{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","Extent":"http:\/\/purl.org\/dc\/terms\/extent","FileFormat":"http:\/\/purl.org\/dc\/elements\/1.1\/format","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Applied Science, Faculty of","@language":"en"},{"@value":"Materials Engineering, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Park, Joong Kil","@language":"en"}],"DateAvailable":[{"@value":"2009-10-01T19:38:23Z","@language":"en"}],"DateIssued":[{"@value":"2002","@language":"en"}],"Degree":[{"@value":"Doctor of Philosophy - PhD","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"Thermo-mechanical phenomena during continuous thin slab casting have been studied\r\nwith the objectives of understanding the mechanism of mold crack formation, and the\r\neffect of mold design upon the mechanical behavior of the stand. To achieve these goals,\r\nseveral finite element models have been developed in conjunction with a series of\r\nindustrial plant trials.\r\nFirst, an investigation of mold crack formation in thin slab casting was undertaken to\r\nelucidate the mechanism by which cracks develop and to evaluate possible solutions to\r\nthe problem. Three-dimensional finite-element thermal-stress models were developed to\r\npredict temperature, distortion, and residual stress in thin-slab casting molds, comparing\r\nfunnel-shaped to parallel molds. Mold wall temperatures were obtained from POSCO in\r\nKorea and analyzed to determine the corresponding heat-flux profiles in thin-slab molds.\r\nThis data was utilized in an elastic-visco-plastic analysis to investigate the deformation of\r\nthe molds in service for the two different mold shapes. The results of a metallurgical\r\ninvestigation of mold samples containing cracks were used together with the results of\r\nthe mathematical models, to determine mechanisms and to suggest solutions for the\r\nformation of mold cracks. Large cyclic inelastic strains were found in the funnel\r\ntransition region just below the meniscus, due to the slightly higher temperature at that\r\nlocation. The cracks appear to have propagated by thermal fatigue caused by major level\r\nfluctuations.\r\n\r\nNext, two-dimensional thermo-elastic-visco-plastic analysis was performed for a\r\nhorizontal slice of the solidifying strand, which moves vertically down the mold during\r\ncasting. The model calculates the temperature distributions, the stresses and the strains in\r\nthe solidifying shell, and the air gap between the casting mold and the solidifying strand.\r\nModel predictions were verified with an analytical solution and plant trials that were\r\ncarried out during billet casting at POSCO.\r\nThe validated model from the billet study was next applied to thin slab casting, using\r\nmold temperature and distortion data from the mold cracking study. An investigation of\r\nthe effect of mold taper on the shrinkage of the solidifying shell, its gap formation, and\r\nstress evolution was carried out for different thin slab mold geometries. The model\r\npredicts that the shell in funnel molds develops a tensile stress at the slab surface in the\r\nfunnel transition region due to funnel retraction. This model also suggests that as the\r\nfunnel depth increases, the possibility of surface cracks at the funnel outside bed position\r\nincreases.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/13475?expand=metadata","@language":"en"}],"Extent":[{"@value":"20805370 bytes","@language":"en"}],"FileFormat":[{"@value":"application\/pdf","@language":"en"}],"FullText":[{"@value":"THERMO-MECHANICAL PHENOMENA IN HIGH SPEED CONTINUOUS CASTING PROCESSES By JOONG KIL P A R K B.Eng. (Metals and Metallurgical Engineering) Seoul National University, Korea, 1986 M.A.Sc (Metals and Metallurgical Engineering) Seoul National University, Korea, 1988 A THESIS SUBMITTED IN PARTIAL F U L F U L M E N T OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Metals and Materials Engineering We accept this thesis as conforming To the required standard THE UNIVERSITY OF BRITISH C O L U M B I A May 2002 \u00a9 Joong kil Park, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M<?K^  \"ft ^ ^ W The University of British Columbia ^ Vancouver, Canada Date 2~ DE-6 (2\/88) A B S T R A C T Thermo-mechanical phenomena during continuous thin slab casting have been studied with the objectives of understanding the mechanism of mold crack formation, and the effect of mold design upon the mechanical behavior of the stand. To achieve these goals, several finite element models have been developed in conjunction with a series of industrial plant trials. First, an investigation of mold crack formation in thin slab casting was undertaken to elucidate the mechanism by which cracks develop and to evaluate possible solutions to the problem. Three-dimensional finite-element thermal-stress models were developed to predict temperature, distortion, and residual stress in thin-slab casting molds, comparing funnel-shaped to parallel molds. Mold wall temperatures were obtained from POSCO in Korea and analyzed to determine the corresponding heat-flux profiles in thin-slab molds. This data was utilized in an elastic-visco-plastic analysis to investigate the deformation of the molds in service for the two different mold shapes. The results of a metallurgical investigation of mold samples containing cracks were used together with the results of the mathematical models, to determine mechanisms and to suggest solutions for the formation of mold cracks. Large cyclic inelastic strains were found in the funnel transition region just below the meniscus, due to the slightly higher temperature at that location. The cracks appear to have propagated by thermal fatigue caused by major level fluctuations. ii Next, two-dimensional thermo-elastic-visco-plastic analysis was performed for a horizontal slice of the solidifying strand, which moves vertically down the mold during casting. The model calculates the temperature distributions, the stresses and the strains in the solidifying shell, and the air gap between the casting mold and the solidifying strand. Model predictions were verified with an analytical solution and plant trials that were carried out during billet casting at POSCO. The validated model from the billet study was next applied to thin slab casting, using mold temperature and distortion data from the mold cracking study. An investigation of the effect of mold taper on the shrinkage of the solidifying shell, its gap formation, and stress evolution was carried out for different thin slab mold geometries. The model predicts that the shell in funnel molds develops a tensile stress at the slab surface in the funnel transition region due to funnel retraction. This model also suggests that as the funnel depth increases, the possibility of surface cracks at the funnel outside bed position increases. iii T A B L E O F C O N T E N T S ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES ix LIST OF FIGURES xi LIST OF SYMBOLS xxii ACKNOWLEDGEMENTS xxvi 1 INTRODUCTION 1 2 LITERATURE REVIEW 6 2.1 Thin slab casting 6 2.1.1 General overview of the thin slab casting process 6 2.1.2 Heat transfer in thin slab casting mold 12 2.1.3 Thermal stress modeling of a slab mold 17 2.1.4 Mold crack formation in continuous casting 21 2.1.5 Thermal stress models of conventional slab casting 22 2.1.6 Thermal stress models of thin slab casting 26 2.2 Billet casting 27 2.2.1 Heat transfer and air gap formation in the mold 28 2.2.1 Longitudinal cracks 29 2.2.2 Mathematical stress models 32 3 SCOPE AND OBJECTIVE OF THE PRESENT WORK 34 3.1 Mold crack formation in thin slab casting 35 3.1.1 Objective 35 iv 3.1.2 Methodology 36 3.2 Mold taper in thin slab casting 38 3.2.1 Objective 38 3.2.2 Methodology 38 4 T H E R M O - M E C H A N I C A L BEHAVIOR OF THE THIN SLAB M O L D 42 4.1 Introduction 42 4.2 Plant trials 43 4.2.1 Mold temperature measurement 43 4.2.2 Other measurements 46 4.2.3 Measured mold temperature profdes 48 4.2.4 Discussion 54 4.3 Thermal-stress computational description 54 4.3.1 Heat flow model 55 4.3.2 Stress model 61 4.4 Heat transfer of parallel mold 64 4.4.1 Heat flux profiles 64 4.4.2 Hot face temperature profiles 65 4.4.3 Validation 65 4.5 Thermal and mechanical behavior of parallel mold 70 4.5.1 Results 70 4.5.2 Validation 74 4.6. Comparison of parallel and funnel mold 78 5 M E C H A N I S M OF M O L D C R A C K FORMATION 85 5.1 Introduction 85 v 5.2 Morphology of mold cracks 86 5.3 Thermal stress model description 94 5.4 Comparison of funnel and parallel mold thermal-mechanical behavior 103 5.5 Funnel mold analysis using temperature and crack measurements 112 5.5.1 Measured temperature profiles 112 5.5.2 Simulations of horizontal profiles 112 5.5.3 Simulations of vertical profiles 115 5.6. Metal level fluctuation study 119 5.6.1 Metal level and meniscus temperature fluctuation measurement 119 5.6.2 Fluctuating temperature simulation 121 5.6.3 Stress-strain fluctuation simulations 124 5.7 Fatigue crack prediction 127 5.7.1 Failure criteria 127 5.7.2 Lifetime prediction 130 5.8 Mechanism of mold crack formation 133 6 SOLID SHELL BEHAVIOR IN THIN SLAB MOLD 136 6.1 Introduction 136 6.2 Geometry and additional length of solidifying shell of funnel slab 138 6.3 Mathematical model description 142 6.3.1 Microsegregation analysis 145 6.3.2 Heat flow analysis 145 6.3.3 Stress analysis 148 6.3.4 Specification of strand domain 157 6.4 Validation of model 163 vi 6.5. Results and discussion.... 163 6.5.1 Temperature, shell and shrinkage behavior 165 6.5.2 Effect of mold taper on shrinkage and air gap in parallel slab casting 170 6.5.3 Effect of funnel on deformation and stress 177 7. CONCLUSION 199 REFERENCE 204 APPENDIX I M O L D T H E R M A L RESPONSE OF THERMOCOUPLE 218 APPENDIX 1.1 Parallel mold 219 APPENDIX 1.2 Funnel mold 224 APPENDIX 1.3 Billet mold 227 APPENDIX II HEAT TRANSFER COEFFICIENT AT THE M O L D WALL\/COOLING W A T E R INTERFACE 229 APPENDiX III SOLID SHELL BEHAVIOR IN BILLET CASTING 232 A3.1 Introduction 232 A3.2 Plant trials 233 A3.2.1 Caster details and nominal operating practice 233 A3.2.2 Mold temperature measurement 234 A3.2.3 Solid shell measurement 234 A3.3 Mathematical model description 234 A3.3.1 Heat flow analysis 237 A3.3.2 Stress analysis 241 A3.3.3 Crack criterion 243 A3.3.4 Strand and mold domain 245 A3.4 Model validation with analytical solution 245 vii A3.5 Model validation with plant trial 248 A3.5.1 Temperature 248 A3.5.2 Heat balance 253 A3.5.3 Solid shell thickness 255 A3.5.4 Bulging below the mold 255 A3.5.5 Stress and crack prediction 262 A3.6 Effect of mold corner radius 265 A3.6.1 Heat transfer 265 A3.6.2 Longitudinal corner surface cracks 270 A3.6.3 Longitudinal off-corner subsurface cracks 275 A3.7 Effect of casting with mold flux 275 A3.8 Mechanism of surface crack formation 279 APPENDIX IV C A L C U L A T I O N OF AIR GAP IN BILLET CASTING 281 viii L I S T O F T A B L E S Table 4.1 Conditions for plant trial measurements of mold temperature, etc 44 Table 4.2 Simulation conditions 56 Table 5.1 Simulation conditions for 2-D slice modelling 95 Table 5.2 The maximum displacement (in mm) at the tension bolt of outer backing plate after 50 sequences of casting (3-D quarter model) 100 Table 5.3 Summary of predicted fatigue life with different thermal cycles 132 Table 5.4 Summary of predicted fatigue life with different hot face temperature 132 Table 6.1 The geometry of funnel slab in this work 144 Table 6.2 Steel composition for plant trial 144 Table 6.3 Specific volume of 5-Fe, y-Fe and liquid steel [107] 155 Table 6.4 Parameters for the constitutive equation [29] 155 Table 6.5 Simulation conditions 162 Table A2.1 Testing of conditions to establish the applicability of the correlation (Eq. A2.1) for obtaining the heat transfer coefficient 231 Table A3.1 Casting conditions in the plant trial 235 Table A3.2 Mold conditions in the plant trial 235 Table A3.3 Conductivity of gap medium (air) with temperature 238 Table A3.4 Temperature dependence of heat transfer coefficient between the mold flux and strand surface 242 ix Table A3.5 Simulation condition for plant trial 247 Table A3.6 Simulation conditions for analytical solution test [139] 249 X L I S T O F F I G U R E S Fig. 1.1 Schematic diagram of tundish and mold of continuous casting of steel with powder lubrication 3 Fig. 1.2 Growth of thin slab casters installed in past decade [1] 3 Fig.2.1 Relation between slab thickness, casting speed and productivity . (lstrand, 1500mm width, operating ratio 95%) [2] 7 Fig.2.2 The SMS concast taper mold for thin slab casting [2] 8 Fig.2.3 Comparison of lay-out between the CSP and ISP caster [3] 10 Fig.2.4 Integral mold heat flux versus casting speed for two cases of thin slab casting with two different mold powder in comparison with conventional slab casting [15] 14 Fig.2.5 Total powder film thickness, liquid and sintered powder film thickness and heat flux as a function of casting speed for 50mm thick slab [16] 14 Fig.2.6 Longitudinal profiles of the temperature in the broad mold plate of a funnel-shaped mold at a distance of 5mm form the hot face [15] 15 Fig.2.7 Transverse profiles of the temperature in a broad mold plate of a funnel-shaped mold at a distance of 5mm from the hot face at (a) 245mm and (b) 755mm from the top of mold [4] 15 Fig.2.8 Transverse profiles of the temperature in the narrow mold plate of a funnel-shaped mold at a distance of 5mm from the hot face [4] 16 Fig.2.9 Temperature contours on inside face of 1 20mm thick mold [20] 18 xi Fig.2.10 Computed variation of mold heat flux with position for the 20mm thick mold [20] 18 Fig.2.11 Heat flux into the mold and the steel speed at the SEN port in the funnel-shaped mold and the rectangular mold [16] 19 Fig.2.12 Crack initiation in the vicinity of brass inclusion and crack propagation due to thermal fatigue [30] 23 Fig.2.13 Appearance of longitudinal corner crack (a) [47] and microstructure of off-corner internal cracks (b) [56] in billet casting 30 Fig.4.1 Back view of wide face of parallel mold showing slot geometry and thermocouple locations in bolt holes 45 Fig.4.2 The profiles of metal level and casting speed (a) Parallel mold (b) Funnel mold 47 Fig.4.3 Mold thermal response recorded at different mold locations around the mold perimeter at 176mm below the top of the parallel mold (typical sequence) 49 Fig.4.4 Mold thermal response recorded at various axial locations down the centerline of parallel mold during steady state 51 Fig.4.5 Time-averaged temperature profile measured along the mold length (Parallel mold) 52 Fig.4.6 Mold temperature profile measured at different positions across the wide face (Parallel mold) 53 Fig.4.7 2-D horizontal section through wide face showing (a) top view of segment model domain, (b) corresponding 3-D section mesh and (c) boundary conditions on vertical section 58 Fig.4.8 Top view of 3-D quarter mold model showing boundary conditions 59 xii Fig.4.9 Stress-strain curve for copper (Cr-Zr alloy) used in this model [23] 60 Fig.4.10 Heat flux profiles of parallel mold with different mold positions (a) Loose face (b) Fixed face 62 Fig.4.11 Hot face temperature distribution down the length of the parallel mold (a) Loose face (b) Fixed face 66 Fig.4.12 Comparison of measured and calculated temperature in the parallel mold (a) Loose face (b) Fixed face 68 Fig.4.13 Comparison of average heat flux calculated from heat flux data and from energy balance on cooling water 69 Fig.4.14 Temperature contours on distorted mold shape during operation for the parallel mold 71 Fig.4.15 End view of distorted mold along the wide face centerline showing the gap between copper plate and water jacket and temperature profiles 72 Fig.4.16 Predicted evolution of thermal distortion on vertical section for the parallel mold (centerline surface) 73 Fig.4.17 Distortion of wide and narrow face along the line where they meet in the parallel mold 75 Fig.4.18 The behavior of back plate distortion for parallel mold during operation (relative to zero along centerline through narrow face) 76 Fig.4.19 Comparison of calculated and measured total mold width contraction during typical casting campaign for the funnel mold 77 Fig.4.20 Comparison of thermo-mechanical behavior of parallel and funnel mold during operation Hot face temperature profile (b) Distortion along different vertical sections 79 xiii Fig.4.21 Mold distortion of parallel and funnel molds across the wide face at 175mm below mold top during operation 81 Fig.4.22 The behavior of back plate distortion during operation according to the mold shape 82 Fig.4.23 Predicted distortion profiles down the narrow face of parallel mold 84 Fig.5.1 Schematic diagram of the funnel mold showing location of mold cracks 87 Fig.5.2 Photograph of (a) mold crack and (b) magnified view 88 Fig. 5.3 Optical micrograph of the cross section of mold crack 90 Fig.5.4 Fractographs of mold cracks just below the mold surface showing the intergranular fracture and wedge cracks at the triple point of grain boundary 91 Fig. 5.5 Fractographs of mold cracks near crack root showing the striation structure .' 92 Fig.5.6 Optical micrograph of (a) mold crack around the top of crack and (b) X-ray spectrum 93 Fig.5.7 Finite element mesh of 2-D horizontal domain for parallel and funnel mold and its thermal and mechanical boundary conditions (a) Parallel mold and stress boundary condition (b) Funnel mold and heat boundary condition 96 Fig. 5.8 Thermal pattern of hot face imposed on the 3-D stress analysis 98 Fig.5.9 The change of displacement at location of 376mm from the center of backing plate with casting sequence at the meniscus of parallel mold 99 Fig. 5.10 Vertical displacement across the mold width at the interface of mold and back plate during the operation for the parallel mold 102 xiv Fig. 5.11 Comparison of heat flux profiles at the meniscus region 104 Fig.5.12 Comparison of hot-face temperature distribution across the wide face with different mold shape (2-D quarter model)'. 105 Fig.5.13 Slot depth profile across the mold width at the meniscus for the funnel mold 106 Fig.5.14 Comparison of Von Mises stress across the hot face with different mold shapes 108 Fig.5.15 Comparison of inelastic strain distribution across the hot face with different mold shapes 109 Fig.5.16 Stress-strain hysteresis loop on the hot face of the parallel mold 110 Fig.5.17 Comparison of stress-strain hysteresis loop after 50 sequences 111 Fig.5.18 Temperature response measured in funnel mold at the meniscus during steady state (100mm below mold top) 113 Fig.5.19 Comparison of calculated and measured mold temperatures and corresponding hot face temperature and heat flux profiles across wide face at the meniscus of funnel mold (2-D quarter model) 114 Fig.5.20 Stress (axx) profile across wide face of funnel mold 116 Fig.5.21 Inelastic strain profile across wide face of funnel mold .....117 Fig.5.22 Effect of heat-flux profile uncertainty on hot face temperature predictions (steady 3-D segment model) 118 Fig.5.23 Power spectra of mold level and temperature response (a) Comparison of metal level between parallel and funnel mold (b) Comparison of metal level and temperature response in the funnel mold 120 xv Fig.5.24 Thermal pattern of funnel mold at the meniscus (a) Measured temperature profile at the thermocouple position (b) Calculated hot face and thermocouple temperature 122 Fig.5.25 Transient hot face temperature response for the various cases of thermal cycle 123 Fig.5.26 Effect of metal level position on the hot face temperature prediction (3-D segment model) 125 Fig.5.27 Stress-strain hysteresis loop at the meniscus hot face surface for the funnel mold with different frequencies of metal level fluctuation 126 Fig.5.28 Inelastic strain profiles with casting time 128 Fig.5.29 Variation of cycle to failure for different fatigue models 131 Fig.5.30 Comparison of stress-strain hysteresis loop with different hot face temperature for the funnel mold 134 Fig.6.1 Longitudinal crack and the associated breakout slab found in thin slab casting 137 Fig.6.2 Change of funnel depth down the mold length (a) Funnel A (b) Funnel B 140 Fig.6.3 Schematic diagram of funnel slab 141 Fig.6.4 Funnel radius and additional length of shell according to the funnel depth 143 Fig.6.5 Calculated solid fraction fs, 5-Fe fraction, y-Fe fraction, and thermal linear expansion as a function of temperature for low carbon (C=0.04wt%) steel 146 Fig.6.6 Enthalpy and conductivity of low carbon steel (C%=0.04) used in this model 147 Fig.6.7 Input and output heat flux profiles down the mold length 149 xvi Fig.6.8 Assumed variation of heat flux profiles across slab surface 150 Fig.6.9 Shape of wide face mold down the mold length, due to mold distortion 152 Fig.6.10 Narrow face taper including mold distortion down the mold length 153 Fig.6.11 Finite element mesh for thin slab strand and mold (a) Parallel (b) Funnel A (c) Funnel B 158 Fig.6.12 Model domains and boundary conditions for thin slab studied in this work (a) Parallel mold and its dimension (b) Funnel A and heat B.C. (c) Funnel B and stress B.C 161 Fig.6.13 Change of additional length of solidifying shell in funnel A (a) and funnel B (b) 164 Fig.6.14 Axial temperature profiles at different locations in parallel slab 166 Fig.6.15 Predicted shell thickness as a function of distance below meniscus in parallel mold 168 Fig.6.16 Profiles of shrinkage down the mold length in parallel slab 169 Fig.6.17 Axial surface temperature profiles down the mold length at slab center... 171 Fig.6.18 Comparison of transverse temperature profiles of slab surface across the wide face with different thin slabs 172 Fig.6.19 Comparison of shrinkage behaviors at the slab corner with different thin slabs.... 173 Fig.6.20 Effect of narrow face taper on shell shrinkage and gap formation in the parallel slab (a) No taper (b) 0.8%\/m taper (c) 1.1%\/m taper 174 Fig.6.21 Measured profile of mold slag thickness at mold exit 176 xvii Fig.6.22 Profile of mold taper down the mold length and the corresponding shrinkage and gap formation in the parallel slab 178 Fig.6.23 Effect of narrow face taper on shell shrinkage and gap formation in the funnel A slab (a) No taper (b) 0.8%\/m taper (c) 1.1%\/m taper 179 Fig.6 24 Effect of narrow face taper on shell shrinkage and gap formation in the funnel B slab (a) No taper (b) 0.6%\/m taper (c) 0.8%\/m taper 180 Fig.6.25 Transverse stress profiles along the funnel A slab surface with different casting times (No thermal stress) 182 Fig.6.26 Evolution of transverse stress through the slab thickness with different locations of funnel A (a) Funnel parallel (b) Funnel inside bend (c) Funnel outside bend 183 Fig.6.27 Transverse stress profiles along the funnel B slab surface with different casting times (No thermal stress) 184 Fig.6.28 Evolution of transverse stress through slab thickness with different locations of funnel B (a) Funnel inside bend (b) Funnel outside bend (c) straight region 185 Fig.6.29 Transverse stress profiles along the parallel slab surface with different casting times (1.1%\/m taper) 186 Fig.6.30 Evolution of temperature (a) and stress distribution (b) through slab thickness for parallel slab (1.1% taper) 187 Fig.6.31 Transverse stress profiles along the funnel A slab surface with different casting times (1.1%\/m taper) 189 Fig.6.32 Evolution of temperature and stress distribution through slab thickness for funnel A slab (1.1% taper) (a) Temperature (b) 100mm from the slab center (c) 200mm from the slab center (d) 350mm from the slab center 191 xviii Fig.6.33 Transverse stress profiles along the funnel B slab surface with different casting times (0.6%\/m taper) 192 Fig.6.34 Evolution of temperature and stress distribution through slab thickness for funnel B slab (0.6% taper) (a) Temperature (b) 100mm from the slab center (c) 350mm from the slab center (d) 520mm from the slab center 194 Fig.6.35 Contours of maximum principle stress for parallel slab (a) 5 sec. (b) lOsec. (d) 15 sec 196 Fig.6.36 Contours of maximum principle stress around 290mm from the slab center for funnel A slab (a) 5 sec. (b) lOsec. (d) 15 sec 197 Fig.6.37 Contours of maximum principle stress around 237.5mm from the slab center for funnel B slab (a) 5 sec. (b) lOsec. (d) 15 sec 198 Fig. A l . 1 Casting condition for the plant trial (parallel mold) 220 Fig.A1.2 Mold thermal response of thermocouple located along the center of parallel mold (Number in parenthesis : distance from the top of mold).... 221 Fig. A 1.3 Mold thermal response of thermocouple located across the mold width at 175mm below the parallel mold top (Number in parenthesis : distance from the enter of mold) 222 Fig.A1.4 Mold thermal response of thermocouple located across the mold width at 305mm below the parallel mold top (Number in parenthesis : distance from the center of mold) 223 Fig.Al .5 Casting condition for the plant trial (funnel mold) 225 Fig. A 1.6 Mold thermal response of thermocouple located across the mold width at 100mm below the parallel mold top (Number in parenthesis : distance from the center of mold) 226 xix Fig. A 1.7 Mold thermal response of thermocouple at different locations during steady state of billet casting 228 Fig. A3.1 Photograph of thermocouple instrumented mold tube 236 Fig.A3.2 Heat transfer coefficient across the strand\/mold for different air gap sizes and surface temperatures 240 Fig. A3.3 Profdes of mold distortion and taper used in this model 244 Fig.A3.4 Finite element mesh of 2-D horizontal quarter domain for billet stand and mold and its boundary condition 246 Fig.A3.5 Comparison of numerical and analytical solution (a) temperature (b) stress250 Fig. A3.6 Comparison of stress profde with analytical solution 251 Fig. A3.7 Comparison of measured and calculated mold temperature 252 Fig.A3.8 Heat flux profiles down the mold for different casting conditions and models 254 Fig.A3.9 Comparison of calculated and measured solid shell thickness (C%=0.04, 285mm below the meniscus, Vc=2.2m\/min.) 256 Fig.A3.10 Comparison of measured and calculated solid shell thickness with casting time '. 257 Fig.A3.11 Evolution of surface temperature profiles at different billet positions for 4mm corner radius (a) Center (b) Off corner (c) Corner 259 Fig.A3.12 Surface temperature profiles along 4mm corner radius billet at different times 260 Fig.A3.13 Evolution of billet surface displacement showing bulging below mold exit with different corner radii 261 XX Fig.A3.14 Temperature and transverse stress profile through the shell thickness at 19s of casting time for 4mm corner radius mold 263 Fig.A3.15 Comparison of crack location and model calculations at 100mm below the mold exit (a) microstructure of off-corner crack (b) hoop stress distribution and (c) equivalent plastic strain contours 264 Fig.A3.16 Variation of shell profiles and temperature contours in the corner region (a) 4mm corner radius (b) 15mm corner radius 266 Fig. A3.17 Evolution of air gap size profiles with different comer radii 269 Fig. A3.18 Contours of hoop stress and hoop plastic strain at 8s of casting time for oil casting (a) 4mm corner radius (b) 15mm corner radius 271 Fig.A3.19 Evolution of hoop plastic strain at 1mm below the billet corner surface for different casting conditions 272 Fig.A3.20 Hoop stress and hoop plastic strain contours for 4mm corner radius (oil casting) (a) mold exit (b) 100mm below the mold 273 Fig.A3.21 Hoop stress and hoop plastic strain contour for 15mm corner radius (oil casting) (a) mold exit (b) 100mm below the mold 274 Fig.A3.22 Comparison of solid shell contours at mold exit with different lubricant and corner radius (a) 4mm corner radius (b) 15 mm corner radius 277 Fig.A3.23 Evolution of calculated stress and strain contours for 15mm corner radius with powder casting (a) Hoop stress (b) Hoop plastic strain 278 Fig.A4.1 Schematic diagram of corner region for 15mm radius billet 283 Fig.A4.2 Schematic diagram of corner region for 4mm radius billet 283 xxi A C K N O W L E D G E M E N T S I would like to express my sincere gratitude to my supervisors Dr. Indira V. Samarasekera and Dr. Brian G. Thomas of Illinois University in USA and for giving me the opportunity to work with them and for their invaluable guidance and support throughout the course of this research. I also wish to thank the POSCO for granting me a scholarship that allowed me to pursue a P h D degree in Canada. The assistance of staff and the discussion with students of the \"Continuous Casting\" research group: Mr. Neil Walker, Gary Lockhart, Winky Lai, Cindy Chow, Julong Fu is deeply appreciated. Besides the aforementioned individuals I am also grateful to Nancy Oikawa, Joan Kitchen, Mary Jansepar. Particularly thanks go to Joydeep Senpta and Joe Shaver for helpful discussion and continued encouragement. I am greatly indebted to many people in the Technical Research Lab., POSCO. In particular I would like to thank Dr. Seongyeon Kim and Dr. U S o k Yoon, who supplied plant trial measurement data and Mr. Wanwook Hur, and Dr. Changhee Yim for their cooperation and assistance. Sincere thanks are due to Dr. Kyuhwan Oh of Seoul National University in Korea, who allowed me to use AMEC2D for calculating the thermo-mechanical behavior of steel shell. I would like to thank Dr. Taejung Yeo (formerly Post Doctorial at UBC), who helped me with the model development. I am very grateful to my parents, whose confidence in me has been a great source of energy. Finally, warm thanks to my wife Junghee, my son Woojin, and my daughter Soyeon for their love, encouragement and understanding throughout this endeavor. xxvi 1 I N T R O D U C T I O N The continuous casting of steel has been widely adopted by the steel industry for the last 50 years because of its inherent advantages of low cost, high yield, and ability to achieve a high quality cast product. This process, as suggested by its name, allows a cross section to be cast continuously, It is a process whereby molten steel from the steelmaking ladle is poured into a \"tundish\", which acts as a reservoir to deliver a controlled flow of liquid to be water-cooled, oscillating copper mold as shown in Fig. 1.1. When mold powder is used as lubricant, the pouring of liquid steel from the tundish into the mold is accomplished through a refractory tube called a Submerged Entry Nozzle (SEN). The mold extracts heat and a solid steel shell is formed. The partially solidified strand is continuously withdrawn and solidification of the steel is completed below the mold where water sprays and radiant cooling take place. The withdrawal of the steel strand is aided by the oscillation of the mold and a constant supply of lubricant that flows between the mold wall and the strand surface. Since continuous casting of steel began, the technology of continuous casting has been a remarkable development. An important recent trend in the steel industry is the development of processes for casting steel closer to the final product size and shape. The continuous casting of thin slabs of only 50 to 75 mm in thickness allows direct hot rolling to be performed in-line with a conventional finishing mill, eliminating the need for a roughing train. The use of advanced continuous thin slab casting technology is increasing in the steel industry owing to the associated savings in capital cost, energy, and manpower. 1 Since the emergence of thin slab casting, the growth of this sector has been dramatic in the past decade with more than 20 thin slab producers currently, or projected to be operating, as shown in Fig. 1.2 [1]. In spite of the remarkable progress, production of superior surface-quality, value-added steels for demanding applications such as exposed automotive body panels by thin slab casting is yet to be achieved. Furthermore, one serious problem that has been nagging the steel industry over the last decade is the formation of mold cracks. Of course, to be efficient, higher casting speeds are necessary in the thin slab casting process, which results in higher temperatures and stresses in the mold. Thus, this process is more prone to mold cracks, particularly in the funnel-shaped mold. In addition to the obvious decrease in mold life, cracks are a serious quality problem. This is because meniscus cracks may locally retard cooling of the steel shell beneath them and lead to longitudinal crack formation and other defects in the steel product. Despite their importance, the thermo-mechanical behavior of the mold and mold crack formation have received little attention and are not fully understood. Furthermore, the influence of design and operation of the funnel-and parallel-molds on quality has not been studied. In addition, the three to fivefold increase in casting speed over conventional slab casting to meet productivity generates problems such as surface defects of slab, especially longitudinal mid-face surface cracks, corner cracks, and frequent breakouts. One of the surface defects that is still issued is longitudinal cracking. Longitudinal cracks are believed to initiate at the initial stage of solidification around the meniscus and grow toward the bottom of mold and below, as in conventional thick-slab casting. To avoid 2 Copper Mold mold Powder layes^^ Submerged Entry Nozzle (SEN) Fig. 1.1 Schematic diagram of tundish and mold of continuous casting of steel with powder lubrication 25 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 Year Fig. 1.2 Growth of thin slab casters installed in past decade [1] 3 longitudinal cracks, the role of mold taper is significant. Mold taper has an important influence on the air gap between the strand and mold, which controls heat transfer. It is one of the few important casting variables that is easy to adjust and control. Over the years, several mathematical models have been developed to investigate the mold taper in conventional molds. However, mathematical analyses of thin slab casting are very limited due to the complex geometry of the funnel mold and the limited understanding of its heat transfer. Furthermore, the effect of different mold shapes, ranging from parallel walls to deep funnels, upon the thermo-mechanical solid shell behavior has not been studied. Thus, the present work was undertaken to improve our understanding of mold cracks with the objective of determining the mechanism behind their formation. Measurements were made on an operating thin slab caster to determine not only the heat extraction rate in the mold but also the effect of geometry upon mold distortion. This information has been employed to perform mathematical analyses on the mold. In addition, samples of mold cracks were metallographically examined to evaluate the cracks themselves. Subsequently, results from plant measurements, mathematical models and metallographic examination were simultaneously analyzed to understand the mold crack formation. In this respect, this work is the first mold cracking study of its kind. This study has led to a better insight on the links between the operating conditions and mold cracking and provided guidelines for the improvement of operating practices. The second part of this research was undertaken to examine the effect of mold taper on the shrinkage of the solidifying shell, gap formation, and stress evolution with different thin slab mold geometries. To achieve this goal, first of all, an existing finite element package was extensively verified using plant measurements of billet casting such as mold 4 temperature, heat balance, solid shell profiles, bulging below the mold, and crack occurrence. The validated model from the billet study was next applied to thin slab casting, using the mold temperature and distortion data from the first study of mold cracking. This work has led to not only a very comprehensive understanding of the thermo-mechanical behavior of the solidifying shell and the susceptibility of surface cracking in different mold configuration, but it has also yielded recommendations for improvement of operating practices. 2 L I T E R A T U R E R E V I E W 2.1 Th in slab casting The following section summarizes the relevant background information necessary for understanding the thin slab casting process and its characteristic feature of heat transfer in the mold. The vast majority of existing literature has been based on conventional slab casting, which operates at low casting speed (less than 2 m\/min.), with very little on thin slab casting operation. The former information provides some foundation for studying mold crack formation in thin slab casters, which operate at high casting speed ranging from 3.5 m\/min. to 5.0 m\/min. 2.1.1 General overview of the thin slab casting process To maintain the same productivity in thin slab casting as in conventional casting, which produces a relatively thick slab ranging from 200 mm to 250 mm, the casting speed must be three to five times higher as shown in Fig.2.1 [2]. The technical barrier of increasing the casting speed lies in developing techniques to feed molten steel into the mold through a submerged entry nozzle (SEN). The breakthrough was achieved in a German foundry by Schloemann-Siemag (SMS) with the adoption of a funnel-shaped mold to accommodate the SEN shown in Fig.2.2 [3]. The upper portion of the mold forms a funnel to accommodate the SEN from the tundish. The taper of the funnel is designed to match the solidification shrinkage of the steel shell. Below the enlarged pouring zone, the broad side-walls are parallel and separated by a distance equal to the slab thickness near the mold exit. In 1989, this new process, termed 6 60 50 100 150 200 250 Slab thickness (mm) Fig.2.1 Relation between slab thickness, casting speed and productivity (1 strand, 1500mm width, operation ratio 95%) [2] 7 Fig.2.2 The SMS concast taper mold for thin slab casting [3] 8 Compact Strip Production (CSP), was commercialized by Nucor at a greenfield plant in Crawfordsville, Indiana, USA as shown in Fig.2-3 (a) [4]. The SMS casting machines, which dominate the emerging thin slab casting scene, are of vertical design with inline bending to a radius of 3.8 m. In the CSP process, 45 m-long thin slab, are fed directly into an inline roller hearth furnace that raises the slab temperature to 1000\u00b0C for rolling. The latter operation consists of five tandem finishing stands preceded by a high pressure descaler. After development of the funnel-shaped mold by SMS, a parallel-walled mold was adopted for thin slab casting at Arvedi\/Italy in 1991 [5]. This process, shown in Fig.2.3 (b), is termed Inline Strip Production (ISP) and developed by Mannesmann Demag Huttentechnik (MDH) of Germany. This process produces a slightly thicker slab of 60 mm cast with a vertical-bow type mold with flat parallel sides, which is then reduced in thickness through inline rolling. The mold is designed with a vertical upper section, which gives way to a curved lower zone having a radius of 5 m. Inline reduction of the thin slabs is effected in two stages, the first to 40 mm by soft reduction (on a liquid core) with rolls below the mold and the second to 15 mm by three shaping stands working on the fully solidified strand. Subsequently, Arvedi changed its mold shape to the funnel type, because of the short service life of SEN. In addition to these processes, four distinct kinds of thin slab casters have been developed to date, which will be described below in brief. One of them is the CONROLL process, developed by Voest-Alpine of Austria, which produces 70-80 mm thick slabs by casting through a straight mold with parallel sides followed by inline bending and straightening [6-8]. The strand is connected to a rolling mill, via a soaking furnace, for 9 (a) Nucor(CSP) (b) Arvedi(ISP) 175 m ISP CSP Maker MDH SMS Mold type Parallel Funnel Slab dimension, mm 75-> 25 50 Casting speed, m\/min. 5.0 6.0 Metallurgical length, m 10.2 5.5 Bending radius, m 5.0 3.0 Vertical length 1.64 5.4 Fig.2.3 Comparison of lay-out between the CSP and ISP caster [4] 10 direct rolling. Molten steel is fed to the mold through a bifurcated submerged entry nozzle. The CONROLL process has been tested at Avesta Sheffield in Sweden, primarily for the production of stainless steels at casting speeds of 2-4 m\/min. It more recently has been tested for the production of carbon steel. This process, which has a maximum design casting speed of 3.7 m\/min., has also been employed on a trial basis at Voest-Alpine Stah! Linz to make 80 x 1285 mm carbon steel slabs. CONROLL has been installed at Mansfield Steel Operations with a startup date of April, 1995 [9]. Sumitomo Metal Industries have developed a high-speed, thin slab casting process based on a straight mold and strand bending, which can produce slabs, 90-120 mm thick by 1000 mm wide, at a maximum casting speed of 5 m\/min. An electromagnetic brake is applied to limit metal level fluctuations to 3 mm [2]. This process is installed at North Star-BHP in Delta, OH in the USA. Danieli installed a thin slab caster with a nominal radius of 5.5 m to produce.plate at the Sabolarie plant in Italy [10,11]. Using a lens-shaped curved mold instrumented with 180 thermocouples, thin slabs of 40 to 140 mm x 900 to 2200 mm can be cast at speeds varying from 0.5 to 6 m\/min., depending on the section dimensions. Design of the SEN, especially the angle of the discharge ports, has been examined with a water model and in plant trials. The stroke length of the mold oscillation cycle is 3.5 mm while the frequency ranges from 3 to 5 Hz, depending on the slab size, casting speed and mold powder. The containment zone has been designed to accommodate future soft-reduction of slabs. Upon leaving the casting machine, the slabs are heated in an induction furnace and then descaled hydraulically before entering a four-high rolling stand. This process was installed at Algoma Steel, Ontario in Canada with a startup date in 1997 [12]. 11 The thin slab casting machines just described rely on mold oscillation to create the relative motion vital for lubrication between the mold wall and the descending strand. In this respect, thin slab casting machine is conventional in concept. Belt casters, on the other hand, minimize friction between the strand and the mold by moving the cooling belts continuously at the same speed as the strand. The technology gained prominence, largely due to Hazlett, for the continuous casting of nonferrous metals like aluminum. Although several different belt designs have been explored for thin slab casting, discussion here will be restricted to the effort by Kawasaki Steel and Hitachi [13,14]. The Kawasaki-Hitachi (KH) caster is of vertical, twin-belt design with \" V bell\" mouth to facilitate liquid metal feeding through a SEN. From large-scale trials, it has been determined that the caster is capable of producing 30 mm x 800 to 1000 mm thin slabs at speeds of 10 to 12.5 m\/min. through a 3700 mm long mold. The slabs are directly rolled without reheating. However, the K H twin-belt caster has not been installed for commercial production. 2.1.2 Heat transfer in thin slab casting mold As mentioned in the previous chapter, the increase in casting speed from 1-2 m\/min. for a conventional 200-250 mm thick slab caster to approximately 4-5 m\/min. for thinner product is well known to increase the mean heat flux in the mold, while it decreases the total heat removed. Wolf [15] has corrected overall mold heat removal with casting speed for both thin and thick slab casters. These are presented in Fig.2.4 as plots of integral mold heat fluxes with casting speed. In this figure, the 50 mm thick slabs were cast from a CSP caster, and the 70 mm slabs from the CONROLL process of Voest-12 Alpine Stahl Linz. He finds that the heat flux increases from 1 MW\/m 2 at 1 m\/min. to 3.5 MW\/m 2 at 5 m\/min. Fleming et al. [16] explain that the higher heat flux associated with increased casting speed is due to the thinner interfacial slag film thickness [15-19]. Fig.2.5 represents the lubricating film thickness subdivided into the mold for a 50 mm strand thickness, as a function of casting speed. With increasing casting speed, the lubrication film thickness decreases and the heat flow increases. With decreasing slag film thickness, which is directly determined by the casting speed, the higher heat flow in turn affects the strand shell cooling and shrinkage. Large differences in heat transfer behavior between funnel-shaped and parallel molds have been reported by Wunneberg and Schwerdtfeger [4]. They measured mold wall temperatures and found that the upper part of the funnel mold has relatively steady heat removal that decreases with increasing distance below the meniscus like the parallel mold as shown in Fig.2.6. However, they also observed a strong time dependence of heat transfer in the lower part of the funnel mold in Fig.2.7. In addition, Fig.2.8 shows the temperature profiles in the narrow sides of the funnel-shaped mold, revealing non-symmetrical heat removal between the two broad faces and in the lower part of the narrow faces. They attribute this to the varying contact between the mold and strand that arises during squeezing out the funnel bulge in the strand. O'Connor and Dantzig [20] conducted a coupled finite-element analysis of fluid flow and heat transfer in the funnel mold of Nucor Steel using FIDAP. According to their calculation, the peak hot face temperature is about 450\u00b0C (Fig.2.9), which is substantially higher than the maximum of 350\u00b0C recommended by Wada et al. [21]. The 13 X X 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Slab thickness (mm) 50 70 Mold powder A (low viscosity) B (high viscosity) Conventional slab casting 0 1 2 3 4 5 Casting Speed (m\/min.) Fig.2.4 Integral mold heat flux versus casting speed for two cases of thin slab casting with two different mold powder in comparison with conventional slab casting [15] i s 53 u n 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 1 1 Casting powder CaO\/SiO,% ZrO,% A B 0.95 1.20 0 3.1 - 6.0 1 1 2 3 4 Vc in m\/min Fig.2.5 Total powder film thickness, liquid and sintered powder film thickness and heat flux as a function of casting speed for 50mm thick slabs [16] 14 ; 'Meniscus )\\ ' \u2022 i I I I I 0 100 200 300 400 500 600 r00 Distance From Top of Mold (mm) Fig.2.6 Longitudinal profiles of the temperature in the broad mold plate of a funnel-shaped mold at a distance of 5mm from the hot face [4] U p S-H S-H B H 300 260 220 180 300 (a) 180 200 AISI grade 1024 4m\/min. 1030 mm x 60mm 10mm concave shape -a-J L min * 25-40 o42-53 \u202285-100 200 400 600 800 1,000 l b ) i i _ i \u2014 j i \u2014 i \u2014 i \u2014 i \u2014 400 600 Distance From Narrow Face (mm) Fig.2.7 Transverse profiles of the temperature in a broad mold plate of a funnel-shaped mold at a distance of 5mm from the hot face at (a) 245mm and (b) 755mm from the top of the mold [4] 15 u o <u S-i I S-i D OH a 03 H 300 250 200 150 100 50 0 Distance from top of mold: \u2022 160 mm A 500 mm \u2022 O 850 mm AISI grade 1015 4m\/min. 1030 mm x 60mm 10mm concave shape 0 10 20 30 40 50 60 Distance From Loose Face (mm) Fig.2.8 Transverse profiles of the temperature in the narrow mold plate of a funnel-shaped mold at a distance of 5mm from the hot face [4] 16 repercussions of running the mold so hot are also seen in the high incidence of mold cracking and reduced working life reported by Nucor Steel [20]. They had also calculated the average and peak hot face heat fluxes to be 3.1 and 5.1 MW\/m 2 , respectively. The peak heat flow took place at the outer bend in the funnel region due to convergent heat flow, while at the inner bend, the heat flow become minimum due to divergent heat flow as shown in Fig.2.10. Uniform heat flow is necessary to avoid longitudinal cracks. Fleming et al. [16] suggested that heat flux in the funnel mold is more uniform than in the parallel mold because the funnel shape centers the strand shell inside the mold as shown in Fig.2.11. This ensures a uniform area between the mold wall and the submerged entry nozzle that encourages uniform slag lubricating film and uniform heat flow between the strand shell and mold walls. 2.1.3 Thermal stress modeling of a slab mold Many mathematical models and plant trials have investigated the thermal and mechanical behavior of conventional slab casting molds [20, 22-27]. Thomas et al. [22,23] applied a three dimensional (3-D) elastic-plastic-creep finite element model to predict temperature and distortion of a conventional slab mold during operation. The wide faces were predicted to bend inward (toward the steel) with a maximum distortion on the order of 1 mm on the wide face centerline between the meniscus and mold mid-height. They also studied the effect of design and operating variables such as slot spacing and shape, copper plate thickness, strand width, clamping force, ferrostatic pressure, pretension stress, water box thickness and copper alloy. Several ways were suggested to 17 Fig.2.9 Temperature contours on inside face of a 20mm thick mould [20] Fig.2.10 Computed variation of mould heat flux with position for the 20mm thick mould [20] 18 1 0 Funnel mold Integral Heat flux Steel s\u00a3eed at SEN port in m\/s 2.5 iMW\/m2! 4.16 T\/min 1.19 ni\/s 0.76 m\/s 2.8 |MW\/m1 2.77 T\/min 75mm 50mm Parallel mold theoretical 2.5 |MW\/m2| 1.78 m\/s 3.5 MW\/m2| 4.16 T\/min 1.78 m\/s 75mm Mold thickness 6 m\/min casting speed \/1200 mm width Fig.2.11 Heat flux into the mould and the steel speed at the SEN port in the funnel-shaped mould and the rectangular mould [16] 19 reduce detrimental residual stress and strain such as designing water slots to lower hot face temperature and optimizing the clamping force. Tada et al. [24] studied the factors affecting the performance and service life of continuous casting slab molds. They emphasized that optimization of the design and maintenance of the cooling water delivery system should improve uniformity of heat transfer, with resulting improvements in mold life and reduction in sticking corner breakouts. Ozgu [25] instrumented a slab mold to measure a wide range of operating parameters, including mold wall temperature and deformation. He found that although the fixed and loose sides of the mold deformed by different amounts, both deformed shapes were convex toward the mold cavity during steady state operation, and the mold cavity decreased by 0.96 mm in the middle of the mold and increased by about 2 mm at the ends of the mold. The measured distortion behavior was consistent with the predictions of Thomas [23]. Salkiewicz et al. [26] measured the effect of copper alloy properties on permanent distortion and wear of copper mold plates. They measured very large width contractions (1.02 mm - 8.38 mm) of the mold plates, with a convex-downward shape across the top and bottom of the plates. Hashimoto et al. [27] have analyzed temperature, distortion, and thermal stress in a fully-constrained slab mold plate including creep, and predicted a permanent 0.2 to 0.3 mm contraction of the wide face after repeated thermal cycling, depending on the copper properties. They found that thinner mold plates have lower temperatures and reduced 20 thermal stresses, resulting in less mold plate distortion. They also observed that to prevent deformation of the mold plate, it should be allowed to move relative to the water-cooling box. O'Connor and Dantzig [20] developed an elastic-plastic-creep finite element model of a funnel mold. The 2-D model predicted 1.75 percent of cyclic inelastic strain in a region below the meniscus along the funnel edge, which resulted from the combination of locally high temperatures with the geometric restraint of the mold. They also calculated locations and time to failure of the mold using the fatigue model. 2.1.4 Mold crack formation in continuous casting A few previous studies have investigated mold defects, including mathematical models and microstructural analysis to study mold crack occurrence [20, 28-31]. Grill [28] investigated mold wear and reported that physical abrasion of the mold is related to the normal force, which acts against the copper faces. Wear occurs in regions of the mold where the air gap is fully closed. Won et al. [29], using a 2-dimensional coupled thermo-elasto-plastic finite element model of conventional slab casting, reported that wear increases with increasing narrow face taper. The width of the worn region was predicted to decrease with increasing carbon concentration in the steel [29]. H. Gravemann [30] analyzed surface cracks in a copper billet-casting mold. These cracks are characterized by transgranular crack propagation. Corrosion fatigue was determined to be the cause and zinc and sulfur to be the corroding media. They also found that in a chrome plated mold, the corrosion of the copper starts under micro-cracks in the chrome layer by forming the brittle intermetallic phases of beta and gamma brass 21 and\/or copper sulfide as shown in Fig.2.12. Fracture of these brittle compounds leaves notches, which act as stress raisers. With alternating thermal stresses from thermal cycling, these notches ultimately develop into cracks that are 10 mm deep. Brimacombe et al. [31] also studied mold cracks in billet molds, and found intergranular cracks combined with the presence of a Zn- and Pb-rich phase, suggesting that the copper plate may have undergone a liquid phase embrittlement when subjected to a tensile stress at very high temperature. O'Connor and Dantzig [20] computed cyclic inelastic strains in a funnel-shaped thin-slab-casting mold, to reach 1.75 percent in a region below the meniscus along the funnel edge. These large strains resulted from the combination of locally high temperatures coupled with geometric constraint of the mold. Longitudinal fatigue cracks on the hot face were attributed to the over-constraint of the copper plates. They suggested cracks could be avoided by loosening the tension bolts connecting the mold with the water jacket, in order to allow the mold to move freely parallel to the broad face. They also predicted the cycles to failure for molds. Important, but complicated phenomena, such as the effect of the water box partial constraint and three-dimensional behavior were not considered. 2.1.5 Thermal stress models of conventional slab casting Over the years, many mathematical models have investigated thermal and mechanical behavior of the solidifying steel shell in the conventional continuous slab casting process. Grill et al. [28] developed an elasto-plastic finite element model to analyze stress distribution and applied it to understand how to avoid corner crack formation by 22 Fig.2.12 Crack initiation in the vicinity of brass inclusion and crack propagation due to thermal fatigue [30] 23 minimizing the tensile stresses associated with surface reheating just below the mold. They studied gap formation and its interaction with heat flow and the effects of operating conditions such as casting speed, slab size, and mold taper on the shell deformation and corner rotation just below mold to predict breakouts and mold wear. Kinoshita et al. [32,33] evaluated the effects of high casting speed, fluid mold slags and mold taper on breakouts and surface cracks in strand cast slabs using a coupled 2-D heat conduction and elasto-plastic stress model. They predicted the air-gap formation and casting conditions necessary to minimize hot spot formation and compressive stresses in the shell. Commercial scale trials on a Voest-type straight mold and a Mannesmann-type curved mold confirmed that 1.1 %\/m of mold taper on the narrow face could prevent breakouts and the development of cracks in the shell. Rammerstorfer et al. [34] used a one dimensional elastic-visco-plastic thermal stress model to simulate the mid-width cross-section through a slab. Their results suggested that reheating of the slab below the mold should be avoided to reduce internal cracks. Their conclusions were based on a maximum strain level of 2-6 % at the solidification front (mushy zone) suggested by Puringer [35]. Ohnaka and Yashima [36] studied the effects of mold taper and mold corner design on stress generation in the off-corner region of the mold using a thermal elasto-plastic finite element model. The stress model included ferrostatic pressure and shell\/mold interaction. Shell deformation due to thermal stress and ferrostatic pressure was shown to affect the shell-mold thermal resistance resulting in tensile stresses that might cause longitudinal corner cracks. Rounded corners and tapers less than 1 % were predicted to reduce the extent of off-corner hot spots and subsequent thermal stresses. 24 Takemoto et al. [37] developed a coupled solidification-deformation model to optimize mold shape to prevent transverse corner cracks in medium carbon steels. In experimental castings, cracks were found to increase with increasing narrow face taper. Model results suggested that high contact pressure and hence large frictional force, are developed in the corner region at tapers exceeding the solidification shrinkage. Thus the tapers should be selected to avoid occurrence of high contact pressure when the shell is relative thin, hot and weak. Okamura and Yamamoto [38] have developed a 2-D slice model of the shell to understand the formation of corner cracks. Their model includes a steady state analysis of the mold heat conduction while tracking the slice in the time domain. They were able to show that by increasing the applied taper, the tensile stress decreases and the location of the high stress region moves away from the corner, thereby reducing chances of crack formation in the off-corner region of the slab. This is obviously contrary to the findings of Takemoto [37]. They also observed that since the whole area of the narrow face shell contacts with the mold wall except for the early stage of the solidification, the large taper is only needed for near the top of the mold where the shell shrinks rapidly. Therefore, they suggested that the mold taper near the top of the mold should be a parabolic curve, which is sufficient to compensate the shrinkage of the wide face shell and to give suitable compression to the wide face shell. Finally, by applying a multiple taper mold, no corner cracks were observed on the narrow or wide faces when the casting speed increased up to 5 m\/min. for a 90 mm thickness slab. Won et al. [29] used a 2-dimensional coupled thermo-elasto-plastic finite element model for the slice of strand to analyze the deformed geometries of the solidifying shell 25 and the air gap near the slab corner for different tapers. Model results suggested that 1.70 %\/m of narrow face taper could compensate for the shrinkage of the solidifying shell and give rise to no air gap on the narrow face for the condition of 0.1wt%C steel, a casting speed of 1 m\/min. and slab width of 1600 mm. Recently, Moitra [39] developed a transient, step-wise-coupled thermo-elasto-visco-plastic finite element 2-D model, which used generalized plain strain condition to estimate the complete 3-D stress state with a 2-D transient model. Using this model together with superheat results from a fluid flow model, they were able to understand and prevent breakouts associated with localized longitudinal shell thinning [40,41] and longitudinal corner depression and subsurface cracking [19]. 2.1.6 Thermal stress models of thin slab casting In the past decade, only a few mathematical models have been reported in the analysis of solidifying shell behavior in thin slab casting process due to limited heat flux data and complexities encountered when modeling the large deformation caused by the funnel change. Ramacciotti et al. [42] treated a horizontal slice of the shell as a beam in conventional structural analysis to simulate thin slab behavior during casting. To match the interdependency between the thermal and mechanical behaviors of the shell, the model operates by alternating thermal simulation cycles with structural analysis. From this analysis, it was suggested that cracking susceptibility when employing the funnel-shaped mold should be very sensitive to small variations in the shape of the curved mold walls, particularly for longitudinal cracking. 26 Cristallini et al. [43] developed the two-dimensional transient thermal and stress analysis assuming elasto-plastic behavior of the shell to design new funnel geometry, although the detailed methodology was not commented on in the paper. In order to find the optimum mold taper, they considered the taper induced by the funnel shape change. Through the calculation of the funnel shape change and shrinkage of solid shell, they suggest that a two-fold narrow side mold taper should be suitable to match the shrinkage of solid shell. Recently, Ridolfi et al. [44] have studied the solidification characteristics of a thin slab mold using the finite element code, M A R C , and finite volume code, PHOENICS, to take into account both the thermo-fluid-dynamics and thermo-mechanical aspects of the in-mold solidification. Using this model, they found that a conventional bifurcated nozzle with a 50 mm thick narrow face gave rise to excess energy of the liquid steel at the meniscus and non-uniformity of thermal and stress distribution, resulting in higher probability of longitudinal crack occurrence, while the open bottom nozzle is more favorable for uniform thermal, stress and strain formation. 2.2 Billet casting This section summarizes initial solidification in the billet mold and its influence on cast billet quality, specifically longitudinal surface cracks associated with the corner radius of mold. Mathematical models used to investigate the thermal and mechanical behavior of the solid shell and air gap formation in con-casting processes are also described. 27 2.2.1 H e a t transfer a n d a ir gap format ion in the m o l d Heat from the solidifying shell is transferred to the mold cooling water by a sequence of steps: conduction through the solid steel shell, conduction and radiation through the mold\/shell gap, conduction through the copper mold and convection to the cooling water at the mold\/cooling interface. The relative importance or each step expressed in terms of resistance to heat flow has been calculated by Samarasekera et al. [45] using mold wall temperature measurements. They found that the mold\/shell air gap provides the largest thermal resistance to heat transfer, particularly near the meniscus. Lower in the mold, the thermal resistance of the solid steel shell may provide a comparable thermal resistance [45], Therefore, the pattern of heat removal in the mold is largely governed by the behavior of air gap formation. With regard to the evolution of the air gap which depends on the ability of the solidifying shell to withstand the ferrostatic pressure, it has been proposed in simple terms as follows: (1) As soon as the superheat in liquid steel is dissipated, solidification starts. At this stage, the solid shell of the billet is very thin and its temperature is very high, which creeps easily by the ferrostatic pressure caused by the liquid steel, makes the solid shell keep contact with the mold. (2) As the solidification proceeds, near the corners, heat is extracted more rapidly, the solid skin becomes thicker and can oppose ferrostactic pressure so an air gap forms. Eventually, the corners of the billet lose contact with the mold, while the center areas of the sides keep contact with the mold. 28 (3) As soon as an air gap forms, heat flow is inhibited in the corners, and the sides solidify more rapidly resulting in greater strength, and eventually the air gap exists around the complete periphery of the casting. Because variables such as temperature and steel composition affect the strength of the solid shell in a complex way [28], gap formation is still poorly understood. Factors which influence the width of the gap such as shrinkage or bulging of the solidifying shell, oscillation-mark depth, shell surface roughness, mold taper and mold distortion also affect mold heat extraction. In addition to this complication, the thermal conductivity of the gap depends on the type of material filling it, which, in turn, is a function of the mold lubricant used. Akimenko and Skvortsov [46] observed that hydrogen content of the gas is a maximum as the lubricant is first introduced (45-55 %), but the average composition during casting is closer to 10-20 %. This drop in hydrogen content lowers the gap conductivity, resulting in decreasing heat transfer across the mold\/strand interface. 2.2.1 Longitudinal cracks Two of the most common mold-related quality problems encountered in billet casting are longitudinal corner surface cracks and longitudinal off-corner internal cracks. Longitudinal corner cracks are located at the corner of billet and are usually 1 to 2 mm deep [47], as pictured in Fig.2.13 (a). They are related to hot tearing close to the solidification front [47]. Although several studies suggest that longitudinal cracks are related to the rhomboid condition of the billet [47, 49-52], these cracks also occur in the absence of rhomboidity, due to improper corner radii [47, 53] or mold distortion and wear [49,50], 29 30 Aketa and Ushijima [53] observed that with a large corner radius, the longitudinal cracks appear along the corner, while with smaller radii, the cracks form more frequently at the off-corner region. They suggest that the corner radius should be 1\/10 of the section size to minimize longitudinal crack formation [48]. However, Samarasekera [47] believes that the modern trend of smaller corner radii such as 3 or 4 mm may solve the longitudinal corner cracking problem, but at the expense of creating more off-corner cracks. Mori [50] observed that the incidence of longitudinal corner cracks increases with the time a mold is in service during a campaign. This suggests that reverse taper due to permanent creep distortion of the mold may be an important contributor. Although corner cracks are believed to form in the mold [50, 54], off-corner internal cracks are believed to form below the mold in the spray-cooling zone [55]. These cracks, as seen in Fig.2.13 (b), are located about 15 mm from a given corner starting at a depth of 4 to 11 mm from the billet surface and extending to a depth of 13-20 mm [55,56]. By analyzing the microstructure of the billet obtained from industrial trials using heat flow calculations, Brimacombe et al. [55] deduced that cracks can form due to bulging of the solid shell in the lower part of the mold. They propose that as bulging occurs, a hinging action develops near the cold and strong corners, causing off-corner tensile stresses near the solidification front and cracking. The cause of the shell bulging was guessed to be thermal distortion or wear in the lower region of the mold. This bulging could arise if improperly set foot rolls or wobbling of the mold during its oscillation cycle caused the strand to move about in the lower region of the mold. 31 2.2.2 Mathematical stress models Solidification of the steel shell during continuous casting in the mold region involves many complex phenomena such as fluid flow, interaction of the shrinkage of the shell and ferrostatic pressure which leads to intermittent contact with the mold and the interaction of the interfacial heat transfer with air gap formation. Over the years, many mathematical models have investigated the thermal and mechanical behavior of the solidifying shell with air gap formation in the continuous casting process [57-62]. Grill et al. [57] applied an elastio-plastic model of the billet strand to study its thermo-mechanical behavior and to explain internal crack formation. They calculated heat transfer coefficients in the corner region and were able to predict corner cracks in the billet by coupling heat flow to the air gap computed from the stress analysis. The model was improved later by Sorimachi and Brimacombe [58] with better material property data. They observed that internal cracks could be caused by surface reheating below the mold. Kristiansson [59,60] simulated billet casting using a step-wise coupled 2-D thermal and mechanical model, which also calculated the size of the shell\/mold gap around each portion of the strand periphery at each time. It was applied to investigate the formation of longitudinal sub-surface cracks in the solidifying shell. They suggested that large air gaps, which may form due to wear or misalignment of the mold, cause large strains in the solidifying shell and a high risk of cracking. Kelly et al. [61] developed a coupled two-dimensional (2-D) axisymmetric thermo-mechanical model for steel shell behavior in round billet casting molds using FIDAP and NIKED2D. Their model was fully coupled through the interface gap, included mold 32 distortion and assumed elasto-plastic mechanical behavior. Their results suggested that thermal shrinkage associated with the phase change from delta-ferrite to austenite in 0.1 %C steel accounts for the decreased heat transfer observed in this alloy, as well as its susceptibility to cracking. Tszeng and Kobayashi [62] calculated billet temperature fields using a temperature-recovery solidification method, followed by an uncoupled stress analysis with plane strain in M A R C . They interpreted the results to get qualitative ideas about possible billet defects. Ohnaka and Yashima [36] studied the effect of mold taper and mold corner radii on the temperature and stress fields in slab casting using an elasto-plastic model, which considered the ferrostatic pressure, mold taper and interaction between the solidifying shell and mold. This model demonstrated that shell deformation due to thermal stress and ferrostatic pressure changes the shell-mold thermal resistance resulting in tensile stress near the slab corner, which may cause longitudinal cracks. They also suggested that a larger mold corner radius should decrease the interfacial gap thickness and tensile stress in the shell and thereby help to prevent cracks. 33 3 SCOPE AND OBJECTIVE OF THE PRESENT WORK Thermo-mechanical phenomena during continuous thin slab casting are studied with the objectives of understanding the mechanism of mold crack formation, and the effect of mold design upon the mechanical behavior of the stand. To achieve these goals, several finite element models are developed in conjunction with a series of industrial plant trials. The mold shape and high casting speed in thin slab casting lead to higher mold temperatures and shorter mold life than in conventional slab casting. This study investigates heat flux profiles and the effect of mold shape on distortion and cracking of a thin-slab mold. Through a series of instrumented mold trials, heat-flux profiles in a thin slab mold are determined. These data are then used in an elastic-visco-plastic analysis to investigate the deformation of the mold in service for the two different mold shapes. Using the mathematical models, together with a metallurgical investigation of mold samples containing cracks, the mechanism of mold cracking is determined and solutions for the prevention of mold cracks are also suggested. A finite-element thermal-stress model to compute the thermo-mechanical state of the solidifying shell in a thin-slab casting mold is developed. Model predictions are verified with both analytical solutions and plant trials during continuous casting of steel in a square billet-casting mold. The model is then applied to study the effect of the mold taper on the shrinkage of the solidifying shell, air gap formation, and stress evolution for different thin slab mold geometries, based on heat-flux profiles extracted from plant measurements. The implications on longitudinal crack formation are discussed. 34 3.1 Mold crack formation in thin slab casting 3.1.1 Objective The previous chapter has shown that there is very little in the literature regarding the thermo-mechanical behavior of the thin-slab casting mold, especially mold crack formation at high casting speed. The main purpose of this part of the research is to characterize the thermo-mechanical behavior of thin slab molds at high casting speeds. This knowledge is then utilized to investigate mold crack formation. The objectives of this research are to conduct a rigorous multi-faceted investigation of mold cracks, which includes the following interlinked tasks: 1) To organize an instrumented mold trial to measure temperature, casting speed, and metal level for the parallel and funnel molds, and to obtain mold crack samples. 2) To formulate, develop, and verify accurate mathematical computer models for calculating heat transfer and stress generation in both funnel and parallel molds used in thin slab casting. 3) To characterize heat flux profiles in a thin slab mold at high casting speeds 4) To conduct a metallurgical investigation of mold crack samples. 5) To clarify the role of stress generation in both mold types and to propose mechanisms for mold crack formation that are consistent with the mathematical model and the plant observations. 35 3.1.2 Methodology This investigation focused on mold crack formation in thin slab casting molds through plant trials, metallographic investigations, and mathematical heat-flow and stress-analysis modeling. While this work was concerned with elucidating the mechanism behind the general problem of mold cracking in the steel industry, it focused primarily on examining the nature of funnel mold cracking as experienced at the Pohang Steel Works (POSCO), in South Korea. This was due to the availability of mold samples containing cracks and temperature and distortion measurement data for validation of the heat-transfer and stress models. The specific methodology used in the present investigation was as follows: 1) To better characterize heat transfer in the thin slab mold, plant trials were organized at POSCO Kwangyang works in South Korea, casting a 1260 mmx70 mm thin slab at 3.6 m\/min. During the plant trial, various other process parameters, such as casting speed, metal level signal (cobalt source method), and the temperature and the flow rate of the cooling water were collected directly from the plant control system. 2) To validate the stress model, mold deformation during steady state casting and permanent contraction of the mold width after a casting campaign were obtained from POSCO for parallel and funnel molds, respectively. 3) To better understand crack occurrence and its relationship to the temperature and stress profiles of the mold, the locations where crack occurred were measured at POSCO for 36 slab molds that contained cracks. 36 4) Several finite-element models such as a complete 3-dimensional (3-D) model of a one-quarter section of the mold, a 3-D segment model of a representative vertical segment of the mold, and a two-dimensional (2-D) horizontal sections through the copper plates were developed using the commercial finite element package, ABAQUS [71]. 5) The models were utilized to characterize heat flux profiles at high casting speed and calculate the temperature and the corresponding distorted shape of the parallel and funnel molds during steady-state operating conditions and after cooling to ambient temperature. 6) The understanding of a cracking problem would be incomplete without a metallurgical examination of the cracks themselves. Thus, the last step was to conduct a metallographic investigation of crack samples supplied by POSCO: a) The polished samples were etched to examine the types of cracks. The etchant used was as follows: 1 g K 2 C r 2 0 7 , 4ml H 2 S 0 4 , 50ml H 2 0 , 2 drops of HC1 b) S E M . and E.D.S. examinations of the fracture surface and concentration profiles around cracks were performed. 7) Finally, the results of the mathematical heat-transfer and stress models were incorporated with the metallurgical investigation and observations by the plant trials to propose mechanism(s) for mold crack formation. 37 3.2 Mold taper in thin slab casting 3.2.1 Objective Although many mathematical models have investigated the thermal and mechanical behavior of the solidifying steel shell in conventional continuous slab casting processes, which have lower casting speed (1 - 2.5 m\/min.), mathematical modeling of stress generation in thin slab casting (usually 3.5 - 5.0 m\/min.) is just beginning. This is because the complexities encountered when modeling the large deformation caused by the funnel shape change, in addition to complex coupled phenomena such as fluid flow, the interaction of shell shrinkage and the mold, ferrostatic pressure, and other highly nonlinear rate-dependent constitutive relationships make this a very computationally demanding task. Furthermore, many models about conventional casting have been applied with coarse grids and without quantitative validation. Also the effect of strand shape on the stress and strain development has not been studied. Further work is needed to understand the thermo-mechanical behavior of the solidifying shell in the thin slab mold, specifically, the shrinkage behavior of the solidifying shell, gap formation and stress evolution, and the influence of mold taper on them for different thin slab mold geometries. These are the aims of this work. 3.2.2 Methodology This investigation focused firstly on validating a finite element package, AMEC2D [29, 114-116], in order to explain the temperature distribution, shape, stress\/strain generation in the solid shell, and the air gap between the casting mold and the solidifying 38 strand. A comparison was made with plant trials in billet casting, performed at Pohang Steel Works (POSCO) in South Korea and analytical solutions used to verify the predictions of the model. By utilizing the AMEC2D and heat flux profiles and mold distortion profiles obtained in the previous study of mold crack formation, an analysis was then performed to understand the thermo-mechanical behavior of the solidifying shell in thin slab casting. Because of the unique characteristics of the funnel slab, there are several funnel geometries that have been applied by steelmaking companies. Among them, this study focused on two kinds of funnel geometry. One is a M D H Concast design used at POSCO and the other one is a SMS Concast design. A parallel shaped mold, which was employed at POSCO, was also studied for comparison. The following is the specific methodology used in the present investigation: 1) A two-dimensional (2-D) transient, coupled thermo-elastic-visco-plastic finite element model has been developed using the existing finite element package, AMEC2D, which was originally developed by Seoul National University in South Korea. This model considers the following solidification phenomenon during casting. a) Microsegregation during solidification b) Ferrostatic pressure by liquid steel c) Mold distortion d) Solid shell\/mold contact 39 2) To validate that AMEC2D is capable of accurate predicting heat transfer and solid shell growth in continuous casting process, a plant trial was conducted with an instrumented billet mold at POSCO, Pohang works, South Korea, for a continuously cast section size of 120 mm square with 0.04 %C steel at a nominal casting speed of 2.2 m\/min. During the plant trials, FeS tracer was added to the liquid pool under steady state casting conditions to investigate solid shell growth. Because FeS cannot penetrate the solid shell, the solid shell front can be clearly recognized after casting by making a sulfur print. Comparisons were made with the following measurements from the plant trial to validate the AMEC2D model: a) Mold temperature b) Heat balance c) Solid shell thickness d) Bulging below the mold e) Internal crack location 3) Funnel shape was characterized from geometric study of two different molds, POSCO, and NUCOR 4) The validated finite element model (AMEC2D) from the previous billet casting study was applied to the thin slab casting process incorporating the following specific boundary conditions from the first part of the project. a) Heat flux, which was extracted from the measurement of mold temperature in the mold cracking study, was imposed as a thermal boundary condition. 40 b) Mold distortion from the previous mold deformation study was incorporated into the mold taper boundary condition. 5) To take account of the slag film thickness into the mold taper, slag film thicknesses across the slab width at mold exit were obtained from POSCO Kwangyang works, South Korea. 6) The shrinkage of the solidifying shell was validated with the crude calculation of the perimeter change due to thermal contraction of the solid shell. 7) Finally, simulations were carried out to study the effect of funnel shape change upon the shrinkage, air gap, and stress development in the solidifying shell. 41 4 THERMO-MECHANICAL BEHAVIOR OF THE THIN SLAB MOLD 4.1 Introduction The mold is the most critical component of the thin slab casting process, which controls initial solidification and determines surface quality. The quantification of heat transfer and distortion of thin slab casting copper mold plates has received relatively little attention in previous literature. Furthermore, the difference between the funnel and parallel mold-design has not been compared. During operation, the mold distorts due to steep thermal gradients. Although this distortion is very small, it may affect the size of the gap between the solidified shell and the mold, which in turn controls heat transfer. Thin slab molds are expected to have higher heat flux and temperature owing to the higher casting speed. The accompanying thermal stress may cause permanent creep deformation near the meniscus, which affects mold life as well. Furthermore, maintaining a reliable, crack-free mold within close dimensional tolerances is also crucial to safety and productivity. This chapter will describe the thermal and mechanical behavior of thin slab casting molds, including the heat flux profiles based on mold temperature measurements and compare the thermal distortion of thin slab molds of two different configurations using three-dimensional (3-D) finite-element models. 42 4.2 Plant trials 4.2.1 Mold temperature measurement A plant trial was conducted with an instrumented parallel mold at POSCO, Kwangyang works in South Korea, casting a 1260 mm x 70 mm thin slab at 3.6 m\/min. The mold was Cu-0.1%Cr-0.15%Zr and 1000 mm long, other relevant details are given in Table 4.1. In this table, the effective wall thickness is defined as the minimum distance from the water channel to the hot face, and was relatively constant on each face. It is noted that to prevent the mold temperature from reaching high temperatures near the meniscus, extra angled water slots were added. Fig.4.1 shows the back side view of the parallel mold for the plant trial, where the mold was instrumented with seven rows of K-type (Chromel\/Alumel) thermocouples along the centerline of the mold for both the loose and fixed broad faces. Each broad face has seven columns of thermocouples for at least the first two rows. Each thermocouple was inserted through a hole drilled along the center of a tension bolt. When the tension bolts were tightened, a spring pushes the thermocouple to the required depth of 22 mm from the hot face. A copper washer was also employed at the tip of each tension bolt in order to prevent water from penetrating to the tip of the thermocouple. An Electro-Magnetic Brake System (EMBR), installed in the mold to control fluid flow, generated electro-magnetic forces during mold oscillation, which caused electrical noise in the temperature data. To avoid this effect, the bare portion of the parallel thermocouple wires were held slightly apart against the mold surface in order to include a small portion of the copper mold in the thermocouple circuit. 43 Table 4.1 Conditions for plant trial measurements of mold temperature, etc. Parallel mold Funnel mold Mold coppers Cr-Zr Slab thickness (mm) 75 Slab width (mm) 1260 1140 Steel grade (Carbon content, wt%) 0.04 Mold powder -Basicity 1.32 - Viscosity at 1300 \u00b0C (poise) 0.52 Meniscus level (mm) 100 Copper plate thickness (wide face) (mm) 60 60 - 85 Effective wall thickness (mm) - wide face 25 25 - 27 - narrow face 22 Mold coating material (thickness) Cr (0.01mm) Cooling water velocity (m\/s) -wide face 10.7 10.5 - narrow face 10.6 Water flow rate (\/\/min.) -wide face 4330 4600 - narrow face 250 Cooling water temperature (\u00b0C) -Inlet 37.8 .39.7 -Outlet 48.1 47.5 Casting speed (m\/min) 3.6 4.2 Number of thermocouples (EA) 42 (Fig.4.1) 7 44 45 The accuracy of the temperature measurement system was assessed by an independent laboratory test. A small scrapped copper plate was instrumented with thermocouples as described previously. The plate was immersed in boiling water, and the measured temperatures were within 1\u00b0C of 100\u00b0C. A second test, on the assembled instrumented mold, found the thermocouple readings between heats to match the mold water temperature. Temperatures were also measured in a funnel mold using similar thermocouples just at the meniscus. Attention was focused on the row of 7 thermocouples near the meniscus, 100 mm below the top of the loose face of the mold. This was used to investigate the temperature profile and thermal fluctuation in the mold and its effect on mold crack occurrence presented in Chapter 5. 4.2.2 Other measurements During the plant trial, various other process parameters, such as the casting speed, the metal level signal (cobalt source method), the temperature and the flow rate of the cooling water were collected directly from the plant control system once per second during casting. Fig.4.2 compares the metal level and casting speed signals during steady state casting for the parallel and funnel mold. The fluctuations are stable within a maximum \u00b14 mm. It seems that the funnel mold has a lower fluctuation frequency of about 0.05 Hz, compared to the higher frequency of 0.3 Hz of the parallel mold. The detailed character of metal level fluctuation, especially its effect on mold temperature, is discussed in Chapter 5 in order to investigate its influence on mold crack formation. 46 8 6 n 1 r-+ 4 \u2014- Metal level Casting speed -o <u Q. m ao 1 6500 6550 6600 Casting time (sec) (a) 6650 6700 + 4 \u2014- Metal level \u2014 Casting speed \u2014I\u2014 6650 6500 6550 6600 Casting time (sec) (b) Fig.4.2 The profiles of metal level and casting speed (a) Parallel mold (b) Funnel mold 6700 'S -o Q. m 60 C 1 47 The temperatures of the inlet and outlet of the cooling water were measured in the parallel mold. Together with the water flow rate, this temperature difference (AT) was used to calculate the total heat removal rate. The AT stayed relatively constant during the entire heat (10\u00b0C), although the inlet temperatures varied from 25\u00b0C to 40\u00b0C on the wide face of the mold. Mold deformation during steady state casting was measured at the junction between the wide and narrow faces of the parallel mold. During assembly, the wide face of the mold contacts continuously along the narrow face edges. During operation, however, formation of a gap between these two faces above the meniscus has been reported [23]. This gap was measured using a simple metal strip gauge during steady state casting. After casting, the mold experiences permanent deformation due to residual inelastic strain [23]. Contraction of the mold width was measured at the top of the funnel mold using a ruler after 350 heats of casting. Finally, a crack sample was obtained from a funnel mold after 345 heats of casting and a metallurgical investigation was conducted. Details are presented in Chapter 5 together with a modelling investigation. 4.2.3 Measured mold temperature profiles The mold thermocouple temperature data gathered during the plant trial were examined. Fig.4.3 shows the typical mold thermal response on the parallel mold for two thermocouples located at 176 mm below the top of the mold for an entire single-heat casting sequence. Both locations show similar amplitude of high-frequency thermal fluctuations (about 0.05 Hz). However, the thermocouple at the off-corner location, 376 mm from the mold center, is always hotter than that of the other thermocouples. 48 300 i . M I I 50 + \u2014 Loose_center \u2014 Loose 376mm End-of-cast 4-0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 Casting time (sec) Fig.4.3 Mold thermal response recorded at different mold locations around the mold perimeter at 176mm below the top of the parallel mold (typical sequence) 4 9 Moreover, the thermocouple located at the center of the mold exhibited significant variations of temperature over a long-time scale (2000-5000s), relative to other thermocouples. These long-time scale variations have also been observed in conventional casting [63-66]. Fig.4.4 shows the thermal response at various distances below the mold top along the centerline of the parallel mold. It can be observed that, as expected, both the temperature and its fluctuation decrease with increasing distance below the meniscus. Thus, at the lower part of mold, the temperatures become close to each other. To calculate the average temperature distribution in the mold wall and corresponding heat flux profiles, mold temperature data obtained in the plant trial were averaged over 5 minutes of steady state casting. Typical examples of the time-average measured mold temperature are presented in Fig.4.5 and Fig.4.6. In these figures, the range of the data falls within the size of the symbols, except where noted with error bars. Figure 4.5 gives the temperature profiles along the centerline of the mold, which decrease with increasing distance from the meniscus as in conventional casting. As shown in this figure, the temperatures on the loose face (inner radius) are 10-50\u00b0C higher than those of the fixed face, especially near the meniscus. The transverse temperature profiles across the broad face are plotted in Fig.4.6 at different distances below the top of the mold. The loose face temperatures are consistently hotter than those of the fixed face, regardless of position around the perimeter. Another striking feature is the deep valley in the temperature profiles in the central region of the wide face near the meniscus. Non-uniform temperature profiles across the mold width, especially around the meniscus, have been reported previously in both conventional slab casting [66-68] and thin slab casting [4]. 50 Fig.4.4 Mold thermal response recorded at various axial locations down the centerline of parallel mold during steady state 51 250 i i i \u2022 > I; 200 4-CJ I 150 OJ Q. s \u20224\u2014\u00bb \"S 100 3 ID 50 i i | i i i i ] i i i i 1 i i i i I i i i i 1 i i ' ' 1 ' ' ' Meniscus Loose face Fixed face 100 200 300 400 500 600 700 Distance from the top of mold (mm) 800 900 1000 Fig.4.5 Time-averaged temperature profile measured along the mold length (Parallel mold) 52 Fig.4.6 Mold temperature profile measured at different positions across the wide face (Parallel mold) 53 4.2.4 Discussion The variation of mold temperature around the submerged entry nozzle (SEN) may occur for numerous reasons. Factors such as turbulence in the liquid slag pool and variations in slag flow into the strand-mold gap [63], argon flow [63], nozzle clogging [64], SEN shape and their associated effects on flow pattern of the molten steel [63-66] may contribute to the variation in the mold temperature. The trends in these measurements are generally consistent with previous measurements in conventional casters. Lower temperatures with greater variation near the meniscus at the central region of the mold width might be explained by several different phenomena. Firstly, the high speed steel flow pattern causes surface contour changes of the molten steel level at the meniscus, and associated variations in the mold flux layer thickness across the width [69]. Birat [70] measured the meniscus profile in a slab mold, which varied from -10 to 15 mm across the slab width with respect to the reference point. The meniscus shape also influences slag infiltration into the gap, which may affect significantly the mold temperature profile as well. Finally, the interaction between steel shell shrinkage and gap formation that is well-known in conventional casting [41] may cause local temperature drops or fluctuation in the funnel region. This awaits study in future research. 4.3 Thermal-stress computational description Finite-element models were developed to calculate temperature and the corresponding distorted shape of parallel and funnel molds during steady operating conditions and after cooling to ambient temperature using the commercial stress-analysis package, ABAQUS 5.8 [71]. Model domains include typical two-dimensional (2-D) horizontal sections 54 through the copper plates, a 3-D segment model of a representative vertical segment of the mold, and a complete 3-D model of one-quarter section of the mold, including the water jackets and bolts (3-D quarter model). Simulation conditions are given in Table 4.2. Geometric domains are given in Figs 4.7 and 4.8, and properties are given in Table 4.2 and Fig.4.9. 4.3.1 Heat flow model The heat flow model solves the transient heat conduction equation [72] for the temperature distribution in the various model domains. Heat flux data were input to the exposed surfaces of the, copper elements on the mold hot faces as a function of position down the mold, as discussed later. The 3-D segment model domain, shown in Figure 4.7b, reproduces the complete geometric features of a typical repeating portion (Fig.4.7a) of the copper wide face discretized into a fine mesh of 18758 nodes and 14622 8-node brick elements. A water-slot heat transfer coefficient hi, was applied to the surface of the water slots using the correlation of Dittus and Boelter [73]. ^ = o . o 2 3 ( D ^ r ^ ^ r ( 4 n kw luw kw v \u2022 ) where, D is the hydraulic diameter of the slot, kw is thermal conductivity of water , \/jw is water viscosity, pw is water density, Cpw is the specific heat of the cooling water, and Vw is cooling water velocity. The errors associated with estimating the above equation and their influence are discussed in Appendix II. Considering the estimated \u00b1 1 5 % errors 55 Table 4.2 Simulation conditions a) Mold geometry Mold length (mm) 1000 Mold width (mm) 1660 Slab width (mm) 1260 Copper plate thickness (mm) - Wide face 60 (Parallel) 80 (Funnel) - Narrow face 75 Water slot depth (mm) 35 Water slot thickness (mm) 5 Distance between slots (mm) 4.6 Nominal cooling water section (mm x mm) 5x10 Bolt length (mm) 445 Bolt diameter (mm) 16 Bolt hole diameter in water jacket (mm) 24 Distance between bolts(mm) 188 Water box thickness (mm) 360 Clamping force (kN) -Top (0.58m from bottom) 19 -Bottom (0.1m from bottom) 44 Tension bolt force - Tightening torque, N-m 120 - Friction coefficient 0.1 56 b) Material properties [22] Copper (plate) Steel (Bolt) Conductivity (W\/mK) 350 49 Density (kg\/m3) 8960 7860 Specific heat (J\/kgK) 384 700 Elastic modulus (GPa) 115 200 Thermal expansion Coeff. (1\/K) 17.7X10\"6 11.7X10-6 Poisson ratio 0.34 0.3 57 (b) Plane of symmetry Hot face Cooling channel <\u2014><\u2014. r-\\ r\u00b1\\ r\u2014i S t e \u00ab bolt' Section to : : : : : :b*: : : : : : modeled '. Metal (Copper) insert ( a) 6Q Mold top h\u2014H q=\u00b0 Metal insert Hot face q=q(z) Water channel h=38KWm2 K 1 Tw=38\u00b0C Mold bottom q=0 ( c ) 2-D horizontal section through wide face showing (a) top view of segment model domain, (b) corresponding 3-D section mesh and (c) boundary conditions on a vertical section 58 Clamping force Cooling water channel convection surface \\ Steel stiffening plate teel water box Copper wide face Steel backing of narrow face Copper narrow face Fig.4.8 Top view of 3-D quarter mold model showing boundary conditions 59 4 5 0 -| Total strain (%) Fig.4.9 Stress-strain curve for copper (Cr-Zr alloy) used in this model [23] 6 0 in the use of equation 4.1 [73], it is expected that heat flux profiles and its associated average heat fluxes, which will be discussed in later, have the \u00b14.0% errors. Cooling water temperatures were imposed as linear functions based on the inlet (top of mold) and outlet (bottom of mold) temperatures given in Table 4.1. Heat flux profiles are given in Fig.4.10 and were chosen to match the measured temperatures. To simplify the geometry for the 3-D quarter-mold model, a pseudo-model was created using a coarse mesh of 759 nodes and 420 elements. The temperature field produced by the pseudo-model was arranged to match that of the 3-D segment model by artificially decreasing the water-slot heat-transfer coefficient and decreasing the hot-face heat flux values. The accuracy of this approach was validated by comparing the results of this approach with the 3-D segment model. The maximum difference between them was 20\u00b0C on the hot face. To simulate thermal cycling of the mold over a complete campaign, it was assumed that the mold was heated from room temperature to the operating temperature by sudden immersion in the high temperature environment for 100 seconds, followed by 4 hours of steady casting for each sequence which includes 5 ladle changes. It was then cooled to room temperature, over 360 seconds after each sequence, which defines a single thermal cycle. 4.3.2 Stress model The thermal-stress model solves for the evolution of displacement and stresses in the mold during a 50-heat casting sequence, based on the previous calculated temperatures. Fig.4.8 shows the top view of the 3-D quarter mold model used to analyze mold deformation. This model includes separate domains for the mold coppers and water 61 I ' 1 1 ' I 100 200 300 400 500 600 700 Distance from the top o f mo ld (mm) (a) 800 900 1000 8 7 -f 6 5 -E 4 3 2 1 -t 0 -HT-o Fixed_center \u2022 Fixed_ 188mm : \u2022 Fixed_376mm A Fixed_564mm \\ Model assumption 0 100 200 300 400 500 600 700 800 900 1000 Distance from the top of mold (mm) (b) Fig.4.10 Heat flux profiles of parallel mould with different mold positions (a) Loose face (b) Fixed face 62 jackets which are coupled mathematically only at those points where they connect mechanically in the caster during operation. The cold side of the copper wide face is mathematically bolted to the back of the water jacket at 40 locations using two-node bar elements. Boundary conditions include clamping forces on the exterior of the water jacket (Table 4.2) and a pre-tension force, Fboit in each bolt determined by the following equation [23]. _ 2r (jvd- juA. I'boii - r \\~T~ r) where, T is bolt tightening torque, d is bolt diameter, X is distance between threads of the bolt (1.5 mm), p is friction coefficient, which varies from 0.2 (greased) to 0.6 (ungreased) [23]. According to the above equation, the pretension force varies from 64.8 kN (greased) to 23.4 kN (ungreased). In the finite-element simulations, an average force between these two of 44 kN was chosen on each bolt. Interface contact elements were used to prevent penetration between the contacting surfaces of the deforming wide and narrow faces of the mold and the water jacket. The maximum friction force between the faces, which depends on the normal pretension force (44 kN) from all of the bolts and the friction coefficient (0.6 max.), is 1.02 MPa. This is negligible relative to the stresses generated by thermal expansion, which are on the order of the yield stress of 280 MPa at 350\u00b0C. Thus, friction forces cannot prevent relatively free expansion of the copper plates, which agrees with findings for conventional molds [23]. In this study, a friction coefficient for relative sliding between contact surfaces of 0.1 was assumed to avoid convergence problems. This condition allows easy sliding, which is reasonable because the bolt holes are oversized and offer no resistance to 63 sliding, as discussed in Chapter 5. Finally, rigid body motion is prevented by constraining the symmetry planes from normal expansion and by fixing displacement at additional points. This elastic-visco-plastic thermal stress model assumes isotropic hardening with a temperature-dependent yield stress function, shown in Fig.4.9. Considering that the total time of operation is less than 100 h, primary creep is assumed, based on the following equation [74]. In this equation, the time variable, t, increase continuously throughout the 50 casting sequence (i.e. It is never reset to be zero). This method is inherently approximate because the plastic strain and creep strain are derived from monotonic experiments and are calculated and act independently. This can only be improved by finding a unified structure parameter from fitting cyclic loading measurements and employing a unified constitutive model, which evolves the structure during the fatigue. sis-') = 2.48x10'4 exp(~^\u00b0V(fe;) - 23.]3[\/(5)r092 (4.3) 4.4 Heat transfer of parallel mold 4.4.1 Heat flux profiles To determine the heat flux distribution on the 3-D mold faces, arbitrary heat flux distributions were first applied as boundary conditions using the 3-D segment model described in section 4.3. Then, these arbitrary values were adjusted until the temperatures calculated by the 3-D segment model at the thermocouple locations match well with the measured ones. 64 Fig.4.10 shows the heat flux distribution over the wide face based on linear interpolation and extrapolation of the heat flux values at the thermocouple positions. The peak values of heat flux at the meniscus were predicted to be about 7 MW\/m 2 , although they vary with mold position. These values are somewhat higher than other measurements reported for thin slab casters, 5.5 MW\/m 2 [20]. There is uncertainty in the peak values extrapolated at the meniscus, however, because there are no thermocouples nearby. 4.4.2 Hot face temperature profiles Axial mold-temperature profiles were calculated using the 3-D segment model based on the above heat flux profiles given in Fig.4.10. Fig.4.11 shows the hot face temperature distribution along the mold length at different positions across the wide face. As seen from this figure, the maximum temperature of the wide face is found about 20 mm below the meniscus, and is 580\u00b0C at the location of 376 mm from the center of the mold. This is 200\u00b0C higher than found in conventional slab casting owing to the higher casting speed. This is also 50\u00b0C higher than previously reported for thin slab casting [20]. These high temperatures may increase the possibility of mold crack occurrence, considering the softening temperature of Cu-Cr-Zr copper alloys of about 500\u00b0C [75]. It is important to note that within about 50 mm of the mold exit, the model predicts an increase in hot-face temperature of almost 50\u00b0C. This surface is expected to be hotter, due to the end of the water slot, 28 mm above the mold exit [23]. 4.4.3 Validation - Temperature profiles 65 \u2014I\u2014I\u2014I\u20141\u2014I\u2014I\u2014I\u2014r-\u2014 Loose_center Loose_188mm \u2022 - - Loose_376mm \u2014 - Loose 564mm 100 200 300 400 500 600 700 800 900 1000 Di s tance f rom top o f mo ld (mm) (a) 600 500 u o \u2022 400 -\\ 3 -ta -tempei 300 -:_ face 200 -\\ o X -100 + + \u2022+-Fixed_center Fixed_188mm - - - Fixed_376mm \u2014 - Fixed 564mm 0 100 200 300 400 500 600 700 800 900 1000 Distance from top o f mo ld (mm) (b) Fig.4.11 Hot face temperature distribution down the length of the parallel mold (a) Loose face (b) Fixed face 66 The degree of fit of the calculated heat flux distribution is demonstrated by comparing the temperatures measured by the thermocouples in the mold wall with the calculations based on the heat flux profiles in Fig.4.10. Fig.4.12 shows complete temperature profilesalong the mid-plane of this parallel mold at the depth below the hot face where the thermocouples are located. The dips are caused by the extra angled water slots (Fig.4.1), which provide extra cooling to the region near the meniscus, except for near the bolt holes which contain the thermocouples. The temperature increase of 75\u00b0C near the mold bottom is caused by the extra distance of the hot face from the cooling water after the water slots end. As can be seen from this figure, although there is some difference between measured and calculated values, on the whole, the model predictions agree reasonably well with the measured temperatures. This comparison confirms that the heat flux curves in Fig.4.10 are calibrated properly. - Heat flux profiles To validate the heat flux profiles, energy balances were performed on the cooling water for each face of the mold. During the process of cooling the mold, the water heats up, and this temperature difference (AT) between the inlet and outlet of the cooling water is commonly used to monitor the total heat removal rate of the mold. Multiplying the temperature rise for each face by the corresponding water flow rate gives the total rate of heat removal from that face. Dividing this by the exposed mold face area gives an average heat flux extracted by that face. These measured values are compared in Fig.4.13 with the total calculated heat fluxes (area under Fig.4.10a and b). Although there is some mismatch (10-13%), considering \u00b14.0% errors from using equation 4.1, the 67 250 0 I \" 1 1 I 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 \" I 1 1 1 1 i \" \" l \" \" I ' 1 \" i 1 1 1 1 I 0 100 200 300 400 500 600 700 800 900 1000 Distance from the top of mold (mm) (a) 250 0 | ' ' ' ' i 1 ' 1 1 I 1 1 1 1 i 1 1 1 1 i 1 1 1 1 I 1 1 1 1 I' 1 1 1 1 I 1 1 1 1 I 1 1 1 1 | 0 100 200 300 400 500 600 700 800 900 1000 Distance from the top of mold (mm) (b) Fig.4.12 Comparison of measured and calculated temperature in the parallel mold (a) Loose face (b) Fixed face (closed circle: time-averaged temperature, bar represents range between max. and min. of measured data) 68 \u2022 Based on heat flux from Fig.4.10 Loose wide face Fixed wide face Fig.4.13 Comparison of average heat flux calculated from heat flux data and from energy balance on cooling water 69 experimental uncertainties, the variation of measured temperatures and the uncertainty of the heat flux value extrapolated at the meniscus, the heat flux distribution down the mold length appears to be reasonable. 4.5 Thermal and mechanical behavior of parallel mold 4.5.1 Results The typical temperature and distorted shape of the parallel quarter mold during operation are shown in Figures 4.14-15 for conditions in Table 4.2. Fig.4.14 shows the mesh, temperature contours on the hot face, and the distorted shape of the mold, exaggerated fifty-fold. At the hot face, the maximum temperature is found to occur approximately 370 mm from the centerline. The results shown in Figures 4.14-15 provide insight into the mechanical behavior of the mold during operation. The copper hot face expands in proportion to its temperature increase, but is restrained by the cold copper beneath it. Consequently, the copper plates bend outward toward the molten steel in a manner similar to the conventional slab mold [23]. The peak distortion of 0.3 mm, which occurs below the meniscus along the center of the wide face, is smaller than that of a conventional caster predicted by Thomas, [23] even though the heat flux is higher for thin slab casting. This is because the water box is 360 mm thick, which is thicker than conventional slab molds. As explained by Thomas, [23] water box rigidity is more important to distortion than hot face temperature. The bending of each copper plate toward the liquid steel stretches the bolts and pulls it away from the front of the water jacket during operation. Despite the bolt pre-stress of 44 kN, a thin gap forms just below the meniscus, as shown in Fig.4.15. The gap between 70 Fig.4.14 Temperature contours on distorted mold shape during operation for the parallel mold 71 Fig.4.15 End view of distorted mold along the wide face centerline showing the gap between copper plate and water jacket and temperature profiles 72 Fig.4.16 Predicted evolution of thermal distortion on vertical section for the parallel mold (centerline surface) 73 the water box and the copper cold face is 0.2 mm in a 60 mm thickness of parallel mold where distortion is greatest. Fig.4.16 shows the evolution of thermal behavior during the first heat and subsequent cooling. The reversal in curvature is similar to that of conventional slab molds [22-23]. 4.5.2 Validation To validate the stress model predictions, comparisons were made with plant trial measurements on the parallel mold. Fig.4.17 compares measured and predicted distortion of the wide and narrow faces along their line of intersection. The gap measured with a simple metal strip gauge at the junction between the wide and narrow face was 0.35 mm. Although the calculated value is about 30% larger than the measured value, this general agreement suggests an accurate prediction of mold behavior. If larger, this gap between the plates above the meniscus may cause problems, since the mold flux might penetrate, freeze, and aggravate mold wear and shell sticking at this critical junction [23]. Fig.4.18 compares the calculated backing plate distortion with measurements from several conventional casting molds [25, 76-77]. The predicted trend is consistent with both the measurements and previous calculations [23]. The distortion depends on hot face temperature and water box rigidity,[23] and is influenced greatly by the casting speed, mold construction, slab width and other parameters, so a quantitative match is not expected. Permanent mold width contraction was also reported after casting [23]. Fig.4.19 shows the calculated evolution of total contraction of the mold width at the top of the funnel mold during a typical campaign of 70 casting sequences. Each sequence is assumed to 74 Measured gap -0.50 -0.40 -0.30 -0.20 -0.10 0.00 Distortion of interface (mm) Fig.4.17 Distortion of wide and narrow face along the line where they meet in the parallel mold 75 s o -1.0 -; -1.2 - : -1.4 0 parallel mould top - - parallel mould bottom [231 X Thomas (1933mm slab, calculated) \u2022 Ozgu (1933mm slab)1\"1 O Kweon (1200mm slab)(7<I A Carlsson (1680mm slab)1771 X X 200 _, , , , , , , , , , 400 600 800 Distance across the wide face (mm) 1000 1200 Fig.4.18 The behavior of back plate distortion for parallel mould during operation (relative to zero along centerline through narrow face) 76 0.8 4 H 1 1 \u2022 Contraction of mold width \u2022 Cu-Ag (by T.Hashimoto)1271 Cu-Cr-Zr (by T.Hashimoto) Measured (after 35 Oheats) 0.4 4 0.2 4 ,-\" H h -'\u2022 :\u2014*=i top Copper wide face bottom 10 20 30 40 50 Thermal cycle 60 70 80 Fig.4.19 Comparison of calculated and measured total mold width contraction during typical casting campaign for the funnel mold 77 include 5 heats (average) or 4 hours, which involves a single thermal cycle of filling and emptying the mold. This is compared with the measured width contraction after 350 heats and with computations by Hashimoto [27] for a conventional mold. As can be seen in this figure, most of the permanent deformation occurs during the first thermal cycles (sequences). It also reveals a very small, but roughly constant decrease in width for each subsequent sequence, leading to a total width contraction of about 0.7 mm (0.042 percent of residual inelastic strain) after the campaign. This is comparable to the measured value of 0.5 mm. Both the measured and calculated values are comparable to the prediction of permanent mold width contraction by Hashimoto [27] and Thomas [23], who predicted that permanent contraction varies from 0.02 percent to 0.08 percent, depending on the constraint of mold. The small contraction for the high temperature mold in this work is due to the lack of constraint offered by the oversized bolt holes in this mold (see Chapter 5). 4.6. Comparison of parallel and funnel mold Next a study was undertaken to compare the thermo-mechanical behavior of parallel and funnel molds using the quarter mold model. To enable a comparison isolating the effect of geometry alone, heat flux, q, in both molds was defined by the following equation [78,79]. q(kWlm2) = 5403 - 990^(sec) (4.4) Fig.4.20 shows the temperature and thermal deflection of the wide face during operation for this heat flux profile for the different mold shapes. The figure also shows 7 8 0 100 200 300 400 500 Tempeature (\u00b0C) (a) -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Displacement toward mold hot face (mm) (b) Fig.4.20 Comparison of thermo-mechanical behavior of parallel and funnel mold during operation (a) Hot face temperature profile (b) Distortion along different vertical sections 79 can be seen from Fig.4.20(b), the deflection behavior for the parallel mold does not change much with position across the mold width. However, the funnel mold deflection varies considerably with position. Specifically, the center of the mold width corresponding to the funnel region has less deflection while the edge region of the funnel has more deflection, relative to the parallel mold. These phenomena can also be observed when distortion of the hot face is plotted across the mold width at the 75 mm position below the meniscus in Fig.4.21. The maximum deflection in both molds is relatively small (<0.3 mm). The effect of mold plate thickness on mold distortion is due mainly to the effect of effective plate thickness on the hot face temperature [23]. These different behaviors of distortion can be explained by the difference of mold shape on plate thickness. In the parallel mold, the plate thickness is constant. However, the funnel mold plate has a variable thickness across the mold width. Specifically, at the center of the funnel mold, the plate thickness is less than that of the parallel mold. It gradually becomes thicker, and, at the edges of the funnel mold, the plate thickness is larger than that of the parallel mold. Although the effective mold wall thickness (distance between the cooling channel and the hot face) is the same for both molds, the plate thickness still has a small effect on distortion. Specifically, the regions with a thick plate have slightly more distortion (for the same hot-face temperature). This agrees with the results of Thomas [23]. Fig.4.22 shows the deflection of the backing plate relative to its original position along the wide face. Distortion of the backing plate for the funnel mold is slightly larger than 80 0.6 i 0.4 -F -0.8 i -i.o r \u2022 1 1 1 i ' 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 ' 1 1 i 1 1 i 1 1 1 1 i 1 0 100 200 300 400 500 600 700 800 Distance across wideface from centerline (mm) Fig.4.21 Mold distortion of parallel and funnel molds across the wide face at 175mm below mold top during operation 81 0.2 Fig.4.22 The behavior of back plate distortion during operation according to the mold shape 82 that for the parallel mold, due to its larger plate thickness at the corner (junction with narrow face). When casting a slab, gaps between the shell and mold tend to form near the narrow face of the mold, because of shrinkage of the solidifying shell on the wide face. The size of the gap is controlled by imposing taper along the mold walls. In addition to the shrinkage, it is also important to know how the mold distorts, because this contributes to the mold taper. Fig.4.23 shows the displacement of the narrow face of the mold, which is qualitatively similar to the wide face. This behavior is affected mainly by the rigidity of the narrow face backing plate. Any differences between the behavior of the funnel and parallel molds should be due to differences in the narrow face backing plate, which were not studied. 83 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Narrow face distortion (mm) Fig.4.23 Predicted distortion profiles down the narrow face of parallel mold 84 5 MECHANISM OF MOLD CRACK FORMATION 5 . 1 Introduction The thin-slab mold for continuous casting of steel is prone to surface crack formation, particularly in funnel-shaped molds. This is likely associated with the higher casting speeds needed for this increasingly popular process to be efficient. Higher casting speeds generate higher heat flux, which cause greater temperatures and stresses in the mold. Cracks pose a serious quality problem, in addition to the obvious loss of mold life. This is because meniscus cracks may locally retard cooling of the steel shell beneath them, and lead to longitudinal crack formation and other defects in the steel product. Despite its importance, the thermo-mechanical behavior of the mold and mold crack formation has received little attention and is not fully understood. Furthermore, most previous research has focused on conventional casting, which has relatively lower mold temperatures, due to the lower heat flux at the lower casting speeds (1 - 2.5 m\/min.) relative to thin slab casting (3.5 - 5 m\/min.). Despite the high level of understanding that has been achieved, several questions of practical importance to mold crack formation remain to be answered. Why does a funnel mold have more cracks than a parallel mold? Why do not cracks always form at the transition area of the funnel mold, where O'Connor found the highest stress? What is the effect of mold alloy? How many cycles to failure can an ordinary mold withstand with no constraint problems? What is the relative importance of thermal cycling due to metal level fluctuation compared with major level changes when emptying and refilling the mold ? This chapter will deal with some of these issues. 85 This work was undertaken to investigate crack formation in thin slab molds. In the previous chapter, the thermal and mechanical behavior of both parallel and funnel thin slab molds were investigated quantitatively based on plant measurements and three-dimensional elastic-visco-plastic finite element models. In this chapter, these models are applied together with a metallurgical analysis of the cracks in order to gain new insight into the mechanism of mold crack formation. 5.2 Morphology of mold cracks Thin slab casting molds still suffer from mold cracks, despite having implemented the suggestion of minimizing constraint by employing oversized bolt holes (see Chapter 4). The first step of this study was to perform a metallurgical analysis of typical mold cracks obtained from an operating thin slab casting funnel mold after 345 heats of casting. Photographs and micrographs of the mold cracks of interest in this study are presented in Figs.5.2-5. Sometimes, cracks were observed after only 2 or 3 casting sequences, but at other times, molds were in service with no cracks for up to 70 sequences. Fig.5.1 shows a schematic funnel mold illustrating the location of the mold cracks. Most of the cracks formed just below the meniscus, and were located about 360 mm from the center of mold, at the transition region of the funnel shape in the same location as observed by O'Connor [20]. A closer examination of the mold surface around the crack reveals two distinct regions, which are identified in Fig.5.2. The top and bottom part of the specimen are dark gray (Fig.5.2 A), which corresponds to the Cr coating layer. The region around the crack, which is found near the meniscus, is yellow (Fig.5.2 B). The composition of 86 Fig.5.1 Schematic diagram of the funnel mold showing location of mold cracks 87 Fig.5.2 Photograph of (a) mold crack and (b) magnified view 88 this thin layer was identified as a brass, using the energy-dispersive spectrum (EDS) on the scanning electron microscope (SEM) (Fig.5.6). Fig.5.3 shows a transverse section through the mold crack, which penetrates to a depth of about 2.8 mm below the mold surface. The crack seems to be intergranular just below the mold surface, but most of its interior length has the characteristic features of transgranular fracture. This is verified from an S E M fracture analysis in Fig.5.4, which shows the intergranular fracture surface found just beneath the mold surface. Fig. 5.4 also shows a wedge crack at the triple point of grains, which is characteristic of creep fracture at high temperature. An examination of the fracture surface at the crack root in Fig.5.5 reveals another fracture morphology that is believed to be fatigue striations. These analyses suggest that the crack is initiated at the surface at high temperature, and is then propagated deeper by alternating thermal stress at lower temperature. This transgranular fatigue crack propagation at lower temperatures is consistent with previous research [30,80-82]. The high-temperature intergranular fracture is consistent with observations of Ashida [81] for all dispersive Cu alloys including Cu-Cr-Zr. However, it contrasts with Gravemann [30] and Collins [80] reports for Cu-Cr-Zr alloy, where zirconium was found to segregate to grain boundaries. This was proposed to be beneficial by forming large harmless precipitates, which decrease stress concentration, avoid grain boundary sliding, prevent void initiation around precipitates, and avoid intergranular fracture [30, 80]. Solid solution strengthened materials such as Cu-Ag alloys are known to have higher fatigue resistance, particularly against intergranular fracture [81]. Another striking feature of the crack occurrence is that near the top surface of the mold, the Cr plated layer is absent from around the crack as shown in Fig.5.6. Instead of 89 Fig. 5 . 3 Optical micrograph of the cross section of mold crack 90 91 92 the Cr layer, a brass layer was formed just beneath the hot face of the mold, in which the Zn concentration reached up to 16% from EDS analysis. The only possible zinc source appears to be the molten metal, which varied from 10 to 20 ppm Zn or higher (up to 60 ppm) depending on the scrap and dust composition in the electric furnace. Zinc absorption into the hottest portion of the mold surface was also reported by O'Connor et al.[20], but only where its temperature exceeds 425\u00b0C. These results suggest that the hottest portion of the hot face of the copper lost its protective Cr layer, exceeded 425 \u00b0C, was severely attacked by Zn in the molten metal, and formed brass. Once zinc attacks the copper plate, it preferentially diffuses along the grain boundaries rather than the matrix. This makes the grain boundaries more susceptible to cracks. Once a short crack is initiated, its tip acts as a stress raiser for crack propagation via transgranular thermal fatigue. 5.3 Thermal stress model description Several models were used to investigate the effect of mold geometry and temperature profiles on the thermal mechanical behavior, its associated crack occurrence and failure criteria for copper thin-slab casting molds. Two-dimensional (2-D) finite-element models were applied to calculate temperature and the corresponding stress and strain fields in the parallel and funnel mold using the commercial stress-analysis package, ABAQUS 5.8 [71]. Simulation conditions are given in Table 5.1 and in Chapter 4. Horizontal sections of one-quarter of the mold (2-D quarter model), including part of the water jacket were taken at the height corresponding to the highest mold temperature, as shown in Fig.5.7. 94 Table 5.1 Simulation conditions for 2-D slice modelling Mold type Parallel mold Funnel mold Copper plate thickness (mm) 60 57-80 Mold width (mm) 1660 Copper alloy 0.5wt% Cr - 0.1 wt% Zr Water slot-hot face distance (mm) 25 25-27 Bolt length (mm) 445 Bolt diameter 16 Bolt hole diameter in water box (mm) 24 Water slot heat transfer coefficient (W\/m2K) 38.45 Cooling water temperature (\u00b0C) 38 95 \u2022\u2022*\u00a5:. S\u00bb8S8:fi|:i SSgSSS'Sii-S mmssr, 96 These slice sections were assumed to deform in plane strain. This section is reasonable because the surface cracks were always longitudinal along the mold length. To obtain accurate predictions from a 2-D model of 3-D phenomena, special boundary conditions were developed in order to match the 3-D model predictions. First, heat flux boundary conditions at the hot face of the 2-D quarter model were chosen to be lower than physically measured (Chapter 4), in order to match the temperature predictions of the full 3-D segment model. To obtain boundary conditions for the stress model, the 3-D quarter model was first run based on the thermal load described in Chapter 4, to reproduce 50 thermal cycles experienced in the real mold shown in Fig.5.8. Symmetry planes at the wide-face centers are mechanically constrained to prevent normal displacements. To realize the pre-tension force into the 2-D quarter model, the x and y displacements of the backing plate at each tension-bolt position from the 3-D analysis (3-D quarter model) described in Chapter 4 were imposed as fixed-displacement boundary conditions on each tension bolt position in the 2-D quarter model. For example, Fig.5.9 shows the displacement boundary condition imposed on the 2-D quarter model at the position of 376 mm from the parallel mold centerline to realize the pretension force. Table 5.2 shows the maximum displacement at other bolts. During heat up at the start of each continuous casting sequence, there is some slight sliding between the mold and the backing plate (water box), which causes the tension bolts to move relative to the bolt holes. Since there are no threads for the tension bolt in the water jacket itself, the constraint condition due to the contact is governed by the clearance between the bolt and the bolt hole in the water jacket. As can be found from Table 5.1, this clearance is 4 mm. This is safely larger than both the dimensional changes 97 600 500 4-400 4-o <D u. C3 u\u00ab ID 9- 300 u o \u2022r 200 o 100 4- A U 412 4 D A M 6 8 10 Casting time (hrs) 12 14 16 Fig. 5.8 Thermal pattern of hot face imposed on the 3-D stress analysis 9 8 -0.10 -I 1 1 1 1 1 1 1 1 1 r 0 5 10 15 20 25 30 35 40 45 50 Casting sequence Fig.5.9 The change of displacement at location of 376mm from the center of backing plate with casting sequence at the meniscus of parallel mold 99 Table 5.2 The maximum displacement (in mm) at the tension bolt of outer backing plate after 50 sequences of casting (3-D quarter model) Bolt position Width displacement Thickness displacement (from center of ( a f t e r c o o l i n S ) ( d u r i n 8 \u00b0 P e r a t i o n ) mold) Parallel Funnel Parallel Funnel 0 -0.002 -0.003 0.136 0.209 188 0.005 0.005 0.169 0.243 376 0.008 0.009 0.225 0.306 564 0.016 0.017 0.285 0.380 752 0.019 0.021 0.348 0.461 100 of the tension bolt diameter, which are less than 0.2 mm, and its horizontal movement, which is a maximum of 1.36 mm at the farthest bolt from the centerline. Thus, the contact surface between the mold plate and the water jacket is relatively free to slide in the width direction as friction was found in Chapter 4 to be negligible. Mathematically, this can be represented with interface elements to simply prevent penetration of the copper plate into the water box. This justifies the boundary conditions used on the 3-D quarter model in Chapter 4. Constraints in the horizontal plane were obtained at each thermal cycle from the results of the 3-D quarter model described in Chapter 4. They were applied as fixed-displacement boundary conditions at the interface of the mold and water jacket of the 2-D quarter model at the bolt locations. Fig.5.10 shows the constraints imposed to restrain expansion in the thickness direction on the parallel mold during heat up at steady state operation. Further details of the elastic-visco-plastic stress model, including the temperature-dependent yield stress function for isotropic hardening and the equation for primary creep of Cu-Cr-Zr are given in Chapter 4. Temperature profiles around the meniscus and crack region were studied using the 3-D segment model, and the heat flux data obtained in Chapter 4. An attempt has been made to refine the heat-flux profile predictions around the meniscus region. There are many popular forms of empirical equations for heat-flux profile as a function of distance in the mold [83-87]. Among them, the following three functions were selected for fitting the heat-flux profile: 0.6 H\u2014I\u2014I\u2014I\u2014i\u2014I\u2014i i\u2014i\u2014i\u2014I\u2014i\u20141\u2014'\u2014i\u2014I\u2014i\u2014i\u2014i\u2014i\u2014I\u2014I\u2014I\u2014i\u2014i\u2014I\u2014i\u2014i\u2014i\u2014i\u2014I\u2014i\u2014i\u2014i\u2014i\u20141\u2014i\u2014i\u2014i\u2014i\u2014I\u2014i\u2014i\u2014i\u2014i\u2014I\u2014r 0.0 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I ' 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 i 1 ' ' ' [ ' ' 1 ' I ' 0 100 200 300 400 500 600 700 800 900 Distance from the mold center (mm) Fig.5.10 Vertical displacement across the mold width at the interface of mold and back plate during the operation for the parallel mold 102 ql(kW\/m2) = 6\\07-&2l-t(s) Linear function (5.1) q2(kW\/m2) = 8420 - 2987\\Ji(s) Square root Function (5.2) q3(kW\/m2) = 6516exp[-0.2f(s)] Exponential function (5.3) Here, q is the heat flux (kW\/m2) down the mold length, t is the residence time (seconds) of the strand below the meniscus. Fig.5.11 compares the heat flux profiles calculated along the centerline of mold for these three equations. Note that all three profiles roughly satisfy the heat flux known at positions where thermocouple measurements were installed (see Chapter 4). The meniscus heat flux is seen to vary from 8 to 6 MW\/m 2 , depending on the equation. Next, this study presents results and discussion of several different studies using these models. 5.4 Comparison of funnel and parallel mold thermal-mechanical behavior To isolate the effect of mold shape alone on temperature, stress, and crack occurrence, the same heat flux (4 MW\/m 2 ) was imposed on 2-D quarter models of the funnel and parallel molds. Fig.5.12 compares the temperature distributions calculated along the hot face of the two molds. As seen in this figure, the funnel mold shows a variation of temperature across the mold width despite the constant heat flux. In particular, the region around the funnel transition region where the cracks occurred is hotter by about 20\u00b0C. This phenomenon is attributed to local variations in the slot geometry for the funnel mold. As shown in Fig.5.13, the distance between the hot face of the mold and the slot root (slot-depth) is not uniform across the funnel mold width. In particular, the funnel 103 T - i \u2014 i \u2014 i I i i\u2014i i 1\u2014i\u2014i\u2014i\u2014i\u2014(\u2014i\u2014i\u2014i\u2014i\u2014f\u2014i\u2014i\u2014i\u2014i\u2014I\u2014i\u2014i\u2014i\u2014r\u2014I\u2014i\u2014i\u2014i\u2014i\u2014\\\u2014i\u2014r\u2014-i\u2014r Distance from the meniscus (mm) Fig.5.11 Comparison of heat flux profiles at the meniscus region 104 500 450 4-0 100 200 300 400 500 600 700 800 Distance from the center of mold (mm) Fig. 5 . 1 2 Comparison of hot-face temperature distribution across the wide face with different mold shape ( 2 - D quarter model) 105 28.5 \u2022+\u2022 28 - : 24.5 +\u20141 1 1 \u2014 1 1 1 1 1 1\u2014I 1 1 1 1 1\u2014' ' ' 1 1\u2014 0 200 400 600 800 Distance from the center of mold (mm) Fig.5.13 Slot depth profile across the mold width at the meniscus for the funnel mold 106 transition region has a slightly greater effective thickness, with slot roots ranging from 0 - 3 mm further from the hot face than the others. This is because the slots are machined to equal depths in groups of 3 or 4 slots. Typical stress and strain results from the 2-D quarter model are shown in Figs.5.14 and 15. Fig.5.14 compares the Von Mises stress profiles across the hot face width of both molds. Fig.5.16 compares the inelastic strain (equivalent plastic strain plus equivalent creep strain), [88] distributions across the hot face together with the region of crack occurrence. Where the hot face temperature is higher in the funnel mold, the stress and strain results are also higher. Thus, both of these fracture criteria correctly indicate the location of cracks and suggest a relationship with the water slot array. To gain insight into the crack formation process, Fig.5.16 plots the evolution of stress and strain over an entire casting campaign at a meniscus position on the mold surface, located where cracks are formed. When the mold temperature increases from room to operating temperature, the mold tries to expand towards the molten steel, but is constrained in x by the cold copper and in y by the bolts on the water jacket, forcing most of the mold into compression. The highest compressive stress occurs on the hot face of the mold where the temperatures are highest. This compressive stress causes plastic yielding accompanied with time by creep strain, leading to residual inelastic compressive strain in the hot face. Upon cooling, this strain causes the mold to try to bend, but is once again restrained from doing so by the bolts against the backing plate. This causes the former hot-face to go into tension, which might initiate a crack. The tensile stress increases after each cycle, encouraging crack propagation. Fig.5.17 compares the stress-strain hysteresis loop after 50 sequences for both mold types. The maximum tensile stress 107 200 100 200 300 400 500 600 700 800 Distance from the center of mold (mm) Fig.5.14 Comparison of Von Mises stress across the hot face with different mold shapes 108 Fig.5.15 Comparison of inelastic strain distribution across the hot face with different mold shapes 109 400 -300 I 1 1 1 1 I 1 1 1 1 i 1 1 1 1 I 1 1 ' 1 I 1 1 1 1 I 1 1 ' 1 I 1 1 ' 1 I -0.20 -0.10 Q.00 0.10 0.20 0.30 0.40 0.50 Total strain (%) Fig. 5.16 Stress-strain hysteresis loop on the hot face of the parallel mold no 350 -t -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 Total strain (%) Fig.5.17 Comparison of stress-strain hysteresis loop after 50 sequences in after cooling down seems to be slightly higher in the funnel mold, where the cracks occur. The magnitude of the differences in both stress and strain between the cracked and non-cracked locations is very small, however. Thus, these results explain only the location where cracks initiate in the funnel mold. They do not completely explain the difference in susceptibility between the funnel and parallel mold shapes. 5.5 Funnel mold analysis using temperature and crack measurements 5.5.1 Measured temperature profiles To investigate the relationship between temperature profile, stress, and cracks near the meniscus, mold temperatures were measured at the meniscus in a funnel mold. Fig.5.18 shows the temperature response at different positions across the hot face at the meniscus during steady state casting. The temperature 376 mm from the mold center is higher than other locations by 50\u00b0C. This was expected, based on the analysis presented in the previous section, due to slot variation. However, the same maximum temperature at this location was observed in the parallel mold as described in Chapter 4, so other phenomena appear to be important also. 5.5.2 Simulations of horizontal profiles Based on the measured temperatures, the 2-D quarter model was used to calculate temperature, stress and strain distributions in the funnel mold. Fig.5.19 shows the temperature distribution across the hot face and thermocouple location, including the 112 C3 S. 60 40 + 20 Center 564mm 4000 5000 6000 7000 Casting time (sec) 8000 9000 Fig.5.18 Temperature response measured in funnel mold at the meniscus during steady state (100mm below mold top) 113 Fig. 5.19 Comparison of calculated and measured mold temperatures and corresponding hot face temperature and heat flux profiles across wide face at the meniscus of funnel mold (2-D quarter model) 114 corresponding artificially low heat flux profile, adjusted for the 2-D quarter model mesh. It can be seen that hot face temperature variation is significantly more severe than at the thermocouple depth. The funnel transition region is more than 70\u00b0C hotter than the center region of the funnel mold. O'Connor [20] suggested that the outer bend of the funnel region is hotter due to convergent heat flow. This work, however, suggests that the peak in this region is greater and wider than can be explained by this phenomenon alone. It is also greater than could be expected from the local variations of water slot depth alone. Stress and plastic strain in the transverse plane follow the temperature profile, and have peak values which coincide with the region of crack occurrence, as shown in Figs.5.20 and 5.21. These results provide even stronger evidence that the cracks initiate in the funnel transition area due to higher temperatures in this region. 5.5.3 Simulations of vertical profiles To investigate why the cracks form at their particular distance vertically down the mold, the hot face temperature profile near the meniscus was simulated with the 3-D segment model using the 3 different heat flux equations. Fig.5.22 shows the effect of the different heat-flux profiles on hot face temperature predictions. Regardless of the exact equation, the peak temperature was always found about 20-30 mm below the molten metal level, which corresponds almost exactly with the region of crack occurrence. Some cracks were observed to extend to above the meniscus, where the temperature is relatively low. This was hypothesized to be crack propagation from cyclic fatigue due to metal level fluctuations. 115 200 150 2 , ioo -50 i i i \u2014 r ~ \\ \u2014 i - i \u2014 i i | i - i \u2014 i \u2014 i \u2014 r - i \u2014 \u2014 I \u2014 i \u2014 I \u2014 i \u2014 \u2014 I \u2014 i \u2014 i \u2014 i \u2014 \u2014 i \u2014 i \u2014 i \u2014 i \u2014 \u2014 r * < Crack locations 100mm below top of mold + _ l I I [_ _ J I I I L_ + 0 100 200 300 400 500 600 700 800 Distance from the mold center (mm) Fig.5.20 Stress (am) profile across wide face of funnel mold 116 0 100 200 300 400 500 600 700 800 Distance from the mold center (mm) Fig.5.21 Inelastic strain profile across wide face of funnel mold Fig.5.22 Effect of heat-flux profile uncertainty on hot face temperature predictions ( steady 3-D segment model) 118 5.6. Metal level fluctuation study 5.6.1 Metal level and meniscus temperature fluctuation measurement Metal level fluctuations are a potential source of cyclic thermal loading on the mold. They can occur in spite of an automatic control system, due to turbulence in the mold related to transient variation in the mold flow pattern. To quantify the magnitude and frequency of these fluctuations on meniscus temperature fluctuations, both metal level signals and meniscus thermocouple measurements were analyzed using Origin [89]. Fourier analysis was applied to determine the frequency distribution of the metal level signals of both the parallel and funnel mold based on the plant data presented in Chapter 4. Fig.5.23 (a) shows the computed intensities, which represent the relative contribution of each frequency to the overall level fluctuation signal. It is evident that the funnel mold has mainly low- frequency level fluctuations averaging about 0.05Hz, compared to the generally high- frequency fluctuations of 0.3 Hz in the parallel mold. This corresponds to fluctuation periods of 20 second (funnel mold) and 3.3 second (parallel mold). The reason may be due to the thicker funnel mold, which contains more fluid volume, and delays the side to side transient sloshing observed by Honeyands et al. [90]. Next, the meniscus temperature variations measured at 100 mm below the top of the funnel mold were analyzed. Fig.5.23(b) compares the power spectra from both metal level and temperature response of the funnel mold. The measured temperature has a peak value at around 0.05 Hz, which corresponds exactly with the frequency of the metal level fluctuations. Each temperature peak is clearly caused by a metal level fluctuation. This agrees with the findings of Lai et al. [64] that mold thermal response provides a good 119 0.2 | 1 1 ' ' I ' 1 1 i I i i i ' 1 1 1 1 1 0.4 fail i i i I i i i i I i i i i I i i i i I i i i i I i i i i I 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency (Hz) (b) Fig. 5.23 Power spectra of mold level and temperature response (a) Comparison of metal level between parallel and funnel mold (b) Comparison of metal level and temperature response in the funnel mold 120 indication of the meniscus level fluctuation. 5.6.2 Fluctuating temperature simulation The transient 2-D quarter model was applied to simulate the effect of metal level fluctuations on temperature, stress and strain variation near the meniscus. First of all, a sinusoidal heat flux equation was adopted, so that the temperature field produced by the 2-D quarter model matches the measured temperature. Specifically, the artificially low heat flux was varied between 2.15 kVvVm2 and 3.39 kW\/m 2. Fig.5.24 (b) shows the calculated temperature histories at both the hot face and thermocouple locations. The thermocouple location shows a temperature range of 100 to 120\u00b0C (an amplitude of 20\u00b0C), which closely matches the measured thermal response in Fig.5.24 (a). The corresponding hot face location temperature varies from 260 to 325\u00b0C (an amplitude of 65\u00b0C), as shown in Fig.5.24 (b). Having calibrated the model, the fluctuation frequency was varied. In order to isolate the effect of level fluctuation alone on temperature, stress and strain, all other conditions were kept constant and both simulations were performed on the funnel mold geometry. Fig.5.25 compares the transient mold temperature response at high frequency (0.3 Hz), low frequency (0.05 Hz), and steady state, respectively. As shown in this Figure, increasing the fluctuation frequency lowers both the peak temperature and its amplitude. Specifically, while the 0.3 Hz frequency has a temperature fluctuation amplitude of 20\u00b0C at the hot face, the 0.05 Hz frequency has an amplitude of 75\u00b0C. The temperature fluctuation at the thermocouple depth is much smaller (0.7\u00b0C for 0.3 Hz and 20\u00b0C for 0.05 Hz). At frequencies of 0.016 Hz or less, the temperature and stress amplitude are virtually the same as at steady state., In this case, hot face 121 20\u00b0C 6400 6450 6500 6550 6600 6650 Casting time (sec) (a) 6700 6750 6800 400 360 320 Q 280 O H 240 | 200 g 3 H 160 120 80 40 i i ' i l \u2014 Hot face temp. Thermocouple temp. 20sec 20 C J AAAAAAAAAAAAAAAI 50 100 150 200 250 Casting time (sec) (b) 300 350 400 Fig.5.24 Thermal pattern of funnel mold at the meniscus (a) Measured temperature profile at the thermocouple position (b) Calculated hot face and thermocouple temperature 122 400 350 4 \u2014 Steady (Max.) \u2014 Freq. =0.05 Hz \u2014 Freq. =0.3 Hz - Steady (Min.) 0 50 100 150 200 250 Casting time (sec) 300 350 400 Fig.5.25 Transient hot face temperature response for the various cases of thermal cycle 123 temperature fluctuation can also be confirmed with calculations using the 3-D segment model. Fig.5.26 shows the effect of metal level on the steady-state hot-face temperature prediction, based on \u00b14 mm level changes. This was achieved by shifting the standard linear heat flux profile in Fig.5.11 vertically by \u00b14 mm for the boundary conditions of the 3-D segment model. As can be seen, the amplitude of hot face temperature variations at the meniscus (100 mm below the mold top) was about 94\u00b0C (range of 228-322\u00b0C), which is similar to the 2-D quarter model predicted amplitude of 110\u00b0C (range of 230-340\u00b0C). 5.6.3 Stress-strain fluctuation simulations Based on the above temperature fields, stress and strain were calculated for the different metal level fluctuation frequencies using the 2-D quarter model. Fig.5.27 shows the stress-strain hysteresis in the hot face of the funnel mold. The x inelastic strain (plastic + creep in the width direction) per thermal cycle is about -0.0024% (0.3 Fiz) and -0.023% (0.05 Hz), respectively. This result suggests that low frequency metal level fluctuations may damage the mold more than those at high frequency. It is noted that the alternating stress and mean stress from the fluctuation of molten steel are both entirely compressive. Relative to detrimental tensile mean stresses, it is well-known that compressive mean stresses are beneficial [91]. For example, compressive stresses increase the fatigue strength of alternating high cycle fatigue by more than 50 percent [92]. Although the level fluctuations create variations that are wholly compressive, tensile forces are created at the end of each sequence, as shown in Fig.5.16. Low cycle fatigue cracks are the result of accumulated inelastic strain damage coupled with tensile forces. Thus, it might be possible that the increasing compressive inelastic strains due to level 124 Fig. 5.26 Effect of metal level position on the hot face temperature prediction (3-D segment model) 125 Fig.5.27 Stress-strain hysteresis loop at the meniscus hot face surface for the funnel mold with different frequencies of metal level fluctuation 126 fluctuations could cause faster propagation of the fatigue cracks. Fig.5.28 shows the trend of plastic and creep strain for the case of low frequency of metal level fluctuation. From this figure, it can be found that almost all of the plastic strain occurs at the first thermal load step. The creep strain increases monotonically with time. Although the increase per cycle is small (only about 0.0002%), the large number of cycles allows a significant amount of creep to accumulate (e.g. up to 0.14% after one 4-hour casting sequence of 720 cycles at 0.05 Hz). During the fluctuation of molten steel level, the mold fatigue life appears to be governed by the creep due to the cyclic thermal loading. 5.7 Fatigue crack prediction In continuous casting, the fatigue load is generated by thermal cycling from both fluctuations of the metal level and major casting transitions, such as SEN changes, tundish and ladle changes, etc. The latter have very low frequency and high amplitude compared to the rapid shallow level fluctuations. 5.7.1 Failure criteria There is a paucity of fatigue-life data available for continuous molds, in particular Cu-0. l%Cr-0.15%Zr composition. One relevant equation is that reported by O'Connor, who used fatigue data for a Cu-2 pet Cr alloy, which was tested at 538 \u00b0C at a strain rate of 2xl0\"3 s\"1 [20j and corrected for the presence of creep, suggesting the following fatigue equation for number of cycles to failure, [20]. 127 0.25 I I I I I ! I I I I I I I 0.24 .5 0.23 ca c 1 0.22 '3 cr tu 0.21 4- \u2022 Plastic strain Creep strain 0.20 100 200 + + 300 400 500 Casting time (sec) 600 700 800 Fig.5.28 Inelastic strain profiles with casting time 128 Nf=l.66xl02x(Aeiny275 ( 5 4 ) where & e . n is the inelastic strain range (percent) of stress-strain hysteresis Another equation was developed by Ashida [81], who conducted thermal fatigue tests for Cu-0.8%Cr-0.2%Zr. Fully-reversed cycling, out-of-phase thermal conditions were employed, ranging from tensile stress at room temperature to compressive stress at 350\u00b0C with a frequency of 4.11 x 10\"3 Hz. Nf =7.08xl0 2 x(Asin)']39 (5.5) This material and loading conditions are quite similar to that in the mold plates of this study. Collins [80] also conducted isothermal low-cycle fatigue for Cu-0.5%Cr-0.1%Zr-0.03%Mg material in strain control, at temperatures up to 400\u00b0C and frequencies of 0.0017 to 0.17 Hz, suggesting the following equation. Nf = 1.43x103 x ( A ^ J - 2 0 . (5.6) These test frequencies roughly correspond to level fluctuation cycles in the present work. Collins states that for transgranular fracture such as mold cracks found in the present study, the fatigue equation is independent of temperature and frequency [80], 129 Fatigue life predictions are plotted logarithmically against the inelastic strain with the different criteria in Fig.5.29. In general, it is known that the fatigue life is influenced greatly by alloy composition, grain size [80,81], temperature [93,94], cycling frequency [95], holding time [96,97], loading pattern [98-100], surface condition of the sample, environment [101,102], test method [103] and other factors. Therefore, the differences in fatigue life are not unexpected. Unfortunately, none include any wholly compressive cycles, so the life cycle cannot be exactly predicted for the present conditions. Nevertheless, a rough estimate of continuous casting mold lifetime was determined by applying the results of these 3 criteria. 5.7.2 Lifetime prediction In order to predict fatigue life, inelastic strain was determined from the stress-strain hysteresis loops for three different cases of metal level fluctuation (0.05 Hz and 0.3 Hz) and SEN change (single thermal cycle defined in Chapter 4). Fatigue life predictions using all 3 equations for the 3 different types of thermal cycles are summarized in Table 5.3. Considering that the mold cracks were observed in the funnel mold within 70 sequences, the predicted number of cycles to failure from SEN changes alone is much longer than the actual life. When applying equations (5.4) - (5.6) to metal level fluctuations, the lifetime is much shorter. This strongly implies that the metal level fluctuations influence mold life significantly. The results in Table 5.3 also reveal that low frequency metal level fluctuations, such as found in the funnel mold are more harmful to the mold service life than the high frequency fluctuations found in the parallel mold despite their fewer numbers. Very low frequencies (i.e. < 0.016 Hz) are expected to begin to decrease fatigue susceptibility again, as the temperature distribution approaches steady 130 1.0E+02 l.OE+03 1.0E+04 l.OE+05 Number of cycle to failure (N\/) Fig.5.29 Variation of cycle to failure for different fatigue model 131 Table 5.3 Summary of predicted fatigue life with different thermal cycles Inelastic Number of thermal cycles to Number of sequence cycles to strain failure failure (1 seq. = 4Hrs) range(%) O'Connor Collins Ashida O'Connor Collins Ashida SEN change* 0.028 3.09 x 106 1.44 x 106 1.02 x 105 3.09 x 106 1.44 x 106 1.02 x 105 Long period (f=0.05Hz) 0.023 5.30 x 106 2.13 x 106 1.34 x 105 7.37 x 103 2.95 x 103 1.86 x 102 Short period (f=0.3Hz) 0.0024 2.65 x 109 1.95 x 108 3.10 x 106 6.14 x 105 4.50 x 104 7.17 x 102 * Assumed to be period of start\/stop of casting transitions (1 sequence) Table 5.4 Summary of predicted fatigue life with different hot face temperature Hot face Inelastic strain Number of sequence cycles to failure temperature , 0 \/ * o range (\/o) O'Conner Collins Ashida 300 0.028 3.09 x 106 1.44 x 106 1.02 x 105 350 0.046 7.88 x 105 5.33 x 10s 5.11 x 104 400 0.065 3.05 x 105 267 x 105 3.16 x 104 450 0.083 1.56 x 105 1.64 x 105 2.25 x 104 500 0.092 1.17 x 105 1.33 x 105 1.95 x 104 550 0.100 9.22 x 104 112x 105 1.73 x 104 132 state and the number of cycles lowers. The above fatigue predictions apply only to the copper base metal. If a coating material such as Cr is applied to the hot face of the copper mold, the predicted cycles to failure are expected to be longer. Once the Cr coating layer no longer protects it, however, these results suggest that mold life will be very short and will be controlled by metal level fluctuations, which is consistent with metallurgical observations in this study. Fig.5.30 shows the stress-strain hysteresis loop for the funnel mold with different hot face temperatures during 5 sequences. Steady state was reached after just one sequence cycle, which is similar to O'Connor [20]. As can be seen in this figure, with the increase of hot face temperature, the inelastic strain range (width of the loop) increases significantly. Table 5.4 summarizes the fatigue life prediction according to the hot face temperatures with different fatigue equations for the case of SEN transition. Increasing temperature is naturally expected to reduce lifetime, both due to higher strain amplitude and from lower fatigue cycles to failure. 5.8 Mechanism of mold crack formation The results of this study suggest that mold cracks begin with the cracking or loss of the protective Cr coating at the hottest portion of the hot face found just below the meniscus in the transition region of the funnel mold. This is followed by attack of the exposed copper by zinc from the molten steel. The Zn forms a brass alloy, which is susceptible to intergranular creep failure. Once a short, shallow crack initiates, it acts as a stress raiser for crack propagation by thermal fatigue. Propagation is greatly 133 400 .300 I 1 1 1 1 I ' 1 1 1 I 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Total strain (%) Fig.5.30 Comparison of stress-strain hysteresis loop with different hot face temperature for the funnel mold 134 aggravated by metal level fluctuations, particularly the more-severe low frequency fluctuations. To avoid cracks, it is important to decrease the hot face temperature, which will not only reduce coating failure and embrittlement by zinc pick-up, but will also lower the maximum stress and inelastic strain. The hot face temperature can be reduced with smaller effective copper thickness, higher thermal conductivity or increased cooling water velocity. It appears to be important to avoid uneven slot depths that increase hot face temperature, particularly in the transition region of the funnel, which is more susceptible to high temperature, stress, and inelastic strain. Metal level fluctuations must be avoided, particularly those at the most detrimental low frequency range (about 0.01 -0.2 Hz). Finally, it is important to minimize residuals such as zinc in the molten steel. 135 6 S O L I D S H E L L B E H A V I O R I N T H I N S L A B M O L D 6.1 Introduction It has been well known that the heat transfer through the interfacial gap, solid shell growth, and evolution of stress\/strain in the solid shell are dependent on the size of the gap between the solidifying shell and mold. The extent of this gap and the associated mechanical behavior of the shell depend, among other things, upon the position of the mold wall, which is determined by the mold distortion, mold taper and mold geometry. Mold taper has an important influence on the early stages of solidification in the mold, and ultimately on the quality of the continuously cast steel. For example, having too little taper leads to decreased heat flow and high surface temperatures, which results in a thin shell. This in turn can lead to bulging both in and below the mold, with generation of longitudinal cracks in crack-susceptible steel grades. Too much taper also produces significant problems such as excessive mold wear, deformation of the shell, and high friction and binding of the shell, which can result in transverse cracks or even breakouts. The thin-slab process is known to be prone to surface crack formation due to high casting speed and small dimensions, which can result in the formation of a thinner and hotter steel shell and serious metal level fluctuations, leading to uneven solidification. Fig.6.1 shows the longitudinal cracks and the associated breakout slab found in the thin slab casting process. These cracks are believed to initiate at the initial stage of solidification around the meniscus and grow toward the bottom of the mold and below, as in conventional thick-slab casting [50,70,104-114]. Around the meniscus, uneven 136 Longitudinal crack Fig.6.1 Longitudinal crack and the associated breakout slab that found in thin slab casting 137 solidification occurs due to uneven heat removal. This may be aggravated by several operational factors such as varying casting speed [70,109,111], overcooling in the upper spray zone [109,1110], insufficient support below the mold [109,110], mold slot design causing non-optimal water cooling [108,109,1114], loss of mold taper [113,114], improper use of mold powder [50,111], and non-optimal liquid flow patterns in the mold [108,111]. Among these parameters, mold taper is one of the few important casting variables that is readily adjustable and has an important influence upon the air gap between the strand and mold. Therefore, mold taper affects the mechanical behavior of the solidifying shell and the formation of longitudinal cracks. In this chapter, the mathematical model, which has been validated rigorously through comparison with an analytical solution and a plant trial described in Appendix III, will be applied to understand the thermo-mechanical behavior of the solid shell in a thin slab casting mold based on heat-flux profiles extracted from plant measurements. The goal is to investigate the influence of mold taper on the shrinkage of the solidifying shell, its gap formation, and stress evolution for three different thin slab mold geometries. 6.2 Geometry and additional length of solidifying shell of funnel slab Because of the unique characteristics of the funnel mold, there are some very interesting aspects of the design, which are discussed below. Since thin slab casters began operation, several mold geometries have been applied, depending on the type of submerged entry nozzle (SEN). To ensure service life of SEN, refractory material must 138 be thick to endure the erosion by molten steel and mold slag. As shown in Fig.2.2, the thin slab caster accommodates the SEN by a pocket shaped bulge (which is called a funnel) in the meniscus region. Generally, the funnel shape can be characterized by the width and depth of funnel. In this work, two kinds of funnel molds have been studied. Funnel A mold with 75 mm slab thickness, has a 25 mm funnel depth, which results in 125 mm between the broad faces at the center of the mold top. The maximum funnel width is 640 mm at the mold top. Fig.6.2 (a) shows the change of funnel depth along the mold length. As can be seen in this figure, the funnel depth linearly tapers from 20 mm at the meniscus (100 mm below mold top) to 6 mm at a point of 650 mm below the meniscus. This 6 mm funnel is present through mold exit. The other mold, funnel B, with a thickness of 25 mm, has a 52 mm funnel depth and a 950 mm funnel width at a point of 100 mm below the mold top. Both the funnel depth and width taper away at 750 mm below the mold top, thereby the broad faces become parallel as in a conventional parallel mold at the exit of mold as shown in Fig. 6.2 (b). In order to understand the shrinkage of the slab wide face, the amount of funnel retraction' as the strand progresses through the mold must be calculated. This is because an additional length of solidifying shell on the wide face has to be deducted from the total shrinkage amount due to the funnel retraction. Fig.6.3 shows a schematic diagram of the funnel slab geometry. From this figure, the additional length of solidifying shell (LA) is found at each distance down the mold from the funnel half width, a, and the funnel half thickness, b by the following equation. LA=2Rd-a (6.1) 139 25.0 20.0 B 15.0 4-10.0 4-100 200 300 \u2014a\u2014 \u2022 Men i s cu s - m - 2 0 0 m m (from meniscus) \u2014o- ' 4 0 0 m m 6 0 0 m m 6 4 5 m m ' 9 0 0 m m ( M o l d exit) 400 500 Distance from the strand center (mm) (a) 600 60.0 50.0 40.0 + 30.0 -+-0.0 20.0 10.0 + ^ - A - - A - - A - - A - - A - - A - - A - . ' ^S. i ft ft ft ft ft 1 # - B \u2014 Meniscus 9 - 200mm (from meniscus) - O - '400mm - A - 600mm -O- - 645mm H 900mm (mold exit) 100 200 300 400 500 Distance from the strand center (mm) (b) Fig.6.2 Change of funnel depth down the mould length (a) Funnel A (b) Funnel B 600 140 a : runnel half width b : funnel half thickness c : funnel depth R : funnel radius Fig. 7.3 Schematic diagram of funnel slab 141 where, the angle 9 (radians) is 9 = arcsin(^^) a2 +b2 and the funnel radius, R, is R = Ab Fig.6.4 shows the relationship between the funnel radius and the additional length of solidifying shell from equation (6.1). It can be seen that, as the funnel depth decreases, the funnel radius increases. At the same funnel depth, the additional length of the solidifying shell decreases with increasing funnel width. The additional lengths of solidifying shell for both molds studied in this work are 1.67 mm and 7.5 mm, respectively. Both values reflect on the effective mold length of 900 mm. Some relevant design details of the funnel thin slabs studied in this investigation are summarized in Table 6.1. 6.3 Mathematical model description To investigate the thermo-mechanical behavior of continuous cast thin slabs, a 2-D transient thermo-elaso-visco-plastic finite element model [29, 114-116] has been developed. This model tracks the thermal and mechanical behavior of a transverse slice through the continuously cast strand as it moves down through the caster. The model includes separate finite element models of heat transfer and stress generation that are step-wise coupled through the size and properties of the interfacial gap. Stresses arise primarily due to thermal strains, while heat transfer across the gap depends on the amount of shrinkage of the solidifying shell. During each step of analysis, the temperature fields 142 8000 4-15 25 35 45 Funnel depth (mm) 55 65 Fig.6.4 Funnel radius and additional length of shell according to the funnel depth 143 Table 6.1 The geometry of funnel slab in this work Funnel A Funnel B Final strand thickness (mm) 75 50 Mould width (mm) 1600 1500 Mould length (mm) 1000 1100 Funnel depth at the top of mould (mm) 25 60 Funnel depth at the bottom of mould (mm) 6 0 Funnel width (mm) 450 475 Funnel length (from top of mould) (mm) 750 750 Table 6.2 Steel composition used in this work Element C Si Mn P s Wt% 0.04 0.2 0.25 0.010 0.015 144 of the mold and strand are calculated simultaneously, extrapolating from the previous step, neglecting axial conduction. Then, the stress analysis calculates deformation of the strand, stress and the air gap size. Iteration continues until the heat transfer coefficient calculated from calculated gap is converged. 6.3.1 Microsegregation analysis Generally, the solidification of steel during continuous casting does not exactly follow the path of the equilibrium binary Fe-C phase diagram due to the rapid cooling and micro-segregation of other solute elements. To determine the variation of liquid, 8-Fe, and y-Fe fraction with temperature, the microsegregation of solute elements of steel was analyzed using the direct finite-difference method of Kim et al. [116] and Ueshima et al. [117] as described elsewhere [118]. Fig.6.5 shows the calculated liquid, 8-Fe and y-Fe fraction as a function of temperature during solidification of the low carbon steel grade in Table 6.2 and the corresponding linear thermal expansion function used in this study. These results were used to determine the thermo-physical properties of steel given later. 6.3.2 Heat flow analysis The heat flow model solves the 2-D transient heat conduction equation for the temperature distribution in the solidifying shell. The effects of solidification and solid-state phase transformation on the heat flow are incorporated through a temperature-dependent enthalpy function as shown in Fig.6.6. This figure also shows the temperature-dependent conductivity function. 145 0.001 j - -0.013 -0.015 1000 1100 1350 1400 1450 1500 1550 Temperature (\u00b0C) Fig.6.5 Calculated solid fraction fs, 5-Fe fraction, y-Fe fraction, and thermal linear expansion as a function of temperature for low carbon (C=0.04wt%) steel 146 Fig.6.6 Enthalpy and conductivity of low carbon steel (C%=0.04) used in this model 147 The following assumptions are used in this calculation 1) The in-coming metal temperature, liquid level and casting speed are constant and axial heat conduction is ignored. 2) Mold oscillation and friction between the shell and the mold are neglected. 3) The effect of convective heat flow in the liquid region is taken into account using the effective thermal conductivity, keff, for molten steel [120]. (62) keJf =27[1 + 6 ( 1 - \/ J 2 ] Thermal boundary conditions at the shell surface are modeled by use of heat flux data, which were obtained from the plant trial (Chapter 5). The boundary condition is described as a function of casting time (equivalent to the distance from the meniscus), as shown in Fig.6.7. The study of billet analysis (Appendix III) has revealed a markedly lower cooling rate in the corner region of the shell than in its mid-face region. This indicates the formation of an insulating air gap at the mold\/shell interface in the corner region. When applying the heat flux value as a boundary condition, surface temperature in the corner region drops significantly due to the two-dimensional heat flow. To avoid the huge temperature drop in the corner region, a relaxation factor of 0.25 was introduced. The assumed variation of the heat flux profile across the strand surface is shown in Fig.6.8. 6.3.3 Stress analysis 148 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 7.0 \" f - 1 \u2014 1 \u2014 i\u2014 1 \u2014 I\u2014 '\u2014 1 \u2014 1 \u2014 1 \u2014I\u2014 '\u2014 1 \u2014 1 \u2014 1 \u2014i\u2014 1 \u2014 1 \u2014 1 \u2014 1 \u2014I\u2014 1 \u2014 1 \u2014 1 \u2014 1 \u2014I\u2014 1 \u2014 1 \u2014 1 \u2014 1 \u2014I\u2014 1 \u2014 1 \u2014 1 \u2014 1 \u2014I\u2014 1 \u2014 r Distance from meniscus (mm) Fig.6.7 Input and output heat flux profiles down the mold length 149 0 0 - i 1 1 r H 1 1 1 r \u2022Heat flux (wide face) Heat flux (narrow face) 20 40 60 80 Distance from the comer of strand (mm) 100 Fig.6.8 Assumed variation of heat flux profiles across slab surface 150 A. Mold taper and distortion Mold distortion due to thermal expansion is added to the mold taper to define the mold wall position. Fig.6.9 and Fig.6.10 show the mold distortions of the wide- and narrow-faces obtained from the stress model of the mold in section 4.5 and 4.6. These distortion values will be incorporated into the mold taper as displacement boundary condition. B. Thermal strain Thermal strain arises from the volume changes caused by changing temperature and phase transformation. This was calculated from the temperature determined in the heat transfer analysis and the thermal linear expansion of steel (TLE), which is found from the phase fractions from the microsegregation analysis and the specific volume, V, of each phase of the steel. TLE(T) = 3 V . v\u20141 (6-3) ' ref V = (fsVs+frVr)fs+V,fl (6.4) where Vref\\s the specific volume at the reference temperature, f&, fp, and\/\/ are the 8 phase fraction in the solid phase, y phase fraction in the solid phase and the liquid fraction, respectively. The Vp and V\\ are the specific volume of 8, \/phase and liquid phase, respectively. The reference temperature is chosen to correspond with the solid fraction of 0.8. The specific volume of various phases is given in Table 6.3, which were obtained from Wray [121]. 151 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.3 I i i i i I i i i i 1 i i i i I I' I i . i i 1 ' i i i I i i I' I i -0.05 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 ' 1 1 I 1 1 1 1 I 1 1 ' 1 1 1 1 1 1 I 1 1 1 1 I 1 1 1 ' I 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) Fig.6.9 Shape of wide face down the mold length, due to mold distortion 152 Casting time (sec) 0 2 4 6 8 10 12 14 9 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 I \u2014 i \u2014 i \u2014 i \u2014 i \u2014 I \u2014 i \u2014 i \u2014 i \u2014 i \u2014 I \u2014 i \u2014 i \u2014 i \u2014 1 \u2014 P -Distance from meniscus (mm) Fig.6.10 Narrow face taper including mold distortion down the mold length 153 C. Effective plastic strain and flow stress in carbon steel At higher temperatures, important to stress development during solidification, inelastic strain from plasticity and creep is important. The following constitutive equation proposed by Han et al. [116, 122-124] is used to describe the flow stress of 8 and y phases at various temperatures and strain rates: sP =Aexp(-Q\/RT)[s\\nh(pK]Um <6-5) a = Ksp\" (6.6) where A and p are constants, Q and R are the activation energy for deformation and the gas constant, m is the strain rate sensitivity, K is the strength coefficient, n is the strain hardening exponent, a is the flow stress, sp is the effective plastic strain. Table 6.4 shows the parameters in the above equation for 8-ferritic and y-austenite phases of steel. D. Elastic Modulus The elastic modulus of steel decreases significantly with increasing temperature. There is still uncertainty concerning the best value of E at high temperatures. The following data by Kinoshita [32] are used in this work. E=1.38xia2T2-225.6T+3.146x10s (kg\/cm2) ' (6.7) 154 Table 6.3 Specific volume of 5-Fe, y-Fe and liquid steel [121] Phase Specific volume (cm3\/g) 5 Phase 0.1234 + 9.38 x IO\"6 (T-20) y Phase 0.1255 + 9.45 x IO\"6 (T-20) + 7.688 x IO*6 Liquid steel 1\/7.035 Table 6.4 Parameters for the constitutive equation [124] Phase A(s_1) (3 (MPa\"1) Q (KJ\/mol) m n 5 Phase 6.754 x 108 0.0933 216.9 0.1028 0.0379 yPhase 1.192 x 1010 0.0381 373.4 0.2363 0.2100 155 E. Treatment of liquid Since elements may be either liquid, solid or mushy, and the volume of liquid in the domain may vary, special care is needed to handle the liquid region. In the present model, negligible (O.SxW4 MPa) stiffness is assigned to those Gaussian integration points whose temperature is above the coherence temperature, assumed to have a solid fraction of 0.7. In addition, thermal expansion is assumed to be zero for temperature above a solid fraction of 0.8. F. Solid shell\/mold contact Interaction between the shell and the mold affects not only the loading on the exterior position of the shell, but also influences the heat transfer significantly. A contact algorithm is applied to restrain the shell elements from penetrating the mold, whose position is defined in Fig.6.9 and Fig.6.10. At each iteration, such penetrations are evaluated, a new global matrix is generated and stresses are resolved. To achieve convergence, penetration parameter is set to 5.0, friction coefficient to 0.2. G. Ferrostatic pressure Ferrostatic pressure from vertical gravity force on the liquid pushes the inside surface of the solidifying shell toward the mold walls and greatly affects gap size and mold heat transfer. It increases in proportion to distance below the meniscus. In AMEC2D, this pressure is applied to every liquid element in the domain at all times. 156 6.3.4 Specification of strand domain Fig.6.11 shows the finite element mesh of a two dimensional horizontal section of the thin slab strand and mold. Symmetry conditions allow a quarter section of the domain to be considered. Model domains consist of 8532 nodes and 8190 four node-elements in a parallel strand, and 216 nodes and 140 elements in the parallel-shaped mold. For the thin slab molds A and B, the F E M mesh contained 8829 and 6867 nodes, and 8476 and 6520 elements in the strand, respectively. The mold mesh is the same as in parallel mold. The element equations are assembled using a single integration point and the equations are solved using Newton-Raphson iteration. The 2-D finite element package, AMEC2D, which developed by Oh et al. at Seoul National University in Korea, was used to construct this model [125]. In this study, only the solidifying strand is modeled in detail. The mold wall is assumed to be rigid and at a constant temperature. The considered slice of the strand, which is one-quarter section of strand, is assumed to be in a plane strain condition, in which the strain along the casting direction is neglected. Symmetry planes at the wide-face centers and the narrow face centers are mechanically constrained to prevent normal displacement. In the funnel strand, symmetry plane at the wide-face centers are applied only at the straight part of strand. This condition enables the strand to experience the strand deformation due to funnel shape change. Detailed dimensions of thin slab shapes and their thermal and mechanical boundary conditions are also shown in Fig.6.12 and simulation conditions are given in Table 6.5. 157 158 159 160 \u2022tf o \u2022 i -H J3 \u2022a 2-3 -2 C \u2022 ^ H J3 C\/3 c o o o cn \u00a33 5 s ^ --3 \u00a7.12 12 fi co \u2014 a o Is !2 \u2122 2 3 \u2022 ^ H \u00a7 -2 S fi J\u00a7 *\u00b0 CO W (D <H -t-< \"~ on < CQ \"3 \"5 fi fi c fi 3 3 PH PH 2\" 'c? 161 Table 6.5 Simulation condition Parallel Funnel A Funnel B Strand thickness (mm) 75 75 50 Slab width (mm) 1260 Meniscus level (mm) 100 Mould length (mm) 900 Casting speed (m\/min.) 3.6 Liquidus temperature [119] 1529 Solidus temperature [119] 1511 Mold taper (%\/m) 0- 1.1 0- 1.1 0-0.8 Funnel depth at the meniscus (mm) 20 52 Funnel depth at the bottom of mold (mm) 6 0 162 6.4 Validation of model Extensive validation of this model is discussed in Appendix III..Further validation of the new feature for funnel slabs was performed as follows. As a strand moves down the funnel mold, an additional solidifying shell took place due to the funnel shape change, which may affect shell shrinkage and associated gap formation. Therefore, this model must predict the same amount of additional shell length as the simple geometric calculation. For this purpose, an analysis is carried out with the assumption of elastic behavior of the solid shell, with elastic modulus of 210 GPa, and Poisson's ratio of 0.35, and no thermal stress. The condition of no thermal stress entails simple imposition of a zero thermal expansion coefficient, so that only the funnel shape change is taken into account. Fig.6.13 compares the change of wide face displacement at the slab corner down the mold length due to the retraction of the funnels for mold A and B, respectively. These figures show that although the additional length is found to be larger than the theoretical results at the initial start of funnel shape change, the calculated final additional length is found to be close to the theoretical results. 6.5. Results and discussion Several complete 2-D model simulations were performed on a 630 mm wide slab with a working mold length of 900 mm. Standard operating conditions assumed a casting speed of 3.6 m\/min. and a superheat of 25\u00b0C. Material properties for plain carbon steel (0.04wt%C) are assumed. 163 0.0 3.0 Casting time (sec) 6.0 9.0 12.0 15.0 0.8 + c 0.0 4.0 - - Geometry calculation \u2022 Model calculation 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) (a) 3.0 Casting time (sec) 6.0 9.0 12.0 + + 15.0 3 - - Geometry calculation \u2022 Model calculation 0 100 200 300 400 500 600 700 800 Distance from meniscus (mm) (b) 900 1.6.13 Change of additional length of the solidifying shell in funnel A (a) and funnel B (b) 164 6.5.1 Temperature, shell and shrinkage behavior Because there are some uncertainties about the material property data, such as the thermal expansion coefficient and the temperature profile in the corner region, which was based on the artificially imposed heat flux as a thermal boundary condition, it is difficult to predict the absolute value of shell shrinkage and the associated gap formation at the slab corner. Therefore, in this study, the qualitative trend of shrinkage and gap formation is discussed. In order to study shrinkage behavior for different strand shapes, an analysis is carried out without the narrow side taper. For the case of funnel mold, the funnel shape is assumed to keep its initial shape through the analysis. A. Parallel slab Fig.6.14 shows axial surface temperature profiles at different locations in a parallel slab. As can be seen in this figure, the temperatures at the center of the wide face and narrow face decrease monotonically to about 1000\u00b0C at mold exit. The temperature rebound at the lower part of the mold is simply due to the lower heat removal by the smaller heat flux that was measured near the mold exit as shown in Fig.6.7. It is noted that the temperature at the slab corner exhibits the higher temperature of 1180\u00b0C compared with the slab center temperature at the exit of mold. Although there is uncertainty about the heat flux at the corner, the assumption of 75% reduction of heat flux at the corner can simulate the effect of the lower cooling rate like billet casting, 165 Casting time (sec) 800 -j :-700 r 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 \" \" \u2022 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) Fig.6.14 Axial temperature profiles at different locations in parallel slab 166 which considered the formation of an insulating air gap at the mold\/shell interface in the corner region. Fig.6.15 shows shell thickness profiles as a function of distance below the meniscus in the parallel mold. The shell thickness at the mold exit is predicted to be about 11 mm, which matches well with other data [126]. Fig.6.16 compares profiles of shell shrinkage at different slab positions as a function of position in the mold. The shrinkage at the corner shows the 9.8 mm, (equivalent to 1.55% shrinkage for the slab width of 630 mm), which is much larger than that of narrow face center, 5.5 mm (equivalent to 0.83% shrinkage for the slab width of 630 mm). This smaller shrinkage in the center of the narrow face is due to ferrostatic pressure, which made the narrow face shell bulge outward toward the mold, resulting in good strand\/mold contact during initial solidification. The shrinkage in the corner is mainly governed by the overall shrinkage of the wide face, which is in good contact with the mold. Therefore, this shrinkage can be simply calculated due to contraction of the wide face from the initial solidification temperature of about 1524\u00b0C to the mold exit surface temperature of about 940\u00b0C, together with the average thermal expansion coefficient of 0.00275 %\/\u00b0C from Fig.6.5. Calculation of crude perimeter calculation results in 10.2 mm shrinkage. The shrinkage depends upon the temperature in the corner, which is uncertain in this study, and the average thermal expansion coefficient is overestimated when compared to the other reported value 0.002 %\/\u00b0C [127]. However, the shrinkage prediction of 9.8 mm by this model seems to be reasonable and matches well with the simple calculation of 10.2 mm. It can be inferred from this figure that more shrinkage occurs near the top of mold. This agrees well with other findings [128]. This result implies that it is difficult to 167 Fig.6.15 Predicted shell' thickness as a function of distance below meniscus in parallel mold 168 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 -11 I 1 1 1 ' I 1 1 1 1 I 1 1 1 ' I 1 1 1 1 1 1 1 1 1 1 1 1 1 ' I 1 1 1 1 I 1 1 1 1 I 1 ' 1 1 I 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) Fig.6.16 Profiles of shrinkage down the mold length of parallel slab 169 compensate for the shrinkage through a simple linear taper. B. Funnel slab Fig.6.17 compares axial surface temperature profiles along the centerline of a slab wide face for the different strand shapes. Since the boundary condition for this case is the same heat flux input as was used for the parallel mold, all the slabs exhibit very similar surface temperature profiles on the wide face of the slab. Fig.6.18 shows the transverse surface temperature profiles across the wide face at the position of 750 mm from the meniscus. Parallel and funnel strands show almost the same transverse temperature profiles, except in the corner region of funnel B slab, which has a strand thickness of 50 mm compared with 75 mm for funnel A slab. Because of the same temperatures profiles of the slab wide face, funnel slabs exhibit similar behavior with regard to wide face shrinkage as shown in Fig.6.19. 6.5.2 Effect of moid taper on shrinkage and air gap in parallel slab casting Fig.6.20 compares shell shrinkages of the parallel slab according to different narrow face tapers. As seen in the figure, the shrinkage of the shell in the corner does not depend on the imposed taper of the narrow face. This is because the shrinkage in the corner is governed mainly by the overall shrinkage of the wide face which is in good contact with the mold and should not be affected by the imposed narrow face taper. The figure also shows the behavior of the air gap formed in the corner region of mold as function of the 170 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 900 - : \" goo r ' 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 ' * i 1 1 1 1 i 1 1 1 1 i 1 1 1 ~ 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) Fig.6.17 Axial surface temperature profiles down the mold length at slab center 171 100 200 300 400 500 600 1400 1300 4-3 1200 2- 1100 + u o 3 1000 -o c 0 0 900 + 800 + + - i \u2014 i \u2014 i i Parallel 'Funnel A \u2022 Funnel B 750mm below meniscus + 100 200 300 400 500 Distance from center of mould (mm) 600 Fig.6.18 Comparison of transverse temperature profiles of slab surface across the wide face with different thin slabs 172 Casting time (sec) _n_0 I 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 ' 1 ' I 1 1 1 1 I 1 ' 1 1 I 1 1 1 1 I 1 1 1 ' 1 1 1 1 ' I 0 100 200 300 400 500 600 700 800 900 \u2022 Distance from meniscus (mm) Fig.6.19 Comparison of shrinkage behaviors at the slab corner with different thin slabs 173 ca 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) (a) 100 200 300 400 500 600 700 Distance from meniscus (mm) (b) 800 900 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) (c) Fig.6.20 Effect of narrow face taper on shell shrinkage and gap formation in the parallel slab (a) No taper (b) 0.8%\/m taper (c) 1.1%\/m taper 174 distance below meniscus for various imposed narrow face tapers. The rate of gap formation is initially very high, due to large shrinkage during initial solidification. As taper increases, the gap tends to decrease. However, at 1.1 %\/m of taper, which is current taper condition, a gap of 3.8 mm formed. In an actual casting process, in addition to the narrow-face mold deformation (incorporated in the linear narrow side mold taper as shown in Fig.6.10), there may be other factors that could contribute to the mold taper. One of them could be the change of slag film thickness, which is zero at the meniscus, and gradually becomes thicker as surface temperature cools. Fig.6.21 shows the profile of measured slag film thickness across the slab width at the mold exit under the same casting conditions of the simulation as described in Table 6.5. The slag film was collected below the mold with a special device and after the casting sequence, the thicknesses of many samples were simply measured with a micrometer and averaged. The slag film thickness varies from 0.2 mm to 0.7 mm. Another factor that could contribute to the narrow side mold taper is the wide face expansion of the mold resulting from the temperature increase of the mold hot face. As studied in Chapter 4 and 5, the wide face of the mold can expand in the transverse direction, which is proportional to the temperature increase. The narrow face can be connected through the clamping force, so that the narrow face can be moved simultaneously with the wide face expansion. According to the results of thin slab mold temperature (Fig.4.11 and Fig.4.12), the temperature difference of the mold hot face between the meniscus region and the bottom of the mold is estimated to be about 270\u00b0C, which corresponds to 0.27% strain, or 1.7 mm for the 630 mm slab thickness (Fig.5.30). Therefore, in this study, 0.5 mm change of mold slag thickness based on the measured 175 176 value at the wide face of strand, and 1.7 mm difference of the wide face of the mold are incorporated into the narrow face taper, which are assumed to change linearly. The profile of mold slag film and wide face expansion of the mold contribute to the narrow side mold taper. The corresponding shrinkage and gap formation are plotted in Fig.6.22. As can be seen in this figure, which includes these factors, the gap amount becomes smaller (1.2 mm) compared with the case of linear narrow side mold taper (3.8 mm). 6.5.3 Effect of funnel on deformation and stress A study was undertaken to compare the effect of funnel shape change upon the shrinkage, air gap and stress development in the solid shell. In this analysis, a linear mold taper is assumed. The slag film thickness and thermal expansion of mold wide face are ignored. Fig.6.23 shows shrinkage and gap formation with various narrow face tapers in the funnel mold A. Although the additional solidifying shell is only 0.78 mm as described in section 7.2, the shrinkage amount is estimated to be about 6 mm in the case of no narrow face taper. This is almost 4 mm less than that for the parallel mold. For funnel B mold, having a large funnel depth of 52 mm and 3.6 mm of additional solidifying shell, the shrinkage amount is only 2.2 mm, as shown in Fig.6.24 (a). This is about 7.5 mm less than that for the parallel mold. These phenomena can be explained by the additional visco- plasticity generated from the large unbending of shell due to the funnel retraction when subjected to both thermal and ferrostatic pressure effects. These effects cause the solid shell to be deformed more than the theoretical prediction of the additional length of 177 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0 100 200 300 400 500 600 700 800 900 Distance from mensicus (mm) Fig.6.22 Profile of mold taper down the mold length and the corresponding shrinkage and gap formation in the parallel slab 178 Cas t ing t ime (sec) 6 9 12 15 u 00 JS 100 200 300 400 500 600 700 800 900 Dis tance from meniscus (mm) (a) Casting time (sec) 6 9 100 200 300 400 500 600 700 Distance from meniscus (mm) (b) Casting time (sec) 6 9 800 900 0 100 200 300 400 500 600 700 800 900 Distance from meniscus (mm) (c) Fig.6.23 Effect of narrow face taper on the shrinkage and gap formation in the funnel A slab (a) No taper (b) 0.8%\/m taper (c) 1.1%\/m taper 179 0 0 ^ -2 1 -4 -t \u00ab - 6 | CO -10 -t -12 Casting time (sec) 6 9 12 15 -r 12 \u2022NFwall Shell \u2022Gap 0 I 1 I 100 200 10 * 8 6 4 * 2 = S, o. CO a 300 400 500 600 700 800 900 Distance from meniscus (mm) (a) Casting time (sec) 6 9 12 15 100 200 300 400 500 600 700 800 Distance from meniscus (mm) (b) Casting time (sec) 6 9 900 700 800 900 I ) I 100 200 300 400 500 600 Distance from meniscus (mm) (c) .6.24 Effect of narrow face taper on the shrinkage and gap formation in the funnel B slab (a) No taper (b) 0.6%\/m taper (c) 0.8%\/m taper 180 solidifying shell. It is noted that the shrinkage amount is different for various taper conditions, ranging from 6 to 7 mm for funnel A mold and 2 to 4 mm for funnel B mold, respectively. The gap is close to zero by mold exit with 1.1 %\/m of narrow face taper for funnel A mold and 0.8 %\/m of taper for funnel B mold, as shown in Fig.6.23 (c) and Fig.6.24 (c). To isolate the effect of funnel shape upon the stress development, the model is run without thermal stress effects. This is achieved by setting the thermal expansion coefficient to be zero. Fig.6.25 shows stress profiles along the slab surface at different positions along the funnel A mold. As the strand moves down the mold, the funnel retraction generates compressive stress at the funnel surface through to the mold exit (15 seconds of casting time). However, the funnel retraction causes the strand to develop different stress levels at different locations. While the funnel inside bend (130-290 mm) develops compressive stress, the funnel outside bend (290-450 mm) develops less compressive stress. These trends of stress development could also be found in stress profiles through the slab thickness as shown in Fig.6.26. Furthermore, as funnel shape changes become larger like funnel B (52 mm funnel depth), tensile stress develops in the funnel outside bend region at 12 seconds (equivalent to 720 mm from meniscus), right after the funnel shape disappeared as shown in Fig.6.27 and 7.28. Finally, the full 2 - D model simulations with the standard taper condition, 1.1 %\/m for parallel and funnel A, and 0.6 %\/m for the funnel B slab were performed. Fig.6.29 and 6.30 show stress profiles along the slab surface and through the slab thickness at different positions in the parallel slab. For the parallel slab, where the shell is in good contact with the wide face and the surface temperatures are uniform over the entire wide face, the 181 Fig.6.25 Transverse stress profiles along the funnel A slab surface with different casting times (No thermal stress) 182 Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s 10 15 20 25 30 35 Distance from the slab surface (mm) (a) Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s -+-10 15 20 25 30 Distance from the slab surface (mm) (c) 35 Fig.6.26 Evolution of transverse stress through slab thickness with different locations of funnel A (a) Funnel parallel (b) Funnel inside bend (c) Funnel outside bend 183 Fig.6.27 Transverse stress profile along the funnel B slab surface with different casting times (No thermal stress) 184 X Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s 5 10 15 20 Distance from tlie slab surface (mm) (c) 25 Fig.6.28 Evolution of transverse stress through slab thickness with different locations of funnel B (a) Funnel inside bend (b) Funnel outside bend (c) straight region 185 20.0 15.0 - : 10.0 -; 5.0 - : -30.0 4 - i \u2014 i\u2014 i\u2014 h H r H \u2014 i \u2014 i \u2014 r -0 Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s 100 1 1 . 1 ' 1 1 ' ' ' ' 1 1 200 300 400 Distance from the slab center (mm) 500 600 Fig.6.29 Transverse stress profile along the parallel slab surface with different casting times (1.1%\/m taper) 186 1600 S -10.0 X Time 5.0s \u2022Time 10.0s Time 12.0s \u2022Time 15.0s -+-10 15 20 25 Distance from surface (mm) (a) 30 35 15 20 25 Distance from surface (mm) (b) Fig.6.30 Evolution of temperature (a) and stress distribution (b) through slab thickness for parallel slab (1.1% taper) 187 main source of stress results from thermal contraction of the solid shell. Therefore, as expected from the theoretical analysis, it is clear that compression appears on the cold surface while tension arises near the solid-liquid interface. However, for funnel slabs, which experience huge strand shape change, stress evolves differently from the parallel slab as shown in Fig.6.31-6.34. For example, as can be seen in Fig.6.31, at 5 seconds, tensile stress develops over the whole wide face surface except for the corner region for funnel A slab. Furthermore, as the funnel shape changes significantly like the funnel B slab, tensile stress becomes larger, especially in the funnel outside bend, which keeps the tensile stress until the mold exit (Fig.6.33). These trends can also found from stress profiles through the slab thickness (Fig.6.32, Fig.6.34). The tensile stress development at the slab surface can be explained by the behavior of solid shell shrinkage and additional solid shell length. During initial solidification, where large shrinkage takes place, the additional increase in solid shell thickness prevents the strand from shrinking. This results in the development of tensile stress at slab surface. It implies that once defects exist at the initial solidification stage, there is high probability of crack occurrence. Calculated maximum principal stress fields for different slab shapes are shown in Fig.6.35-6.37 at various casting times. For a parallel slab, where the stress is uniform over the whole wide face the contours, maximum principal stress contours are plotted at the 100 mm from the center of strand as shown in Fig.6.35. For funnel slab, where stress fields vary with strand position, the contours of maximum principal stress at the transition area of funnel inside and funnel outside bend, which is 290 mm and 237.5 mm from the center of funnel slab, are plotted in Fig.6.36 and 6.37, respectively. As expected from the 188 Fig.6.31 Transverse stress profile along the funnel A slab surface with different casting times (1.1%\/m taper) 189 190 T\u2014(-10 15 20 25 Distance from surface (mm) (c) Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s -+-30 35 -i\u2014i\u2014h Time 5.0s -Time 10.0s Time 12.0s \u2022Time 15.0s 15 20 25 30 35 Distance from surface (mm) (d) Fig.6.32 Evolution of temperature and stress distribution through shell thickness for funnel slab A (a) Temperature (b) 100mm from the slab center (c) 200mm from the slab center (d) 3 50mm from the slab center (1.1%\/m taper) 191 Fig.6.33 Transverse stress profile along the funnel B slab surface with different casting times (0.6%\/m taper) 192 193 Q-S -10.0 + 10 15 Distance from sirface (mm) (c) 15.0 + H r a, X Time 5.0s Time 10.0s Time 12.0s Time 15.0s 10 15 Distance from sirface (mm) (d) Fig. 6.3 4 Evolution of temperature and stress distribution through shell thickness for funnel slab B (a) Temperature (b) 100mm from the slab center (c) 350mm from the slab center (d) 520mm from the slab center (0.6%\/m taper) 194 transverse stress result, the parallel slab has uniform stress profiles across the slab with the peak value at the front of solid shell. However, for the funnel slab, as shown in Fig.6.36, 6.37, the maximum peak of principal stress is found not only at the inside of solid shell but also at the slab surface around the transition area of funnel inside and funnel outside bend. 195 7.4mm I 9.2mm i n I E 2.67 2.67 0.0 (a) 4.0 UL 2&L J U L 12.3mm (b) X2T 4.0 - 1 6 X . 31. 0.0 (c) 20x20mm Unit: M P a Fig.6.35 Contours of maximum principle stress for parallel slab (a) 5 sec. (b) 10 sec. (c) 15 sec. 196 80mm (C) Fig.6.36 Contours of maximum principle stress around 290mm from the slab center for funnel slab A (a) 5 sec. (b) 10 sec. (c) 15 sec. 197 80mm 2.67 1.33 0.0 4.0 (b) i i UI a m r\u00bb ^ r 0.0 4.0 4.0 0.0 (c) Fig.6.37 Contours of maximum principle stress around 237.5mm from the slab center for funnel slab B (a) 5 sec. (b) 10 sec. (c) 15 sec. 198 7. CONCLUSION Thermo-mechanical phenomena during thin-slab continuous casting have been studied with the objectives of understanding the mechanism of mold cracking, and the effect of mold design upon the air gap formation between the strand and mold and stress generation in the strand. To achieve these goals, several finite element models have been developed along with a series of industrial plant trials. First, a comprehensive study was undertaken to understand the mechanism of mold crack occurrence in thin slab casting. Extensive measurements such as temperature, deformation and permanent distortion of mold were obtained from an operating thin slab caster of POSCO, to determine not only the heat extraction rate in the mold but also the distortion of the mold for different mold geometries. This information has been utilized to develop mathematical models of molds. Metallographic studies of mold crack samples were also performed. Results from plant measurements, mathematical models and metallographic examination were simultaneously analyzed to understand mold crack formation. The second part of this research was undertaken to examine the effect of mold taper on the shrinkage of the solidifying shell, its gap formation and stress evolution with different thin slab mold geometries. Measurements such as mold temperature, heat flux, thickness of the solidifying shell, bulging deformation and location of longitudinal crack formation were made on an operating billet-casting machine to verify a two-dimensional thermo-elastic-visco-plastic finite element model of a slice through the continuous casting strand. 199 Using the validated finite-element mathematical model from the billet casting analysis together with heat flux data and distortion data obtained from the first study on the thin slab mold, the mathematical model has been successfully applied to obtain temperature distribution, shrinkage, air gap, and stress evolution in the solidifying shell for different slab mold geometries. The major conclusions of this work are presented under two major headings as follows: A. Mold crack formation in thin slab casting 1) Heat flux profiles, based on measured temperatures for 3.6 m\/min. casting speed, show that thin slab casters have much higher heat extraction than conventional casters, ranging from 2.4 to 2.6 MW\/m 2 , with peak meniscus heat flux approaching 7 MW\/m . 2) During operation, the hot face temperature reaches 580\u00b0C and is highest 20 mm below the meniscus towards the mold edges 37 0 mm from the centerline. 3) The copper plates bend toward the steel as in a conventional caster, with a maximum distortion toward the liquid steel of only 0.3 mm. This occurs just above the center of the wide faces, and is generally less than the distortion of a conventional mold, owing to the increased thickness and corresponding rigidity of the water box. 200 4) Residual distortion is smaller than seen in previous measurements, owing to oversized bolt holes in the water box, which do not over constrain the thermal expansion and contraction. 5) The thickness variations of the funnel mold lead to a different shape distortion profile than in a parallel mold. The distortion magnitudes are quite similar, however, with maximum values of 0.25 (funnel) and 0.22 mm (parallel mold). 6) Initiation of the cracks is associated with intergranular embrittlement caused by brass formed by zinc attack from the molten steel, after the loss of the protective Cr coating at the hottest portion of the hot face, just below the meniscus. 7) SEM examination of the cracks reveals a transgranular fracture surface and striation structure beneath the intergranular surface portion of the crack, indicating fatigue propagation. 8) The cracks are formed just below the meniscus in the transition region of the funnel mold, because this location has the highest temperature, stress, and inelastic strain. 9) The slot depth of the funnel mold varies with distance across the hot face, ranging from 25-28 mm from the hot face in the funnel transition area, which contributes to higher stress and strain fields relative to the parallel mold. 10) The funnel mold was found to have metal level fluctuations with low frequency (0.05 Hz) relative to the parallel mold, which had high frequency (0.3 Hz). 201 1 l)The low frequency metal level fluctuations are predicted to cause higher cyclic stresses and lower fatigue life, thus helping to explain why cracks were found only in the funnel mold. 12) Cracks might be avoided by avoiding Zn residuals in the steel, lowering hot face temperature, reducing level fluctuations, and preventing coating failure. B. Thermo-mechanical behavior of solidifying shell in thin slab mold 1) The two-dimensional transverse slice model (AMEC2D) for simulating the continuous casing process was validated by comparison with measurements from the billet casting plant trial. The following are specific findings from the billet casting analysis: a) The AMEC2D model ignores axial heat conduction so that the predicted mold temperature is not expected to exactly match with measured ones, but still agrees reasonably well. The corresponding predicted heat flux profile shows the classic monotonically decreasing profile, which is commonly observed in casting processes. b) The model prediction of an average heat flux of 1.80 MW\/m 2 is comparable to the measured value of 1.84MW\/m2 based on the temperature increase of cooling water. c) The model can predict the local thinning effect of the solidifying shell formation around the corner regions. 202 d) The calculated bulging amount of 1 - 4 mm below the mold exit appears to be consistent with the maximum measured value of 2 mm. e) The peak effective plastic strain occurs in a region of tensile hoop stress and corresponds roughly to the position of observed longitudinal crack occurrence. 2) Regardless of slab geometries, the shrinkage at the slab corner is estimated to be 1.55% for a slab width of 630 mm. 3) The shell is predicted to shrink more near the top of the mold than near the bottom of the mold, which suggests that it is difficult to compensate for the shrinkage through a simple linear taper. 4) The taper in the funnel mold should be less than in the parallel mold due to both the additional length supplied by flattening out the funnel and additional visco-plasticity due to the large unbending of the shell while it is subjected to both thermal and ferrostatic pressure forces 5) The parallel thin slab has uniform compressional stress developed across the cold surface and tension near the solid-liquid interface. 6) The shell in the funnel mold develops tensile stresses at the slab surface of the funnel transition region due to the funnel retraction. 7) This model suggests that as funnel depth increases, the possibility of surface cracks at the outside bend of the funnel increases. 203 R E F E R E N C E 1. 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University of Illinois at Urbana-Campaign, 1996 217 APPENDIX I MOLD THERMAL RESPONSE OF THERMOCOUPLE 218 APPENDIX 1.1 Parallel mold 219 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Fig. A l . 1 Casting condition for the plant trial (Parallel mold) 220 300 i 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 300 i 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Casting time (sec.) Fig. A l .2 Mold temperature profiles with the position of mold length (at the center of parallel moi (Number in parenthesis : distance from the top of mold) 221 150 H h 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 150 4 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20( Casting time (sec.) Fig.A1.3 Mold temperature profiles with the position of mold width (175mm down the parallel mold top) (Number in parenthesis : distance from center of mold) 222 H 1 1 1 1 1 1 1 * 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 200 u o I I 150 + & p H (Fixed) (-564) (-188) (0) (188) (376) (564) (-376) 3*m 1 i. * * I ioo -P 1 1 1 1 1 \u2014\u2022 1 1 1 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Casting time (sec) Fig. A 1.4 Mold temperature profiles with the position of mold width (305mm down the mold top) (Number in parenthesis : distance from center of mold) 223 A P P E N D I X 1.2 Funnel mold 224 4.5 -.s S I 4 -\u00ab IS-3.5 \u2022 \u2022I 3 -3 2.5 \u2022 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 0 2000 4000 6000 8000 10000 12000 14000 16000 .18000 20000 22000 24000 4800 4 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 50 -j CJ 45 \u2022 1 40 \u2022 o 35 -1 30 \u2022 H 25 -20 \u2022 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 Casting time (sec.) Fig.Al .5 Casting condition for the plant trial (funnel mold) 225 Fig. A 1.6 Mold temperature profiles with the position of mold width (100mm below the funnel mold top) (Number in parenthesis : distance from center of mold) 226 A P P E N D I X 1.3 Billet mold 227 3 0 '3jmej9dui9i 3 O 'sjmejaduiai 228 APPENDIX II HEAT TRANSFER COEFFICIENT AT THE MOLD WALL\/COOLING WATER INTERFACE Heat transfer between the mold wall and the cooling water takes place by forced convection which had been characterized with the aid of a heat-transfer coefficient determined from the following dimensionless correlation [73]. w \\0.4 X \" ( ~ } ( ~ } ( A 2 1 ) The above correlation is applicable only in the absence of nucleate boiling in the cooling channel. Nucleate boiling phenomena were not considered in this study. However, it may be noted that near the entrance region of the cooling channel the flow is not expected to be fully developed which can affect the heat transfer coefficient. It has been reported that in the case of turbulent flow through a pipe the length required to establish a fully developed flow is approximately equal to 10 times the hydraulic diameter of the pipe [129]. Based on this rationale, in the present case (the effective hydraulic diameter is about 5.7 mm), the flow may not fully establish within a region, which is approximately 6% of the length of the cooling channel, 944 mm, (equivalent to 85 mm of the mold length). As the entrance is located at the top as the direction of the water flow is from top to the bottom of the mold and thus, only near the top of the mold the flow may not fully established. Therefore, the entrance effect will affect the boundary 229 conditions above the meniscus, which is 100 mm below the top of mold. Consequently the predictions of the heat flux in the upper region of the mold particularly near the meniscus, which is of prime importance to this study will not be influenced. Besides, the employment of the above correlation depends on the following conditions [130]. (a) 0.7<Pr< 120 (b) 1000 < Re < 120000 (c) L \/ D H > 60 Calculations were performed to ensure that the above criteria were met and the results from the calculations are listed in Table A2.1. Thus, equation A2.1 has been utilized to characterize the heat-transfer coefficient with the assumption that there is no boiling in the whole cooling channel. 230 Table A2.1 Testing of conditions to establish the applicability of the correlation (Eq. A2.1) for obtaining the heat transfer coefficient. Water slot dimension (m x m) (mm2) 5 x 7 , 47.3 Hydraulic diameter of the cooling channel (mm) 5.7 Velocity of water (m\/s) (wide face) 10.7 Velocity of water (m\/s) (narrow face) 10.6 Specific heat of water (J\/KgK) 4182 Density of water (kg\/m3) 998.2 Thermal conductivity of water (W\/mK) 0.597 Viscosity of water (Ns\/m2) 993 x 10\"6 Prandtl number 7.0 Reynolds number (wide face) 61309 Reynolds number (narrow face) 60736 Ratio of length and hydraulic diameter of cooling channel 156.6 231 APPENDIX III SOLID SHELL BEHAVIOR IN BILLET CASTING A3.1 Introduction During the continuous casting of steel billets, the corner regions of the cast section often experience local thinning. This phenomenon, referred to as \"re-entrant corners\", is due to the complex behavior of the air gap between the mold and the solidifying shell. This common occurrence can lead to problems such as longitudinal cracks near the billet corner, especially at high casting speed [48, 55, 131]. In extreme cases, the corners may be so thin that a breakout occurs, even though the average shell thickness is easily large enough to withstand the ferrostatic pressure at mold exit. Two decades of operating experience have shown that reducing the corner radius from 12-16 mm down to 3 or 4 mm is beneficial in minimizing longitudinal corner cracks [132]. In addition to lessening crack frequency, decreasing the corner radius also tends to move the crack location from the corner itself to the off-corner region [133], Unfortunately, billets with sharp edges tend to \"fold over\" during the rolling process [54], Therefore, mold designers must struggle to satisfy these two conflicting requirements. It would be very desirable to find a better way to solve the longitudinal corner crack problems. An important step towards this end is the achievement of an accurate, quantitative understanding of the crack formation mechanism(s). This understanding would aid mold design optimization, especially for high-speed casting. 232 Over the years, many mathematical models have been developed to help to understand the origin of defects in complex processes such as continuous casting [57-62]. However, quantitative understanding of the re-entrant corner phenomena of the solidifying shell in the billet mold has received relatively little attention. Furthermore, the effect of the billet mold corner radius on the temperature, corner gap, and stress development has not been studied. In the present work, a thermo-elastic-visco-plastic finite-element model was developed to simulate temperature and stress in a transverse slice through the solidifying shell of a typical billet caster. The evolution of the air gap is calculated from the deformation of the strand and the tapered and distorted mold. Its coupled effect on the temperature distribution is taken into account with a distance-dependent heat transfer coefficient between the mold and strand. The accuracy of the 2-D slice model formulation in this analysis has also been investigated through comparison with both an analytical solution and a plant trial test. Finally, the model is applied to the specific problem of how the corner radius of the mold affects the thermal, deformation and stress fields of a low carbon steel billet continuous-cast using both oil lubricant and mold-powder practices. The implications on longitudinal crack formation are discussed. A3.2 Plant trials A3.2.1 Caster details and nominal operating practice A plant trial was conducted at POSCO, Pohang works, South Korea, for a 120 mm square section of 0.04% C steel continuously cast at 2.2 m\/min. The mold was relatively 233 pure DHP copper with a wall thickness of 6 mm, a corner radius of 4 mm and a single \"linear\" taper of 0.75 %\/m. Other operating parameters and mold geometry details are provided in Tables A3.1 and A3.2. A3.2.2 M o l d temperature measurement The mold tube was instrumented with 12 K-type thermocouples on the inside-radius face as shown in Fig.A3.1. They were arranged in three columns along the centerline, and +45 mm from the centerline, and in four rows located at 120, 170, 400 and 700 mm below the top of the 800 mm long mold. The thermocouples were embedded in the mold wall to a depth of 3 mm from the hot face. The mold water temperature increase was not recorded at the time, but is estimated to be 30\u00b0C based on recent measurements for the same conditions. A3.2.3 Solid shell measurement To investigate the solid shell growth, FeS tracer was suddenly added into the liquid pool during steady state casting. Because FeS cannot penetrate the solid shell, the position of the solid shell front at that instant can be clearly recognized after casting using a sulfur print. A3.3 Mathematical model description To investigate the thermo-mechanical behavior of the continuous billet and mold, a 2-D finite element package was used (see Chapter 6). In this study, both the solidifying strand and the mold are modeled in detail, which are coupled through the size and properties of the interfacial gap. 234 Table A3.1 Casting conditions in the plant trial Billet Size 120mm sq. Nominal casting speed 2.2 m\/min. Meniscus level 100mm Oscillation type Sinusoidal Stroke length 8mm SEN Open pouring Machine radius 8m Table A3.2 Mold conditions in the plant trial Material DHP-Cu Mold length 800mm Thickness 6mm Construction Tube Taper (linear) 0.75%\/m Corner radius 4mm Cooling water flow rate 1100 1\/min. Cooling water velocity 9.2m\/sec. 235 Fig. A3.1 Photograph of thermocouple instrumented mold tube 236 A3.3.1 Heat flow analysis The heat flow model solves the 2-D transient heat conduction equation for the temperature distribution in the solidifying shell and mold. The effects of solidification and solid-state phase transformation on the heat flow are incorporated through a temperature-dependent enthalpy function as described in section 6.3.2. The detailed assumptions used in this calculation were described in Chapter 6. A. Oil-casting interface heat transfer Heat extraction from the solid shell surface in the mold is primarily controlled by heat conduction across the interface between the mold and the solidifying steel shell. This is modeled as an internal boundary condition, using an interfacial heat transfer coefficient, he, as a function of air gap thickness and surface temperature of the strand, according to the relations of Kelly et al [61]. K =h d + \u2014 = h. +kjdaa ( A 3 1 ) C rad T-J raa g gap Here RT is thermal resistance, kg is the thermal conductivity of the gap medium assumed to be 100% air in this study and given in Table A3.3, dgap is the thickness of the gap, and hrad is the heat transfer coefficient for radiative heat flow when an air gap exists between the strand and the mold: Kd=o-SBe(Ts + Tm)(T2+T2m) (A 3- 2) 237 Table A3.3 Conductivity of gap medium (air) with temperature [61] Temperature (\u00b0C) Conductivity (W\/mK) 200 0.032 400 0.039 600 0.045 800 0.051 1000 0.057 1200 0.063 1400 0.068 238 where OS B is the Stefan-Boltzman constant, Ts is the shell surface temperature and Tm is the mold hot face temperature. The average emissivity of shell and mold surface was assumed to be 0.8 [134]. If the value of he computed from Eq.(A3.1) exceeds the value associated with direct contact, it was replaced by that value. The he for direct contact was taken to be 2500 W\/m 2K, which represents a minimum contact resistance or average gap due to oscillation marks of 0.02 mm depth [61]. Fig.A3.2 shows plots of this heat transfer coefficient function with air gap size, assuming strand surface temperatures of 1500\u00b0C, 1000\u00b0C and mold hot face temperatures of 300\u00b0C, 200\u00b0C. B. Powder-casting interface heat transfer To study the effect of using mold powder as a lubricant, simulations were also performed using the following thermal resistance between the solidifying shell surface and the mold, consisting of four terms: RT = . - L + ^ + ^ L + - i - (A3.3) The first thermal resistance is the contact resistance between the mold wall surface and the mold flux, where hm is the contact heat transfer coefficient set to 2500 W\/m 2K. The second resistance is conduction through the air gap, which is the same as calculated for oil casting. The third resistance is conduction through the mold flux film, with a thermal conductivity, kjiux, of 1.0 W\/mK [114]. The thickness of the mold flux layer, w a s assumed to be 0.1 mm [135]. The final term is the contact resistance between the mold flux and the strand surface, where hSheii depends greatly on temperature, due the large 239 Fig.A3.2 Heat transfer coefficient across the strand\/mould for different air gap sizes and surface temperatures 240 change in viscosity of the mold flux over the temperature range of strand surface. The temperature dependency of hsheiiis given in Table A3.4 [136]. C. Spray cooling To investigate bulging of the billet below the mold, thermal calculations were extended to 200 mm below the mold exit, assuming 500 W\/m 2 K for the heat transfer coefficient at the billet surface and ambient temperature of 30\u00b0C. This value was chosen to represent a typical spray cooling coefficient, which ranges 200 to 600 W\/m 2 K in the literature [137]. D. Mold temperature Temperature in the mold was assumed to be steady within each time step and slice It was calculated in AMEC2D by applying the water heat transfer coefficient to the cold face of the mold based on the correlation of Dittus and Boelter [73]. This analysis ignores axial heat conduction. Thus, a second model, CON1D [138], was applied to validate the heat flux profile. C O N ID takes into account axial heat conduction in the mold, so has more accurate mold temperature predictions than AMEC2D. A3.3.2 Stress analysis The stress and strain distributions associated with temperature change in the transverse slice of the solidifying shell were calculated with the assumption that the slice is in a plane strain condition, in which strain along the casting direction was neglected. The temperatures calculated by the thermal model were input to the incremental thermal stress model. 241 Table A3.4 Temperature dependence of heat transfer coefficient between the mold flux and strand surface [136] Temperature (\u00b0C) h4 (W\/m 2K) Mold flux crystalline temperature, 1030\u00b0C 1000 Mould flux softening temperature, 1150\u00b0C 2000 Metal solidus temperature, 1511\u00b0C 10000 Metal liquidus temperature, 1529\u00b0C 20000 I 242 Mold distortion due to thermal expansion, which is added to the mold taper to define the mold wall position, is calculated using the following equation. ^mold ~ amold fmoldwidthVTcold+Thotc rp \\ 2ref V ^ J V 2 j (A3.4) where, ctmoid = mold thermal linear expansion coefficient (1.6xlO^K\"1) TCoid = mold cold face temperature (\u00b0C) Thotc = mold hot face temperature (\u00b0C) T r e f = average mold temperature at meniscus (\u00b0C) For equation A3.4, the mold temperature was based on the results of the CON ID model [138], which matches with the measured temperature well. Fig.A3.3 shows profiles of mold distortion, 0.75 %\/m linear profile of mold taper and the actual mold wall shape adopted in this work, respectively. Detailed description of thermal strain, constitutive equation, ferrostatic pressure, and solid shell\/mold contact were described in Chapter6. A3.3.3 Crack criterion In order to study the susceptibility of corner crack occurrence, \"hoop stress\" (an) and \"hoop strain\" (eh) components were calculated to show the transverse stress\/strain component oriented parallel to the perimeter of the shell. To calculate hoop values, first of all, the angle of heat flux direction, <|), with respect to the global x and y-axis can be 2 4 3 0.35 -t .0.1 f 1 1 1 1 I 1 1 1 1 I ' 1 1 1 1 ' 1 1 ' 1 1 ' 1 ' I 1 ' 1 1 I ' 1 1 1 1 0 100 200 300 400 500 600 700 Distance from meniscus (mm) Fig. A3.3 Profiles of mold distortion and taper imposed used in this model 244 obtained from the temperature results. Stress and strain component perpendicular to that direction, 6=90-d>, is then derived by the following equation. cr + cr cr - c r ah = x y+ ' y cosie + T^smlO (A3.5) where, ax is x stress, ay is y stress and t x y is shear stress, respectively. Note that along the horizontal shell, 9=0, so the hoop stress becomes a x . The hoop stress becomes a y along the vertical shell as 9=90. Similar calculations are applied to find hoop strain, eh. A3.3.4 Strand and mold domain Fig. A3.4 shows the finite element mesh of a 2-D horizontal section of the billet strand and mold and its boundary conditions. A two-fold symmetry assumption allows a quarter transverse section of the billet to be modeled. This domain consists of 5273 nodes and 5135 4-node iso-parametric quadrilateral elements in the billet, and 207 nodes and 136 elements in the mold for the 4 mm radius mold. For the 15 mm radius mold, the F E M mesh contained 8947 nodes and 8775 elements in the billet, and 243 nodes and 160 elements in the mold. The boundary conditions used are also shown in Fig.A3.4. Further simulation conditions for the plant trial are described in Table A3.5. A3.4 Model validation with analytical solution The internal consistency of the finite-element model developed in this work using AMEC2D [29, 114-116] has been validated with analytical solutions under the conditions of plane strain using an element mesh size of 0.3 mm as shown in Fig.A3.4. Weiner and 245 246 Table A3.5 Simulation condition for plant trial Steel grade (see Table 6.2) C=0.04 wt% Liquidus temperature [119] 1529\u00b0C Solidus temperature [119] 1511\u00b0C Superheat 25\u00b0C Contact heat transfer coefficient [61] 2500 W\/m 2K Mould-water heat transfer coefficient 29400 W\/m 2 K Casting speed 2.2 m\/min. Taper 0.75 %\/m 247 Boley [139] developed an exact analytical solution of thermal stress during one-dimensional solidification of a semi-infinite elasto-perfectly-plastic body after a sudden decrease in surface temperature. Table A3.6 shows the detailed condition for verification of analytical solution. Fig.A3.5 compares this solution with numerical calculations for different solidification times. Although the temperature profile of AMEC2D agrees closely with the analytical solution, Fig A3.5(a), the maximum tensile stress and compressive stresses are 6.5 MPa and -22.9 MPa, which differ from the analytical solution by 34% and 11.5%, respectively (Fig.A3.5b). This discrepancy is due to the assumption of plane strain in AMEC2D, which is different from the true state of generalized plane strain in the analytical solution. However, when compared with CON2D model results [39,41] with a fine mesh size of 0.1 mm, as seen in a Fig.A3.6, both results show almost the same stress profiles, which implies that the mesh size adopted in this work should be adequate. A3.5 Model validation with plant trial The 2-D transverse slice model for simulating billet casting under the plane strain condition described in the previous section was validated by comparing with measurements from the plant trial, based on conditions in Table A3.5. A3.5.1 Temperature Axial mold-temperature profiles were calculated using both the AMEC2D and CON ID models. Fig. A3.7 compares the predictions with the measured mold temperature 248 Table A3.6 Simulation conditions for analytical solution test [143] Density 7400 kg\/m3 Specific heat 700 J\/kgK Thermal conductivity 33 W\/mK Latent heat 272 kJ\/Kg Initial temperature 1469\u00b0C Liqudius temperature 1469\u00b0C Solidus temperature 1468\u00b0C Surface temperature 1300\u00b0C Young's modulus 40 GPa Poisson's ratio 0.35 Thermal expansion coefficient 20x10\"6 1\/K Yield stress at surface temperature 20 MPa 249 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 n n H g i g f l f f l f f l r a H i S B f f l n Analytic (1 sec.) Analytic (5 sec.) \u2022Analytic (10 sec.) \u2022 AMEC2D (1 sec.) o AMEC2D (5 sec.) o AMEC2D (10 sec.) 3.0 4.0 5.0 6.0 7.0 Distance from cold surface (mm) (a) 8.0 9.0 10.0 15.0 4 I i i i i I H B B B B B B B B B Analytic (1 sec.) Analytic (5 sec.) Analytic (10 sec.) \u2022 AMEC2D (1 sec.) A AMEC2D (5 sec.) O AMEC2D (10 sec.) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Distance from cold surface (mm) (b) Fig. A3.5 Comparison of numerical and analytical solution (a) temperature (b) stress 250 Fig.A3.6 Comparison of stress profile with analytical solution 251 Fig.A3.7 Comparison of measured and calculated mold temperature 252 profile down the mold, found by averaging the thermocouple values across each of the four rows. The heat flux profile in the CON ID model was adjusted carefully in order to match the temperatures accurately. The AMEC2D model ignores axial heat conduction so is not expected to exactly match, but still agrees reasonably well. Fig.A3.7 also includes the hot and cold face temperature. The corresponding heat flux profiles predicted by both models are compared in Fig. A3.8. The accurate CON ID model curve shows a slight dip and rebound in heat flux somewhere from 20-100 mm below the meniscus. This is due to the unexpected lower temperature measured by the highest thermocouple. It is interesting to note that this drop corresponds roughly with the region of negative mold distortion, suggesting that this negative taper at the meniscus might play a role. This heat-flux dip phenomenon has been observed by others [56, 140, 141]. The AMEC2D curve is the classic monotonically decreasing profile, which is more commonly observed. Heat flux for the mold power casting case is also included in Fig.A3.8. Its overall heat flux is much lower than the oil casting case. This result agrees with the finding of others [142]. This lower heat flux is due to the insulating effect of the mold flux layer between the mold and strand. A3.5.2 Heat balance To validate the heat flux profiles, a comparison was made with an energy balance performed on the cooling water. The model predictions of average heat flux, found from the area under the curves in Fig.A3.8, are 1.84 and 1.80 MW\/m 2 for CON1D and AMEC2D respectively. The measured cooling-water temperature increase of 8\u00b0C 253 Casting time (sec) 0 5 10 15 3.5 0.5 4-0 100 200 300 400 500 600 700 Distance from meniscus (mm) Fig.A3.8 Heat flux profiles down the mold for different casting conditions and models 254 corresponds to an average heat flux of 1.84 MW\/m 2 , which agrees quite well with both of the model predictions. A3.5.3 Solid shell thickness Fig.A3.9 compares the measured solid shell thickness in a transverse section through the billet with the corresponding model prediction. The transverse section was taken at 285 mm below the meniscus, which corresponds to a simulation time of 7.8 second. The deformed shape of strand is superimposed with temperature contours in the same figure. Shell thickness was defined in the model as the isotherm corresponding to the coherency temperature, assumed to be 70% solid. The general shapes of the predicted and measured solid shell match quite reasonably. It is noted that the model also can predict the re-entrant corner effect that is observed in the sulfur print with the assumption of 100% air in the gap between the mold and strand. If hydrogen content existed in the gap, the thermal conductivity of the gap medium is expected to increase, leading to more uniform solidifying steel shell than that obtained in this study. This agreement appears to validate the remaining features of this model, including the air gap formation in the corner region. The shell thickness was plotted in Fig.A3.10 as a function of residence time in the mold with plant trial measurements by tracer test. As seen in Figure A3.10, the predicted solid shell growth is quite reasonable, considering the uncertainty about the penetration depth of the tracer into the mushy zone of the solidifying shell. A3.5.4 Bulging below the mold Bulging below the mold depends on the temperature and strength of the shell at mold 255 Fig. A3.9 Comparison of calculated and measured solid shell thickness (C%=0.04, 285mm below the meniscus, Vc=2.2m\/min.) 256 Casting time (sec) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 10 I i i \u2022 i I i i i i I . ' ' ' I > \u2022 > > 1 i i ' > I i ' ' ' I ' i i ' I i ' i < I ' ' ' ' 1 1 Distance frm meniscus (mm) Fig. A3.10 Comparison of measured and calculated solid shell thickness with casting time 257 exit. In the mold, the surface temperature of the strand is governed by the contact between the strand and the mold, which defines the gap between them. This is influenced by the mold taper, so a simulation was also performed for the extreme case of no mold taper. Fig.A3.11 shows axial profiles of the surface temperature at the strand center, corner and 5 mm off from the corner. Regardless of taper, the centerline surface temperature has the same profile, decreasing monotonically to 900\u00b0C at mold exit. This is because the billet strand is always in good contact with the mold at the strand center. The temperature rebound below the mold is simply due to the slower rate of heat removal by the sprays. At the corner region, the temperature rebounds after about 1 second for both cases due to the air gap formation. This time corresponds to initial formation of the air gap and is delayed by applying the taper as shown in Fig.A3.11 (b). An air gap still forms because the taper of 0.75 %\/m is not sufficient to match the shrinkage of the shell. Fig.A3.12 depicts transverse temperature profiles along the billet surface at different casting times with taper. After the initial solidification stage (time=0.5 sec), the temperature around the corner region is shown to remain higher throughout casting. This was not observed by Brimacombe et al [55], who did not simulate air gap formation during the calculation. They attributed off-corner internal cracks to a hinging action around a cold, strong corner. However, Fig.A3.12 implies that the corner region has a higher surface temperature, which enhances hinging below the mold. The strand shell exiting the mold is weak and hot, so the internal liquid pressure causes the shell to bulge below the mold. Although it might be supposed that this bulging in billet casting is small compared with slab casting, the higher casting speeds and lack of support can make it significant. This bulging can cause internal strain in the shell, 258 1600 + u 1550 I ire(c 1500 1 mperati 1450 -\\ mperati 1400 u H 1350 -\\ 1300 Casting time (sec) 10 15 20 \u2022 Without taper \u2022 With taper 200 400 600 Distance from the meniscus (mm) (a) Casting time (sec) 5 10 15 800 20 100 200 300 400 500 600 700 800 Distance from the meniscus (mm) (b) Casting time (sec) 5 10 15 20 \u2022 Without taper \u2022 With taper 0 200 400 600 800 Distance from the meniscus (mm) (c) Fig. A3.11 Evolution of surface temperature profiles at different billet positions for corner radius (a) Center (b) Off corner (c) Corner 259 Fig.A3.12 Surface temperature profiles along 4mm corner radius billet at different times 260 Casting time (sec) -i.4o r1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 , 1 i 1 1 1 1 ii 1 1 1 1 1 0 100 200 300 400 500 600 700 800 Distance from meniscus (mm) Fig. A 3 . 1 3 Evolution of billet surface displacement showing bulging below mold exit with different corner radii 261 depending on the billet geometry features such as corner radius and taper. Fig.A3.13 shows the evolution of displacement at the center and corner of the billet surface. The bulging of the billet during plant trial was also measured based on the distance from the billet center to the non-bulged line extended between the two billet off-corner locations (4 mm from each edge). These measurement were made on the cold section and ranged greatly from 0 to over 2 mm. Considering the uncertainties when evaluating the bulging, the calculated bulging amount seems consistent with the measured value. A3.5.5 Stress and crack prediction To illustrate the stress state through the solidifying shell, transverse stresses, a x, are plotted at different strand positions at 19 seconds of casting time (mold exit) in Fig.A3.14. The peak tensile stress is about 3 MPa, and is found beneath the surface. In the mold, it is similar around the billet perimeter except near the corner. The peak compressive stress is found at the surface and is much higher at the center region than the off-corner and corner. This is due to the huge drop of surface temperature resulting from good contact between the strand and the mold, which increases the shell strength. The superimposed temperatures through the shell show that the peak stress clearly corresponds to the 8-ferrite region, as indicated by the horizontal lines. This agrees with the findings of Thomas [41] that the sudden shrinkage from the 5 to y phase produces these tensile peaks, which may cause subsurface cracks. During the plant trial, billet samples were also taken under the same casting condition as described in Table A3.1 and A3.2 and their micro structure was investigated. Fig.A3.15 compares the typical microstructure for off-corner cracks that were found in this plant 262 Fig.A3.14 Temperature and transverse stress profile through the shell thickness at 19s of casting time 4mm comer radius mold 263 30mm Fig.A3.15 Comparison of crack location and model calculation at 100mm below the mold exit (a) microstructure of off-corner crack (b) hoop stress distribution (c) equivalent plastic strain contours 264 trial with stress and strain development at 100 mm below the mold. Usually solidification cracking or hot tearing may occur when the steel in the mushy zone is under tension beyond the some critical limit, due to the existence of a liquid film. Peak hoop tensile stresses, which pull apart dendrites and result in hot tears, take place both at the center and at the off corner of the billet as shown in Fig.A3.15 (b). Effective plastic strain is highest only at the off corner location (Fig. A3.15c). It is interesting to note that the peak strain occurs in a region of tensile hoop stress and corresponds roughly to the position of crack occurrence. The exact location of this crack obviously matches the surface depression. A 3 . 6 Effect of mold corner radius Next the model was applied to compare the thermo-mechanical behavior of steel cast in 4 mm (small) and 15 mm (large) corner radius molds. The results are evaluated according to the effects on heat transfer and gap formation, longitudinal corner surface cracks and longitudinal off-corner subsurface cracks. A 3 . 6 . 1 Heat transfer Fig. A3.16 shows temperature contours with the deformed shapes of both billets near the corner region at four locations down the mold. Both billets experience increasing solid shell thinning at the corner and its associated evolution of an air gap with casting time. During initial solidification, a uniform solidifying shell forms as a result of good contact between the strand and mold. After less than 1 second, the shell starts to shrink 265 (a) Fig.A3.16 Variation of peripheral shell profile in the vicinity of corner regi (a) 4mm corner radius (b) 15mm corner radius 266 267 away from the billet and an air gap forms near the corner. This reduces the local heat flow from the strand to the mold. This raises the temperature of the corner regions 22 mm below the meniscus, as shown in Fig.A3.11. Closer examination of the temperature profile around the corner reveals that the 15 mm corner radius billet develops both higher surface temperature at the corner and more severe non-uniform temperature contours along the billet surface as solidification proceeds. This reentrant-corner effect persists even below mold exit. The air gap size for both molds is plotted at different casting times in Fig.A3.17. As time progresses with increasing distance below the meniscus, the gap spreads further around the corner. By mold exit, the gap size extends to about 1.3 mm around the corner of the 4 mm radius billet and 1.4 mm for the 15 mm radius billet. In fact, the gap size in the 15 mm radius mold is larger than the 4 mm radius at every time. This leads to a higher surface temperature as shown in Fig.A3.16. The air gap grows with thermal contraction of the circumference of the long, thin shell. The circumferential length along the billet surface is 118.3 mm and 113.6 mm for the 4 mm and 15 mm radius mold, respectively. The shell shrinkage is 2.38 mm for the 4 mm and 1.95 mm for the 15 mm radius billet. The 15 mm radius billet shrinks a little less because its shell is slightly hotter. However, this study shows that the 4 mm corner radius has a smaller air gap size. This result is opposite to that of Ohnaka [36], who simulated slab casting and reported that the air gap size decreased with increasing corner radius, resulting in lower stress near the corner. The larger air gap size predicted for the large corner radius in this work is consistent with simple analysis of the strand geometry given in Appendix IV. For a given amount of 268 30 1.40 1.20 1.00 1 \u00b0 - 8 0 V n. ffl 00 I 0.60 - : 0.40 - : 0.20 0.00 30 35 Distance from center of billet (15mmR) (mm) 40 45 50 H 1 r + - 0 - - 4mrnR (5 sec.) - o- 4mmR(10 sec.) - a 4rnmR (19 sec.) \u2022 15mmR (5 sec.) -15mmR(10 sec.) M5mmR(19 sec.) 55 \u2014I\u2014 35 .0 40 45 50 Distance from center of billet (4mmR) (mm) 55 Fig. A 3 . 1 7 Evolution of air gap size profiles with different corner radii 269 shrinkage, the shell around the large radius corner must pull further away from the wall, than the small radius case, which generates \"slack\" more easily. This larger air gap can also be guessed from the extreme case of large corner radius: a round-section billet, where an air gap tends to form around the entire perimeter. A3.6.2 Longitudinal corner surface cracks Fig. A3.18 compares contours of hoop stress and hoop plastic strain of both billets near the corner region at the casting time 8 sec. As can be seen in this figure, both hoop values are much higher in the 15 mm radius billet. The development of hoop plastic strain with time is shown in Fig.A3.19 at a critical corner location, 1 mm deep beneath the corner surface where longitudinal corner cracks were found. This figure reveals that the large corner radius billet develops tensile plastic strain from 4 to 14 seconds in the mold (150 -520 mm below meniscus). This is consistent with breakout shell observations that corner cracks begin some distance below the meniscus. Compression is found both before and after this time. Below the mold, bulging makes the shell hinge around the corner, forcing the corner surface into compression. The small radius billet experiences compressive plastic strain at this location throughout casting, owing to two-dimensional cooling at the corner. This finding of higher susceptibility of surface cracks with large corner radius matches well with other plant observations [131]. It is also noted from this figure that using the mold powder as a lubricant can reduce the plastic strain due to the formation of a more uniform shell, resulting in less crack occurrence. 270 I M i l I 4 \u2022 uiuioe 2 7 1 Casting time (sec) 0.0 5.0 10.0 15.0 20.0 100 200 300 400 500 600 700 800 Distance from meniscus (mm) Fig. A3.19 Evolution of hoop plastic strain at the 1mm below the billet comer surface for different casting conditions 272 273 274 A3.6.3 L o n g i t u d i n a l off -corner subsurface cracks Figures A3.20 and 21 compare contours of hoop stress and hoop plastic strain of both billets near the corner region at the mold exit and 100 mm below the mold. All the results indicate compression at the surface, which implies that no surface cracks can form at mold exit or below. Both billets develop similar maximum tensile hoop stresses of about 3 MPa located near the solidification front everywhere except near the corner. Although the stress changes little between mold and below, the hoop plastic strain changes dramatically. At 100 mm below the mold, bulging of the billet causes the face to hinge around the corner. This causes subsurface tensile strain, increasing from a peak of only 0-0.1% at mold exit to over 0.4% at 100 mm for both billets. The location of the peak strain also moves from the corner to off-corner with decreasing corner radius. This movement of the peak strain location is directly related to the shell behavior at the corner. The shell is thicker at the corner than at the off-corner for the small radius billet, while the shell in the large radius billet is thinnest, hottest and weakest at the exact center of the corner. Therefore, the small radius billet is more susceptible to off-corner subsurface cracks than the large radius billet, as suggested by Samarasekera [133]. Furthermore, the high peak strain beneath the corner of large radius billets below the mold suggests that surface corner cracks, which initiate easily in the mold as shown previously, may grow more severe below the mold, as shown in Fig.2.12 (a). A3.7 Effect o f cast ing w i t h m o l d flux Finally, a simulation was performed to study the effect of mold powder lubrication on 2 7 5 the thermo-mechanical behavior of steel cast in the two different corner radius molds but with the same inadequate linear taper. Fig.A3.22 compares the solid shell contours at mold exit. Both billets show more uniform solid shell formation, leading to smaller air gap size, despite having a thinner average shell due to the lower heat flux associated with thicker gap. The smaller air gap size is due to less shrinkage of the hotter shell. In oil casting, this extra uniformity could be achieved by increasing taper. Changing the lubricant from oil to powder does not change the nature of the stress and strain development or susceptibility of large and small corner radius of billets to corner and off-corner cracks, respectively. The 15 mm corner radius billet develops peak hoop stress and strain at the corner and the 4 mm comer radius billet generates both peaks at the off-corner region. Fig.A3.23 shows the evolution of hoop stress and strain with casting time for 15 mm corner radius billet. With powder, the heat flux is lower and the solidifying shell is hotter and weaker. Thus, all of stresses and strains, and the associated surface defect are slightly worse. In this analysis, flux layer was assumed to maintain constant thickness during gap formation. In reality, it is likely that liquid flux will build up to fill the gap. This would increase corner heat flux relative to the predictions here, which would give rise to more uniform shell thickness. Therefore, for the same average heat flux and shell thickness at mold exit, powder-casting practice is expected to be less susceptible to cracks, due to better uniformity of solidifying shell. 276 30mm n (b) Fig. A3.22 Comparison of solid shell contours at the exit of mould with different lubricant and mold radius (a) 4mm corner radius (b) 15mm corner radius 277 with powder casting (a) Hoop stress (b) Hoop plastic strain 278 A3.8 Mechanism of surface crack formation The numerical analysis performed in the present study indicates two distinct mechanisms to generate longitudinal corner cracks or longitudinal off-corner internal cracks in casting steel billets with inadequate linear taper. Longitudinal corner cracks are predicted to arise only in large corner radius billets, due to tension developing across the hotter and thinner shell along the exact center of the corner during solidification in the mold. Such surface cracks could extend deeper due to solid shell bulging both in the mold, due to mold wear, or below, due to poor alignment of guide rolls. On the other hand, small corner-radius billets allow formation of a thinner shell at the off corner region inside the mold. This exacerbates hinging action accompanying bulging below the mold. This causes high plastic tensile strain across the dendrites in the off-corner region, leading to longitudinal subsurface off-corner cracks in billets cast in these molds. Although this analysis ignores the important effects of asymmetry, rhomboidity phenomena and lower ductility from copper pick-up on these defects, these mechanisms suggest more about mold operation. Applying mold powder as the lubricant allows the shell to solidify more uniformly, which could potentially reduce both of these cracks. Employing an optimized parabolic mold taper could achieve the same benefit. Mold wear effectively lowers the taper and likely worsens both cracking problems. Mold wear at the corner would cause a more severe gap, leading to a hotter and thinner shell there, which would increase the susceptibility to corner surface cracks. Mold wear at the center would allow billet bulging to occur inside the mold. This could allow the hinge action inside the mold, and increase the susceptibility to off-corner subsurface cracks: Misaligned or missing guide rolls also aggravate the below-mold bulging and hinging mechanism. 2 7 9 Finally, this work suggests that mold corner radius controls how longitudinal cracks are manifested, but is not the root cause of the problem. This means that large-corner radius molds could be used effectively to improve smooth rolling operations while still maintaining quality billets free of longitudinal cracks, so long as other casting parameters are optimized. Specifically, an optimized parabolic mold taper should be employed together with a well-maintained mold shape (free of wear and permanent distortion), mold powder lubrication, and adequate aligned foot rolls. More study is needed to achieve these requirements for different casting speeds, section sizes and mold lengths. 280 APPENDIX IV CALCULATION OF AIR GAP IN BILLET CASTING The gap size of 4 mm and 15 mm corner radius molds for casting 120 mm square billets was approximated geometrically assuming: 1) Shrinkage is 0.5%. 2) Circumferential length of the gap is 11.78 mm (TC*15\/4 mm) Fig.A4.1 shows the schematic diagram of corner region of a 15 mm radius billet, BF is initial radius, 15 mm, AF = AD is new radius, EF is initial half-perimeter of 15 mm radius DF is new half-perimeter of new radius From the Fig.A4.1, the following relationship can be obtained. where, 9 EF(\\-0.5%) (AAA) AF = AD = \u00a3Fsin45 sine? (A4.2) 281 AB=AFcos0-BFcos45 (A4.3) From equations (A4.1) - (A4.3), gap size for a 15 mm radius billet, D~E = AB + 15-AD, is 1.3 mm. Fig.A4.2 shows the corner region of. a 4 mm radius billet. Assuming the same circumferential gap length, the billet perimeter can be divided into three parts, a straight part ( F G ) , an angled part (FH ) and the 4 mm radius part as shown in this figure. The new half perimeter, FG + FI + ID is 0.5% less than the initial half perimeter, GHE, which is expressed by the following equation FG+^FH+W+^(BE-DE) = FHEKX - 0.5%) ( A 4 . 4 ) Solving equation A4.4, the gap size (DE = HI) for a 4 mm radius billet is 0.79 mm. 282 A4.1 Schematic diagram of billet corner region for 15mm radius billet A4.2 Schematic diagram of billet corner region for 4mm radius billet 283 ","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"GraduationDate":[{"@value":"2002-11","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0078652","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Materials Engineering","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Thermo-mechanical phenomena in high speed continuous casting processes","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/13475","@language":"en"}],"SortDate":[{"@value":"2002-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0078652"}