes in the temperatures of phases can be described by the simplified equation

cliq ⎫⎪ dT = Q (1 − ε ) . ⎬ × Jf dx cv ⎪⎭

(6.52)

7. Compression and Rarefaction Waves in Granular Bed Unsteady processes in infiltrated disperse systems, that are caused by sharp changes in the inlet pressure, may occur in operation of industrial apparatuses in different transient regimes and in off-optimum situations and emergencies [44]. Therefore, modeling of such processes is of undeniable practical interest. This is particularly true of the heat-releasing granular bed in which failures in filtration of the heat-transfer agent may produce uncontrolled changes in its temperature. The main regularities of compression and rarefaction waves at small times can be studied in the bed without heat release.

Formulation of the Problem We consider a one-dimensional two-temperature model of perfect gas flow through a granular bed of spherical particles. The resistance force is calculated from the Ergun formula (3.9). The system of equations describing such a process has the following form:

(1 − ε ) μf u − 1.75sgn v 1 − ε ρf u 2 + μ ∂ 2v , (7.1) ∂v ⎞ ∂p ⎛ ∂v ρ f (1 + Κ (1 − ε ) ) ⎜ + v ⎟ = − − 150 ( ) 3 f ∂x ⎠ ∂x ε3 d2 ε d ∂x 2 ⎝ ∂t 2

∂T ⎛ ∂Tf +v f ∂x ⎝ ∂t

ρf ε cf ⎜

∂ρf ∂ + ( ρf v ) = 0 , ∂t ∂x

(7.2)

p = ρ f R*Tf ,

(7.3)

∂p ⎞ ∂ ⎛ x ∂Tf ⎞ ⎛ ∂p ⎟ = ε ⎜ + v ⎟ + ⎜ ελf ∂x ⎠ ∂x ⎝ ∂x ⎠ ⎝ ∂t

⎞ ⎟+ ⎠

6 (1 − ε ) α 1 − ε ) μf u 2 ( 1 − ε ρf u + + 1.75 , (Ts − Tf ) + 150 3 ε ε3 d d2 d 2

3

(7.4)

430

Yu.S. Teplitskii and V.I. Kovenskii

ρs (1 − ε ) cs

∂Ts ∂ ⎛ ∂T ⎞ 6 (1 − ε ) α = ⎜ (1 − ε ) λsx s ⎟ + (Tf − Ts ) ∂t ∂x ⎝ ∂x ⎠ d

(7.5)

System (7.1) – (7.5) has been solved with the following boundary conditions: the initial conditions

p ( 0, x ) = p 0 ( x ) , Tf ( 0, x ) = Tf0 ( x ) , Ts ( 0, x ) = Ts0 ( x ) , v ( 0, x ) = v 0 ( x ) , ρ f ( 0, x ) = ρ f0 ( x ) , where p

0

( x ) , Tf0 ( x ) , Ts0 ( x ) , v 0 ( x )

(7.6)

and ρ f0 ( x ) are the known functions determined

by solution of the corresponding stationary problem at the pressure at entry into the bed

p 0 ( 0 ) = p00 (atmospheric pressure was prescribed at exit from the bed in all cases); the boundary conditions

x = 0 , p = p0 ,

∂T ∂v = 0 , f = 0 (v < 0) ∂x ∂x

v0 ρf0 cf T0 = ε vρ f cf Tf − ελfx

(1 − ε ) λsx x=H ,

The condition

p = patm ,

∂T ∂Tf − (1 − ε ) λsx s ∂x ∂x

(v > 0) ;

∂Ts = α 0 (Ts − T0 ) ; ∂x

∂T ∂T ∂v = 0 , f = 0 . (1 − ε ) λsx s = α H (Tf − Ts ) ∂x ∂x ∂x

(7.7)

∂v = 0 for x = 0 and x = H has been taken in accordance with the ∂x

recommendations of [51]. It is necessary for solving the problem for the gas flow rate

J f ( t , x ) not known in advance and determined in the process of solution. The term with an

effective viscosity

μf , which makes the order of the equation higher, has been introduced in

a standard manner into (7.1).

Transfer Processes in a Heat Generating Granular Bed

431

Dimensionless Form The procedure of making system (7.1) – (7.7) dimensionless is nontrivial because of the arbitrariness in selection of the time and gas-velocity scales. These scales are established in *

the following manner. We write Eq. (7.1) in dimensionless form without specifying t and

v* :

ρ f0 H patm

⎛ * ′ ( v* ) 2 ⎞ ′ p −p v ∂v ∂v′ ⎟ ⎜ ′ ′ (1 + Κ (1 − ε ) ) ρf ⎜ t * ∂t ′ + H v ∂ξ ⎟ = − 0 p atm ∂∂pξ − atm ⎝ ⎠

* 2 * 1 − ε ρ f0 ( v ) H μ f v * ∂ 2 v′ 2 ⎛ 1 − ε ⎞ μf v H ′ ′ ′ ′ , v − 1.75sgn ( v ) ρf ( v ) + −150 ⎜ ⎟ 2 dpatm Hpatm ∂ξ 2 ε ⎝ ε ⎠ d patm 2

(7.8)

It is necessary that the left-hand side of (7.8) has the simplest form

′

⎛

′⎞

(1 + κ (1 − ε ) ) ρ ′ ⎜ ∂∂vt ′ + v′ ∂∂vξ ⎟ f

⎝

⎠

This yields two conditions

ρf0 Hv* patmt * *

= 1,

ρ f0 ( v* )

2

patm

= 1,

(7.9)

*

enabling us to determine t and v as follows:

t* =

ρf0 patm

H=

H *

R T0

,v = *

patm

ρf0

= R*T0 .

(7.10)

As is seen, the characteristic velocity determined by (7.10) is equal, accurate to the factor

γ , to the velocity of sound in the gas vsd at the temperature T0 and atmospheric pressure. Then, with account for (7.10), we write system (7.1) –(7.7) in dimensionless form as

p − patm ∂p′ ⎛ ∂v′ ∂v′ ⎞ + v′ =− 0 − ⎟ patm ∂ξ ∂ξ ⎠ ⎝ ∂t ′

ρ f′ (1 + Κ (1 − ε ) ) ⎜ 2

2

1− ε H μ f 1 ∂ 2 v′ 2 ⎛ 1 − ε ⎞ μf 1 ⎛ H ⎞ ′ , ′ ′ ′ v 1.75sgn v v ρ −150 ⎜ − + ( ) ( ) ⎟ ⎜ ⎟ ε d f μf0 Re ∂ξ 2 ⎝ ε ⎠ μf0 Re ⎝ d ⎠

(7.11)

432

Yu.S. Teplitskii and V.I. Kovenskii

∂ρf′ ∂ + ( ρf′v′) = 0 , ∂t ′ ∂ξ ⎛

⎛ p0 ⎞ ⎞ − 1 ⎟ + 1⎟ ⎝ patm ⎠ ⎠

ρ f′ = ⎜ p′ ⎜ ⎝

⎛ ∂θ f ∂θ + v′ f ∂ξ ⎝ ∂t ′

ρ f′ε ⎜

⎞ ⎛ ∂p′ ∂p′ ⎞ ∂ ⎟ = ε P ⎜ ′ + v′ ⎟+ ∂ξ ⎠ ∂ξ ⎠ ⎝ ∂t

(7.12)

(θf + 1) ,

(7.13)

⎛ 1 ∂θ f ⎞ ⎜ ⎟+ ⎝ Pe f ∂ξ ⎠

(1 − ε ) μf 1 H P v′ 2 + 1.75 1 − ε P ρ ′ v′ 3 , (7.14) 1 + f (θ s − θ f ) + 150 ( ) ( ) f Pe ε μf0 Re d 2

(1 − ε )

∂θs ∂ ⎛ 1 ∂θs ⎞ 1 = (θf − θs ) ⎜ ⎟+ ∂t ′ ∂ξ ⎝ Pes ∂ξ ⎠ Pes

(7.15)

with the initial conditions

Tf0 (ξ ) − T0 Ts0 ( ξ ) − T0 p 0 (ξ ) − patm , θ f ( 0, ξ ) = , θ s ( 0, ξ ) = , p′ ( 0, ξ ) = T0 T0 p0 − patm v′ ( 0, ξ ) =

v 0 (ξ ) ρ f0 (ξ ) ′ , 0, , = ρ ξ ( ) f v* ρf0

(7.16)

and the boundary conditions

ξ = 0 , p′ = 1 ,

∂θ ∂v′ = 0 , f = 0 ( v′ < 0 ) ∂ξ ∂ξ

∂θf 1 − ε λsx ∂θs = Pef ερf′v′θf − ∂ξ ε λfx ∂ξ

( v′ > 0 ) ;

∂θs α0 H = θs ; ∂ξ (1 − ε ) λsx

ξ = 1 , p′ = 0 ,

∂θ ∂v′ ∂θ αH H = 0, f = 0. s = (θ f − θs ) ∂ξ ∂ξ ∂ξ (1-ε ) λsx

(7.17)

Transfer Processes in a Heat Generating Granular Bed

433

Parameters of the Theoretical Model The thermal conductivities of the gas and the bed sceleton have been calculated from the formulas (2.8), (2.16), (2.18) and (2.19). The heat-exchange coefficients α and α 0 have been calculated from the dependences (4.10), (4.11). In the calculations, we took air as the gas and silica as particles.

Analysis of the Results Obtained Figure 31 shows the characteristic profiles of p, v,

ρf , J f , and Tf at different

instants of time for the rarefaction wave. The initial and final stationary distributions of these quantities are given here. The appearing return flow is characterized by the sharp increase in the velocity on the inlet portion. Also, we observe a considerable cooling of the heat-transfer agent behind the wave front, which is due to the conversion of the internal energy of the gas to a kinetic one. For the minimum gas temperature to be calculated we have obtained the following approximation:

θ min

⎛d ⎞ = −0.28 ⎜ ⎟ ⎝H ⎠

0.20

⎛ p00 − p0 ⎞ ⎜ ⎟ ⎝ p0 − patm ⎠

0.12

⎛ p00 − p0 ⎞ ⎜ ⎟ ⎝ cf ρ f0T0 ⎠

0.24

,

(7.18)

which reflects the influence of the controlling factors. The time of reaching a new stationary state can be evaluated from the dependence

⎛ p00 − p0 ⎞ Δt = 12.3 ⎜ ⎟ t* ⎝ p0 − patm ⎠ Figure 32 gives the characteristic profiles of p, v,

0.18

.

(7.19)

ρf , J f , and Tf at different instants of

time for the compression wave. As is seen, the behavior of the curves in this case and in figure 31 is different. A characteristic feature of the compression wave is sharp increase in the velocity and temperature of the gas behind its front. For the maximum gas temperature to be calculated we have obtained the following approximation:

θ max

⎛d ⎞ = 0.19 ⎜ ⎟ ⎝H ⎠

0.11

⎛ p0 − p00 ⎞ ⎜ ⎟ ⎝ p0 − patm ⎠

0.16

⎛ p0 − p00 ⎞ ⎜ ⎟ ⎝ cf ρ f0T0 ⎠

0.37

.

(7.20)

The correlation

Δt ⎛d ⎞ = 5.7 ⎜ ⎟ * t ⎝H⎠

−0.40

⎛ p0 − p00 ⎞ ⎜ ⎟ ⎝ cf ρf0T0 ⎠

0.70

has been obtained for the time of reaching a new stationary state.

(7.21)

434

Yu.S. Teplitskii and V.I. Kovenskii Since the entire process of reaching the new stationary state was very short and its 0

duration did not exceed 1 sec in the investigated range of variation in the quantities p0 and

p0 (for the compression wave, 1.1 ≤ p00 ≤ 3 atm and 3 ≤ p0 ≤ 11 atm; for the rarefaction wave, 3 ≤ p0 ≤ 11 atm and 1.1 ≤ p0 ≤ 3 atm), the particle temperature remained constant, 0

in practice.

Figure 31. Pressure (a, b, and c), velocity (d, e, and f), density (g, h, and i), mass flux (j, k, and l), and temperature (m, n, and o) profiles of the gas for the rarefaction wave at different instants of time for

p00 = 11 atm and p0 = 1.1

atm; a, d, g, j, and m – t = 0.002 sec; b, e, h, k, and n – 0.0082; c, f, i, l,

and o – 0.041; 1 – d = 0.4 m and 2 – 0.04; 3 – final states; 4 – initial ones (the upper curve, d = 0.04 m, the lower curve, 0.4 m). p, atm; v, m ⁄ sec; ρf, kg ⁄ m3; Jf, kg ⁄ (m2·sec); Tf, K.

Transfer Processes in a Heat Generating Granular Bed

435

Figure 32. Pressure (a, b, and c), velocity (d, e, and f), density (g, h, and i), mass flux (j, k, and l), and temperature (m, n, and o) profiles of the gas for the compression wave at different instants of time for

p00 = 1.1 atm and p0 = 11

atm. Notation a–o is the same as in figure 31. p, atm; v, m ⁄ sec; ρf, kg ⁄

m3; Jf, kg ⁄ (m2·sec); Tf, K.

Figure 33 shows results of calculation of the wave-front velocity from the formula

vw =

xw tw

(7.22)

436

Yu.S. Teplitskii and V.I. Kovenskii

where xw is the minimum value of the coordinate x at the instant of time t w for which the running value p ( t , x ) differs from the initial value p

0

(t, x )

by no more than 1%. As is

seen, the wave velocities were dependent just on the particle diameter.

Figure 33. Wave-front velocity vs. dimensionless coordinate: solid curve – rarefaction wave; dashed curve – compression wave; 1 – d = 0.4 m and 2 – 0.04.

Conclusion The physically justified boundary conditions, which are based on the account of the fact that the phases on the boundaries are isolated, are formulated. The third-kind boundary condition, which allows for the turbulence degree of the flow of heat carrier issuing from the granular bed, is formulated. The formulas obtained for the thermal-conductivity coefficients: meet the requirements of similarity of the processes of convective heat and mass transfer; allow for all the basic mechanisms of heat transfer in the granular bed. A generalized Brinkman equation has been proposed. It represents a superposition of the well-known Ergun equation and the equation of viscous flow in a channel. It has been shown that, with allowance for the actual porosity distribution in the wall zone, the generalized Brinkman equation enables one to describe the regularities of filtration in the granular bed near the wall in different regimes of flow (including nonlinear filtration). The concepts of the filtration boundary layer, the viscous sublayer and thermal sublayer have been introduced. Approximation relations have been obtained for calculation of the thicknesses. It was shown that the calculated values of the coefficient of heat exchange with the wall are in good agreement with the relation obtained within the framework of the two-layer model of heat transfer. An approximation relation was found for calculation of the inlet thermal section. In the isothermal case, the account of the gas compressibility has made it possible to establish the range of applicability of the known Ergan formula, outside which an essentially nonlinear pressure profile is formed. For the isobaric case where the pressure dependence of

Transfer Processes in a Heat Generating Granular Bed

437

the gas density can be neglected, the formula taking into account the influence of the heat release on the bed resistance has been obtained. A universal criterion for quasi-homogeneity of a heat- releasing bed is formulated. It allows estimating the thermal state of the two-phase system under the given conditions. Mathematical simulation of processes occurring in a heat generating bed of spherical particles in the presence of a first-order phase transition (liquid evaporation) is carried out. For water evaporation, the profiles of temperature and pressure have been calculated for different conditions of the occurrence of the processes of heating and evaporation of a heat carrier. Based on this, a simple engineering method of calculation of the system of evaporative cooling has been developed. It includes dimensionless conditions of the existence of one, two, and all three zones of cooling; approximating dependences for calculating the pressure drop in a granular bed; formulas for calculating the position and magnitude of the liquid evaporation zone The numerical investigation carried out has shown that the character of the compression and rarefaction waves formed in the disperse medium is substantially dependent on the structural parameters of the fill and the value of the initial jump in the pressure at the system entry. The influence of particles on the character of gas dynamic processes in the gas–solid two-phase system is allowed for by the simplex d h which represents the relation of the micro- and macroscale of the system. The obtained approximations have a simple form and are convenient for use in engineering practice.

References [1] Ergun, S. ChEP. 1952, 48, 89–94. [2] Dankwerts, P. V. ChES. 1953, 2, 1–13. [3] Levin, V. A., Lutsenko, N. A. In Heat and Mass Transfer–MIF-2004; Zhdanok, S. A., Ed.; HMTI: Minsk, By, 2004; Vol. 2, pp 219–220. [4] Matros, Yu. Sh., Lugovskoi, V. I., Ogarkov, B. L., Nakrokhin, V. B. TOKhT. 1978, 12, 291–294. [5] Polezhaev, Yu. V., Seliverstov E. M. TVT. 2002, 40, 922–930. [6] Luikov, А. V. Theory of heat conduction; Vysshaya Shkola Press: Moscow, SU, 1967; pp 181–273. [7] Isachenko, V. P., Osipova, V. A., Sukomel A. S. Heat Transfer; Energoizdat Press: Moscow, SU, 1981; pp 266–272. [8] Gol’dshtik, M. A., Processes of Transfer in a Granular Bed. IThPh SO AN SSSR: Novosibirsk, SU, 1984, pp 14–18. [9] Aerov, M. A., Todes, O. M., Narinskii D. A. Apparatuses with a Stationary Granular Bed; Khimiya Press: Leningrad, SU, 1970, pp 111–127. [10] Gel’perin, N. I., Ainshtein V. G. In Heat Transfer; Davidson I. F., Harrison, D., Eds.; Khimiya: Moscow, SU, 1974; pp 428. [11] Kovenskii, V. I., IFZh. 1980, 38, 983–988. [12] Borodulya, V. A., Kovenskii, V. I. IJHMT. 1983, 26, 277–287.

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Yu.S. Teplitskii and V.I. Kovenskii

[13] Siegel, R., Howell J. R. Thermal Radiation Heat Transfer. Mir Press: Moscow, SU, 1975; pp 137–140. [14] Adzerikho, K. S., Nogotov, E. F., Trofimov V. P. Radiation Heat Transfer in Two-Phase Media; Nauka i Tekhnika Press: Minsk, SU, 1987; pp 46–49. [15] Botterill, J. S. M., Salway, A. G., Teoman Y. IJHMT. 1989, 32, 595–609. [16] Schotte, W. AIChEJ. 1960, 6, 63–67. [17] Chudnovskii, A. F. The Thermophysical Characteristics of Disperse Materials; GIFML Press: Moscow, SU, 1962; pp 35–37. [18] Popov, Yu. A. IFZh. 1978, 34, 703–706. [19] Pavlyukevich, N. V. IFZh. 1990, 59, 606–608. [20] Kharlamov, A. G. Thermal Conductivity of High-Temperature Thermal Insulators; Atomizdat Press: Moscow, SU, 1979; pp 64–65. [21] Godbee, H. W., Zirgler W. T. APh. 1966, 37, 55–65. [22] Laubitz, M. J. CJPh. 1959, 37, 798–808. [23] Imura, S., Takegoshi E. HTJR. 1974, 3, 13–19. [24] Wakao, N., Kato, K. JChEJ. 1969, 2, 24–32. [25] Bauer, R., Schlunder, E. U. IChE. 1978, 18, 189–198. [26] Zehner, P. VDIV. 1973, 558–565. [27] Kunii, D., Smith, J. M. AIChEJ. 1960, 6, 71–83. [28] Vortmeyer, D. GChE. 1980, 3, 124–135. [29] Vortmeyer, D., Kasparec, G. TJSMESIS. 1967, 27–32. [30] Nasr, K., Viskanta, R., Ramadhyani, S. JHT. 1994, 116, 829–837. [31] Vortmeyer D. In Proc. 6th Int. Heat Transfer Conf., Toronto, Ca, 1978; Vol. 6, pp. 525– 539. [32] Shonnard, D. R., Whitaker, S. IJHMT. 1989, 32, 503–512. [33] Korolev V. N. Structural Gasdynamic Conditions and External Heat Transfer in Fluidized Media (Author’s Abstract of Doctoral Dissertation in Engineering); IThPh SO AN USSR: Novosibirsk, SU, 1989; pp 80–83. [34] Brinkman, H. C. ASR. 1947, 1, 21–28. [35] Gortyshov, Yu. F., Popov, I. A. In Proc. 1st Russian Nat. Heat Transf. Conf.; MEI: Moscow, Ru, 1994; Vol. 7, pp. 59–64. [36] Volkov, V. I., Mukhin, V. A., Nakoryakov, V. E. ZhPKh. 1981, 54, 838–842. [37] Teplitskii, Yu. S. Hydrodynamics and heat transfer in free and frozen fluidized beds (Author’s Abstract of Doctoral Thesis in Engineering); Novosibirsk, SU, 1991; pp 65–67. [38] Teplitskii, Yu. S. IFZh. 2004, 77, 86–92. [39] Narinskii, D. A., Kagan, A. M., Gelperin, I. N., Aerov, M. E. TOKhT. 1979, 13, 748– 755. [40] Dekhtyar, R. A., Sikovskii, D. F., Gorin, A. V., Mukhin, V. A. TVT. 2002, 40, 748–755. [41] Mukhin, V. A., Smirnova, N. N. Study of Heat and Mass Transfer Processes in Filtration in Porous Media; IThPh SO AN SSSR: Novosibirsk, SU, 1978; pp 6–8. [42] Fand, R. M., Kim, B. H. K., Lam, A. C. C., Phan, R. T. JFE. 1987, 109, 268–277. [43] Aerov, M. E., Todes, O. M. Hydraulic and Thermal Principles of Operation of Apparatuses with a Stationary Fluidized Bed; Khimiya Press: Leningrad, SU, 1968; pp 382.

Transfer Processes in a Heat Generating Granular Bed

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[44] Maslov, V. P., Myasnikov, V. P., Danilov, V. G. Mathematical Modeling of the Accident Unit of the Chernobyl Nuclear Power Station; Nauka Press: Moscow, SU, 1988; pp 93–97. [45] Levin, V. A., Lutsenko, N. A. IFZh. 2006, 79, 35–40. [46] Gel’perin, N. I., Ainshtein, V. G., Kvasha V. B. Principles of Fluidization Techniques; Khimiya Press: Moscow, SU, 1967; pp 256. [47] Vargaftik, N. B. Handbook of Thermophysical Properties of Gases and Liquids; Nauka: Moscow, SU, 1972; pp 586–631. [48] Polyaev, V. M., Maiorov, V. A., Vasil’ev, L. L. Hydrodynamics and Heat Transfer in the Porous Elements of the Constructions of Aircraft; Mashinostroenie Press: Moscow, SU, 1988; pp 127–158. [49] Kelley, J. B., L’Ecnyer, M. R. Transpiration Cooling — Its Theory and Application, JPC 422, Report No. TM-66-5, 1966, pp 38–39. [50] Krylov, V. I., Bobkov, V. R., Monastyrnyi, P. I. Principles of the Theory of Computational Methods. Differential Equations; Nauka i Tekhnika Press: Minsk, SU, 1982; pp 253–257. [51] Rozhdestvenskii, B. L., Yanenko, N. N. Systems of Quasilinear Equations; Nauka Press: Moscow, SU, 1978; pp 448. Reviewed by professor N. V. Pavlyukevich

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 441-471

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 6

THE INFLUENCE OF ND LASER IRRADIATION PARAMETERS ON DYNAMICS OF METAL CONDENSED PHASE PROPAGATING NEAR TARGET V.K. Goncharov, K.V. Kozadaev and M.V. Puzyrev Sevchenko Scientific-Research Institute of Applied Physical Problems of Belarussian State University, Minsk, Belarus

Abstract Erosion laser jet of metals with presence of condensed phase particles ejected from the target was investigated experimentally. Jets were generated during interaction of intensive laser irradiation with metal surfaces. Optimal intensities for precision and rapid laser processing were empirically determined for several metals. Mechanisms of drop-liquid phase formation were studied for the metal targets. Time delay between the start of bulk vaporization and the beginning of laser action was experimentally defined. Observed data allowed to reveal the fact of metal drops fragmentation near the target surface. The possibility of liquid phase parameters control with an implementation of electrical and electrical-magnetic fields was showed. Real-time way of nanoparticle separation during their formation and method of metal nanoparticle suspensions producing were described.

Introduction The vast circle of technological problems can be solved due to implementation of powerful lasers. The intensive application of powerful lasers began both for various scientific investigations and industrial purposes. At industry the powerful lasers are widely extended for realizing of different technologies of metal processing (hardening, welding, cutting, drilling). Usually for such purposes the solid laser on the base of neodymium (wavelength λ=1.06 μm) is used. That laser can generate both in regimes of continuous lasing and pulses. The laser allows to obtain the necessary power density for the metal processing. Moreover the technical arrangement on the base of that laser is characterized with the high level of reliability.

442

V.K. Goncharov, K.V. Kozadaev and M.V. Puzyrev

However, the industrial implementation of that laser would be impossible without the advanced scientific basement. The present work is dedicated to the description of the processes which take place during the interaction of neodymium laser radiation (moderate power density) with metal targets.

Discussion Action of laser radiation of moderate intensity (106-108 W/cm2) on metal targets generates damage products which travel opposite to the laser beam. The laser radiation incident on a target then interacts not only with the surface but also with the damage products and these products may affect the transmission of the laser radiation traveling toward the target surface. This effect was studied by us in an experiment involving measurements of the energy balance of radiation of an auxiliary laser which was used to probe an erosion jet formed by the radiation of a second more powerful laser. We used the experimental arrangement shown in figure 1. A zinc target was located at the center of an integrating sphere and it was subjected to focused radiation from a neodymium laser operating in the free-running regime; the total energy of this radiation was up to 1.5 kJ and the duration of the neodymium laser pulses was 10-3 sec. These pulses were focused by a lens with a focal length of 25 cm to form a spot 7.5 mm in diameter and the power of the neodymium laser radiation was varied by neutral optical filters. An erosion jet formed in this way was probed by radiation from an auxiliary ruby laser operated in the regular spiking regime. The total duration of emission from the ruby laser was 1.5×10-3 sec, the diameter of the probe beam was 2 mm, and the erosion jet was probed at a distance of 1.5 mm from the target surface. The power density in the probe radiation did not exceed 104 W/cm2 in order not to perturb the parameters of the probed medium. The energies of the radiations from both lasers were monitored employing IKT-1N meters. The time characteristics of the neodymium laser radiation were recorded using a photocell (figure 1) and a storage oscilloscope. The power of the ruby probe laser radiation incident on the erosion jet was measured by a photocell 11 (figure 1) and the transmitted power was measured by a photocell 13. The ruby laser radiation power scattered by the erosion jet was determined with a photocell 12. All the measuring devices were first calibrated. The influence of background radiation on the photocells was avoided by placing glass filters in front of them. However, this was insufficient in the case of the photocells recording the transmitted and probe radiations. The interference of the radiations emitted by the erosion jet and by the neodymium laser made it necessary to place interference filters in front of these photocells and these filters were tuned to pass only the ruby laser radiation. In this way we recorded simultaneously the power of the incident, transmitted and scattered ruby laser (probe) radiation. This probe radiation clearly satisfied the relationship Pin(t)=Ptr(t)+Psc(t)+Pab(t)

(1)

i.e., the power of the ruby laser radiation incident on the erosion jet was equal to the sum of the powers transmitted, scattered, and absorbed by the erosion jet. Normalization to unity gave

The Influence of ND Laser Irradiation Parameters on Dynamics…

443

1=Ktr(t)+Ksc(t)+Kab(t)

(2)

where Ktr(t)=Ptr(t)/Pin(t); Ksc(t)=Psc(t)/Pin(t); Kab(t)=Pab(t)/Pin(t). In this way by measuring the Pin(t), Ptr(t) и Psc(t) powers we were able to find the transmission and scattering coefficients of the probe radiation and then to calculate from Eq. (2) the absorption coefficient. Ktr(t), Ksc(t) and Kab(t) curves are plotted in figure 2.

Figure 1. Schematic diagram of the apparatus: 1) target; 2) erosion jet; 3) integrating sphere; 4) heliumneon lasers used for alignment; 5) neodymium laser; 6) amplifiers; 7) glass optical filters; 8) energy meters; 9) dual-trace storage oscilloscope; 10) ruby laser; 11)-14) photocells- 15) interference optical filters. 1,0

Ktr

a

0,8 0,6 3 0,4

2 1

0,2 0,0 0,0

0,2

0,4

0,6

0,8

1,0

t,ms

Figure 2. Continued on next page.

1,2

1,4

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V.K. Goncharov, K.V. Kozadaev and M.V. Puzyrev

0,6

Ksc

b

0,5

1

0,4 0,3 2

0,2 3

0,1 0,0 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

t,ms

0,7

Kabs

c

0,6 0,5

1

0,4 0,3

2

3

0,2 0,1 0,0 0,0

0,2

0,4

0,6

0,8 t,ms

1,0

1,2

1,4

Figure 2. Dependences of the transmission coefficient (a) scattering coefficient (b) and absorption coefficient of laser jet created by neodymium laser radiation of power densities 3 MW/cm2 (1), 2 MW/cm2 (2), and 0.8 MW/cm2 (3).

Simultaneously with a probing of an erosion jet by the ruby laser radiation we carried out spectroscopic measurements of the temperature and density of electrons in the jet plasma, which were determined by comparing the measured and theoretically calculated coefficients of the plasma and the ratio of the coefficients for the two spectral ranges. The emission coefficients of the investigated plasma were found by using an SFR-1 image-converter system operated as a high-speed cine camera with a two-row lens insert. Two interference filters with transmission maxima at the wavelengths 432 and 686 nm and with half-width 5 and 17 nm, respectively, were placed at the entry stop of the camera. Preliminary spectroscopic investigations of the plasma revealed the absence of spectral lines within the transmission bands of the interference niters. The entry stop was modified first to record simultaneously the plasma images at different wavelengths. The plasma jet image was focused on a film in the SFR-1 camera so that the jet axis was perpendicular to the optic axis of the camera. The angular velocity of the SFR-1 mirror was 7500 rpm and the exposure time of one frame was then 16μsec. In this way we recorded on the SFR-1 film two series of images of the plasma jet in two ranges of wavelengths. The absolute values of the emissivity of the plasma at these

The Influence of ND Laser Irradiation Parameters on Dynamics…

445

wavelengths were determined by recording under the same conditions the streak patterns of the radiation from an ISP-5000 flashlamp passed through a nine-stage attenuator. The ISP5000 flashlamp radiation was first calibrated against an EV-45 standard light source. These measurements made it possible to determine the beam-average transmission coefficient of the plasma for two spectral ranges with a resolution both in time and space. The emission of the continuous spectrum was calculated for postulated parameters of the plasma on the assumption that the radiation was due to recombination and deceleration of electrons. The total recombination emissivity was found by summing the contributions of the individual quantum levels on the assumption of an equilibrium distribution of electrons in respect of the velocities and of ions in respect of the energy levels, employing a method described in Ref. [3]. The recombination at levels with the orbital quantum number l<3 was calculated by the quantum defect method [4] and the recombination at levels with l ≥3 was studied employing a formula for hydrogen-like atoms allowing for the Gaunt factor [5]. The reduction in the ionization potential was allowed for using the Inglis-Teller formula [6]. The intensity of bremsstrahlung in the ion field was calculated in the quasi-classical approximation subject to the Gaunt correction given by an analytic expression taken from Ref. [7]. A calculation of the continuum resulting from the deceleration of the electrons by collisions with atoms was carried out employing the relevant elastic scattering cross section and a calculation formula taken from Ref. [8]. The results of spectroscopic measurements carried out at the probe height in the axial zone of the erosion jet are presented in figure 3. Figure 4 gives the radial distributions of the electron temperature and density in the jet at the probe height. The nature of the curves Ktr(t), Ksc(t) and Kab(t) cannot be accounted by the principal processes of the absorption of radiation by a plasma, because (for the postulated plasma parameters) the absorption should be slight. An estimate of the absorption of the probe (λ= 693 nm) radiation by the plasma was obtained by a calculation similar to that performed in respect of the emissivities. Since in our case the energy of the probe radiation photons was low compared with the ionization potential, estimates obtained using an approximate Kramers-Unsold formula [7] gave similar results. When the power density of the radiation used to form the plasma was 0.8 MW/cm2, the total absorption coefficient amounted to less than 0.002 in our case, but at the power density of 2 MW/cm2 it raised to 0.03. Even at the maximum parameters of the plasma jet (figure 3) the plasma absorption did not exceed 0.09, which was considerably less than the value found experimentally. The only mechanism which can account for Ktr(t), Ksc(t) and Kab(t) is the absorption and scattering of the ruby laser probe radiation by particles of a liquid phase formed from the target material. The dependences Ktr(t), Ksc(t) and Kab(t) demonstrate (figure 2) that during the first 50 μsec the coefficient Ktr(t) decreases, whereas Ksc(t) and Kab(t) increase, reaching steady-state values, whereas during the next 500 μsec the changes are small. After 500-600 μsec from the beginning of the plasma formation (depending on the actual degree of radiation used for form the plasma) all these coefficients change drastically reaching extremal values at the end of the neodymium laser pulse (800-900 μsec). These changes in the coefficients can, in our opinion, be accounted for by the appearance of a large number of liquid drops of the target material at the end of the neodymium laser pulse. These drops appear because a crater is formed in the region of interaction of the neodymium laser radiation with the target and this crater is created not only by evaporation, but also by a number of secondary processes such as heating of the walls of the crater by the vapor and by

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the scattered radiation, followed by melting and hydrodynamic flow driven by the escaping gas stream [10]. At the end of the laser pulse a pressure pulse ejects the liquid phase from the crater in the form of fairly large drops [11]. However, the absorption and scattering by an erosion jet of the incident probe radiation occurs throughout the period of action of the neodymium laser radiation (figure 2) and this happens at a rate which is only slightly lower than the end of the neodymium laser pulse. Such absorption and scattering of the probe radiation can be explained by the appearance, right from the beginning of the plasma formation process, of the target material particles generated as a result of volume vaporization [12], since the formation of particles by hydrodynamic erosion of the crater walls (particularly during the initial stage of the interaction) is difficult in view of the shallowness of the crater. After the end of the laser radiation pulse the depth of the crater does not exceed 0.3 mm and its diameter is 7.5 mm. Volume vaporization is facilitated greatly by gases dissolved in the metal and by the spatial inhomogeneity of the power density of the laser radiation, which is typical of the free-running regime [13, 14].

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Taking into account the experiment of laser irradiation screening by the products of erosion for wide circle of metals (Cd, Bi, Mg, Al, Cu, Ni, Ti, Mo, W) [15], we can conclude that during the interaction of laser radiation of moderate power density with metals the erosion jet is consist of vapor, plasma and drop-liquid phase particles of target material. These particles play significant role in laser radiation weakening when it’s passing through the erosion products to the surface of target. Since the drop-liquid particles formed due to action of hydrodynamic mechanism appear in erosion jet only at recession of laser pulse density, its influence on the jet dynamics is not very noticeably in comparison with particles producing by the volume vaporization. In connection with that fact it seem to be interesting to investigate the processes which takes place during interaction of laser irradiation with drop-liquid phase particles of target material, that moving toward the radiation. But for salvation of this problem it’s necessary to obtain the measures of particle sizes and concentration during their formation and movement. One of the ways of control of condensed particles sizes in plasma fluxes at condition of real-time measurements is laser probing. This method is based on dependence of absorption and scattering characteristics of small particles on their sizes. When we compare the empirically measured ratio of absorbed and scattered components of probing radiation with theoretically calculated one, we can define the particle sizes. Moreover, the knowing of investigating medium length allows determining the numerical and volume concentration of particles. During simple laser probing only incident and passed components of radiation are usually measured. According this way it’s possible to define the total losses of radiation cause of absorption and scattering. But in our case it’s necessary to divide the probing light losses due to absorption and scattering separately. It can be realized by the placing of investigated object into the integrating sphere. The details of this method can be found at [16]. As it was mentioned above, all previous experiments were carried out with an employment of neodymium laser generating in regime of free-running. This way of generating is characterized by its time-scale shape: consecution of separately spikes with 1-3

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μs duration and 1-6 μs time delays between them. These spikes have random distribution of beam intensity cross-section, time of generation and amplitude. The typical time-scale shape of such laser pulse is placed at figure 5. It is reasonable that such space-time heterogeneity of laser irradiation complicates the treatment of investigations of interaction laser irradiation with a substance. We create the laser plant based on neodymium glass which allows generate the laser pulses possessing the rectangular time-scale shape (see figure 6) and changeable duration (50500 μs). Energy of pulse reaches 500 J and maximal heterogeneity of radiation is ~3%. The details of this equipment are described at [17]. To research the beginning of drop-liquid phase appearing due to acting of volume vaporization during the laser erosion of metal targets it’s necessary to have maximal sensitivity of measuring equipment during the control of the particle concentration. In this certain case we reached the minimal measuring numerical concentration of particles (liquid drops) close to value of 108 cm-3. These investigations were carried out with an empoyment of the special shaped targets. The acting laser irradiation was directed at the top of metal cylinder, the diameter of which was a little less than the beam diameter. It excluded the formation of erosion crater and in that way it was possible to realize cross-cut probing very close to target surface (down to 100 μm) by the subsidiary ruby laser irradiation. It allows surely detect the beginning of drop-liquid phase formation in products of laser erosion. In this project, we studied the erosion jets of Zn, Sn, Pb, and Ni targets. The experiments showed that, starting with a certain power density of the active radiation, particles of the fine dispersed liquid-drop phase, formed by volume vaporization, appear in the erosion jet virtually right after the onset of the action. The time resolution of the measurements in our case is determined by the interval between separate spikes of the probe radiation from the ruby laser, which lasts about 10 μs. The onset of the appearance of the particles because of volume vaporization is rather intense, and determining the time in this case is not difficult, but the accuracy is determined by a time of about 10 μs.

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t,μs Figure 7. Concentration of liquid drops of a lead target vs time for the following power densities of the active radiation: 1 – 0.46 MW/cm2, 2 – 0.23 MW/cm2, 3 – 0.15 MW/cm2.

Of the metals studied here, lead had the lowest threshold for the appearance of volume vaporization. When the power density of the active radiation of the neodymium laser is 0.46 MW/cm2, volume vaporization begins virtually right after the laser begins to act (curve 1, figure 7). When the power density of the active radiation is reduced, the volume vaporization begins with a certain delay, which is larger, the lower the power density is (curves 2 and 3, figure 7). It should be pointed out that the onset of counting on the figures coincides with the onset of lasing of the neodymium laser. In recording the time of the appearance of volume vaporization, when its delay with respect to the onset of action is observed, it is necessary to detect the time of onset of plasma formation. For this purpose, the radiation of the plasma in a wide spectral range was extracted from the zone of interest by means of a quartz capillary and was recorded by means of an FZU-68. The dynamics of the development of the erosion laser jet of metals can be tracked, using the lend target as an example. The experiments showed that there are conditions of laser

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action for each metal for which, at a certain time after the onset of the irradiation, the breakdown products consist of luminous vapor transparent for the radiation (figure 8a). A certain time after these vapors form in the erosion jet, fine liquid drops of the target material begin to appear in it because of volume vaporization (figures 8 b and c). After about 400 μS, larger particles with a significantly smaller concentration appear in the torch because of a hydrodynamic mechanism (figures 8 b and c). The same regularity appears for the other metals that were studied, except that the power density of the active radiation of the neodymium laser at which it occurs is characteristic of each metal. This is clearly seen in figure 9. Such behavior of the onset of volume vaporization is associated with differences of the optical and thermophysical characteristics of the metals. Rel.units

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t,μs Figure 8. Time variations of the luminescence of the erosion jet (a), size of the liquid drop (b), and their concentration (c), when radiation with a power density of 0.38 MW/cm2 acts on a lead target.

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Thus, experiments with a cylindrical target made it possible to carry out transverse laser probing close to the surface, which made it possible to determine the onset of volume vaporization in metallic laser targets from measurements of the concentration of fine liquid drops of the target material. The erosion jet initially consists of transparent luminous vapor of the target material; after a certain delay, fine liquid drops appear in the erosion jet because of volume vaporization. The delay of the appearance of volume vaporization depends on the power density of the active laser radiation and on the thermo physical characteristics of the target material. As the power density of the active radiation increases, the delay of the appearance of volume evaporation decreases.

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From certain for each metal power density the fine dispersed drop-liquid phase of target material formed due to volume vaporization appears almost without time delay (less than time resolution of experiment ~10 μs). In such conditions the sufficiently smooth changes of probing irradiation absorption and scattering can be observed during the whole laser acting. Increasing the power density of acting laser beam results the intensification of volume vaporizing process so the quantity of drop-liquid phase particles also increases. In the upshot, the absorption and scattering of probing irradiation grows in the same case and their timescale changes preserves smooth shape. But from some power density of acting laser radiation the time-scale curve of absorption coefficient gains sharp pulsations. The time-scale curve of scattering remains sufficiently smooth shape (as it can be seen at figure 10). K

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t,μs Figure 10. Time dependences of the scattering (curves 1) and absorption (curves 2) coefficients of a zinc target for plasma-forming laser radiation power densities 1 MW/cm2 (a), 2 MW/cm2 (b), and 4 MW/cm2 (c).

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To verify this, we, simultaneously with the probing of the laser erosion jet formed upon acting radiation of the neodymium laser on a zinc target with power density 3.7 MW/cm2, performed spectroscopic measurements of the jet plasma parameters in the probing zone with a time resolution of 5 µsec. The plasma absorption coefficient χpl was estimated using the values of the temperature T and electron concentration in the plasma Ne that were determined by spectroscopic techniques. The temperature was determined from the absolute intensity at the maximum of the self-converted spectral line ZnI 468.01 nm by Bartels’ method. The value of Ne was calculated from the obtained plasma temperature values and measured intensities of the continuous spectrum in the vicinity of the line indicated. The experimental procedure is described in more detail in [18]. Figure 11 presents the curves of the time dependences of the temperature T, electron concentration Ne, and of the absorption coefficient χpl of the jet plasma. Also, we give the absorption coefficient χj of the erosion jet that was measured by the probing technique. 3

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Figure 11. Time dependence of temperature (a), electron concentration (b), and absorption coefficients of plasma (c) and erosion jet (d) for the zinc target at a power density of plasma-forming laser of 3.7 MW/cm2.

Analysis of the curves in figure 11 shows that in transverse probing of the erosion jet, at the instant of the increase in the absorption coefficient χj, the plasma temperature and electron concentration increase; this is also accompanied by an increase in the absorption coefficient χpl. From this it follows that at the instant of a sharp increase in the absorption coefficient (see figure 11 c), a sharp plasma absorption or a plasma flash appears. Thus, at the instant of the plasma flash, laser radiation losses are principally caused by the basic mechanisms of absorption and scattering in the plasma. At smaller laser radiation energies, its losses are determined by the absorption and scattering at the particles of the liquid-drop phase of the target material which enter into the laser-induced erosion jet due to volume vaporization [19]. Experiments on simultaneously monitor the absorption coefficient in the erosion jet and the diameter of the target-material particles have showed that particle diameter decreases when the absorption coefficient increases. This confirms that the mechanism of low-threshold breakdown (absorption bursts) is operative in the erosion jet due to additional particle vaporization. Figure 12 shows results of an individual experiment involving the irradiation of a nickel target by a neodymium laser with a power density of 4.8 MW/cm2. It is apparent that numerous absorption bursts occur in the erosion jet as particle diameter decreases.

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t,μs Figure 12. Change over time in the absorption coefficient (curve 1) and particle diameter (curve2) in the erosive jet of a nickel target with a power density of 4.8 MW/cm2 for the plasma-forming laser.

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As it can be seen in the erosion laser jet there are a lot of sharp increases of absorption (due to plasma breakdown) with decreasing at these moments the diameter of particles. The effect of low-threshold absorption burst (plasma breakdown) is inspired by the vaporizing of the drop-liquid phase of target material. At this work that effect was for the first time obtained and explained. The effect of low-threshold plasma burst (plasma breakdown), which is inspired by evaporating drop-liquid phase of target material, was for the first time discovered and explained at [18]. Based on literature data and experimental investigations, we can represent the processes of interaction of laser radiation with metals as follows. At small laser radiation power densities, only heating of the metal occurs up to its melting. Here, an erosion jet is practically lacking. The target surface begins to evaporate as soon as the acting-radiation power density reaches a certain value characteristic of each metal. Starting at this instant, a noticeable erosion jet starts to appear which consists of the vapors and plasma of the target material. Since the jet parameters are small, there takes place adiabatic expansion of a practically transparent erosion plume. These regimes have been investigated in detail in [1, 12 and 20]. At moderate laser radiation power densities, a fine dispersed target-material liquid-drop phase produced by volume vaporization begins to enter into the laser-induced erosion jet. In the regimes of formation of laser-induced erosion jet with volume vaporization occurring in the melt zone, we deal with two-phase flows involving plasma and liquid drops. The liquid drops, while moving opposite to the laser beam, scatter and absorb its energy. Experiments have shown [2,15] that in this case the laser radiation losses are determined not by the basic processes proceeding in the plasma, but rather by the target material particles which enter into the erosion jet due to the volume vaporization. While moving, the particles overheat, and their residual evaporation occurs, thus creating a denser medium around themselves than is the case in adiabatic expansion of practically transparent vapors. Therefore, when the acting radiation reaches a certain power density, which is also its own for each metal, a lowthreshold plasma flash initiated by residual evaporation of liquid drops appears followed by a sharp increase in the plasma parameters. From this time on, laser radiation losses are essentially determined by the basic mechanisms of losses in plasma. At theoretical articles [21-26] the regimes of laser irradiation and matter interaction were in details discussed for the case of absence of a drop-liquid phase. So the base losses of laser irradiation were defined by the absorption in plasma. For a wide variety of metals, experiments were carried out to establish the ranges of the acting laser radiation power densities of different space-time forms, where radiation losses are determined by the liquid-drop particles of the target material, which are formed due to volume vaporization. The experiments have shown that the entry of the liquid-drop particles into the erosion jet after exposure of the metals to laser radiation is a common phenomenon. The results obtained are listed in table 1; it also presents the energy ranges in which the main losses of laser radiation occur due to scattering and absorption by the liquid-drop particles formed in erosion plumes because of volume vaporization. The experiments have shown that in the case of smooth-topped pulses the volume vaporization and plasma breakdown set in at higher power density of the acting laser radiation. This can be attributed to two reasons. First, for the regime of irradiation by laser pulses with 100% modulation, the radiation intensities q are determined as average values, but individual spikes can have an instantaneous power density which exceeds the mean one

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by two to three times. Second, as indicated above, in this regime it is observed that a larger number of particles enter into the erosion jet. Although the most exact measurement results correspond to the case of a rectangular pulse, they were performed for a small number of metals. The reason is that it is rather difficult to obtain a high power density in a big spot; however, the measurements indicated provide answers to the basic questions concerning the formation of erosion laser jets. For a wide range of metals the ranges mentioned were determined (see table 1) in the case of acting by a laser pulse with 100% modulation, since in the majority of cases laser processing employs the most widespread free-running mode of generation. Its characteristic feature is that the radiation pulse of duration 10–3 sec involves a collection of separate spikes of duration 10–6 sec. The proviso should be made concerning the data on copper and titanium listed in table 1. The high power densities of the laser radiation incident on copper are attributed to the fact that in the spectral domain of the neodymium laser generation (1.06 µm) the radiation reflection coefficient is very high. The low power densities in the case of titanium target exposure are attributable to the involvement of considerable exothermic reactions of oxidation in the process of plasma formation. Thus, investigations on the physics of interaction of laser radiation with metals enable one to optimize laser technologies of metal machining, in particular, cutting and drilling. The results listed in table 1 make it possible to determine the laser radiation power density at which precision machining of metals is possible. In this case, it is necessary to use the radiation power density below the level corresponding to the onset of volume vaporization. On the other hand, in view of the fact that the mass of the ejected material in the form of liquid-drop particles is an order of magnitude larger than the mass of vapors and plasma, the use of the energy ranges listed in table 1 allows one to obtain high-speed techniques of machining. Table 1. Energy Ranges (q, MW/cm2) of the Formation of Laser-Induced Erosion Plumes with a Small Dispersed Liquid-Drop Phase of the Target Material Target material Zinc Tin Lead Cadmium Magnesium Bismuth Aluminum Nickel Copper Titanium Tungsten Molybdenum Zirconium

Laser pulse with 100% modulator I II 0.9 2.65 – – 0.3 0.9 0.7 1.9 0.7 2.2 0.25 2.15 1.0 3.0 1.6 3.3 4.0 12 0.7 1.1 2.6 4.2 1.7 4.8 2.6 5.6

Comment. I – onset of volume vaporization, II – plasma burst.

Laser rectangular smoothtopped pulse I II 2.0 5.4 0.7 3.0 0.9 3.2 1.0 3.5 – – – – – – – – – – – – – – – – – –

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With an appearance of low-threshold plasma burst (plasma breakdown) effect due to metal drop-liquid phase vaporization, the absorption of laser irradiation in plasma significantly grows. In this case only slightly part of acting radiation energy reaches the target surface, so the implementation of those laser radiation parameters for drilling and cutting of metals possesses the low effectiveness. Moreover, it’s necessary to note, that presence of drop-liquid phase particles in erosion plasma jets significantly decreases the effectiveness of laser processing of metals in a number of cases. For example, the sputtering of quality microfilms, producing of laser-plasma light emitters, formation of laser generation media etc. But on the other hand the availability of drop-liquid phase particles in plasma jets allows improving the technologies of rapid laser processing, laser synthesis of nano- and microparticles, formation of compound materials etc. Let us discuss the problems of implementation of a drop-liquid phase forming during action of prolonged (~10-3 s) neodymium laser pulses on metals. As it was mentioned above, in this case the fine particles (10-100 nm) of drop-liquid phase initially arrive in erosion laser jet. These particles are formed due to acting of volume vaporization. Later, at the recession of laser radiation intensity the comparatively large particles (1-100 μm) appear in laser jet, the hydrodynamic mechanism is responded for their obtaining. Under the action on metals of a laser pulse in the free-running mode fairly large liquiddrop particles (dozens of microns) are formed by the hydrodynamic mechanism practically from the very beginning of the laser pulse. These particles can be registered in high-speed photographs in the light of either the plasma jet [27] or the laser (if the laser radiation wave length is in the spectral sensitivity area of the apparatus [28]). Particles of the liquid-drop phase of the target material not only scatter, but also absorb the laser radiation transmitted through the erosion jet. However, if metal targets are subjected to the action of a squared laser radiation pulse with a sufficiently high spatial-temporal homogeneity and with a sufficiently large irradiation area (when the diameter of the erosion hollow is much larger than its depth), then it is possible to separate in time the small particles formed by volume vaporization and the large ones formed by the hydrodynamic mechanism, which only appear at the end of the laser pulse. Since at this time there is no illumination by the laser or plasma radiation, these particles are usually invisible for experimenters. To study the dynamics of such particles, it is expedient to make use of the radiation from an auxiliary laser. The basic requirements for studying such probe radiation are that the radiation wavelength is in the spectral sensitivity area of the recording apparatus and the laser pulse duration is much shorter than the characteristic dynamic processes in the erosion laser jet (10−6 sec). The general scheme of the experiment on studying the dynamics of liquid-drop particles formed by the hydrodynamic mechanism in the light of the gating radiation from an auxiliary laser is shown in figure 13. In the present work, a target from aluminum alloy D16T was exposed to the radiation from a high-energy laser facility based in neodymium-activated glass working substances. The basis of the facility is a master laser with a confocal cavity that permits obtaining a quasi-stationary pulse of duration up to 1.5 msec, from which a squared pulse of duration from 50 to 500 μsec is cut out by a mechanical shutter. Then this pulse is amplified by two optical quantum amplifiers. The laser and the amplifiers are based on

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standard GOS-1001 M heads. The time form of the acting laser pulse is shown in figure 6. The high-energy laser facility is described in more detail in [17].

Figure 13. General scheme of the experiment: 1, 34) alignment laser; 2) diaphragm; 3, 4, 26, 29) deflecting mirrors; 5, 7) spherical mirrors; 6) working substance of the driving generator; 8) system of two cofocusing lenses; 9) rotating diaphragm; 10) electric motor; 11) lock-in tube; 12) firing photodiode; 13, 14) amplifiers; 15, 17, 18, 28) rotating plates; 19, 21, 24) lenses; 16, 27) photodiodes; 20) calorimeters; 22) target; 23) optical filters; 25) CCD matrix camera; 30–33) ruby laser components; 32) optical filter; 35–38, 41) pulse generators; 40) computer; 42) controlling generator.

A ruby laser in the monopulse mode of duration ~ 50 nsec was used as a probe laser. Qswitching was carried out by means of a phototropic shutter. Image recording in the light of the ruby laser radiation (λ= 694.3 nm) was performed with the use of a digital camera based on a CCD matrix. The remaining radiation was cut off spectrally by a set of glass and interference filters placed before the camera lens. Data acquisition, storage, and processing are automated and computer-controlled. To provide the required operating time of a particular unit of the experimental facility, we used a rather complex timing system based on a multichannel generator of delayed sync pulses. The purpose of all components of the general scheme of the experiment is given in the caption to figure 13. Varying the delay time of the ruby laser pulse permits obtaining images of the erosion laser jet at different instants of time and thus makes it easy to follow the evolution of the jet. The action has been made on a target from aluminum alloy D16T to a squared pulse of the radiation from a neodymium laser. The pulse duration was 450 μsec (see figure 14). The diameter of the irradiation spot on the target surface was 10 mm, and the power density of the neodymium laser radiation in the irradiation spot was 1.4 MW/cm2. The hollow depth upon irradiation was ~ 0.3 mm. Figure 14 shows the images obtained in the light of the gating ruby laser radiation. The acting time of each image was ~50 nsec. The zero time was assumed to be the instant at which the intensity of the squared pulse of the neodymium laser radiation began to decrease.

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Figure 14. Photographs of the liquid-drop phase of the erosion laser jet (irradiation spot diameter 10 mm) taken at different instants of time after the moment the laser pulse intensity began to decrease: a) Δt = 0; b) 170; c) 260; d) 350; e) 450 μsec.

As is seen from figure 14a, at the moment that the intensity of the laser radiation pulse begins to decrease, liquid drops formed by the hydrodynamic mechanism do not get into the erosion jet yet. It is also seen from the photograph that, unlike [27], we excluded background lights (from the plasma, the pump lamps, etc.) from the experiments by using not only glass but also interference filters. Liquid drops formed by the hydrodynamic mechanism appear only some time after the neodymium laser radiation pulse intensity begins to decrease, and, as is seen from the sequence of images, the jet widens with time. It should be noted that the target destruction products in the form of rather large particles escape at a small angle with the target surface in the form of a cone, and the opening of the cone therewith (~155°) practically remains unchanged in the course of time. This indicates that during the recording time the shape of the solid edges of the hollow and its depth also remain unaltered, i.e., under the action of the recoil momentum, when the laser radiation intensity decreases, the whole of the liquid layer of the metal begins to move from the center of the hollow to its edges and splashes out at an angle determined by the depth of the hollow and the shape of its edges. Processing of the photographs shown in figure 14 has made it possible to determine the velocity of motion of drops. At the beginning of motion they have a velocity of ~50 m/sec, and then it decreases to 25 m/sec. These results are in good agreement with the data of [27]. To elucidate the dynamics of the liquid phase from erosion crater in which the diameter and the depth are commensurable, we varied the focusing of the acting neodymium laser. In so doing, a spot of diameter ~ 5 mm was illuminated on the target surface. In this case, the radiation power density reached 17 MW/cm2. The hollow depth upon irradiation was ~0.6 mm. At such power densities of the laser pulse the intensity of evaporation and volume vaporization is much higher than in the case of focusing the laser radiation into a spot of diameter 10 mm. Due to this, in the course of time the erosion hollow has a greater depth. Therefore, liquid droplets formed by the hydrodynamic mechanism should escape at a larger angle with the target surface.

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Figure 15. Photographs of the liquid-drop phase of the erosion laser jet (irradiation spot diameter 5 mm) taken at different instants of time after the moment the laser pulse intensity began to decrease: a) Δt = 0; b) 30; c) 250; d) 450 μsec.

The results of the experiments are presented in figure 15. As follows from figure 14a, at the instant of time at which the intensity of the neodymium laser radiation pulse begins to decrease large drops do not yet appear, but after some time (see figure 15b) it is seen that large drops get into the erosion laser jet, and the angle between the direction of their escape and the target surface is greater than in the previous experiments. These points to a stronger influence of the crater walls on the mechanical trajectory of liquid drops in this experiment. Processing of the experimental results has made it possible to determine the velocities of motion of particles. Initially, they reach ~100 m/sec and then decrease to 25 m/ sec. This points to the fact that inside the hollow a higher pressure is realized due to the increase in the laser radiation intensity. As is seen from figure 15, in the course of time the angle between the trajectory of escaping particles formed by the hydrodynamic mechanism and the target surface increases. It should be emphasized that now neither the laser radiation nor the plasma radiation are present; however, the hollow depth increases with time. This point to a gradual ousting of the liquid phase formed during the laser action by the recoil momentum upon a sharp decrease in the laser pulse intensity. Such dynamics of the laser erosion products can be explained by the liquid metal viscosity. Thus, with the aid of a monopulse of the illuminating (probe) pulse of a ruby laser we have visualized the nonluminous laser erosion products and studied the dynamics of their expansion. The experiments on the investigation of the dynamics of the liquid-drop phase formed under the action of a squared laser radiation pulse on metals have shown that under our conditions liquid drops formed by the hydrodynamic mechanism begin to get into the erosion torch only after a decrease in the laser radiation intensity. With large irradiation spots, when the erosion crater diameter is much larger than its depth, liquid drops escape at a small angle

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with the target surface (varying slightly with time), and the recoil momentum therewith ousts the liquid throughout the thickness. With smaller irradiation spots, when the diameter of the erosion crater is commensurable to its depth, the liquid-drop phase of the target material formed by the hydrodynamic mechanism also appears only after a decrease in the laser radiation intensity. However, it escapes at a larger angle with the target surface and this angle grows with time. This points to a gradual deepening of the hollow in the absence of both the radiation and the plasma, which is probably connected with the value of the liquid metal viscosity in the irradiation area. Taking into account the fact that small (nanosized) liquid-drop particles of the target material getting into the erosion torch during the acting laser pulse move perpendicularly to the target surface, the results of the present investigation can be used for spatial separation of small particles formed by volume vaporization and large particles formed by the hydrodynamic mechanism by with an employment a diaphragm. This effect can be applied for producing of water suspensions with nickel nanoparticles. Nowadays several physical methods of nanoparticles suspensions formation are known: electrical discharge in liquids and laser erosion in liquids. The metal electrodes (Ni, W, steel) placed in liquid medium (water, ethanol) are used for the metal nanoparticles suspension production by the electrical discharge method. The electrodes destroy and great amount of micro- and nanoparticles form during the electric discharge (ark [29] or spark [30]). They consist of every possible sorts and kinds of chemical compounds of electrode and liquid medium materials. The applying of spark discharge [30] provides more effective electrode material entry to discharge zone in compare with ark discharge. The employment of alternating current discharge is governed by the necessity to have uniform erosion of both electrodes. The WC nanoparticles with the diameter from 2.5 to 35 nm were obtained [30]. The possibility of metal (Ag, Au, Ti, Cu and their alloys) nanoparticle production by the laser erosion is under recognition at the work [31]. It was showed that during laser erosion of metal target placed in liquid medium the significant quantity of nanosize objects (with dimensions of 10-100 nm) is formed. The second harmonic of Nd:YAG laser (λ=532 nm) or irradiation of copper vapor laser (λ=510 nm) were applied for the laser acting. Both of these methods possess significant disadvantage: the chemical purity of obtained suspensions seems to be very questionable. The chemical composition of given nanosize objects includes all chemical spectrum (and every possible sorts and kinds of their combinations) both the target (electrode) material and liquid medium substance as the processes of particle formation pass in liquid medium. Scheme of the experiment of water suspensions with nickel nanoparticles getting is placed at figure 16. We want note, that laser irradiation possesses the property of sterility, and fine drop-liquid phase moves in atmosphere of target vapor. So metal particles penetrate into water having low temperature in comparison with case of laser ablation of metals in water medium. The installation base [17] on Nd-glass laser with λ=1.06 μm. The laser radiation has been focused in a diameter 10 mm. The water suspension of nanoparticles was produced by the multi action of the laser irradiation on nickel target. Particles characteristics in water medium were controlled with the employment of laser probing method.

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Figure 16. Scheme of the laser erosion mechanism; 1) Acting laser irradiation, 2) The particles formed by hydrodynamic mechanism, 3) The particles formed by the volume evaporation, 4) Target, 5) Mask, 6) Pan with a liquid.

At the present work the analysis of two examples was made. The first one – water suspension of nickel particles, obtained with an employment of a diaphragm (small particles was in water). The second one – water suspension has nickel particles, obtained without this procedure (i.e. small and large particles flow move in water). It has been determined parameters by the method laser probing of the sample obtained using a diaphragm: the average particles diameter 85 nm, average particles number concentration 1.2×109 cm-3. The probing of the sample obtained without a diaphragm shows that the large particles ~ 1 µm are in a water suspense.

Figure 17. Picture of the surface relief of a substrate with nickel particles obtained: a) without the procedure of diaphragming after evaporation, the electron microscope; b) the same, the atomic-force microscope; c) with the procedure of diaphragming after evaporation, the atomic-force microscope. (x and y, coordinates on the sample’s surface in scanning by the atomic-force microscope, μm; z, coordinate reflecting the level of the sample’s relief in scanning by the atomic-force microscope, nm;).

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The comparative analysis of suspense’s (by the atomic-power and electronic microscopy) was made to prove the results, obtained with laser probe method. The water samples were evaporated and drifted on substrate. Relief surface image of substrate with nickel nanoparticles, obtained without diaphragm is given in figure 17a. This image was taken with an employment of the electron microscope. Average particles size at the image is 3-5 µm. This fact is in agreement with data obtained with an employment the probing laser method. However we can see at this image that there are a lot of submicron particles with the relatively large particles. They are in the resolution limit of the electron microscope. Submicron particles of this sample have been found with an employment of the atomic-power microscope (figure 17b). At this image we can clear see the relatively large particles ~2-4 µm and small particles ~40-70 nm. The investigating results of the particles obtained with an employment of diaphragm are given in figure 17 c. This image has been made with an employment of the atomic-power microscopy. The nickel particles were prepared similar previous case. Even the primary relief surface investigation of the substrate shows a significant reduction of the large particles quantity. Image analysis (figure 17 c) shows that effective particles diameter is 80 nm with dispersion 40% . Effective particles diameter measured by the laser probe method is in agreement with results obtained with an employment of atomic-power microscopy. Thus it’s possible to produce nanosize nickel particles with effective diameter ~70-80 nm and their suspense’s using the metal laser erosion. The main advantage of given method is independence the process of nanoparticles suspense production on the type of penetrating medium and its physical-chemical properties. So it’s possible to produce metallic suspense’s of nanoparticles in different media apart from corrosive mediums to material nanoparticles. The method of laser probing allows real-time obtaining of average sizes and concentration of drop-liquid phase particles in transparent media with high level of accuracy. So we have discussed some questions of laser technologies dealing with implementation of drop-liquid particles of target material forming at erosion laser jet. In cases when the presence of that particles is harmful we’ll concern the possibility of controlling particle size and concentration. The efforts to obtain erosion plasma flows of metal targets, which would be free of drops, by increasing the power density of the acting radiation led to the fact that smaller particles began to enter the erosion flame; however, the concentration of particles increased. In this connection, it is very important to find methods that would allow one to control the diameters and concentrations of the liquid-drop phase of erosion laser jets of metals and to find methods of producing erosion jets free of drops. In such experiments, the target was placed between two plates to which an external electric field perpendicular to the axis of the flame was applied. The action was performed by a rectangular pulse of the neodymium laser radiation with a power density of 0.46 MW/cm2 . The size and concentrations of liquid drops were monitored by the method of probing of the jet by the radiation of an auxiliary ruby laser. The probing was performed at a height of 1 mm from the target surface. The electric field was changed from 0 to 4 kV/cm. Figure 18 a shows the time variation of the diameter of particles in the erosion jet of the lead target. For a strength of the external electric field of 1 kV/cm, the curve of time variation of the diameter of the particles 2 reaches the maximum after 400 µsec, while without an external electric field the maximum is observed at the 200th µsec (curve 1) after the

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beginning of the action. At the strength of 4 kV/cm, the maximum of curve 3 corresponds to a time of 700 µsec. The number concentration of the particles increases with increase in the strength of the external electric field (figure 18 b). It may also be suggested that as the strength of the external electric field increases further, the diameters of the liquid-drop-phase particles will decrease and the number concentration will increase. Such a time behavior of the parameters of the liquid-drop-phase particles can be explained by the spatial redistribution of electric charges on the surface of the particles, which stimulates their crushing [32, 33].

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The investigations have shown that in the case where neodymium-laser radiation acts on a lead target exposed to the external electric field, the concentration of particles in the erosion laser flame is higher and the particle size is smaller as compared to the case where the action is performed without an electric field. This phenomenon can be used for control of the size and concentrations of particles in two-phase flows formed as a result of the action of laser radiation on metals. We now consider the kinetics of particles of the condensed phase of the metal-target material in crossed laser beams when the acting laser radiation is directed perpendicularly to the surface of the target and the auxiliary laser radiation is directed parallel to the surface at a

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certain distance from it. The auxiliary laser radiation interacts with the erosion products without changing the conditions on the target surface [34]. As the acting radiation, we used nearly rectangular pulses of a neodymium laser with a 400 – 500 µsec duration. In all the experiments, the power density of the acting radiation was 1.4 MW/cm2 for a diameter of the irradiated spot of 0.6 cm. As the additional vaporizing radiation, we used the radiation of a pulsed neodymium laser operating in the free-running mode. The pulse duration was ∼10−3 sec and the diameter of the laser beam in the additional vaporization zone was 0.8 cm. The axis of the additional vaporizing laser beam was positioned at a distance of 2 mm from the target surface. The investigations were carried out for different power densities of completely vaporizing radiation. The general pattern of formation of the liquid-drop phase of the target material in the absence of additional vaporizing laser radiation is presented in figure 19. It should be noted that the size of large particles is somewhat understated in our measurements, since, on the one hand, the procedure of monitoring of the particle size by the ratio between the scattered and the absorbed components of the probing ruby-laser radiation (λ = 0.7 µm) is difficult to use for a lead target for a particle size of more than 10 µm. In this case, monitoring of large particles was performed with a large error, because of the ambiguity of the ratio between the scattered and the absorbed components for some diameters and because of the weak dependence of the particle size on this ratio. On the other hand, because of the insignificant amount of very large particles in the erosion jet, the probability exists that they fly past the probing beam, whose diameter is 1 mm in our case. However, in the present work, most of the attention is concentrated on monitoring the size of particles that become smaller in the process of their additional vaporization. In this case, the use of the procedure proposed is quite justified.

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t,μs Figure 19. Time behavior of the laser radiation intensity (1) (relative units) and time variation of the size of liquid-drop phase particles (2) in the erosion jet of the lead target.

The kinetics of the condensed phase in crossed laser beams can depend strongly on the particle size. Therefore, it makes sense to consider the cases where the following particles are present in the erosion jet: (1) small particles owing to volume vaporization (for the time of action of the plasma-forming pulse of a neodymium laser); (2) mainly large particles formed

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through the hydrodynamic mechanism (as is seen from figure 19, this occurs 650–700 µsec after the beginning of laser action on the target); (3) particles formed through the two mechanisms (according to figure 19, this occurs 450–500 µsec after the beginning of laser action). The experiments have shown that additional vaporization of condensed phase particles formed owing to volume vaporization begins for a power density of the additionally vaporizing laser of ∼0.1 MW/cm2, while for ∼0.5 MW/cm2 particles become so small that the probing ruby laser radiation scattered by them is beyond the sensitivity limits of the measuring system. Figure 20 shows the results of experiments on the action of neodymium laser radiation with a power density 1.4 MW/cm2 on a lead target in the absence of additionally vaporizing laser radiation and in the presence of such radiation of power density q = 0.23 MW/cm2 . It is seen that even for a low intensity of the additionally vaporizing laser radiation (as compared to the acting radiation) the size of the condensed-phase particles and their concentration markedly decrease. For a higher power density of the additionally vaporizing laser radiation, particles of the liquid-drop phase of erosion laser jet are no longer detected.

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Figure 21. Variation of the size (a) and the number (b) and volume concentrations (c) of particles of the liquid-drop phase of the erosion flame of the lead target with increase in the power density of the additionally vaporizing laser. The moment of detection is 500 (1) and 700 (2) μsec after the beginning of the action.

Figure 21 (curves 1) shows results of experiments for an instant of time of 500 µsec after the beginning of the action of a radiation pulse of a neodymium laser on a target, when condensed phase particles formed through the two mechanisms of formation are present in the erosion jet. In this case, the acting radiation also had a power density of q = 1.4 MW/cm2. The power density of the additional vaporizing radiation changed from 0 to ∼0.1 MW/cm2. The

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increase in the dependence d = f(q) to a certain maximum, which is shown in figure 21 a (curve 1), can be explained by the fact that the smallest particles formed owing to the process of volume vaporization are completely vaporized and a larger fraction of large particles formed through the hydrodynamic mechanism remains in the jet. As the power density of the additionally vaporizing radiation increases further, the size of the large particles decreases. The foregoing is well illustrated by figure 21 b (1), where the behavior of the number concentration of particles as a function of the change in the power density is shown. The decrease in the volume concentration, which is shown in figure 21 c (1), is evidence in favor of a fairly effective additional vaporization of the condensed phase of the target material for an increased power density of the additionally vaporizing radiation. If the energy of the additionally vaporizing laser radiation increases further, the liquid-drop phase particles are vaporized completely. At an instant of time of 700 µsec after the beginning of the action of laser radiation on the target, when fairly large particles formed through the hydrodynamic mechanism are present in the erosion flame (see figure 21a (2)), the particle size decreases with increase in the power density of the additionally vaporizing radiation. However, in this case, the number concentration increases initially (see figure 21b (2)), which can be explained by the crushing of certain particles due to the superheating, and then, because of the competition of the processes of crushing of particles and their vaporization, additional vaporization becomes predominant and the number of particles decreases. The behavior of the volume concentration (see figure 21c (2)) points to the fact that the mass of the material contained in the particles of the condensed phase decreases significantly with increase in the power density of the additionally vaporizing radiation, and particles can be vaporized completely for a certain power density. Thus, investigations of the kinetics of particles of the condensed phase of erosion flames in crossed laser beams have shown that, using the action of auxiliary-laser radiation, one can exert efficient control of the parameters of the liquid-drop-phase particles and in so doing of the parameters of the erosion laser flames themselves. It is shown that to do this it is necessary that the power density of the additionally vaporizing laser radiation be lower than the power density of the acting radiation and much lower for particles formed owing to volume vaporization. The power densities necessary for additional vaporization of particles formed through the hydrodynamic mechanism can also be decreased if an erosion laser flame is exposed to an external electric field: liquid-drop phase particles crushed under the action of electric forces are easier to completely vaporize by auxiliary- laser radiation. The results of the investigations carried out can be used to obtain controlled model twophase flows in which it would be possible to change the parameters of the liquid-drop phase (diameters of the particles and their concentration) using an external electric field and an additional completely vaporizing laser radiation and to obtain a "sterile" erosion plasma free of drops, which is necessary in solution of a number of technical problems.

Conclusion So the interaction of laser irradiation with metals can be represented by the following way. The part of laser electromagnetic waves energy is absorbed by the electrons of conductivity. These electrons transmit energy to the crystal lattice points due to the process of

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non-elastic collisions. When the laser irradiation intensity is comparatively low, a metal target only heats. But during further increasing of light power density the metal melts with following boiling and intensive vaporizing. At that time the laser erosion jet begins to form, it consists of almost transparent metal vapors. Below the part of vapors ionizes and there erosion jet consists of vapors and plasma at this case. The concentration of charged particles is relatively low cause of low level of ionization. So that particles can’t have a significant influence on the transport of laser irradiation to the target surface. In according with described regime of interaction the velocity of vaporizing front propagation is less than velocity of melting front. So at the bottom of interaction zone the volume of liquid metal is formed. Increasing of power density inspires the process of volume evaporation in zone of liquid phase and the fine (1-100 nm) liquid drops begin to propagate to erosion laser jet. Experiments show a delay of volume evaporation appearance in dependence on power density of laser irradiation. The increasing of laser irradiation intensity results the decreasing of delay of fine drop-liquid phase appearance due to volume evaporation. For the definite values of power density (distinctive for each type of metal) the process of volume evaporation starts very close to beginning of laser action (in comparison on laser pulse duration). The fine drop-liquid phase of target material moves toward the laser beam, absorbs and scatters of acting laser irradiation. Initially the losses of incident radiation are negligible. But increasing of power density leads to losses growth. So the fine particles of metal drop-liquid phase fly in laser beam, absorb the part of its energy, divide into pieces and form significantly dense medium around itself (in comparison on adiabatically spreading transparent vapors). At definite values of power density of acting irradiation (distinctive for each type of metal) in erosion laser jet the effect of low-threshold plasma bursts (plasma breakdown) has taking place. This effect is inspired by drop-liquid phase vaporizing. The temperature and charged particle concentration rapidly grow and lead to sharp increasing of absorption of laser irradiation in plasma. In this case only slightly part of acting irradiation energy reaches the target surface, the residuary part of radiation only enlarges the plasma parameters. So the implementation of that laser irradiation for drilling and cutting of metals possesses the low effectiveness. Except the fine drop-liquid phase of target material appearing due to volume evaporation at erosion laser jet there are more large-scale (1-100 μm) drops. It formed by the hydrodynamic mechanism. In case of short (~100 ns) laser pulses acting at metals the dropliquid phase of target material forms due to condensation at the end of action. At present work we discussed some questions of laser technologies dealing with implementation of drop-liquid particles of target material forming at erosion laser jet. In cases when the presence of those particles is harmful we’ll concern the possibility of controlling (with help of electric and magnetic fields) particle size and concentration. So the drop less sterile laser plasma can be finally observed.

References [1] S. I. Anisimov, Ya. A. Imas, G. S. Romanov, and Yu. V. Khodyko, Effect of HighPower Radiation on Metals [in Russian], Moscow (1970).

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[2] V. K. Goncharov, V. I. Karaban', and A. V. Kolesnik Quantum Electronics. 15 (4), 498 (1985). [3] H. Griem, Plasma Spectroscopy, McGraw-Hill, New York (1964). [4] Burgess and M. J. Seaton, Mon. Not. R. Astron. Soc. 120, 121 (1960). [5] W. J. Karzas and R. Latter, Astrophys. J. Suppl. 6, 167 (1961). [6] D. R. Inglis and E. Tуller, Astrophys. J. 90, 439 (1939). [7] Optical Properties of Hot Air [in Russian], Moscow (1970). [8] V.A. Kas'yanov and A.N. Starostin, Zh. Eksp. Teor. Fiz. 48, 295-302 (1965) [Sov. Phys. JETP 21, 193 (1965). [9] Ya. B. Zel'dovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, 2 vols., Academic Press, New York (1966, 1967). [10] P.I. Ulyakov, Zh. Eksp. Teor. Fiz. 52, 820 (1967) [Sov. Phys. JETP 25, 537(1967)]. [11] O.I. Putrenko and A.A. Yankovskii, Zh. Prikl. Spektrosk. 15, 596 (1971). [12] N.N. Rykalin, A.A. Uglov, and A.N. Kokora, Laser Processing of Materials [in Russian], Mashinostroenie, Moscow (1975). [13] N.N. Rykalin and A.A. Uglov, Fiz. Khim. Obrab. Mater. No. 2, 30 (1970). [14] B.M. Zhiryakov, N.N. Rykalin, A.A. Uglov, and A. K. Fannibo, Zh. Tekh. Fiz. 41, 1037 (1971) [Sov. Phys. Tech. Phys. 16, 815 (1971)]. [15] V K Goncharov, V I Karaban', V A Ostrometskiĭ Quantum Electronics, Volume 16 (1986), Number 6, Pages 808-811. [16] V. K. Goncharov, V. L. Kontsevoi, M. V. Puzyrev, and A.S. Smetannikov, Instruments and Experimental Techniques. Vol 38, No. 5, part 2 661–666 (1995). [17] V.K. Goncharov. J. Opt. Technol. 67, 968- (2000). [18] A P Byk, V K Goncharov, V V Zakhozhiĭ, V I Karaban', A V Kolesnik, V V Revinskiĭ, A F Chernyavskiĭ Role of particles of target material in the dynamics of plasma formation Quantum Electronics, Volume 18(1988), Number 12, Pages 1605-1609. [19] V. K. Goncharov Action of High-Energy Neodymium Laser Radiation Pulses Having a Different Space-Time Shape on Metals // Journal of Engineering Physics and Thermophysics. Volume 74, Number 5 / Сентябрь 2001 г 1053-1355 [20] J. F. Ready, Effects of High-Power Laser Radiation [Russian translation], Moscow (1974). [21] G. G. Vilenskaya and I. V. Nemchinov, Zh. Prikl. Mekh. Tekh. Fiz., No. 6, 3–19 (1969). [22] G. G. Vilenskaya and I. V. Nemchinov, Dokl. Akad. Nauk SSSR, 186, No. 5, 1048–1051 (1969). [23] G. G. Vilenskaya and I. V. Nemchinov, Zh. Prikl. Spektrosk., 11, No. 4, 637–643 (1969). [24] V. Nemchinov and S. P. Popov, Pis’ma Zh. E′ksp. Teor. Fiz., 11, No. 9, 459–462 (1970). [25] V. Nemchinov and S. P. Popov, Zh. Prikl. Mekh. Tekh. Fiz., No. 5, 35–45 (1971). [26] V. I. Bergel’son, A. P. Golub’, T. V. Loseva, I. V. Nemchinov, T. I. Orlova, S. P. Popov, and V. V. Svetsov, Kvantovaya E′lektron., 1, No. 5, 1268–1271 (1974). [27] V. I. Nasonov, Journal of Applied Spectroscopy 73 No 2, 234-244 (2006). [28] В.К.Гончаров, Л.Я.Минько, С.А.Михнов, В.С.Стрижнев, Квантовая электроника. №5, 112-116 (1971). [29] N. Parkansky, B. Alterkop, R. L. Boxman, S. Goldsmith, Z. Barkay, and Y. Lereah, Powder Technol., 150, No. 36, 123–128 (2005).

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[30] В.С. Бураков, Н.А. Савастенко, П.Я. Мисаков, Н.В. Тарасенко. Труды ИМАФ НАНБ, С. 435-437. 2005 г. [31] А.В. Симакин, В.В. Воронов, Г.А. Шафеев. Труды института общей физики им. Прохорова, 60, С.83-107. (2004). [32] V. E. Badan, V. V. Vladimirov, and V. Ya. Poritskii, Zh. Tekh. Fiz., 57, No. 6, 1197– 1198 (1987). [33] V. E. Badan, V. V. Lisitchenko, and V. Ya. Poritskii, Zh. Tekh. Fiz., 59, No. 8, 141–142 (1989). [34] V K Goncharov, M V Puzyrev Quantum Electronics. 27 (4) 319-321 (1997).

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 473-492

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 7

THERMODYNAMIC AND KINETIC STUDY OF OIL SHALE PROCESSING G.Y. Gerasimov1, E.P. Volkov2 and E.V. Samuilov2 1

Institute of Mechanics, Moscow State University, Moscow, Russia. 2 G. M. Krzhizhanovski Power Engineering Institute (ENIN), Moscow, Russia.

Abstract This chapter describes the main features of oil shale transformation under thermal processing. Thermodynamic investigation of oil shale gasification in oxygen was performed using the TETRAN computational software and associated database. In this way, adiabatic gasification temperatures were calculated as a function of oxygen excess. Optimal conditions of synthesis gas generation were determined. Kinetic model of oil shale thermal decomposition (pyrolysis) was constructed on the base of analysis of available experimental data. Model includes the kinetics of organic matter (kerogen) decomposition at high temperatures, the processes of heat and mass transfer inside of single oil shale particle, polydispersity of the particles, their fragmentation, and the secondary chemical reactions (cracking, condensation, and polymerization of the shale oil inside of the particles with coke deposition). The model was applied to engineering procedure that simulates the Galoter process (pyrolysis of oil shale in a horizontal rotary drum-type reactor in contact with fine-grained solid heat carrier, namely, hot ash obtained from solid residue combustion).

Introduction Limited nature of the world reserves of conventional fossil fuels such as crude oil, natural gas, and coal raises a problem of alternative energy sources search. In this connection, oil shale can be considered as a substitute of the conversional fuels [1]. The largest oil shale resources in the world are concentrated in the United States with total amount of 3,340 billion tones or 62% of the world’s known recoverable oil shale reserves [2]. The use of oil shale

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represents for many countries a valuable potential source for production of liquid hydrocarbons and energy [3]. Oil shale is characterized by a high content of mineral components, which are closely connected with organic matter (kerogen). Therefore, the direct combustion of oil shale in boilers is confronted with some technical difficulties. Nevertheless, oil shale is used in some countries (Estonia, Israel, China, and Germany) for electricity generation [3–6]. The character of the kerogen is a high content of hydrogen (the mass ratio C/H is approximately equal to eight and is close to corresponding value for raw oil) [1]. In this connection, the thermal processing of oil shale allows to transform up to 90% of its organic mass into vapour products and to receive not only high quality boiler fuel, but also motor fuels [7] and valuable chemical raw materials [8]. Oil shale processing is a complex approach to solid fuels utilization, at which purely power processes (fuel combustion and heat release) are coupled with technological ones (gasification and pyrolysis) [5]. The main purpose of this approach is an increase of the efficiency of low-grade fuel conversion to the heat and electricity with simultaneous minimization of the environmental impact. The research efforts and industrial applications of oil shale processing are concentrated now on the improvement of the existing technologies as well as on the development of new methods of oil shale use. Gasification of solid fuels is one of the promising technologies for power generation [9]. The effects of reaction medium, residence time, temperature, pressure, etc. on conversion levels and product gas composition were studied in numerous investigations as applied to gasification of coal [10-11], oil shale [12-13], biomass [14-15], and waste [16-17]. Considerable research attempts were recently made in study of oil shale pyrolysis. It was shown that thermal conversion of kerogen depends on kerogen type, reaction temperature, heating rate, particle size, etc. [18-22]. Various decomposition models were proposed to explain the experimental data. This study is further advancement in understanding of the main features of oil shale processing leading to effective extraction of organic matter. Two theoretical methods were used for the description of oil shale behaviour at high temperatures, namely, thermodynamic approach as applied to oil shale gasification in oxygen atmosphere, and kinetic approach as applied to the Galoter process (pyrolysis of oil shale particles in a horizontal rotary drum-type reactor in contact with fine-grained solid heat carrier) [5].

Description of Model Approaches The oil shale transformation at heating conditions depends on multitude of thermochemical, mass-transfer, and kinetic processes. The high temperature gasification of oil shale can be considered as thermodynamically favored process with limiting state of the reacting system realized at sufficiently long its stay under given medium parameters [23]. More reliable data especially in the range of low temperatures can be received with use of kinetic methods. First of all this relates to consideration of oil shale pyrolisis, which includes the kinetics of kerogen decomposition, heat and mass transfer inside of single oil shale particle, the secondary chemical reactions, etc. The common description of these processes is impossible and requires some simplifications and model approaches.

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Thermodynamic Equilibrium Calculation Procedure The description of complex, multi-component, multi-phase chemically reacting system such as oil shale gasification products involves great difficulties associated with the necessity of taking into account a large number of chemical reactions, many of which practically are not understood. Therefore, the thermodynamic equilibrium analysis is till date one of the main computational methods to investigate the chemistry of solid fuels gasification process. Equilibrium study, however, has several limitations connected with ignoring of any head- and mass-transfer processes, chemical kinetics, surface phenomenon, and physical changes of the fuel during heat up [24]. The results of an analysis of different thermodynamic calculations show that the obtained equilibrium composition of the chemically reacting system is very sensitive to the selection of the initial information [25]. Calculations must take into account all possible compounds for considered system. Thermodynamic data for these compounds must be correct and consistent. Considerable disagreement in the final results can also occur when different computational procedures and convergence criteria are employed [26]. Chemical composition of oil shale gasification products was determined using the TETRAN software system [27] based on reliable Gibbs free energy minimization algorithm and consistent thermodynamic database. Continuous operation of the system showed its reliability, a stable convergence of iteration process, and the adequacy of obtained results to experimental data [28]. A distinctive feature of the system is direct calculation of multi-phase chemically reacting systems without preliminary analysis of the possibility for the condensed compounds appearance. The TETRAN thermodynamic database includes characteristics of about 2000 different compounds as applied to thermal conversion of solid fuels, which are taken from various sources, particularly, from [29, 30] or calculated by the authors. Our knowledge about properties of all substances formed during thermal conversion of solid fuels doesn’t allow carrying out of accurate thermodynamic calculation of such complicated chemical systems. Therefore, the model of ideal solutions and the model of pure phases were used the TETRAN calculating complex for the description of properties of condensed phases. Table 1. Composition of the oil shale considered in the study As received (wt%) Moisture Ash Carbon Hydrogen Oxygen Nitrogen Sulphur

10.7 43.7 24.8 2.5 16.7 0.1 1.5

Bulk ash (wt% as oxide) SiO2 Al2O3 Fe2O3 CaO MgO K2O Na2O TiO2

25.5 6.5 5.4 55.2 4.4 1.8 0.7 0.5

The investigation of oxygen gasification process was carried out for Baltic kukersite oil shale from “Vivikond” open-pit mine (Estonia). Table 1 shows the characteristics of the oil shale that are used in the predictions [31]. The elements C, H, O, N, and S in table 1 represent the oil shale organic matrix, whereas such elements as Si, Al, Fe, Ca, Mg, K, Na, and Ti represent the mineral matter. The assumed composition of products, which are formed under

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gasification, consists of 180 chemical components including 85 condensed ones. In accordance with calculating procedure, all condensed components must be distributed among phases. The total number of phases is determined by Gibbs rule of phases. The concrete phases composition can be received using data, which represent forms of various substances being in initial oil shale. One can use also, for example, the following considerations. The oil shale contains considerable proportion of calcium carbonate, CaCO3. This component at temperature increase gives rise to formation of CaO, CaC2, Ca3N2, and CaSO4 in the condensed state. These substances are localized in the vicinity of CaCO3, which leads to formation of the calcium phase. It is clear that some inaccuracies are possible at this procedure. But, energies of substances interaction in solution, as a rule, are noticeably lower of their chemical energy that is determinant at the formation of substance in the system.

Kinetic Model of Oil Shale Pyrolysis The oil shale kerogen is a homogeneous molecular compound insoluble in organic solvents and resistant to the action of the majority of chemicals. The molecular matrix of the kerogen can be represented as the complicated three-dimensional polymer, which consists of aromatic, hydroaromatic, and heterocyclic structures (clusters) connected with each other by aliphatic chains and contained various functional groups as substitutes [32]. The chemical structure of the kerogen is studied using a combination of macroscopic, spectroscopic, and pyrolytic methods, which allow identifying of its main units [33]. Kerogen of Baltic kukersite is highly aliphatic organic compound [34] belonged to kerogen of II/I type in classification scheme [35]. It is abundant in oxygenated aromatic compounds such as alkylphenols, alkylresorcinols, and alkylhydroxybenzofurans. All phenolic structure units contain long (up to C19), linear alkyl side-chains with low branching level [36-37]. The modeling of the kerogen molecular structure is based on analysis of experimental data, and is a useful tool for the description of the main steps of the thermal decomposition of this complex macromolecule [38]. Numerous attempts were made to give the theoretical description of the processes occurring during the thermal decomposition of oil shale kerogen. Some models operate with parallel firstorder reactions with different activation energies and frequency factors to include the rupture of various chemical bonds with different energies in kerogen macromolecule [39-40]. Others consider first-order kinetics in terms of consecutive chemical reactions describing the main degradation steps of kerogen at high temperatures [41-43]. The kinetic parameters of kerogen pyrolysis for various kinds of oil shale were determined from experimental data by using of manifold analysis techniques including: the Direct Arrhenius Method, Integral Method, Differential Method, Friedman Procedure, Maximum Rate Method etc. [43-44]. The thermal conversion of kerogen in the general case depends on kerogen type, reaction temperature, heating rate, particle size, etc. [45]. Its dynamics is described by effective rate constant that depends from process conditions. Flash pyrolysis of fine-grained oil shale particles (less than 100 μm in size), which have a low internal thermal resistance and heated with rates of the order of 104 – 105 K/s, allows reducing to minimum the influence of the secondary processes and singling out in pure form the kinetics of the kerogen decomposition [46-47]. The rate constant of the process becomes independent of the particle diameter, heating rate, ambient conditions, etc.

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Thermal Decomposition of Kerogen It is assumed in the following consideration that oil shale pyrolysis can be described by first-order rate process that proceeds through various intermediate stages before the final products are produced. Various kinetics schemes have been proposed to explain the process for different kerogen types [45]. In the general case, the kerogen (KER) under heating conditions transforms into a plastic state followed by the release of tar (TAR) and subsequent disintegration of the last into the mixture of liquid hydrocarbons, gas (GAS), and coke (COKE). The mixture of liquid hydrocarbons is usually separated on shale oil (OIL) and lighter component, i.e., gasoline (BEN). The analysis of the experimental data [48] on the flash pyrolysis of kukersite kerogen allows proposing the following kinetic scheme of this kerogen decomposition: k1 k2 KER ⎯ ⎯→ TAR ⎯⎯→ OIL ⎯⎯→ GAS + BEN,

(1)

where k1 and k2 are the rate constants of the tar and the shale oil thermal cracking. The dynamics of the decomposition process and the release of volatiles as a gas phase in accordance with this kinetic scheme under the assumption of first-order reactions relatively to the concentrations of the components are described by the equations: dci/dt = wi,

(2)

wtar = -k1 xtar, woil = k1 ctar – k2 coil, wgas = k2,gas coil, wben = k2,ben coil, where ci is the concentration of ith component, kg/(kg of dry oil shale). The values of k1 for kukersite kerogen restored on the basis of the experimental data [48] can be approximated by the dependence: k1 = 6.31 × 1013 exp(-25600/T) s-1, which is close to data [49] for kerogens of II/I type [35] (see figure 1). The corresponding expression for k2 was received in [50] as a least-square fit to experimental data [48]: k2 = k2,gas + k2,ben = 9.0 × 103 exp(-7400/T) s-1, k2,gas/k2,ben = 4. Figure 2 shows the calculated curves ci = ci(T) obtained using Eqs. (2) in comparison with measured products concentrations of kukersite pyrolysis [48].

Secondary Chemical Reactions The kinetic scheme (1) is true for small oil shale particles having a low internal thermal resistance. Large particles (more than 100 μm in size) are heated not instantaneously, and considerable temperature gradients appear in them. Pyrolysis products formed in the interior of the particle migrate outside, which is accompanied by different secondary reactions such as cracking, condensation, and polymerization of the tar with the deposition of a certain amount of coke and release of the corresponding amount of gas. In this case, the kinetic scheme (1) must be supplemented with the competitive mechanism: k3 TAR ⎯⎯→ COKE + GAS.

(3)

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The rate constant k3 for this process can be described by the temperature dependence: k3 = 5.5 × 1011 exp(-23000/T) s-1 received on the base of calculated curves fit to experimental data [50] taking into account that k3,coke/k3,gas = 1.5.

102

Rate constant, s-1

101 100 10-1 10-2

10-3

0.0012 0.0014 Inverse temperature, K-1

0.0016

Figure 1. Rate constant k1 of kerogen decomposition as a function of T. Solid and dotted lines are model predictions [48] and [49], accordingly, for kerogen of II/I type; points are experimental data: high-temperature (●) for kukersite [48] and low-temperature (O, , Δ) for three different oil shales [51].

Concentration, wt.%

80 OIL 60

40 20 0 750

KER GAS BEN

800 850 900 Temperature, K

950

Figure 2. Comparison of calculated and experimental data [48] on the yield of pyrolysis products for Baltic kukersite oil shale.

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The volatiles released as a result of the oil shale pyrolysis are filtered through the layer of solid particles (the mixture of fuel and solid head carrier, namely, shale ash) before they arrive in the zone of free space in the upper part of the reactor. Since the mass flow rate of the heat carrier is, as a rule, much higher than the flow rate of the fuel, a considerable contribution to the kinetics of secondary chemical reactions can be made by the carbonization of shale oil on the heat carrier particles. The carbonization kinetics of shale oil on ash particles has been considered in [52], where the following kinetic equation has been proposed to describe a change in the coke concentration ccoke, kg/(kg of dry shale):

dccoke / dt = wcoke ,

wcoke = (Ga / G f )k coke y oil (1 + Bcoke k coke y oil t ) , −1

(4)

where Ga and Gf are the mass flow rates (kg/s) of the solid head carrier (ash) and fuel, accordingly; kcoke is the rate constant of carbonization of the shale oil vapor on ash particles (m3/kg s); yoil is the concentration of the shale oil in the gas (kg/m3); Bcoke is deactivation constant; t is the process time. The quantity yoil can be represented in the form yoil = ρgasGoil/Ggas, where Goil and Ggas are the rates of shale oil and gas output in reactor volume (kg/s). The gas density ρgas can be described by the following approximation expression: ρgas = 1.10 (300/T) kg/m3. The quantities Goil and Ggas are determined as a result of the modeling study of the pyrolysis process. The carbonization kinetic constants are equal: Bcoke = 215, kcoke = 0.0104 m3/kg s in accordance with the experimental data [52].

Heat and Mass Transfer in Drum-Type Pyrolysis Reactor As stated above, oil shale particles soften under heating and pass through stage of the plastic state. The elimination of particles agglomeration in these conditions can be reached owing to continuous motion of the mixture of fuel and solid head carrier in the reactor. The simple construction that can realize this requirement is horizontal rotating drum. Despite the wide use such reactors in various industries (cement, silicate, chemical, metallurgical, and others), their investigations are not universal. Most of the works deal with inclined open-end drums having a relatively large length and low coefficient of the working volume filling. The special property of the drum-type pyrolysis reactor is that it belongs to short (the relation between the length L and the diameter D is equal to 2 – 3) horizontal drums rotating with a low velocity (1 – 2 rpm) and operating with a high degree of the volume filling with moving material (ϕ = 0.3 – 0.6) at a relatively short time of stay of the solid phase here (τr = 600 – 1000 s) and a comparatively high temperature (T = 700 – 900 K) [53].

Motion of Fine-Grained Material in Reactor The simplest way of the reactor volume filling with the solid phase is by installing of supporting rings at the drum ends. One welds connections to them on which end seals of special structure are installed. The solid phase is fed through these connections. The finegrained material moves in the axial direction due to the difference in the heights of the layer at the inlet and outlet of the drum. The design of the outlet end of the reactor according to such scheme is determined by the supporting device in the form of a ring with a hole diameter

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d (relatively small as compared to D) and cylindrical connection of length l. The slot between the exterior surface of the rotary connection and the stationary system is blown with water vapor to keep the vapor-gas mixture in the reactor from the escape. The scheme of the pyrolysis reactor of considered type is shown in figure 3. The velocity of solid phase motion ur in the horizontal direction (or time of particles stay in the reactor τr = L/ur) and degree of filling of the reactor with the solid phase ϕ depend on the flow rate of the fuel Gf and the flow rate of the solid head carrier (ash) Ga, the angular rotational velocity of the reactor ω, and its diameter D and length L. They also depend on the dimensions of the supporting device (diameter d and length l) in the reactor with end seals. It was established on the base of experimental data [53] that:

ϕ = ARsp Fr 0.05 (d / D ) (1 + l / D ), q

τ r = ρ s ϕV / (G f + Ga ),

(5)

where Rs = 2π (Gf + Ga)/(ρsωV); Fr = (ω/2π)2D/g; ρs is average density of solid phase; V = 0.25πD2L is the reactor volume; and g is the acceleration of free fall. At Rs = 0.01 – 0.20; Fr = (3 – 110) × 10-5; L/D = 1.8 – 5.0; d/D = 0.24 – 0.64; and l/D = 0.005 – 0.5 the parameters A, p, and q in (5) are equal: A = 0.066 exp(0.95 L/D), p = 0.18 (L/D – 1) d/D, and q = 0.5 (L/D – 3.1).

L D

l

d

Figure 3. The scheme of the horizontal rotary drum-type pyrolysis reactor.

Heat Transfer between Particles of Fuel and Heat Carrier One of the main stages in elaboration of the mathematical model of solid fuel thermal decomposition is carrying out of the submodel of individual fuel particle behavior in the high temperature region. To ascertain the basic regularities of the heat transfer between particles of solid fuel and the heat transfer agent in the pyrolysis reactor one can use the model of plug flow reactor, in which each volume element of a medium is considered as an open chemical system moving along the reactor axis with the velocity ur and exchanging by the matter and energy with ambient medium. Let us assume that the size of head carrier particles is sufficiently small to be considered as isothermal and that their mixing with fuel particles occurs instantaneously and uniformly at the inlet of the reactor. To take into account the polydispersity of the fuel, the fuel particles are divided into fractions in each of which the particle size is the same. The equation of

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481

energy balance between the particles of the solid head carrier and the fuel for constant heat capacities of the particles and in absence of heat loss can be written as: N

Rk

сa Ga [Ta ,0 − Ta (t )] = c f G f ∑ f k (4 / 3πRk3 ) −3 ∫ [T (r ,t ) − T f ,0 ]4πr 2 dr . k =1

(6)

0

Here ca and Ta are the heat capacity and temperature of the heat carrier particles; N is number of fractions; fk is mass part of the particles of the kth fraction in the total mass of the fuel; Rk is the radius of the particle in the kth fraction. It is assumed that the thermal effect as applied to thermal decomposition reactions of the fuel organic mass is small [54]. The equation of heat conductivity describing the heating of the fuel particles can be represented in the spherical coordinate system as:

ρ f c f ∂T (r ,t ) / ∂t = λ f r −2 ∂ / ∂[r 2 ∂ / T (r ,t ) / ∂r ].

(7)

Here ρf and λf are the density and thermal conductivity of the fuel particle. The boundary conditions on the particle surface and at its center have the form:

− λ f ∂T (r ,t ) / ∂r

r = Rk

= β[T ( Rk ,t ) − Ta (t )],

∂T (r ,t ) / ∂r

r =0

= 0.

(8)

The heat from fine-grained material to the fuel particle is transferred mainly by conduction through the gas interlayer [50]. This process can be represented as the heat exchange between two surfaces separated by distance δ with the Nusselt number Nu = βδ/λgas = 1.0, where β is the coefficient of heat exchange between the fuel particle and the heat carrier. It can be taken δ ≅ 0.13 Rk for the effective gas-interlayer thickness at Rk = 5 – 10 mm [50]. Numerical solution of the system of heat exchange equations simultaneously with the system of kinetic equations allows calculating the dynamics of the solid fuel thermal decomposition in its motion along the reactor axis. The rate of release Gi of the ith product component from the oil shale particles into the free volume of the reactor is equal: τr Rk ⎡N ⎤ Gi = G f ∫ ⎢∑ 3 f k Rk−3 ∫ r 2 wi (r ,t )dr + wcoke ,i (t )⎥dt . ⎥⎦ 0 ⎢ 0 ⎣ k =1

(9)

It is assumed that the volatiles released from the oil shale particles seep rapidly into the free volume of reactor, where the decomposition of OIL into GAS and BEN begins. Temporal integration in (9) from the process beginning up to t gives the rate of ith product component realization on the interval from the beginning of fuel feeding to the coordinate z = urt. Oil shale represents a finely granular porous structure consisting of an inorganic skeleton and organic disseminations with size up to 100 μm. The initial porosity of oil shale particles ε0 is fairly high and is equal 0.19 for kukersite [50]. The porosity of the particles increases during the process of thermal destruction of their organic mass attaining its maximum value εmax, when total conversion of the organic mass to volatiles is occurred. As a result of the

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mechanical action of the solid heat carrier (ash), the inorganic skeleton of the oil shale particle is destroyed. The thermal destruction of the organic mass of large oil shale particle begins from the periphery gradually moving into the particle. Its porosity has the maximum value near the particle surface. Let us assume that the process of the inorganic skeleton destruction begins when the local porosity ε(r,t) near the particle surface attains a certain critical value εcrit. As a result, the particle radius decreases, while the outer shell disintegrates and mixes with the ash mass acquiring its temperature and rapidly losing the remains of the organic mass. Mathematically this appears in a procedure of interrupting of the heat conductivity equation solution at the radius Rcrit when the porosity ε = εcrit. The boundary condition (8) is also shifted into the particle interior, which leads to an increase of the heating rate and to decrease of the characteristic time of the basic decomposition process. On the other hand, the integration in Eq. (9) with respect to r is continued to the radius Rk, which gives the possibility to take into account the release of volatiles from the disintegrated particle shell.

Chemical Reactions in Free Volume of Reactor The realized products of the thermal destruction of oil shale particles do not leave the reactor immediately. They move for a certain time in its volume that free of the fuel and solid heat carrier. During this time, the reaction of OIL decomposition into GAS and BEN occurs. To take into account this process, the model of well-stirred reactor was used in which a homogeneous composition of the reacting gas mixture is attained due to intense mixing. The initial mixture with constant mass velocities of the components Gi,0 goes into the reactor volume. The values Gi,0 are a result of the solution of the problem considered above (see Eq. (9) in which the values Gi are replaced by Gi,0). The temperature in the volume is assumed to be equal to the steady-state temperature of the mixture consisting of the fuel and solid heat carrier. The system of kinetic equations that describes the temporal change of the concentrations ci can be represented in the form:

dci / dt = −(ci − ci 0 ) / τ + wi ,

(10)

where τ = ρgas(1 - ϕ)V/Ggas is the characteristic time of gas mixture stay in well-stirred reactor; ϕ is the degree of filling of the reactor with the solid phase; Ggas is the rate of gas output in reactor volume V during the pyrolysis process. The chemical reaction rates wi for individual components include only the decomposition of shale oil (OIL) into gas (GAS) and gasoline (BEN). The most important feature of the processes in well-stirred reactor at constant rate of mass transfer is the establishment of the steady state, in which dci/dt = 0 and ci = ci0 + τwi as it follows from (10). Since the reaction rates wi are linear relative to the concentrations, the concentrations ci in the latter expression can be replaced by the corresponding mass flow rates. In this way, the following system of algebraic equations can be obtained with account for expressions (2) for the quantities wi:

Thermodynamic and Kinetic Study of Oil Shale Processing

G gas = G gas ,0 + τk 2 ,gas Goil ,

Goil = Goil ,0 − τk 2 Goil ,

483

Gben = Gben ,0 + τk 2 ,ben Goil .

This system reduces to a quadratic equation for the mass flow rate Ggas using simple transformations. Thus, somewhat changed values of the quantities Gi as compared to Gi0 are obtained at the outlet from the pyrolysis reactor.

Results and Discussion The described above mathematical models allow reproducing the total behaviour of oil shale under conditions of gasification and pyrolysis processes. In spite of some limitations connected with model assumptions, the received results can be useful for estimation of the oil shale products yields in wide range of the process parameters. The calculations were carried out as applied to Baltic kukersite oil shale.

Equilibrium Analysis of Oil Shale Gasification The conversion of oil shale particles during oxygen gasification process proceeds in the oxygen atmosphere. An important process parameter is the excess oxygen factor α, the ratio of oxygen quantity used to its stoichiometric quantity for complete oxidation of oil shale organic matter. The calculation of the equilibrium composition of the chemically reacting system oil shale/oxygen was performed for two values of this parameter: α = 0.4 and 0.3, which are typical for oxygen gasification of solid fuels.

Yield, mole/(kg of oil shale)

20 H2

10 8 6

H2O CO2

4 CH4 2 1 400

CO

C*

800 1200 1600 Temperature, K

2000

Figure 4. Main products of oil shale gasification at oxygen excess α = 0.4.

The equilibrium composition of gasification products was calculated using the distribution of components along the following phases: gas phase, phase of liquid

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hydrocarbons including condensed carbon C*, calcium phase, phase of silicon oxides, phase of aluminosilicates, and phase of iron oxides. The results of calculations are given in figure 4 and 5, where the composition of the generated gas is represented as a function of the process temperature. It is observed the increase in the amount of fuel gas under temperature growth with CO and H2 being the major components in temperature range 900 – 1600 K. Simultaneously with calculation of equilibrium composition of oil shale gasification products the enthalpy of the system h(T, p) was calculated, which allows to determine the adiabatic temperature of gasification TA from the correlation: h(TA, p0) = hinit(T0, p0). The initial enthalpy of the system is calculated from the expression:

(

)

hinit (T0 , p 0 ) = Δh° f 298.15 1 + αν O2 μ O2 , where Δh° f 298.15 is enthalpy of oil shale formation; ν O 2 is the number of oxygen moles necessary for combustion of 1 kg of oil shale at α = 1; μ O 2 is molecular mass of oxygen. The enthalpy of oil shale formation for Baltic kukersite is equal: Δh° f 298.15 = - 8947.6 kJ/kg. Calculations show that adiabatic temperatures of gasification in cases α = 0.4 and 0.3 are equal 1500 and 1075 K, accordingly. The composition of generated gas at adiabatic temperature is equal: H2 = 9.57, CO = 15.24, H2O = 8.72, CO2 = 5.37 in case α = 0.4; and H2 = 14.31, CO = 14.41, H2O =3.88, CO2 = 4.64 mole/(kg of oil shale) in case α = 0.3. It is obvious that the second choice is preferable. Figure 6 presents the calculated equilibrium concentrations of the calcium phase at oxygen excess α = 0.4. As it follows from analysis of calcium components yields, calcium carbonate decomposes practically completely at adiabatic temperature TA = 1500 K for this case with formation of CaO and CaS.

Yield, mole/(kg of oil shale)

20 10 8 6

CO H2 H2O

C*

4 2 1 400

CO2 CH4 800 1200 1600 Temperature, K

2000

Figure 5. Main products of oil shale gasification at oxygen excess α = 0.3.

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485

Yield, mole/(kg of oil shale)

102 CaCO3 100 CaS 10-2 CaO 10-4 CaSO4 10-6 400

800 1200 1600 Temperature, K

2000

Figure 6. Yields of components for calcium phase at oxygen excess α = 0.4.

Yield, mole/(kg of oil shale)

20

CO H2

10 8 6

H2O

4 2 1 400

CO2 CH4 800 1200 1600 Temperature, K

2000

Figure 7. Main products of oil shale gasification at oxygen excess α = 0.3 without taking into consideration of condensed C*.

The determination of the equilibrium composition of the generated gas needs taking into account of various representatives of hydrocarbons both gaseous (CH4, C2H6, HCO, etc.) and liquid (C5H12O, C7H16, C8H18, etc.). As calculations show, concentrations of all these substances except methane are small or equal zero. The steady components are only condensed carbon and methane (see figures 4 and 5). It is possible that the consideration of hydrocarbons as a separate thermodynamic system will give somewhat changed results. Figure 7 presents the equilibrium composition of the generated gas received at the assumption that the concentration of condensed carbon C* is equal zero. As it was in the previous case (see figure 5), methane remains the predominant hydrocarbon component.

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Oil Shale Pyrolysis in Drum-Type Reactor Calculation of the pyrolysis process begins with the determination of the degree of filling of the reactor with the solid phase ϕ and the characteristic time of stay of a solid phase in the reactor τr in accordance with Eq. (5). As data input, the flow rates of the fuel Gf and the solid head carrier (ash) Ga, the angular rotational velocity of the reactor ω, and its geometric parameters are prescribed. The drum-type reactor with oil shale productivity Gf = 200 tons/day = 2.32 kg/s has been considered as an example. At ratio Ga/Gf = 3; angular rotational velocity of the reactor ω = 0.15 s-1 (1.43 rpm); and reactor geometric parameters D = 2 m, L = 4 m, d = 1 m, l = 0.5 m the degree of filling of the reactor is ϕ = 0.369; the characteristic time is τr = 600 s; and velocity of motion of the solid material in the horizontal direction is ur = 6.667 × 10-3 m/s. 800 360

700 Temperature, K

240 600 120 500 t = 60 s

400 300

4 6 8 2 Particle radial distance, mm

0

10

Figure 8. The dynamics of oil shale particle warming-up under fuel/solid heat carrier motion along the reactor axis: Ga/Gf = 3, R = 10 mm, Ta,0 = 850 K, Tf,0 = 293 K.

The subsequent computational procedure involves the solution of the heat conductivity equation (7) and energy balance correlation (6) that enables to describe the process of fuel particles heating and corresponding cooling of solid heat carrier. To take into account the polydispersivity of the fuel, the oil shale particles were divided into a certain number of fractions with set fraction size calculated as the average size within the limits of each fraction. As an example, the following fractional composition of the fuel was chosen: fk Rk, mm

0.07 1.25

0.23 2.50

0.40 5.00

0.25 7.50

0.05 10.0

where fk is mass part of the particles of the kth fraction in the total mass of the fuel and Rk is the radius of the particle in the kth fraction. It is assumed that the initial temperature of dry

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fuel particles is Tf,0 = 293 K, and the initial temperature of the heat carrier is Ta,0 = 850 K. The values of the thermo-physical characteristics for solid particles were chosen as following: cf = 1.0 kJ/(kg K) for the heat capacity, ρf = 1200 kg/m3 for the density, and λf = 0.3 W/(m K) for the heat conductivity. The initial mass content of kerogen in the dry mass of the oil shale is cker,0 = 0.376 kg/(kg of oil shale) and the initial porosity is ε0 = 0.19. The calculation results of warming-up of fuel particles with Rk = 10 mm under fuel/heat carrier motion along the reactor axis are shown in figure 8. As one can see, the heating of particles is completed on a reactor length z/L of the order of 0.6. Smaller particles are heated more rapidly, and their thermal destruction comes to the end at a shorter distance from the site of feeding of the fuel and the solid heat carrier.

Normalized concentration

1.0 0.8 0.6

0.4

3

0.2

1

2

0 0

0.8

1.6 2.4 Axis distance, m

3.2

4.0

Figure 9. The influence of particles fragmentation on the dynamics of kerogen decomposition under fuel/solid heat carrier motion along the reactor axis. Critical porosity εcrit is equal 0.25 (1); 0.3 (2); 0.4 (3); Ga/Gf = 3, L = 4 m, Ta,0 = 850 K, Tf,0 = 293 K.

A series of calculations with different values of the critical porosity εcrit was carried out to evaluate the influence of fuel particles fragmentation on their heating and the rate of volatiles release. The particle porosity for Baltic kukersite changes from the initial porosity ε0 = 0.19 to the maximum porosity εmax = 0.50 when the entire organic mass is released into gas phase in the form of volatiles. Figure 9 presents the results of calculations describing the dynamics of kerogen destruction at various values of εcrit. It is seen that a change of this parameter in the interval from 0.25 to 0.50 has small influence on the behaviour of cker. The results of calculations at εcrit equal to 0.4 and 0.5 (maximum value of porosity for kukersite) are practically coincident. Therefore, the consequent calculations were carried out at the most real value of εcrit equal to 0.4 that corresponds to a residual kerogen concentration ccrit equal to 0.117 kg/(kg of oil shale).

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The dynamics of the kerogen decomposition inside of particles with Rk = 10 mm is shown in figure 10. It is seen that the kerogen concentration in particles decreases as the mixture of fuel and solid heat carrier moves along the reactor axis, and when it reaches the value equal 0.117 that corresponds accepted critical porosity, the external shell of the particle disintegrates and mixes with ash mass acquiring its temperature. At the same time, the shell fragments lose quickly the remainder of organic mass, which shown by dashed line.

Concentration, kg/(kg of oil shale)

0.5 t = 180 s

0.4

300 0.3

360

0.2

420

0.1 0 0

2

4 6 8 Radial distance, mm

10

Figure 10. The dynamics of the kerogen decomposition inside of the oil shale particle. Ga/Gf = 3, R = 10 mm, Ta,0 = 850 K, Tf,0 = 293 K, cker,0 = 0.376 kg/(kg of oil shale), εcrit = 0.4.

Normalized concentration

1.0 0.8 KER 0.6 OIL 0.4 COKE GAS

0.2 0 0

1

2 3 Axis distance, m

4

Figure 11. The dynamics of kerogen transformation in products of its decomposition under fuel/solid heat carrier motion along the reactor axis. Ga/Gf = 3, L = 4 m, Ta,0 = 850 K, Tf,0 = 293 K.

Thermodynamic and Kinetic Study of Oil Shale Processing

489

Normalized concentration

1.0 0.8 0.6 OIL

0.4 0.2

COKE

0 0

1

2 3 Axis distance, m

4

Figure 12. The dynamics of shale oil coking under fuel/solid heat carrier motion along the reactor axis. Dashed lines show corresponding data calculated without taking into account the OIL deposition on the ash particles.

10 GAS

0.4

8

t

6

0.3 0.2 0.1

0 1.0

4

OIL

2

BEN

1.5

2.0

Residence time, s

Normalized concentration

0.5

2.5

0 3.0

Flow rates ratio, Ga/Gf Figure 13. The composition of vaporized pyrolysis products at the reactor exit and their residence time in free volume of the reactor as a function of Ga/Gf at G = 9.27 kg/s and final system temperature 710 K.

The dynamics of the kerogen conversion into the products of its thermal destruction at the plug flow conditions (without taking into account of shale oil decomposition in the free volume of the pyrolysis reactor) is presented in figure 11. An analysis of the figure shows, that the concentration of the shale oil as well as the concentration of the condensed carbon (coke) in the particles of the fuel increases as the kerogen is destroyed. The role of the

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kinetics of OIL deposition on the ash particles in the general process of the pyrolysis products carbonisation is shown in figure 12. One can see that the carbonisation on the ash particles yields nearly half of the total amount of coke removed with solid phase from the pyrolysis reactor. Figure 13 presents the kinetics of shale oil decomposition in the volume of the pyrolysis reactor free of the fuel and the solid heat carrier (well-stirred conditions). The calculations were carried out for different ratios Ga/Gf at constant total flow rate of the mixture fuel/heat carrier and final temperature of the mixture. The results of calculations show that the mass concentration of the gas (GAS) and gasoline (BEN) weakly depends on the ratio Ga/Gf. The decrease of the OIL mass concentration with increase of Ga/Gf is explained by the growth of the OIL carbonisation on the ash particles with increase of the flow rate Ga. The effect of the kinetics of oil shale decomposition in the free volume of the reactor is distinctly revealed at the comparison of the mass concentrations in figure 11, where the data of calculation for Ga/Gf = 3 are given, and the mass concentrations in figure 13. The GAS concentration is doubled due to the decomposition of OIL, and the OIL concentration decreases nearly three times.

Conclusion The purpose of this study is revealing of the main features of high temperature processing of oil shale leading to effective extraction of its organic matter. Thermodynamic investigation of oil shale gasification in oxygen atmosphere as applied to Baltic kukersite shows that the amount of the generated gas increases at temperature growth with CO and H2 being the major components in temperature range 900 – 1600 K. The content of these components at adiabatic temperature of gasification is higher for lower oxygen excess in the reacting system. The developed engineering procedure, which simulates the oil shale thermal decomposition in the Galoter process, allows estimating of the influence of main process parameters on the yield of gaseous and liquid products. In particular, calculations show that secondary chemical reactions lead to a considerable carbon removal from the reactor owing to shale oil coking on particles of solid carrier. The reactions of decomposition of pyrolysis products in the superlayer space of the reactor also decrease the shale oil fraction in the total mass of the volatile pyrolysis products. Such results are very important in determination of optimal process conditions leading to the maximum thermal conversion of the oil shale organic matter into liquid shale oil.

References [1] Russell, P. L. Oil Shales of the World, Their Origin, Occurrence and Exploitation; Pergamon Press: Oxford, 1990. [2] Altun, N. E.; Hicyilmaz, C.; Hwang, J.-Y. et al. Oil Shale. 2006, 23, 211-227. [3] Bredow, K. Oil Shale. 2003, 20, 81-92. [4] Fainberg, V.; Hetsroni, G. Energy Sources. 1996, 18, 95-105. [5] Volkov, E.; Stelmakh, G. Oil Shale. 1999, 16, 161-185. [6] Jiang, X. M.; Han, X. X.; Cui, Z. G. Prog. Energy Combust. Sci. 2007, 33, 552-579.

Thermodynamic and Kinetic Study of Oil Shale Processing [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

491

Abu-Quidais, M.; Al-Widyan, M. I. Energy Conver. Manag. 2002, 43, 673-682. Akar, A.; Ekinci, E. Fuel. 1995, 74, 1113-1117. Minchener, A. J. Fuel. 2005, 84, 2222-2235. Harris, D. J.; Roberts, D. G.; Henderson, D. G. Fuel. 2006, 85, 134-142. Gräbner, M.; Ogriseck, S.; Meyer, B. Fuel Process. Technol. 2007, 88, 948-958. Rubel, A. M.; Robl, T. L.; Carter, S. D. Fuel. 1990, 69, 992-998. Jaber, J. O. Energy Conver. Manag. 2000, 41, 1615-1624. Weerachanchai, P.; Horio, M.; Tangsathitkulchai, C. Biores. Technol. 2009, 100, 14191427. Xu, G.; Murakami, T.; Suda, T. et al. Fuel Process. Technol. 2009, 90, 137-144. Yassin, L.; Lettieri, P.; Simons, S. J. R.; Germana, A. Chem. Eng. J. 2009, 146, 315327. Dalai, A. K.; Batta, N.; Eswaramoorthi, I.; Schoenau, G. J. Waste Manag. 2009, 29, 252-258. Jaber, J. O.; Probert, S. D.; Williams, P. T. Energy. 1999, 24, 761-781. Williams, P. T.; Ahmad, N. Fuel. 1999, 78, 653-662. Arro, H.; Prikk, A.; Pihu, T. Fuel. 2003, 82, 2179-2195. Aboulkas, A.; El Harfi, K. Oil Shale. 2008, 25, 426-443. Nazzal, J. M. Energy Conver. Manag. 2008, 49, 3278-3286. Jarungthammachote, S.; Dutta, A. Energy Conver. Manag. 2008, 49, 1345-1356. Sandelin, K.; Backman, R. Environ. Sci. Technol. 1999, 33, 4508-4513. Thompson, D.; Argent, B. B. Fuel. 2002, 81, 345-361. Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; Wiley Interscience: New York, 1982. Gerasimov, G. Y.; Zhegulskaya, N. Y.; Rozhdesrvenski, I. B. et al. Mat. Model. 1998, 10, 3-16. Gerasimov, G. Y. J. Eng. Phys. Thermophys. 2003, 76, 98-103. Glushko, V. P.; Gurvich, L. V.; Bergman, G. A. et al. Thermodynamic Properties of Individual Substances; Nauka: Moscow, 1978-1982; Vol. 1-4. JANAF Thermochemical tables; 2nd ed.; NBS: Washington, 1971; Suppl. 1974-1982. Vdovchenko, V. S.; Martynova, M. I.; Novitski, N. V.; Yushina, G. D. PowerGenerating Fuel of the USSR: Handbook; Energoizdat: Moscow, 1991. Vandenbrouke, M.; Largeau, C. Organic Geochem. 2007, 38, 719-833. Riboulleau, A.; Derenne, S.; Sarret, G. et al. Organic Geochem. 2000, 31, 1641-1661. Mastalerz, M.; Schimmelmann, A.; Hower, J. C. et al. Organic Geochem. 2003, 34, 1419-1427. Ganz, H.; Kalkreuth, W. Fuel. 1987, 66, 708-711. Derenne, S.; Largeau, C.; Casadevall, E. et al. Organic Geochem. 1990, 16, 873-888. Bajc, S.; Ambles, A.; Largeau, C. et al. Organic Geochem. 2001, 32, 773-784. Lille, Ü.; Heinmaa, I.; Pehk, T. Fuel. 2003, 82, 799-804. Li, S.; Yue, C. Fuel. 2003, 82, 337-342. Aboulkas, A.; El-Harfi, K.; El-Bouadili, A. J. Mater. Process. Technol. 2008, 206, 1624. Skala, D.; Kopsen, H.; Neumann, H. J.; Jovanović, J. Fuel. 1989, 68, 168-173. Khraisha, Y. Energy Convers. Manag. 1998, 39, 157-165. Torrente, M. C.; Galán, M. A. Fuel. 2001, 80, 327-334.

492 [44] [45] [46] [47] [48] [49] [50] [51]

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Li, S.; Yue, C. Fuel Process. Technol. 2004, 85, 51-61. Karabakan, A.; Yűrűm, Y. Fuel. 1998, 77, 1303-1309. Khraisha, Y. Energy Convers. Manag. 2000, 41, 1729-1739. Olukcu, N.; Yanik, J.; Saglam, M.et al. J. Analyt. Appl. Pyrolysis. 2002, 64, 29-41. Gerasimov, G.; Ter-Oganesyan, G. Comb. Explos. Shock Waves. 2001, 37, 309-314. Behar, F.; Vandenbroucke, M.; Tang, Y. et al. Organic Geochem. 1997, 26, 321-339. Gerasimov, G. J. Eng. Phys. Thermophys. 2003, 76, 1310-1317. Sresty, G. C.; Dev, H.; Snow, R. H.; Brides, J. E. Proc. 15th Oil Shale Symp.; Colorado School of Mines: Golden, 1982; pp 411-423. [52] Udaja, P.; Duffi, G. J.; Chensee, M. D. Fuel. 1990, 69, 1150-1154. [53] Krasnovski, G. A. Study of Drum-Type Reactors for Thermal Processing of FineGrained Oil Shale and Methods of Their Design; PhD Thesis; ENIN: Moscow, 1970. [54] Saxena, S. C. Prog. Energy Combust. Sci. 1990, 16, 55094.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 493-508

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 8

RADIATION INDUCED SYNTHESIS AND MODIFICATION OF CARBON NANOSTRUCTURES G.Y. Gerasimov Institute of Mechanics, Moscow State University, Michurinsky Avenue 1, 119192 Moscow, Russia

Abstract This chapter gives a brief review of the recent progress in application of radiation techniques for synthesis and modification of carbon nanostructures. The review includes the examination of available experimental data as applied to radiation-induced formation of carbon nanostructures such as fullerenes, concentric-shell carbon clusters (onions), single- and multiwall carbon nanotubes, and graphene sheets. The considerable part of these experiments deals with modification of initial carbon materials under electron beam irradiation in an electron microscope, which is of paramount importance because it allows in situ observation of dynamic processes on an atomic scale. It is also discussed the modification of carbon nanostructures under irradiation: polymerization of fullerenes films, modification of carbon nanotubes structure, junction of crossing carbon nanotubes, etc. The research part of the work contains the theoretical estimations of carbon nanostructures stability under electron beam irradiation, which use the analytical approximation of the cross-section for the Coulomb scattering of relativistic electrons by carbon atom nuclei as well as results of classical molecular dynamics simulations.

Introduction Nanotechnology is one of the most developing directions in science and engineering arose on the crossing of physics, electronics, chemistry, biology and material science [1]. Nanotechnology deals with materials, which size or structure are controlled at the nanometer scale (less than 100 nm). Carbon nanostructures (CNSs) such as fullerenes, carbon nanotubes (CNTs), and graphene sheets occupy an important place among nanomaterials due to their unique properties, which form the basis of new technological applications. These applications include electronic devices [2, 3], CNSs-based polymers [4, 5], composite materials [6, 7],

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hydrogen storage systems [8], biosensors [9], devices with highly focused ion beams [10], etc. Fullerenes represent clusters of carbon atoms arranged in a closed cage of spherical or more complicated 3D form [11]. The most common member of the fullerenes family is C60 structure, which attracts the greatest attention of researchers. In addition to the laser ablation technique first used for synthesis of C60 [12], fullerenes are also produced in macroscopic amounts by the arc discharge method [13] and oxidative combustion of hydrocarbons [14, 15]. All these methods typically give a mixture of fullerenes of different sizes with dominant production of C60. The unique hollow molecular structure of C60 fullerenes results in its unusual physical, chemical, and biological properties. Carbon nanotubes can be represented as cylinders rolled-up from graphene sheets [16]. Depending on the formation conditions, they can be both single-walled and multiwalled. There are many other CNSs closely related to CNTs. These structures include end cups, tapered rods, springs, filled tubes including CNTs with fullerenes inside, T-, X-, and Yjunctions, etc. The most common methods for CNTs production, as in the case of fullerenes, are arc discharge, laser ablation, and chemical vapor deposition. CNTs are considered as excellent reinforcements for a new generation of composites with superior mechanical properties [6]. Graphene is an isolated monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be considered as an ideal realization of two-dimensional material [17]. Recent advances in micromechanical extraction and fabrication techniques for graphite structures [18] enable to investigate such exotic system experimentally. As measurements show, individual graphene sheets have extraordinary electronic transport properties, making this material a good candidate for future chips instead of silicon [19]. The use of radiation technique plays a significant role in investigation and fabrication of the nanostructured systems with high resolution as the radiation beams can be focused into few nanometer scales or less and scanned with quite high speed. The use of radiation, namely, X-rays, electron beams, and focused ion beams is essential in nanolithography, 3D fabrication, production of nanopores and nanowires, cross-linking of nanopolymers and nanocomposites, etc. [20]. This paper gives a brief review of the recent progress in application of radiation techniques for synthesis and modification of carbon nanostructures. The review includes the examination of available experimental data and theoretical estimations. The most part of considered experiments deals with modification of initial carbon materials under electron beam irradiation in an electron microscope, which is of paramount importance because it allows in situ observation of dynamic processes on an atomic scale [21]. The research part of the paper contains the results of theoretical estimations of carbon nanostructures stability under electron beam irradiation.

Carbon Nanostructures Behavior under Irradiation The interaction of energetic particles such as electrons or ions with solid carbon and CNSs can result in their modification or new structures formation. The most important primary radiation effects that leads to transformation of target structure are: (a) electronic excitation or ionization of individual atoms of target; (b) collective electronic excitation or plasmon formation; (c) breakage of bonds or cross-linking; (d) generation of phonons, leading

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to heating of the target; (e) displacement of atoms in the bulk of the target; (f) sputtering of atoms from the target surface [21]. The influence of these effects on the behavior of carbon structures under irradiation is considered below.

Radiation-Induced Formation of Carbon Nanostructures There is a moderate quantity of experiments concerning CNSs formation under irradiation, because this method at present doesn’t allow receiving of considerable amounts of these products. As experiments show, the electron beam irradiation of carbon specimens such as soot, graphite, carbon films, etc. leads to the formation of various single-shell and multishell cages with shapes and sizes corresponding to fullerenes or onions [22-26]. This process can be represented as separation of graphene fragments from the specimen surface and their subsequent closure into small spherical shells as it shown in figure 1. The formation of fullerene molecules can also include so-called “zipper” mechanism [27], started with a sandwich-like arrangement of two graphene fragments followed by the arising of C – C bonds between edges of graphene fragments [28]. This leads to formation exactly 12 pentagons independently of the size of fragments, which are placed into the most energetically favorable positions. The formation of fullerene molecules under electron beam irradiation occurs more rapidly with increasing beam energy. A threshold electrons energy, below which no structural changes occur, lies between 40 and 80 keV [22]. The role of the electron beam in the formation and stability of fullerene molecules consists in the sputtering or displacement of carbon atoms from graphite surface by knock-on collisions of electrons with atom nuclei. Due to this displacement, vacancies and interstitials are generated leading to a rupture of bonds in the graphitic planes. Broken bonds terminate lattice fragments, which are able to leave the graphite surface overcoming the van der Waals attraction to the surface.

Electron beam irradiation

Figure 1. Fullerene-like cages formation over the surface of graphitic particle under electron beam irradiation.

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The fullerene-like cages formed on graphitic surface possess higher irradiation stability in comparison with C60 molecules within fullerite crystals or fullerene derivatives films on silicon surface [29]. The process of cages formation in [22] was carried out in high-resolution electron microscope operated at 1.25 MV of acceleration voltage and flux density of 200 A/cm2. On the other hand, the decay of the C60 crystals was observed at electron energies 80 kV and above. The stability of the fullerene cages attached to graphitic surface can be explained in terms of a pick-up mechanism. In this mechanism vacancies in cages formed under the action of knock-on collisions of electrons with the nuclei of carbon atoms are filled by the capture of weakly bound carbon atoms from the defective graphitic surface on which the cages are attached by van der Waals attraction. The process is facilitated by the rapid rotation of the cages. The formation of fullerene molecules are also observed in the wake of highly energetic ions passed through specimen material [30, 31]. A simple theoretical model of the process, which based on latent ion-track formation in condensed carbonic matter, considers a short thin core of highly ionized carbon in the track. This core is the dense non-ideal “gas”, which gives rise to homogeneous “nucleation and grows” of fullerene molecules at the outer track wall in the absence of heterogeneities and impurities in specimen material. The important members of the fullerene family are onion-like fullerenes, which have the unique hollow cage and concentric shell structure [32]. The radiation-induced formation of onions was observed in experiments with electron irradiation of various carbon modifications (soot, amorphous carbon, and graphite) in electron microscope [22-24, 33]. The irradiation dose required for the complete transformation of the amorphous carbon into onions decreases with increasing of electrons energy. The transformation time averages 45 min at electron energy 400 keV and beam current density 180 A/cm2 [23]. Onions can also be produced under ion irradiation [34]. The formation of multiwalled CNTs under electron beam irradiation was observed in experiments with polyyne-containing carbon ((-C≡C-)n) prepared on the surface of epoxy resin by electrochemical reduction [35, 36]. The obtained specimen was heated to temperature 870 – 1070 K and irradiated in the transmission electron microscope with accelerating voltage 100 kV and current density about 1 A/cm2. The process proceeds in two steps: quick formation of rod-like carbons in accordance with growth-from-the-bottom mechanism in a solid phase, and slow formation of hollows inside the rods accompanied by graphitization of the walls. The reaction rate is dependent on the temperature; namely, the reaction is finished in 1 min at 1070 K and in 30 – 60 min at 870. At more than 1170 K all quantity of carbon is evaporated, and at less than 870 K the reaction does not proceed. The products are almost the same in length (of the order of 1 μm), diameter (from 10 to 50 nm), and number of graphene layers (about five) regardless of the reaction temperature. The extension of irradiation leads to the deformation of the nanotubes caused by cross-linking between the graphene layers.

Mechanisms of Carbon Nanostructures Destruction As was mentioned above, the interaction of energetic particles with CNSs causes various radiation effects such as excitation, ionization, atoms displacement, phonons generation, etc. These effects designate possible mechanisms, which can explain the CNSs destruction under irradiation. The most significant of them are: (1) the direct energy transfer from energetic

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particles to the internal motion of the molecular cage (knock-on mechanism), and (2) excitation of cage electronic structure followed by partition of the excitation energy into internal vibrational modes of the macromolecule leading to its fragmentation (excitation mechanism) [37]. The second mechanism as applied to electron beam destruction of solid structures is minimal in metals, semimetals, and small-gap semiconductors because beaminduced electronic excitations are generally not sufficiently localized for effective change into kinetic energy of a single atom [38]. Since the most CNSs are metals or small-gap semiconductors [3, 39], it is assumed that the primary cause of their destruction under electron beam irradiation is knock-on collisions of electrons with atomic nuclei of corresponding lattice [40] (see figure 2).

Electron

Atom

Ion

Fullerene (a)

(b)

Figure 2. Possible mechanisms of CNSs destruction under irradiation: (a) knock-on mechanism; (b) excitation mechanism.

The knock-on destruction of CNSs under electron irradiation was investigated in the variety of experiments [37, 40-44]. One of the main purposes of these investigations is the determination of the energy threshold. As experiments show, the destruction of C60 molecules within fullerite crystals begins at electron energies above 80 kV [21]. The decay of the crystalline structure occurs simultaneously with the decay of the single molecules. It was found that C60 molecules on the first stage of the process coalesce by forming large cages of irregular form, and under further irradiation the graphitization and final formation of onionlike clusters occurred. The threshold electron energy for the knock-on destruction of single-walled CNTs is similar to the corresponding value for C60 molecules. Transmission electron microscopy observations of carbon nanotube behavior under electron beam irradiation show that modification of its structure begins at electron energies below 100 kV [45]. Theoretical estimations utilizing known ejection threshold energies [40] predict that an isolated nanotube will damage preferentially, when the electron beam if normal to its surface. Minimum incident electron energy of 86 kV is required to remove a carbon atom from the lattice by a knock-on collision for this geometry. Higher electron energies are required for any other geometry, and at energies exceeding 139 kV every atom of nanotube lattice is susceptible to

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ballistic ejection. This results cut against the unsubstantiated conventional opinion that carbon nanostructures are stable over relatively long period of time at 200 keV [41]. The formation of vacancies owing to knock-on damage of CNSs under the electron beam irradiation is also observed for graphene membranes [43]. As observations show, the graphene membranes are highly stable in the electron beam, when irradiated during a few hours at 100 kV of acceleration voltage and flux density of 7 A/cm2. In comparison, singlewalled CNTs show strong deformations under the same energy of electrons and dose [40, 42], probably because the cylindrical geometry allows beam-induced defects to relax via local deformations more easily. The vacancies in graphene membrane can disappear by interaction with mobile small molecules that absorbed onto the membrane surface from the vacuum contamination [43]. The excitation mechanism can explain the fragmentation of CNSs under exposure of high-energy ion beams [46-52]. In the case of fullerene C60 molecules contained at low density in the gas phase, electronic or, more specifically, multiple plasmon excitation by the electromagnetic field of the incident ion leads to catastrophic disintegration of the molecule structure [48] (see figure 2). Coupling of electronic excitations to molecular vibrational excitations and sharing of the deposited energy among the fullerene vibrational modes precede this disintegration. From computational point of view, the ion-fullerene collision system is ideal model system to study electronic and vibrational excitation mechanisms including the fundamental process of electron-vibrational coupling in atomic many-body systems with a large, but still finite number of degrees of freedom [49]. The excitation mechanism, which is able to explain the single fullerene destruction under ion irradiation in the gas phase, is equally applicable to the case of electron beam irradiation of a solid fullerene films [37]. The irradiation process as applied to decomposition of ultrathin (1-4 monolayers) films of C60 grown on hydrogen-passivated Si substrates proceeds at relatively low electron energies of the order of 3 kV. As in the case of gas phase C60 fragmentation by ion beam, electronic excitation of C60 molecule by the incident electron impact can cause multiple energy transitions. Part of the energy in this process may become partitioned into vibrational modes as the molecule equilibrates. The formation of a highly excited vibrational state is equivalent to a local heating of an individual C60 molecule. Under these conditions, the fragmentation process involves sequential dimmer emission or a complete disintegration of the carbon cage. The low-energy electron irradiation was found to damage single-walled CNTs [53], meaning that electronic excitations are responsible for the defects formation.

Modification of Carbon Nanostructures under Irradiation The modification of CNSs is the most perspective application of the radiation techniques for these nanomaterials processing. It includes welding of crossing CNTs [54], polymerization of C60 layers [55], formation of new nanoscale structures [56], radiation chemistry of CNSs [57], etc. As an initiation step, the irradiation gives rise to formation of vacancies and other defects in the CNSs lattice. The subsequent transformation of nanobjects depends both on their internal structure and reciprocal arrangement leading to their amorphization at high irradiation doses.

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The electron beam irradiation of crossing single-walled CNTs at high temperatures induces structural defects, which lead to the formation of stable junctions of various geometries involving seven- and eight-membered carbon rings at the surface between the tubes [58]. The welding and controllable preparation of some special nanostructures such as nanospindles and nanodumbbells at electron beam irradiation of multiwalled CNTs were reported in [59]. The structural transformation of fluorinated single-walled CNTs bundles into multiwalled-like nanotubes was observed in [60]. This transformation can be explained by the local strains induced by fluorination. Nanotubes welding is also induced by the ion beam irradiation [54, 61-62]. The ion beam irradiation of multiwalled CNTs can be an effective way for the formation of nanowire junctions [63]. The observations of crossing CNTs modification were supported by molecular dynamics simulation [42, 58, 64-65]. The polymerization of C60 films attaches lately increase attention because this new form of carbon material exhibits physical and chemical properties of both diamond and graphite [55]. In the case of irradiation-induced polymerization, the unifying influence of energetic particles is accompanied by the destruction of fullerene-cage structure, which makes the polymerization phenomenon more complicated [66]. The structural change of C60 films caused by irradiation of 0.5 and 3.3 keV electron beam is polymerization at low dose and fragmentation at high one [37]. The long irradiation of fullerite C60 film with 1.5 keV electrons leads to its gradual transformation towards respective structure of amorphous carbon [67]. The formation of giant fullerenes from crystalline C60 films epitaxially grown on various substrates is observed under action of electron injection from the probe tips of scanning tunneling microscopes [68-70]. The electron beam irradiation of C60 intercalated graphite at 200 keV leads to its conversion to diamond without the application of high pressure and temperature [71]. The thermal activation and electron beam irradiation of hexanitro(60)fullerene film deposited on Au substrate produce a new polymeric material with unique characteristics [72]. The formation of new carbon materials is bound up with previous consideration. The electron beam irradiation of solid C60 layers epitaxy on a graphene sheet leads to fabrication of new nanostructure consists of C60 clusters, which can open new opportunities in the carbon nanoelectronics [56]. Experiments with single-walled CNTs filled with ZrCl4 by a catalytic arc synthesis reveal clusters formation under the electron beam irradiation [73]. This phenomenon may represent a route to the synthesis of one-dimensional arrays of quantum dots within CNTs, if it will be extended in a controlled way to bulk CNTs samples containing semiconducting or metallic species. Analogous structural evolution of single-walled CNTs filled with fullerenes or metal-doped fullerenes was observed in [74]. The graphitisation of glassy carbon film by electron beam results in formation of graphitic nanowire structure [75]. The irradiation of single-walled CNTs with B and N ions leads to substitution of carbon atoms, which is an alternative way to introduce boron and nitrogen impurities into nanotubes in addition to the chemical substitution and arc-discharge methods [76, 77]. The controlling of graphene film shapes using ion beam machining was studied in [78]. The appreciable set of investigations in the viewed field of action belongs to radiation chemistry of fullerenes [57], which considers chemical reactions of dissolved fullerene derivatives with active species generated in solvents under radiolysis [79, 80]. The potential participation of water-soluble fullerenes in biological applications initiated the study of radiolytic formation of such fullerene derivatives as polyhydroxylated fullerenes [81]; polyfunctionalized fullerenes containing carboxyl, ammonium, and sulphite groups [82];

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negatively charged fullerene clusters [83], etc. Some applications of radiation chemistry of fullerenes belong to other effective solvents. In particular, radiolysis of C60 in polar chlorinated methane results in production of C60-CHCl2 and C60-CCl3 adducts [84, 85]. The radiation-induced synthesis of fullerene-silica hybrid nanomaterials in toluene solution was considered in [86].

Theoretical Estimation of CNSs Stability The brief review of the recent progress in investigation of CNSs behavior under irradiation shows that the main process parameters, which are responsible for their modification, are the acceleration voltage and flux density. These parameters determine limits of lattice stability for the given CNS, after which the process of lattice modification or its destruction begins. The present consideration of CNSs stability under electron beam irradiation includes the analytical approximation of the cross-section for the Coulomb scattering of relativistic electrons by carbon atom nuclei [21] as well as results of classical molecular dynamics simulations.

Displacement Cross-Section As it was discussed above, the interaction of the electron beam with CNSs surface leading to its damage is described by the knock-on mechanism of energy transfer from incident electron to carbon nuclei of the corresponding CNS lattice. The typical electron energy E for such process is in the range 100 – 1000 keV at which the electron motion must be treated as relativistic. It means that parameter β = v/c, where v and c are the velocities of electron and light, is closed to unit and must be calculated by the correlation: β2 = 1 – (E/mc2 + 1)-2, where m denotes the electron rest mass. The relationship between the electron kinetic energy E and energy ΔE transferred to atom of mass M in binary collision can be written as:

ΔE = ΔE max cos 2 Θ ,

ΔE max = 2 E ( E + 2mc 2 ) / Mc 2 ,

(1)

where Θ is the angle between the initial electron direction and that of the knock-on carbon atom (see figure 2), and ΔEmax is the maximum energy that may be transferred to atom by a head-on collision with electron. One of the main physical characteristics in the study of radiation damage of crystals is the threshold energy of atom displacement Ed [87]. In its simplest form, it is defined as the minimum energy required for displacing the atom from the regular lattice site in such way that a stable interstitial-vacancy (Frenkel) pair produces without its spontaneous recombination. As it will be seen from the following consideration, the threshold energy of carbon atom displacement from CNS surface depends on its scattering direction with respect to surface lattice. The minimum energy of incident electron Emin at which the atom displacement from CNS surface occurs can be obtained from (1) by substitution of ΔE on Ed at Θ = 0.

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The collision of the relativistic electron with carbon atom nuclei in CNS lattice can treated by using a simple Coulomb potential because the screening effect of the surrounding electrons is negligible [21]. An analytical approximation to the well known “Mott crosssection” of the Coulomb scattering of relativistic electrons by atomic nuclei with charge Z that valid for α = Z/137 < 0.2 was obtain in [88]. The displacement cross-section, which represents the probability for carbon atom to be displaced from CNS lattice under action of the relativistic electron with energy E, can be written in the case of isotropic displacement energy as [21]:

σ = πr02 Z 2 (1 − β 2 )β −4 F (ΔE max / E d ) ,

(2)

F ( x) = x + 2παβ x1 / 2 − [1 + 2παβ + (β 2 + παβ)lnx], where r0 = e2/mc2 is classical electron radius. The value of ΔEmax is calculated at given value of incident electron energy E from Eq. (1). The dependence of the displacement cross-section σ from the threshold energy of carbon atom displacement Ed at various energies E of incident electron is given in figure 3. It is observed the decrease of σ with increase of the threshold energy Ed for all electron energies E. The difference between calculated curves at high values of E (400 keV and more) is not so sensitive as at low ones. The behavior of the cross-section σ as a function of E at fixed value of Ed has more complicated character. The increase of the cross-section with growth of the incident electron energy is observed for used values of E at threshold energies Ed ≥ 20 eV, which are typical for CNSs lattice.

Displacement cross section, barns

50 40 30

20 E = 800 keV 10

400 100

200

0 10

30 40 50 60 20 Displacement threshold energy, eV

Figure 3. Displacement cross section σ of carbon atom as a function of displacement threshold energy Ed at various energies E of incident electron.

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Threshold Energy of Carbon Atom Displacement Classical molecular dynamics simulation is the most realistic method available within the limits of current computer capacity for studying of CNSs destruction process dynamically and for determination of the threshold energy of carbon atom displacement Ed [89]. The practical use of this method requires of appropriate CNSs lattice identification. Several empirical interatomic potentials were developed for description of the structural properties and energetics of carbon lattice such as Tersoff potential [90], modified Brenner potential [91] and others. Most of these potential models are classical in nature and cannot account for quantum mechanical effects of the bonding in carbon systems. An alternative approach is to include the effects of directional covalent bonding through the underlying electronic structure described by an empirical tight-binding Hamiltonian [92, 93]. Such scheme allows taking into account the quantum mechanical nature of the covalent bonding in the potential naturally. The tight-binding molecular dynamics simulation of the carbon nanotubes behavior under electron beam irradiation was performed in [41]. The threshold energy of the carbon atom displacement as a function of the transferred impulse direction was determined. As calculations show, the impulse to the escaped atom within the local tangent plane to the nanotube surface causes large distortions of approximately ten neighboring atoms, which can explain the large threshold energy for this impulse. The radial impulse allows the carbon atom to escape the nanotube with minimal distortion of the local environment. This gives the threshold energy of carbon atom displacement only moderately large than the sum of the nearest neighbor bond energies. Figure 4 represents the dependence of Ed from escape angle α, the angle between impulse of ejected atom and the local normal to the nanotube surface. It is observed the increase of Ed from 17 to 37 eV with the deflection of the escape direction from normal to tangent. It must be noted that the displacement threshold within the tangent plane has variations from 33 to 43 eV resulted from curvature and local orientation effects. 50

Threshold energy, eV

40 Fullerenes

30 20

Carbon nanotubes 10

0

0

20

40 80 60 Escape angle, degrees

100

Figure 4. Dependence of threshold energy Ed of carbon atom displacement from angle α between impulse of ejected atom and the local normal to the CNSs surface.

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Displacement cross section, barns

25 20 15 800 10

200 E = 100 keV

5

0

400

0

20

40 60 80 Escape angle, degrees

100

Figure 5. Dependence of displacement cross section σ from escape angle α for carbon nanotube.

The molecular dynamics simulation of fullerene C60 damage under electron beam irradiation was realized in [94]. The C-C atomic interactions in fullerene cage were described by using of the Tersoff bond-order potential. As in the case of carbon nanotubes damage [41], the threshold energies of carbon atom displacement Ed as a function of the escape angle α were determined. Received results for one of the representative atoms on the surface C60 cage are given in figure 4. Calculations show that at energies less than Ed the tested carbon atom and its neighbors underwent a strong disturbance around their equilibrium sites and gradually came to rest. In the case of energies exceeded Ed, the target atom leaves the defected cage of fullerene. It is interesting that the escaped atom in calculations always has a certain final kinetic energy even if its initial kinetic energy is equal to Ed. The difference between threshold energies for nanotube and fullerene, which one can observe in figure 4, may be explained by use of various potential approaches because geometrical differences should not have the important influence on the behavior of this parameter. The displacement cross-section σ vs. escape angle α at various electron energies E is given in figure 5. Calculations were performed using Eq. (2) for σ with dependence Ed = Ed(α) for carbon nanotubes taken from figure 4. The analysis of figure 5 shows that values of this parameter for E ≥ 200 keV differ faintly practically all over the interval of α change. The electron energy E = 100 keV is critical for carbon atom displacement from nanotube lattice, which proves to be true by experimental data [40].

Characteristic Time of CNSs Destruction The theoretical estimation of CNSs stability under the electron beam irradiation can be considered using simple correlations, which include the main process parameters described above. The displacement cross-section σ divided on square S, which fits one carbon atom in CNS lattice, gives the probability of atom displacement by electron impact. Multiplying this

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value by the frequency jS of electron incidence on the square S, one can receive the probability of carbon atom displacement per unit time. For given current density j, the characteristic time τ of CNS destruction under the electron beam influence is expressed by the correlation: τ = (σj)-1. The typical current density of the electron beam in experiments with carbon nanotubes and graphene sheets is of the order of 1 - 10 A/cm2 [43, 95-96]. The previous consideration of threshold energy behaviour under various geometrical configurations of the scattering process shows that this lattice parameter for carbon nanotubes is equal approximately Ed = 17 eV. In this conditions, as it seen from figure 5, the displacement cross-section σ for electron energies greater than 200 keV is of the order of σ = 20 barns. Figure 6 represents the dependence of characteristic time τ from the current density j calculated for these conditions and its comparison with available experimental data for carbon nanotubes and graphene sheets [40, 42, 43, 95-96]. The open triangle corresponds to observation of single-walled CNTs modification in experiment [42], which is focused on the behaviour of individual nanotubes irradiated in a 200 keV high-resolution transmission microscope (TEM) with the approximate electron flux of 1 A/cm2. In about half an hour of irradiation, tubes shrink from an original diameter of 1.4 nm to 0.4 nm. The overall shape of the tube remains cylindrical, even for the smallest tube observed. Continued irradiation results in the breakage of separate tubes. One can speculate that limiting case of nanotubes modification is the formation of atomic chains made of linear arrays of carbon atoms. Similar results with single-walled CNTs were received in TEM experiments [40] operated with 200 keV electrons and current density j = 3.7 A/cm2 (see open circle in figure 6), where the microstructure of tubes after 26 min of irradiation collapses and becomes uniformly amorphous-like. The combination of defects formation on the surface of CNTs via electron irradiation (E = 100 keV, j = 1 A/cm2) and simultaneous resistive heating (the maximum CNT temperature is at least 1200 K) and electromigration in vacuum causes the continuous shrinking of nanotube [96]. The process can be repeated in a highly controlled fashion yielding a highquality CNT of any preselected and precise diameter. The dynamics of the shrinking process is described by exponential decay with time constant τ = 1000 s, which marked in figure 6 by reverse triangle. The estimated threshold energy under these conditions is equal 5.5 eV, a much smaller value, which approaches the Ed of CNSs at room temperature. The corresponding value of the displacement cross-section σ is equal 160 barns. This explains the excess of calculated characteristic time over experimental in figure 6. The characteristic time τ of structural transformation of fluorinated single-walled CNTs induced by electron beam irradiation (E = 300 keV, j = 2.46 A/cm2) during TEM observations [95] is marked in figure 6 by full circle. Time evolution of the TEM images in this experiment shows that the original CNTs bundle with a diameter of about 4 nm was heavily distorted particularly in the middle due to the topological presence of fluorine atoms, which leads to its transformation into multiwall-like nanotube. Such structural transformation can be explained by the local strains induced by fluorination, when fluorine-attached C-C bonds became 0.153 nm, a single bond similar to that of a diamond. The corresponding lowering of C–C bond energy gives the increase of Ed and σ, which in one’s turn decrease the characteristic time of fluorinated CNTs bundle reconstruction in comparison with usual CNTs (see figure 6).

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Characteristic time, s

104

103

102

0

2

4 8 6 Current density, A/cm2

10

Figure 6. Dependence of characteristic time τ of carbon nanotube destruction from current density j of electron beam.

The dynamics of defects generation in graphene membranes owing to knock-on damage by the electron beam irradiation with E = 100 keV and j = 7 A/cm2 in TEM was investigated in [43]. The vacancies appearance is described by the characteristic time marked in figure 6 by open square. The excess of experimental data as compared to calculated curve one can explain by low electrons energy E that gives half of used in calculations displacement crosssection σ at threshold energy Ed = 17 eV (see figure 3). It must be noted that graphene membranes are highly stable in the electron beam at E = 100 keV. At least the observed in [43] vacancies generation doesn’t signify the whole graphene membrane destruction.

Conclusion The brief review of the recent progress in application of radiation techniques for synthesis and modification of carbon nanostructures shows that this direction of scientific investigations is of the fundamental importance due to unique electronic, mechanical and chemical properties of considered materials, which make them attractive possible candidates for a variety of uses ranging from molecular electronics to composite materials. The modification of carbon nanostructures includes such radiation-induced processes as the welding of crossing carbon nanotubes, polymerization of fullerenes layers and carbon nanotubes bundles, formation of new nanoscale structures, synthesis of various fullerene derivatives under irradiation of water-soluble fullerene systems, etc. The theoretical consideration of carbon nanostructures stability under electron beam irradiation was performed using simple correlations, which include the main process parameters such as acceleration voltage and flux density. These parameters determine limits

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of lattice stability for the given carbon nanostructure, after which the process of lattice modification or its destruction begins. The calculated characteristic time of lattice structural transformation for carbon nanotubes and graphene membranes was compared with available experimental data.

Acknowledgments This work was performed in the framework of the IAEA Technical Cooperation Project “Supporting Radiation Synthesis and the Characterization of Nanomaterials for Health Care, Environmental Protection and Clean Energy Applications”, Project No. RER/8/014.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Springer Handbook of Nanotechnology; Bhushan, B.; Ed.; Springer: Berlin, 2007. Anantram, M. P.; Leonard, F. Rep. Prog. Phys. 2006, 69, 507-561. Son, Y.-W.; Cohen, M. L.; Louie, S. G. Nature 2006, 444, 347-349. Wang, C.; Guo, Z.-X.; Fu, S. et al. Prog. Polym. Sci. 2004, 29, 1079-1141. Ahir, S. V.; Huang, Y. Y.; Terentjev, E. M. Polymer 2008, 49, 3841-3854. Hu, Y.; Shenderova, O. A.; Hu, Z. et al. Rep. Prog. Phys. 2006, 69, 1847-1895. Ramanathan, T.; Abdala, A. A.; Stankovich, S. et al. Nature Nanotech. 2008, 3, 327331. Hirscher, M.; Becher, M. J. Nanosci. Nanotech. 2003, 3, 3-17. Shim, M.; Kam, N. W. S.; Chen, Y. et al. Nanolett. 2002, 2, 285-288. Biryukov, V. M.; Bellucci, S.; Guidi, V. Nucl. Instr. Meth. 2005, B231, 70-75. Perspectives of Fullerene Nanotechnology; Osawa, E.; Ed.; Kluwer: New York, 2002. Kasuya, D.; Kokai, F.; Takahashi, K. et al. Chem. Phys. Lett. 2001, 337, 25-30. Alekseyev, N. I.; Dyuzhev, G. A. Carbon 2003, 41, 1343-1348. Nakamura, E.; Tahara, K.; Matsuo, Y. et al. J. Am. Chem. Soc. 2003, 125, 2834-2835. Takehara, H.; Fujiwara, M.; Arikawa, M. et al. Carbon 2005, 43, 311-319. Monthioux, M; Serp, P.; Flahaut, E. et al. In Springer Handbook of Nanotechnology; Bhushan, B.; Ed.; Springer: Berlin, 2007; pp 43-112. Katsnelson, M. I. Mater. Today 2007, 10, 20-27. Novoselov, K. S.; Geim, A. k.; Morosov, S. V. et al. Science 2004, 306, 666-669. Van Noorden, R. Nature 2006, 442, 228-229. Chmielewski, A. G.; Chmielewska, D. K.; Michalik, J.; Sampa, M. H. Nucl. Instr. Meth. 2007, B265, 339-346. Banhart, F. Rep. Prog. Phys. 1999, 62, 1181-1221. Füller, T.; Banhart, F. Chem. Phys. Lett. 1996, 254, 372-378. Zwanger, M. S.; Banhart, F.; Seeger, A. J. Crystal Growth 1996, 163, 445-454. Banhart, F.; Ajayan, P. M. Nature 1996, 382, 433-435. Burden, A. P.; Hutchison, J. L. Carbon 1997, 35, 567-578. Takeuchi, M.; Muto, S.; Tanabe, T. et al. J. Nucl. Mater. 1999, 280-284. Richter, H.; Howard, J. B. Prog. Energy Combust. Sci. 2000, 26, 565-608. Radovic, L. R.; Bockrath, B. J. Am. Chem. Soc. 2005, 127, 5917-5927.

Radiation Induced Synthesis and Modification of Carbon Nanostructures [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68]

507

Robinson, A. P. G.; Palmer, R. E.; Tada, T. et al. Chem. Phys. Lett. 1998, 289, 586-590. Chadderton, L. T.; Fink, D.; Gamaly, Y. et al. Nucl. Instr. Meth. 1994, B91, 71-77. Gamaly, E. G.; Chadderton, L. T. Proc. Roy. Soc. 1995, A449, 381-409. Xu, B.-S. New Carbon Mater. 2008, 23, 289-301. Xu, B.-S.; Tanaka, S.-I. Acta Mater. 1998, 46, 5249-5257. Wesolowski, P.; Lyutovich, Y.; Banhart, F. et al. Appl. Phys. Lett. 1997, 71, 1948-1950. Kawase, N.; Yasuda, A.; Matsui, T. et al. Carbon 1999, 37, 522-524. Yashuda, A.; Mizutani, W.; Shimizu, T.; Tokumoto, H. Surfase Sci. 2002, 514, 216-221. Hunt, M. R. C.; Schmidt, J.; Palmer, R. E. Phys. Rev. 1999, B60, 5927-5937. Was, G. S.; Allen, T. R. In: Radiation Effects in Solids; Sickafus, K. E.; Kotomin E. A.; Uberuaga, B. P.; Eds.; Springer: Dordrecht, 2007; pp 65-98. Qin, L.-C. Rep. Prog. Phys. 2006, 69, 2761-2821. Smith, B. W.; Luzzi, D. E. J. Appl. Phys. 2001, 90, 3509-3515. Crespi, V. H.; Chopra, N. G.; Cohen, M. L. et al. Phys. Rev. 1996, B54, 5927-5931. Ajayan, P. M.; Ravikumar, V.; Charlier, J.-C. Phys. Rev. Lett. 1998, 81, 1437-1440. Meyer, J. C; Girit, C. O.; Crommie, M. F.; Zettl, A. Nature 2008, 454, 319-322. Rauwald, U.; Shaver, J.; Klosterman, D. D. et al. Carbon 2009, 47, 178-185. Mølhave, K.; Gudnason, S. B.; Pedersen, A. T. et al. Ultramicroscopy 2007, 108, 52-57. Cheng, S.; Berry, R. W.; Dunford, R. W. et al. Phys. Rev. 1996, A54, 3182-3194. Schlathölter, T.; Hadjar, O.; Hoekstra, R. et al. Phys. Rev. Lett. 1999, 82, 73-76. Campbell, E. E. B.; Rohmund, F. Rep. Prog. Phys. 2000, 63, 1061-1109. Kunert T.; Schmidt, R. Phys. Rev. Lett. 2001, 86, 5258-5261. Todorović-Marković, B.; Draganić, I.; Marković, Z. et al. Full. Nanot. Carb. Nanostruct. 2007, 15, 113-125. Mathew, S.; Bhatta, U. M.; Ghatak, J. et al. Carbon 2007, 45, 2659-2664. Singhal, R.; Kumar, A.; Mishra, Y. K. et al. Nucl. Instr. Meth. 2008, B266, 3257-3262. Suzuki, S.; Kobayashi, Y. Chem. Phys. Lett. 2006, 430, 370-374. Krasheninnikov, A. V.; Nordlund, K. Nucl. Instr. Meth. 2004, B216, 355-366. Onoe, J.; Nakayama, T.; Aono, M. et al. J. Phys. Chem. Solids 2004, 65, 343-348. Hashimoto, A.; Terasaki, H.; Yamamoto, A. et al. Diam. Relat. Mater. 2009, 18, 388391. Guldi, D. M. Studies Phys. Theor. Chem. 2001, 87, 253-286. Terrones, M.; Banhart, F.; Grobert, N. et al. Phys. Rev. Lett. 2002, 89, 075505. Yang, Q.; Bai, S.; Wang, G.; Bai, J. Mater. Lett. 2006, 60, 2433-2437. An, K. H.; Park, K. A.; Heo, J. G. et al. J. Am. Chem. Soc. 2003, 125, 3057-3061. Adhikari, A. R.; Bakhru, H.; Ajayan, P. M. et al. Nucl. Instr. Meth. 2007, B265, 347351. Kim, B.; Kim, E.; Lee, J. et al. Thin Solid Films 2008, 516, 3474-3477. Wang, Z.; Yu, L.; Zhang, W. et al. Phys. Lett. 2004, A324, 321-325. Salonen, E.; Krasheninnikov, A. V.; Nordlund, K. Nucl. Instr. Meth. 2002, B193, 603608. Krasheninnikov, A. V.; Nordlund, K.; Keinonen, J.; Banhart, F. Nucl. Instr. Meth. 2003, B202, 224-229. Yogo, A.; Majima, T.; Itoh, A. Nucl. Instr. Meth. 2002, B193, 299-304. Shnitov, V. V.; Mikoushkin, V. M.; Gordeev, Yu. S. Microel. Eng. 2003, 69, 429-434. Nakamura, Y.; Mera, Y.; Maeda, K. Appl. Surface Sci. 2008, 254, 7881-7884.

508 [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96]

G.Y. Gerasimov Zhao, Y. B.; Poirier, D. M.; Pechman, R. J. et al. Appl. Phys. Lett. 1994, 64, 577-579. Nakaya, M.; Nakayama, T.; Aono, M. Thin Solid Films 2004, 464-465, 327-330. Gupta, V.; Scharff, P.; Miura, N. Mater. Lett. 2005, 59, 3259-3261. Zhai, R.-S.; Das, A.; Hsu, C.-K. et al. Carbon 2004, 42, 395-403. Brown, G.; Bailey, S. R.; Sloan, J. et al. Chem. Commun. 2001, 845-846. Gloter, A.; Suenaga, K.; Kataura, H. et al. Chem. Phys. Lett. 2004, 390, 462-466. Takai, K.; Enoki, T. Physica. 2007, E40, 321-323. Kotakoski, J.; Pomoell, J. A. V.; Krasheninnikov, A. V.; Nordlund, K. Nucl. Instr. Meth. 2005, B228, 31-36. Paredez, P.; Marchi, M. C.; da Costa, M. E. H. M. et al. J. Non-Crystal. Solids 2006, 352, 1303-1306. Terdalkar, S. S.; Zhang, S.; Rencis, J. et al. Int. J. Solids Struct. 2008, 45, 3908-3917. LaVerne, J. A.; Araos, M. S. Radiat. Phys. Chem. 1999, 55, 525-528. O’Shea, K. E.; Kim, D. K.; Wu, T. et al. Radiat. Phys. Chem. 2002, 65, 343-347. Lu, C.-Y.; Yao, S.-D.; Lin, W.-Z. et al. Radiat. Phys. Chem. 1998, 53, 137-143. Guldi, D. M.; Asmus, K. D. Radiat. Phys. Chem. 1999, 56, 449-456. Guldi, D. M.; Hungerbühler, H.; Asmus, K. D. J. Phys Chem. 1997, A101, 1783-1786. Lian, Z. R.; Yao, S. D.; Lin, W. Z. et al. Radiat. Phys. Chem. 1997, 50, 245-247. Cataldo, F.; Gobbino, M.; Ragni, P. Full. Nanot. Carb. Nanostruct. 2007, 15, 379-393. Cataldo, F.; Angelini, G.; Lilla, E.; Ursini, O. Full. Nanot. Carb. Nanostruct. 2007, 15, 445-463. Hohenstein, M.; Seeger, A.; Sigle, W. J. Nucl. Mater. 1989, 169, 33-46. McKinley, W. A.; Feshbach, H. Phys. Rev. 1948, 74, 1759-1763. Nordlund, K.; Keinonen, J.; Mattila, T. Phys. Rev. Lett. 1996, 77, 699-702. Tersoff, J. Phys. Rev. Lett. 1988, 61, 2879-2882. Brenner, D. W.; Shenderova, O. A.; Harrison, J. A. et al. J. Phys. Condens. Matter. 2002, 14, 783-802. Xu, C. H.; Wang, C. Z.; Chan, C. T.; Ho, K. M. J. Phys. Condens. Matter. 1992, 4, 6047-6054. Goedecker, S.; Colombo, L. Phys. Rev. Lett. 1994, 73, 122-125. Cui, F. Z.; Li, H. D.; Huang, X. Y. Phys. Rev. 1994, B49, 9962-9965. An, K. H.; Park, K. A.; Heo, J. G. et al. J. Am. Chem. Soc. 2003, 125, 3057-3061. Yuzvinsky, T. D.; Mickelson, W.; Aloni, S. et al. Nano Lett. 2006, 6, 27

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 509-533

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 9

MONITORING A 22-STORY BUILDING UNDER SEVERE TYPHOONS WITH BAYESIAN SPECTRAL DENSITY APPROACH Ka-Veng Yuen* and Sin-Chi Kuok Department of Civil and Environmental Engineering, University of Macau, China.

Abstract Typhoon is a frequently-occurred natural phenomenon. It is valuable to investigate the behaviour of the infrastructures in coastal cities under this severe aerodynamic condition. Two severe typhoons, namely Nuri and Hagupit, attacked the southern China coast in August and September 2008. Nuri and Hagupit passed by Macao from the northeast side and south side, respectively, generating significantly different aerodynamic condition to the infrastructures. The East Asia Hall, which is a 22-story 64.70 m reinforced concrete building in Macao, is investigated in this study. The floor layer is in L-shape with unequal spans of 51.90 m and 61.75 m and the height-to-width aspect ratio of the building is close to unity. Due to the particular geometry of this building, the structural response is sensitive to both the wind speed and the wind attacking angle. Its acceleration time histories were measured for the complete duration of these two typhoons. The Bayesian spectral density approach is applied to identify the modal parameters of the building and the excitation, such as the modal frequencies of the structure and the spectral intensity of the modal forces. Moreover, the associated uncertainties of these estimated parameters can be quantified by Bayesian inference to reflect the reliability of the identification results. It is important to distinguish whether the changes are due to statistical uncertainty or other factors. During the two typhoons, substantial changes appeared in the modal frequencies and damping ratios, but these changes recovered almost immediately after the typhoon dissipated. Aerodynamic effects such as vortex shedding will also be investigated. In order to develop a reliable structural health monitoring system, it is important to identify the factors, except for structural damage, that influence the modal parameters.

Keywords: Bayesian inference, Modal updating, Spectral analysis, Structural health monitoring, Typhoon. *

E-mail: [email protected]. Tel: (853) 83974960 Phone: (853) 28838314 (Corresponding author)

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1. Introduction Tropical cyclone is a common natural phenomenon formed by unstable overlying atmosphere. If the intensity of a tropical cyclone is sufficiently high, an ‘eye’ with low atmospheric pressure will be developed as the core of the rotating air flow. A severe tropical cyclone can cause considerable economic damage and human life loss to the attacked coastal cities. There were on average 17 tropical cyclones per year in the Northwest Pacific Basin and most of them occurred between July and October [1]. Meanwhile, with the rapid growth of urban population and commercial development in the coastal cities in East Asia, high-rise buildings and other infrastructures have been constructed more and more compactly in this region. Therefore, it is valuable to investigate the structural behaviour under this severe aerodynamic phenomenon. In the recent decades, with the remarkable revolution of data acquisition hardware, fullscale field monitoring for structural systems has become possible [2]. The health condition of a structure can be evaluated by monitoring any abnormal changes of the identified model parameters. In practice, the modal frequencies, damping ratios and mode shapes are popular choices as the diagnostic parameters [2-6]. Abnormal drop of the modal frequencies implies loss of structural stiffness and hence possible damage of the structure. A literature review on damage detection summarized that 5% of change in modal frequencies could be interpreted as structural damage [4]. Furthermore, modal damping and mode shape may be used as indirect damage detectors for evaluation of the structural health status [5, 6], because degradation of structural integrity induces notable changes of these physical parameters. For example, an abnormally high value of the damping ratio indicates possible hysteretic behaviour of the structure. Since the modal parameters provide important information for the structural health status, many methods have been developed for the identification of these parameters [7-12]. In the present study, the Bayesian spectral density approach [13] is applied to identify the modal frequencies and damping ratios of the East Asian Hall, which is a 22-story reinforced concrete building in Macao. An appealing feature of the Bayesian method is that it provides both the optimal estimation and the associated uncertainty, which is in contrast to most existing identification techniques. Therefore, the reliability of the estimation can be quantified and confidence interval can be constructed for further comparison. The Bayesian model updating framework for structural dynamics problems was presented in [14]. In the original presentation, it focused on input-output measurement. The Bayesian spectral density approach was developed for output-only measurements of linear dynamical systems. It is a frequencydomain method, and the modal/model parameters are estimated through the information carried by the spectral density estimator. Yuen and Beck (2003) extended the Bayesian spectral density approach for nonlinear dynamical systems [15]. In this study, it is necessary to adopt a method that utilizes output-only measurements since the complete time-varying wind pressure field is not available. There have been a number of studies on full-scale monitoring of landmarked infrastructures under typhoons in recent years. Li et al. (2000) compared the structural response of a hollow truss system tall building under four typhoons [16]; Li et al. (2002) evaluated the fatigue damage of a long-span steel-deck bridge under typhoon excitation and traffic loading [17]; Xu et al. (2003) analyzed the structural parameters of a high-rise

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reinforced concrete building under typhoon York, which was the strongest typhoon with the longest duration since 1983 [12]. All these previous studies showed that typhoons produce temporary or even permanent effects to the attacked structures. Furthermore, the changes of modal frequencies and damping ratios could be more than 5% even though there was no damage of the structure. Hence, structural health monitoring during this level of violent wind excitation and further investigation are needed to determine whether immediate repair of the structure is necessary. In this study, the aerodynamic condition and structural response time histories were recorded for the whole process of two severe typhoons, namely Nuri and Hagupit. The structural modal frequencies, damping ratios, and spectral intensity of the modal excitation were identified in an hourly basis. Discussion will be made for the time-varying parameters and their correlation with the wind speed and wind attacking angle. Effects of ambient conditions are also discussed.

2. Instrumentation of the East Asia Hall In 2005, the East Asia Hall was inaugurated for lodging athletes during the 4-th East Asian Games hosted in Macao. Since then, it has been serving as a dormitory of University of Macau. It is a 22-story reinforced concrete building of 64.70 m height, and its floor layer is in L-shape with unequal spans of 51.90 m and 61.75 m. A typical floor plan is shown in figure 1. Unlike most selected buildings for full-scale monitoring with regular configurations, such as rectangular or circular floor layer, and large height-to-width ratio [12, 16, 18, 19], the East Asian Hall has an L-shape floor section with aspect ratio close to unity. Due to the particular geometry of the building, the aerodynamic effects to the structural behaviour are more complex.

Figure 1. A typical floor plan of the East Asian Hall.

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Acceleration time histories were measured by two state-of-the-art servo-accelerometers which operate based on standard exploration geophone spring-mass system with sensitivity 50 V/g. These accelerometers were installed orthogonally on the 18-th story of the overall twenty-two storys, and they are located at the conjunction of the two spans as indicated in figure 1. Meanwhile, ambient conditions such as the temperature and relative humidity were recorded by digital thermo-hygrometers installed at the same location.

3. Bayesian Spectral Density Approach 3.1. Formulation A structural system can be discretized by finite-element methods [20], and the vibratory motion can be adequately modeled by a dynamical system with Nd degrees of freedom (DOFs) of the generalized coordinates: is given by

x = [ x1 (t ), x2 (t ),..., x N d (t )]T

. Its equation of motion

Mx + Cx + Kx = F(t )

(1)

where x and x are the velocity and acceleration vector, respectively; M, C, and K are the mass, damping and stiffness matrix of the structure. The unmeasured excitation F is assumed to have wide band and it is adequately modeled as band-limited white noise. Using modal analysis, the generalized coordinates are transformed to the modal coordinates through the following linear operation:

x(t ) = Φ ⋅ q(t ) q = [q (t ), q (t ),..., q (t )]T

(2)

] Nd 1 2 where is the modal coordinates and Φ = [φ , φ ,..., φ is the mode shape matrix containing Nd linearly independent mode shape vectors. These mode shape vectors are normalized to be unity for the jr -th DOF, which is an arbitrarily chosen observed DOF: φ (j rr ) = 1,

(1)

r = 1,2,..., N d

( 2)

(Nd )

(3)

This is done instead of the popular mass normalization so that the mass matrix is not required in the identification process. Also, the complete mode shape vector is not available unless all DOFs are observed. Therefore, mass normalization is usually not possible in modal identification. Note that jr can be different for different modes. Then, the equation of motion in Eq. (1) is decoupled to Nd ordinary differential equations of the modal coordinates:

Monitoring a 22-Story Building under Severe Typhoons… 2 qr (t ) + 2ζ r ωr q r (t ) + ωr qr (t ) = f r (t ),

r = 1,2, ... , N d

513

(4)

where ωr , ζ r and f r are the modal frequency, damping ratio and modal force of the r-th mode, respectively. The power spectral density matrix of the modal forces is constant due to the whiteness assumption of the excitation so

S f (ω ) = S f 0

in the specified frequency band

f (t ) = [ f (t ), f (t ),..., f (t )]

T

1 2 Nd . for the modal forces Let No be the number of the measured DOFs. Note that No is usually much smaller than Nd, so the selected DOFs have to be properly chosen for the observation of the structural

behaviour [21]. The structural response is measured with sampling time step Δt and the measurement is assumed to be contaminated by measurement noise ε . Specifically, the measurement at time t = nΔt is given by

y (nΔt ) = L 0x(nΔt ) + ε(nΔt )

(5)

L 0 ∈R N o × N d is the observation matrix comprised by zeros and ones, and ε , with ε(nΔt ) ∈ R N o , is the measurement noise vector process. The stochastic process ε is

where

modeled as zero-mean discrete Gaussian white noise and it satisfies the following correlation structure:

E[ε(nΔt )ε(n' Δt ) T ] = Σε δ nn ' where E[.] denotes the mathematical expectation; and which is equal to unity if n = n' and zero otherwise. of the measurement noise.

(6)

δ nn ' is the Kronecker delta function,

Σε ∈ R N o × N o is the covariance matrix

N

Suppose only the first m modes have significant contribution to the structural response. The uncertain parameter vector a is comprised of:

a = [a1 , a 2 , a 3 ]T where

a1 = [ω1 , ω2 ,..., ωN m , ζ 1 , ζ 2 ,..., ζ N m ]

and damping ratios of these

(7)

is the vector including the modal frequencies

N m modes; a = a row vector containing the elements of the first 2

N m columns of the matrix L Φ, except the elements used for normalization as mentioned in 0 Sf0

Eq. (3); and a3 = a row vector containing the elements in the upper right triangle of

and

Σε , because the lower triangular parts can be determined by the upper triangles. Note that if

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Ka-Veng Yuen and Sin-Chi Kuok

different DOFs are chosen for the mode shape normalization, the identification results of L0Φ and

Sf0

are affected. In particular, if the mode shape components are scaled by a constant cr

for the r-th mode shape, the element

S (fr0,s )

will be scaled by (cr cs)-1,

r , s = 1,2,..., N m .

3.2. Spectral Density Matrix Estimator Consider M sets of independent and identically distributed discrete response measurements:

YN( m ) = {y ( m ) (nΔt ), n = 0,1,..., N − 1}

(8)

To transform the discrete time histories to the frequency domain, the discrete Fourier transform is applied. For a single set of measurement scaled discrete Fourier transform is given by

YN (ωk ) =

Δt 2πN

N −1

YN = {y (nΔt ), n = 0,1,..., N − 1} , the

∑ y(nΔt ) e ω -i

k nΔt

n=0

(9)

ω = kΔω , k = 0,1,..., N1 − 1 with N1 = INT( N / 2), Δω = 2π / T , and T = NΔt . where k Then, the spectral density matrix estimator is defined as follows: S y , N (ω k ) = YN (ω k )YN T (ω k )

(10)

where z is the complex conjugate of a complex variable z. The structural response x and the measurement noise ε are assumed statistically independent, so the second derivative of the



modal coordinates q and the measurement noise ε are also independent. Then, by substituting Eq. (2) and (5) to Eq. (10) and taking expectation, one obtains:

E[S y , N (ωk ) | a] = (L 0Φ) E[S q, N (ωk ) | a](L 0Φ) T + E[Sε , N (ωk ) | a]

S

(ω ) and S

(11)

(ω )

ε ,N k where q, N k are defined in the same manner as that described by Eq. (9) and (10). It can be easily shown that the expected value of the spectral density matrix estimator of the measurement noise is given by

E[Sε , N (ωk ) | a] =

Δt Σε ≡ Sε 0 2π

(12)

Monitoring a 22-Story Building under Severe Typhoons… On the other hand, the matrix

E[S q, N (ω k ) | a]

S q(r, N, s ) (ωk ) =

Δt 2πN

on the right hand side of Eq. (11) can be

S q, N (ω k )

easily obtained by noting that the element in

has the form:

N −1

∑ q (nΔt )q (n' Δt )e ω -i

n, p =0

515

r

s

k

( n − n ') Δt

(13)

By grouping the terms with same value of (m－n’) and taking expectation yields:

[

]

E S q(r, N, s ) (ωk ) =

where the coefficients

Δt 4πN

∑ b [R N −1 n=0

(r,s) q

n

(nΔt )e-iω k nΔt + Rq(s , r ) (nΔt )eiω k nΔt

(14)

bn are given by n=0 ⎧N , bn = ⎨ ⎩ 2( N − n), n ≥ 1

and

Rq(r , s )

]

(15)

represents the correlation function which can be computed by

R

(t ) = ∫

(r ,s) q

ω 4 S (f r , s ) (ω )e iωt dω [(ω r2 − ω 2 ) + 2iωω rς r ] [(ω s2 − ω 2 ) − 2iωω sς s ]

(16)

that integrates over the frequency band of the excitation. With the M sets independent and identically distributed discrete time histories, the

Y ( m ) (ω ), m = 1,2,..., M

k are statistically independent and corresponding random vectors N follow a zero-mean complex Ns–variate normal distribution [22]. The averaged spectral density matrix estimator can be obtained by

S My , N (ωk ) =

Note that

S My , N (ωk )

1 M

M

∑S m =1

( m) y,N

(ωk ) =

1 M

M

∑Y m =1

( m) N

T

(ωk )YN(m) (ωk ) (17)

follows a central complex Wishart distribution of dimension No with M-

E[S M (ω )] = E[S

(ω )]

y,N k y,N k DOFs and mean [22]. The M sets of independent and identically distributed measurements may be obtained approximately by partitioning the dataset into several subsets with equal length. The corresponding probability density function (PDF) of the averaged spectral density matrix estimator is given by

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Ka-Veng Yuen and Sin-Chi Kuok

(

)

p S My , N (ωk ) =

π −N

o ( N o −1) / 2

M N o ( M − N o ) | S My , N (ω k ) | M − N o

No

(∏ ( M − p )!) | E[S y , N (ω k )] | M

exp(−M tr[ E[S y , N (ω k )]−1 S My , N (ωk )])

(18)

p =1

where |.| and tr [.] are the determinant and trace of a matrix, respectively. Note that Eq. (18) is valid only if

M ≥ No . S M (ω )

ω ≠ω

S M (ω )

y, N k l and large N, the random matrices and y , N l are In addition, for k statistically independent in a properly selected range of frequencies [23], so the joint probability density can be written as

p[S My , N (ωk ), S My , N (ωl )] = p[S My , N (ωk )] p[S My , N (ωl )]

(19)

3.3. Identification Using the Bayesian Spectral Density Approach Assume that M independent sets of discrete time histories available. The corresponding spectral density matrix estimators computed by Eq. (9) and (10), and the averaged spectra

Sˆ

M y, N

Sˆ

ˆ ( m ) , m = 1,2,..., M Y N ( m) y,N

(ωk )

, m = 1,2,..., M

, are

, can be

can be obtained by Eq.

[ω , ω ] = [k Δω, k Δω]

k1 k2 1 2 , the corresponding (17). For a properly selected frequency range spectral density matrix estimators are grouped to form the data set for Bayesian inference:

Sˆ My , N, k1 , k 2 = [Sˆ My , N (kΔω ), k = k1 , k1 + 1,..., k2 ]

. Note that a proper choice of k1 and k2 not only enhances the accuracy of the approximations in Eq. (18) and (19), but also improves the computational efficiency. According to [13], one choice of this frequency range is just to cover all the peaks of the modes for identification. Based on Eq. (18) and (19), the likelihood function can be easily obtained: k2 | Sˆ My ,N (ωk ) |M − No M ,k1 ,k2 ˆ p S y ,N | a ≈ c∏ exp − M tr{E[S y ,N (ωk )]−1 Sˆ My , N (ωk )} M k =k1 | E[S y , N (ωk )] |

(

)

(

⎡ − No ( No −1) / 2 No ( M − No ) No ⎤ M / ∏ ( M − p )!⎥ where c = ⎢π p =1 ⎣ ⎦

) (20)

k2 − k1 +1

. By Bayes’ Theorem, the posterior

PDF is proportional to the product of the prior distribution and the likelihood function [14]:

p (a | Sˆ My , N, k1 , k 2 ) ∝ p (a) p (Sˆ My , N,k1 ,k 2 | a)

(21)

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517

where p(a) is the prior PDF representing the previous knowledge or engineering judgment of the uncertain parameters before the data is obtained. In the current situation, since the prior information of the parameters is not available, p(a) is taken to be an improper prior and it is absorbed into the normalizing constant. Therefore, the parametric identification depends

p (Sˆ M ,k1 ,k 2 | a)

y, N , that expresses the degree of data fitting. A solely on the likelihood function better fit to the data associates with a larger value of the likelihood function. Finally, the objective function is defined as the negative logarithm of the likelihood function in Eq. (20) without taking the constant that does not depend on the uncertain parameters:

J (a) = M ln | E[S y , N (ωk )] | + M tr{E[S y , N (ωk )]−1 Sˆ My , N (ωk )}

(22)

By minimizing the objective function with respect to the modal parameters in a , the optimal modal parameters aˆ can be obtained. For modal identification, the updated PDF

p (a | Sˆ My , N, k1 , k 2 )

can be well approximated by a Gaussian distribution with mean aˆ and

Σ

covariance matrix aˆ [13, 14]. This is done instead of maximizing directly the likelihood function since the proposed optimization problem offers better computational conditions. The covariance matrix can be approximated by the inverse of the Hessian of J (a) , evaluated at the optimal parameters aˆ :

Σ aˆ = H(aˆ ) −1

(23)

The elements of the Hessian matrix are given by the second derivatives of the objective function:

∂ 2 J (a) H αβ (a) = ∂a α ∂a β

(24)

This can be computed accurately by finite difference method since the objective function is approximately quadratic [13, 14].

4. Meteorological Information One-minute-average wind speed and wind direction are recorded regularly by the nearest meteorological station, Direcção dos Serviços Meteorológicos e Geofísicos (SMG), which is 1.25 km east-south from the East Asian Hall. It is worth emphasizing that the Bayesian spectral density approach requires only response measurements of the structure. The information of the wind excitation was utilized only for reference and for interpretation of the results but it was not involved in the identification process.

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Ka-Veng Yuen and Sin-Chi Kuok

4.1.Typhoon Nuri Nuri was the fourth typhoon that affected Macao in 2008. It was formed as a tropical depression on August 17 and it was rapidly intensified to a tropical storm on the next day. After making the first landfall in the Philippines, it entered the South China Sea and moved straight to Hong Kong. The maximum 10-minute sustained wind speed and the lowest pressure of Nuri were 140 km/hr and 955 hPa, respectively. With six-day duration, Nuri landed in Hong Kong. The track of Nuri is shown in figure 2, and it can be seen that Nuri passed by Macao from the northeast side with core distance less than 100 km. The hoisting signals from SMG with the corresponding wind speed magnitude and hoisting time are listed in table 1.

Figure 2. Track of Nuri [provided by SMG (http://www.smg.gov.mo/c_index.php) and the time corresponds to GMT +08:00)].

Table 1. Announced typhoon signals by SMG Typhoon Signal

Mean Wind Speed near Typhoon Core (km/hr)

Gust Speed (km/hr)

1 3 8 3 0

< 41 [41, 62] [63, 117] [41, 62] < 41

-110 180 110 --

Hoisting Time (mm/dd GMT+08:00) Nuri Hagupit 20:00 08/20 22:00 09/22 00:30 08/22 13:00 09/23 12:00 08/22 19:15 09/23 00:30 08/23 09:30 09/24 11:00 08/23 16:00 09/24

The 10-minute-average wind speed and wind direction of the two typhoons observed in SMG of Macao are shown in figure 3. The wind speed shown in this figure is the value measured by SMG in Macao but not the core wind speed of the typhoon. The wind direction angle counts from the true north clockwise. For example, an angle of 90o denotes the easterly wind. Even though the track of Nuri was fairly smooth, its wind direction was fluctuating significantly because its track was very close to Macao. Since the rotation of the typhoon surrounding air flow in the North Hemisphere is always counter-clockwise, the wind direction depends highly on its position to the city of concern. Before the 30-th hour, the wind speed was in a relatively calm range and it was mainly controlled by the background wind field.

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519

After the 30-th hour, Nuri was approaching Macao and the wind speed was accelerating rapidly from less than 20 km/hr to approximately 55 km/hr within 10 hours. The wind direction remained north before the typhoon reached the coastline. After the typhoon arrived at the point with the shortest distance to Macao, the wind direction changed sharply from north to south because it almost passed through Macao. 360

Wind speed Direction

60

270

40

180

20

90

0

0

10

20

30 Time ( hr )

40

50

60

Direction ( deg )

Mean wind speed ( km/hr )

80

0

Figure 3. Wind speed and wind direction of Nuri.

4.2. Typhoon Hagupit On September 14, a tropical disturbance was formed to the northeast of Guam and moved towards the Philippines. Being continuously intensified in the following days, the disturbance became a tropical storm and it was named Hagupit on September 19. On the same day, it was upgraded to a tropical storm and it was progressively approaching southern China. Hagupit had the highest 10-minute sustained wind speed of 165 km/hr and the lowest pressure of 935 hPa. More than 60 people were killed and the economic loss was estimated to be no less than one billion US dollars. Hagupit was dissipated after the landfall was made between the Guangdong and Guangxi province. Afterwards, the typhoon signal was cancelled at 16:00 on September 24. In contrast to Nuri, Hagupit passed by Macao from the south side as shown in figure 4 and its track was very straight. Hagupit was the strongest typhoon affected Macao since year 2000 and it generated the largest wind loading on the East Asian Hall in the history of the building. The magnitude of the maximum wind speed of Hagupit was much larger than that of Nuri. Figure 5 shows the wind speed and wind direction observed in SMG of Macao and its high-wind-speed duration (10-minute-average wind speed over 40 km/hr) was also much longer even though its total

520

Ka-Veng Yuen and Sin-Chi Kuok

signalled period was shorter. One can also observe that between the 20-th to 35-th hour, the wind direction rotated gradually from north to east and then to south. As shown on the right tails of the figure, the typhoon-effect was mitigated after the landfall was made and the major wind direction was recovered to the background prevalent offshore wind direction.

Figure 4. Track of Hagupit [provided by SMG (http://www.smg.gov.mo/c_index.php) and the time corresponds to GMT +08:00)]. 80

400

60

300

40

200

20

100

0

0

5

10

15

20 25 Time ( hr )

30

35

40

Direction ( deg )

Mean wind speed ( km/hr )

Wind speed Direction

0

Figure 5. Wind speed and wind direction of Hagupit.

In general, when a typhoon is approaching, the wind speed increases gradually while the wind direction changes. Once it reaches the closest point to a city, the wind speed achieves the highest magnitude but the wind direction changes drastically if the typhoon is very close to the city. Therefore, the wind loading generated by a typhoon, and hence the structural

Monitoring a 22-Story Building under Severe Typhoons…

521

response, is considered as a non-stationary stochastic process, especially when the typhoon is close to the city [24]. Special treatment has to be imposed in the identification process [25].

5. Analysis with the Field Measurements 5.1. Acceleration Response Measurements Acceleration time histories of the East Asian Hall were recorded under the whole duration of the two typhoons, Nuri and Hagupit, with sampling time interval 0.002 sec. The full recorded durations were 64 and 46 hours, respectively. In addition, for each typhoon, six sets of measurements were taken with 3 of them before and the other 3 after the typhoon. Nuri (08/22 13:00)

-2

Sy11 (m2 sec -3)

Sy11 (m2 sec -3)

10

-6

10

-10

10

1

2

10

3

-2

0

1

2

3

0

1 2 Freq. (Hz)

3

-1

10 Sy22 (m2 sec -3)

Sy22 (m2 sec -3)

-5

10

-9

0

10

-6

10

-10

10

Hagupit (09/23 23:00)

-1

10

-5

10

-9

0

1 2 Freq. (Hz)

3

10

Figure 6. Averaged acceleration spectra under hoisting signal no. 8 of Nuri and Hagupit.

In the present study, the first ten-minute record of each hour is used to trace the hourly modal parameters of the structure and excitation. The averaged spectra are obtained as follows: 1. partition the ten-minute response into four sets (M=4) with equal time duration, so each set contains a record of 150 sec and N = 75000; 2. use Eq. (9) and (10) to compute the individual spectral density matrix estimator of each set; 3. compute the averaged spectra by Eq. (17).

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Ka-Veng Yuen and Sin-Chi Kuok

Two representative averaged spectra under hoisting signal no. 8 of the two typhoons are shown in figure 6. They correspond to the diagonal elements of the averaged spectral density matrix estimator. In the spectra of the second channel (the lower subplots), the first mode is absent for both cases, implying that the first mode does not have notable contribution to the motion of the DOF associated with direction 2 at the mounting location. Nevertheless, four peaks clearly show up in the spectra of the first channel, so these modes are to be identified (Nm =4). The spike at the zero frequency is proportional to the temporal average of the signal, which is commonly observed and is removed for the analysis.

5.2. Aerodynamic Effects to the East Asian Hall Instead of rectangular or circular floor plan, the L-shape geometry of the East Asian Hall increases the complexity of analyzing the aerodynamic effects to the structural behaviour. Some possible major aerodynamic effects are discussed in this section.

5.2.1. Drag Force Forces are generated by pressure gradient when a fixed object is submerged in a flow. The net force can be decomposed into the along-flow and across-flow directions. The alongflow component is referred as the drag force and it can be expressed as

1 FD = C D ρU 2 A 2

(25)

where ρ is the air density, U is the far field wind speed, A is the characteristic area of the building, and CD is the drag coefficient. Due to the particular geometry of the building, the values of CD and A are time-varying. Table 2 lists the values of CD for an L-shape object under three different flow directions [26, 27]. The largest difference, which occurs between the first two cases, is 22.5%. According to Eq. (25), the drag force is proportional to the drag coefficient, so the resultant force for different wind attacking angles could be significantly different for the same wind speed. Therefore, as will be shown in the following paragraphs, larger structural response may be associated with smaller wind speed. Table 2. Drag coefficient for different flow angle ([26, 27]) Case 1

Case 2

Case 3

1.55

2.0

1.83

Profile and wind direction

Drag coeff. CD

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523

-3

RMS acc. (m/sec 2)

1.5

x 10

Dir. 1 Dir. 2

1

0.5

0

0

10

20

0

10

20

30

40

50

60

40

50

60

Wind Speed (km/hr)

60

40

20

0

30 Time (hr)

Figure 7. RMS acceleration and mean wind speed under Nuri. -3

RMS acc. (m/sec 2)

5

x 10

Dir. 1 Dir. 2

4 3 2 1 0

0

5

10

15

0

5

10

15

20

25

30

35

40

20 25 Time (hr)

30

35

40

Wind Speed (km/hr)

80 60 40 20 0

Figure 8. RMS acceleration and mean wind speed under Hagupit.

Figure 7 shows the root-mean-square (RMS) acceleration and the 10-minute-average wind speed observed in Macao by SMG during typhoon Nuri. The rectangular window

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Ka-Veng Yuen and Sin-Chi Kuok

encloses the region with wind speed exceeding 40km/hr and the two arrows point out the peaks of the wind speed and structural response. It is observed that even the two peaks did not occur at the same time, the maximum response was still achieved when the wind speed was close to its maximum value. Besides, in this severe wind speed region, the wind direction of Nuri maintained in north (figure 3) and the attacking angle is similar to Case 1 of table 2. On the other hand, figure 8 presents an interesting phenomenon of Hagupit. The highwind-speed window was much longer than that of Nuri. It is observed that the structural acceleration magnitude on the right boundary of the high-wind-speed window is about twice of the left boundary, but they are associated with the same wind speed (40km/hr). Moreover, the arrows indicate that the peaks of the RMS acceleration and wind speed do not coincide. This is caused by the wind attacking angle of the typhoon. From figure 5, the wind direction changed gradually from north to east. By figure 1, the wind direction of Hagupit on the East Asian Hall was transformed from Case 1 to Case 2 in table 2, so the drag coefficient was increasing in this process. Under the same wind speed, larger resultant force was created in the later stage so the corresponding structural response was larger. It turned out that the peak of the structural response occurred 6 hours after the peak of the wind speed. When the maximum response was achieved, the wind speed was 20% lower than its maximum value.

5.2.2.Vortex-Shedding Effects When a building is immersed in an unsteady flow, separation including diverging wake and vortices will be produced by the alternating pressure. The periodic alternation of the low pressure zone induces vibration of the body. If the frequency of this oscillation is closed to any of the modal frequencies of the building, resonance will occur and this will create catastrophic failure to the structure. The frequency of the vortex shedding induced oscillation can be estimated by a well-known dimensionless parameter, namely the Strouhal number [28]:

St =

f VSl U

(26)

where fVS is the vortex shedding frequency, l is the characteristic length and U is the flow speed. According to [26], the Strouhal number is in the range from 0.07 to 0.15 for an object with L-shape cross section. For the present case of study, the characteristic length may be taken as 60 m for the building, the maximum wind speed of both typhoon is around 55 km/hr. Therefore, the vortex shedding frequency is roughly in the range of 0.02 – 0.04 Hz. Since the vortex shedding frequency is proportional to the wind speed by Eq. (26), it is in general smaller than 0.04 Hz. The modal frequencies of the building are 30 times larger than the vortex shedding frequency so aerodynamic resonance is not likely to occur. As shown in both cases of figure 6, there was no obvious oscillation under 1 Hz. The relative larger values near the zero frequency are observed because the wind spectrum is well unknown to have such characteristics [29].

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525

5.3. Identification of Modal Parameters 5.3.1. Spectral Intensity of Excitation The spectral intensity of the modal forces describes the input energy for a mode and they are shown in figure 9 and 10 for typhoon Nuri and Hagupit, respectively. Compared with the wind speed shown in figure 7 and 8, it is observed that all these curves had a similar trend but the peaks of the spectral intensities did not occur at the same time of the wind speed. In figure 9, the peaks of spectral intensities occurred at approximately the 40th hour but the peak of the wind speed occurred 4 hours later (figure 7). This is even more obvious for Hagupit in figure 10 that shows the peaks of spectral intensities at the 25-th hour, which is five hours after the peak of the wind speed (figure 8). This confirms that not only the wind speed but also the wind direction affects the magnitude of the structural response. It can be further verified by the fact that the peaks of the spectral intensities (figure 9 and 10) and structural response (figure 7 and 8) occurred at the same time. Note that it is meaningless to compare the spectral intensities of different modes since the values depend on the mode shape normalization as discussed in Section 3.1. However, its variation of a particular mode indicates the energy variation against time since the normalization is fixed. -10

10

S(1,1) f0 S(2,2) f0

-11

10

S(3,3) f0 S(4,4) f0

-12

diag ( Sf0 )

10

-13

10

-14

10

-15

10

-16

10

0

10

20

30 Time ( hr )

40

50

Figure 9. Spectral intensities of different modes of the excitation by Nuri.

60

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Ka-Veng Yuen and Sin-Chi Kuok -9

10

S(1,1) f0 S(2,2) f0

-10

10

S(3,3) f0 S(4,4) f0

-11

diag ( S f0 )

10

-12

10

-13

10

-14

10

-15

10

0

5

10

15

20 25 Time (hr)

30

35

40

Figure 10. Spectral intensities of different modes of the excitation by Hagupit.

5.3.2. Structural Parameters Figure 11 shows the identified modal frequencies with the confidence interval of plus and minus three standard derivations from the identified values. The identified modal frequencies are referred to the equivalent modal frequencies since the building may not behave perfectly linear, especially under the severe wind pressure. The ± 3σ confidence interval includes a probability of 99.7% that the actual effective value falls inside. They are all narrow, implying that the statistical uncertainty of the identification results is small so there are other factors for the fluctuation of the modal frequencies. Therefore, it is statistically evident to state that when the spectral intensity of the modal force increased, the corresponding modal frequency decreased. The first and the fourth modes had relatively significant reduction. Under the passage of Nuri, the frequencies of these two modes were reduced by 5% and 3%, respectively. For Hagupit, the modal frequency of the first mode was reduced by 6% and the modal frequency of the fourth mode exhibited the largest reduction of 9%. However, such reduction was recovered almost immediately after the typhoon was dissipated in the case of Hagupit. Nevertheless, it was not the case of Nuri and this will be further discussed in Section 5.4. Figure 12 shows the modal frequencies versus the spectral intensity of the corresponding modal forces in the semi-logarithmic scale. In all four modes, the modal frequencies decreased as the spectral intensity of the corresponding modal force increased, and the relationship is approximately linear in the semi-logarithmic scale. The structural stiffness was degraded when the excitation was at high level. However, there is no evidence for structural damage since such loss was recovered after the typhoon was dissipated. One possible explanation is that the structure went through hysteretic behaviour so the identified modal frequencies of the equivalent linear system were smaller.

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1.45

1.42

1.42 1.39 20

40

1.7

1.675

ωn2 (Hz)

1.7

1.68 1.66

10

20

30

40

0

10

20

30

40

0

10

20

30

40

0

10

20 30 Time ( hr )

40

1.625 1.6

0

20

40

60 1.87

1.86

1.85

1.85 1.84 0

20

40

1.83 1.81 1.79

60

2.46

2.4

2.43

2.34

2.4 2.37 2.34

0

1.65

1.87

1.83

-3 σ

1.36 1.33

60

+3 σ

1.39

1.72

1.64

ωn3 (Hz)

0

ωn3 (Hz)

ωn2 (Hz)

1.36

ωn4 (Hz)

ωn1 (Hz)

1.45

ωn4 (Hz)

ωn1 (Hz)

ωn

1.48

0

20 40 Time ( hr )

(a)

60

2.28 2.22 2.16

(b)

Figure 11. Estimated modal frequencies with +/- 3σ confidence interval: (a) Nuri and (b) Hagupit.

Figure 13 shows the relationship between the damping ratios and the spectral intensity of the corresponding modal forces. The damping ratio of the first mode varied in a wide range from 0.3 % to 3.0 %. Although the data points are more scattered than those in figure 12, it remains clear to conclude that the damping ratios increased as the spectral intensity of the corresponding modal force increased. The excitation dependent damping ratios indicate again possible hysteretic behaviour of the building. Even though the equivalent modal frequency of the building was reduced during severe wind pressure, the increase of the damping ratio is much larger than the decrease of the modal frequency. Therefore, the equivalent damping coefficient exhibited significant increase. These two opposite trends of the modal frequency and damping ratio are consistent to the possible hysteretic behaviour of the building that the equivalent stiffness was degraded and the energy dissipated by the hysteretic loops would contribute to equivalent damping.

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Ka-Veng Yuen and Sin-Chi Kuok 1.5

1.75 Nuri

ωn2 (Hz)

ωn1 (Hz)

1.45 1.4 1.35 1.3 -15 10

Hagupit

1.7 1.65 1.6 -15 10

-10

10

-10

10 S (2,2) f0

1.88

2.5

1.86

2.4

ωn4 (Hz)

ωn3 (Hz)

S(1,1) f0

1.84

2.3

1.82 1.8 -14 10

2.2 -12

10

2.1

-10

10

-15

-10

10

10

S (3,3) f0

S(4,4) f0

Figure 12. Estimated modal frequencies and spectral intensity of the modal forces.

3

Nuri

2

2

ζ2 (%)

ζ1 (%)

Hagupit

1.5

1

1 0.5

0 -15 10

0 -15 10

-10

10

-10

10

S(1,1) f0

S(2,2) f0 3

ζ4 (%)

ζ3 (%)

1

0.5

0

-10

10 S(3,3) f0

2 1 0

-15

-10

10

10 S(4,4) f0

Figure 13. Estimated damping ratios and spectral intensity of the modal forces.

Table 3. Ambient conditions with estimated modal frequencies and damping ratios Modal Frequency (Hz)

Ambient Cond. Typhoon

Date Time mm/dd

Temp. °C

R.H. %

23:00 08/17

27.3

78

23:00 08/18

27.8

76

23:00 08/19

26.9

77

[22.2, 32.7]

[61, 95]

23:00 08/23

24.4

87

23:00 08/24

26.4

84

23:00 08/25

25.4

84

ω1

ω2

ω3

Damping Ratio (%) ω4

ζ1

ζ2

ζ3

ζ4

1.4409 (0.0013) 1.6840 (0.0009) 1.8587 (0.0008) 2.4101 (0.0017) 0.863 (0.094) 0.644 (0.057) 0.447 (0.041) 0.875 (0.070) 1.4352 (0.0017) 1.6828 (0.0011) 1.8574 (0.0007) 2.3823 (0.0014) 1.272 (0.131) 0.901 (0.066) 0.331 (0.039) 0.702 (0.054) 1.4338 (0.0021) 1.6898 (0.0009) 1.8540 (0.0009) 2.4032 (0.0015) 1.514 (0.155) 0.690 (0.056) 0.549 (0.046) 0.699 (0.059)

19:30 08/20 Nuri

~

[1.3801, 1.4552] [1.6532, 1.6970] [1.8407, 1.8640 [2.3588, 2.4430] [0.370, 2.918] [0.344, 1.361] [0.197, 0.696] [0.297, 1.472] (0.0016) (0.0010) (0.0008) (0.0015) (0.119) (0.059) (0.043) (0.058)

11:30 08/23

1.4047 (0.0011) 1.6694 (0.0010) 1.8497 (0.0008) 2.4045 (0.0012) 0.808 (0.086) 0.764 (0.065) 0.406 (0.039) 0.581 (0.052) 1.4156 (0.0013) 1.6838 (0.0010) 1.8556 (0.0004) 2.3853 (0.0011) 0.823 (0.094) 0.706 (0.059) 0.348 (0.030) 0.524 (0.046) 1.4124 (0.0012) 1.6810 (0.0009) 1.8597 (0.0007) 2.4028 (0.0013) 0.702 (0.087) 0.597 (0.051) 0.405 (0.036) 0.679 (0.058)

Table 3. Continued

Typhoon

Date Time mm/dd

Modal Frequency (Hz)

Ambient Cond. Temp. °C

R.H. %

23:00 09/19

27.3

80

23:00 09/20

27.7

74

23:00 09/21

28.5

70

[22.8, 33.4]

[44, 88]

23:00 09/24

26.5

85

23:00 09/25

26.5

81

23:00 09/26

27.1

76

ω1

ω2

ω3

Damping Ratio (%) ω4

ζ1

ζ2

ζ3

ζ4

1.4145 (0.0031) 1.6689 (0.0009) 1.8426 (0.0007) 2.3254 (0.0011) 0.658 (0.091) 0.589 (0.054) 0.345 (0.045) 0.495 (0.047) 1.4113 (0.0012) 1.6741 (0.0009) 1.8400 (0.0006) 2.2842 (0.0012) 0.570 (0.076) 0.640 (0.054) 0.288 (0.042) 0.583 (0.049) 1.4170 (0.0016) 1.6744 (0.0009) 1.8463 (0.0006) 2.3086 (0.0015) 1.196 (0.136) 0.633 (0.055) 0.252 (0.042) 0.772 (0.058)

21:30 09/22 Hagupit

~

[1.3566, 1.4402] [1.6102, 1.6824] [1.8029, 1.8534] [2.1725, 2.3845] [0.518, 2.654] [0.435, 2.119] [0.247, 0.964] [0.500, 2.373] (0.0016) (0.0011) (0.0009) (0.0020) (0.133) (0.068) (0.047) (0.080)

16:30 09/24

1.3964 (0.0015) 1.6623 (0.0010) 1.8461 (0.0007) 2.3358 (0.0013) 1.250 (0.107) 0.728 (0.059) 0.331 (0.043) 0.588 (0.049) 1.4049 (0.0014) 1.6762 (0.0009) 1.8514 (0.0007) 2.3664 (0.0014) 1.095 (0.111) 0.592 (0.052) 0.354 (0.041) 0.745 (0.053) 1.4060 (0.0013) 1.6790 (0.0009) 1.8508 (0.0007) 2.3805 (0.0013) 0.928 (0.095) 0.599 (0.055) 0.359 (0.041) 0.662 (0.052)

() indicates the associated standard deviation; [] indicates the range of the parameter.

Monitoring a 22-Story Building under Severe Typhoons…

531

5.4. Ambient Conditions Besides the wind speed and wind attacking angle, ambient conditions are also important factors that affect the properties of a structure. For example, temperature and relative humidity are the ones which influence directly the Young’s modulus, material thermal expansion, and internal lubricity of concrete, etc [2]. Throughout the duration of a typhoon, substantial changes of the temperature and relative humidity are possible. Temperature is normally high before the typhoon and it will drop when the typhoon is approaching. Meanwhile, the relative humidity normally increases due to precipitation. These two ambient parameters along with the structural modal parameters are summarized in table 3. Identification was carried out for an expanded period with six additional sets of measurement obtained daily at 23:00 with three of them before and the other three after each typhoon. During the typhoon period, the temperature and relatively humidity fluctuated substantially, and so did the identified modal frequencies and damping ratios. If the modal frequencies are compared immediately before and after the typhoon, they did not recover completely, especially in the case of Nuri. However, the temperature immediate after the typhoon was lower than that before the typhoon in both cases so the comparison is unfair. In the case of Hagupit, the temperature rose gradually after the typhoon was dissipated and the modal frequencies also increased approximately back to the original level. Therefore, direct comparison of the structural properties is meaningful only under the same ambient conditions. Further investigation is needed to examine the influence on modal parameters due to changes of ambient conditions.

4. Conclusion This manuscript demonstrated the potential of Bayesian inference for civil engineering [30]. The Bayesian spectral density approach was applied for the identification of the modal parameters of the East Asia Hall, which is a 22-story reinforced concrete building in Macao. Its acceleration time histories were recorded during the passage of two severe typhoons in August and September 2008. It is valuable to investigate the structural behaviour under such violent wind excitation. Due to the special configuration of the monitored building, the aerodynamic effects to the structure are significantly more complex than the buildings with rectangular or circular floor plan. As shown by the acceleration response, the structural behaviour was influenced by the wind speed and also the wind direction. By monitoring the modal frequencies through the complete process of the two typhoons, it is concluded that the modal frequencies decreased with increasing spectral intensity of the excitation while the damping ratios had an opposite trend with the spectral intensity. Furthermore, substantial changes of the modal frequencies and damping ratios occurred when the excitation level was severe. This indicates that the building could have undergone nonlinear hysteretic behaviour and the identified modal parameters are only the value of the equivalent linear system. After the typhoons dissipated, the reduction recovered almost immediately. However, full recovery of the modal frequencies was achieved when the ambient conditions became similar to that before the typhoon event. Further work is on progress for the effects on modal parameters due to different ambient conditions.

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References [1] Atlantic Oceanographic and Meteorological Laboratory, Hurricane Research Division. (2006). Frequently Asked Questions: When is Hurricane Season? [2] Sohn, H., Farrar, C.R., Hemez, F.M., Shunk, D.D., Stinemates, D.W. & Nadler, B.R. (2003). A review of structural health monitoring literature: 1996-2001. Los Alamos national laboratory report. [3] Smith, B.S. & Coull, A. (1991). Tall Building Structures: Analysis and Design. John Wiley & Sons: New York. [4] Salawu, O.S. (1997). Detection of structural damage through changes in frequency: a review. Engineering Structures, 19(9), 718-723. [5] Kareem, A. & Sun, W.J. (1989). Dynamic response of structures with uncertain damping, Engineering Structures, 12(1), 2-8. [6] Holmes, J. D. (1987). Mode shape corrections for dynamic response to wind, Engineering Structures, 9, 210-212. [7] Hoshiya, M. & Saito, E. (1984). Structural identification by extended Kalman filter. Journal of Engineering Mechanics, 110(12), 1757-1770. [8] Ibrahim, S.R. (1986). Double least squares approach for use in structural modal identification. AIAA Journal, 24(3), 499-503. [9] Juang, J.N. & Suzuki, H. (1988). An eigen-system realization algorithm in frequency domain for modal parameter identification. Journal of Vibration, Acoustics, Stress and Reliability in Design, 110(1), 24-29. [10] Beck, J.L. (1990). Statistical system identification of structures. In Structural Safety and Reliability, ASCE, New York, 1395-1402. [11] Ewins, D.J. (1995). Modal Testing, Theory and Practice. John Wiley & Sons. [12] Xu, Y.L., Chen, S.W. & Zhang, R.C. (2003). Modal identification of Di Wang building under typhoon York using the Hilbert-Huang transform method. The Structural Design of Tall and Special Buildings, 12(1), 21-47. [13] Katafygiotis, L.S. & Yuen, K.V. (2001). Bayesian spectral density approach for modal updating using ambient data. Earthquake Engineering and Structural Dynamics, 30(8), 1103-1123. [14] Beck, J.L. & Katafygiotis, L.S. (1998). Updating models and their uncertainties. I: Bayesian statistical framework. Journal of Engineering Mechanics, 124(4), 455-461. [15] Yuen, K.V. & Beck, J.L. (2003). Updating properties of nonlinear dynamical systems with uncertain input, Journal of Engineering Mechanics, 129(1), 9-20. [16] Li, Q.S., Wong, C.K., Fang, J.Q., Jeary, A.P. & Chow, Y.W. (2000). Field measurements of wind and structural responses of a 70-storey tall building under typhoon conditions. The Structural Design of Tall Buildings, 9(5), 325-342. [17] Li, Z.X., Chan, T.H.T. & Ko, J.M. (2002). Evaluation of typhoon induced fatigue damage for Tsing Ma Bridge. Engineering Structures, 24(8), 1035-1047. [18] Zhang, L., Tamura, Y., Yoshida, A., Cho, K., Nakata, S. & Naito S. (2002). Ambient vibration testing and modal identification of an office building. Proc. of the 20th IMAC, Los Angeles, USA. [19] Li, Q.S., Xiao, Y.Q., Wu, J.R., Fu, J.Y. & Li, Z.N. (2008). Typhoon effects on super-tall buildings. Sound and Vibration, 313(3-5), 581-602.

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[20] Friswell, M.I. & Mottershead, J.E. (1995). Finite Element Model Updating in Structural Dynamics, Boston: Kluwer Academic Press. [21] Yuen, K. V., Katafygiotis, L.S., Papadimitriou, C. & Mickleborough, N.C. (2001). Optimal sensor placement methodology for identification with unmeasured excitation. ASME Journal of Dynamical Systems, Measurement and Control, 123(4), 677-686. [22] Krishnaiah, P.R. (1976). Some recent developments on complex multivariate distributions. Journal of Multivariate Analysis, 6(1), 1–30. [23] Yuen, K.V., Katafygiotis, L.S. & Beck, J.L. (2002). Spectral density estimation of stochastic vector processes. Probabilistic Engineering Mechanics, 17(3), 265-272. [24] Chen, J. & Xu, Y.L. (2004). On modelling of typhoon-induced non-stationary wind speed for tall buildings, The Structural Design of Tall and Special Buildings, 13(2), 145163. [25] Bierens, H.J. & Guo, S. (1993). Testing stationarity and trend stationarity against the unit root hypothesis. Econometrics Review, 12(1), 1-32. [26] ASCE (1961). Wind forces on structures Part II, Transactions of the American Society of Civil Engineering, 126, 1124-1198. [27] Blevins, R.D. (1984). Applied Fluid Dynamics Handbook, Wiley: New York. [28] White, F.M. (1999). Fluid Mechanics (4th edition), McGraw Hill. [29] Davenport, A.G. (1961). The spectrum of horizontal gustiness near the ground in high winds. Quarterly Journal of the Royal Meteorological Society, 87(372), 194-211. [30] Yuen, K.V. (2010). Bayesian Methods for Structural Dynamics and Civil Engineering, Wiley: Singapore.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 535-546

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 10

CHARACTERISTICS OF OHM LAW FOR METAL AT LOW TEMPERATURE A.N. Volobuev* and V.V. Galanin Department of Medical and Biological Physics, Samara State Medical University, Box 1423, Samara 443079, Russia

Abstract The characteristics of Ohm law for a metal at low temperature are considered on the basis of research kinetics of charge carriers in the conductor, which can transit to a superconducting state. In work has shown that at decrease of absolute temperature to zero in the conductor some current can be conserved. According to quantum representations this current has identified with the current of superconductivity. The relation of the researched model with Ginzburg-Landau superconductivity theory is investigated.

1. Introduction Observable experimental linear dependence of the specific resistance of metal and his absolute temperature ρ ~ T is usually correct in a wide enough interval of temperatures. But at low temperature the sharp nonlinear decrease of specific resistance occurs and at attainment of some critical temperature the resistance of the metal practically disappears. The metal transits to superconducting state. The superconduction phenomenon was discovered by Dutch physicist Kamerling-Onnesom in 1911. It consist that the resistance of mercury sharply decreased up to zero at about temperature of Kelvin zero. Now the theory of superconductivity can be presented at two methods: the phenomenological approach (Landau, Ginzburg) and the quantum-microscopic theory (Bardeen, Cooper, Schrieffer). The special charge carriers in a substance – Cooper pairs – are assumed in quantummicroscopic theory. The Cooper pair is two bound electrons with oppositely directed spins. *

E-mail address: [email protected]

536

A.N. Volobuev and V.V. Galanin

As total spin is zero the Cooper pair is boson from the point of view of quantum statistics. Cooper pairs unlike electrons, which are fermions, do not submit to the Pauli exclusion principle and can be in a same energy state. If temperature of a conductor is close to zero, all Cooper pairs occupy an identical lowest energy state. When Cooper pairs can not transmit energy of directed movement to ions of a crystal lattice, resistance of a conductor become zero. From the point of view of the band theory an impossibility of transfer of energy from pairs to a crystal lattice indicates that forbidden band Δ existence in a quantum energy spectrum of a crystal lattice. This forbidden band exceeds Cooper pair’s energy at the lowest energy level.

2. Two Kinds of Charge Carries in Metal and Ohm Law for Metal In the common theory of superconductivity the general current density in a conductor is accepted to represent as the sum normal current, associated with Joule heat release, and superconductivity current [1]:

j = jn + js .

(1)

The purpose of this work is to analyze Ohm law on the basis of study kinetics of current carrier in a conductor, which can transit to superconducting state. At the beginning let us assume that conductor’s temperature is not super low, but in it also are generated Cooper pairs. Cooper pairs are generated due to dipole magnetic interaction (attraction) electrons. The formed pairs can be attracting each other due to quadrpole interaction. The formed quadrpole can be attracting each other due to octopole interaction etc. Therefore, general current in a conductor is determined by different current carriers: free electrons are magnetic dipoles, Cooper pairs are magnetic quadrpoles, magnetic octopoles etc. j = ∑ ji , where the summation is conducted for all types current carriers in a conductor. i

Assume the considerable contribution to an electrical current introduces only free electrons and Cooper pairs in a considered temperature range. Note that the strength of Cooper pairs, and furthermore even more complex electronic structures, is small in consequence of powerful Coulomb repulsion of electrons. Therefore, concentration of these pairs is determined by dynamic equilibrium of their continuous decay and formation at that temperature. Thus, the current in a conductor transport both fermions are free electrons and bosons are Cooper pairs. Therefore, we represent the general current density in a conductor as:

j = j1 + j 2 , where the index 1 refer to fermion, and index 2 to boson, Fig. 1. For such conductor Ohm law can be written as:

(2)

Characteristics of Ohm Law for Metal at Low Temperature

j=−

1 ∂ ϕ1 1 ∂ ϕ 2 − . ρ1 ∂ X ρ 2 ∂ X

537

(3)

In equation (3) ρ1 and ρ 2 are specific resistances conditioned, accordingly, electrons and Cooper pairs, X is coordinate along a conductor. The equation (3) should be complemented by balance increments of potential:

Δ ϕ = Δ ϕ1 + Δ ϕ 2 ,

(4)

where Δ ϕ is increment of potential along conductor Δ X , Δ ϕ1 is member of increment potential, producing electronic current, Δ ϕ 2 is member of increment potential, producing Cooper current. Note increment potential has a mark opposite to a potential difference. Assume conductor is homogeneous, round in cross-section a metal wire by the area S. Then, drop of potential along a conductor is linear and equation (3) is can be written as:

j=−

1 Δ ϕ1 1 ∂ ϕ 2 − . ρ1 Δ X ρ 2 ∂ X

(5)

Transform the equation (5) to form:

j ρ 2 +ψ

Δ ϕ1 ∂ϕ =− 2 , ΔX ∂X

(6)

where ψ = ρ 2 ρ1 is relation Cooper and electronic specific resistance. Introduce:

Z = j ρ2 +ψ

Δ ϕ1 . ΔX

(7)

Then, the equation (6) is given as follow

Z =−

∂ ϕ2 . ∂X

Using the method of separation of variables, we find: ΔX

ϕ2 X

0

ϕ20

∫d X = −

∫

dϕ , Z

(8)

where ϕ 20 is Cooper potential on the left end of a site of a conductor, ϕ 2 X is Cooper potential on the right end of a site of a conductor of length Δ X , as shown in figure 1.

538

A.N. Volobuev and V.V. Galanin

Figure 1. Electron and Cooper current density in a conductor.

Pass to variable Z in the equation (8). For this purpose, using equation (7), we shall find differential:

dZ =

ψ ΔX

d (Δ ϕ1 ) .

(9)

Here have assumed, that conductors temperature T does not change. Values ρ 2 and ψ are functions of conductor’s temperature. Really, as a first approximation specific resistance of a conductor is in inverse proportion to numerical concentration of current carriers [2], i.e. quantity of charges in unit of volume of conductor. Hence,

ρ1 ~ 1 N 1 , ρ 2 ~ 1 N 2 , ψ = ρ 2 ρ1 = N1 N 2 , where N1 and N 2 are numerical concentrations of electrons and Cooper pairs. At reduction of temperature of a conductor numerical concentration of free electrons, transporting the charge in a conductor is reduced at the expense of increase of concentration of Cooper pairs. The decay probability of a Cooper pair is proportional exp − W k T , where k is

(

Boltzmann constant, W

)

is binding energy of electrons in a Cooper pair [3]. Hence,

probability of pair production

⎛ W ⎞ N2 ⎟ = 1 − exp ⎜⎜ − ⎟ N1 + N 2 k T ⎝ ⎠ and −1

⎛ ⎛ W ⎞⎞ N ⎟⎟ . ψ (T ) = 1 = ⎜1 − exp ⎜⎜ − ⎟⎟ N 2 ⎜⎝ k T ⎝ ⎠⎠ All free electrons are united in Cooper pairs at zero temperature T → 0 , i.e. N1 → 0 , and

N 2 → N 2 max . Therefore, we can consider:

ψ → 0 at T → 0 .

(10)

Characteristics of Ohm Law for Metal at Low Temperature

539

Using equation (4), we obtain:

dZ = −

ψ ΔX

d (Δ ϕ 2 ) = −

ψ ΔX

d (ϕ 2 − ϕ 20 ) = −

ψ ΔX

d (ϕ 2 ) .

(11)

Founding d ϕ 2 from equation (11) and substituting this differential in the equation (8), we obtain

ΔX =

ΔX

ψ

ZX

∫

Z0

dZ , Z

(12)

where Z 0 and Z X are value of function Z at the boundaries of a conductor of length Δ X , as shown in figure 1. Integrating equation (12), we derive:

Z X = Z 0 expψ .

(13)

Using Eq. (7), we replace to potentials in equation (13):

j ρ 2 +ψ

Δ ϕ1 X

Δ ϕ1 0 ⎞ ⎛ ⎟ expψ , = ⎜⎜ j ρ 2 + ψ ΔX Δ X ⎟⎠ ⎝

(14)

where Δ ϕ10 = ϕ10 and Δ ϕ1 X = ϕ1 X − ϕ10 are increment of electronic potential at the boundaries of a conductor of length Δ X , as shown in figure 1. Taking into account that equation (4), we obtain:

Δ ϕ10 = Δ ϕ − Δ ϕ 2 0 , Δ ϕ1 X = Δ ϕ − Δ ϕ 2 X ,

(15)

where Δ ϕ 2 0 = ϕ 2 0 and Δ ϕ 2 X = ϕ 2 X − ϕ 2 0 are increment of boson potential at the boundaries of a conductor of length Δ X . Substituting equation (15) into (14), we derive general density of a current in a conductor: Δϕ2 X − Δϕ20 ⎞ ψ Δϕ ψ ⎛⎜ ⎟. + Δ − (16) j=− ϕ 2 0 ρ 2 Δ X ρ 2 Δ X ⎜⎝ expψ − 1 ⎟⎠ The formula (16) shows, that at ψ → 0 , i.e. at T → 0 , some density of a current is kept:

js = −

Δϕ2 X − Δϕ20

ρ2 Δ X

.

Resulting density of a current is caused by only Cooper pairs.

(17)

540

A.N. Volobuev and V.V. Galanin We designate summand in equation (16) as:

j3 = −

1 Δϕ 1 Δ ϕ1 1 Δ ϕ 2 ψ Δϕ =− =− − , ρ2 Δ X ρ1 Δ X ρ1 Δ X ρ1 Δ X

j4 =

Δϕ2 X − Δϕ20 ⎞ ⎛ ⎜ Δϕ20 − ⎟. ρ 2 Δ X ⎜⎝ expψ − 1 ⎟⎠

ψ

(18)

(19)

Density of a current j3 is not associated to the superconduction phenomenon. It is determined by transport of the fermions and bosons at enough high temperatures. At that the resistance to carry of Cooper pairs depends only on resistance to transport of the individual electrons of components a pair. The summand j 4 is more interesting. The occurrence of potential difference, casing movement of Cooper pairs, is associated with unification electrons into a pair, therefore we can write: W W Δϕ20 = 0 , Δϕ2 X = X , (20) 2e 2e where W0 and W X are binding energy of electrons in Cooper pair in the beginning and the end of a conductor of length Δ X , 2е is charge of Cooper pair. In spite of the fact that potential increases are at the left in equation (20), instead of their difference, mark before binding energy is positive, since in a denominator a charge of pair is negative. We substitute equation (20) in (19):

j4 =

ψ 2 e ρ2 Δ X

⎛ W − W0 ⎞ ⎜⎜W0 − X ⎟. expψ − 1 ⎟⎠ ⎝

(21)

At a derivation equations (16) and (21), despite of use of a quantum terminology (fermions, bosons etc.), quantum representations about charge transport in a conductor practically were not used. Therefore it is impossible to find value of critical temperature of transition of a conductor in superconductivity state, using spent analysis.

3. Quantum Analysis of Charge Transport at Ultralow Temperatures In order to show, that received density of a current in equation (17) is superconductivity current, it is necessary to use to quantum representations. Let us consider band energy structure of a conductor at the lowest energy levels, i.e. at ultralow temperatures, as shown in figure 2.

Characteristics of Ohm Law for Metal at Low Temperature

541

As it has already noted above the existence of energy gap Δ is characteristic for substance in this range. At temperature of substance T = 0 the energy gap separates decay energy of Cooper pair ε K + ε 0 from bound state Е, so [4]:

ε K + ε 0 − E = Δ = W0 .

(22)

In symbols (20) energy gap Δ = W0 . In equation (22) ε 0 is Fermi energy, ε K is energy of system of two unbound electrons above Fermi level. By distance passage Δ X , the binding energy of Cooper pairs has changed, as shown in figure 2:

ε K + ε X − E = WX ,

(23)

where ε K + ε X is energy of two unbound electrons before formation Cooper pairs on the end of conductor. Substituting equation (22) and (23) into (21), we obtain:

j4 =

ψ 2 e ρ2 Δ X

⎛ ε −ε0 ⎞ ⎜⎜ ε K + ε 0 − E − X ⎟. expψ − 1 ⎟⎠ ⎝

(24)

ε X −ε0 = Δ. expψ − 1

(25)

If current j 4 = 0 then

ε K + ε0 − E =

The similar result is received in [2] and [4] on the base of the solution of Schrodinger equation for Cooper pairs in momentum representation.

Figure 2. Band energy structure of a conductor by Fermi energy.

542

A.N. Volobuev and V.V. Galanin

Figure 3. Process of union electrons in Cooper pairs in a field of wave numbers.

In the process of increase density of Cooper pairs the value ε K → 0 at lowering of a temperature. In [4] there is a successful illustration of the given process, as shown in figure 2. In figure 3 process of union electrons in Cooper pairs in a field of wave numbers is shown. The total electron momentum of pairs is proportional to a wave vector K , joining the centers of spheres. The momentum is connected to wave number a ratio p = = k , where = is reduced Planck constant. Vectors k 0 characterize momentums of two electrons at Fermi levels. Momentums of separate electrons belong to area δ k 0 . Blackened areas of interaction of spheres are proportional to density Cooper pairs N 2 in a conductor. From the figure 3, one can see that, the density Cooper pairs maximize N 2 max at superposition of spheres, i.e. as already noted above, at temperature T → 0 . Its results in quantity of the total momentum K = 0 that is equivalent to a case ε K = 0 . On figure 2 dotted line shows a case when in the process of union of the increasing quantities electrons in Cooper pairs, i.e. at ε K → 0 , the bound state energy E ' lowers below Fermi level ε 0 due to big energy gap:

εK + ε0 − E < Δ .

(26)

In this case the Cooper pair cannot transfer energy to ions of a crystal lattice at the motion and there is a superconducting state. Note that equation (17) made possible a little in other respects to interpret incipient of a superconducting state. Possibly, that in some cases the superconducting state is associated with the appearance of interior electromotive force (EMF), supporting a current, and not to disappearance of resistance to a current of Cooper pairs at T → 0 . From equation (24), at ψ → 0 , it is possible to estimate the quantity of that electromotive force:

js = − Hence:

ε X −ε0 . 2e ρ2 Δ X

(27)

Characteristics of Ohm Law for Metal at Low Temperature

EMF = j s ρ 2 Δ X = −

ε X −ε0 2e

≈ 0.1 V .

543

(28)

To calculate we was used [4] ε X − ε 0 ≈ 0.2 eV . Possibly, that last representation about superconductivity is true for so-called hightemperature superconductors to which the formal classical part of the developed analysis completely is applied. Let us construct dependence of a part of Cooper pairs in the general number of free charge carriers in metal on his temperature by the example of lead Pb as:

β=

⎛ W ⎞ N2 ⎟⎟ . = 1 − exp ⎜⎜ − N1 + N 2 ⎝ kT ⎠

(29)

The energy gap width Δ can be found from the ratio at temperature near absolute zero:

Δ =ξ , k Tk

(30)

where value ξ = 2.15 for lead [2]. For critical temperature of transition of lead in a superconducting state Tk = 7.19 K we shall find value of a energy gap Δ = 21.3 ⋅ 10 −23 J . As it has already noted above superconductivity will appear in case binding energy of Cooper pairs less energy gap width. In this case, taking with the account equation (26), is true energy ratio: W = ε K + ε0 − E ≤ Δ . (31) Setting, that the energy gap width of a crystal lattice of metal weakly depends on temperature, according to equation (29), we can write:

Δ ⎞ ⎟⎟ . ⎝ kT ⎠ ⎛

β = 1 − exp ⎜⎜ −

(32)

In figure 4 dependence of the quantity β in percentage of temperature for lead is shown. As follows from figure 4, at room temperature ( T = 300 K ) Cooper pairs make a little bit more than 5 % of all charge carriers. At temperature of transition of lead in a superconducting state Tk = 7.19 K their part increases to 88 %. The deviation from 100 % is connected to limitation of application of classical representations transition of charges in metals at low temperatures.

544

A.N. Volobuev and V.V. Galanin

Figure 4. Part of Cooper pairs in the general number of free charge carriers in metal as a function of temperature by lead Pb.

4. Association of Researched Model with Ginzburg-Landau Theories of Superconductivity Let's find association of parameters of considered model and parameters of Ginzburg – Landau theory [1]. Setting, that a Cooper pairs current as:

j2 = js =

1 ∂ϕ2 e h ψ = ρ2 ∂ X m

2

∂Φ , ∂X

(33)

where е is electron charge, m is its weight, ψ

2

so ψ = ψ exp (i Φ ) , Φ is its phase, and ψ

= N 2 . Taking into account, that in Cooper pair

2

is a square of the module of order parameter,

are two electrons. Using equation (33), we can write:

Δϕ2 =

e h ρ2 2 e h ρ2 ψ ΔΦ = N2 Δ Φ = θ h Δ Φ , m m

(34)

e ρ2 N2 . m Taking into account, that changes of a phase Δ Φ = 2 π n , where n is an integer [1, 2], we shall obtain:

where it is called θ =

Δϕ 2 = 2π hθ n .

(35)

Thus, a part of an increment of the potential, causing Cooper current, is quantized. The minimal value of its increment of potentials is Δ ϕ 2 min = 2 π h θ .

Characteristics of Ohm Law for Metal at Low Temperature

545

Let's calculate the minimal value Δ ϕ 2 . According to Drude-Lorentz theory [2] complex

(

)

ρ 2 N 2 = m 2 e 2 τ . In this case, quantity τ can be treated as lifetime of Cooper pairs at the given temperature. According to Frenkel theory [5] for a Bose condensate of Cooper pairs we can be written:

⎛W ⎞ ⎟⎟ , ⎝ kT ⎠

τ = τ 0 exp ⎜⎜

(36)

where τ 0 = 1 ν 0 is the oscillation period of electron in pair, ν 0 is frequency of his oscillations. Hence:

ρ2 N2 =

m 2e τ 2

=

mν 0

⎛ W ⎞ ⎟⎟ . exp ⎜⎜ − 2e ⎝ kT ⎠ 2

(37)

Substituting the given complex in the formula for θ , we receive:

θ=

ν0

⎛ W ⎞ ⎟⎟ . exp ⎜⎜ − 2e ⎝ kT ⎠

(38)

Thus,

Δ ϕ 2 min = π

⎛ W ⎞ hν 0 ⎟⎟ . exp ⎜⎜ − e ⎝ kT ⎠

(39)

The value of quantum of the potential, that make a Cooper current, tent to zero at decrease of temperature of substance. Setting, as before, energy of oscillating electron hν 0 ≈ 0.2 eV , we shall find for lead at T = 7.19 K the quantity Δ ϕ 2 min ≈ 0.07 V . Under order of magnitude the last value corresponds to value (28) which has been calculated above, that specifies validity of estimations.

5. Conclusions Studding analysis of kinetics of charges carry in a conductor which can pass in a superconducting state, we showed that the fundamental laws of transport deduce from classical representations in the assumption, that electrons can be united in Cooper pairs which concentration is determined by dynamic balance of their formation and decay. At usual temperature substance of a conductor puts up resistance to a current of Cooper pairs. The quantum analysis of transport process enables to identify a current in equation (17) which is conserved in a conductor at ultralow temperature, with a superconducting current. The formula (17) and following of it equation (28), probably enables to interpret formation of a superconducting state as formation internal electromotive force, supporting a current, at T → 0 .

546

A.N. Volobuev and V.V. Galanin

The potential difference causing Cooper current, is quantized, and the value of quantum of the potential tent to zero at decrease of temperature of substance.

References Lifshitz E M and Pitaevskii L P 1980 Statistical Physics. (Landau and Lifshitz Course of Theoretical Physics) 9 Pergamon. Press Ashcroft N W and Mermin N D 1976 Solid State Physics Brooks Cole. Feynman R, Leighton R and Sands M 2005 Feynman Lecture on Physics. 2 AddissonWesley Levich V G, Vdovin Yu A and Myamlin V A 1969 Course of Theoretical Physics. Moscow Fizmatizdat (in Russian) Yavorsky B M and Detlaf A A 1974 Handbook of Physics Moscow Science. (in Russian)

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 547-557

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 11

CALCULATING AND EXPERIMENTAL RESEARCHES OF FREE-FLOWING SUBSTANCE AXISYMMETRIC MOVEMENT AS QUASI-NEWTON LIQUID V.V. Lozovetsky1, F.V. Pelevin2 and S.N. Leontiev1 1

Moscow State University of Forest, 1ya Institutskaya Str., Mytishchi, Moscow Region, 141005, Russia 2 N. É. Bauman Moscow State Technical University, Moscow, Russia

Abstract The chapter contains brief analysis of existing calculating and theoretical models describing free-flowing (granular) substance movement. There is given a survey of advantages and disadvantages of the analyzed models. In the paper, it is noted that the model suggested by V.N. Krymasov provides satisfactory results, which were proved by experimental data. Based on the model there are suggested the analytical expressions for the calculation of free-flowing substance movement and is justified the choice of numerical calculating scheme. Based on the data of the experiments the authors suggest the differential equation set. The authors justify the choice of boundary conditions describing two variants: one that takes into account the slipping on the boundary and the other – the so-called universal boundary conditions. The calculations made using the boundary conditions mentioned above are compared with the test results; the coincidence is satisfactory. The authors conclude that the suggested method may be used successfully for the calculation of hoppers of axisymmetric form where there is freeflowing substance movement.

At present there are very few works in which existing theoretical models of free-flowing (granular) substance movement (chart 1) are accomplished with applicable calculation results. Evidently, it complicates the estimation of these models’ adequacy for the calculation of freeflowing substance movement, which can be carried out by comparing the results of the calculations with the test data.

Chart 1. Free-flowing substance and spherical elements movement modeling # 1

Brief description of the flow model 2

Mathematical interpretation 3

ρ+

1

Model of D. Druker and W. Prager. The free-flowing substance is an incompressible homogeneous material [8].

Commentary 4

dW+ = − F + divσ μν ; divW+ = 0; dt

(σ x − σ y )2 + 4σ xy2 = (σ x + σ y + 2kctgδ + )2 sin2 δ + ; −1

⎛ dWx ⎞ ⎛ dWx dW y ⎞ ⎟ ; ⎟⎟ ⋅ ⎜⎜ + tgδ + = ⎜⎜ dx ⎟⎠ ⎝ dy ⎠ ⎝ dy

The closing equation is the constraint of Coulomb-Mohr.

δ+ − angle of internal friction; ρ+ – density of free-flowing material; σ − tension tensor; W+ − vector of absolute velocity of free-flowing substance; F − vector of mass forces; t − time; k − proportionality coefficient; x, y − coordinate axes.

2

Model of W. Jenike, T. Shield and Radenkovich. Homogeneous incompressible free-flowing substance [2].

dW+ = − F + divσ μν ; divW+ = 0; dt (σ x − σ y )2 + 4σ xy2 = (σ x + σ y + 2k ⋅ ctgψ )2 sin 2 ψ ;

ρ+

ψ=

3

Model of R. Hill, A.Y. Ishlinsky. Homogeneous incompressible freeflowing substance [6].

π

+

δ+

. 4 2 dW+ ρ+ = − F + divσ μν ; divW+ = 0; dt

(σ x − σ y )2 + 4σ xy2 = (σ x + σ y + 2k ctgδ + )2 sin 2 δ + ; 2σ xy (σ x − σ y )

−1

−1

⎛ dWx dWy ⎞ ⎛ dWx ⎞ ⎟⋅⎜2 = ⎜⎜ + ⎟ . dx ⎟⎠ ⎝ dx ⎠ ⎝ dy

The closing equation is the constraint of Coulomb-Mohr.

The closing equation is the constraint of Coulomb-Mohr. There are the calculations of the field of velocities under uniform loading.

Chart 1. Continued # 1

4

Brief description of the flow model 2 Model of G.A. Geniev and Josseling de Iong. Homogeneous incompressible free-flowing substance [4; 5].

5

Model of M.A. Goodman and S.L. Cowin. The free-flowing substance is binary viscous substance [7].

6

Model of D. Bedenig. Single-phase incompressible viscous free-flowing substance [9].

7

Model of D. Bedenig. Single-phase incompressible viscous free-flowing substance [9].

Mathematical interpretation 3

ρ+

dW+ = − F + divσ μν ; divW+ = 0; dt

2σ xy

σ x −σ

ρ+

y

1 ⎛ dW y dW x ⎞ dW x ⎜ ⎟± + tg δ + 2 ⎜⎝ dx dy ⎟⎠ dx . = dW x 1 ⎛ dW y dW x ⎞ ⎟⎟ tg δ + ± ⎜⎜ + 2 ⎝ dx dx dy ⎠

dW+ = − F + divσ μν ; divρ +W+ = 0; ρ + = ρ + (1 − ε ) dt

are given as reologic ratios k = k (ρ+); δ+ = δ (ε); μ+ = μ+ (ε). Limiting balance condition: max(τ - σ⋅tgδ∗) = k, ε − porosity of free-flowing substance.

ρ+

dW+ = − F − ∇p + divσ μν ; σ μν = μ + η μν ; divW+ = 0, dt

where p – gas pressure, μ+ = μ (tgδ+).

ρ+

dW+ = − F − ∇p + divσ μν ; σ μν = μ + η μν ; divW+ = 0, dt

where p – gas pressure, μ+ = μ (tgδ+).

dW+ = − F − ε∇p + divσ μν ; σ μν = μ + η μν ; dt divρ +W+ = 0; ρ + = ρ (1 − ε );

Commentary 4

There is given the solution of the problem of free-flowing substance rod movement.

k, δ+, μ+ – experimental characteristics.

There is given an engineering procedure for the calculation of the field of velocities for spherical filling in a hopper. The problem of shifting is solved.

ρ+

8

Model of V.N. Krymasov. Singlephase compressible quasi-newton liquid [1].

1 dρ + (cos γ e1 + sin γ e2 ); + divρ +W+ = 0; − F − ρ + g dt cos δ + γ = θ + δ+, (p – gas pressure; e1, e2 – orts of the natural coordinate system).

There is found the solution of plastic flow.

550

V.V. Lozovetsky, F.V. Pelevin and S.N. Leontiev

In some cases, experimental verification of the theoretical models may bring positive results. In particular, from the point of calculation the model suggested by V.N. Krymasov [1], which reflects the processes of spherical filling movement with physical adequacy, is of high interest. It is confirmed with the results of calculations and experiments, which are displayed on figure 1 as limiting sliding lines for the two values of the coefficient of internal friction f + = 0.268 and f + = 0.364.

Figure 1. The boundary between movable and immovable zones in free-flowing substance (the spherical filling). Here Ο, Δ − experimental data: d+ = 10.2 mm, f+ = 0.364; calculation based on model [1]: f+ = 0.268 ( ), f+ = 0.364 ( ).

From the comparison it follows that the calculated sliding line for f + = 0.364 coincides with the results of our experiments, which were obtained using the model with flat bottom and polished steel spheres of 10.2 mm in diameter. Experimental researches [10; 11] carried out in axisymmetric hoppers with conic bottom testify that the flow of filling consisting of spherical elements when the ratio of the diameters of the hopper and the spheres is equal or more than 40 is similar to the flow of the continuous substance in laminar mode. In consequence of this fact the spherical filling may be considered as a certain quasi-newton liquid, for describing the flow of which there can be used the following equations set [12]:

ρ+

dW __ = F − ε∇p + divσ ; dt

ρ + = (1 − ε )ρ ;

dρ + + divρ +W = 0; dt

(1)

Calculating and Experimental Researches of Free-Flowing Substance…

ρ + F − ε∇p = ρ +

551

g (cos γ e1 − sin γ e2 ); γ = θ + δ + . cos δ +

Here θ – angle between direction of a spherical element’s absolute speed W and free fall acceleration g. Supposing that the value of pressure does not depend on the direction in the flow of G spherical elements and there is linear relation between the components of tension tensor σ and deformation speed η, where apparent viscosity coefficient μ+ is used as proportionality coefficient, we get:

σ = μ кη .

(2)

In this case, the equation of flow in the system (1) may be brought to a form similar to Navier-Stokes equation with viscosity corresponding to apparent viscosity of spherical filling. After several simple transformations, we may get [12]:

ρ+

g (cos γ e1 − sin γ e2 ) = −∇П. cos δ +

Hence, the equation of movement of free-flowing substance in a hopper may be represented in the following form:

ρ+

dW = −∇П + divσ . dt

In view of expression (2), supposing μ+ = const, we may get the following dependence:

divσ = μ + div∇W = μ + ∇ 2W . Than the spherical filling flow equation takes the following form:

ρ+

dW = −∇П + μ + ∇ 2W . dt

(3)

After applying rot operator to both parts of the equation (3) we transform it for the case of settled axisymmetric flow of spherical filling by allowing new variables of flow function ψ and vortex function ω (in cylindrical coordinates z, r):

⎡ ∂ ⎛ ω ∂ψ ⎜ ⎣ ∂z ⎝ r ∂r

ρ+r 2 ⎢

⎞ ∂ ⎛ ω ∂ψ ⎞⎤ ∂ ⎡ 3 ∂ ⎛ μ +ω ⎞⎤ ∂ ⎡ 3 ∂ ⎛ μ + ω ⎞⎤ ⎟⎥ . ⎜ ⎟⎥ + ⎢r ⎜ ⎟− ⎜ ⎟⎥ = ⎢ r ⎠ ∂r ⎝ r ∂z ⎠⎦ ∂z ⎣ ∂z ⎝ r ⎠⎦ ∂r ⎣ ∂r ⎝ r ⎠⎦

(4)

552

V.V. Lozovetsky, F.V. Pelevin and S.N. Leontiev

In order to find the unknown flow function and vortex function there may be used the following dependence as the second equation:

1 ⎛ ∂ 2ψ ∂ 2ψ 1 ∂ψ + − r ⎝ ∂r 2 ∂z 2 r ∂r

ω = − ⎜⎜

⎞ ⎟⎟. ⎠

(5)

The equations (4) and (5) are of the second order and in view of boundary conditions’ complexity, which will be mentioned further, to find the solution there was used an iterative algorithm based on successive integration of the two coherent equations of the second order for vortex function and flow function. The so-called counter-flow oriented model, which stabilizing influence on the calculation by finite differences method is known [13], was chosen as a numerical model. Unlike a viscous liquid, for which the velocity on the firm wall bounding the flow is equal to zero, for the spherical filling this condition is not suitable. Depending on the roughness of the wall’s surface and spherical elements, which is taken into account through the coefficient of external friction f_ the speed of the spherical elements’ slipping on vertical and inclined walls of a conic hopper may vary from zero up to some final value. It is necessary to note that as it was proved by our researches the presence of the spherical elements’ slipping on a wall does not lead to any alteration in the form of the flow equation, which can be used for this specific case of substance flow as well. Let Wz = Wzo ± WN,

(6)

where Wzo is current value of speed at the slipping speed WN = 0. Than the equations, describing the flow projected on axis z will take the following form:

ρ + (W zo ± W N ) divW z =

∂ (W zo ± W N ) ∂ (W zo ± W N ) ∂p ∂τ zz 1 ∂ + ρ +Wr = −ρ+ z + + rτ rz + ∂z ∂r ∂z ∂z r ∂r ∂ (Wzo ± WN ) 1 ∂ ∂W z 1 ∂ + rWr = + rWr = divWzo = 0 ∂r r ∂r ∂z r ∂r ⎛ ∂W ∂Wzo ⎞ ⎡ ∂Wr ∂ (Wzo ± WN ) ⎤ = μ+ ⎜ r + + ⎟ = τ rzo ⎥ ∂r ⎠ ∂r ⎝ ∂z ⎣ ∂z ⎦

τ rz = μ + ⎢ 2 3

τ zz = divWz + 2μ +

(7)

∂ (Wzo ± WN ) 2 ∂Wzo = divWzo + = τ zzo . ∂z ∂z 3

Hence, we get:

ρ +Wzo

∂Wzo ∂Wzo ∂Wzo ∂p ∂τ zzo 1 ∂ + ρ +Wr + ρ +WN = −ρ+ z + + rτ rz . + ∂z ∂r ∂z ∂z r ∂r ∂z

(8)

Calculating and Experimental Researches of Free-Flowing Substance…

553

Let us set:

−

∂p ∂Wzo ∂p − ρ кWN =− ∂z ∂z ∂z o

(9)

Than

ρ +Wzo

∂Wzo ∂Wzo ∂τ ∂p 1 ∂ + ρ +Wr = −ρ+ z − + zzo + zτ rzo , ∂z ∂r ∂z o ∂z r ∂r

(10)

QED When solving the equations set (4) and (5) the value of vortex function on the immovable boundary (vertical and inclined walls) was determined in two forms. In the first case there was obtained the following expression for the vortex function, which takes into account the slipping speed:

3(ψ ij −ψ N ) 1 rij 3WN+ ωN = − ωij − , − 2 rN rN Δn 2 Δn +

where WN – speed of slipping on a wall attributed to average flow rate; ψN, ωN – values of correspondingly flow function and vortex function at point N on hopper’s wall; ψij, ωij – values of correspondingly flow function and vortex function in node ij in the flow of spherical filling; Δn – distance between hopper’s axis and point N on its wall (figure 2).

Figure 2. Determination of vortex function on an inclined wall.

When vortex function is determined in this way, the spheres’ slipping speed is determined through empirical dependences that were obtained through experimental data processing, one of which has the following form:

554

V.V. Lozovetsky, F.V. Pelevin and S.N. Leontiev

WN+ = WN+

⎛ d+ ⎞ ⎜ D⎟ ⎠ ⎝

1− z +

(d D )

0 , 02 (1− z + )

,

r+3

z + =1

+

where WN | z+ =1 − relative slipping speed of spherical elements on a wall in the section, which corresponds to the free surface of spherical filling (at relative altitude z + = 1 ); D, d+, d – correspondingly diameters of the hopper’s cylindrical part, unloading port and spherical element, r+ – relative radius. Experimental researches which were carried out on a hopper with one unloading port and the main diameters ratio D/d+ = 4.0; 5.0; 6.66; D/d ≈ 40; 55, inclination angle of the conic part α = 30°; 45°; 60° at the flow of steel spheres 7.2⋅10-3 m и 10.2⋅10-3 m in diameter, proved +

that there is linear dependence between WN | z+ =1 and coefficient of external friction f− in sufficiently wide range of its change, which is defined by the expression

WN+ | z+ =1 = 1 – 1,335 f−. As the wall is impenetrable, the flow function ψN = 1, which corresponds to the total flowrate through the considered area.

Figure 3. Spherical elements’ speed profiles in section on height z = D from the unloading port: o − f− = 0.218, f+ = 0.36; □ − f− = 0.268, f+ = 0.68; ∇ − f− = 0.3, f+ = 0.36; • − f− = 0.36, f+ = 0.68.

At the input boundary there was set the speed allocation (figure 3) corresponding to physical and mechanical properties of the spherical filling, which may be described by the following empirical dependence:

W (r+ ) z

+ =1

+ = A( f + )⎛⎜Wmax − WN+ ⎝

z + =1

(

)

⎞⎟ ⋅ 1 − r 2 + W + N + ⎠

z + =1

,

Calculating and Experimental Researches of Free-Flowing Substance… +

555 kr

where Wmax – relative maximal speed (related to average flow rate speed); A(f+) = f + + ; k =

2r − current value of relative radius. D At the output boundary it was supposed that ∂ψ ∂r = ∂ω ∂r = 0. Coefficient of

0.1 – empirical coefficient; r+ =

apparent viscosity of spherical filling and coefficients of internal and external friction used for numerical calculations were determined according to the recommendations [12]. The calculation results made with the specified boundary conditions are compared (figure 4) with the results of the experiments received in a hopper with one central unloading port (here ○ – experiment, ● – calculation of potential flow model, – calculation with taking into account apparent viscosity, z+ = z/D – relative height). Calculated and experimental velocity and flow line profiles coincide satisfactorily with each other in the most part of the hopper. At approaching the unloading port ( z + ≤ 0,05 ) the experimental data may differ significantly from the calculation results as the step-type behaviour of the spherical filling at its gravitational unloading is revealed in this area. The boundary conditions for the vortex function on vertical and inclined walls used in these calculations have particular nature since they are true in rather a narrow alteration range of geometrical, physical and mechanical parameters.

Figure 4. Comparison of experimental and calculating data.

556

V.V. Lozovetsky, F.V. Pelevin and S.N. Leontiev

A more universal boundary condition was obtained from the hypothesis [8] according to which for the spherical filing, as a free-flowing substance a generalized rule, which is defined by the differential equation set for the case of axisymmetric flow, is true:

∂W z ⎞ ⎛ ∂W r ⎛ ∂W z ⎞ − ⎜ ⎟ sin 2 γ − ⎜ ⎟ cos 2 γ = 0 , ∂z ⎠ ⎝ ∂r ⎝ ∂z ⎠

∂W r (sin δ + − cos γ ) − ∂ W z (sin δ + + cos 2γ ) = 0 , ∂r ∂z

(11)

where Wr, Wz – speed components; γ – inclination angle of algebraically maximum of main tension to axis r, which according to [8]

γ=

π

4

+

δ

2

.

After several transformations [3] there may be got the expression connecting the angle of external friction δ− with derivatives of speed projections on the hopper’s walls:

tg δ _ =

2W z ∂ z ∂W r / ∂z

N

N

+ ∂W z / ∂r

, N

from which the dependence for vortex function on the hopper’s walls follows:

ωN =

⎛ 1 ∂ 2ψ 2 / r ∂ 2ψ / ∂ r ∂ z 1 ∂ψ ⎞ . ⎟ − 2 ⎜⎜ − 2 2 tg δ _ r ∂ r ⎟⎠ ⎝ r ∂r

Figure 5. Alteration of spherical elements’ speed profile depending on height of the model with dimension ratios D/d = 40; D/d+ = 4; d = 10.2 mm; f+ = 0,36; f− = 0.3: ○ – experiment; ⎯ − calculation using universal boundary conditions.

Calculating and Experimental Researches of Free-Flowing Substance…

557

The other boundary conditions are similar to the considered above. The calculation results made with the universal boundary condition for the vortex function are compared (figure 5) with experimental data, received in a hopper with central unloading port (here ○ – experiment; ⎯ – calculation using universal boundary conditions). The coincidence of the calculated and experimental data is satisfactory. The suggested calculation method may be used successfully when designing hopper equipments of axialsymmetric form with one unloading port where there is free-flowing substance movement.

References [1] Крымасов, В.Н. Сыпучая среда как модель неньютоновской жидкости. Вопросы атомной науки и техники. Серия «Атомно-водородная энергетика и технология». Москва. 1980, с. 138-141. [2] Дженике, Э.В. и др. Нагрузка на бункеры. Часть 2. Основные понятия. Труды Американского общества инженеров-механиков. Конструирование и технология машиностроения. Мир, Москва, 1973; № 2, c. 254-258. [3] Николаевский, В.Н. Определяющие уравнения пластического деформирования сыпучих сред. ПММ. 1971. т. 35, № 6, с. 411-420. [4] Гениев, Г.А. Вопросы динамики сыпучей среды. Госстройиздат. Москва. 1958, с. 178. [5] Josseling de Iong, G. The daible sliding free rotating model for granular assemblies. Geotechnique. 1971. Vol. 21, pp 155-163. [6] Hill, R. The mathematical theory of plasticity clareden. Oxford. 1956. p.347. [7] Goodman, M.A.; Cowin, S.L. A continuum theory for granular materials. Arch., Rat. Mech. Anal. 1972. Vol. 44. pp 249-260. [8] Druker, D.; Prager, W. Soil mechanics and plastic analysis or limit design. Quart. Appl., Math. 1952. Vol. 10. pp 157-165. [9] Bedenig, D. Theoretisches Model zur Beschreibung des Kugelhausen Fliebverhaltens im Core eines Kugelhaufenreaktors. Nucl. Enging. and Design. 1967. № 6. pp 479-488. [10] Лозовецкий, В.В.; Крымасов, В.Н. Гидромеханические и тепловые процессы в ядерных реакторах с микротвэльным топливом. ВИНИТИ РАН. Москва. 2003. с. 326. [11] Bedenig, D. Untersuchungen zum Stromungsverhalten eines Kugehaufebs im Core eines Kugel haufenreaktors. Dissertation Technische Hochschule. Wien. EUROATOM. 1966. № 3284d. s. 171. [12] Крымасов, В.Н.; Лозовецкий, В.В.; Мордвинцев, В.М. Расчет движения шаровых твэлов в активной зоне ВТГР. Вопросы атомной науки и техники. Сер. атом.водородн. энергет. и технол. М.1990. Вып. 2. c. 44-46. [13] Jilly, D.K. On the computational Stability of numerical Solution of time-dependent nonlinear geophysical fluid dynamics problems. U.S. Weather Bureau Monthly Weather Review. 1965. pp 93.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 559-575

ISBN: 978-1-60876-227-9 © 2010 Nova Science Publishers, Inc.

Chapter 12

VIBRATION ANALYSIS OF NON-UNIFORM BEAMS USING SPLINE COLLOCATION METHOD Ming-Hung Hsu* Department of Electrical Engineering, National Penghu University, Penghu, Taiwan

ABSTRACT The natural frequencies of non-uniform beams are numerically obtained using the spline collocation method. The spline collocation method is an effective numerical approach for solving partial differential equations. The boundary conditions accompanied the spline collocation procedure to convert the partial differential equations of nonuniform beam vibration problems into a discrete eigenvalue problem. The beam model considers the taper ratio, α , β , inertia, boundary conditions, and other factors, all of which affect the dynamic behavior of non-uniform beams.

Keywords: vibration analysis, non-uniform beam, spline collocation method.

INTRODUCTION The dynamic characteristics of non-uniform beams are of considerable importance in many designs. Abrate et al. [1—3] solved vibrations in non-uniform rods and beams using the Rayleigh-Ritz method. Hodges et al. [4] computed the fundamental frequencies and the corresponding modal shapes from a discrete transfer matrix method. Lee and Kuo [5] solved the problem of bending vibrations in non-uniform beams with an elastically restrained root. In this study, the spline collocation method is employed to formulate the discrete eigenvalue problems of different non-uniform beams. The spline collocation approach is easy to implement and should be of interest to designers. Simulated results are compared with *

E-mail:[email protected]

Ming-Hung Hsu

560

numerical results obtained using the finite element method and with other relevant results in the literature.

FORMATION Figure

(

1

shows

the

A = A0 1 + x / L + x / L 2

2

)

sketch

of

(

a

clamped-free

and I = I 0 1 + x / L + x / L 2

2

).

non-uniform

beam

with

The corresponding kinetic

energy of a beam with non-uniform cross-section is

T=

ρA 2

⎛ ∂w ( x, t ) ⎞ ⎜ ⎟ dx ⎝ ∂t ⎠ 2

∫

L

0

(1)

where L is the length of the non-uniform beam, w is the transverse bending deflection,

A = A0φ ( x ) , A is the cross-sectional area of the beam, and ρ is the density of beam

material. Function

φ ( x ) is dependent upon the shape of the cross-section. Parameter Ao is

the area of the cross section at the end x = 0 . The strain energy of non-uniform beam is 2 1 L ⎛ ∂ w ( x, t ) ⎞ U = ∫ EI ⎜ ⎟ dx 2 2 0 ⎝ ∂x ⎠ 2

(2)

where I is the inertia of the non-uniform beam, I = I 0ψ ( x ) , and E is Young’s modulus of non-uniform beam material. Function

ψ ( x ) is dependent upon the shape of the cross-

section. Parameter I o is the moment of inertia at the end x = 0 . Hamilton’s principle is given by

∫ (δ T − δ U + δ W ) dt = 0 t2

t1

(3)

where δ W is the virtual work. Substituting Eqs. (1) and (2) into Eq. (3) yields the equations of motion. The transverse motion w of a non-uniform beam is governed by

∂2 ⎛ ∂2w ⎞ ∂2w EI ρ A + =0 ⎜ ⎟ ∂x 2 ⎝ ∂x 2 ⎠ ∂t 2

(4)

Consider the clamped-free beam to be clamped at the end x = 0 and the corresponding boundary conditions are

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

w ( 0, t ) = 0

∂w ( 0, t ) ∂x

EI ( L )

(5)

=0

∂ 2 w ( L, t ) ∂x 2

561

(6)

=0

∂ 2 w ( L, t ) ⎞ ∂ ⎛ EI L ( ) ⎜ ⎟=0 ∂x ⎜⎝ ∂x 2 ⎟⎠

(7)

(8)

Consider the following pinned-pinned non-uniform beam and the corresponding boundary conditions are

w ( 0, t ) = 0

∂ 2 w ( 0, t ) ∂x 2

=0

w ( L, t ) = 0

∂ 2 w ( L, t ) ∂x 2

=0

(9)

(10)

(11)

(12)

Consider the following clamped-clamped non-uniform beam and the corresponding boundary conditions are

w ( 0, t ) = 0

∂w ( 0, t ) ∂x

=0

w ( L, t ) = 0

∂w ( L, t ) ∂x

=0

Let the displacement response be

(13)

(14)

(15)

(16)

Ming-Hung Hsu

562

w ( x, t ) = W ( x ) exp ( iω t ) , where yields

(17)

ω is the natural frequency of non-uniform beam. Substituting Eq. (17) into Eq. (4)

d 2 EI ( x ) d 2W ( x ) dx 2

dx 2

+2

dEI ( x ) d 3W ( x ) dx 3

dx

+ EI ( x )

d 4W ( x ) dx 4

= ω 2 ρ A ( x )W ( x )

(18)

Equation (18) can be rewritten as EI 0

d 2ψ ( x ) d 2W ( x ) dx

2

dx

2

+ 2 EI 0

dψ ( x ) d 3W ( x ) dx

dx

3

+ EI 0ψ ( x )

d 4W ( x ) dx

4

= ω 2 ρ A0φ ( x )W ( x ) (19)

The corresponding boundary conditions of the clamped-free non-uniform beam are

W ( 0) = 0

dW ( 0 ) dx EI ( L )

(20)

=0

d 2W ( L ) dx 2

(21)

=0

d 2W ( L ) ⎞ d ⎛ ⎜⎜ EI ( L ) ⎟=0 dx ⎝ dx 2 ⎟⎠

(22)

(23)

The corresponding boundary conditions of the pinned-pinned non-uniform beam are

W ( 0) = 0

d 2W ( 0 ) dx 2

=0

W ( L) = 0

d 2W ( L ) dx 2

=0

(24)

(25)

(26)

(27)

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

563

The corresponding boundary conditions of the clamped-clamped non-uniform beam are

W ( 0) = 0

dW ( 0 ) dx

(28)

=0

(29)

W ( L) = 0

dW ( L ) dx

(30)

=0

(31)

MODELING OF SPLINE COLLOCATION The solutions to numerous complex non-uniform beam problems have been efficiently obtained using fast computers and the range of available numerical methods, including the Galerkin method, the finite element technique, the differential quadrature approach, the differential transform, the boundary element method, and the Rayleigh-Ritz method [6-9]. In this study, the spline collocation method is employed to formulate discrete eigenvalue problems of various non-unifrom beams. Prenter et al. [10-12] investigated spline and variation methods. Bert and Sheu [13] presented a static analysis of beams and plates using the spline collocation method. El-Hawary et al. [14] discussed quartic spline collocation methods for solving linear elliptic partial differential equations. Archer [15] investigated odddegree splines using high-order collocation residual expansions and adopted nodal collocation methods to solve the problem with one-dimensional boundary values. Patlashenko and Weller [16] applied the spline collocation approach to solve two-dimensional problems, and determine the post bucking behavior of laminated panels subjected to mechanical and heating induced loadings. In this work, the knots xk ,i are considered as follows.

xk ,i = ak + ihk for i = −2, −1, 0,..., N + 1, N + 2 where

xk ,0 , xk ,1 , xk ,2 ,..., xk , N −1 , xk , N

are

the

(32) abscissas

of

the

knots

and

xk ,−2 , xk ,−1 , xk , N +1 , xk , N + 2 are the abscissas of the extended fictitious knots. hk =

bk − ak N

(33)

where the distance hk between two adjacent knots remains constant. The spline function is given as follows [10-13].

Ming-Hung Hsu

564

Bk ,i ( xk ) = ⎧ (xk − xk ,i−3 )5 xk ∈ ⎡⎣ xk ,i−3, xk ,i−2 ⎤⎦ ⎪ hk 5 ⎪ ⎪ (xk − xk.i−3 )5 − 6(xk − xk ,i−2 )5 ⎪ xk ∈ ⎡⎣ xk ,i−2 , xk ,i−1 ⎤⎦ hk 5 ⎪ ⎪ (xk − xk ,i−3 )5 − 6(xk − xk ,i−2 )5 +15(xk − xk ,i−1)5 ⎪ xk ∈ ⎡⎣ xk ,i−1, xk ,i ⎤⎦ ⎪ hk 5 ⎪ (xk − xk ,i−3 )5 − 6( xk − xk ,i−2 )5 +15(xk − xk ,i−1)5 − 20( xk − xk ,i )5 ⎨ xk ∈ ⎡⎣ xk ,i , xk ,i+1 ⎤⎦ ⎪ hk 5 ⎪ ⎪ ( xk − xk ,i−3 )5 − 6(xk − xk,i−2 )5 +15( xk − xk,i−1)5 − 20( xk − xk,i )5 +15( xk − xk,i+1)5 ⎪ xk ∈ ⎡⎣ xk ,i+1, xk ,i+2 ⎤⎦ hk5 ⎪ ⎪ 5 5 5 5 5 5 ⎪( xk − xk ,i−3 ) − 6( xk − xk ,i−2 ) +15(xk − xk ,i−1) − 20(xk − xk,i ) +15(xk − xk,i+1) − 6(xk − xk,i+2 ) x ∈ ⎡ x , x ⎤ k ⎣ k,i+2 k,i+3 ⎦ ⎪ hk 5 ⎪ 0 otherwise ⎩

(34) where

k

is

element

number,

and

Bk , −2 ( xk ) ,

Bk , −1 ( xk ) ,

Bk ,0 ( xk ) ,…,

Bk , N +1 ( xk ) , Bk , N + 2 ( xk ) form a basis for the function defined over the region ak ≤ xk ≤ bk . The deflection of the k

th

beam element at the knots is given by the following equation.

N +2

Wk ( xk ) = ∑ ak ,i Bk ,i ( xk ) for k = 1, 2, ..., M

(35)

i =−2

where M is the total number of elements, ak ,i is a coefficient to be determined, and

Bk ,i ( xk ) is the spline function. The domain contains N + 5 collocation points. The equations of motion of a non-uniform beam can be rearranged into the spline collocation method formula, yielding,

⎡ d 2 EI ( xk ,i ) d 2 Bk ,−2 ( xk ,i ) ⎢ 2 dxk 2 ⎢⎣ dxk

d 2 EI ( xk ,i ) d 2 Bk , −1 ( xk ,i ) dxk 2

d 2 EI ( xk ,i ) d 2 Bk , N + 2 ( xk ,i ) ⎤ ⎥ ⎡⎣ ak ,−2 dxk 2 dxk 2 ⎥⎦

⎡ dEI ( xk ,i ) d 3 Bk ,−2 ( xk ,i ) ⎢2 dxk dxk 3 ⎢⎣

2

T

ak , −1 ... ak , N + 2 ⎤⎦ +

dEI ( xk ,i ) d 3 Bk ,−1 ( xk ,i ) dxk

...

dxk 2

dxk 3

...

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

dEI ( xk ,i ) d 3 Bk , N + 2 ( xk ,i ) ⎤ 2 ⎥ ⎡⎣ ak ,−2 dxk dxk 3 ⎥⎦

⎡ d 4 Bk ,−2 ( xk ,i ) ⎢ EI ( xk ,i ) dxk 4 ⎢⎣

EI ( xk ,i )

EI ( xk ,i )

T

ak ,−1 ... ak , N + 2 ⎤⎦ +

d 4 Bk , −1 ( xk ,i )

...

dxk 4

d 4 Bk , N + 2 ( xk ,i ) ⎤ ⎥ ⎡⎣ ak ,−2 dxk 4 ⎥⎦

T

ak ,−1 ... ak , N + 2 ⎤⎦ =

⎡ω 2 ρ A ( xk ,i ) Bk ,−2 ( xk ,i ) ω 2 ρ A ( xk ,i ) Bk , −1 ( xk ,i ) ⎣

...

ω 2 ρ A ( xk ,i ) Bk , N + 2 ( xk ,i ) ⎤⎦ ⎡⎣ ak ,−2 ak , −1 ... ak , N + 2 ⎤⎦

T

for i = 0,1,L , N and k = 1, 2,L , M

⎡ Bk , −2 ( xk , N ) Bk , −1 ( xk , N ) ⎣ ⎡⎣ ak ,−2

(36)

T

ak ,−1 ... ak , N + 2 ⎤⎦ −

⎡ Bk +1, −2 ( xk +1,0 ) Bk +1,−1 ( xk +1,0 ) ⎣ ⎡⎣ ak +1,−2

Bk , N + 2 ( xk , N ) ⎤⎦

...

Bk +1, N + 2 ( xk +1,0 ) ⎤⎦

...

ak +1,−1 ... ak +1, N + 2 ⎤⎦ = [ 0] T

for k = 1, 2,L , M − 1

⎡ dBk , −2 ( xk , N ) ⎢ dxk ⎢⎣ ⎡⎣ ak ,−2

dBk ,−1 ( xk , N )

dBk , N + 2 ( xk , N ) ⎤ ⎥ dxk ⎥⎦

...

dxk T

ak ,−1 ... ak , N + 2 ⎤⎦ −

⎡ dBk +1,−2 ( xk +1,0 ) ⎢ dxk +1 ⎢⎣ ⎡⎣ ak +1,−2

(37)

dBk +1,−1 ( xk +1,0 )

...

dxk +1

dBk +1, N + 2 ( xk +1,0 ) ⎤ ⎥ dxk +1 ⎥⎦

ak +1,−1 ... ak +1, N + 2 ⎤⎦ = [ 0] T

for k = 1, 2,L , M − 1

⎡ d 2 Bk ,−2 ( xk , N ) ⎢ dxk 2 ⎢⎣

(38)

d 2 Bk ,−1 ( xk , N ) dxk 2

...

d 2 Bk , N + 2 ( xk , N ) ⎤ ⎥ dxk 2 ⎥⎦

565

Ming-Hung Hsu

566

⎡⎣ ak ,−2

T

ak ,−1 ... ak , N + 2 ⎤⎦ −

⎡ d 2 Bk +1, −2 ( xk +1,0 ) ⎢ dxk +12 ⎢⎣ ⎡⎣ ak +1,−2

d 2 Bk +1,−1 ( xk +1,0 )

...

dxk +12

d 2 Bk +1, N + 2 ( xk +1,0 ) ⎤ ⎥ dxk +12 ⎥⎦

ak +1,−1 ... ak +1, N + 2 ⎤⎦ = [ 0] T

for k = 1, 2,L , M − 1

⎡ d 3 Bk ,−2 ( xk , N ) ⎢ dxk 3 ⎢⎣ ⎡⎣ ak ,−2

(39)

d 3 Bk ,−1 ( xk , N )

...

T

ak ,−1 ... ak , N + 2 ⎤⎦ −

⎡ d 3 Bk +1,−2 ( xk +1,0 ) ⎢ dxk +13 ⎢⎣ ⎡⎣ ak +1,−2

dxk 3

d 3 Bk , N + 2 ( xk , N ) ⎤ ⎥ dxk 3 ⎥⎦

d 3 Bk +1,−1 ( xk +1,0 ) dxk +13

...

d 3 Bk +1, N + 2 ( xk +1,0 ) ⎤ ⎥ dxk +13 ⎥⎦

ak +1,−1 ... ak +1, N + 2 ⎤⎦ = [ 0] T

for k = 1, 2,L , M − 1

(40)

The spline collocation method can be used to rearrange the boundary conditions of a clamped-free non-uniform beam into matrix form,

⎡ B1,−2 ( x1,0 ) B1,−1 ( x1,0 ) B1,0 ( x1,0 ) ... B1, N +1 ( x1,0 ) B1, N + 2 ( x1,0 ) ⎤ ⎣ ⎦ ⎡⎣ a1,−2

a1,−1 ... a1, N + 2 ⎤⎦ = [ 0]

⎡ dB1, −2 ( x1,0 ) dB1,−1 ( x1,0 ) ⎢ dx1 dx1 ⎣⎢

T

...

(41)

dB1, N + 2 ( x1,0 ) ⎤ T ⎥ ⎡⎣ a1,−2 a1,−1 ... a1, N + 2 ⎤⎦ = [ 0] (42) dx1 ⎦⎥

⎡ d 2 BM ,−2 ( xM , N ) d 2 BM ,−1 ( xM , N ) EI ( xM , N ) .... ⎢ EI ( xM , N ) dxM 2 dxM 2 ⎢⎣ d 2 BM , N + 2 ( xM , N ) ⎤ T EI ( xM , N ) ⎥ ⎡⎣ aM ,−2 aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] 2 dxM ⎥⎦ ⎡ d 3 BM , −2 ( xM , N ) d 3 BM ,−1 ( xM , N ) EI ( xM , N ) .... ⎢ EI ( xM , N ) dxM 3 dxM 3 ⎢⎣

(43)

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

d 3 BM , N + 2 ( xM , N ) ⎤ EI ( xM , N ) ⎥ ⎡⎣ aM ,−2 dxM 3 ⎥⎦

567

T

aM ,−1 ... aM , N + 2 ⎤⎦ +

⎡ dEI ( xM , N ) d 2 BM ,−2 ( xM , N ) dEI ( xM , N ) d 2 BM , −1 ( xM , N ) .... ⎢ dxM 2 dxM dxM 2 ⎢⎣ dxM dEI ( xM , N ) d 2 BM , N + 2 ( xM , N ) ⎤ T ⎥ ⎡⎣ aM ,−2 aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] 2 dxM dxM ⎥⎦

(44)

The spline collocation method can be used to rearrange the boundary conditions of a pinnedpinned non-uniform beam into matrix form,

⎡ B1,−2 ( x1,0 ) B1, −1 ( x1,0 ) ⎣ ⎡⎣ a1,−2

B1, N + 2 ( x1,0 ) ⎤⎦

...

a1,−1 ... a1, N + 2 ⎤⎦ = [ 0] T

(45)

⎡ d 2 B1,−2 ( x1,0 ) d 2 B1,−1 ( x1,0 ) EI ( x1,0 ) ⎢ EI ( x1,0 ) dx 2 dx 2 ⎢⎣ d 2 B1, N + 2 ( x1,0 ) ⎤ T ... EI ( x1,0 ) ⎥ ⎡⎣ a1,−2 a1,1 ... a1, N + 2 ⎤⎦ = [ 0] 2 dx ⎥⎦ ⎡ BM ,−2 ( xM , N ) BM ,−1 ( xM , N ) ⎣ ⎡⎣ aM ,−2

...

(46)

BM , N + 2 ( xM , N ) ⎤⎦

aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] T

(47)

⎡ d 2 BM ,−2 ( xM , N ) d 2 BM ,−1 ( xM , N ) EI ( xM , N ) ... ⎢ EI ( xM , N ) dx 2 dx 2 ⎢⎣ d 2 BM , N + 2 ( xM , N ) ⎤ T EI ( xM , N ) ⎥ ⎡⎣ aM ,−2 aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] 2 dx ⎥⎦

(48)

The spline collocation method can be used to rearranged the boundary conditions of a clamped-clamped non-uniform beam into matrix form,

⎡ B1,−2 ( x1,0 ) B1,−1 ( x1,0 ) ⎣ ⎡ dB1,−2 ( x1,0 ) dB1,−1 ( x1,0 ) ⎢ dx1 dx1 ⎢⎣

...

...

B1, N + 2 ( x1,0 ) ⎤⎦ ⎡⎣ a1,−2 a1,−1 ... a1, N + 2 ⎤⎦ = [ 0] (49) T

dB1, N + 2 ( x1,0 ) ⎤ ⎥ ⎡⎣ a1,−2 dx1 ⎥⎦

a1,−1 ... a1, N + 2 ⎤⎦ = [ 0] (50) T

Ming-Hung Hsu

568

⎡ BM ,−2 ( xM , N ) BM ,−1 ( xM , N ) ⎣ ⎡⎣ aM ,−2

...

aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] T

⎡ dBM ,−2 ( xM , N ) dBM ,−1 ( xM , N ) ⎢ dx dx ⎣⎢ ⎡⎣ aM ,−2

BM , N + 2 ( xM , N ) ⎤⎦

...

(51)

dBM , N + 2 ( xM , N ) ⎤ ⎥ dx ⎦⎥

aM ,−1 ... aM , N + 2 ⎤⎦ = [ 0] T

(52)

The following figures summarize the obtained results.

RESULTS Figure 2 shows the non-dimensional natural frequencies of the non-uniform beam with

A = A0 (1 + x / L + x 2 / L2 ) and I = I 0 (1 + x / L + x 2 / L2 ) . The non-dimensional natural frequency is defined as Ω = ω

ρ A0 L4 EI 0

. Results computed using the spline collocation

method are compared successfully with the numerical results of Abrate [1]. The difference between the present numerical results and the Rayleigh-Ritz method data [1] is 0.003 0 0 . Figure 3 displays the clamped-free non-uniform beam with A = Ao (1 − 0.5 x / L ) and

I = I o (1 − 0.5 x / L ) . Figure 4 plots the non-dimensional natural frequencies of the clamped–free non-uniform beam with A = Ao (1 − 0.5 x / L ) and I = I o (1 − 0.5 x / L ) . The frequencies are computed using the finite element package FEMLAB and the spline collocation method. The finite element mesh that is adopted to model the non-uniform beam consists of 3036 eight-noded solid elements. The curve obtained using the spline collocation method closely follows the curve that is obtained using the finite element method. The computational times required to apply the spline collocation method with 13, 16, 23, 26 and 33 collocation points are 0.313, 0.344, 0.453, 0.547 and 1.172 seconds, respectively. However, a computational time of over 10 seconds is required when the finite element method is used to model the non-uniform beam. Figures 5—7 plot the non-dimensional natural

frequencies

of

the

non-uniform

beams

with

A = Ao (1 + α x / L ) and

I = I o (1 + α x / L ) . The modes are affected by tapering. The first and second non-uniform beam frequencies decrease as α increases. Figures 8—10 list the non-dimensional natural frequencies of the non-uniform beams with A = Ao (1 + β x / L ) and I = I o (1 + β x / L ) . 3

The first and second frequencies of the clamped-free non-uniform beam decrease as

β

increases. The first and second frequencies of the clamped-clamped and pinned-pinned non-

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

569

β increases below 0.5 . The first and second frequencies of the clamped-clamped and pinned-pinned non-uniform beams increase as β increases above 0.5 . uniform beams decrease as

Figure 1. A clamped-free non-uniform beam with

(

I = I 0 1 + x / L + x 2 / L2

).

A = A0 (1 + x / L + x 2 / L2 )

Figure 2. Non-dimensional natural frequencies of non-uniform beams with

(

A = A0 1 + x / L + x 2 / L2

) , I = I (1 + x / L + x 0

2

/ L2

and

) and various boundary conditions.

570

Ming-Hung Hsu

Figure 3. A clamped-free non-uniform beam with

I = I o (1 − 0.5 x / L ) .

A = Ao (1 − 0.5 x / L )

Figure 4. Calculated frequencies of non-uniform beams with

I = I o (1 − 0.5 x / L )

and various boundary conditions.

and

A = Ao (1 − 0.5 x / L ) ,

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

Figure 5. Non-dimensional natural frequencies of clamped-free non-uniform beams with

A = Ao (1 + α x / L )

and

I = I o (1 + α x / L ) .

Figure 6. Non-dimensional natural frequencies of pinned-pinned non-uniform beams with

A = Ao (1 + α x / L )

and

I = I o (1 + α x / L ) .

571

Ming-Hung Hsu

572

Figure 7. Non-dimensional natural frequencies of clamped-clamped non-uniform beams with

A = Ao (1 + α x / L )

and

I = I o (1 + α x / L ) .

Figure 8. Non-dimensional natural frequencies of clamped-free non-uniform beams with

A = Ao (1 + β x / L )

and

I = I o (1 + β x / L )

3

.

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

Figure 9. Non-dimensional natural frequencies of pinned-pinned non-uniform beams with

A = Ao (1 + β x / L )

and

I = I o (1 + β x / L )

3

.

Figure 10. Non-dimensional natural frequencies of clamped-clamped non-uniform beams with

A = Ao (1 + β x / L )

and

I = I o (1 + β x / L )

3

.

573

574

Ming-Hung Hsu

CONCLUDING REMARKS This work develops an efficient algorithm for solving eigenvalue problems of nonuniform beams based on the spline collocation scheme, Euler-Bernoulli beam theory and Hamilton’s principle. Appropriate boundary conditions and the spline collocation method are applied to transform the partial differential equations of non-uniform beams into discrete eigenvalue problems. Numerical results revel that α and β significantly affect the frequencies of the non-uniform beams. Smaller α values produce larger frequencies of the non-uniform beams. The effectiveness of the spline collocation method in elucidating the dynamic behavior of non-uniform beams is demonstrated.

REFERENCES [1] [2] [3] [4]

[5]

[6]

[7] [8] [9]

[10] [11] [12] [13]

Abrate, S. (1995). Vibrations of non-uniform rods and beams. Journal of Sound and Vibration, Vol. 185, No. 4, pp. 703-716. Gottlieb, H. P. W. (1996). Comments on vibrations of non-uniform beams and rods. Journal of Sound and Vibration, Vol. 195, No. 1, pp. 139-141. Naguleswaran, S. (1996). Comments on vibration of non-uniform beams and rods. Journal of Sound and Vibration, Vol. 195, No. 2, pp. 331-337. Hodges, D. H., Chung, Y. Y., and Shang, X. Y. (1994). Discrete transfer matrix method for non-uniform rotating beams. Journal of Sound and Vibration, Vol. 169, pp. 276283. Lee, S. Y., and Kuo, Y. H. (1992). Bending vibrations of a rotating non-uniform beam with an elastically restrained root. Journal of Sound and Vibration, Vol. 154, No. 3, pp. 441-451. Hsu, M. H. (2006). Mechanical analysis of non-uniform beams resting on nonlinear elastic foundation by the differential quadrature method. Structural Engineering and Mechanics, Vol. 22, No. 3, pp. 279-292. Hsu, M. H. (2003). Vibration Analysis of Non-uniform Beams Using the Differential Quadrature Method, Doctoral Dissertation, National Sun Yet-Sen University, Taiwan. Hsu, M. H. (2004). Free vibration analysis of non-uniform beams. Journal of Penghu Institute of Technology, Vol. 8, pp.179-200. Ho, S. H., and Chen, C. K. (2006). Free transverse vibration of an axially loaded nonuniform spinning twisted Timoshenko beam using differential transform. International Journal of Mechanical Sciences, Vol. 48, pp. 1323-1331. Prenter, P. M. (1975). Spline and Variational Methods, John Wiley & Sons, Inc., New York. Greville, T. N. E. (1969). Theory and Applications of Spline Functions, Academic Press, New York. Schumaker, L. (1980). Spline Functions: Basic Theory, Wiley-Interscience, New York. Bert, C. W., and Sheu, Y. (1996). Static analyses of beams and plates by spline collocation method. Journal of Engineering Mechanics, Vol. 122, No. 4, pp. 375-378.

Vibration Analysis of Non-Uniform Beams Using Spline Collocation Method

575

[14] El-Hawary, H. M., Zanaty, E. A., and El-Sanousy, E. (2005). Quartic spline collocation methods for elliptic partial differential equations. Applied Mathematics and Computation, Vol. 168, pp. 198-221. [15] Archer, D. A. (1973). Some Collocation Methods for Differential Equations, Philosophy Doctor Thesis, Rice University, Houston, TX, USA. [16] Patlashenko, I., and Weller, T. (1995). Two-dimensional spline collocation method for nonlinear analysis of laminated panels. Computers & Structures, Vol. 57, No.1, pp. 131-139.

In: Engineering Physics and Mechanics Editors: M. Sosa and J. Franco, pp. 577-598

ISBN 978-1-60876-227-9 c 2010 Nova Science Publishers, Inc.

Chapter 13

M ULTICLASS F UZZY C LASSIFIERS BASED ON K ERNEL D ISCRIMINANT A NALYSIS Ryota Hosokawa and Shigeo Abe Graduate School of Engineering, Kobe University Nada, Kobe, Japan

Abstract In this chapter, we discuss a fuzzy classifier based on kernel discriminant analysis (KDA) for two-class and multiclass problems. For two-class problems, in the one-dimensional feature space obtained by KDA we define, for each class, a onedimensional membership function and generate a classification rule. To improve classification performance of the fuzzy classifier, we tune the membership functions based on the same training algorithm as that of a linear support vector machine (SVM). A data sample is classified into the class with the maximum degree of membership. For multiclass problems, we define a membership function for each pair of classes and then tune the membership functions of each pair of classes by the same method as that for two-class problems. A data sample is classified as follows: calculate the membership degree of the sample for each class by taking the minimum value among the membership degrees associated with the class, and classify the sample into the class with the maximum degree of class membership. Through computer experiments, we show that the performance of the proposed classifier is comparable to that of SVMs and least squares (LS) SVMs, and show that we can easily analyze the behavior of the proposed classifier using the membership functions.

Keywords: fuzzy classifiers; kernel discriminant analysis; multiclass problems; support vector machines

1.

Introduction

Support vector machines (SVMs) [1, 2] are one of the classifiers widely used in the area of pattern recognition because of their high generalization ability. However, one of the disadvantages of SVMs, is that it is difficult to analyze their behavior because the input space is mapped into a high-dimensional, in some cases infinite, feature space. There are

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roughly two approaches to solve this problem. One is to extract rules from the trained SVM [3, 4, 5, 6, 7] and the other is, instead of using SVMs, to define fuzzy rules in the feature space [8, 9, 10]. In [3], the SVM is visualized by using ellipsoidal regions generated by combining the prototype of each class with support vectors. In [4], the classification boundary of the weighted linear SVM whose performance is comparable to that of a nonlinear SVM is discussed by combining some linear classifiers in the input space. In [5, 6], an explanation capability is added to SVMs by extracting fuzzy rules from SVMs. In [8], fuzzy rules are defined using the kernel Mahalanobis distance. To improve the generalization ability, the membership functions are tuned by counting the number of misclassifications and allowing the increase of misclassifications so long as the total recognition rate is improved. In [10], a two-class fuzzy classifier with an explanation capability based on kernel discriminant analysis (KDA) is proposed. In this chapter, extending [10] to multiclass problems, we discuss a KDA-based kernel fuzzy classifier with high generalization ability and whose behavior is easily analyzable. For two-class problems, using KDA we obtain the axis onto which the projections of training data maximally separate two classes. Because this one-dimensional axis well separates two classes, it is suitable for generating a powerful classifier whose behavior is easily analyzable. We call this one-dimensional space obtained by KDA KDA space. Then, in the KDA space we define a one-dimensional membership function for each class based on the Euclidean distance from the class center. Using these membership functions, we generate classification rules in the KDA space, and classify a data sample into the class whose degree of membership is larger. To improve the performance of the two-class fuzzy classifier we need to tune fuzzy rules, in other words, we need to tune the slope and the bias term of each membership function as discussed in [8]. We show that the tuning procedure can be performed based on the same training algorithm as that of linear SVMs. In this tuning procedure, we use the two-dimensional feature space whose axes are the distances from the centers of the membership functions. The training of linear-SVMs in this two-dimensional feature space is equivalent to the fuzzy rule tuning in the KDA space. To extend the two-class fuzzy classifiers to multiclass problems, we may use generalized KDA algorithms for multiclass problems [11, 6, 12]. But this method obtains the orthogonal axes, which is equal to the number of classes minus one. Thus we cannot generate a class membership. Therefore, we extend the two-class fuzzy classifiers to multiclass classifiers based on pairwise methods [2]. In pairwise methods, we construct two-class classifiers for all the combinations of class pairs, and define the class membership function using the minimum operation for the membership functions associated with the class. And we classify a data sample into the class with the highest class membership degree. In Section 2. we summarize KDA, and in Section 3. we describe how to construct the proposed fuzzy classifiers for two-class problems. In Section 4., we show that the tuning procedure of membership functions reduces to the same training algorithm as that of linear SVMs, and in Section 5. we explain how to extend our two-class fuzzy classifiers to those for multiclass problems. In Section 6., we evaluate the proposed method using benchmark data sets, and finally in Section 7. we conclude our work.

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Kernel Discriminant Analysis

In this section we summarize KDA for two-class problems, which calculates the axis on which the projections of the training data maximally separate two classes. Let the sets of m-dimensional data belong to class i (i = 1, 2) be {xi1, ..., xiMi}, where Mi is the number of data belonging to class i, and data x be mapped into the l-dimensional feature space by the mapping function g(x). The aim of KDA is to find the l-dimensional vector w onto which the projections of the training data maximally separate the two classes. The projection of g(x) on w is obtained by wT g(x)/kwk. In the following we assume that kwk = 1, thus we calculate the projection by wT g(x). We find such w that maximizes the distance between the class centers, and minimizes the variances of the projected data. The square projected difference of the class centers, d2c , is calculated as d2c = wT (c1 − c2)(c1 − c2 )T w,

(1)

where ci are the centers of class i data: M 1 Xi g(xij ) for i = 1, 2. Mi

(2)

QB = (c1 − c2 )(c1 − c2 )T

(3)

ci =

j=1

We define

and call QB the between-class scatter matrix. Here, in order to simplify matters, we redefine the data sets as {x11, ..., x1M1 , x21, ..., x2M2 } = {x1, ..., xM }, where M is the number of all the training data and M = M1 + M2 . In the following if there is no confusion, we use both representations for data sets. The variance of all the projected data, s2 , is s2 = wT QT w, where QT =

1 (g(x1), ..., g(xM ))(IM M

(4) 

 gT (x1)   .. − 1M )  . . T g (xM )

(5)

Here, IM is the M × M unit matrix and 1M is the M × M matrix with all elements being 1/M . We call this matrix, QT total scatter matrix. Now, in KDA, we maximize the following criterion: J(w) =

wT QB w d2c , = s2 w T QT w

(6)

but since w, QB , and QT are defined in the feature space, we cannot calculate them explicitly. Here we need to use kernel tricks. Any solution w in the feature space can be written as an expansion of the form w = (g(x1), ..., g(xM ))α,

(7)

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where α = (α1 , ..., αM )T and α1 , ..., αM are scalars. Substituting (7) into (6), we can rewrite the KDA criterion, J, as J(α) =

αT KB α , αT KT α

(8)

where KB = (kB1 − kB2 )(kB1 − kB2 )T ,   M 1 Xi  H(x1, xij )    Mi   j=1   kBi =  ...  for i = 1, 2,   Mi   1 X i   H(xM , xj ) Mi

(9)

(10)

j=1

KT

=

1 K(IM − 1M )K. M

(11)

Here H(x, x0) = gT (x)g(x0) is a kernel function, and K = {H(xi, xj )} is a kernel matrix constructed by using all the training data. Thus KT is a positive semi-definite matrix. If KT is positive definite, the solution of (8) is given by α = KT−1 (kB1 − kB2 ).

(12)

If KT is positive semi-definite, the inverse KT−1 does not exist. One way to overcome singularity is to add positive values to the diagonal elements [13]: α = (KT + εIM )−1 (kB1 − kB2 ),

(13)

where ε is a small positive parameter, and determines the structure of projections in the KDA space. From the assumption that kwk = 1, we can calculate the projection of g(x) on w, p, with kernel tricks as follows: p = wT g(x) = gT (x)w = (gT (x)g(x1), ..., gT (x)g(xM ))α = (H(x, x1), ..., H(x, xM ))α.

(14)

Using (14), training data {xi1, ..., xiMi } for class i are expressed by one-dimensional features {pi1, ..., piMi }, where pij = (H(xij , x1), ..., H(xij, xM ))α for j = 1, ..., Mi. We call this one-dimensional space, obtained by (14), KDA space.

3. 3.1.

Two-class Fuzzy Classifiers Based on KDA Concept

In this section, we discuss fuzzy classifiers based on KDA for two-class problems (KDA-FCs). KDA is a powerful tool to obtain the one-dimensional feature that well sepa-

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rates two classes in the feature space. Hence, using KDA we can easily construct a powerful fuzzy classifier whose behavior is easily analyzable. In our method, using (14), we calculate the class i one-dimensional features for training data. Then, in the KDA space, for each class we define the following fuzzy rule: Ri : if p is µi , then x belong to class i,

(15)

where µi is the center of class i in the KDA space: Mi 1 X pij for i = 1, 2. µi = Mi

(16)

j=1

3.2.

Definition of Membership Functions

In the KDA space, for the centers µi , we define one-dimensional membership functions mi (x) that define the degree to which p belongs to µi . We consider two types of mi (x), i.e., without a bias term and with a bias term. 3.2.1. Membership Functions without Bias Terms Based on the Euclidean distance di (p) of p from the center µi in the KDA space, we define mi (x) as follows (see Fig. 1): mi (x) = 1 −

di(p) , βi

(17)

where di (p) is the Euclidean distance from the center µi , di(p) = |p − µi |, and p is the projection of g(x) on w calculated by (14). The parameter βi is a tuning parameter of the slope for mi (x). We allow negative values of mi (x) so that any point p can be classified into a definite class. We call this type of two-class fuzzy classifier KDA-FC1. We calculate the degree of each membership function for input datum x, mi (x), and classify it into the class whose membership is maximum. This is equivalent to finding the minimum Euclidean distance when βi in (17) is equal to 1. If mi (x) is equal to 1, the projection p is at the center of class i, µi , in the KDA space. Instead of allowing negative values of membership, we can use the following Gaussian membership function:  2  d (p) . (18) mi (x) = exp − i βi However because there is not much difference of generalization abilities between the classifier using (17) and (18), in the following we discuss the classifier using (17) as KDA-FC1. 3.2.2. Membership Functions with Bias Terms In order to consider a more general form of membership functions, we add a bias term, bi , to (17) as follows: mi (x) = 1 −

di (p) − bi . βi

(19)

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m 1(x)

m 2 (x)

µ1

µ2

1

0

p

Figure 1. KDA-FC1 without bias terms.

D egree ofm em bership

Figure 2 shows membership functions with bias terms in the KDA space. We call this type of two-class fuzzy classifier KDA-FC2. Because the classification of KDA-FCs is performed in the KDA space whose dimension is one, we can easily analyze its behavior by visualizing each membership function. When we use the membership functions given by (17) or (19), however, we need to determine the values of βi or those of βi and bi . This process is called fuzzy rule tuning and is described in Section 4.

m 1(x)

m 2 (x)

1

b1

0

b2

µ1

µ2

p

Figure 2. KDA-FC2 with bias terms.

4.

Fuzzy Rule Tuning

We tune membership functions of KDA-FCs using the same training algorithm as that of linear SVMs. By training a linear SVM, the decision function is determined for classification. Hence we first derive the decision function of KDA-FCs, and then we optimize it based on the same training algorithm as that of Least Square SVMs (LS-SVMs) [14] which are the variants of SVMs. Optimization of the decision function of KDA-FCs is equivalent to tuning membership functions. In the following, we describe the tuning procedure.

4.1.

Decision Function of KDA-FCs

First, we consider the case of KDA-FC2, which involves the membership functions with bias terms given by (19). In KDA-FCs, the data sample x is classified based on the

Multiclass Fuzzy Classifiers Based on Kernel Discriminant Analysis following classification rule:  m1(x) > m2(x) =⇒ Class 1, m1(x) < m2(x) =⇒ Class 2.

583

(20)

If the data sample x satisfies m1(x) = m2 (x), it is on the classification boundary and unclassifiable. Here, we define the function L(x) as the difference of m1 (x) and m2 (x): L(x) = m1 (x) − m2 (x) d1 (p) d2(p) = − + + (b2 − b1 ). β1 β2

(21)

Now we set β = (− β11 , β12 )T , d = (d1(p), d2(p))T , and b = b2 − b1 . Then (21) is rewritten as follows: (22) L(d) = β T d + b. Based on the classification rule (20), if the data sample x satisfies L(d) > 0, it is classified into Class 1, and if L(d) < 0, it is classified into Class 2. When L(d) = 0, it is unclassifiable. From the above discussion, L(d) is the decision function of KDA-FC2 whose input is a two-dimensional vector d = (d1(p), d2(p))T , and its form is the same as that of the decision function of linear SVMs. Hence, by calculating the weight vector β and bias term b based on the same training algorithm as that of linear SVMs in the two-dimensional space (d1(p), d2(p)), we can determine the slope parameter βi and bias term bi of each membership function. (But in this formulation, we cannot determine the values of b1 and b2 uniquely. However, because the classification boundary is invariant so long as b = b2 − b1 is constant, we can assume that either bi is equal to 0.) In the following, we call the twodimensional space, (d1(p), d2(p)), 2-D tuning space. On the other hand, in the KDA-FC1, because its membership functions include no bias terms as shown in (17), the decision function L(d) is represented as L(d) = βT d.

(23)

In this case, by the same training as that of linear SVMs without a bias term in the 2-D tuning space, we can determine the slope parameter βi . If we use Gaussian membership functions shown in (18), the components of the 2-D tuning space become the square distances (d21(p), d22(p)).

4.2.

2-D Tuning Space

As we discussed before, the fuzzy rule tuning of our KDA-FCs can be performed by the same training method as that of linear SVMs in the 2-D tuning space whose axes are di(p), each of which is the distance from the center of the ith membership function to p. In this section, we discuss the structure of this 2-D tuning space. If the point p is between µ1 and µ2 in the KDA space, the following equation is satisfied: d1(p) + d2(p) = µ

for µ1 ≤ p ≤ µ2 ,

(24)

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where µ is the Euclidean distance between µ1 and µ2 . Namely, µ = |µ2 − µ1 |. Similarly, if point p is outside of the interval [µ1 , µ2], the following equation is satisfied: |d1(p) − d2(p)| = µ

for p < µ1

or

µ2 < p.

Hence, the relationship between d1(p) and d2(p) is described as follows:  −d1 (p) + µ for µ1 ≤ p ≤ µ2 , d2(p) = for p < µ1 or µ2 < p. d1(p) ± µ

(25)

(26)

Therefore any point p in the KDA space is mapped on the reflexed dashed line whose slopes are ±1 in the 2-D tuning space as shown in Fig. 3. (In the case of using Gaussian membership functions (18), the mapped region shown as the reflexed dashed line becomes the quadratic curve.) In this space, we determine the boundary L(d) = 0 by training a linear SVM. Particularly in the KDA-FC1, because membership functions include no bias terms bi , the boundary, L(d) = 0, goes through the origin in the 2-D tuning space.

d2 (p)

Boundary ofthe LinearSVM

L(d)= 0

0

d1(p) Figure 3. 2-D tuning space.

4.3.

Training of Decision Functions in the 2-D Tuning Space

In this section, we describe the training method of the decision function L(d) in the 2-D tuning space. Here, we consider using the same training method as that of LS-SVMs whose training is done by solving a set of linear equations, instead of a quadratic programming problem. First, we discuss the case of KDA-FC2. The LS-SVM in the 2-D tuning space is trained by minimizing M CX 2 1 2 kβk + ξi (27) 2 2 i=1

subject to the equality constraints: βT di + b = yi − ξi

for i = 1, ..., M,

(28)

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where yi = 1 and −1 if the ith datum di belongs to Class 1 and Class 2, respectively, ξi is the slack variable for di , and C is the margin parameter. In LS-SVMs, ξi can be negative. Introducing the Lagrange multipliers λi into (27) and (28), we obtain the unconstrained objective function: M M CX 2 X 1 2 ξi − λi (βT di + b − yi + ξi ), Q(β, b, λ, ξ) = kβk + 2 2 i=1

(29)

i=1

where λ = (λi, ..., λM )T and ξ = (ξi, ..., ξM )T . Taking the partial derivatives of (29) with respect to β, b, and ξ and equating them to zero, we obtain the optimal conditions as follows: β=

M X

λi d i ,

(30)

i=1 M X

λi = 0,

(31)

i=1

λi = Cξi

for i = 1, ..., M,

(32)

T

β di + b − yi + ξi = 0 for i = 1, ..., M.

(33)

Substituting (30) and(32) into (33) and expressing it in matrix form and (31) in vector form, we obtain a set of linear equations as follows: Ωλ + 1b = y,

(34)

T

(35)

1 λ = 0, where Ω ij =

dTi dj

δij , δij = + C



1 for i = j, 0 for i 6= j,

y = (y1 , ..., yM )T ,

(36)

and 1 is the M -dimensional vector with all elements being 1. The original minimization problem (27) and (28) is solved by solving (34) and (35) for λ and b as follows. Because of 1/C (> 0) in the diagonal elements, Ω is positive definite. Hence, λ = Ω−1 (y − 1b).

(37)

Substituting (37) into (35), we obtain b=

1Ω−1 y . 1Ω−1 1

(38)

Thus, substituting (38) into (37), we obtain the optimal solution of λ. Similarly, in the KDA-FC1, the LS-SVM without a bias term in the 2-D tuning space is trained by minimizing M CX 2 1 kβk2 + ξi (39) 2 2 i=1

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subject to the equality constraints: β T di = yi − ξi

for i = 1, ..., M.

(40)

Introducing the Lagrange multipliers λi into (39) and (40), we obtain the unconstrained objective function: M M 1 CX 2 X Q(β, λ, ξ) = kβk2 + ξi − λi(β T di − yi + ξi ). 2 2 i=1

(41)

i=1

In this case, the optimal conditions are represented as follows: β=

M X

λi d i ,

(42)

i=1

λi = Cξi

for i = 1, ..., M,

T

β di − yi + ξi = 0

for i = 1, ..., M,

(43) (44)

and a set of linear equations becomes Ωλ = y.

(45)

Multiplying both terms of (45) by Ω−1 from the left, we obtain the optimal solution of λ.

5.

Extension to Multiclass Problems

There are several types of SVMs that handle multiclass problems. Among them, pairwise SVMs and one-against-all SVMs are widely used. Since their generalization abilities are comparable but training by pairwise SVMs is faster, we use pairwise methods to extend KDA-FCs to multiclass problems.

5.1.

Extension Based on Pairwise Methods

In the KDA-FC2 bias terms are included. But since the values of bias terms are not uniquely determined, they need to be adjusted among membership functions. To avoid this, in the following we consider extending only the KDA-FC1, which has no bias terms, to multiclass problems. In pairwise methods, we construct two-class classifiers for all the combination of class pairs. Thus, the number of two-class classifiers is n(n − 1)/2 for an n-class problem. Now, let the membership function for class i against class j in the kth KDA-FC1 among all KDA-FC1s constructed in pairwise methods be mij (pk ) (i 6= j) (see Fig. 4). Here, the superscript k (k = 1, ..., n(n − 1)/2) shows the kth KDA-FC1 trained using the training data belonging to classes i and j, and pk shows the projection of g(x) on vector wk of the kth KDA-FC1. In each KDA-FC1, expressing the vector wk as an expansion of the training data belonging to classes i and j: wk = (g(xi1), ..., g(xiMi ), g(xj1), ..., g(xjMj ))αk

(46)

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Boundary

m 21(p1) KD A space ofthe 1stKD A-FC 1

m 12 (p1)

C lass 1 C lass 2 C lass 3

Figure 4. Membership functions in each KDA-FC1. with coefficient vector αk = (α1, ..., αMi , αMi+1 , ..., αMi+Mj )T , and calculating αk using (13), we obtain pk as follows: pk = wkT g(x) = (H(x, xi1), ..., H(x, xiMi ), H(x, xj1), ..., H(x, xjMj ))αk .

(47)

We define the class i membership function of pk associated with the original input x by the minimum operation for mij (pk ) (j 6= i, j = 1, ..., n): mi (x) = min mij (pk ). j6=i j=1,...,n

(48)

Now, a data sample x is classified into the class arg

max mi (x).

i=1,...,n

(49)

Here, arg max returns the index for the maximum value of mi (x). We call this type of proposed multiclass classifier KDA-FC M . The classification boundary of the KDA-FC M is determined by the hyperplane on which the degree of each class membership is the same.

5.2.

Behavior Analysis of KDA-FCM s

We can analyze the behavior of the KDA-FC M in the similar way as that of the twoclass KDA-FCs. In this case we visualize the KDA-FC1 in which we need to analyze the classification among all KDA-FC1s constructed by pairwise methods. At the first step, we prepare the membership graph as shown in Fig. 5, which shows the example of membership graph for three-class problems. The horizontal axis of the graph shows the index i allotted to each input x, and the ordinate axis shows the degree of each class membership x. By using this graph, we can visually understand the degree of each class membership for all the data. Then, by examining this graph, we choose two target classes we need to analyze, and we visualize the KDA-FC1 trained by the training data belonging to them. From Fig. 5, we examine the degree of each membership (m1 (xi ), m2(xi ), m3 (xi)) the data sample xi has. From the figure, we can see that m2 (xi) and m3 (xi)

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(shown as the symbols  and ×, respectively) take the closest values to each other, on the other hand, m1(xi ) (shown as ) takes the value far from them for almost all the data. Hence in this classification problem, we can see the regions of Classes 2 and 3 are close to each other, and thus are difficult to classify. In this case, by visualizing the KDA-FC1 trained by the training data belonging to Classes 2 and 3, we can analyze classification in the region where classification is difficult.

6.

Performance Evaluation

In this section, we evaluate our proposed methods through the computational experiments for two-class and multiclass classification problems. In each classification problem, we first compare the performance of our proposed classifiers to that of SVMs and LS-SVMs, and then we show that we can easily analyze the behavior of our proposed classifiers. Throughout the experiments, we use RBF kernel functions: H(x, x0) = exp(−γkx − x0 k2),

(50)

where γ is a positive parameter for controlling the radius. For performance comparisons in multiclass classification problems, we use the pairwise SVM and pairwise LS-SVM [2].

6.1.

Benchmark Data Sets

In two-class classification problems, we use the benchmark data sets [15] shown in Table 11 , which lists the numbers of input dimensions, training data, and test data. In addition, as shown in the column “Sets” in the table, each data set consists of 100 (or 20) training and test data sets. For all data sets, we normalize the input ranges into [0, 1]. In multiclass classification problems, we use iris, numeral, blood cell, and hiragana-13 [16] data sets shown in Table 2. Each data set consists of one training data set and one test data set, and the column “Classes” in the table shows the number of classes of each data set. As is the case for two-class problems, we normalize the input ranges into [0, 1] for all data sets. http//ida.first.fraunhofer.de/projects/bench/benchmarks.htm

D egree ofm em bership

1

1

:m 1(xi) :m 2 (xi) :m 3(xi) 0

5

10

index i

Figure 5. Membership graph (Example for a three-class problem).

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Table 1. Two-class benchmark data specification Data Banana B. cancer Diabetes German Heart Image Ringnorm F. solar Splice Thyroid Titanic Twonorm Waveform

Inputs 2 9 8 20 13 18 20 9 60 5 3 20 21

Train 400 200 468 700 170 1300 400 666 1000 140 150 400 400

Test 4900 77 300 300 100 1010 7000 400 2175 75 2051 7000 4600

Sets 100 100 100 100 100 20 100 100 20 100 100 100 100

Table 2. Multiclass benchmark data specification Data Iris Numeral Blood cell Hiragana-13

6.2.

Inputs 4 12 13 13

Train 75 810 3097 8375

Test 75 820 3100 8356

Classes 3 10 12 38

Parameter Setting

In order to compare the performance of the proposed classifiers to that of SVMs or LS-SVMs fairly, we need to optimize the user parameters of each classifier using the same methods. In SVMs or LS-SVMs, we need to determine the value of kernel parameter γ and margin parameter C [1, 2]. In our proposed classifiers, in addition to γ and C (shown in (27), (39)), we need to determine the value of the positive parameter ε in KDA, which is used to overcome singularity of the matrix KT (see (13)). In order to determine the values of these parameters, we perform fivefold crossvalidation, changing γ = {0.1, 0.5, 1, 5, 10, 15}, C = {1, 10, 50, 100, 500, 1000, 2000, 3000, 5000, 8000, 10000, 50000, 100000}, ε = {10−2, 10−3, 10−4, 10−5, 10−6, 10−7 , 10−8}. As shown in Table 1, each two-class data set consists of 100 (or 20) training and test data sets. We use first five training data sets for parameter settings in two-class problems. In SVMs and LS-SVMs, we perform fivefold cross-validation for the first five training data sets and determine the values of γ and C. In KDA-FC1 or KDA-FC2, we previously set ε = 10−3 as used in [13] and determine the values of γ and C as is the case with SVMs

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and LS-SVMs, and then we perform fivefold cross-validation for ε for the first five training data sets under the determined values of γ and C. Similarly, in multiclass problems we perform fivefold cross-validation using each training data set, and determine the values of γ and C. In KDA-FCM s, from our computational experiments, we know that their generalization abilities for multiclass problems are improved when we set the smaller values of ε. Hence in KDA-FCM s, we initially set ε = 10−8 and determine the values of γ and C, and then we optimize the value of ε by fivefold crossvalidation under the determined γ and C values. Tables 3 and 4 list the selected parameter values by the above procedure. Table 3. Parameter settings for two-class problems

Data Banana B. cancer Diabetes German Heart Image Ringnorm F. solar Splice Thyroid Titanic Twonorm Waveform

SVM γ C 15 100 0.1 500 0.1 3000 0.1 50 0.1 50 15 500 15 1 0.5 10 10 10 15 100 0.5 10 0.5 1 10 1

LS-SVM γ C 15 2000 1 1 0.1 3000 0.1 500 0.1 50 15 500 15 10 0.1 100 10 50 15 100 0.1 50 0.5 100 5 1

KDA-FC1 γ C ε 15 1 10−7 0.1 1 10−8 5 10 10−3 10 1 10−3 1 10 10−3 15 10 10−8 0.1 1 10−3 1 10 10−3 10 1 10−6 15 10 10−4 5 1 10−3 0.5 1 10−4 10 10 10−3

γ 15 0.1 5 1 1 15 0.1 1 10 15 0.1 1 5

KDA-FC2 C ε 1 10−5 100 10−6 10 10−3 10 10−3 10 10−2 1 10−7 100 10−5 10 10−3 1 10−6 10 10−4 100 10−2 1 10−3 10 10−3

Table 4. Parameter settings for multiclass problems

Data Iris Numeral Blood cell Hiragana-13

6.3.

SVM γ C 0.1 100 0.1 500 5 100 10 500

LS-SVM γ C 0.1 100 5 100 5 500 5 50000

KDA-FCM γ C ε 1 10 10−8 5 1 10−8 5 50 10−8 15 500 10−8

Evaluation for Two-class Problems

In this section, we discuss the experimental result for two-class problems. We first compare the performance of the proposed KDA-FC1 and KDA-FC2 to that of SVMs and

Multiclass Fuzzy Classifiers Based on Kernel Discriminant Analysis

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LS-SVMs, and then we show that we can easily analyze the behavior of KDA-FCs using one-dimensional membership functions in the KDA space. 6.3.1. Generalization Ability for Two-class Problems We trained the classifier for 100 or 20 training data sets and calculated the average recognition rate and the standard deviation for the test data sets using parameter settings shown in Table 3. Table 5 lists the average recognition rates and the standard deviations for each data set. We statistically analyzed the difference of the recognition rates by Welch t-test, and if there was a significant difference between the performance of each classifier, we highlighted the best result in the row with boldface in Table 5. In addition, if there was a significant difference between the performance of the KDA-FC1 and that of the KDA-FC2, we highlighted the better result with the symbol “ ∗” in the right shoulder in the numeral. From Table 5, we can see that the performance of KDA-FCs is comparable to that of SVMs or LS-SVMs for the banana, diabetes, german, titanic data sets. Especially for the waveform data set, the KDA-FC1 shows the best performance among all classifiers. On the other hand, there are no significant differences of recognition rates among all classifiers for the heart, splice, and thyroid data sets, and for other data sets, the SVM or LS-SVM shows the best performance. Hence, from Table 5, we can see that our proposed KDA-FCs shows performance comparable to that of SVMs and LS-SVMs in many cases, and depending on data sets, their performance is better than that of SVMs and LS-SVMs. However, we can see that the performance of the KDA-FC2 for the ringnorm data set is poor. This is because the parameter settings: (γ, C, ε) determined by cross-validations does not work well for several test data sets among 100 test data sets. (We will discuss later this ill-posed problem for user parameters using the 2-D tuning space.) The number of data sets with significant difference of the performance between the KDA-FC1 and the KDA-FC2 is 7 among 13 data sets. Of them, the KDA-FC1 shows better performance than the KDA-FC2 for the banana, image, ringnorm, and waveform data sets, and the KDA-FC2 shows better performance than the KDA-FC1 for the diabetes, german, and titanic data sets. Therefore, we can say that there is a data dependence between the KDA-FC1 and the KDA-FC2 for their performance. 6.3.2. Behavior Analysis of KDA-FCs Here, we show that we can easily analyze the behavior of KDA-FFCs by visualizing each class membership functions in the KDA space or the 2-D tuning space. Since we cannot determine the values of b1 and b2 uniquely in the KDA-FC2, here we analyze the behavior of the KDA-FC1. Figure 6 shows classification of the KDA-FC1 in the KDA space for the first heart test data set among 100 sets. In the figure, for example, m1 (x) and m2(x) show the membership functions for Classes 1 and 2, respectively, and the classification boundary in the KDA space is defined as their intersecting point. Using the figure, we can easily and visually examine the behavior of the KDA-FC1. The region shown as A in the figure is the region in which the input is classified into Class 2, however the degree of its membership function, m2 (x), is low, or almost 0. Hence, we can consider such region as that far from training data and in which the reliability of the classification is considered to be low. On the other

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Ryota Hosokawa and Shigeo Abe Table 5. Comparison of the average recognition rates (%) and the standard deviations of the rates for two-class problems Data Banana B. cancer Diabetes German Heart Image Ringnorm F. solar Splice Thyroid Titanic Twonorm Waveform

SVM 89.3 ± 0.52 72.4 ± 4.67 76.3 ± 1.83 76.2 ± 2.27 83.7 ± 3.41 97.3 ± 0.41 97.8 ± 0.30 67.6 ± 1.74 89.2 ± 0.71 96.1 ± 2.08 77.2 ± 1.12 97.6 ± 0.14 90.0 ± 0.44

LS-SVM 89.4 ± 0.45 74.0 ± 4.69 76.9 ± 1.67 76.4 ± 2.23 83.8 ± 3.13 97.5 ± 0.33 96.3 ± 0.41 66.7 ± 1.57 89.4 ± 0.72 95.9 ± 2.13 77.3 ± 1.13 97.4 ± 0.18 89.9 ± 0.49

KDA-FC1 89.3 ± 0.47∗ 71.0 ± 4.40 76.0 ± 1.83 75.2 ± 2.06 83.7 ± 3.44 97.2 ± 0.38∗ 96.5 ± 0.81∗ 66.8 ± 1.56 89.4 ± 0.67 95.5 ± 2.24 76.4 ± 2.10 97.5 ± 0.19 90.3 ± 0.38∗

KDA-FC2 89.0 ± 0.55 71.7 ± 4.34 76.6 ± 1.79∗ 76.5 ± 2.40∗ 83.5 ± 3.62 96.1 ± 0.51 92.7 ± 7.07 66.9 ± 1.52 89.3 ± 0.67 95.6 ± 2.20 77.6 ± 1.25∗ 97.5 ± 0.17 90.1 ± 0.41

hand, because in the region near the classification boundary the degree of each membership function, m1 (x) and m2 (x), is almost the same, we can consider such region as that in which the classification is very difficult. m 2 (x)

m 1(x)

A Boundary

Figure 6. Classification of the KDA-FC1 in the KDA space for the first heart test data set. Figure 7 shows the 2-D tuning space corresponding to the KDA space shown in Fig. 6. Because the classification in this 2-D tuning space is equivalent to that in the KDA space, we can also analyze the classification behavior using this 2-D tuning space. However, the 2-D tuning space is more suitable for analyzing the training of KDA-FCs than examining its classification. For example, Fig. 8 shows the 2-D tuning space of the KDA-FC2 for the second ringnorm test data set whose classification does not work well because of ill-posed parameters as discussed in Section 6.3.1.. In Fig. 8, the solid line shows the classification boundary obtained by setting C = 100 determined by cross-validation (see Table 3), and the dashed line shows that obtained by setting C = 1000. For other parameter settings, we use (γ, ε) = (0.1, 10−5) as shown in Table 3. From Fig. 8, we can obviously see that the setting of C = 1000 is more suitable than C = 100 determined by cross-validation for the training

Multiclass Fuzzy Classifiers Based on Kernel Discriminant Analysis

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L(d)= 0

Figure 7. 2-D tuning space of the KDA-FC1 for the first heart test data set.

Figure 8. Difference of the boundaries of the KDA-FC2 in the 2-D tuning space depending on C for the second ringnorm data set.

in the 2-D tuning space. Therefore, we can consider that the reason of the performance degradation for the ringnorm data set in the KDA-FC2 is concerned with the selection of the margin parameter C. In addition, from Fig. 8 we can see that the data belonging to Class 2 have a wider distribution than Class 1. Because in the 2-D tuning space, we use the same

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training algorithm as that of linear LS-SVMs in which we regard all the data as support vectors [1, 2], such difference of the data distribution may greatly influence the generation of the classification boundary, L(d) = 0, and cause the degradation of robustness for C.

6.4.

Evaluation for Multiclass Problems

In this section, we discuss the experimental result for multiclass problems. We first compare the performance of the proposed KDA-FC M to that of SVMs and LS-SVMs. Then, we show that we can easily analyze the behavior of the KDA-FC M in the same way as proposed for two-class KDA-FCs. 6.4.1. Generalization Ability for Multiclass Problems We trained each classifier using parameter settings shown in Table 4, and calculated the recognition rates of the test and training data sets. Table 6 lists the recognition rates of the test data sets, and the parenthetical numerals show the recognition rates of the training data sets. For each data set, the highest recognition rate of the test data set is shown in boldface. Table 6. Comparison of the recognition rates (%) for multiclass problems Data Iris Numeral Blood cell Hiragana-13

SVM 97.33 (100) 99.63 (100) 93.10 (96.84) 99.72 (99.96)

LS-SVM 97.33 (100) 99.88 (100) 94.32 (97.22) 99.88 (100)

KDA-FCM 97.33 (100) 99.39 (100) 94.45 (96.90) 99.84 (99.96)

From Table 6, we can see that the KDA-FCM shows better performance than the SVM and LS-SVM for the blood cell data set. For the hiragana-13 data set, the performance of KDA-FCM is better than that of the SVM, however it is not better than that of the LS-SVM. On the other hand, the performance of the KDA-FC M is equivalent to that of the SVM and LS-SVM for iris data set, and for numeral data set, the SVM and LS-SVM show better performance than the proposed method. Except for the result for numeral data set, because the performance of the KDA-FCM is comparable to, or depending on the data sets better than that of the SVM and LS-SVM, we can see that the extension of two-class KDA-FCs to multiclass classifiers based on pairwise methods works well. 6.4.2. Behavior Analysis of KDA-FC M s In this section, using the iris and blood cell data sets we show that we can analyze the behavior of the KDA-FC M as is the case with two-class KDA-FCs based on the methods as discussed in 5.2.. Figure 9 shows the membership graph for the iris test data set using parameter settings as shown in Table 4. The iris data set consists of three classes with 75 test samples: {xs1, ..., xs75}, and each class includes 25 samples. Here, in order to simplify matters, we sort all the test samples and let {xs1, ..., xs25} belong to Class 1, {xs26, ..., xs50} to Class2,

Multiclass Fuzzy Classifiers Based on Kernel Discriminant Analysis

s x32

s x57

595

:m 1(xi) :m 2 (xi) :m 3 (xi)

Figure 9. Membership graph for the iris data set

m 32 (pk )

s x32

C lass 3

m 23 (pk )

s x57

C lass 2

Figure 10. KDA-FC1 trained by the Classes 2 and 3 iris training data set

{xs51, ..., xs75} to Class 3, respectively. The recognition rate of the iris test data is 97.33% as shown in Table 6, hence we can see 2 test samples are misclassified among 75 samples. These 2 samples are shown as the 32nd sample: xs32 and the 57th sample: xs57 in Fig. 9. From the figure, we can see that xs32 belonging to Class 2 is misclassified into Class 3, and at the same time xs57 belonging to Class 3 is misclassified into Class 2. In addition, we can see that m2(xi ) and m3 (xi) take the close values to each other, and m1 (xi) take the value far from the other two for almost all the data. Hence we can consider that the regions of Classes 2 and 3 are close to each other, and are especially difficult for classification. In such a case, we visualize the KDA-FC1 trained by the training data belonging to Class 2 and Class 3 as shown in Fig. 10. From the figure, we can see that xs32 locates near the classification boundary in the KDA space, and xs57 almost locates on the boundary. Hence, we conclude that since both xs23 and xs57 locate in the region near the boundary, their classification is difficult. The classification of blood cell data set is known to be very difficult because the boundaries of some classes are ambiguous. Figure 11 shows the membership functions of the KDA-FC1 for Classes 2 and 3 and the distribution of the training data for Classes 2 and 3. From the figure, we can see that the training data of Classes 2 and 3 are widely overlapped, and the classification in the overlapped region is very difficult. Fig. 12 shows the member-

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ship functions of the KDA-FC1 for Classes 2 and 3 and the distribution of the test data for Classes 2 and 3. From the figure, we can see that the test data gather near the classification boundary, and thus the classification of Classes 2 and 3 test data is very difficult. m 23 (pk )

m 32 (pk )

Boundary

Figure 11. KDA-FC1 for Classes 2 and 3 trained by the blood cell training data set and the distribution of the training data belonging to Classes 2 and 3 (Limited to 399 samples)

m 32 (pk )

m 23 (pk )

Boundary

Figure 12. KDA-FC1 for Classes 2 and 3 trained by the blood cell training data set and the distribution of the test data belonging to Classes 2 and 3 (Limited to 400 samples)

7.

Conclusions

In this chapter we discussed a fuzzy classifier based on KDA with high generalization ability and whose behavior is easily analyzable for multiclass problems. For each pair of classes in an n-class problem, by KDA we obtain the one-dimensional space called KDA space onto which the projections of the training data maximally separate in the feature space. Then, in the one-dimensional space, we define a one-dimensional membership function for each class and generate classification rules. In this way we generate fuzzy rules for n(n − 1)/2 pairs of classes. In classification of a data sample, we calculate each class membership by taking the minimum of membership degrees associated with the class and classify the sample into the class with the maximum degree of class membership. From the computer experiments using two-class and multiclass benchmark data sets, we showed that the performance of the proposed fuzzy classifiers was comparable to that of SVMs and LS-SVMs. In addition by visualizing the membership functions in the KDA space, we showed that we could easily analyze the behavior of the proposed fuzzy classifiers.

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References [1] V. N. Vapnik. The Nature of Statistical Learning Theory . Springer-Verlag, New York, 1995. [2] S. Abe. Support Vector Machines for Pattern Classification . Springer-Verlag, London, 2005. [3] H. N´un˜ez, C. Angulo, and Catal`a. Rule extraction from support vector machines. In Proceedings of the Tenth European Symposium on Artificial Neural Networks (ESANN 2002), pages 107–112, Bruges, Belgium, 2002. [4] D. Caragea, D. Cook, and V. Honavar. Towards simple, easy-to-understand, yet accurate classifiers. In Proceedings of the Third IEEE International Conference on Data Mining (ICDM 2003), pages 497–500, Melbourne, FL, 2003. [5] Ad. C. F. Chaves, M. M. B. R. Vellasco, and R. Tanscheit. Fuzzy rule extraction from support vector machines. In Proceedings of the Fifth International Conference on Hybrid Intelligent Systems (HIS 2005) , pages 335–340, 2005. [6] J. Yang, A. F. Frangi, J.-Y. Yang, D. Zhang, and Z. Jin. KPCA plus LDA: A complete kernel fisher discriminant framework for feature extraction and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence , 27(2):230–244, 2005. [7] J. Diederich (Ed.). Rule Extraction from Support Vector Machines . Springer-Verlag, Berlin, 2008. [8] K. Kaieda and S. Abe. KPCA-based training of a kernel fuzzy classifier with ellipsoidal regions. International Journal of Approximate Reasoning , 37(3):145–253, 2004. [9] S. Abe. Training of kernel fuzzy classifiers by dynamic cluster generation. In Proceedings of the IEEE ICDM 2005 Workshop on Computational Intelligence in Data Mining, pages 13–20, 2005. [10] R. Hosokawa and S. Abe. Fuzzy classifiers based on kernel discriminant analysis. In J. Marques de S´a, L. A. Alexandre, W. Duch, and D. Mandic, editors, Artificial Neural Networks (ICANN 2007)—Proceedings of the Seventeenth International Conference, Porto, Portugal, Part II, pages 180–189. Springer-Verlag, Berlin, Germany, 2007. [11] G. Baudat and F. Anouar. Generalized discriminant analysis using a kernel approach. Neural Computation, 12(10):2385–2404, 2000. [12] D. Cai, X. He, and J. Han. Efficient kernel discriminant analysis via spectral regression. In Proceedings of the Seventh IEEE International Conference on Data Mining (ICDM 2007), pages 427–432, Omaha, NE, 2007. [13] S. Mika, G. R¨atsch, J. Weston, B. Sch¨olkopf, and K.-R. M¨uller. Fisher discriminant analysis with kernels. In Y.-H. Hu, J. Larsen, E. Wilson, and S. Douglas, editors, Neural Networks for Signal Processing IX—Proceedings of the 1999 IEEE Signal Processing Society Workshop, pages 41–48, 1999.

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[14] J. A. K. Suykens, T. Van Gestel, J. De Brabanter, B. De Moor, and J. Vandewalle. Least Squares Support Vector Machines . World Scientific Publishing, Singapore, 2002. [15] K.-R. M¨uller, S. Mika, G. R¨atsch, K. Tsuda, and B. Sch¨olkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181– 201, 2001. [16] S. Abe. Pattern Classification: Neuro-Fuzzy Methods and Their Comparison . Springer-Verlag, London, 2001.

INDEX A absolute zero, 543 absorbents, 6, 8, 22, 26, 27, 29, 31, 36, 66, 74, 77, 99, 135 absorption, 3, 4, 5, 6, 8, 9, 10, 22, 23, 24, 48, 66, 74, 93, 96, 100, 101, 103, 122, 123, 126, 135, 141, 142, 143, 144, 145, 146, 182, 217, 365, 375, 376, 410, 443, 444, 445, 446, 447, 452, 453, 454, 455, 457, 469 absorption coefficient, 365, 375, 443, 445, 452, 453, 454 accuracy, 20, 39, 45, 56, 87, 97, 98, 150, 280, 448, 463, 516 activation, 239, 240, 242, 243, 260, 264, 269, 274, 286, 476 activation energy, 239, 240, 242, 243, 260, 264, 269, 274, 286 actuators, 296, 298, 299, 333 additives, 22, 29, 74, 142 adducts, 500 adiabatic, x, 145, 248, 249, 250, 251, 254, 255, 265, 268, 276, 365, 455, 473, 484, 490 Ag, 461 agent, 429, 433, 480 aggregation, 312 aggregation process, 312 aid, 296, 426, 460 aircraft, 439 algorithm, ix, xii, 92, 150, 295, 296, 297, 298, 305, 316, 317, 318, 323, 327, 331, 332, 335, 346, 350, 351, 475, 532, 552, 577, 578, 582, 583, 594 alloys, 299, 461 alternative, 2, 6, 7, 8, 9, 11, 13, 15, 30, 31, 64, 88, 92, 93, 95, 96, 99, 103, 108, 111, 112, 115, 120, 122, 123, 124, 125, 126, 129, 131, 134, 135, 136, 138, 144, 473, 499, 502 alternative energy, 473 aluminium, 12, 22, 49, 73 aluminosilicates, 484 aluminum, 242, 263, 265, 268, 269, 273, 276, 277, 279, 456, 457, 458 ambient air, 16, 109, 110

amino acids, 217 ammonia, 138 ammonium, 499 amorphization, 498 amorphous, 496, 499 amorphous carbon, 496 amplitude, 47, 151, 167, 195, 197, 198, 199, 200, 201, 202, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 448 AMS, 292 analytical models, 195, 298 angular velocity, 444 anisotropy, 381, 390 annealing, 262, 264, 274, 275 anode, 218 antithesis, 164 application, vii, xi, 2, 8, 9, 22, 48, 140, 152, 159, 178, 179, 197, 209, 212, 213, 390, 441, 493, 494, 498, 499, 505, 543 aqueous solution, 23, 29, 33, 74, 226 aqueous solutions, 23, 29, 33, 74 Arrhenius law, 266 ARs, 480 ASEAN, 144, 146 ash, xi, 454, 455, 473, 475, 479, 480, 482, 486, 488, 489, 490 aspect ratio, xi, 509, 511 assessment, 375, 376 assignment, 327 assimilation, 106 assumptions, viii, 147, 170, 218, 242, 280, 419, 420, 421, 483 asymmetry, 205, 209, 277 asymptotic, 249, 250, 253, 324 asymptotically, ix, 295, 297, 320, 323, 325, 335, 336 Atlantic Ocean, 532 atmosphere, 1, 2, 11, 112, 114, 122, 158, 461, 474, 483, 490, 510 atmospheric pressure, 5, 364, 366, 430, 431, 510 atoms, 245, 246, 247, 252, 255, 256, 258, 260, 274, 275, 280, 287, 445, 494, 495, 496, 499, 502, 503, 504 Au substrate, 499 automata, 239

600

Index

B batteries, 3 battery, 299 Bayesian, xi, 509, 510, 512, 516, 517, 531, 532, 533 beams, xii, 464, 465, 468, 494, 498, 559, 563, 568, 569, 570, 571, 572, 573, 574 BED, 361 benchmark, 343, 346, 348, 578, 588, 589, 596 bending, 253, 559, 560 bias, 578, 581, 582, 583, 584, 585, 586 binding, 538, 540, 541, 543 binding energy, 538, 540, 541, 543 biomass, 3, 474 biosensors, 494 blood, 216, 588, 594, 595, 596 boilers, 7, 474 boiling, 469 Boltzmann constant, 538 bonding, 502 bonds, 494, 495, 504 Bose condensate, 545 bosons, 536, 539, 540 boundary conditions, x, xii, 45, 70, 74, 77, 79, 80, 81, 84, 95, 149, 161, 201, 219, 222, 252, 255, 348, 361, 369, 370, 373, 384, 390, 394, 398, 400, 406, 422, 424, 425, 426, 430, 432, 436, 547, 552, 555, 556, 557, 559, 560, 561, 562, 566, 569, 570, 574 Boussinesq, 218 branching, 476 breakdown, 450, 454, 455, 457, 469 bremsstrahlung, 445 bubbles, 143 buffer, 217, 218, 219, 220, 222, 224, 226, 227 buildings, 3, 144, 510, 511, 531, 532, 533

C calcium, 21, 23, 476, 484, 485 calcium carbonate, 476 calibration, 49 cancer, 589, 590, 592 candidates, 505 capacitance, 48 capacity, 7, 9, 11, 15, 18, 21, 24, 25, 26, 34, 55, 64, 77, 85, 87, 89, 92, 98, 110, 124, 125, 135, 239, 249, 276, 299, 346, 362, 481, 487, 502 capillary, 51, 197, 198, 199, 200, 201, 226, 230, 449 capsule, 159 carbide, 241, 243, 244, 245, 246, 247, 250, 253, 254, 255, 256, 257, 259, 261, 280, 286, 287 carbides, viii, 237, 238, 246, 260, 286, 287 carbon atoms, 247, 258, 494, 495, 496, 504 carbon dioxide, vii, 1 carbon film, 495, 499 carbon materials, xi, 493, 494, 499 carbon nanotubes, xi, 493, 502, 503, 504, 505, 506

carbonization, 479 carboxyl, 217, 499 Carnot, 18 casting, 169, 238 cathode, 218 cavities, 52, 61, 70, 137, 218, 234 C-C, 503, 504 CEC, 10, 11, 16, 17, 64, 74, 99, 103, 110, 111, 112, 113, 114, 115, 116, 117, 138 cell, viii, 147, 157, 204, 206, 207, 208, 216, 217, 218, 221, 222, 223, 224, 225, 226, 227, 231, 240, 241, 244, 245, 248, 249, 250, 253, 254, 257, 258, 286, 588, 589, 590, 594, 595, 596 cement, 479 centralized, 298, 333 CH4, 483, 484, 485 channels, 9, 10, 30, 31, 33, 37, 43, 47, 48, 52, 59, 60, 62, 63, 65, 71, 72, 73, 74, 75, 76, 93, 111, 112, 114, 124, 143 charge density, 226 charged particle, 469 chemical bonds, 476 chemical composition, 461 chemical energy, 476 chemical interaction, 246, 247 chemical kinetics, 475 chemical properties, 146, 499, 505 chemical reactions, x, 248, 473, 474, 475, 476, 479, 490, 499 chemical vapor deposition, 494 chemicals, 476 Chernobyl, 439 Chevron, 341, 345 chloride, 21, 22, 23 circulation, 77, 79, 82, 83, 84, 85, 86, 89, 98, 165, 188, 204 civil engineering, vii, 333, 531 civil structures, 297, 299, 327, 333 classes, xii, 577, 578, 579, 581, 586, 587, 588, 594, 595, 596 classical, xi, 5, 238, 239, 243, 263, 264, 388, 493, 500, 501, 502, 543, 545 classification, xii, 390, 476, 577, 578, 582, 583, 587, 588, 591, 592, 594, 595, 596 cleaning, 31 climate change, vii, 1, 142 closure, 495 clustering, ix, 295, 297, 307, 316, 317, 318, 319, 350 clusters, xi, 476, 493, 494, 497, 499, 500 CNS, 500, 501, 503, 504 CNTs, 493, 494, 496, 497, 498, 499, 504 CO2, 2, 483, 484, 485 coal, 2, 473, 474 coefficient of performance (COP), 18, 19, 20, 21, 35, 86, 87, 88, 89, 98, 109, 110, 137 coil, 169, 477 coke, x, 473, 477, 478, 479, 481, 489, 490 collaboration, 218 collisions, 445, 469, 495, 496, 497

601

Index combustion, xi, 238, 239, 240, 242, 243, 248, 249, 265, 275, 276, 285, 287, 473, 474, 484, 494 comfort zone, 103 community, vii, 1 compaction, 257 compensation, 7, 231, 297 competition, viii, 237, 267, 268, 277, 281, 287, 468 complexity, 174, 322, 522, 552 components, 5, 8, 9, 10, 22, 25, 36, 38, 71, 74, 77, 83, 85, 114, 135, 144, 150, 156, 169, 170, 177, 198, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 247, 262, 373, 447, 458, 465, 474, 476, 477, 482, 483, 484, 485, 490, 514, 540, 551, 556, 583 composites, 141, 494 composition, 103, 110, 111, 151, 217, 238, 240, 242, 243, 250, 256, 264, 274, 278, 461, 474, 475, 476, 482, 483, 484, 485, 489 compounds, viii, 237, 238, 240, 243, 246, 253, 260, 286, 287, 461, 475, 476 compressibility, x, 361, 408, 436 computation, 320, 333, 426 condensation, x, 47, 66, 69, 71, 469, 473, 477 conditioning, 5, 7, 8, 9, 11, 15, 119, 122, 123, 125, 126, 128, 131, 135, 136, 138, 140, 141, 143, 145 conductance, 48, 169 conduction, 25, 31, 49, 52, 54, 78, 169, 170, 218, 371, 381, 389, 390, 394, 395, 406, 437, 481 conductive, 366, 374, 381 conductivity, x, 73, 136, 159, 170, 219, 239, 361, 365, 374, 376, 378, 381, 389, 393, 394, 399, 424, 468, 481, 482, 486, 487 conductor, xi, 218, 535, 536, 537, 538, 539, 540, 541, 542, 545 confidence interval, 510, 526, 527 configuration, 15, 36, 111, 181, 197, 200, 204, 205, 206, 210, 243, 336, 341, 345, 531 conformity, 4, 5, 9, 10, 31, 55, 66, 74, 99 conservation, 80, 146, 372 constant rate, 168, 482 constraints, 323, 584, 586 construction, 6, 7, 8, 10, 12, 31, 32, 55, 59, 74, 79, 87, 93, 124, 135, 479 construction materials, 6, 74 consumption, 4, 5, 7, 8, 9, 10, 11, 59, 62, 63, 64, 74, 103, 126, 136, 143, 149, 238, 260, 272, 278, 281 contamination, 299, 498 convective, viii, x, 66, 147, 149, 152, 157, 158, 159, 164, 169, 170, 174, 177, 188, 201, 204, 206, 207, 208, 216, 220, 231, 361, 366, 369, 374, 436 convergence, 150, 475 convergence criteria, 475 conversion, 198, 238, 239, 254, 255, 262, 276, 278, 284, 285, 286, 331, 418, 433, 474, 475, 476, 481, 483, 489, 490, 499 convex, 62, 63, 323, 324, 327 cooling process, 21, 73, 74, 95, 100 Cooper pairs, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545 copper, 456, 461

correlation, 5, 8, 52, 54, 124, 156, 433, 484, 486, 500, 504, 511, 513, 515 correlation function, 515 correlations, 503, 505 corrosion, 7, 31, 262 corrosive, 22, 24, 29, 30, 74, 111, 135, 463 costs, 31, 139 Coulomb, xi, 300, 493, 500, 501, 536 coupling, 498 covalent, 247, 502 covalent bonding, 502 cracking, x, 473, 477 critical temperature, 535, 540, 543 critical value, 138, 482 cross-linking, 494, 496 cross-sectional, 59, 560 cross-validation, 589, 590, 592 crude oil, 473 crystal growth, viii, 147, 148, 150, 151, 152, 158, 160, 162, 164, 167, 168, 169, 170, 177, 178, 179, 180, 181, 182, 191, 194, 195, 201, 204, 210, 211, 216, 226, 228, 229, 230, 231, 232, 233, 234, 235 crystal lattice, 259, 468, 536, 542, 543 crystal structure, 151, 494 crystalline, 259, 497, 499 crystallization kinetics, 240, 241, 285 crystals, 148, 151, 152, 164, 172, 181, 191, 197, 216, 228, 233, 496, 497, 500 Cybernetics, 356, 357, 358, 359 cycles, 3, 8, 18, 21, 139, 142 cyclones, 510

D damping, xi, 198, 300, 303, 304, 337, 339, 340, 344, 345, 348, 509, 510, 511, 512, 513, 527, 528, 529, 531, 532 Darcy, 383, 385, 386, 402 data distribution, 594 data set, 251, 255, 261, 274, 275, 276, 277, 278, 279, 281, 282, 283, 284, 285, 316, 516, 578, 579, 588, 589, 590, 591, 593, 594, 595, 596 data structure, 316 database, x, 473, 475 decay, 2, 159, 496, 497, 504, 536, 538, 541, 545 decomposition, vii, x, 1, 473, 474, 476, 477, 478, 480, 481, 482, 487, 488, 489, 490, 498 decoupling, 350 defects, 498, 504, 505 deformation, 52, 195, 197, 201, 226, 227, 234, 252, 337, 496, 551 degradation, 476, 510, 593, 594 degrees of freedom, 498, 512 density values, 59 Department of Energy (DOE), 4 deposition, x, 473, 477, 489, 490, 494 derivatives, 218, 220, 372, 373, 496, 499, 505, 517, 556, 585

602

Index

desorption, 23, 24, 74, 135, 141 destruction, 459, 482, 487, 496, 497, 498, 499, 500, 502, 504, 505, 506 detection, 467, 510 deviation, 46, 47, 198, 242, 379, 543 dew, 5, 9, 10, 20, 74, 103, 116, 137 diamond, 499, 504 diaphragm, 458, 461, 462, 463 differential equations, 149, 304, 305, 337, 338, 342, 345, 559 diffusion process, 227 diffusion rates, 270 diffusion time, 278 diffusivity, 150, 219, 245, 248, 256, 274 dipole, 536 discordance, 223 discriminant analysis, xii, 577, 578, 597 dislocation, 149, 152, 191 dislocations, 181 disperse systems, ix, 361, 429 dispersion, 45, 220, 234, 463 displacement, 174, 221, 252, 255, 256, 258, 272, 277, 280, 286, 287, 300, 301, 302, 303, 304, 306, 337, 340, 342, 345, 346, 350, 495, 496, 500, 501, 502, 503, 504, 505, 561 distortions, 502 division, 114, 181, 362, 363, 364, 366, 367, 371, 424 DNA, 216 Doppler, 387 duration, xi, 79, 160, 218, 434, 442, 448, 456, 457, 458, 465, 469, 509, 511, 518, 519, 521, 531 dust, 87 dynamic systems, 314 dynamic viscosity, 136, 247, 365, 366 dynamical system, 510, 512, 532

E earthquake, 320, 333, 334, 335, 337, 340, 346, 350 ecological, 3, 4, 5, 7, 8 ecology, 4 ecosystems, 3 elaboration, 287, 480 elastic deformation, 252 electric charge, 464 electric current, 169, 218, 227 electric field, viii, 147, 217, 218, 220, 222, 227, 229, 463, 464, 468 electric potential, 218, 219 electric power, 2, 3, 4, 15, 34 electrical conductivity, 170 electrical power, 7 electrical resistance, 48 electricity, 15, 474 electrodes, 48, 49, 50, 54, 218, 219, 226, 461 electrolyte, 218 electromagnetic, 232, 468, 498 electromagnetic waves, 468

electromigration, 504 electromotive force, 542, 545 electron, xi, 445, 446, 447, 453, 454, 462, 463, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 542, 544, 545 electron beam, xi, 493, 494, 495, 496, 497, 498, 499, 500, 502, 503, 504, 505 electron charge, 544 electron density, 446, 447 electron microscopy, 497 electronic structure, 497, 502 electrons, xi, 444, 445, 468, 493, 494, 495, 496, 497, 498, 499, 500, 501, 504, 505, 535, 536, 537, 538, 540, 541, 542, 544, 545 electrophoresis, viii, 147, 148, 216, 217, 218, 219, 222, 226, 229, 234 EMF, 542, 543 emission, 363, 442, 444, 445, 498 emitters, 457 employment, 2, 447, 461, 462, 463 encapsulated, 191, 233 encapsulation, viii, 147, 191, 228 energy consumption, 5, 7, 8, 9, 10, 11, 62, 63, 64, 74, 126, 136, 238 energy transfer, 496, 500 environment, x, 3, 260, 285, 361, 502 environmental impact, 474 environmental protection, vii, 1 epitaxy, 499 epoxy, 49, 496 equality, 38, 49, 206, 584, 586 equating, 585 equilibrium, ix, 150, 156, 160, 162, 163, 182, 238, 240, 242, 244, 245, 250, 258, 260, 263, 266, 267, 268, 269, 270, 271, 272, 276, 280, 286, 287, 325, 445, 475, 483, 484, 485, 503, 536 equipment, vii, 3, 4, 7, 8, 23, 24, 25, 26, 34, 140, 143, 151, 216, 234, 448 erosion, x, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469 estimating, 243, 415, 437, 490 estimator, 298, 319, 320, 329, 330, 351, 510, 514, 515, 521, 522 ethanol, 461 evaporation, x, 5, 64, 66, 69, 73, 114, 136, 361, 419, 422, 423, 437, 445, 451, 455, 459, 462, 469 evolution, 183, 188, 268, 388, 458, 499, 504 excitation, xi, 335, 494, 496, 497, 498, 509, 510, 511, 512, 513, 515, 517, 521, 525, 526, 527, 531, 533 exclusion, 536 expansions, 563 expenditures, 8, 10, 26, 34, 35, 36, 103, 131, 136 experimental condition, 384 exposure, 444, 455, 456, 498 external influences, 228 extraction, 86, 474, 490, 494, 597 extrapolation, 247

603

Index extrusion, 238

F feedback, ix, 295, 296, 297, 298, 322, 323, 327, 329, 330, 334, 335, 336, 351 feeding, 9, 13, 64, 481, 487 FEM, 349 Fermi energy, 541 Fermi level, 541, 542 fermions, 536, 540 Feynman, 546 film, 6, 7, 9, 10, 30, 31, 39, 41, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 61, 66, 67, 68, 73, 74, 94, 111, 112, 114, 116, 138, 143, 164, 165, 200, 252, 258, 259, 264, 268, 270, 272, 280, 284, 285, 363, 396, 444, 499 film thickness, 31, 37, 41, 48, 52, 137, 258, 280 films, xi, 263, 493, 495, 496, 498, 499 filters, 442, 443, 444, 458, 459 filtration, ix, 55, 361, 362, 365, 367, 381, 382, 383, 384, 385, 389, 390, 391, 392, 393, 398, 404, 406, 429, 436 finite differences, 70, 77, 85, 552 finite element method, 559, 568 flame, 238, 239, 240, 463, 464, 467, 468 flame propagation, 240 flexibility, 182 flight, 148, 157, 162, 163, 167 floating, viii, 147, 148, 151, 157, 164, 167, 169, 170, 177, 179, 181, 182, 183, 195, 197, 201, 204, 210, 216, 226, 228, 229, 233, 234, 299 flooding, 41, 43, 55, 59 flow rate, 21, 26, 33, 34, 43, 44, 55, 66, 67, 68, 71, 77, 81, 86, 89, 92, 95, 98, 99, 103, 104, 124, 125, 135, 136, 365, 403, 404, 408, 412, 418, 419, 430, 479, 480, 483, 486, 490, 553, 555 fluctuations, 99, 213 fluidized bed, 143, 438 fluorinated, 499, 504 fluorination, 499, 504 fluorine, 504 fluorine atoms, 504 focused ion beam, 494 focusing, 148, 216, 217, 218, 219, 224, 225, 459 foils, 262, 263, 264, 285 food industry, 142 fossil, vii, 1, 2, 473 fossil fuel, vii, 1, 2, 473 Fourier, 36, 514 fractional composition, 486 fractionation, 148 fragmentation, x, 441, 473, 487, 497, 498, 499 free energy, 475 free volume, 481, 489, 490 freedom, 498, 512 freezing, 143 fresh water, 11, 16, 114

friction, 300, 351, 368, 548, 550, 552, 554, 555, 556 FTA, 89, 91 fuel, vii, ix, 1, 2, 4, 361, 367, 474, 475, 479, 480, 481, 482, 484, 486, 487, 488, 489, 490 fulfillment, 150 fullerene, 494, 495, 496, 498, 499, 500, 502, 503, 505 fullerenes, xi, 493, 494, 495, 496, 499, 500, 505 fusion, 240, 249 fuzzy logic, 296, 297, 298, 310, 315, 333 fuzzy sets, 296, 310, 311, 312, 313, 316, 321, 322

G GaAs, 152, 153, 154, 155, 156, 157, 158, 160, 161, 191, 199, 233 gallium, 161, 163, 165, 166 gas phase, 149, 477, 483, 487, 498 gases, 446 gasification, x, 473, 474, 475, 476, 483, 484, 485, 490 gasoline, 477, 482, 490 Gaussian, 297, 318, 341, 513, 517, 581, 583, 584 Gaza, 230, 234 gels, 6, 299 generalization, 46, 152, 170, 197, 228, 383, 385, 396, 402, 577, 578, 581, 586, 590, 596 generation, x, 4, 9, 92, 145, 418, 448, 456, 457, 473, 474, 494, 496, 505, 594, 597 generators, 458 genetic algorithms, 296, 297 geophysical, 557 germanium, 161, 165, 166 Gibbs, 265, 475, 476 Gibbs free energy, 475 glass, 12, 78, 90, 91, 92, 98, 136, 397, 442, 443, 448, 457, 458, 459 glasses, 262 global warming, vii, 1, 2, 3, 4, 7 grades, 322 grain, 285 grains, 241, 257, 258, 260, 263, 264, 265, 268, 280, 284, 285, 286, 287 graph, 587, 588, 594, 595 graphene sheet, xi, 493, 494, 499, 504 graphite, 244, 246, 254, 256, 260, 494, 495, 496, 499 Grashof, 151, 175, 179, 219 gravitation, 156 gravitational waves, 200 gravity, viii, 38, 41, 147, 151, 153, 155, 158, 159, 161, 175, 179, 197, 220, 228, 229, 230, 231, 233 greenhouse, vii, 1, 138 greenhouse gas, vii, 1, 138 greenhouse gases, vii, 1, 138 grouping, 316, 325, 515 groups, 48, 217, 316, 476, 499 growth mechanism, 254, 262, 278 growth rate, 161, 167, 240, 258, 267, 268, 272, 278

604

Index

guidelines, 320, 333, 351

H H1, 431 H2, 483, 484, 485, 490 Hamilton’s principle, 560, 574 Hamiltonian, 502 hanging, 33 hazards, 7, 296, 320, 333 health, xi, 509, 510, 511, 532 health status, 510 heat capacity, 21, 26, 77, 87, 89, 239, 249, 276, 481, 487 heat conductivity, 73, 365, 481, 482, 486, 487 Heat exchangers, 123 Heat Exchangers, 31 heat loss, 78, 89, 92, 98, 137, 285, 481 heat pumps, 4, 138, 139, 144 heat release, ix, x, 218, 219, 239, 240, 243, 248, 249, 250, 251, 254, 257, 260, 265, 276, 286, 361, 363, 365, 398, 408, 409, 411, 412, 416, 419, 423, 424, 429, 474, 536 heat removal, 18, 265, 278 heat transfer, 17, 26, 27, 28, 31, 33, 34, 36, 64, 71, 77, 137, 141, 218, 232, 239, 240, 264, 286, 287, 365, 367, 370, 381, 389, 390, 393, 396, 397, 398, 406, 415, 417, 436, 438, 480 heating rate, 241, 242, 260, 261, 262, 264, 268, 272, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 474, 476, 482 height, 30, 37, 47, 49, 54, 55, 59, 64, 68, 93, 112, 125, 137, 151, 160, 218, 222, 224, 225, 347, 445, 463, 511, 554, 555, 556 helium, 380, 443 hemoglobin, 226, 227 Hessian matrix, 517 heterogeneity, 448 heterogeneous, 239, 240, 241, 260, 263, 275, 279, 282, 283, 286, 287 high power density, 145, 456 high pressure, 257 high resolution, 494 high temperature, 24, 30, 238, 257, 262, 264, 280, 310, 473, 474, 476, 479, 490, 499, 540 high-speed, 240, 444, 456, 457 HIS, 597 Hm, 249, 254, 265 homogeneity, 228, 457 Hong Kong, 518 horizon, 87 host, 182 hot water, 3, 9, 12, 88, 92 humidity, 20, 21, 56, 74, 92, 99, 136, 145, 512, 531 hybrid, 146, 296, 307, 500 hydrides, 6 hydro, 474, 477, 484, 485, 494 hydrocarbon, 485

hydrocarbons, 474, 477, 484, 485, 494 hydrodynamic, viii, 43, 47, 49, 59, 62, 77, 147, 167, 168, 217, 222, 223, 226, 227, 229, 232, 389, 390, 391, 446, 447, 450, 457, 459, 460, 461, 462, 466, 468, 469 hydrodynamics, x, 143, 149, 223, 361, 367, 398 hydrogen, 145, 217, 474, 494 hysteresis loop, 301, 302, 303, 304, 305

I IAEA, 506 IEA, 4, 145 illumination, 457 images, 444, 458, 459, 504 implementation, x, 333, 334, 441, 442, 457, 463, 469 imports, 2 impurities, 149, 238, 496, 499 in situ, 493, 494 incidence, 92, 504 incompatibility, 176 incompressible, 149, 199, 252, 548, 549 indium, 160 induction, 159, 169, 178, 187, 188, 192, 211, 213 industrial, vii, 1, 2, 3, 5, 22, 144, 229, 429, 441, 442, 474 industrial application, 5, 474 industry, 2, 142, 169, 441 inequality, 79, 323, 324, 329 inert, 240 inertia, xii, 164, 197, 559, 560 infrastructure, 3 inhibitors, 22, 74 inhomogeneity, 153, 157, 160, 163, 169, 175, 177, 178, 179, 188, 189, 190, 192, 193, 194, 195, 264, 446 initial state, 221 inorganic, 146, 481, 482 InP, 151 instabilities, 226, 229 instability, viii, 46, 147, 167, 168, 181, 226, 227, 229, 232 insulation, 5, 8, 78, 89, 90, 91, 92 integration, ix, 254, 295, 298, 299, 327, 341, 350, 372, 400, 403, 422, 481, 482, 552 interaction effect, 296 interface, 66, 138, 152, 155, 157, 158, 159, 160, 163, 167, 168, 169, 170, 178, 182, 183, 185, 188, 189, 194, 232, 241, 242, 245, 246, 247, 250, 255, 256, 258, 260, 264, 265, 266, 268, 269, 272, 276, 368, 369, 372, 426 interfacial tension, 187, 188, 189 interference, 442, 443, 444, 458, 459 intermetallic compounds, 240, 266 intermetallics, viii, 237, 238, 241, 260, 267, 286, 287 International Energy Agency, 4 International Trade, 4 interphase, 364, 365, 387, 399, 425

605

Index interpretation, 10, 517, 548, 549 interstitial, viii, 237, 246, 255, 260, 286, 287 interstitials, 495 interval, 25, 28, 30, 224, 248, 267, 270, 448, 481, 487, 503, 521, 535, 584 intrinsic, 239, 241, 264, 286 ion beam, 498, 499 ionization, 445, 469, 494, 496 ions, 152, 157, 158, 163, 216, 217, 445, 481, 494, 496, 499, 536, 542 iris, 588, 594, 595 iron, 239, 484 irradiation, x, xi, 441, 447, 448, 450, 452, 454, 455, 457, 458, 459, 460, 461, 462, 468, 469, 493, 494, 495, 496, 497, 498, 499, 500, 502, 503, 504, 505 island, 3 isoelectric point, 217, 224, 225 isothermal, 242, 252, 256, 260, 267, 272, 375, 401, 402, 403, 436, 480 isotherms, 23, 24, 99, 103 isotropic, 501 iteration, 77, 220, 317, 426, 475

J Jacobian matrix, 319 Jc, 372, 373, 423 JOR, 143 Joule heating, 169

K Kalman filter, ix, 295, 323, 331, 532 kernel, vii, xii, 577, 578, 579, 580, 588, 589, 597 kerogen, x, 473, 474, 476, 477, 478, 487, 488, 489 kinematics, 136 kinetic constants, 479 kinetic energy, 497, 503, 560 kinetic equations, 481, 482 kinetic model, 240, 243, 276, 286 kinetic parameters, 476 kinetics, vii, viii, x, xi, 237, 239, 240, 241, 242, 264, 265, 267, 269, 270, 274, 284, 285, 287, 464, 465, 468, 473, 474, 475, 476, 477, 479, 490, 535, 536, 545 knots, 563, 564 Kyoto Protocol, vii, 1

L L1, 45, 51, 71, 81, 83, 85 L2, 41, 45, 81, 83, 84, 85, 428 Lagrange multipliers, 317, 585, 586 lamellar, 55, 262

lamina, 26, 31, 49, 50, 52, 66, 79, 83, 165, 170, 173, 174, 175, 179, 202, 203, 204, 205, 208, 210, 211, 212, 216, 390, 391, 392, 550 laminar, 26, 31, 49, 50, 52, 66, 79, 83, 165, 170, 173, 174, 175, 179, 202, 203, 204, 205, 208, 210, 211, 212, 216, 232, 390, 391, 392, 550 laminated, 262, 563, 575 Landau theory, 544 large-scale, 3, 86, 297, 298, 299, 327, 333, 334, 351, 469 laser, vii, x, 387, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 494 laser ablation, 461, 494 laser radiation, 442, 444, 446, 447, 451, 452, 454, 455, 456, 457, 458, 459, 460, 461, 463, 464, 465, 466, 468 lasers, 441, 442, 443 lattice, 256, 259, 280, 495, 497, 498, 500, 501, 502, 503, 504, 506, 536 law, xi, 48, 170, 201, 202, 242, 249, 252, 266, 267, 276, 322, 331, 338, 383, 535, 536 laws, x, 361, 545 lead, 149, 173, 216, 229, 280, 402, 449, 450, 463, 464, 465, 466, 467, 469, 490, 499, 543, 544, 545, 552 learning, 296, 297, 598 lens, 442, 444, 458 lenses, 458 limitation, 158, 543 limitations, 475, 483 linear dependence, 535, 554 linear function, 223, 313, 324 linear model, 296, 297 linear regression, 308 linguistic, 307, 318 linguistic information, 307 liquid film, 10, 30, 31, 36, 37, 43, 47, 48, 49, 51, 54, 55, 65, 66, 67, 137, 139, 140 liquid metals, 169, 239, 247, 273 liquid phase, x, 46, 167, 235, 265, 272, 273, 279, 280, 285, 441, 445, 446, 459, 460, 469 liquids, 31, 77, 232, 461 lithium, 21, 22, 29, 138, 142, 144 long period, 498 losses, 21, 26, 29, 31, 32, 64, 78, 82, 83, 89, 98, 114, 137, 285, 447, 454, 455, 469 low power, 9, 299, 456 low temperatures, 272, 281, 374, 474 low-temperature, 22, 144, 242, 248, 249, 478 luminescence, 450 Lyapunov, 297, 298, 320, 324, 325, 357

M M1, 579 Macao, xi, 509, 510, 511, 518, 519, 523, 531

606

Index

machines, 142, 577, 597 macromolecules, 217 magnet, 299 magnetic effect, 218 magnetic field, viii, 147, 148, 157, 159, 160, 164, 165, 166, 167, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 197, 210, 211, 212, 213, 214, 215, 216, 228, 230, 231, 232, 233, 299, 300, 469 maintenance, 30, 298 manifold, 476 mapping, 579 market, 2, 3, 5, 8 mass transfer, viii, x, 36, 66, 135, 137, 143, 147, 149, 164, 177, 197, 218, 228, 229, 232, 268, 278, 361, 374, 382, 436, 473, 474, 482 mass transfer process, 228 material sciences, 149, 228, 229 materials science, vii, 148 mathematics, 169 matrix, ix, 295, 297, 316, 317, 319, 320, 324, 325, 326, 327, 328, 329, 339, 340, 341, 344, 345, 348, 458, 475, 476, 512, 513, 514, 515, 516, 517, 521, 522, 559, 566, 574, 579, 580, 585, 589 Maxwell equations, viii, 147, 170 MB, 327, 329 measurement, 216, 456, 510, 513, 514, 531 measures, vii, 1, 2, 4, 447 mechanical properties, 253, 554 media, 4, 6, 7, 8, 11, 43, 99, 149, 216, 228, 234, 235, 369, 398, 457, 463 melting, 157, 181, 182, 191, 192, 193, 194, 195, 200, 231, 238, 239, 240, 241, 242, 243, 244, 245, 247, 249, 250, 252, 253, 254, 260, 261, 262, 263, 264, 265, 266, 267, 268, 274, 275, 277, 278, 280, 286, 287, 446, 455, 469 melting temperature, 239, 243, 253, 260, 261, 265, 267, 277, 280, 286 melts, 199, 260, 263, 267, 268, 270, 278, 280, 281, 285, 469 membranes, 498, 505, 506 memory, 239, 299 mercury, 535 metal carbides, 287 metal nanoparticles, 461 metals, x, 22, 169, 239, 247, 273, 276, 284, 441, 447, 449, 450, 455, 456, 457, 460, 461, 463, 464, 468, 469, 497, 543 meteorological, 517 methane, vii, 1, 2, 485, 500 microclimate, 6, 99 microelectronics, 151 microgravity, viii, 147, 148, 149, 151, 152, 158, 159, 160, 164, 167, 169, 195, 216, 228, 229, 231, 232, 233 microorganisms, 22 microscope, xi, 462, 463, 493, 494, 496, 504 microscopy, 463, 497

microstructure, 257, 504 mixing, 80, 150, 152, 165, 182, 216, 222, 480, 482 mobility, 217, 219, 223, 225 model reduction, 348 model system, 498 modeling, viii, ix, 8, 86, 90, 140, 143, 149, 232, 237, 238, 239, 240, 241, 243, 259, 262, 263, 264, 267, 269, 272, 273, 274, 275, 276, 278, 280, 285, 286, 300, 302, 310, 314, 319, 321, 369, 429, 476, 479, 548 modernization, 226 modulation, 455, 456 modulus, 252, 253, 531, 560 moisture, 5, 6, 8, 9, 10, 18, 19, 20, 21, 22, 23, 56, 66, 67, 93, 95, 97, 99, 103, 106, 108, 110, 111, 114, 116, 122, 131, 135, 136 moisture content, 5, 6, 8, 9, 10, 18, 19, 20, 21, 22, 23, 67, 93, 95, 97, 99, 103, 106, 108, 110, 111, 114, 116, 131, 135, 136 molar ratio, 243 molecular dynamics, xi, 493, 499, 500, 502, 503 molecular mass, 484 molecular structure, 476, 494 molecules, 217, 495, 496, 497, 498 momentum, 149, 170, 459, 460, 461, 541, 542 monograph, 149 monolayer, 494 monolayers, 498 morphology, 239 motion, 26, 82, 158, 159, 164, 165, 183, 188, 194, 197, 202, 220, 221, 222, 224, 298, 337, 338, 339, 341, 343, 345, 382, 384, 419, 459, 460, 479, 480, 481, 486, 487, 488, 489, 497, 500, 512, 522, 542, 560, 564 movement, vii, xi, xii, 6, 7, 8, 10, 20, 30, 31, 33, 38, 55, 59, 62, 65, 66, 67, 70, 77, 78, 84, 85, 86, 93, 98, 111, 114, 115, 122, 123, 447, 536, 540, 547, 548, 549, 550, 551, 557 MR fluids, 299, 301 multiphase flow, 143 multiplicity, 93, 125 multivariate, 533

N nanocomposites, 494 nanoelectronics, 499 nanolithography, 494 nanomaterials, 493, 498, 500 nanometer, 493, 494 nanoparticles, 461, 463 nanoscale structures, 498, 505 nanostructures, 493, 494, 495, 496, 497, 498, 499, 501, 503, 505, 507 nanotubes, xi, 493, 496, 499, 502, 503, 504, 505, 506 nanowires, 494 NASA, 195, 232, 233 Nash, 292

607

Index National Academy of Sciences, 147, 237 natural, xi, xii, 2, 7, 9, 13, 48, 77, 79, 82, 83, 84, 85, 86, 89, 98, 197, 198, 296, 320, 333, 473, 509, 510, 549, 559, 561, 568, 569, 571, 572, 573 natural gas, 2, 9 natural hazards, 296, 320, 333 Navier-Stokes, 81, 150, 170, 218, 220, 222, 226, 551 Navier-Stokes equation, 81, 218, 220, 222, 226, 551 Nd, 461, 512, 513 neglect, 157, 401, 403 neodymium, vii, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 453, 454, 456, 457, 458, 459, 460, 463, 465, 466, 467 neon, 443 neural network, 296, 297 neural networks, 296, 297 Newton, 338 Ni, viii, 237, 239, 240, 241, 247, 262, 263, 264, 265, 266, 267, 268, 269, 270, 272, 273, 274, 275, 276, 277, 278, 280, 281, 284, 285, 287, 316, 447, 448, 461 nickel, 240, 241, 262, 263, 264, 265, 267, 268, 273, 274, 275, 286, 451, 454, 461, 462, 463 NiTi shape memory, 239 nitrate, 138, 142 nitrides, 238, 246 nitrogen, 287, 499 nodes, 30, 348 noise, 2, 341, 345, 512, 513, 514 nonlinear, vii, ix, 91, 143, 180, 203, 207, 295, 296, 297, 298, 300, 301, 302, 304, 305, 306, 307, 310, 314, 315, 316, 319, 320, 321, 322, 323, 325, 326, 329, 333, 346, 350, 351, 382, 384, 385, 401, 403, 436, 510, 531, 532, 535, 557, 578 non-linear, 76, 249, 259 nonlinear dynamic systems, 321 nonlinear dynamics, 143 nonlinear systems, 296, 310 non-uniform, xii, 264, 285, 559, 560, 561, 562, 563, 564, 566, 568, 569, 570, 571, 572, 573, 574 normal distribution, 515 normalization, 512, 513, 514, 525 norms, 87, 93, 99 nuclear, ix, 361, 367 nuclear power, 367 nuclear power plant, 367 nucleation, 242, 260, 496 nuclei, xi, 242, 493, 495, 496, 497, 500, 501 nucleic acid, 216 nucleus, 259, 280 numerical analysis, 69 Nusselt, 137, 363, 481

O obligation, 1 observations, 62, 257, 497, 498, 499, 504 observed behavior, 277

offshore, 520 oil, vii, x, xi, 49, 473, 474, 475, 476, 477, 478, 479, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490 oil shale, vii, x, xi, 473, 474, 475, 476, 477, 478, 479, 481, 482, 483, 484, 485, 486, 487, 488, 490 operator, 97, 551 opposition, 188 optical, 98, 137, 366, 376, 442, 443, 450, 457, 458 optimization, 140, 204, 210, 211, 212, 215, 216, 229, 316, 317, 318, 323, 324, 517 orbit, 158, 167 ordinary differential equations, 512 organic, x, 473, 474, 475, 476, 481, 482, 483, 487, 488, 490 organic matter, x, 473, 474, 483, 490 organization, 4, 8, 13, 103, 131, 136 orientation, 89, 103, 189, 208, 229, 502 oscillations, 164, 167, 168, 172, 173, 175, 176, 178, 180, 195, 198, 202, 203, 204, 205, 206, 208, 210, 211, 212, 213, 214, 215, 233, 240, 545 oxidation, 456, 483 oxidative, 494 oxide, 164, 165, 475 oxides, 238, 484 oxygen, x, 473, 474, 475, 483, 484, 485, 490 ozone, 7, 146

P pacing, 49, 67 packaging, 102 parabolic, 42, 218, 220, 223, 242, 267, 276, 394, 395 partial differential equations, xii, 559, 563, 574, 575 particle density, 366 particle shape, 382 particle temperature, 434 partition, 316, 497, 521 passive, 3, 299, 333 pathways, viii, ix, 237, 238, 241, 261, 286 pattern recognition, 316, 577 Pb, 448, 543, 544 PDC, 297, 320, 322 Peclet number, 156, 157, 160, 167, 363 percolation, 240, 257 percolation theory, 257 perfect gas, 373, 419, 429 performance, ix, xii, 140, 141, 143, 144, 148, 295, 297, 298, 299, 301, 302, 303, 306, 310, 320, 323, 327, 329, 332, 333, 335, 336, 337, 346, 351, 577, 578, 588, 589, 590, 591, 593, 594, 596 periodic, 36, 41, 164, 524 permeability, 362 permittivity, 226 perturbation, 44, 228 perturbations, 46, 50, 229 PG, 43, 325 pH, 217, 220, 222, 224

608

Index

phase boundaries, ix, 182, 238, 256, 263, 266, 267, 269, 270, 272, 276, 287 phase diagram, ix, 238, 240, 242, 243, 244, 245, 250, 258, 263, 266, 267, 277, 280, 286 phenol, 217 phenolic, 476 phonons, 494, 496 phosphorus, 210 photocells, 442 photographs, 457, 459 photons, 445 physical and mechanical properties, 554 physical exercise, 153 physical properties, 161, 163, 181, 228 physics, vii, 145, 230, 235, 456, 493 pI, 421, 426 piezoelectric, 351 pipelines, 77, 80, 83, 138 pitch, 37, 112, 137, 202 PL, 38, 39 planar, 240, 242, 265, 268, 270, 286 Planck constant, 542 planning, 2, 59 plants, 4, 10, 74, 123, 126, 136, 141 plasma, 444, 445, 446, 447, 449, 453, 454, 455, 456, 457, 459, 460, 461, 463, 468, 469, 470 plastic, 55, 477, 479, 549, 557 plasticity, 252, 557 play, 2, 64, 204, 239, 447 Poisson, 252, 253 polarization, 48 pollutants, 2 pollution, 8, 149 polydispersity, x, 473, 480 polymer, 299, 476 polymerization, x, xi, 473, 477, 493, 498, 499, 505 polymers, 493 polynomial, 298, 300, 301, 302, 305, 306, 313, 314, 331, 352 polynomials, 302 pore, 256, 257 porosity, 169, 255, 256, 257, 258, 260, 365, 371, 382, 383, 436, 481, 482, 487, 488, 549 porous, 481 powder, 238, 262, 285, 286 power, vii, ix, 1, 2, 3, 4, 5, 7, 10, 15, 34, 47, 48, 103, 145, 228, 298, 299, 302, 333, 361, 363, 365, 367, 412, 415, 416, 419, 441, 442, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 458, 459, 463, 465, 466, 467, 468, 469, 474, 513 power generation, 145, 474 power plants, 367 power stations, ix, 3, 361 powers, 442, 443 PPA, 326 Prandtl, 137, 150, 219, 363 precipitation, 257, 260, 280, 284, 285, 531 preference, 59, 226 prices, 34

probability, 23, 465, 501, 503, 504, 515, 516, 526, 538 probability density function, 515 probe, 442, 443, 445, 446, 448, 457, 458, 460, 463, 499 production, 3, 11, 15, 30, 149, 232, 461, 463, 474, 494, 500, 538 productivity, 486 program, 4, 59, 85, 148, 229 propagation, x, 243, 260, 361, 376, 469 property, 26, 220, 461, 479 proportionality, 548, 551 propulsion, 231 protection, vii, 1 protective coating, 262 protein, 217 proteins, 216 prototype, 299, 340, 578 pseudo, 170 public, 2, 99, 146 pulse, 445, 446, 447, 448, 449, 456, 457, 458, 459, 460, 461, 463, 465, 467, 469 pulses, 441, 442, 448, 455, 457, 458, 465, 469 pumping, 4, 6, 9, 15, 412 pumps, 4, 7, 15, 88, 114, 122, 138, 139, 144, 145 pyrolysis, x, xi, 473, 474, 476, 477, 478, 479, 480, 482, 483, 486, 489, 490

Q QED, 553 quadratic curve, 584 quadratic programming, 584 qualitative concept, 242 quantum, xi, 445, 457, 499, 502, 535, 536, 540, 545, 546 quartz, 449 quasi-equilibrium, 244, 258, 259, 260, 262, 264, 265, 268, 276, 284, 285, 286, 287

R radial distance, 486 radial distribution, 445 radiation damage, 500 radical, 398 radius, 148, 149, 153, 169, 199, 201, 210, 216, 228, 245, 247, 250, 251, 255, 259, 261, 285, 327, 364, 390, 481, 482, 486, 501, 554, 555, 588 random, 198, 229, 317, 448, 515, 516 random matrices, 516 RAS, 1 raw materials, 474 Rayleigh, 151, 152, 153, 157, 559, 563, 568 RBF, 588 reactants, viii, 237, 240, 241, 242, 254, 255, 260, 262, 264, 265, 276, 285, 286, 287

609

Index reaction mechanism, 240, 262 reaction medium, 474 reaction order, 239 reaction rate, 239, 482, 496 reaction temperature, 474, 496 reactivity, 239 reality, 23, 310 reasoning, 311, 312, 313 recession, 447, 457 recognition, 316, 461, 577, 578, 591, 592, 594, 595, 597 recombination, 445, 500 reconstruction, 504 recovery, 531 recrystallization, 149 recurrence, 375 redistribution, 60, 62, 64, 464 reduction, 3, 4, 7, 20, 51, 55, 59, 62, 64, 110, 111, 148, 152, 153, 158, 181, 190, 194, 197, 198, 214, 216, 228, 297, 321, 348, 445, 463, 496, 526, 531, 538 reflection, 364, 375, 456 reflectivity, 376 refractory, viii, 237, 238, 240, 243, 253, 262, 286, 287 refrigeration, 1, 7, 9, 64, 124, 138, 140, 141, 144, 145 regeneration, 4, 5, 6, 7, 8, 9, 10, 12, 13, 24, 30, 88, 103, 106, 121, 122, 124, 135, 136, 141 regional, 2 regression, 308, 309, 597 regression equation, 309 regular, 10, 30, 37, 47, 49, 50, 51, 52, 53, 54, 56, 59, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 75, 112, 138, 139, 143, 271, 442, 500, 511 regulators, 351 rejection, 182 relationship, 11, 23, 25, 45, 46, 47, 49, 51, 53, 62, 67, 69, 79, 97, 99, 102, 111, 156, 167, 168, 179, 300, 302, 307, 308, 328, 370, 382, 442, 500, 526, 527, 584 relationships, 20, 70, 194, 223, 240, 271 reliability, xi, 23, 24, 25, 30, 31, 333, 358, 441, 475, 509, 510, 591 renewable energy, 2, 4, 139, 141 research, vii, xi, 49, 135, 139, 143, 218, 296, 297, 300, 305, 320, 333, 351, 448, 474, 493, 494, 535 researchers, 51, 343, 346, 494 reserves, 473 residential, 99, 143 residential buildings, 143 resin, 49, 496 resistance, x, 6, 7, 28, 30, 48, 61, 62, 82, 83, 104, 111, 123, 262, 264, 361, 382, 389, 396, 398, 401, 402, 403, 412, 415, 429, 437, 476, 477, 535, 536, 537, 538, 540, 542, 545 resistive, 504 resolution, 445, 448, 452, 453, 463, 494 resources, 2, 473

response time, 299, 511 Reynolds number, x, 31, 33, 137, 151, 169, 218, 361, 364, 387 rings, 479, 499 Ritz method, 559, 563, 568 robustness, 320, 594 rods, 494, 496, 559, 574 room temperature, 265, 273, 274, 504, 543 root-mean-square, 46, 523 roughness, 10, 30, 37, 47, 49, 50, 51, 52, 53, 54, 56, 59, 60, 61, 62, 64, 65, 70, 71, 73, 112, 137, 138, 139, 140, 143, 240, 552 rural areas, 2

S SAC, 346 safety, 2 salt deposits, 23 salts, 74, 218 sample, xii, 224, 234, 238, 239, 248, 250, 257, 265, 308, 462, 463, 577, 578, 582, 583, 587, 595, 596 sampling, 221, 225, 513, 521 saturation, 362, 364, 367, 419, 424 scalar, 220 scaling, 243 scanning tunneling microscope, 499 scatter, 455, 457, 579 scattering, xi, 376, 443, 444, 445, 446, 447, 452, 454, 455, 493, 500, 501, 504 scheduling, 297, 320 Schmidt number, 150, 160, 219, 223 SCs, 77, 85, 98 seals, 479, 480 search, 4, 52, 473 searching, 49 Seattle, 356, 359 sediments, 31 segregation, viii, 147, 148, 152, 158, 159, 160, 161, 164, 165, 166, 167, 168, 177, 179, 183, 187, 189, 190, 191, 194, 197, 204, 205, 206, 207, 210, 211, 212, 213, 214, 215, 228, 229 seismic, 342, 346, 348 self-organizing, 297 semiconductor, 152, 153, 154, 156, 157, 160, 163, 168, 202, 228 semiconductors, 158, 169, 226, 497 semimetals, 497 sensitivity, 3, 448, 457, 466, 512 sensors, 48, 298, 299, 333 separation, viii, x, 46, 62, 89, 103, 147, 148, 216, 217, 218, 220, 221, 222, 229, 234, 441, 461, 495, 524, 537 series, 15, 36, 139, 140, 216, 232, 444, 487 shape, xi, 30, 43, 217, 232, 280, 299, 303, 304, 382, 447, 448, 449, 452, 459, 504, 510, 512, 514, 525, 532, 560 shape memory alloys, 299

610

Index

shear, 165, 252, 253, 278 Shell, 142, 252 Sherwood number, 137 sign, 82, 182, 217 signals, 300, 305, 307, 315, 331, 334, 335, 346, 518 signs, 79 silica, 6, 433 silicate, 479 silicon, 3, 210, 231, 232, 484, 494, 496 similarity, 28, 33, 34, 156, 436 simulation, viii, x, 139, 141, 147, 149, 152, 164, 183, 191, 192, 193, 216, 228, 232, 234, 239, 264, 267, 268, 270, 272, 274, 275, 278, 280, 283, 361, 437, 499, 502, 503 simulations, ix, xi, 296, 493, 500 Singapore, 533, 598 single crystals, 151, 181 sintering, 248, 257, 258, 260 SiO2, 475 sites, 258, 503 skeleton, 365, 378, 381, 481, 482 Slovenia, 4 smoothing, vii, 1 software, x, 135, 473, 475 solar, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 22, 23, 24, 85, 87, 88, 89, 90, 91, 92, 93, 98, 122, 124, 126, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 589, 590, 592 solar collection, 136 solar collectors, 5, 7, 8, 9, 12, 88, 93, 135, 139 solar energy, 3, 4, 5, 7, 9, 15, 22, 23, 139, 141, 142, 143 solar system, 4, 6, 7, 8, 126, 135 solid phase, 240, 265, 273, 479, 480, 482, 486, 490, 496 solidification, 159, 232 solid-state, viii, 237, 239, 240, 241, 242, 243, 244, 245, 259, 261, 262, 264, 265, 266, 267, 268, 269, 272, 276, 278, 284, 285, 286, 287 solubility, 22, 24, 29, 30, 241, 245, 250, 263, 267, 272, 274, 277 solutions, viii, 4, 6, 8, 21, 24, 25, 26, 27, 28, 29, 33, 34, 36, 74, 77, 95, 122, 135, 142, 146, 151, 220, 221, 237, 315, 318, 324, 386, 387, 395, 402, 475 solvents, 217, 476, 499, 500 soot, 495, 496 sorbents, 4, 6, 8, 135 sorption, 5, 6, 139, 142 space station, 229 space-time, 448, 455 spatial, 218, 243, 446, 461, 464 species, 217, 246, 247, 499 specific heat, 136, 219, 265, 362, 363 specific surface, 364 spectrum, 153, 203, 208, 445, 453, 461, 524, 533, 536 speed, xi, 6, 15, 243, 494, 509, 511, 517, 518, 519, 520, 522, 523, 524, 525, 531, 533, 551, 552, 553, 554, 555, 556

SPF, 91 spheres, 364, 542, 550, 553, 554 spin, 239, 536 springs, 494 sputtering, 457, 495 stability, viii, xi, 6, 11, 30, 31, 43, 49, 50, 55, 61, 112, 124, 147, 150, 173, 174, 175, 178, 180, 181, 197, 201, 202, 204, 206, 207, 210, 216, 228, 231, 234, 297, 298, 320, 323, 324, 325, 326, 327, 328, 351, 493, 494, 495, 496, 500, 503, 505, 506 stabilization, 390 stages, 9, 21, 77, 216, 272, 301, 310, 477, 480 standard deviation, 530, 591 standard error, 427, 428 standards, 99 Standards, 288 State Department, 140 stationary distributions, 433 statistics, 536 steady state, 97, 367, 482 steel, 169, 461, 550, 554 sterile, 468, 469 stiffness, 301, 302, 303, 304, 306, 337, 339, 340, 344, 345, 510, 512, 526, 527 stochastic, 513, 521, 533 stoichiometry, 256, 273 storage, 142, 143, 144, 145, 442, 443, 458, 494 strain, 252, 560 strains, 499, 504 strategies, 142, 143, 298, 299 stratification, 89 streams, 116 strength, 217, 253, 262, 299, 305, 306, 463, 464, 536 stress, 41, 43, 81, 252, 253, 306 stroke, 213 structural characteristics, 241, 260 structural defects, 499 structural health monitoring, xi, 509, 511, 532 structure formation, ix, 238, 262, 283 structuring, 240 substances, 6, 21, 22, 23, 29, 30, 111, 135, 146, 239, 249, 457, 475, 476, 485 substitutes, 476 substitution, 153, 198, 249, 256, 332, 402, 499, 500 substrates, 498, 499 superalloys, 262 superconducting, xi, 535, 536, 542, 543, 545 superconductivity, xi, 535, 536, 540, 543 superconductors, 543 superimposition, 278 superposition, 385, 436, 542 supervisor, 333, 336, 351 supplements, 408 supply, 2, 3, 9, 12, 18, 48, 88, 92, 96, 99, 124, 131, 139, 140, 145, 220, 222, 223, 224, 246 suppression, 175 surface area, 144 surface diffusion, 245, 246, 252 surface layer, 198

611

Index surface roughness, 37, 50 surface tension, 41, 44, 70, 136, 164, 183, 185, 186, 187, 188, 194, 197, 201, 228, 232 surface wave, viii, 147, 197, 200, 204, 228, 229, 233, 234 suspense, 462, 463 suspensions, x, 441, 461 switching, 6, 211, 226, 227, 458 symbols, 428, 541, 588 symmetry, 150, 182, 204, 206, 228, 240, 242, 243, 245, 252, 258, 286, 287, 371 synthesis, vii, viii, x, xi, 237, 238, 243, 248, 261, 262, 283, 284, 286, 287, 306, 457, 473, 493, 494, 499, 500, 505

T tanks, 88 TAR, 477 targets, vii, x, 441, 442, 448, 451, 457, 463 TE, 238, 260, 261, 264, 265, 272, 274, 275, 285 technology, 149, 151, 169, 181, 197, 228, 229, 241 Tel Aviv, 125, 133 TEM, 504, 505 temperature dependence, 23, 250, 254, 269, 270, 273, 478 temperature gradient, 152, 168, 238, 389, 394, 477 temporal, 482, 522 tensile stress, 253 tension, 187, 188, 189, 548, 551, 556 territory, 3 test data, 547, 588, 591, 592, 593, 594, 595, 596 theory, xi, 30, 43, 50, 140, 144, 145, 234, 239, 242, 263, 296, 383, 535, 536, 545, 557 thermal activation, 499 Thermal Conductivity, 373, 374, 375, 378, 438 thermal decomposition, 473, 476, 481 thermal destruction, 481, 482, 487, 489 thermal energy, 144 thermal expansion, 151 thermal properties, 142 thermal resistance, x, 62, 64, 264, 361, 382, 389, 476 thermodynamic, 7, 18, 134, 182, 268, 474, 475, 485 thermodynamic calculations, 475 thermodynamic equilibrium, 182, 475 thermodynamics, 242 thin films, 258, 274, 275, 286 third order, 150 three-dimensional, 170, 210, 223, 476 threshold, 2, 50, 449, 455, 495, 497, 500, 501, 502, 503, 504, 505 thyroid, 591 time increment, 270 time resolution, 448, 453 tin, 22, 451 TiO2, 475

titanium, 241, 243, 244, 245, 246, 247, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 280, 286, 287, 456 toluene, 500 topological, 504 total energy, 2, 136, 442 traffic, 510 training, xii, 297, 577, 578, 579, 580, 581, 582, 583, 584, 586, 587, 588, 589, 590, 591, 592, 594, 595, 596, 597 trajectory, 460 transfer performance, 143 transformation, vii, x, 15, 50, 61, 305, 348, 473, 474, 488, 494, 496, 498, 499, 504, 506 transformations, 483, 551, 556 transition, viii, ix, 26, 49, 52, 59, 136, 159, 160, 161, 168, 170, 172, 175, 179, 180, 205, 206, 207, 214, 216, 228, 232, 237, 238, 241, 263, 264, 280, 281, 374, 418, 437, 540, 543 transition period, 160, 161 transitions, 287, 498 translation, 234, 470 transmission, 366, 375, 442, 443, 444, 445, 496, 504 transparent, 55, 78, 90, 92, 450, 451, 455, 463, 469 transpiration, 367 transport, 26, 30, 54, 140, 149, 152, 153, 158, 160, 161, 163, 170, 211, 376, 382, 469, 494, 536, 540, 545 transport processes, 153, 158, 170, 211 transportation, 111 trend, 8, 140, 387, 525, 531, 533 trial and error, 302, 320 turbulence, x, 50, 169, 205, 207, 232, 361, 364, 370, 436 turbulent, 26, 27, 31, 33, 43, 49, 52, 64, 66, 172, 174, 181, 205, 206, 208, 214, 216, 391, 392 turbulent flows, 206 two-dimensional, 49, 77, 149, 158, 175, 181, 210, 221, 494, 563, 578, 583 typhoon, xi, 509, 510, 511, 518, 519, 520, 521, 523, 524, 525, 526, 531, 532

U Ukraine, vii, 1, 2, 3, 4, 140 UNEP, 146 uniform, 55, 160, 186, 222, 228, 245, 461, 548, 559, 560, 561, 562, 563, 564, 566, 568, 569, 570, 571, 572, 573, 574 urban population, 510

V validation, 144, 589, 590 validity, 243, 286, 378, 545 van der Waals, 495, 496

612

Index

vapor, 143, 144, 362, 363, 365, 366, 367, 419, 420, 423, 445, 447, 450, 451, 461, 479, 480, 494 variables, 40, 150, 177, 218, 221, 233, 296, 303, 304, 305, 310, 312, 321, 326, 339, 537, 551, 585 variance, 579 variation, 97, 167, 225, 241, 267, 271, 272, 434, 451, 463, 464, 465, 466, 525, 563 vector, xii, 151, 153, 155, 158, 163, 175, 179, 226, 229, 308, 312, 313, 314, 315, 321, 324, 339, 340, 341, 344, 512, 513, 533, 542, 548, 577, 579, 583, 585, 586, 587, 597 ventilation, 5, 10, 87, 93, 99, 103, 119, 123, 125, 128, 130, 136, 138 versatility, 157 vibration, viii, xii, 147, 148, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 167, 195, 197, 198, 199, 200, 201, 203, 206, 226, 228, 229, 231, 233, 297, 320, 333, 351, 524, 532, 559, 574 vibrational modes, 497, 498 viscosity, 22, 26, 27, 136, 149, 181, 183, 217, 219, 247, 275, 366, 384, 430, 460, 461, 551, 555 vision, 50 visualization, 55 voids, 258 volatilization, 238 Volume of Fluid (VOF), 197, 234 vortex, xi, 509, 524, 551, 552, 553, 555, 556, 557

vortices, 524

W water evaporation, 437 water heater, 56 water vapour, 23, 66, 69, 135, 142 water-soluble, 499, 505 wave number, 46, 542 wave power, 3 wave propagation, 238, 284, 285 wave vector, 542 wavelengths, 444, 445 wavelet neural network, 296 welding, 262, 441, 498, 499, 505 wet-dry, 11, 41, 55, 64 wind, xi, 2, 3, 92, 297, 509, 510, 511, 517, 518, 519, 520, 522, 523, 524, 525, 526, 527, 531, 532, 533 wind turbines, 3 wires, 265, 333

Z zeolites, 6 zinc, 239, 442, 451, 448, 452, 453, 454, 456