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Sticking-type breakouts during the continuous casting of steel slabs Mimura, Yoshihito 1989

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Sticking-type Breakouts during the Continuous Casting of Steel Slabs By Yoshihito Mimura B.Eng. (Applied Physics), The University of Tokyo, Tokyo, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES METALS AND MATERIALS ENGINEERING We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1989  ©  Yoshihito Mimura, 1989  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.  Metals  Department of  and  Engineering  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Materials  Abstract Sticking of the shell in the mold, which often occurs in a high-speed continuous slab casting machine, can be detected with thermocouples in the mold copper plates and prevented from developing into a breakout by reduction of the casting speed. However, a rapid reduction of the casting speed causes some quality problems and a low slab temperature. Thus, sticking-type breakouts remain a concern to the steel industry, and it is still not clear how and why the sticking initiates at the meniscus. The objectives of this study were to identify the causes of sticking at the meniscus, to elucidate the mechanism of sticking and finally to propose methods to reduce the occurrence of sticking. In order to identify the causes of sticking, it was necessary to examine a sticking-type breakout shell metallurgically, especially the dendrite structure at the shell surface. To link the metallurgical information to the casting conditions, the validity of a correlation in the literature between secondary arm spacing and local cooling rate has been examined. The secondary dendrite arm spacing in the subsurface of a slab has been measured and linked to a local cooling rate calculated from the measured mold heat-flux with this correlation. From this analysis, it was confirmed that Suzuki's correlation between secondary dendrite arm spacing and local cooling rate can be applied for a high cooling rate such as in continuous casting. A sticking-type breakout slab exhibiting five sticking events of 0.08% carbon steel, has been studied and it has been found that small holes exist at the surface in the sticking shells (most likely the site of entrapment, of solid mold flux). The shell which initially sticks  ii  exhibits a coarse dendrite structure and, in a longitudinal section, the shape of the initial sticking shell is parabolic. Moreover, with one exception, segregation lines typically 1-3 mm below the surface and almost parallel to the surface have been found in most of the sticking shell. From secondary ion mass spectroscope studies, the solutes concentrating in these segregation lines were determined to be Mn and S. Apparently, the sticking occurs at the meniscus where heat extraction is greatest and molten mold flux flows between the shell and solid mold flux rim oscillating with the mold. Therefore, to explain these meniscus phenomena, mathematical models of heat transfer at the meniscus and fluid flow in the mold flux channel have been formulated. To analyze the initial sticking event, the meniscus level has been changed in the computer simulations and the following mechanism has been proposed to explain the initiation of a sticking-type breakout. If the meniscus level rises, a deep notch forms in the shell due to the interaction between the mold flux rim and the shell. When a thick mold flux rim moves downward, it contacts the shell above the notch and the shell sticks to the mold flux rim. During the upstroke motion of the mold, tensile forces on the shell cause a rupture at the deep notch which is the hottest and weakest. The predicted solid flux rim profile agrees well with the parabolic shell shape measured in a longitudinal section of the sticking shell. Since the hot spot is the most likely point to be ruptured, conditions which minimize the hot spot were sought with the models. It was found that most of the conditions required to reduce hot-spot formation are exactly opposite to those required to minimize oscillation mark depth. Notwithstanding this, there are a few techniques to reduce the occurrence of  iii  sticking and to improve the surface quality: use a low melting point mold flux and, probably, maintain a deep mold flux pool. An interesting finding with respect to oscillation mark formation is that, if the mold flux rim is thick, the oscillation mark is caused by the interaction of the flux rim with the solidifying shell, while the fluid pressure development in the molten flux film dominates the mark formation in the case of a thin flux rim. For the analysis of the segregation line, a mass transfer model has been formulated based on a consideration of 8 — y transformation. From this analysis, it was found that the segregation observed in the sticking shell is a band of interdendric segregation enhanced by enlarged primary dendrite arm spacing which, probably, is caused by the appearance of an air gap due to the shell shrinkage.  iv  Table of Contents  Abstract  ii  Table of Contents  v  List of Tables  x  List of Figures  xii  List of Symbols  xxii  Acknowledgement  xxxii  1. Introduction  1  2. Literature Review  3  2.1 Sticking-type Breakouts  3  2.2 Dendrite Arm Spacing  9  v  2.3 Oscillation Mark Formation  13  2.4 Lubrication in the Mold and Behavior of Solid Flux Rim  17  2.5 Heat Transfer Models near the Meniscus  18  2.6 Segregation due to Solidification Followed by 8 - y Transformation  19  3. Scope of Present Work  22  4. Secondary Dendrite Arm Spacing in Continuous Cast Slabs and Its  24  Relationship to Local Cooling Rate 4.1 Heat Flux Calculation Data and Measurement of Arm Spacing  24  4.2 Mathematical Model with Finite-Difference Method  29  4.3 Definition of Cooling Rate  31  4.4 Calculation Accuracy of Mathematical Model Based on Finite-Difference Method  34 35  4.4.1 Effect of Time Step  35  4.4.2 Effect of Mesh Size  36  vi  4.4.3 Effect of Superheat in the Mold 4.5 Analytical Solution to Link between Secondary Dendrite Arm Spacing and  38  Mold Heat Flux 4.6 Prediction of Secondary Dendrite Arm Spacing with the Value of Heat Flux  5. Metallurgical Observation of Sticking-type Breakout Shells  40  42  5.1 General Aspects  42  5.2 Measurement of Oscillation Marks  48  5.3 Micro-Structure of Breakout Shells  51  5.4 Metallography with Secondary Ion Mass Spectroscope (S.I.M.S.)  59  6. Mathematical Model Heat Transfer near the Meniscus with Mold Flux  66  Lubrication 6.1 General Flow Sheet  66  6.2 Flow Rate of Mold Flux  71  vii  6.3 Meniscus Shape  '78  6.4 Introduction of Energy Equation  83  6.5 Heat Transfer in the Mold Flux Film  86  6.6 Slab Temperature Calculation  90  7. Model Results and Discussion  92  7.1 Response from the Initial Conditions  92  7.2 Effect of Convection on the Heat Transfer near the Meniscus  93  7.3 Heat Flux to the Mold and Heat Flux from the Slab  96  7.4 Oscillation Mark Formation  102  7.5 Mechanism of the Initiation of Sticking-type Breakout  110  7.6 Prevention of Sticking  117  8. Segregation below the Surface in a Low Carbon Steel 8.1 Cooling Rate Change due to Solid Flux Rim Remelting  viii  121 121  8.2 Cooling Curves for the Sticking Shells  125  8.3 Mass Transfer Model during Solidification Followed by 8 - y  129  Transformation: An Explanation for the Segregation Lines 8.4 Solute Redistribution after 8 - y Transformation  139  8.5 Segregation below the Surface in the Sticking Shell  144  8.6 Non-equilibrium Phase Diagram  146  9. Summary and Conclusions  149  Bibliography  153  ix  List of Tables 2.1  Correlation between Secondary Dendrite Arm Spacing and Cooling Rate,  12  Proposed by Suzuki et al. 4.1 (a)  Steel Composition, Liquidus and Solidus Temperature of Sample R  26  4.1(b)  Operating Conditions of Sample R  26  4.2  Average and Standard Deviation of Secondary Dendrite Arm Spacing  27  Measurement of Sample R 5.1 (a)  Steel Composition, Liquidus Temperature and Solidus Temperature of the  43  Sticking-type Breakout Slab 5.1 (b)  Operating Conditions of the Sticking-type Breakout Slab  43  5.2  Average of the Mark Pitch and Depth  48  5.3  Measuring Conditions of S.I.M.S.  61  6.1  Calculation Conditions and Operating Conditions  69  6.2  Steel Properties and Pouring Temperature  70  6.3  Mold Flux Properties and Pool Depth  71  x  6.4  Estimation of Terms in Equation (6.35)  86  8.1  Boundary Conditions for Mass Transfer Model  135  8.2  Distribution Coefficient and Diffusion Coefficient Based on Ueshima et al.  137  8.3 (a)  Primary and Secondary Dendrite Arm Spacing and Cooling Rate for Sample  138  IC1 8.3 (b)  Primary and Secondary Dendrite Arm Spacing and Cooling Rate for Sample IC7  xi  139  List of Figures 2.1  Sticking-type Breakout Shells  4  2.2  Shell Profile on Section a - a ' i n Figure 2.1  4  2.3  Process of Sticking in the Mold  5  2.4  Prevention of Breakouts due to Reduced Casting Speed  6  2.5  Breakouts with Excessive Heat Removal or with Insufficient Heat Removal  9  2.6  Schematic Representations for Isothermal Coarsening of Dendrite  11  2.7  Schematic Diagram Showing the Longitudinal Section (Parallel to the Broad  14  Face) of a Continuous Casting Mold 2.8  Oscillation Mark Formation due to Emi's and Kawakami's Models  16  4.1  Heat Flux Data of Sample R  25  4.2 (a)  Section at the Surface of Sample R, Etched with 3.5 % Picric Acid  28  4.2 (b)  Section 20 mm below the Surface of Sample R, Etched with 3.5 % Picric  28  Acid 4.3 (a)  Enthalpy as a Function of Temperature  xii  31  4.3 (b)  Modified Temperature as a Function of Temperature  31  4.4 (a)  Temperature Calculated with Time  32  4.4 (b)  Cooling Rate Calculated with Time  32  4.5  Secondary Dendrite Arm Spacing Prediction and Measurement (Average and  33  Standard Deviation) with Distance from the Slab Surface 4.6  Effect of Time Step on the Predicted Cooling Rate  35  4.7  Effect of Mesh Size on the Predicted Cooling Rate  36  4.8  Effect of Super Heat on the Predicted Cooling Rate  37  4.9  Variation of Non-dimensional Temperature with rj  40  4.10  Secondary Dendrite Arm Spacing Prediction  41  5.1  Casting Speed Variation after the Nozzle Change  43  5.2  Photograph of Broadface (Outside Radius) of Sticking-type Breakout Slab  44  5.3  Slab with Sticking Shell and Sample Locations  45  5.4  Appearance of Sound Area IC5 in the Breakout Slab  46  5.5  Appearance of Initial Sticking Area OC2 in the Breakout Slab  47  xiii  5.6  Appearance of Rupture Zone OC3 in the Breakout Slab  47  5.7  Mark Depth vs. Mark Pitch in Sound Shell  49  5.8  Mark Depth vs. Mark Pitch in Sticking Shell  50  5.9  Mark Depth vs. Mark Pitch in Ruptured Shell  50  5.10  Cast Structure in IC5, Sound Shell, Etched with 2.5 % Picric Acid  52  5.11  Cast Structure in OC2, Sticking Zone, Etched with 2.5 % Picric Acid  52  5.12  Cast Structure in OC3, Rupture Zone, Etched with 2.5 % Picric Acid  53  5.13  Cast Structure without Segregation Line in Sample IC7 Etched with 2.5 %  53  Picric Acid 5.14  Locations in the Breakout Shell Exhibiting Segregation Lines  54  5.15 (a)  Sketch of a Cross Section Including the Initial Sticking Region  55  5.15 (b)  Photograph of Cast Structure in the Initial Sticking Area  55  5.16  Parabolic Shape of the Initial Sticking Shell  56  5.17  Change of Secondary Dendrite Arm Spacing with Distance from the Surface  57  in Sound Shell IC3  xiv  5.18  Change of Secondary Dendrite Arm Spacing with Distance from the Surface  58  in Ruptured Shell with the Segregation Line IC1 5.19  Change of Secondary Dendrite Arm Spacing with Distance from the Surface  58  in Ruptured Shell without the Segregation Line IC7 5.20  Change of Secondary Dendrite Arm Spacing with Distance from the Surface  59  in Initial Sticking Shell OC2 5.21  Cast Structure Examined with S.I.M.S., Sample IC1  61  5.22  Carbon Distribution in the Area Shown in Figure 5.21  62  5.23  Manganese Distribution in the Area Shown in Figure 5.21  62  5.24  Sulphur Distribution in the Area Shown in Figure 5.21  63  5.25  Phosphorus Distribution in the Area Shown in Figure 5.21  63  5.26  Distribution of Pearlite in Sample IC1  64  5.27  Manganese and Sulphur Distribution with Line Analysis  65  6.1  Relationship among Four Parts of Heat Transfer Model at the Meniscus  67  6.2  General Flow Chart of Heat Transfer Model at the Meniscus  68  6.3  Physical System of Mold Flux Row  73  xv  6.4  Physical System in the Calculation by Takeuchi and Brimacombe or Anzai et  75  al. 6.5  Channel between Solid Mold Flux Rim and Shell with Oscillation Marks  78  6.6  Physical System at the Meniscus  80  6.7  Meniscus Shape Calculated by Analytical Solution and Numerical Method  82  6.8  Physical System of Heat Transfer in the Mold Flux  87  6.9  Tracking System of the Slab Temperature Distribution  91  7.1  Response of the Average of the Heat Flux to the Mold and the Average of  93  the Heat Flux from the Slab in the Region 0 ~ 1 cm below the Meniscus from the Initial State 7.2  Profile of Heat Flux to the Mold  94  7.3  Profile of Heat Flux from the Slab  94  7.4  Comparison of the Heat Flux to the Mold with or without Convection  95  7.5  Comparisons of the Heat Flux from the Slab with or without Convection  95  7.6  Heat Balance in the Region 0 ~ 1 cm below the Meniscus  xvi  96  7.7  Effect of Flux Viscosity on the Heat Fluxes near the Meniscus  97  7.8  Effect of Casting Speed on the Heat Fluxes near the Meniscus  97  7.9  Effect of Mold Oscillation Frequency on the Heat Fluxes near the Meniscus  98  7.10  Effect of Mold Oscillation Half Stroke on the Heat Fluxes near the Meniscus  98  7.11  Effect of Flux Melting Point on the Heat Fluxes near the Meniscus  98  7.12  Effect of Flux Viscosity on the Heat Balance near the Meniscus  100  7.13  Effect of Casting Speed on the Heat Balance near the Meniscus  100  7.14  Effect of Mold Oscillation Frequency on the Heat Balance near the Meniscus  100  7.15  Effect of Mold Oscillation Half Stroke on the Heat Balance near the  100  Meniscus 7.16  Flux Velocity Distribution with a Low or High Mold Flux Viscosity  101  7.17  Flux Velocity Distribution with a Low or High Casting Speed  101  7.18  Shape of Oscillation Mark Calculated in the Model  103  7.19  Effect of Flux Pool Depth on Predicted Oscillation Mark Depth  104  7.20  Effect of Flux Viscosity on Predicted Oscillation Mark Depth  104  xvii  7.21  Effect of Casting Speed on Predicted Oscillation Mark Depth  104  7.22  Effect of Mold Oscillation Frequency on Predicted Oscillation Mark Depth  104  7.23  Effect of Mold Oscillation Half Stroke on Predicted Oscillation Mark Depth  105  7.24  Effect of Flux Melting Point on Predicted Oscillation Mark Depth  105  7.25  Effect of Flux Viscosity on Average Distance of the Shell from the Mold  106  Wall 7.26  Effect of Casting Speed on Average Distance of the Shell from the Mold  106  Wall 7.27  Schematic Change of Oscillation Mark Shape from Kawakami-type to  109  Emi-type Mark 7.28  Effect of Mold Oscillation Frequency on Emi-type Oscillation Mark Depth  110  7.29  Meniscus Level Change in the Computer Simulation  111  7.30  Shell Profile and Solid Flux Rim Profile with a Constant Meniscus Level  112  7.31  Slab Surface Temperature Profile with a Constant Meniscus Level  112  7.32  Shell Profile and Solid Flux Rim Profile after a Rise in the Meniscus  113  7.33  Slab Surface Temperature Profile after a Rise in the Meniscus  113  xviii  7.34  Comparison between the Real Sticking Shell and the Shell Profile or the  114  Solid Flux Profile 7.35  Comparison between the Real Sticking Shell and the Meniscus Shape  114  without Solid Flux Rim and Liquid Flux Film 7.36  Mechanism of the Occurrence for the Initial Sticking of the Shell  115  7.37  Shell Profile and Solid Flux Profile after a Fall in the Meniscus  117  7.38  Slab Surface Temperature Profile after a Fall in the Meniscus  117  7.39  Effect of Flux Viscosity on the Predicted Hot Spot Temperature  119  7.40  Effect of Casting Speed on the Predicted Hot Spot Temperature  119  7.41  Effect of Mold Oscillation Frequency on the Predicted Hot Spot Temperature 120  7.42  Effect of Flux Melting Point on the Predicted Hot Spot Temperature  120  7.43  Effect of Mold Oscillation Half Stroke on the Predicted Hot Spot  120  Temperature 7.44  Effect of Flux Pool Depth on the Predicted Average of Solid Flux Rim  120  Thickness above the Meniscus 8.1  Schematic Diagram of the Rupture in the Sticking Shell  xix  123  8.2  Comparison of the Secondary Dendrite Arm Spacing between Measurement  125  and Calculations Incorporating Solid Flux Rim Remelting 8.3  Heat Flux Pattern for Sample IC1  126  8.4  Heat Flux Pattern for Sample IC7  127  8.5  Comparison of Measured Secondary Dendrite Arm Spacing for Sample IC1  127  to Values Calcualted by the Trial-and-Error Method 8.6  Comparison of Measured Secondary Dendrite Arm Spacing for Sample IC7  128  to Values Calculated by the Trial-and-Error Method 8.7  Cooling Curves Calculated for Sample IC1  128  8.8  Cooling Curves Calculated for Sample IC7  129  8.9  Solidification Model Used (a) Position of Volume Element in Mushy Zone  130  (b) Magnified Volume Element with Concentration Profile 8.10  Control Volume in Primary Arm Spacing for Mass Transfer Model  133  8.11  General Flow Chart for Mass Transfer Model Based on the Model of  134  Ueshima et al. 8.12  Solute Concentrations after Solidification for Sample IC1  xx  141  8.13  Solute Concentrations after 8 - YTransformation for Sample IC1 (at 1300 °C)  142  8.14  Solute Concentrations after 8 - YTransformation for Sample IC7 (1300 °C)  143  8.15  Mechanism for Appearance of Air Gap during the Sticking  145  8.16  Non-equilibrium Transformation Temperature Calculated for IC1  147  8.17  Equilibrium and Non-equilibrium Phase Diagram (a) with Different Cooling  148  Rate (b) Different Primary Arm Spacing  xxi  List of Symbols a  the capillary constant (-)  2  a  mold oscillation half stroke (cm)  C  solute concentration (wt%)  C°  initial solute concentration (wf %)  C7  solute concentration in liquid far from solid interface (wt%)  s  C, °, C\ r  x  solute concentration at the interface of dendrite with radius r and r , 0  respectively (wt%) C*  solute distribution at liquid-solid interface (wt%)  C  L  solute concentration in liquid (wt%)  C  5  solute concentration in 5-phase (yvt%)  C  Y  solute concentration in y-phase (wr%)  xxii  x  CR  local cooling rate (°C/s)  Cp  specific heat of steel (Jlg°C)  Fe  Cp  D  D  P  specific heat of mold flux (Jig °C)  diffusion coefficient in solid (cm Is) 2  s  &  D  y  D  slab  diffusion coefficient in 5-phase (cm Is) 2  diffusion coefficient in y-phase (cm Is) 2  slab thickness (cm)  d  mold copper plate thickness (cm)  dp  mold flux film thickness (cm)  F  friction stress between shell and mold (dyne I cm )  Cu  f  2  t  mold oscillation frequency (cycleIs)  f  s  fraction of solid (-)  xxiii  g  gravitation constant (cm/s ) 2  H,  enthalpy of steel (Jig)  F  h  thickness of molten flux film (cm)  h ~ /z {  3  meniscus dimension in the through thickness direction defined in Figure 6.4 (cm)  h h h  h  B  h  T  h  c  N  distance of the shell from the mold wall defined in Figure 6.5 (cm)  the largest distance of shell from the mold wall defined in Figure 6.5 (cm)  the smallest distance of shell from the mold wall defined in Figure 6.5 (cm) a constant heat-transfer coefficient (between steel and mold cooling water) (W/cm C) 2o  h^  average heat-transfer coefficient between mold hot face and cooling water (W/cm C) 2o  h  PM  heat-transfer coefficient between solid flux rim and mold  xxiv  (W/cm °C) 2  h  heat-transfer coefficient between mold cold face and cooling water (Wlm °C)  k  distribution coefficient (-)  2  w  k*  distribution coefficient between liquid steel and 5-phase (-)  k  distribution coefficient between liquid steel and y-phase (-)  11  rL  k  ih  k  Cu  distribution coefficient between 5-phase and y-phase (-)  thermal conductivity of mold copper plate (kW/m°C)  k,  thermal conductivity of steel (W/cm C)  k  P  thermal conductivity of mold flux (W/cm°C)  k  w  thermal conductivity of mold cooling water (W/m°C)  L  characteristic length for heat transfer (cm)  L  H  latent heat (Jig)  L  Fe  latent heat of steel (Jig)  F  2o  xxv  Lp  latent heat of mold flux (Jig)  l  mold flux pool depth (cm)  P  l ~/ x  meniscus dimension in the casting direction defined in Figure 6.4 (cm)  2  l l h  distance of points on the shell surface from the meniscus defined in Figure 6.5  N  (cm)  M  the number of control volume at the liquid-solid interface in mass transfer model (-)  M  the number of the outer control volume (middle point between two primary  c  dendrites) in mass transfer model defined in Figure 8.10 (-) m  liquidus slope  N  the number of control volume at the 8 - y interface in mass transfer model (-)  P or P  P^  (°C/wt%)  pressure in the mold flux film (dyne I cm ) 2  P  pressure in the air (dyne I cm ) 2  xxvi  AP  pressure difference between steel and mold flux (dyne I cm )  Pt  the Prandd number of the mold cooling water (-)  2  w  Q  mold flux flow rate (cm}Is)  q  heat generation (Jig)  q  heat flux to mold (W/cm )  L  2  u  q  heat flux from slab (W/cm ) 2  s  Re  the Reynolds number of mold cooling water (-)  w  r, r 0  x  r  x  radius dendrite defined in Chapter 2 (cm)  distance of the meniscus from the top of mold defined in Chapter 6 (cm)  r  distance of mold flux pool from the top of mold defined in Chapter 6 (cm)  s  distance along the meniscus (cm)  2  S  0  thickness of solid mold flux rim (cm)  xxvii  T  A4  orT  T  AAe  A4l  temperature at which the 8 - y transformation starts (°C)  temperature at which the 8 - y transformation completes (°C)  T,  temperature in steel slab (°C)  T°  poring temperature of steel in mold (°C)  F  Fe  T,  equilibrium liquidus temperature (°C)  T  mold hot face temperature (°C)  M  T  T  P  PM  T,  temperature in mold flux (°C)  melting point of mold flux (°C)  equilibrium solidus temperature (°C)  T  slab surface temperature (°C)  TSJ-  slab surface temperature at the meniscus (°C)  T  mold cooling water temperature (°C)  slab  w  xxviii  t  time  t*  ti  t  non-dimensionalized time defined in Equation (6.40) (s)  time  when the temperature decreases to liquidus temperature (s)  negative strip time of mold oscillation (s)  N  t  (s)  time when the temperature decreases to solidus temperature (s)  s  u  mold flux velocity in the fixed system (cmIs)  ii  mold flux velocity in the moving system (cmls)  u*  non-dimensionalized mold flux velocity defined as Equation (6.37) (-)  V  mold oscillation velocity (cmls)  M  V  V  z  casting speed (cmls)  w  velocity of mold cooling water (m/s)  x  distance in the casting direction (cm)  xxix  x*  non-dimensionalized distance in the casting direction defined in Equation (6.39) H  y  distance in the through thickness direction (cm)  y"  non-dimensionalized distance in the through thickness direction defined in Equation (6.38) (-)  Z  distance from the top of mold to carbon-enrich flux layer (cm)  0  Zj  <x  system length concerned in the heat transfer model at the meniscus (cm)  thermal diffusivity of steel (cm Is) 2  Ft  a?  thermal diffusivity of mold flux (cm Is)  T\  mold flux viscosity (poise)  9  non-dimensionalized temperature defined in Equation (6.36) (-)  9/  local solidification time (s)  X  primary dendrite arm spacing (cm or\xm)  x  2  XXX  %2  secondary dendrite arm spacing (cm or \un)  v  kinematic viscosity of mold flux (cm Is) 2  v  kinematic viscosity of mold cooling water (m /s) 2  w  p  density of steel (glcm?)  Ft  p  p  P  density of mold flux (g/cm )  w  density of mold cooling water (kglm?)  3  a  o"  y  interfacial tension between liquid steel and molten mold flux (dyne I cm)  s  liquid-solid interfacial tension (dyne I cm)  <}>  contact angle (radian)  ¥  modified temperature defined in Equation (4.3) (°C)  Fe  co  angular velocity of mold oscillation (s~ ) l  xxxi  Acknowledgement I would like to express my sincere gratitude to my supervisor, Professor J.K. Brimacombe for his continuous assistance and guidance during the course of this research. The assistance of Associate Professor I.V. Samarasekera is gratefully acknowledged. I wish to thank Nippon Steel Corporation and the Natural Science and Engineering Research Council of Canada for the provision of a graduate scholarship and research support, respectively. Without the help of LTV Steel Company, this study would not be possible. I want to express my gratitude to Dr. P.H. Dauby and Mr. W.H. Emling of LTV Steel Company for their permission and support. It is my obligation and pleasure to note here that many friends in the Department of Metals and Materials Engineering help this study.  xxxii  1  1. Introduction In the last fifteen years, the productivity of continuous casting machines has increased substantially. The production of a slab casting machine has changed from one hundred thousand tons per month to two hundred thousand tons per month largely by increasing the casting speed [1]. More recently, as a result of directly connecting the continuous casting process and the hot strip mill, maintenance of high casting speed has become an important technology not only for an increase in productivity but also for energy reduction in the reheating furnace [2]. Unfortunately, breakouts, especially sticking-type breakouts, occur more often in a high casting speed machine [1]. Breakouts, in which liquid steel flows out below the mold, result in significant repair costs and disturbance of production. Many researchers and engineers have investigated the prevention of sticking-type breakouts. As a result of this work, sticking-type breakouts can be detected with thermocouples in the mold copper plate [3]~[6] or by measurements of the friction force during mold oscillation [3][6]. Moreover, through these studies, lubrication in the mold as a function of mold flux properties is becoming understood. With the development of detection systems for sticking-type breakouts, the steel industry has succeeded in minimizing the breakout itself; however, this success does not mean the elimination of the sticking. For the prevention of these breakouts, the casting speed must be reduced [4]~[6], which causes some quality problems and a low slab temperature. Thus, the sticking-type breakouts remain a concern to the steel industry and it is still not clear how and why sticking-type breakouts initiate at the meniscus in the mold. For an analysis of initial sticking, information about events near the meniscus is important. Measurements of the mold copper plate temperature and the mold friction force,  2  however, do not always fully reflect meniscus phenomena. Other information that is invaluable in elucidating the causes of sticking is the dendrite structure at the sticking shell surface. However, previous studies of dendrite structure have been usually done on a laboratory scale and, therefore, it is difficult to apply the information gained from such studies to actual continuous casting operations. For example, correlations between secondary dendrite arm spacing and cooling rate have been found under conditions of a uniform temperature gradient and a low cooling rate. Therefore, it should be investigated whether or not information on secondary dendrite arm spacing in the literature could be employed for analyzing the meniscus phenomena in a continuous casting mold, where the temperature gradient is not uniform and the cooling rate is high. Moreover, in this region, there are mechanical interactions between the solidified shell and oscillating mold. Many previous studies have been done solely from the viewpoint of heat transfer or simply by considering mechanical aspects, but not both. In the region near the meniscus, these phenomena are related and, consequently, the study of meniscus phenomena should involve both a heat transfer analysis and a mechanical investigation. The primary purpose of this study is to identify the causes of sticking at the meniscus, particularly by examining the dendrite structure at the surface of a sticking-type breakout shell and relating it to casting conditions. The applicability of correlations in the literature between secondary dendrite arm spacing and cooling rate to continuous casting has been examined. Links have been established between mold heat flux and dendrite structure at the slab surface. Mathematical models of fluid flow in the mold flux channel and heat flow at the meniscus have been employed to elucidate the mechanism of sticking. The final objective of the work is to propose methods to eliminate sticking-type breakouts from a fundamental understanding of how they initiate.  3  2. Literature Review 2.1 Sticking-type Breakouts Sticking-type breakouts are induced by shell sticking which starts at the meniscus with the resulting shell rupture moving successively down the mold. Shells of sticking-type breakouts have a characteristic appearance as shown in Figure 2.1 (based on the figure by Itoyama et al. [3]). The shell that underwent sticking, which usually remains in the mold after the breakout, can be seen as (A) in Figure 2.1, while (B) is a sound shell which is withdrawn with the strand. The ruptured part (C) usually moves with the sound shell (B). Between the sticking part (A) and the ruptured part (C), there is a distinct line (D) which has an inclination of 30-45 ° from the horizontal which is the site of maximum shear stress. An actual breakout occurs at the bottom of the line (D) when the line moves down below the mold. The sound shell (B) has oscillation marks which are horizontal, while the sticking part (A) and the ruptured part (C) exhibit marks which are not horizontal but V-shaped [4]. The marks in the sticking and ruptured parts are described frequendy as psuedo-oscillation marks [4] or ripple marks [3]. The pitch of the marks both in parts (A) and (C) are smaller than that in part (B). If the shell profiles are examined at the longitudinal section a - a' in Figure 2.1, the section would appear as shown in Figure 2.2. The sound shell (B) has a profile in which the thickness increases with increasing distance downward in the casting direction. On the other hand, the sticking part (A) has the opposite shell profile, in which the  4  thickness increases towards the top of the mold (towards the meniscus). Thus, it can be concluded that the shell near the meniscus has stuck initially because it has a thicker shell.  a  Figure 2.1 Sticking-type Breakout Shells  Figure 2.2 Shell Profile in Section a - a'in Figure 2.1  Based on the above observations of the sticking shells, Itoyama et al. [3] and Tsuneoka et al. [4] have explained how the rupture proceeds in the mold, which is illustrated in Figure 2.3. Initially, the shell near the meniscus has stuck for reasons undetermined and the shell rupture occurs a little below the initial sticking point. Thus, liquid steel flows into this area and forms the new shell. Part (A) remains stuck and moves up during the upward stroke of the mold oscillation, while part (C) is withdrawn with the strand. As a result, the shell is placed in tension which causes another rupture  5  at the weakest point where the shell is newly forming. These processes continue in each oscillation cycle and, finally, when the rupture line reaches the bottom of the mold, liquid steel flows out to create a breakout.  Figure 2.3 Process of Sticking in the Mold [3]  If the mold temperature is measured at several positions in the casting direction, when the sticking occurs, first, in the top of the mold, the temperature increases and gradually decreases. Because the newly formed thin shell has a high temperature, the hot spot moves down according to the movement of the rupture. Usually, the velocity of the hot spot is lower than the casting speed [4]. Thus, sticking in the mold can be detected by mold temperature measurements [3]~[6]. In order to release the sticking, the operator reduces the casting speed quickly and as shown in Figure 2.4, if the casting speed is reduced to below 1.0 m/min, the sticking breakouts can be prevented in this particular machine [4]. From a consideration of the sticking process and release of sticking due to reduced casting speed, Tsuneoka et al.[4]  6  and Itoyama et al.[6] have proposed a mechanism for preventing the rupture from progressing. In both proposed mechanisms, it is suggested that an increase in the shell thickness during the negative-strip time stops the rupture. During the negative-strip time, the rupture does not occur owing to the compressive force exerted on the shell by the downward motion of the mold. Thus, if the shell solidifies to sufficient thickness during the negative-strip time, the sticking part can be removed from the mold without rupture, when the mold begins to move upward and tensile forces begin to act on the shell. 0.6 a = 3  = 0.75 L = 450  0.5  L = 400  L = 350 ^)  0.3 / 1 1 '  111  i HI/I  a  if -V  1 III  0.2  L = 300  1 i,  , , 'I'II  V ^•^Shel 1 recovery curve  Ac tua1 oscilati on cor.di tion curve 0.1  L: Distance from meniscus to quasi-meniscus j • Occurrence of breakout Q Prevention of breakout by reducing casting speed ! 0.5  Casting  2.0  1.0  Speed  (m/  m i n  )  Figure 2.4 Prevention of Breakouts due to Reduced Casting Speed [4]  Although the sticking movement or the sticking release is relatively well understood, it is not clear why and how the initial sticking occurs near the meniscus. Many previous studies of initial sticking have been based on analyses of friction force  7  between the shell and the mold [3],[6]~[10]. The equation often used for the friction force analysis is: V -V M  F  Z  > = T \ - ^ a  (2.D  P  where F, : friction stress between the shell and the mold rj : mold flux viscosity V : casting speed z  V : mold oscillation velocity M  d : mold flux film thickness P  These studies have indicated that a large friction force brings about a shell rupture near the meniscus and the ruptured shell is stuck to the mold. Therefore it has been concluded that, for prevention of the sticking, the friction force should be reduced with a low viscosity of mold flux [3], or an increase of flux film thickness (an increase of flux consumption) [7]. The effect of Al 0 2  3  [11] or gas bubbles [9] on the flux viscosity also  have been discussed. It has been reported that fluxes containing Li and Mg [12], mold flux with a low melting point [12] and non-sinusoidal mold oscillation [7] improves the flux consumption and reduces the friction force. Another approach for the prevention of sticking is the elimination of disturbance to the shell near the meniscus. The importance of mold level control [14] and of an adequate mold flux pool depth [15] has been emphasized. However, these studies cannot explain what the shell is stuck to and what causes the initial sticking. Considering the strength of steel at high temperature, the initial rupture must occur at the hottest part of the shell, which means that a hot spot must exist below the sticking point before the initial sticking occurs. No mechanism for appearance of the hot spot has been proposed. In the previous studies, the causes of the initial sticking have not been determined, but the findings can be classified into two categories. The first is related to the direct  8  contact of liquid steel to the mold plate since Itoyama et al.[3] and Tsuneoka et al.[4] have found copper or coating material on the sticking shell. In the second case, metal is not found on the sticking shell but, instead, there is a carbon-rich region in the vicinity of the initial sticking [4] [16]. Data which is useful in considering these two categories regarding the initial sticking are reported by Ogibayashi et al.[16], as shown in Figure 2.5. When the breakout occurs, the mold heat-transfer coefficient calculated is excessively large or small, relative to normal operation. Because the mold flux represents the largest thermal resistance, it is considered that more heat transfer is caused by a thin mold flux film while less heat transfer implies a thicker flux film. The first cause (direct contact of liquid steel to the mold plate) is, then, related to the smaller flux film thickness. It can be said that poor lubrication causes the sticking, because liquid steel easily contacts and sticks to the mold plate. On the other hand, it can be regarded that the rich carbon in the vicinity of the initial sticking comes from the mold flux [17]. Therefore, the second cause may be explained by phenomena involving a thicker flux film. However, the actual causes of the initial sticking, especially in the second case (less heat transfer), are still unknown.  9  —i 0.8  Figure 25  1  •  Breakout slab  O  Normal slab  i i 1.4 1.6 Casting speed (m/min)  1.0  :  I  1.2  1.8  Breakouts with Excessive Heat Removal or with Insufficient Heat Removal or with Insufficient Heat Removal [16]  2.2 Dendrite A r m Spacing Measurements of the dendrite arm spacing are useful for understanding the thermal history of metal during solidification. Many correlations between the primary or secondary dendrite arm spacings and the thermal conditions have been proposed theoretically and empirically [18]~[31]. For the analysis of the thermal history of continuously cast steel, the measurements of secondary dendrite arm spacings are more often made than those of primary arm spacings, because the latter is affected by convection which is always present in the continuous casting mold [28]. Consequently, in this study, secondary dendrite arm spacings have been investigated.  10  The secondary dendrite arm spacing can be determined by the coarsening and coalesence processes, which essentially are Ostwald ripening [33]. Kattamis et al. have proposed two models for dendrite arm coarsening, as shown in Figure 2.6 [22]. From the Gibbs-Thomson relationship, for part "0" of dendrites with radius r : 0  T-T=——  (2.2)  'v-'h  where  T,>T>T  S  T : equilibrium liquidus temperature t  T : equilibrium solidus temperature s  T : a constant temperature (i.e. isothermal field is assumed) <j"' : liquid-solid interfacial tension L : latent heat H  If the liquidus slope is m, Equation (2.2) is:  c  -_ ;o _5_ roL m c  =  (2  .3)  H  Similarly, for part "1" with radius r , x  Cr-CN-^—  rL m x  (2.4)  H  If r <r , C/' < Ci°. Thus, solute is transferred from the liquid-solid interface of part "0" 1  x  0  to part "1". The solute transferred decreases the melting point temperature of part "1" and, therefore, part "1" becomes thin and diminishes (Model-1) or melts-off and moves away (Model-2). The critical time for part "1" to survive depends on the solute diffusion and the reduction of the melting temperature below the field temperature T. The critical  11  time, which equals to the local solidification time if the dendrite survives, in both Model-1 and Model-2 is: tcrUical^Ad,  (2.5)  where A = mC(k - 1)  m = liquidus slope C = concentration k = distribution coefficient d - distance betw Therefore, the secondary dendrite arm spacing (^ = d) is: (2.6)  Model-l  Model-2  Figure 2.6 Schematic Representations for Isothermal Coarsening of Dendrite [22] Many researchers have experimentally studied the correlation between the cooling rate and the secondary dendrite arm spacing or that between the local solidification time and the arm spacing [24],[27]~[31]. There are some good reviews about these correlations for steel [32][34]. For a plain carbon steel, Suzuki's equations listed in Table 2.1 [27] are recognized and are often referred to by other researchers. The  12  exponents of the cooling rate (0.35-0.44) are a little larger than that of Kattamis' theory. Together with other experiments, Suzuki's tests are based on a thermal analysis involving thermocouple measurements. Suzuki and his colleagues have calculated the average cooling rate between the liquidus and solidus temperatures based on the equilibrium phase diagram. Table 2.1 Correlations between the Secondary Arm Spacing and the Cooling Rate, Proposed by Suzuki et al., where %2 = Secondary Dendrite Arm Spacing (\un), CR = Cooling Rate (C°/s)  and C, Si, Mn, P, S and Al are Chemical Composition of Steel (wt%).  c  Si  Mn  0.14  0.46  0.65  0.014 0.015 0.011  157.6(0?)-*  0.27  0.46  0.60  0.016 0.015 0.015  152.9(CR)-*  0.04<CR<5.0  0.43  0.45  0.65  0.016 0.016 0.018  150.1(C#y*  0.04<CR<11.7  0.62  0.36  0.64  0.015 0.016 0.016  U3.5(CRy*  0.04<CR<11.7  0.76  0.46  0.70  0.015 0.016 0.021  i33.5(a?y*  0.88  0.46  0.73  0.015 0.016 0.030 140. ICC/?)"**  P  5  Al  Range ofCR 36  3S  37  M  39  4  0.05<CR<6.7  0.04<CR<6.7 0.05<CR<11.7  A limitation, both in the theory and in the experiments, is whether or not the correlation between cooling rate and secondary dendrite arm spacing can be applied to high cooling rates realized in the continuous casting process. Huang and Glicksman have concluded that the coarsening should occur even during high-speed solidification, based on their experiments with a high-purity succinonitrile specimen [26]. This problem, as it is applied to plain carbon steel, will be discussed further in Chapter 4.  13  Another problem, when a correlation between the cooling rate and the secondary dendrite arm spacing is applied to continuously cast steel, is how the cooling rate should be calculated, because it cannot be directly measured in the process. Wolf and Kurz have calculated the cooling rate from an analytical solution which is derived under such assumptions as planar solid-liquid interface at the melting point, constant liquid temperature far from the interface and constant slab surface temperature [28]. They have fitted a K value (shell thickness X =K^solidification s  time t), which is also derived under  the above assumptions, to their measured data of shell thickness. Cramb has also used this parabolic freezing law for the calculation of the cooling rate and shell thickness [31]. dX  t  K  However, to avoid an infinite value of the solidification rate( ~i~=^-) at t=0, he assumes:  constant.  Application of the parabolic freezing law for the calculation of the cooling rate may give a good agreement with the measurements in slabs, but it cannot give rise to an adequate correlation between the cooling rate and the secondary arm spacing near the surface because the temperature cannot be assumed to be constant and the shell thickness does not grow with V T [35]. This problem also will be discussed in Chapter 4.  2.3 Oscillation Mark Formation One consequence of the meniscus phenomena in continuous casting is oscillation mark formation. The depth of the marks affects the surface quality of the slabs, eg.transverse cracks and pin holes- and, therefore, the effect of casting conditions on the mark depth has been well studied experimentally [16],[36]~[39]. The depth of oscillation marks decreases with increasing mold flux viscosity [16] [36], casting speed [37] and oscillation frequency (decreasing negative-strip time) [38] and it increases with  14  increasing oscillation stroke length [37] [39]. Here, some words related to the oscillation mark formation, which are often used in this investigation, should be defined. Figure 2.7, based on Branions's description [40], shows the appearance of a mold with a lubricant (mold flux) and a tundish-to-mold tube (submerged nozzle). The mold flux above the liquid steel meniscus generally consists of three layers: (1) An unmelted, dark, powder layer on the top (2) A carbon-enriched or perhaps sintered layer in the middle (3) A liquid molten flux pool directly over the steel The molten flux flows down between the solidified shell and the water-cooled copper mold wall, and is thereby consumed while the solid flux rim is formed from the molten flux due to the mold cooling.  ir-fli _  Strand Withdrawal  Figure 2.7 Schematic Diagram Showing the Longitudinal Section (Parallel to the Broad Face) of a Continuous Casting Mold [40]  15  Several mechanisms have been proposed to explain the oscillation mark formation [38][39],[41]~[45]. The mechanisms proposed can be classified into two concepts: The first mechanism has been proposed in a qualitative manner by Emi et al.[39] and explained theoretically with a mathematical model by Takeuchi and Brimacombe [43] and Anzui et al.[44]. Based on this concept, the top edge of the shell is pushed into the liquid steel during the negative strip period of the mold oscillation due to pressure that develops in the flux channel. At the end of the negative strip period, the flux pressure is released and ferrostatic pressure either causes liquid steel to overflow the partially solidified meniscus to form a hook, or the meniscus is pushed back toward the mold wall without a hook by a reversal in the pressure. According to the second idea proposed by Kawakami et al.[38], the top edge of the shell is bent by interaction of a solid powder film which moves with the mold during the negative-strip period, while liquid steel is prevented from overflowing. Because the solid flux rim has a negative taper in the casting direction , when the mold moves upward, the shell is pushed back to the mold wall and liquid steel overflows from the top edge of shell. The difference between the two mechanisms is shown schematically in Figure 2.8. Thus, the first mechanism indicates that the bottom of the oscillation mark is formed when the downward velocity of the mold is maximum while, according to the second mechanism, its occurrence coincides with the lowest point of mold oscillation (the bottom of the mold stroke).  16  Figure 2.8 Oscillation Mark Formation due to Emi's and Kawakami's Models  Samarasekera et al. have proposed a mechanism for oscillation mark formation in billet casting where the lubricant is oil, not mold flux [45]. Based on an analysis of mold distortion, they calculate the negative taper of the mold wall at the meniscus, and conclude that the oscillation marks form as a result of mechanical interaction between the mold and the strand, similar to Kawakami's idea. However, Samarasekera et al. have suggested that the bottom of the oscillation mark is formed at the maximum downward velocity of the mold oscillation, on considering the force on the shell. Fundamentally, the difference between the two mechanism is as follows. Irrespective of when the bottom of the oscillation mark forms, based on the second mechanism in which the negative taper of the flux rim or the negative taper of the mold  17  wall causes the oscillation mark, the mark depth depends on the taper value. On the other hand, the first mechanism suggests that the flux pressure dominates the oscillation mark depth.  2.4 Lubrication in the Mold and Behavior of Solid Flux Rim Mold flux is used to prevent liquid steel from directly contacting the mold copper plate and to reduce the friction force on the shell. Near the meniscus, the surface temperature of the shell is about 1500 °C, the mold hot face temperature is close to 300 °C and the melting point of the mold flux is approximately 1000 °C. Therefore, the mold flux between the shell and the mold has two phases, solid and liquid, as shown in Figure 2.7. There are two concepts of lubrication in the mold. The first concept, termed "solid lubrication", suggests that the lubrication occurs at the interface between the solid flux rim and the mold wall. On the other hand, the second concept, "liquid lubrication", implies that the solid flux rim is attached to the oscillating mold while the liquid flux provides lubrication between the shell and the solid flux rim. From the view point of mold flux consumption, solid lubrication suggests that both phases of the flux move with the strand. In the case of liquid lubrication, the flux is consumed in the liquid flux film between the strand and the solid flux rim. Many studies of lubrication in the mold have been done with friction force measurements [7][10][11][36][46]. However, all that can be measured is the total force which includes the mold inertia; an estimate of the friction force both in solid lubrication  18  and liquid lubrication is based on simplifying assumptions employed in the deductions. Therefore, a variety of conclusions have been reached for both complete liquid lubrication [11] and superior solid lubrication [16]. Based on experiments to measure the residence time of the flux in the mold, Ogibayashi et al. have concluded that the flux rim adheres to the mold and liquid lubrication is dominant [16]. Upon termination of a heat, the flux rim adhering to the mold was analyzed and a flux employed 20 minutes before the end of the casting was found to be present. Therefore, it can be regarded that the solid flux rim remains in the mold for atimeconsiderably in excess of the residence time of the shell in the mold and does not move with the strand as has been postulated by some researchers.  2.5 Heat Transfer Models near the Meniscus Although current understanding suggest heat flux in the mold is affected by the mold flux properties and the mold oscillation conditions [8] [40] [47], there are few mathematical models in which these conditions are considered. The reason is due to the complex nature of the phenomena such as oscillation mark formation and shell shrinkage, which in turn are related to the mold flux film thickness, i.e. the thermal resistance. Although the purposes of the calculations are different, some researchers have calculated the heat flux or the temperature distribution in the mold [42][43],[48]~[50]. Usually, for the calculation of the overall thermal field in the mold, the following simplifications are often made: constant thermophysical properties of steel steady-state one-dimensional heat flow empirical heat-flux profile  19  The consequences of these simplifications are discussed in Section 6.4. Under the condition of constant casting speed, the casting direction is considered as the second dimension and a two-dimensional steady-state model which includes the casting direction and the through-thickness direction is developed for the calculation. However, heat transfer during casting of slabs can also be modelled with the one-dimensional unsteady state heat-conduction equation with the time-axis instead of the casting direction valid only for constant casting speed. In both approaches, conduction in the casting direction and the width direction is ignored due to the small temperature difference, compared to the through-thickness direction. Ackermann et al. have calculated the shell shape at the meniscus, according to the above method [42]. Samarasekera and Brimacombe have indicated that a two-dimensional model is needed for the calculation of the temperature field in the meniscus region [50]. Takeuchi and Brimacombe [43], and Laki et al.[49] have used two-dimensional unsteady-state models for this purpose. In each model, however, the mold flux properties and the mold oscillation conditions are not explicitly considered. They have used empirical heat flux data which is generally back calculated from mold temperature measurements which implicitly include effects of the mold flux properties and the mold oscillation.  2.6 Segregation due to Solidification Followed by 8 - y Transformation Solute redistribution during solidification affects micro- and macro-segregation, which are related to the dendrite structure [51]. Therefore, it is useful for the analysis of the sub-surface structure of slabs to predict the solute redistribution quantitatively.  20  Usually, the following conditions are assumed for directional solidification: complete mixing in the liquid local equilibrium at the interface between the liquid and the solid Additionally, if "no diffusion in the solid" is assumed, Scheil's equation can be applied: [52] c;=kc\\-a  (2.7)  C* = solute distribution at the liqid - solid interface k = distribution coefficient C° = initial solute content f - fraction of solid  However, in Equation (2.7), the effect of the microstructure cannot be represented. Brody and Flemings have proposed a model which considers solute diffusion in the solid [53]. Their results are: for a constant solidification velocity (2.8) for a parabolic solidification velocity with time C; = * C ° { l - ( l - 2 o * ) / , }  where a =  D - diffusion coeffitient Q = local soldification time s  f  X = primary dendrite arm spacing t  (2.9)  21  These equations are often referred to in the analysis of segregation, but there are limitations. For example, if k = 0.5 and / , = 1 are assumed in Equation (2.8), the result is: (2.10) If a > 0.67, C (f = 1) < C°. Thus, Equations (2.8) and (2.9) indicate incorrectly that the s  s  solute is redistributed less than the initial alloy composition in the whole region. Therefore, the application of these equations is limited to slow diffusion, i.e. small a. To improve Brody's equations (2.8) and (2.9), Clyne and Kurz [54] and Ohnaka [55] have proposed other models. However, these analytical solutions cannot be applied for the case of 8 - y transformation following solute redistribution after solidification. With a numerical method, Chuang et al. have calculated the carbon redistribution due to 8 - y transformation for a 0.39 % carbon steel [56]. For steel with multiple alloy elements, Ueshima et al.[57] and Kobayashi et al.[51] have proposed models concerning solute redistribution. They have indicated that interdendric segregation is enhanced by 8 - y transformation, if the solute has a diffusion coefficient above unity. Moreover, their models imply that solidification and the 8 - y transformation are affected by cooling rate, as discussed further in Section 8.6.  22  3. Scope of Present Work From the previous work, development of the rupture due to shell sticking in the mold is reasonably well understood and moreover the sticking can be detected with thermocouples in the mold copper plate or by measurements of the frictional forces on the mold during its oscillation cycle. A reduction in casting speed can prevent the sticking from developing to create a breakout. Still unknown, however, is the mechanism of initial sticking, especially in the case in which a carbon-rich region is found in the vicinity of the initial sticking, and which may be related to a thicker mold flux film. The primary objective of the present work is: (1) to find the causes of initial sticking. Apparently, causes of the initial sticking are related to phenomena in the meniscus region. The most dependable source of information on meniscus phenomena is the dendrite structure near the slab surface. To accomplish this, the following methodology was followed: (i) sticking-type breakout shells were examined metallurgically and links have been established between the characteristics of these shells and casting conditions.  (ii) relationships between secondary dendrite arm spacing and mold heat flux have been established and compared to those measured in the subsurface of a slab.  (iii) to elucidate the mechanism of initial sticking, mathematical modeling of heat transfer and lubrication at the meniscus have been conducted.  23  With the value of heat flux calculated from the secondary dendrite arm spacing at the surface of initial sticking shell, the mathematical models of heat transfer and lubrication at the meniscus has been evaluated. In the meniscus region, a large heat extraction from the liquid steel to the mold occurs and the oscillating mold interacts mechanically with the solidified shell. Therefore, in the present work, heat transfer phenomena and lubrication in the mold have been studied and link between both phenomena has been established in these mathematical models. Additionally, other meniscus phenomena such as oscillation mark formation which is related to slab defects (transverse cracks and pin holes) have been explained consistently. Without slab quality, there is no reason for continuous casting operation. Therefore, the other objectives are: (2) to clarify the effects of mold flux properties and operating conditions on oscillation mark depth. (3) to propose methods for both the elimination of sticking-type breakouts and improvement of slab surface quality.  24  4.  Secondary Dendrite Arm Spacing in Continuous Cast  Slabs and Its Relationship to Local Cooling Rate 4.1 Heat Flux Calculation Data and Measurement of Arm Spacing Examination of the dendrite structure at the slab surface is a technique that provides insight on meniscus phenomena. For plain carbon steels, Suzuki's equations relating secondary dendrite arm spacings and cooling rates [27] have often been referred to, but application of these equations to continuous casting is beset by two problems. First, it is not clear whether these equations can represent such a high cooling rate and, second, because the cooling rate is not easily measured in the production process, particularly during a sticking-type breakout, it has to be determined indirectly through other related process data such as mold heat flux. As has already been discussed in Section 2.2, the correlation between shell thickness and secondary dendrite arm spacing may be applicable inside slabs, but it often gives incorrect information near the surface region. For the prediction of the secondary dendrite arm spacing near the surface, the heat flux is relatively easily measured and can be translated to local cooling rate with a mathematical model. Therefore, in this study, it has been necessary to establish whether secondary dendrite arm spacing can be predicted from the heat flux data. In plant trials on an operating caster, Mahapatra [58] has measured mold copper plate temperature with thermocouples and calculated heat flux from the slab with a three-dimensional heat-transfer model of the mold plate, as shown in Figure 4.1. Figure 4.2 (a) and (b) show sections of Sample R, etched with 3.5 % picric acid in a hot water  25  bath, near the surface and 20 mm below the surface, respectively at precisely the location corresponding to the heat flux in Figure 4.1. The steel composition and the operating conditions are listed in Table 4.1 (a) and (b). Measurements of secondary dendrite arm spacing (Xj were made by counting the number of arms over a given length of primary arm. All measurements contained at least 5 arms and were recorded as a function of distance from the surface. The average and the standard deviations of the measurements are given in Table 4.2.  300  c?200-  E o  2 X 3  u-100(0  I  0  0  20  40  Time (s) Figure 4.1 Heat Flux Data of Sample R  60  26  Table 4.1 (a) Steel Composition (wt%), Liquidus and Solidus Temperature (°C) of Sample R  c  Mn  Si  P  S  Al  T,  T.  0.29  1.25  0.19  0.016  0.005  0.041  1506.8  1446.9  Table 4.1 (b) Operating Conditions of Sample R  Slab Width (mm)  Tundish Temperature (°C)  Casting Speed (cm/min)  1530  1550  80.0  27  Table 42 Average and Standard Deviation of Secondary Dendrite Arm Spacing Measurement of Sample R  Distance from the Surface (mm)  Secondary Arm Spacing (\vn)  Deviation (\\m)  0 ~1  50  21.8  1-2  68  21.4  2-3  63  18.9  3-4  77  16.3  4-6  82  14.2  6-8  85  19.8  8-11  97  7.5  11 - 14  98  15.8  14- 17  101  22.0  17 - 20  119  13.3  28  I mm  Figure 42 (a) Section at the Surface of Sample R, Etched with 35 % Picric Acid (x!  I mm  Figure 42 (b) Section 20 mm below the Surface of Sample R, Etched with 35 % Pic Acid (x!6)  29  4.2 Mathematical Model with Finite-Difference Method Except for the corner, a one-dimensional heat transfer calculation is adequate to estimate the slab temperature distribution, as will be discussed further in Section 6.4. However, the release of latent heat and the temperature dependence of the thermal conductivity of steel complicate the temperature calculation. To minimize the complexity, enthalpy and modified temperature have been adopted in the governing equation as follows. The governing equation is:  (4.1)  The definitions of enthalpy and modified temperature are:  (4.2)  (4.3)  where kp = thermal conductivity atT = T e  0  Equation (4.1) is integrated with respect to temperature:  (4.4)  Therefore, the governing equation becomes:  30  dt  using:  Fa  — K,  ay  2  (4.5)  9 9 — = V —dt  The enthalpy and the modified temperature for a 0.29 % carbon steel are shown as a function of temperature in Figure 4.2. The boundary conditions are: 9T -k -^- = q (x)  <*ty=0  Fe  s  (4.7)  9T aty=y  Dltabl2  ^ f =0  (4.8)  T = T°  (4.9)  and: atx=0,0<y<y ,  D 2  Fe  Equation (4.6) was solved, subject to the boundary conditions, using the explicit finite-difference method. A time step of 0.02 seconds and mesh size of 0.1 cm was initially employed. In Section 4.4 the effect oftimestep, mesh size and superheat on the accuracy of the computed temperature will be examined.  31  0  800 Temperature (C)  1600  Figure 4.3 (a) Enthalpy as a Function of Temperature [59]  4.3  0  800 Temperature (C)  1600  Figure 4.3 (b) Modified Temperature as a Function of Temperature [59]  Definition of Cooling Rate As an example, Figure 4.4 shows the model predicted temperature and cooling rate  with time. Obviously the cooling rate is not constant during the solidification, eg. at the position 5 mm below the surface, the cooling rate just above the liquidus temperature the cooling rate below the liquidus temperature above the solidus temperature  \T=T^J  it is 9.15  and  below the solidus temperature the cooling rate  For  correlation to the secondary dendrite arm  \T=TI)  \T TI) =  is 3.64  is 1-28  °C/s;  °C/s;  °C/s; |r=r») is 38.75  °C/s.  spacing, consideration must be given to  theoretical aspects and to the relationship established earlier.  32  Surface  10 r " i — i — i — i — i — i — r -  "o  14  28  Time(s) Figure 4.4 (a) Temperature  Figure 4.4 (b) Cooling Rate  Calculated with Time  Calculated with Time  The most popular definition of the cooling rate is: (4.10)  CR = ts-tl  Usually T and T, are based on the equilibrium phase diagram. By this definition, the t  cooling rate is 6.96 °C/s at a position 5 mm below the surface. The secondary dendrite arm spacing can be calculated with this definition and Suzuki's empirical correlation [27]: -0-35  ^=152.9(0?)  (4.11)  The predictions are shown in Figure 4.5 together with the measured data (average and standard deviation) as a function of distance from the surface. The calculated results agree well with the measurements.  33  Distance from the Surface (mm)  Figure 45 Secondary Dendrite Arm Spacing Prediction and Measurement (Average and Standard Deviation) with Distance from the Slab Surface  These results have two important implications. Firstly, Suzuki's empirical correlation can be applied for the high cooling rate (at least 60 °C/s) and secondly, if Equation (4.11) is used, the definition of cooling rate should be the average cooling rate during solidification. If — | f  =7i  is used as the cooling rate (this is the definition of  cooling rate applied when the arm spacing is represented as a function of the shell thickness), it yields a small dendrite arm spacing. However, owing to the large deviation in the measurement of secondary dendrite arm spacing, this definition also seems to be acceptable. The predictions of the secondary dendrite arm spacing based on the other definitions of the cooling rate appear to be unreasonable because they lie outside the standard deviation.  34  As will be shown in Chapter 8, the solidus temperature in continuously cast slabs is not the equilibrium solidus temperature. The temperature at which liquid phase is completely changed to the solid phase is usually decreased as a result of interdendritic segregation. Therefore, it may be preferable to use the reduced solidus temperature (non-equilibrium solidus temperature T ), instead of the equilibrium solidus temperature s  in defining the cooling rate. Indeed, based on the Ostwald ripening theory of the secondary dendrite arm spacing, the non-equilibrium solidus temperature should be used for the cooling rate calculation. However, many empirical relationships between the secondary dendrite arm spacing and the cooling rate have been derived with the equilibrium solidus temperature. Because the non-equilibrium solidus temperature itself is affected by the cooling rate, and also because it is changed by other factors such as convection in the liquid, it appears that the use of a non-equilibrium solidus temperature for the calculation of the cooling rate under different thermal conditions would be difficult. Therefore, in this study, the equilibrium solidus temperature is used for the calculation of the cooling rate.  4.4 Calculation Accuracy of Mathematical Model Based on Finite-Difference Method Although the mathematical model based on the finite-difference method for the calculation of the cooling rate, together with Suzuki's equation, gives better agreement with the measured secondary dendrite arm spacing, it is necessary to check the accuracy of the model prediction. Thus, calculations have been made with different time steps, mesh sizes and superheats in the mold. With respect to superheat, it is a variable that normally is not measured in the mold and, therefore, in almost all the cases, is unknown.  35  4.4.1 Effect of Time Step The time step for the calculation was changed from 0.01 to 0.04 second and the results are shown in Figure 4.6. The effect of time step is negligibly small under such conditions.  0.01s • 0.02 s A 0.04 s  X  ~\  0  i  i  i  i  i  i  i  i  i  i  r  12  Distance from the Surface (mm)  Figure 4.6 Effect of Time Step on the Predicted Cooling Rate  4.4.2 Effect of Mesh Size Figure 4.7 shows the effect of the mesh size on the cooling rate. Except for the surface, this effect is also negligible. However, at the surface, the cooling rate increases with a decrease in the mesh size. Thus, even though the mathematical model based on the finite-difference method gives good agreement with the  36  secondary dendrite arm spacing data measured inside slabs, the cooling rate or the local solidification time at the surface must be independently calculated by another method.  Q "1  0  l  i  l  l  l  l  l  I  l  l  7 Distance from the Surface (mm)  l  l  I  14  Figure 4.7 Effect of Mesh Size on the Predicted Cooling Rate  4.4.3 Effect of Superheat in the Mold The  superheat in the mold gives rise to a remarkable effect on the cooling rate  except at the surface. Figure 4.8 shows that a higher temperature of the liquid steel in the mold increases the cooling rate several millimeters in the slab. At the surface, on the other hand, a higher temperature seems to give a lower cooling rate. However, Babu has estimated the effect of superheat on the cooling rate at a billet surface with a similar mathematical model and found a higher temperature gives a  37  larger cooling rate both inside and at the surface [60]. The difference between these two calculations lies in the heat flux pattern employed. In Babu's calculation, the heat flux increases in the first few seconds after starting heat extraction in the mold, while, in this calculation, the heat flux simply decreases with time until 4.5 seconds, as shown in Figure 4.1. Although it is not clear which heat flux pattern is true in reality, if the constant heat flux is assumed, the cooling rate calculated at the surface is scarcely affected by super heat in the mold, because of the large latent heat of steel.  38  4.5 Analytical Solution to Link between Secondary Dendrite Arm Spacing and Mold Heat Flux A mathematical model such as that presented here can predict the cooling rate inside slabs if the super heat in the mold is known. However, the finite-difference technique is generally poor for the prediction of cooling rate at the surface of the slab close to the meniscus owing to the finite nature of the calculations. Refining the mesh to obtain a better estimation of the cooling rate has limitations. Since the problem is limited to the cooling rate at the surface, an analytical solution can be found because the phases considered in the model are in the mushy zone. i.e. the temperature range considered is solely between the solidus and liquidus. In the mushy zone, constant properties of steel such as thermal conductivity (k ), density (p ), and Ft  Ft  specific heat (Cp ) can be assumed. The latent heat of steel (L ) can be included in the Ft  Ft  specific heat as follows:  Cp =Cp +-^Ft  where Cp  Fe  (4.12)  Ft  = the specific heat including the latent heat  An additional assumption is a constant heat-transfer coefficient (h ) between the steel and c  the cooling water (water temperature T ). Under these conditions, there is an analytical w  solution [61] for the transient temperature distribution in the steel.  39  T(y,t)-T,  —  _  — = 1 - erf<& - {exp(5,. + (3) (1 - erf& + VP))}  where $ = \]]?-  B^^-  \ 4oa  p =——;  k Cp p Ft  Fe  Fe  a=  (4.13)  ^  PF,Cp  Fe  y = distance from the surface  Thus, at y=0, Equation (4.13) can be reduced to:  r(o,o-r,  7V-7-, = l-expP{l-er/Vp}  (4.14)  r(o,»)-r Figure 4.9 shows the non-dimensional value ——— as a function of Ti. If the heat flux (  'W~'l  (q ) is measured, it can be converted to a heat-transfer coefficient (h ) with the following s  c  equation:  c=T7T~  (- )  h  I  t  4  15  rp  The calculation procedure for the cooling rate at the surface involves the following steps: (1) Substitute the solidus temperature T for T(0,t) in the left hand side of Equation s  T  ,- i T  (4.14) and calculate the non-dimensional value -—-. 'W~'l  (2) Read the P-value corresponding to it from the Figure 4.9. (3) Calculate the time t  s  40  (4.16) (4) Calculate the cooling rate CR Tl-Ts  CR =  I.  (4.17)  Figure 4.9 Variation of Non-dimensional Temperature with |3  4.6 Prediction of Secondary Dendrite Arm Spacing with the Value of Heat Flux Although an objective of this study is application of the relationship between the secondary dendrite arm spacing and heat flux in the literature to the heat transfer phenomena near the meniscus, another result can be predicted with the analytical solution introduced in the previous section. The result is a correlation between heat flux and  41  secondary dendrite arm spacing for 0.08 % carbon steel, as shown in Figure 4.10. The procedure of the calculation is almost the same as described in the previous section. Suzuki's empirical equation has been employed to calculate the secondary dendrite arm spacing from a value of cooling rate, which in turn has been obtained for a specific value of heat flux. Figure 4.10 indicates that a heat flux of 2.5 MWIm (heat flux in a typical 2  slab casting mold) gives a secondary dendrite arm spacing of 28.9 u. and the upper limit of 7 MWIm measured at the meniscus of billet casters corresponds to a value of 13.7 |i. 2  These results agree well with data reviewed by Brimacombe and Samarasekera [32].  100 T </) 2 80o  "o o |  0~"— — — — — — — — — — — — — — — 0 10 20 30 1  1  1  1  1  1  1  1  1  1  1  1  1  1  Heat Flux (MW/m"2)  Figure 4.10 Secondary Dendrite Arm Spacing Prediction  42  5 Metallurgical Observation of Sticking-type Breakout Shells 5.1 General Aspects It is important to examine mechanical and thermal abnormality in a sticking-type breakout slab, because the sticking occurs at the meniscus where the moving flux rim attached to the oscillating mold mechanically interacts with the solidifying shell resulting in a high heat extraction rate. Therefore, before the breakout shell was examined metallurgically to obtain thermal information from the dendrite structure, it was inspected to confirm the locations of sticking in the slab and its appearance. The steel composition and operating conditions of the sticking-type breakout slab are listed in Table 5.1 (a) and (b). It should be noted that, before the breakout, an emergency nozzle change (with a cold nozzle) was performed due to a blockage of the submerged nozzle. After the nozzle change, the actual breakout occurred, accompanied by five incidences of sticking. Three of them and the sticking causing the actual breakout occurred at the center of one broadface (outside radius) and the other two were also located near to the center of the other broadface (inside radius). The casting speed variation after the nozzle change is shown in Figure 5.1. The breakout detection system sounded alarms for all the sticking events including the actual breakout, but, except for the first alarm, all of the alarms were ignored. The breakout slab is shown in Figure 5.2.  43  Table 5.1 (a) Steel Composition (wt%), Liquidus Temperature (T, ; °C)and Solidus Temperature (T, ; °C) of the Sticking-type Breakout Slab  c  Mn  S  P  Al  N  T  0.081  0.32  0.010  0.010  0.033  0.0039  1531  T.  t  1498  Table 5.1 (b) Operating Conditions of the Sticking-type Breakout Slab  Casting Speed  Oscillation  Oscillation Half  (cnVmin)  Frequency  Stroke (mm)  Melting Point of  Mold Flux  Viscosity (poise) Mold Flux (°C)  (cycle/min) 76.2  4  60  1125  1.7  120  0 ~F 0  1  1  1  1  1  1  2 4 6 Time after Nozzle Change (min)  1  r—  8  Figure 5.1 Casting Speed Variation after the Nozzle Change  44  In the operating sequence (time sequence), each sticking region was labelled as A - l - A-4 on the outside radius broadface and B-l - B-2 on the inside radius broadface of the breakout slab shown in Figure 5.3 (sticking zone A - l has already been removed). It should be noted that successive sticking events on a single face occurred at the same position. The actual breakout occurred as a result of the sticking event A-4. When the breakout occurred, the sticking shells of A-4 and B-2, which often remain in the mold, fell and adhered to the slab surface (around sticking part A-3), as shown in Figure 5.2, and was withdrawn with the strand. As already described in Chapter 2.1, the sticking shell and ruptured shell exhibit marks which are V-shaped and have a smaller pitch than sound shells. The marks in the sticking parts have an inclination of approximately 40  0  with the horizontal.  Figure 5.2 Photograph of Broadface (Outside Radius) of Sticking-type Breakout Slab  45  Figure 5.3 Slab with Sticking Shell and Sample Locations  46  For metallurgical observation, small samples (15 cm x 15 cm x 2 cm) named 0C1 - 0C6 and IC1 - IC7 in Figure 5.3 were cut from the slab. Unfortunately, the sticking shell surfaces of OC5 and OC6 could not be taken out successfully because significant amounts of steel which had overflown had covered the surfaces. The cut samples were cleaned by sand blasting and were photographed. Figure 5.4 ~ 5.6 show the surfaces of the sound shell, the initial sticking zone and the rupture area, respectively. The distinct line in Figure 5.5 was caused by the overflow of liquid steel when sticking A-2 was released. In Figure 5.5 and also in Figure 5.6, small holes can be observed. These were caused by entrapment of unmelted mold flux on the shell and subsequent scouring out during sand blasting. Note also that, in the pre-rupture area, which is the lower part in Figure 5.6, the oscillation mark is lost (right hand side) or deeply notched (left hand side). These appearances of the sticking shells are considered evidence of the interaction between the shell and the flux rim. These phenomena will be discussed in Chapter 7.  Figure 5.4 Appearance of Sound Area ICS in the Breakout Slab  47  48  5.2 Measurement of Oscillation Marks The pitch and depth of the oscillation marks in the sound shells, the sticking and ruptured shells have been measured with an automated profilometer [62]. In the sticking and rupture areas, the center region where the oscillation marks were horizontal was chosen for the measurements. The averages of the pitch and depth of oscillation marks in the three zones are listed in Table 5.2. From the operating conditions listed in Table 5.1 (b), the oscillation mark pitch expected (= Vy/) is 12.7 mm, which is relatively larger than the measured value of 9.94 mm in the sound area of the breakout shell. Addition of the pitch values in the rupture zone and sticking zone gives a value of 10.46 mm, which is close to the measured value in the sound area. If a shell rupture occurs in each oscillation cycle during sticking and the shell is separated into a sticking and a rupture zone, as described in Chapter 2.1, the number of oscillation marks in the sticking and rupture zones are the same as in a sound area and the addition of these two average pitches should equal to the oscillation mark pitch in the sound zone. These measurements, although not precise, support the mechanism of rupture occurring in each oscillation cycle. Table 52  Average of the Mark Pitch and Depth  Zone  Pitch (mm)  Depth (mm)  Sound  9.94  0.497  Sticking  6.90  0.261  Rupture  3.56  0.191  49  The averages of the oscillation mark depth in the sticking zone and rupture zone are smaller than in the sound area. Figures 5.7 ~ 5.9 show the local value of oscillation mark depth plotted against pitch measured in the three zones of the breakout shell. Usually, if the casting speed and oscillation frequency are constant, an increase of the oscillation mark pitch due to metal level fluctuation causes an increase in the mark depth, as shown in Figure 5.7. This mechanism will be discussed in Chapter 7. However, in the sticking area and in the rupture area, the relationship between the oscillation mark pitch and depth, as shown in Figure 5.8 and 5.9. is less clear. Marks in the sticking and rupture zone are formed due to the liquid steel flowing into the rupture point below the meniscus, as described in Chapter 2.1. Thus, the mark depth depends on how the liquid steel flows, which is related to the viscosity of the steel and the temperature of the flux rim and, therefore, it may not exhibit a clear relationship to the mark pitch.  1.0-1  1  0H 0  1  1  1  1  1  1  1  1  r 1  1  1  1  1  1  1  1  1  1  10 Oscillation Mark Pitch (mm)  Figure 5.7 Mark Depth vs. Mark Pitch in Sound Shell  1  20  1.0 •  0.5-  0H 0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  r  10 Oscillation Mark Pitch (mm)  20  Figure 5.8 Mark Depth vs Mark Pitch in Sticking Shell  0  i  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  10 Oscillation Mark Pitch (mm)  Figure 5.9 Mark Depth vs. Mark Pitch in Ruptured Shell  r  20  51  5.3 Micro-structure of Breakout Shells The sample shown in Figure 5.3 was cut longitudinally or transversely into smaller sections, which were polished and etched with 2.5 % picric acid. The microstructure was examined and photographed with an optical microscope. Typical photographs from the three zones- sound part, sticking and rupture zones - are shown in Figures 5.10 ~ 5.12, respectively. Interestingly in Figures 5.11 and 5.12, distinct segregation lines are seen 1-3 mm below the surface and the dendritic structure becomes coarse on the segregation lines. Usually segregation such as is seen in Figure 5.11 or 5.12 is macro-segregation caused by liquid flow [63] but the segregation lines are almost parallel to the surface and do not reflect the sticking shell profile which is shown in Figure 2.2. Therefore, the segregation may not have been caused by liquid flow. The segregation lines existed characteristically in the sticking and rupture zones, but, as shown in Figure 5.13, were not found in sample IC7 taken out from sticking B-2 where the shell thickness was less than 10 mm. This is because the actual breakout A-4 occurred while sticking B-2 was in progress. The regions where the segregation lines were observed are shown in Figure 5.14. A sketch of a cross section including the initial sticking region is shown in Figure 5.15 (a) and the microstructure in this region is presented in Figure 5.15 (b). A. distinct line separating the initial sticking shell from the overflowing steel is seen continuously from the surface exhibited on the left hand side of Figure 5.15 (a). Probably, when the sticking was released, liquid steel overflowed and adhered to the sticking shell. The initial sticking shell has a parabolic shape on the top (right hand side in Figure 5.15 (a) and precisely shown in Figure 5.15 (b)). This parabolic shape was recorded by measuring the distance from the line extending from the straight part in the sticking shell profile and the result is shown in Figure 5.16.  52  Figure 5.12 Cast Structure in 0C3, Rupture Zone, Etched with 25 % Picric Acid (xl6)  54  S  Figure 5.14 Locations in the Breakout Shell Exhibiting Segregation Lines  55  Initial Sticking Figure 5.15 (a) Sketch of a Cross Section Including the Initial Sticking Region  Figure 5.15 (b) Photograph of Cast Structure in the Initial Sticking Area  56  Distance from Extending Line (mm)  Figure 5.16 Parabolic Shape of the Initial Sticking Shell Dendrite arm  spacing measurements were undertaken by counting the number of  arms over a given primary arm  length. Most measurements were made over 10 arms and  recorded as a function of distance from the surface. However in some cases, owing to the randomness of the primary arms, spacings were measured only over 5 arms. Average values of secondary dendrite arm  spacing are shown in Figure 5.17 (sound shell, IC3),  Figure 5.18 (ruptured shell with the segregation line, IC1), Figure 5.19 (ruptured shell  57  without the segregation line, IC7) and Figure 5.20 (initial sticking shell, OC2). In the sound shell, secondary dendrite arm spacing increases continuously with distance from the surface, while the spacing has a discontinuous change in the vicinity of the segregation line in the ruptured shell. In sample IC7 without the segregation line, secondary dendrite arm spacing increases quickly just below the surface and remains relatively constant with increasing distance from the surface. A large secondary dendrite arm spacing appears at the surface in the initial sticking shell (63 |i) and, by the calculation described in Section 4.5, it corresponds to heat flux of 85.6 W/cm which is 2  much smaller than the 141.6 W/cm calculated for the sound shell surface. The dendrite 2  structure in the initial sticking shell becomes more coarse a few millimeters below the surface, probably because the overflowing steel offered a large thermal resistance after the sticking was released, drastically decreasing the cooling rate.  150  <x>  Distance from the Surface (mm)  CO  Figure 5.17 Change of Secondary Dendrite Arm Spacing with Distance from the Surface in Sound Shell IC3  58  150  2 o  ^ 1 0 0 o as  a.  CO  E < a> a  50  Segregation Line  •o  CO  10  Distance from the Surface (mm)  Figure 5.18 Change of Secondary Dendrite Arm Spacing with Distance from the Surface in Ruptured Shell with the Segregation Line IC1  Figure 5.19 Change of Secondary Dendrite Arm Spacing with Distance from the Surface in Ruptured Shell without the Segregation Line IC7  59  150  co Figure 520  Distance from the Surface (mm) Change of Secondary Dendrite Arm Spacing with Distance from the Surface in Initial Sticking Shell OC2  5.4 Metallography with Secondary Ion Mass Spectroscope (S.I.M.S.) The segregation line found in almost all the sticking and rupture zones except for sample IC7 is accompanied by an enlarged dendrite structure. This means that, when solute segregated below the surface, the cooling rate was extremely low. Generally speaking, when the solidification rate is reduced by fluid flow such as due to electro magnetic stirring, the solute (with distribution coefficients under unity during solidification) negatively distributes and forms a "white band" [64]. This concept suggested that the segregation found in this study might be caused not by redistribution during solidification but during 8 - y transformation, because carbon and manganese have distribution coefficients over unity for 8 - y transformation. The sample studied is a 0.08 % carbon steel and the solidification is followed immediately by the 5 - y transformation.  60  Thus, it was expected that, even though solute negatively distributed during the solidification with a low solidification rate, the following 8 - y transformation might make the solute segregates positively. With the S.I.M.S., the solute in the segregation line was determined under the conditions listed in Table 5.3 for the sample cut from section IC1. Figure 5.21 shows the segregation line and the area which was examined by S.I.M.S.. Figure 5.22 ~ 5.25 show the distributions of carbon, manganese, sulphur, and phosphorus, respectively. Carbon, manganese and sulphur appear to correspond to the segregation line whereas phosphorus has an almost uniform distribution. Because carbon easily diffuses at a relatively low temperature, it seemed strange that the carbon rich area existed inside the shell. Therefore, with 2 % nital, the locations of pearlite in the sample were identified, as shown in Figure 5.26. It is thus evidence that the carbon concentrates in the pearlite phase, whose distribution is almost uniform, and it is concluded that the carbon does not relate to the segregation line. The distribution of sulphur corresponds to that of manganese. This is also confirmed with line-analysis by S.I.M.S., as shown in Figure 5.27. Therefore, it can be concluded that the solute in the segregation is manganese accompanied by sulphur. From these considerations and the results, it was expected that the mechanism of manganese segregation was attributed to the redistribution by 8 - y transformation following the solidification. The mechanism of the segregation will be discussed in Chapter 8.  61  Table 5-3 Measuring Conditions of SI.M.S.  Size of Analysis Spot  20 \un  Electron Beam Irradiation Time  40-2580 sec  Acceleration Voltage  8 KV  Probe Current  20 nA  Vacuum Condition  \0- mbar  Figure 521  w  Cast Structure Examined with SJ.M.S. Sample IC1 (xl6)  (rectangle in photograph is the area examined with SJ.M.S. and arrow indicates the segregation line)  Figure 5.23 Manganese Distribution in the Area Shown in Figure 5.21 (xl6)  Figure 524  Figure 525  Sulphur Distribution in the Area Shown in Figure 521  (xl6)  Phosphorus Distribution in the Area Shown in Figure 521 (x!6)  Figure 5.26 Distribution of Pearlite in Sample ICl (x!6) (parallel lines (center of photograph) are sites ofSJM.S. and pin-link spots are marks to indicate the location of segregation line)  65  Manganese  k—  Segregation  Sulphur  100  I  Line  ^  !  Am 1  Figure 527  Manganese and  Sulphur Distribution with Line Analysis  66  6 Mathematical Model of Heat Transfer near the Meniscus with Mold Flux Lubrication 6.1 General Flow Sheet As discussed in Sections 2.3 and 2.5, the meniscus phenomena include not only heat transfer but also mechanical movement of the mold flux. Therefore, to examine the effects of mold oscillation and mold flux properties on meniscus phenomena, a mathematical model of heat transfer in which the mold flux movement is considered must be formulated. It is difficult, however, to consider the heat-transfer problem and the mechanical movement simultaneously. Thus, the model is divided into parts and, in each part, only a single phenomenon is considered and the results from each part are subsequently combined. As shown in Figure 6.1, the mathematical model is actually divided into four parts: mold flux velocity calculation (A), meniscus shape calculation (B), heat transfer analysis in the mold flux film (C) and slab temperature calculation (D). Each part can be combined with the others by common process data. For example, pressure gradient in the mold flux (dP ldx) which is calculated from flux flow rate (Q) is related to both part P  (A) and part (B). The result from part (B), thickness of molten mold flux film (h), is in turn used as the boundary conditions in part (A) and part (C). In part (C), using mold flux velocity (u) from part (A), slab surface temperature (T ) from part (D) and other slah  process data, the mold flux temperature is calculated and heat flux from the slab (h ) and s  thickness of the solid flux rim (5 ) can be introduced. The general flow chart for the 0  model is shown in Figure 6.2. Cyclical steady state is assumed and is achieved  67  computationally by an interative technique in which initial conditions are given a priori to start the calculation. The conditions employed in the calculation are listed in Tables 6.1 - 6.3.  h  dP/dx^  Mold Flux Velocity in Liquid Film  ® q  s  T  Lab  Slab Temperature  il  11 Vz  Figure 6.1 Relationship among Four Parts of Heat Transfer Model at the Meniscus  68  Start Initial Data Set t=0  Geometrical Shape I  Mold Flux Velocity  f  Mold Flux Film Temperature  T  Slab Temperature  t=t+  At  Figure 6.2 General Flow Chart of Heat Transfer Model at the Meniscus  Table 6.1 Calculation Conditions and Operating Conditions  Symbol  Condition  Initial Value  dt  calculation time step  12/  s  f  mold oscillation frequency  1  cycle/s  a,  mold oscillation half stroke  0.4  cm  casting speed  1.267  cm/s  meniscus position from the top of  10  cm  0.702  Wlcrn^C  300  °C  i  Units  the mold (in fixed coordinate system)  hpM  heat-transfer coefficient between the solid flux rim and the mold mold hot face temperature  Table 6.2 Steel Properties and Pouring Temperature  Symbol  property / condition  Initial Value  Units  pouring temperature of liquid  1536  °C  steel in the mold T,  liquidus temperature  1531  °C  T  solidus temperature  1498  °C  k  Fe  thermal conductivity of steel  0.514  W/cm°C  PF,  density of steel  7  g/cm  specific heat of steel  0.6  Jlg°C  s  Cp  Ft  2  71  Table 6.3 Mold Flux Properties and Pool Depth  symbol  kp  property / condition  Initial Value  Units  melting point of the mold flux  1125  °C  thermal conductivity of the mold  0.0072  W/cm°C  density of the mold flux  2.6  gi'cm  specific heat of the mold flux  0.9  Jig°c  mold flux pool depth  0.45  cm  mold flux viscosity  1.7  poise  interfacial tension between the  1200 [65]  dyne/cm  flux  PP  Cp  IF  a  P  3  liquid steel and the molten mold flux  6.2 Flow Rate of Mold Flux In this section, to calculate the velocity of molten mold flux in the flux channel, the Navier-Stokes equation has been employed and solved analytically, but the analytical solution includes two undetermined values: thickness of the flux film which is obtained in the meniscus shape calculation (in the next section) and pressure gradient in the flux  72  film. To calculate the pressure gradient, the mold flux flow rate (Q), which is introduced by integration of the velocity distribution in the flux film, has been estimated from the process data reported in the literature. The density of the mold flux is 2.6 g/cm while its viscosity at operating 3  temperatures is at least 0.5 poise. The velocity of the molten mold flux in the flux channel can be estimated to be up to 3 cm/s (~ casting speed) and the characteristic length of the mold flux flow is the mold flux film thickness of about 0.1 cm (~ estimated from mold flux consumption). Therefore, the Reynolds number for the powder flow is: „ Re=  2.6x3x0.1 — =1.56 t  (6.1)  Usually, turbulent flow occurs if the Reynolds number is larger than 1000, and von Karman's vortex street appears in the flow with the Reynolds number over 50 [66]. Consequendy, the flow of molten mold flux is laminar without a vortex. Under these conditions, the Navier-Stokes equation can be reduced to a simple form based on the following assumptions: (1) constant properties of the fluid (2) one-dimensional flow However, the geometry near the meniscus is two-dimensional, so that the second assumption may not be valid. This problem is taken up again in Section 6.5. Subject to the above assumptions, the Navier-Stokes equation is: du  du 2  f  dP ^ P  PPS-  V  dx  (6.2)  The details of the coordinate system in Figure 6.3 and the meaning of " ~ " are discussed in the next section.  73  V z* v  Figure 6.3 Physical System of the Mold Flux Flow  If the left hand side of Equation (6.2) (p r:) and the first term in the right hand P  d* 2  side C n  are compared, the left hand side can be neglected. In other words, the inertial  term is small, compared to the viscous term as follows: p -—160 at  <  P  rj—;~500gcm/s dy  2  i „ , , i du 2 , where p = 2.6 glcm , — < a.u) , T) = 1 poise at P  dy  2  d  , a = 0.4cm, co = 47t, s  d -0Acm p  p  Therefore, the governing equation is: d u dPp 2  (6.3)  74  The time dependency of the mold oscillation can be incorporated into the boundary conditions, which are (refer to Figure 6.3): u(y=0) = V -V M  (6.4)  z  (6-5)  u(y=h) = 0  The solution is: u=  dP  2n  P  ~dx  ~PP8 iy -hy)  +  2  (6.6)  ^(V -V )(y-h) z  M  J  With the equation of continuity, the mold flux flow rate Q can be defined as: r*  1  dP  M  P  ~dx  (6.7)  ~PP8  Takeuchi and Brimacombe [43] or Anzai et al. [44] have calculated the mold flux pressure P from Equation (6.7) assuming a fixed meniscus shape as shown in Figure P  6.4. The mold flux pressure gradient — can be solved with the equation of continuity subject to the following boundary conditions: P (x=0) = P  (6.8)  P (x=l )=P  (6.9)  p  p  air  2  air  It should be noted that implicit in Equation (6-9) is that the air gap appears at x = l . 2  In reality, the meniscus shape during the mold oscillation is not fixed. Therefore, the mold flux pressure cannot be determined explicitly from Equation (6.7). The pressure gradient — or the mold flux flow rate Q really needs to be predicted from another equation or from process data.  75  Figure 6.4 Physical System in the Calculation by Takeuchi and Brimacombe [43] or Anzai et al. [44]  The factors which have an effect on the mold flux flow rate are considered next. As shown in Figure 6.5, the mold flux channel between the mold and the solid steel shell is assumed to consist of a series of notches corresponding to the oscillation marks. Similar to the method of Takeuchi and Brimacombe [43] or Anzai et al. [44], the mold flux pressure (P, at JC = A _ i) can be calculated as follows: 6T\(YM-V )  6r\Q  Z  h \hi  (6.10)  + 1  where  6r)(V -V ) M  P  N  P  N-l  (6.11)  hi ht  i +  z  l<i<N  6n<2 ('*-'*-i)-7r^(** i  If Equations (6.10) to (6.12) are added,  +  +  Afv)('*-'Ar-i)  + P^('*-'*-i)  (6-12)  76  0=6 T i ( y - y ) ^ ^ - 6 T Q ^ % l ^ i ( / . - / _ ) M  z  1  1  1  i  i  1  +  M  /  (6.13)  w  If A/ = / , - / , _ i (2 < / < N) is assumed,  hh 2  hh  x  3  '  2  Each value is: h ~lcm , l ~\cm , x  x  h^-hj, hj~h  l -l ~Al 2  x  (see Figure 6.5)  B  (W-2)A/~80c//i Then,  Similarly,  Substituting Equations (6.14) and (6.15) into Equation (6.13), 1 , h +h 0 = 6ri(V - V )—6T)Q +pg B"-T nn B  w  z  T  (6.16)  p  1  B  T  Then, „ Q  (V -Vz)h h TTT n +n M  B  B  T  T  p gtiti /wi. i_ u \ or\(n +n ) p  +  B  (6.17)  T  Equation (6.17) does not include the values of the meniscus shape such as h , h and l x  2  u  and, therefore, it implies that the flow rate of the mold flux Q does not directly depend on the meniscus shape. In other words, the flow rate of the mold flux is a function of the oscillation mark depth (h - h ), minimum distance between the shell and the mold B  T  77  (h ) and the other conditions (y , T  z  V, U  Therefore, the flow rate of the mold flux can  be calculated independently of the meniscus shape. However, the oscillation mark depth cannot be estimated without measurements and the minimum distance between the shell and the mold is related to the shell shrinkage and mold distortion which cannot be evaluated without a stress-strain analysis. Nonetheless, from the measured data reported by Ogibayashi et al. [16], the oscillation mark depth can be determined as: h - h = -0.003857TIV + 0.051 B  h -h B  T  (f\V  Z  = -O.0004375T[V + 0.02701  T  Z  < 7)  (6.18)  (7 < T\V )  (6.19)  Z  Z  Thus, under the conditions where rj = 1 poise and V = Icmls, h is calculated to be Z  0.0157  T  cm, so that the powder consumption would be 0.35 kg/tonne-steel. Here, it is  assumed that an air gap does not appear in the mold; but, for the case of a low casting speed (V < 1 cmls) with a large shell shrinkage, the value of h should be re-evaluated. z  T  With the values of h and h determined by the above method, mold flux flow rate T  B  Q is calculated from Equation (6.17); then, if Q is substituted into Equation (6.7), the pressure gradient can be calculated as a function of h. dP  P  1211(2 ,  6T\(V -V ) M  Z  The value of h at the meniscus can be computed from the meniscus shape as shown in the next section and from the thickness of the solid mold flux rim as shown in Section  78  Figure 65 Channel between Solid Mold Flux Rim and Shell with Oscillation Marks  6.3 Meniscus Shape Meniscus shape, which is related to the interfacial tension between liquid steel and molten mold flux and pressure in both liquids, has to be determined to obtain the domain of the mold flux channel. Before pursuing the meniscus shape calculation, it is worth considering what is meant by pressure in the mold flux channel. In a thermodynamical sense, pressure is, of course, a scalar state variable. However, in hydrodynamics, pressure means inner stresses acting on the fluid, which are not always constant in each direction. The pressure in hydrodynamics is the same as the pressure in thermodynamics, only when the fluid is stagnant [67]. Therefore, in the case where the pressure within the fluid must be considered, the pressure should be measured in the  79  moving system with the fluid. In other words, when the pressure on the shell is calculated, the coordinate system should move with the shell as in the calculation of Takeuchi and Brimacombe [43]. On the contrary, in the case of the heat-transfer calculation in the mold, the coordinate system is the fixed system, because the mold does not have the same motion as the shell. Therefore, for the heat-transfer calculation as discussed in Sections 6.4 and 6.5, the velocity of the mold flux must be converted from a moving coordinate system on the shell to a fixed coordinate system. u=  a+v  z  (6.21)  where u = the mold flux velocity in the fixed system ii = the mold flux velocity in the moving system For the meniscus shape calculation, the moving coordinate system should be used. The meniscus shape can be determined from interfacial tension between the shell and the molten mold flux. The governing equation is : d<5>  * ~ T  AP <6  -  22)  where <]) is the contact angle as shown in Figure 6.6, s is the distance along the meniscus, AP, pressure difference between the steel and the mold flux, is defined as: AP =  p gx-P (x) Fe  p  (6.23)  80  Figure 6.6 Physical System at the Meniscus  From geometry, dx  = sin<|)  (6.24)  = -cos<(>  (6.25)  ds dh ds  Therefore, d<]>_dcos<|> ds  and  (6.26)  dx  dh  dx  1 tan<J>  (6.27)  Combining Equations (6.20) and (6.23) dAP  l2r\Q  dx  h'  6T](V -V ) M  Z  +  (pFe-p )8 P  (6.28)  81  With the above equations, the meniscus shape (=h(x)) can be calculated numerically. In finite-difference form, from Equation (6.28) and the definition of the pressure gradient, AP' = AP+^-Ax ax  (6.29)  From Equations (6.22) and (6.26), cos<b' = cos(b  AP  a  Ax  (6.30)  Expressing Equation (6.27) in finite difference form, h' — h ~~~~7 tan 9  (6.31)  Near x=0, where ^ is infinite, the meniscus shape can be calculated from an analytical solution for a static meniscus, because h is large there and AP ~ (p -p )gx. The Fe  p  analytical solution is taken from the work of Takeuchi and Brimacombe [43] and is expressed as follows:  V2a 2 / z = - V 2 V ^ - ^ l n ^2a +^2a -x £  2  2  + 0.3768a  +  *  V  (6.32)  J  .2  where a , the capillary constant, is defined as 2  a =;  2a  2  r  (6.33)  In the numerical calculation of the meniscus shape, the length increment Ax is 0.01 cm. Figure 6.7 exhibits the accuracy of this numerical calculation for a static meniscus comparing to the analytical solution in the whole region.  82  0  0.4 Distance from the Meniscus (cm)  0.8  Figure 6.7 Meniscus Shapes Calculated by Analytical Solution and Numerical Method If this calculation is performed neglecting the mold flux rim and a following boundary condition is added, the results of Takeuchi and Brimacombe's work [43] can be introduced. 0=^  aty=0  (6.34)  If the meniscus shape is directly related to the oscillation mark, however, their results disagree with some of the empirical data obtained in plant. Their model indicates that the factors increasing the mold flux pressure increase the oscillation mark depth, but, in reality, high-frequency of the mold oscillation [38] and high viscosity of the mold flux [16] [36] result in shallow oscillation marks. Takeuchi and Brimacombe's model is based on Emi's mechanism, which attributes the oscillation mark formation to the pressure  83  change in the molten flux film and is less concerned about the movement of the solid flux rim. Therefore, in this model, not Emi's mechanism but Kawakami's mechanism is focused on, in which oscillation mark formation is related to the interaction of the solid flux rim with the newly forming shell. The meniscus shape based on Kawakami's mechanism is defined as the sum of the solid flux rim and liquid flux film thickness (S (x) + h(x)) while , in Emi's mechanism, the solid flux rim (S ) is ignored, i.e. the 0  0  profile of liquid flux film (h(x)) equals to the meniscus shape. For actual calculations of the meniscus shape based on Kawakami's mechanism, Equation (6.31) should be rewritten as: {S (x) + h(x)}' = {S (x) + h(x)}tan<t> 0  0  (6.310  The results of the meniscus shape calculation provide the domain of the mold flux channel where large heat transfer occurs, as shown in the following two sections.  6.4 Introduction of Energy Equation For the purpose of developing a heat-transfer model at the meniscus which incorporates the effects of the mold flux properties and mold oscillation, both conduction and convection need to be considered as well as the geometry of the boundary conditions. In other words, a lubricant (mold flux) should be considered to be a fluid, not a stagnant material. For the calculation of the heat transfer in the mold flux, the following energy equation must be solved, (6.35)  where 7> = temperature of the molten mold flux film u = velocity of the mold flux in the x-direction a = thermal diffusivity of the mold flux P  Cp = specific heat of the mold flux P  v = kinematic viscosity of the mold flux Equation (6.35) can be non-dimensionalized as follows: 6=  Tp —TST TPM — TST  I* = —  V  z  v* = — dp  y  x*=-  L  LIV  where  Z  = slab surface temperature at the meniscus (=1500 °C) T  PM  = melting point of the mold flux (=1000 °C)  V = casting speed (=2 cm/s) z  dp = molten mold flux film thickness (=0.1 cm) L = characteristic length for heat transfer ( cm ) The energy equation with these dimensionless variables is  La \dt* P  +  U  dx*\  %\  CppL a AT 2  P  where AT =  T -T ST  PM  2  ye dx* ' 2  85  With consideration of the system size and the temperature difference in the system, the characteristic length L in the x direction should be chosen so that the order of —, is unity. For example, if the overall mold is focused on, the length L should be derived from the slab surface temperature profile. (Note that the mold flux temperature adjacent to the shell is assumed to be the same as that of the shell surface.) Usually, the slab surface temperature at the bottom of the mold is approximately 1200 °C; consequently, 6 changes from 0 to 0.6 in the mold. The mold length is 80 cm and, if — is unity, the dx  length L may be 80/0.6 = 133 cm. This estimation is changed by the definition of 0. If 9 is defined as 6 =  f  " (T = slab temperature at the bottom of the mold), the SB  'SB~'ST  characteristic length L will be the mold length, but, in this case, the system concerned is only slab surface, not including the mold flux film. With these values, the order of each term in Equation (6.35) can be estimated and the results are listed in Table 6.4. From Table 6.4, if the thermal field in the overall mold is of concern, the energy equation can be reduced to a one-dimensional steady-state heat-conduction equation. 0 = aP—i  (6.42)  Therefore, a simple mathematical model can give reasonable results for the macroscopic heat-transfer from the slab surface. However, if the meniscus region is of interest, the length L can be determined with the derivation involving the cooling rate which can be calculated based on the secondary dendrite arm spacing at the slab surface. As shown in Chapter 4, the cooling rate of slabs near the meniscus is approximately 100 °Cls and, divided by the casting speed V , z  86  *L is 50 °Clcm.  In the channel at the meniscus (length 1 cm), AT = 50 °C and A0 is 0.1.  Ax  Therefore, the typical length L may be 1/0.1 = 10 cm. The order of each term in Equation (6.35), in this case, are also listed in Table 6.4. Then, the energy equation is: dT  dT  P  P  <?Tp  cfT  f  This means that the convection term, u —, cannot be neglected in the meniscus area but that two-dimensional heat-conduction model need not be considered. The reason why several researchers have proposed a two-dimensional heat-transfer model in the past has been simply due to the two-dimensional geometry of the meniscus. Table 6.4 Estimation of Terms in Equation (6.35) Term  for L=133 cm 0.0376 « 1  Cp L a AT 2  F  1.256 x l 0  _ 1 3  for L= 10 cm 0.5  «l  2.22xl0  _ u  «l  r  d  JL L  7.519 x K T * ! 4  0.01«1  6.5 Heat Transfer in the Mold Flux Film As mentioned in the previous section, the governing equation in the mold flux film at the meniscus is Equation (6.43). However, if the thermal field in the mold flux pool  87  above the meniscus is considered as well, where, the primary direction of heat transfer is in the x-direction, the governing equation is:  > a7, 2+  (6.44)  The velocity of the molten mold flux (u) can be provided from Equation (6.6) and (6.21). Equation (6.44) can be applied both in the solid mold flux rim and in the molten flux pool as well as in the liquid mold flux film. The physical system and the coordinate system are shown in Figure 6.8. The domain of the molten mold flux channel is determined by the meniscus shape calculation, as shown in Section 6.3, and the thickness of the solid mold flux rim (S ) which can be obtained from the locus of calculated 0  melting point temperatures (T ) in the mold flux film. PM  If-  Z  1  Mold  Figure 6.8 Physical System of Heat Transfer in the Mold Flux  88  The boundary conditions are: ftp  aty=0,  k —  = h (T - T )  P  PU  P  (6.45)  M  where T = mold hot face temperature (= 300 °C) u  h  PM  = heat transfer coefficient between the mold flux rim and the mold hot face (0.702  W/cm C) forZ <x<r , 2o  0  T =T  2  P  v  r,<x<Z  0  (6.46)  3  dT -^\ =0  r <x<r 2  (S ^y<D )  PU  (6.47)  y=D  T (y=0) = T  v  p  (6.48)  slab  The value of slab surface temperature (T ) is provided from the slab surface calculation, ltab  as shown in the next section. dT - ^=0 P  0<y<S , Z <x<Z 0  0  u  (6.49)  37"  a7 . Uz  atx=z , l  The value of h  PM  =0  (6  -  50)  in Equation (6.45) was estimated as follows. The mold cooling water  is in turbulent flow and the Nusselt number N is [68], w  N , = 0.023i?4 ^^ 8  (6.51)  33  w  where  is the Nusselt number of mold cooling water (=-^-^), Re is the Reynolds w  V  W  D  W  number (——); h is the heat-transfer coefficient between the mold cold face and the w  cooling water, and the properties of water (at 40 °C) [69] for the Prandd number Pt  w  (4.3) are the thermal conductivity k (0.633 W/m°C), the density p (992.2 kg/m ) and 3  w  the kinematic viscosity v  w  (0.658 x l(T m /s). The characteristic dimension of the mold 6  w  2  cooling slit D is 0.0018 m and the cooling water velocity V is 6 m/s. Thus, w  w  89  Re = 1.64 x 10 , h = 3.08 x 10 W/m °C- T h average heat-transfer coefficient between 4  w  4  2  e  w  the mold hot face and the cooling water / i  w  is, (6.52)  1  where the mold copper plate thickness d mold plate k  Cu  is 0.02 m and the thermal conductivity of the  is 0.32 kW/m C. Thus, 2o  Cu  h = 1.053 W/cm C 2o  ltw  Nukai et al. have estimated that the percentage of the contact area of the mold flux rim to the mold hot face is 40 % [70]. Then, the average heat-transfer coefficient between the mold flux rim and the mold hot face h  h x0A w  Then, h  PM  PM  is,  hiw  +-— h  (6.53)  PM  = 0.702 W/cm C. 2o  Among the boundary conditions (6.45) ~ (6.50), Equation (6.46) is based on a major assumption, that is, in the carbon-enriched layer above the molten flux pool (as shown in Figure 2.7), the temperature is uniform and equal to the melting point of the mold flux. Also it is assumed that heat transfer occurs from the carbon-enriched layer to the solid mold flux rim but that the solid flux rim thickness is not affected by the carbon-enriched layer. These assumptions are related to the mold flux melting behavior which depends on a particular operating situation such as the powder layer thickness above the carbon-enriched layer. Here, it is assumed that the powder layer is sufficiently thick to prevent thermal radiation from the carbon-enriched layer. Additionally, it is also assumed that the mold flux in the pool is stagnant (i.e. u = 0 for x < r ). This assumption, which means that heat transfer in the mold flux pool t  90  occurs only by conduction, corresponds to the "dead meniscus" coined by operators. This assumption is, therefore, meaningful but causes a problem in which the mass of the mold flux is not conserved at the boundary (JC = r ). Owing to this assumption and x  one-dimensional mold flux flow in the channel below the meniscus, as mentioned in Section 6.2, the heat transfer from the mold flux pool to the channel between the shell and the solid mold flux may be underestimated by this model. The numerical calculation was undertaken with an implicit finite-difference method and each control volume was assumed to be rectangular. In each calculation step, the thickness of the solid mold flux rim moving with the mold was calculated in the fixed coordinate system. This term became the boundary condition for the meniscus shape calculation in the next calculation step.  6.6 Slab Temperature Calculation Although the geometrical shape of the slab is two-dimensional, a one-dimensional unsteady state formulation suffices for the slab temperature calculation if only the heat flux normal to the surface is considered. Because the slab descends with a velocity V  z  and the coordinate system is fixed, the thermal history in each control volume must be tracked as shown in Figure 6.9. In a different manner to the calculation in Chapter 4, the slab temperature calculation has been done following "the specific heat method" in which the latent heat of steel is incorporated into the specific heat between the liquidus and solidus temperatures. Thus, the governing equation is: Ft  Ft 'Ft  3y  2  (6.54)  91  where constant properties of steel are assumed and s = distance along the shell. The boundary conditions are: (6.55)  (6.56) The value of the heat flux from the slab (q ) is provided by the calculation of the heat s  transfer in the mold flux film, as shown in the previous section. The heat flux from the slab, q , is calculated from the temperature gradient of the liquid mold flux film in the s  normal direction to the slab surface. The initial condition is: T =T° , Ft  F  ats=0  (6.57)  The mesh size of the control volume in the implicit finite-difference method was 0.1 cm near the surface of the slab and 1 cm inside the slab where heat flow was much smaller.  T(1,j)  JM. T(3,j) Flux Rim  Jdjl T(2,j) T(3,j)  ~i  T(ij)  VZ At  JML  Figure 6.9 Tracking System of the Slab Temperature Distribution  92  Model Results and Discussion 7.1  Response from the Initial Condition Because a two-dimensional unsteady-state model has been used for the calculation  of the heat transfer near the meniscus, the flux temperature distribution, solid flux rim thickness profile and slab temperature distribution must be given as the initial conditions. Therefore, if steady state is sought from this initial state, the time to attain the latter should be sought. For the slab temperature, this effect may remain only in the residence time of the slab at the meniscus, because the slab is withdrawn with a constant velocity V and is continuously renewed at the meniscus. In contrast, the flux moves downward z  intermittently in the casting direction. Thus, the flux temperaturetimeresponse relative to the initial conditions must be checked. From the initial state, changes in the heat flux to the mold and the heat flux from the slab, which are representative of the temperature distribution in the mold flux, are shown in Figure 7.1. The calculation conditions, steel properties and flux properties are the same as the conditions listed in Tables 6.1 ~ 6.3. After one mold oscillation, the change in the heat fluxes become relatively small, but the transient still continues. As discussed in Section 2.4, the solid flux rim adheres to the mold and it causes a thermal hysteresis [16]. Therefore, strictly speaking, there is no steady state of the heat transfer in the mold. This is the reason why the effects of the initial conditions cannot be eliminated completely in a mathematical model. In the following discussion, therefore, a fully steady state at the meniscus cannot be pursued. As a pseudo-steady state, the behavior of the meniscus during the third  93  oscillation from the initial conditions will be analyzed. During the third oscillation, the average heat flux from the slab in the region between the liquidus and solidus temperature of the steel is 116.6 W/cm and it agrees reasonably to the heat flux (141.6 2  W/cm ) calculated from the secondary dendrite arm spacing with Equations (4.14) and 2  (4.15).  210 - i  —  1  —  Oscillation Cycle from the Initial State  Figure 7.1 Response of the Average of the Heat Flux to the Mold (q ) and the Average M  of the Heat Flux from the Slab (q ) in the Region 0-1 cm below the Meniscus from the s  Initial State  7.2  Effect of Convection on Heat Transfer near the Meniscus Figures 7.2 and 7.3 show the profiles of the heat flux to the mold (q ) and the heat M  flux from the slab (q ). It should be noted that the two heat fluxes are different. The s  average heat flux from the slab in the region 0~1 cm below the meniscus (the meniscus  94  channel) is 78.7 w/cm while the average heat flux to the mold is 62.2 w/cm - The heat 2  2  flux from the slab has a maximum relatively high in the meniscus channel. In contrast, the heat flux to the mold has a maximum lower in the meniscus region. c a.  Heat Flux to the Mold (W/cm"2)  Figure 72 Mold  Profile of Heat Flux to the  Heat Flux from the Slab (W/cm"2)  Figure 73 Profile of Heat Flux from the Slab  There are two reasons for this difference between the heat flux from the slab and the heat flux to the mold. One of them is the effect of the convection in the liquid flux film. In other words, the flux flow carries heat from the slab downward. The heat flux to the mold is seen to be unaffected by convection, but the heat flux from the slab decreases when the mold flux is stagnant, as shown in Figure 7.4 and 7.5. Both during the upstroke and downstroke motion of the mold, the flux flowing into this channel is  95  cooler and it removes a large amount of heat from the slab. Because the heat flux to the mold with or without convection is the same, it can be regarded that the excess heat is carried down from this area by the liquid flux flow. The other cause is the movement of the solid flux rim with the mold. Independently of convection in the liquid flux film, the solid flux carries heat into the meniscus channel and removes heat from there, depending on the movement of the mold during its oscillation cycle. Figure 7.6 demonstrates the heat balance in this case. Almost 79 % of the heat flux from the slab passes through the liquid and solid flux rims and flows to the mold, 15 % of the heat flux is carried out by the liquid flux, and the rest moves with the solid flux rim. Therefore, as discussed in Section 6.4, the effect of the convection in the liquid flux on the heat transfer near the meniscus cannot be ignored. If the velocity difference between the mold flux and the slab increases or the cooling rate of the slab increases, the effect of convection will be more significant. 150  150  § 100  |100  CM  ll  1  2  3 Mold Oscillation Phase  4  1  2  3 Mold Oscillation Phase  4  Figure 7.4 Comparison of the Heat Flux  Figure 75  Comparison of the Heat Flux  to the Mold with or without Convection  from the Slab with or without Convection  Left Hand Side : with Convection  Left Hand Side : with Convection  Right Hand Side : Stagnant  Right Hand Side : Stagnant  96  iI I  Slab  Solid  Flux  Rim  Figure 7.6 Heat Balance in the Region 0 ~ 1 cm below the Meniscus V = 1.27 cm/s z  7.3  f=\  cycle Is a = Amm s  r\ = 1.7 poise 7> = 1125°C M  Heat Flux to the Mold and Heat Flux from the Slab Not only for the prediction of sticking in the mold but also for judging slab quality,  measurements of the heat flux or the mold copper plate temperature are useful. Especially the magnitude of the heat flux near the meniscus where almost all surface defects of slabs are initiated will be important. Therefore, the effects of the flux properties and the operating conditions on the heat flux near the meniscus have been  97  analyzed with the mathematical model, which include the calculation of mold flux velocity, meniscus shape, mold flux temperature and slab temperature, as described in Chapter 6. Figures 7.7 ~ 7.11 show the effects on the heat flux to the mold (q ) and the heat M  flux from the slab (q ). Here, the heat flux is the time average in the region 0 ~ 1 cm s  below the meniscus, the same as in the previous discussion. The primary conditions in the calculation are noted in the figures and the remaining conditions are given in Tables 6.1 ~ 6.3. It has been found that the average heat flux to the mold and the average heat flux from the slab responds similarly with changes to operating conditions and mold flux properties. An increase of the flux viscosity (in Figure 7.7), the casting speed (in Figure 7.8) and the flux melting point (in Figure 7.11) decreases the heat flux. A high oscillation frequency (in Figure 7.9) and a long mold oscillation stroke (in Figure 7.10) enhance the heat flux.  0  OH—i—i—i—i—i—t—i—i—i—i—i—i—i—i—i—i—i—i—r-  2 4 Mold Flux Viscosity (poise)  1.5  2 2.5 Casing Speed (cm/s)  3  Figure 7.7 Effect of Flux Viscosity on  Figure 7.8 Effect of Casting Speed on  the Heat Fluxes near the Meniscus  the Heat Fluxes near the Meniscus  V = 1.27cmls f=\cyclels  n = 1.7poise f=\cyclels  z  a, =4mm 7> = 1125°C M  a, = 4mm  T =U25°C PM  120  OH  1  i  1  1  1  1  3 5 Mold Oscillation Frequency (cycle/s)  1 7  Q-i  1  1  1  1  1  1  1  I  1  1  2  1  1  1  1  1  3  1 4  Mold Oscillation Half Stroke (mm)  Figure 7.9 Effect of Mold Oscillation Frequency on the Heat Fluxes near the Meniscus  Figure 7.10 Effect of Mold Oscillation Half Stroke on the Heat Fluxes near the Meniscus  11 = 1.7poise V = 2.54cmls a, = 4mm z  T =U25°C PM  T| = 1.7poise V = Z54 c/s f = 6.67cycle/s Z  T =\\2S°C PU  120  Figure 7.11 Effect of Flux Melting Point on the Heat Fluxes near the Meniscus  0| 750  850 950 Mold Flux Melting Point (C)  1050  1150  T) = 1.7 poise V =1.26 cmls f - Icydels a, = 4 mm z  The effect of casting speed as shown in Figure 7.8 is different from that on the overall mold heat transfer. In the overall mold, an increase in the casting speed increases the slab surface temperature and, therefore, the heat flux from the slab is enhanced. However, a few millimeter below the meniscus, the slab temperature is not significantly  99  changed because of the large latent heat and is less affected by the casting speed. Thus, the value of the heat flux near the meniscus depends on the thermal resistance (i.e. flux film thickness) and the convection in the molten flux, not the slab surface temperature. The effect of the flux melting point is easily understood. A high melting point of the flux enhances the solid flux rim thickness and a thicker solid flux rim pushes the shell away from the mold. Therefore, an increase of the flux melting point results in reduced cooling near the meniscus. As discussed in the previous section, convection in the liquid flux film has an effect on the heat transfer near the meniscus. Figures 7.12-7.15 show the heat balance changes with the flux viscosity, the casting speed, the oscillation frequency and the oscillation stroke respectively. Although the heat flux to the mold and from the slab have the same tendency with respect to the above conditions, the mechanisms are not always the same, as shown in Figure 7.12 and Figure 7.13. The average heat flux from the slab is largely changed by the effect of the convection in the liquid flux film with changes of the flux viscosity and the casting speed. In the stagnant case, the heat flux from the slab is almost constant with these changes. On the other hand, the average heat flux to the mold is almost independent of the convection. Therefore, the heat flux to the mold is mainly affected by conduction. In other words, the meniscus shape, which changes the thermal resistance in the liquid flux, causes a change of the solid flux rim thickness. The change of the meniscus shape will be discussed in the next section.  100  120  120  1.27  2.54 Casting Speed (cm/s)  Mold Flux Viscosity (poise)  Figure 7.12 Effect of Flux Viscosity on  Figure 7.13 Effect of Casting Speed on  the Heat Balance near the Meniscus  the Heat Balance near the Meniscus  V = \.21cmls  i\-1.7poise  z  f=\cyclels  a,=4mm  r  w  = 1125°C  f=\cyclels  a, = 4mm  T =\\25°C PU  1 4 Mold Oscillation Half Stroke (mm)  Mold Oscillation Frequency (cycle/s)  Figure 7.14 Effect of Mold Oscillation  Figure 7.15 Effect of Mold Oscillation  Frequency on the Heat Balance near the  Half Stroke on the Heat Balance near the  Meniscus  Meniscus  T| = 1 . 7 poise  V = 2 . 5 4 cmls z  a, = 4 mm  T „=U25 C 0  P  T| = 1.7powe V = Z54c/s z  /=6.67cyclels  T =W25°C PU  101  The effect of the flux viscosity or the casting speed on the heat flux from the slab is, therefore, related to the flux velocity distribution. Figures 7.16 and 7.17 schematically show the flux velocity distributions with changes in mold flux viscosity and casting speed. With a high flux viscosity or with a high casting speed, the flux velocity in the vicinity of the slab is almost the same as the casting speed and, therefore, the heat transfer situation is similar to the stagnant case.  Figure 7.16 Flux Velocity Distribution with a Low or High Mold Flux Viscosity  LOUJ  V  Z  High  V-  Figure 7.17 Flux Velocity Distribution with a Low or High Casting Speed  102  With a change in mold oscillation frequency, it is apparent that a change in heat flux is caused by a change in meniscus shape, because the heat flux from the slab in the stagnant case increases with an increase in mold oscillation frequency. In this case, the effect of convection reduces the change. If the oscillation stroke is reduced, the heat carried by the solid flux, as well as the change in the meniscus shape, affects the heat balance near the meniscus. In the region 0 ~ 1 cm below the meniscus, the solid flux rim usually receives heat from the slab through the liquid flux film and, above the meniscus, it transfers the heat to the mold wall. As a result, above the meniscus, the solid flux rim become thicker. However, with short-stroke oscillation, the molten flux pool in the region affected by the movement of the solid flux rim has a high temperature, because it is close to the meniscus. Therefore, the heat in the solid flux rim is not significantly reduced above the meniscus and is carried again into the meniscus channel region.  7.4  Oscillation Mark Formation As described in Section 2.3, there are two mechanisms for oscillation mark  formation. One of them, termed Emi's model [39], links oscillation mark formation to flux pressure change. Kawakami [38] associates the change of the solid flux rim thickness near the meniscus with oscillation mark formation. In this study, with a mathematical model based on the Kawakami's concept, the meniscus shape is calculated with consideration of the solid flux rim as mentioned in Section 6.3. One result of a calculated oscillation mark is shown in Figure 7.18. The bottom of oscillation mark ("2" in Figure 7.18) is formed at the lower point of the mold oscillation displacement, and the top of the oscillation mark ("4" in Figure 7.18) corresponds to the  103  upper point of the mold oscillation displacement. If this deformation of the shell near the meniscus corresponds to the oscillation mark formation, the predicted depth is larger than the measured values.  4  3 -c3  1  o  \  *  ^  - « ^  \ 2  8 °<  —  "Co  I  3.8  I  I I  -0.3  -0.1  C a s t i n g Direction i  0.1  0.3  0.8  i 0.7  i  D i s t a n c e o n the S l a b S u r f a c e (cm)  i 0.9  i  i 1.1  i  i 1.3  Figure 7.18 Shape of Oscillation Mark Calculated in the Model rj = 1.7poise V = 1.27cmls f=\cyclels z  a, = 4mm  T =ll25°C PM  The effects of the flux properties and of the operating conditions on the oscillation mark depth near the meniscus are shown in Figures 7.19 ~ 7.24. A high viscosity of the flux, a high casting speed, a high frequency and a short stroke of the mold oscillation decrease the oscillation mark depth. These tendencies agree with data reported in previous works [16][36]~[39]. Although no data has been reported, it is understandable that an increase in flux pool depth or reduction in melting point of the flux would reduce the oscillation mark depth.  104  0.3  0.3 ••  fo,; o.  <s • •s 10  3 § 0.1 "  1  OH  1  4.5  1  1  1  1  1  1  6.5 Mold Flux Pool Depth (mm)  8.5  Figure 7.19 Effect of Flux Pool Depth on Predicted Oscillation Mark Depth r\ = 1.7poise V -\.27cmls z  / = 1 cyclels  Mold Flux Viscosity (poise)  Figure 720 Effect of Flux Viscosity on Predicted Oscillation Mark Depth / =4.5 mm V = 1.27 cmls f=\cyclels P  z  a, = Amm T  a, = Amm 7> = 1125°C  PU  =  \\25°C  M  0.3  |  1  T  0.! •  S o.i;  1  I ; O-l  0.3-1  .2  ,, ,11 11 11 •1 . . , . •1 1r 1.5  2  2.5  'S o  !  o-l  3  1  Casting Speed (cnVs)  1  1  1  1  3 5 Mold Oscillation Frequency (cycle/s)  1  7  Figure 721 Effect of Casting Speed on  Figure 7.22 Effect of Mold Oscillation  Predicted Oscillation Mark Depth  Frequency on Predicted Oscillation Mark  l = 4.5mm TJ = 1.7 poise f = 1 cyclels  Depth  a = 4mm  l = 4.5 mm TJ = 1.7 poise V = 2.54 cmls  P  s  T  m  = 1125°C  P  z  a = Amm 7> = 1125°C s  M  105  0.3  0.3  0-1 750  Mold Oscillation Half Stroke (mm)  Figure 7.23 Effect of Mold Oscillation Half Stroke on Predicted Oscillation Mark Depth l = 4.5 mm r\ = 1.7 poise V = 2.54 cmls P  z  f= 6.67 cycle/s T  PM  = 1125 °C  ,  ,  ,  1  ,  ,  ,  850 950 1050 Mold Flux Melting Point (C)  1 1150  Figure 724 Effect of Flux Melting Point on Predicted Oscillation Mark Depth l = 4.5 mm TJ = 1.7 poise V =l .27 cmls P  z  / = 1 cycle Is a = 4mm s  The effect of the oscillation frequency or of the oscillation stroke on the mark depth can be explained by a heat balance on the solid flux, as mentioned in the previous section. If the period or the distance of exposure of heated solid flux to a cooler flux pool is reduced, the solid flux rim thickness is reduced and the oscillation mark depth become shallow. In contrast to the above thermal analysis, the effects of the flux viscosity and the casting speed on the oscillation mark depth are mechanical in origin. Figures 7.25 and 7.26 show the average distance of the shell from the mold wall in the region 0 ~ 1 cm below the meniscus. A high mold flux viscosity or a high casting speed moves the shell farther from the mold and, therefore, the shell is less affected by the movement of the solid flux rim. As mentioned in the previous section, the conductive heat transfer near the meniscus is affected by these meniscus shape changes. In the case of a high oscillation frequency the convection heat transfer is much larger because the shell is  106  closer to the mold than in the case with a low oscillation frequency. In spite of that, a small change of the solid flux rim thickness during the oscillation, because of heat remaining in the flux rim, makes the oscillation mark shallow, as mentioned previously. 0.25  0.25  ||0.20  2  o  I  .2 °-15 o  ""'0  E  0.1  2 4 Mold Flux Viscosity (poise)  I  i — i I  1.5  I  I—r-i—I—I  I  I—i—r—i—i—i—r  2 2.5 Casting Speed (cm/s)  3  Figure 7.25 Effect of Flux Viscosity on  Figure 726 Effect of Casting Speed on  Average Distance of the Shell from the  Average Distance of the Shell from the  Mold Wall  Mold Wall  l = 4.5mm V = \.21cmls f= 1 cycle/s  l = 4.5mm rj = 1.7 poise / = 1 cycle/s  a = 4mm 7> = 1125°C  a = 4mm T  P  s  z  M  P  s  PM  = U25°C  Here, it should be noted how the meniscus behaves during mold oscillation. Takeuchi and Brimacombe [43] have indicated, based on Emi's model, that, at the maximum downward velocity of the mold oscillation, the shell is pushed away farther than at the maximum upward velocity of the mold oscillation, as shown in Figure 2.6. However, the calculation results in this study show that the shell is located close to the mold at the maximum downward velocity of the mold oscillation, if the solid flux rim does not exist. For the meniscus shape calculation, Equation (6.22) is used in the same way as in the study of Takeuchi and Brimacombe. Only one difference exists, in the boundary condition, Equation (6.34). In this calculation, the condition (<t> = j at y =0) is achieved only when the flux is stagnant, as shown in Figure 6.7. In other words, Takeuchi and Brimacombe have calculated the meniscus shape from the contact point  107  between the shell and the mold toward the meniscus, while, in this study, the calculation starts from the meniscus. This difference in the calculation step, however, brings about opposite results in the distances of the shell from the meniscus and from the mold wall. For example, a large radius of the meniscus shape means a large capillary length in Takeuchi and Brimacombe's study, but, in this study, it results in a small displacement (of the contact point between the shell and the mold) from the meniscus. In Takeuchi and Brimacombe's study, the change of the meniscus shape is explained by the capillary theory: that is a small pressure difference and a large interfacial tension between the steel and molten flux make the meniscus shape radius large and push the shell far from the mold. Although the wave equation is not used in this model, the result is closer to the surface wave phenomena than to the capillary phenomena. In other words, a small pressure difference and a large interfacial tension reduce the displacement from a flat meniscus. Matsushita et al. [71] have succeeded directly in observing the meniscus shape in the mold. Although they did not explicitly indicate the wave phenomena at the meniscus, their figures of the meniscus shape show that the meniscus is closer to the mold during downward movement than during upward movement. Therefore, it is a reasonable result that, at high frequency mold oscillation which introduces a high pressure in the liquid flux, the shell is closer to the mold and a large heat transfer occurs. The effects of the flux viscosity and casting speed on the heat transfer near the meniscus can be explained by the same mechanism. A high viscosity of the flux brings about a low internal flux pressure because of a small flow rate, as suggested in Equation (6.17). If the casting speed increases, the mold flux flow near the slab surface  108  increases, then, the pressure is reduced (Bernoulli's theorem). Therefore, with a high viscosity mold flux and a high casting speed, the shell is farther from the mold wall and the heat flux decreases. Under normal casting conditions, oscillation marks are formed by the movement of the solid flux rim at the meniscus. However, it is interesting that, if the flux rim thickness is reduced and its taper value becomes small, the mechanism of the oscillation mark formation changes from Kawakami's concept to that of Emi. For example, in the case of 1 mm mold osculation stroke in Figure 7.23, the bottom of the oscillation mark is formed at the maximum upward velocity of the mold oscillation and the top of the oscillation mark corresponds to the downward motion. The above finding permits prediction of the shape of the oscillation mark. If the solid flux rim is thicker, the oscillation mark is deep and its shape has a steep slope from the top to the bottom and a relatively mild slope from the bottom to the top in the casting direction, as shown in Figure 7.27. In the transient state from Kawakami's to Emi's concept, because the top of the oscillation mark will correspond to the downward velocity, its shape has a relatively steep slope from the bottom to the top in the casting direction. If the mechanism of the mark formation is changed completely to Emi's model, the mark shape becomes a sine curve. Therefore, if the solid flux rim thickness is largely reduced, the effect of the flux properties or the operating conditions on the oscillation mark depth is not always consistently explained by Kawakami's concept. Figure 7.28 shows, for example, the effect of the mold oscillation frequency on the mark depth for T  PM  = 919°C.  A low  109  frequency causes a shallow oscillation mark, contrary to the tendency in Figure 7.22. This result indicates that the oscillation mark formation, in the case of a thin flux rim, is based on Emi's mechanism.  Time  Casting Direction  c o )  Kamakami-type  \///////////////////////////////,  CP)  Transient-  type  V//////////////////////////////  (C)  Emi - type  Figure 727 Schematic Change of Oscillation Mark ShapefromKawakami-type Mark to Emi-type Mark  110  3  5  Mold Oscillation Frequency (cyde/s)  Figure 7.28 Effect of Mold Oscillation Frequency on Emi-type Oscillation Mark Depth l = 4.5 mm P  7.5  T\ = 1.5 poise V = 2.54 cm/s z  a = 4mm s  T  PM  = 914°C  Mechanism of the Initiation of Sticking-type Breakout Even in a high casting speed machine, the rare occurrence of breakouts makes the  analysis of sticking-type breakouts difficult. The same conditions under which sticking-type breakouts occur do not always bring about sticking. Rather, analysis of the sticking-type breakouts is a study which elucidates the possibility of sticking. Therefore, in the computer simulation, there is a need to enhance the possibility of sticking. In other words, in the computer simulation, another factor - noise, disturbance - should be included. At the meniscus, this noise is usually caused by meniscus level fluctuation. In this simulation, the meniscus level is allowed to rise and to fall during the mold oscillation, as shown in Figure 7.29. In the case in which the meniscus level remains constant, the predicted solid flux rim profile and the shell profile are shown in Figure 7.30, while the surface temperature profile is exhibited in Figure 7.31. It can be seen that, at the bottom of the oscillation  Ill  mark, the surface temperature of the slab is higher than in the other area, but the increase is not significant. Although the meniscus shape is affected by the solid flux profile, in this case, there is a wide meniscus channel for the liquid flux and it reduces the friction force between the solid flux and the shell.  Mold  0) i a) o  Displacement  6  9.6  § <b o c  10.0 10.4  <5  the  ((£>> fi)  e o  Case-l  Meniscus  leuel  rises  " 9.6 10.0 10.4  o c  o rr  0  <5 co  Case-2  Meniscus  2n  3n leuel  4n fails  Figure 729 Meniscus Level Change in the Computer Simulation  112  Distance from the Mold Wall (cm) 1500  ^4-1  1  Slab Surface Temperature (C) 1520 1  1—-J-—i  1  1540 1  1  S2J0-  0  Figure 7.30 Shell Profile and Solid Flux  Figure 731 Slab Surface Temperature  Rim  Profile with a Constant Meniscus Level  Profile with a Constant Meniscus  Level Figure 7.32 shows the solid flux rim  profile and the shell profiles after the  meniscus level rises. Over a long region from the meniscus, the channel is squeezed and it may cause a large friction force on the shell. It should be noted that, as shown in Figure 7.33, below this interaction area between the solid flux and the shell a remarkable hot spot exists, corresponding to a deep notch on the shell (as shown in Figure 7.32), which forms due to the interaction of a thicker flux rim  when the meniscus rises. At  such high temperatures, the solidified steel is easily ruptured by friction between the oscillating mold and descending strand.  113  Distance from the Mold Wall (cm) 1500  Slab Surface Temperature (C) 1520  »2.0  0  Figure 7.32 Rim  Shell Profile and  Solid Flux  Profile after a Rise in the Meniscus  Figure 7.33  Slab Surface Temperature  Profile after a Rise in the Meniscus  Figure 7.34 indicates that the real sticking shell profile, as previously shown in Figure 5.16, is close to the solid flux rim solid flux rim  and  profile calculated below the meniscus. If the  the molten flux film do not exist, the shell profile is precisely the  same as the static meniscus which can be calculated using surface tension (1900 dyne/cm) [65] instead of the interfacial tension between the flux and the steel. The  result  in the case without flux is shown in Figure 7.35. At the meniscus, its shape is close to the sticking shell but below the meniscus it is completely different from the sticking shell profile. Therefore, this sticking does not occur to the mold wall when liquid and solid mold fluxes are not present. The above finding indicates that the shell sticks to the solid flux  rim.  114  Distance from the Mold Wail (cm)  Detance from the Mold Wall (cm)  Figure 7.34 Comparison between the  Figure 7.35 Comparison between the  Real Sticking Shell and the Shell Profile  Real Sticking Shell and the Meniscus  or the Solid Flux Profile  Shape without Solid Flux Rim and Liquid Flux Film  With consideration to the above finding, a mechanism for initial sticking can be proposed, as schematically shown in Figure 7.36. During casting, if the meniscus level rises, the meniscus shape (shell profile) has a deep notch due to the interaction of a thicker flux rim (b). During the upward motion of the mold oscillation, the thicker flux rim moves away from the meniscus, then, the meniscus moves toward the mold (c). The shell close to the mold becomes cooler than the deep notch area. This is accompanied by the appearance of a hot spot below the meniscus (d). During the downward motion of the mold oscillation, the thicker flux rim moves down and interacts with the shell (e)  115  and the shell sticks to the flux rim. When the mold begins to move upward again, tensile friction force acts on the shell and a rupture occurs at the weakest (hottest) point of the shell.  Hot  Spot  Figure 7.36 Mechanism of the Occurrence for the Initial Sticking of the Shell In the case in which the meniscus level falls, there is no interaction between the solid flux and the shell and no hot spot exist below the meniscus, as shown in Figures  116  7.37 and 7.38. So, the sticking is not likely to occur after the meniscus level falls. It is interestingly seen in Figure 7.37 that an oscillation mark does not form, because the interaction of the mold flux rim with the newly forming shell is reduced after the fall of the meniscus level. As mentioned in Section 5.1, it was found in the pre-rupture area of the sticking-type breakout shell that the oscillation mark was lost or deeply notched at the same shell surface. These appearances of the shell surface, therefore, are considered as an evidence of the mold level fluctuation, which means that, if the meniscus level rises, the shell is deeply notched (even though sticking does not occur) and, in the case of the meniscus level fall, the oscillation mark is shallow or lost. It also can be predicted from Figure 7.32 and 7.37 that, if the meniscus level rises (not significantly), the pitch and depth of the oscillation mark increase and a relatively small fall of the meniscus level brings about a short and shallow oscillation mark; this is the relationship between the mark pitch and depth found by oscillation mark measurement for the sound shell, as mentioned in Section 5.2.  117  Figure 7.37 Shell Profile and Solid Flux  Figure 7.38 Slab Surface Temperature  Profile after a Fall in the Meniscus  Profile after a Fall in the Meniscus  7.6  Prevention of Sticking It is worthwhile to analyze the friction force when sticking occurs, because it is  responsible for rupture of the shell. However, this analysis must be conducted along the strand, because the rupture does not occur at the sticking point. In other words, the initial sticking point is different from the initial rupture point. Therefore, the likelihood of the occurrence of sticking depends on the likelihood of rupture below the sticking point. Without a complicated stress-strain analysis of the sticking shell, the best manifestation of the rupture is the hot spot temperature. Figures 7.39 ~ 7.41 show the hot spot temperature after the meniscus level rises as a function of the viscosity of the flux, the casting speed and the frequency of the mold  118  oscillation. The likelihood of rupture as indicated by the hot spot temperature agrees with the result of the friction force analysis. A high viscosity flux, a high casting speed and a high cycle mold oscillation is likely to cause the sticking. Because a high viscosity flux and a high casting speed reduce the heat flux from the slab as mentioned in the previous section, the hot spot temperature increases. In the meanwhile, in the case of high mold oscillation frequency, the period from the rise of the meniscus level to the sticking of the shell significantly affects on the hot spot temperature. In a short period (high frequency), the temperature of the hot spot scarcely decrease. In addition to the above conclusions, the notch depth of the shell, when the meniscus level rises, is related to the hot spot temperature. Figure 7.42 shows the effect of the flux melting point on the hot spot temperature. A low melting point flux reduces the solid flux rim thickness and makes the notch shallow. As shown in Figure 7.43, the effect of the oscillation stroke on the hot spot temperature is not clear. Although a short stroke brings about a thin solid flux rim in the meniscus channel region (below the meniscus) as mentioned in the previous section, the solid flux rim far from the meniscus (above the meniscus) is thicker than in the case of long stroke of the mold oscillation. This is because the heat near the meniscus which is carried by the solid flux rim is small at the location far from the meniscus, if the mold oscillation stroke is short. Therefore, a short stroke mold oscillation is not always adequate for a large meniscus level fluctuation. The effect of the flux pool depth on the hot spot was not analyzed, because the control system in this mathematical model has a limited length above the meniscus for the calculation of a case of large flux pool depth accompanying a meniscus level rise.  119  Instead of that, the average solid flux rim thickness above the meniscus shows in Figure 7.44. A deep flux pool reduces the occurrence of the sticking, because the solid flux rim above the meniscus is thin. Almost all methods to reduce the hot spot temperature are opposite to those for adopted shallow oscillation marks which minimize pin holes and transverse cracks. Only possible methods to improve both the occurrence of the sticking and the surface quality of slabs are to use a low melting point flux and probably to keep a deep flux pool. However, it should be noted that too low a melting point flux may cause the first type of sticking with an excess heat removal as mentioned in Section 2.1. 1530  1530  O CD  1520  Mold Flux Viscosity (poise)  Casting Speed (cm*)  Figure 7.39 Effect of Flux Viscosity on  Figure 7.40 Effect of Casting Speed on  the Predicted Hot Spot Temperature  the Predicted Hot Spot Temperature  l = 4.5 mm V = 1.27 cmls f= 1 cyclels  lp = 4.5 mm TJ = 1.5 poise / = 1 cyclels  a, = 4 mm T  a, = 4 mm 7> = 974°C  P  z  PM  = 974°C  M  120  1530  1530  o 3  1525  a CO  3 5 Mold Oscillation Frequency (cycle/s)  1520  7  Figure 7.41 Effect of Mold Oscillation  750  850 950 1050 Mold Flux Melting Temperature (C)  1150  Figure 7.42 Effect of Flux Melting Point on the Predicted Hot Spot Temperature  Frequency on the Predicted Hot Spot  l = 4.5 mm T) = 1.5 poise V = l.27 cmls  Temperature  P  l = 4.5 mm Tj = 1.5 poise V = 2.54 cmls a, = 4mm 7> = 974°C P  z  f=\cyclels  a, = 4mm  z  M  1530  O  s i1525 -  1520  1  1  1  i  1  1  1  1  1  1  1  2 3 Mold Oscillation Half Stroke (mm)  1  1  1  1  4  0H 4.5  1  1  1  1  1  1  1  5.5 6.5 7.5 Mold Flux Pool Depth (mm)  1 8.5  Figure 7.43 Effect of Mold Oscillation  Figure 7.44 Effect of Flux Pool Depth  Half Stroke on the Predicted Hot Spot  on the Predicted Average of Solid Flux  Temperature  Rim Thickness above the Meniscus  l = 4.5 mm r\ = 1.5 poise V = 2.54 cmls  rj = 1.7 poise V - 1.27 cmls f= 1 cyclels  f= 6.67 cycle/s T  a, = 4 mm 7> = 1125°C  P  z  PM  = 914°C  z  M  121  Segregation below the Surface in a Low Carbon Steel 8.1 Cooling Rate Change due to Solid Flux Rim Remelting Although the mechanism of initial sticking has been clarified, as discussed in Chapter 7, the segregation line with enlarged dendrite arm spacing found in almost all the samples from the sticking-type breakout shell (with one exception) has not been explained. In this chapter, the causes of the segregation, which probably is a characteristic of the sampled slab because it has not been reported previously, have been investigated. It was proposed that a low cooling rate which was equivalent to enlarged dendrite structure could bring about negative segregation if the distribution coefficient was under unity; and therefore the segregation found in the sticking-type breakout shell (0.08 % carbon steel) was not caused by solute redistribution during solidification but possibly due to the 8 - y transformation in which some elements had a distribution coefficient over unity. Before the calculation of solute redistribution in the dendrite, it was necessary to determine the thermal history of the dendrite in the sticking-type breakout shell, which can be related to the diffusion time of solute. Therefore,first,the mechanism of the discontinuous change of secondary dendrite arm spacing at the segregation line has been considered in an attempt to link it to characteristic phenomena of sticking below the meniscus. It is reasonable to suppose that, when the sticking occurs and the rupture is proceeding in the mold, liquid steel flows into the gap created by the rupture, as described in Section 2.1, and it comes into contact directly with the solid flux rim as shown schematically in Figure 8.1. A large heat transfer from the liquid steel to the solid flux rim occurs in the moment after the contact, and the heat conducted through the mold  122  flux subsequently decreases if the thermal field become steady. Therefore, it is expected that the cooling rate for the sticking shell is reduced discontinuously after the contact of the liquid steel. To estimate the cooling rate, a mathematical model of heat transfer in the flux rim and the steel, as shown in Figure 8.1, has been formulated. The governing equations are: For the mold flux rim, =a —f;  (8.1)  ?  For the steel, the same as Equation (4.5),  The boundary conditions are: aty=0,  q = -k u  P  P  F  (8.3)  T =T ,  (8.4)  0  aty=d ,  <*y=d  |,. = -h (T _ - T ) Py  P  Q  M  F  dTp dTp  dHp dHp4  q, = -kp-^\ top,  = ——  t  y=dp  (8.5)  dy aty=D Jl The initial condition is: s  0=  Tp  t  =  —  (  8  -  T , d <y < —^— t  P  2  and the initial temperature distribution in the mold flux rim is assumed to be:  6  )  (8.7)  123  q° Tp(y)=l~y+T , 0<y<d M  w/iere q =k  (8.8)  P  (8.9)  P  d  P  The properties of the mold flux and the steel are listed in Table 6.2 and 6.3. The mold flux rim thickness (d ) was changed as a parameter representative of the location of the P  rupture in the casting direction, because it was supposed that the flux rim thickness increased with the distance from the meniscus due to decreasing heat flux. The numerical calculation of temperature distribution in the mold flux and steel after the liquid steel contacts the flux rim was conducted with an implicit finite difference method.  Steel  Figure 8.1 Schematic Diagram of the Rupture in the Sticking Shell As shown in Figure 8.2, the results from this calculation were combined with Suzuki's equation (^= 157.6(C/?)~° ) [27] to yield a value of secondary dendrite arm 36  spacing which has been compared with the measured value in Sample IC1 (as previously shown in Figure 5.18) with segregation line (the location of samples in the sticking-type breakout shell are shown in Figure 5.3). Figure 8.2 indicates that the secondary dendrite  124  arm spacings calculated are not discontinuously changed in the direction of shell growth. Mills et al. [72] have reported that the mold flux has a large latent heat (L = 576 P  J/g) and this could have an effect on the cooling rate. As in the case of Equation (4.12), the latent heat of the mold flux can be included in the specific heat at a temperature between T  PU  - AT/2 and T  PM  + AT/2. Cp = Cp +^ P  P  (8.10)  where AT is the temperature difference between the solidus and the liquidus of the mold flux. In this model it is a parameter, because the liquidus and solidus temperature of the mold flux employed during the casting of sticking-type breakout slab are not well defined. Considering the mold flux latent heat, the model predicts a high cooling rate at the surface of the shell and a low cooling rate inside of the shell. However, the cooling rate again does not change discontinuously as expected from the measurements of the dendrite arm spacings.  125  (*)  Distance from the Surface (mm)  Figure 82 Comparison of Measured Secondary Dendrite Arm Spacing to Values Calculated by Incorporating Solid Flux Rim Remelting  8.2 Cooling Curve for the Sticking Shell Since the cooling rate change cannot be explained by the remelting of the solid flux rim, as described in the previous section, the cooling curves of the sticking shells IC1 and IC7 were calculated by a trial-and-error method in which the heat flux was altered until good agreement was obtained between measured and calculated cooling rates. The mathematical model employed for this computation was presented in Section 4.2. The heat fluxes so obtained are presented in Figure 8.3 for Sample IC1 and in Figure 8.4 for Sample IC7. Comparison of the measured secondary dendrite arm spacings with values calculated from Suzuki's equation are shown in Figures 8.5 and 8.6 and it is evident that the agreement is good. The cooling curves calculated are further  126  exhibited in Figure 8.7 and 8.8. Cooling curves of locations from the surface to 2 mm below the surface in Sample IC1 indicate the occurrence of reheating, corresponding to a drop in the heat flux as shown in Figure 8.3, while Sample IC7 experienced reheating only at the surface. The causes of the discontinuous changes in the cooling rates will be reconsidered in Section 8.5.  Figure 83 Heat Flux Pattern for Sample IC1 Location of Sample is shown in Figure 53  127  Figure 8.4 Heat Flux Pattern for Sample IC7 Location of Sample is shown in Figure 53 120-  Calculation  20-  c  o  Distance from the Surface (mm)  10  Figure 85 Comparison of Measured Secondary Dendrite Arm Spacing for Sample IC1 to Values Calculated by the Trial-and-Error Method  128  Figure 8.6 Comparison of Measured Secondary Dendrite Arm Spacing for Sample IC to Values Calculated by the Trial-and-Error Method  Time (s)  Figure 8.7 Cooling Curves Calculated for Sample IC1  129  1200 -I 0  1  1  1  1  1  1  1  1  1  1  '  12 Time (s)  24  Figure 8.8 Cooling Curves Calculated for Sample IC7  8.3 Mass Transfer Model during Solidification Followed by 8 - y Transformation: An Explanation for the Segregation Lines To explain the cause of the segregation, the cooling curves computed in the previous section in Figure 8.7 and 8.8 will be employed in this section for the calculation of mass transfer in a section of a primary dendrite arm. The latter approach is being taken, because the segregation line found in the sticking shell does not have the appearance of macro-segregation caused by fluid flow, as discussed in Section 5.3. Therefore, it can be considered that solute may segregate interdendritically and exhibit metallographical macro-segregation lines. The 5 - y transformation also enhances the segregation with a enlarged dendrite arm spacing, as mentioned in Section 5.4.  130  Therefore, with reference of Kobayashi et al.'s [51] and of Ueshima et al.'s [57] models, the following mass transfer model has been formulated. The system considered is a section of the primary dendrite, as shown in Figure 8.9, where the secondary arms are neglected.  C(tJ) cs  b)  c/  c,» e  t  §E l m#/f.  Figure 8.9 Solidification Model Considered (a) Position of Volume Element in Mushy Zone, (b) Magnified Volume Element with Concentration Profile [56] Additional assumptions are: (1) Uniform temperature in the volume element (located on the radius of primary dendrite arm, as shown in Figure 8.9); and the temperature is given by the cooling curves, as shown in Figure 8.7 or Figure 8.8. (2) During solidification, the 8 phase first appears at the center of the dendrite and grows outward and the y phase always appears the outer control volume of the 8 phase and grows inward. (3) Without supercooling, solidification in the control volume occurs, when the temperature given is lower than the liquidus temperature (T,) which is calculated for  131  solute contents in the control volume. Similarly, the transformation occurs when the temperature is lower than the transformation temperature (T ). A4  T, = 1538 - {/(C) + 13.0(%5i) + 4.8(%Af/i)}  (8.11)  /(C) = 55(%C) + 80(%C) , C <0.5% 2  /(C) = 44-21(%C) + 52(%C) , 0.5% <C 2  T =/,(C) - 60(%Si) + \2(%Mn) - 140(%/>) M  (8.12)  f (C) = 1392 +1122(%C), C < 0.09% r  / (C) = 1495, 0.09% <C r  (4) Local equilibrium at each interface. Cf = k C 6IL  (8.13)  L  or C] = k^ C  L  (8.14)  C] = k^C-  (8.15)  L  and  where i = the number of control volume (5) Complete mixing in the liquid. (6) After solidification, the solute cannot transfer between the liquid and the solid. That means, if the outer control volume is changed from 8 to y, the excess solute or the deficit solute in the control volume due to the transformation must be moved only in the solid phase without a change of the solute content in the liquid.  132  The governing equation is: dC dC 37=^TT 2  n  (8-16)  where C represents the solute concentration such as carbon, manganese, silicon, phosphorus and sulphur, and D, is the diffusion coefficient of the solute. The boundary conditions are: at the center of primary dendrite  dC -r-\ ox  at the solid - liquid interface  C = k* C  =0  x=0  &  L  (8.17) (8.18)  L  or C =k C y  at the 8 - Y interface in the liquid  YL  C =* C Y  C =^— L  Y 8  (8.19)  L  (8.20)  8  J^  C d x  }  (8.21)  ~2~  X  where C° is the initial composition of the solute in the liquid. Under these conditions, the governing equation (8.16) can be solved by an implicit finite difference method, as shown in Figure 8.10. In the actual calculation step, when a phase transformation occurs, the mass balance must be considered. Therefore, the expressions of the above conditions in the calculation are further complicated. For example, during solidification (from liquid to 8-phase), if the y-phase appears at the solid-liquid interface, then, in this model, the phase of control volume M - l is changed from 8 to y, and the excess (Si, P, S, etc.) or insufficient solute (C, Mn) due to the transformation diffuses in the solid. Hence, simultaneously with the transformation in M - l , the excess or sufficient solute flows into or out of M-2, respectively, and the mass  133  in M-l  and M-2 is conserved during the transformation. Expressions representing these  events are listed in Table 8.1. The general flow chart for this model is shown in Figure 8.11, based on the model of Ueshima et al. [57]; and the equilibrium distribution coefficients and the diffusion coefficients, also taken from Ueshima et al., are listed in Table 8.2. With this model, solute redistributions during solidification followed by the 8 - y transformation have been calculated every 1 mm from the surface of the slab for Sample IC1 and IC7.  Dendrite arm spacing data are shown in Table 8.3 while the  cooling rate is based on Figures 8.7 or 8.8. The calculated solute concentration distributions are presented in the next section.  Center of Primary  Arm  M-2 M-l  N  /M  Mt  X Figure 8.10 Control Volume in Primary Arm  Spacing for Mass Transfer Model  Start »• Move soild/liquid interface by 1 node  Increase tim e by dt and decrease Tern perature by dT  Solve the diffusion equation  Calculate T  a dT i n  L  M  Figure 8.11 General Flow Chart for Mass Transfer Model based on the Model Ueshima et al. [57]  135  Table 8.1 Boundary Conditions for Mass Transfer Model where ['] indicates new concentration in the calculation time step  Conditions  Equations  i = 1 M= 2 N < 1  for 5  C' =  k C t,L  l  or for y  u  C '= ^ C ' L  L  X  i= 1 M = 3 N = 2  i= 1 M > 3 N = 2  C/ 2 { l ^ i - 2 S c 3 ' « C +  i = 1 M= 4 N =3  +  1  +  2C  a  C ' + 2C ' = C +2C l  2  l  2  i = 1 others  or for y  2<i^N-2,  { ^ 2 ^ - 2 ^  N>4  i = N - 1, M = 4 N = 3  C '-—  c  = 0,  136  i = N - 1, others ( N > 3 )  {i+^lc  +°Z£L\C  i = N, N = M - 1 ( N > 2 )  i = N, others  ' - ^ L r  '  C ' = k**C  —r  '-r  +r  u  N  (N>2)  N+l < i < M-2 (M < M ) c  i = M-1  for 6  C .( = k C ' VL  L  H  or for y  C _( = k^ C ' L  L  M  c:=c ' L  M <i < M  c  where C '= L  „  1  ' '  ( V, is the volume of the control volume i)  |  137  Table 82 Distribution Coefficient and Diffusion Coefficient based on Ueshima et al. [57]  Element k*  L  k~*  D\cm ls)  D\cm ls)  L  D\cm ls)  D\cm ls)  2  2  2  2  1400-1500 °C 1300-1400 °C 1400-1500 °C 1300-1400 °C C  0.19  0.34  1.79  4.368 xlO"  2.799 x 10" 0.6489 x 10  -5  Si  0.77  052  0.68  2.459 x 10  0.588 x 10  0.0775 x 10  -7  Mn  0.76  0.78  1.03  1.275 xlO"  0.0164 x l O  -7  P  023  0.13  057  3.28 x 10~  S  0.05 0.035 0.70  5  -7  7  7  5  -7  0.364 x 10  -7  0.907 x 10~ 0.2991 x 10  1.506x10"* 0.4545 x 10  7  -6  -7  0.309 x 10"  5  0.019 x 10'  7  0.0041 x 10~  7  0.108 xlO"  7  0.4314x10^ 0.1249x10^  138  Table 8.3 (a) Primary and Secondary Dendrite Arm Spacings and Cooling Rate fo Sample IC1  Distancefromthe Surface (mm)  Kim)  Cooling Rate CCIs)  0-1  76.6  44.9  32.7  1-2  118.2  46.4  29.9  2-3  132.3  53.5  20.1  3-4  170.9  71.9  8.8  4-5  191.7  75.6  7.7  5-7  196.1  86.8  5.2  7-9  200.4  101  3.4  9-11  230.5  101.6  3.4  139  Table 8.3 (b) Primary and Secondary Dendrite Arm Spacing and Cooling Rate for Sample IC7  Distancefromthe Surface (mm)  K (m)  Kim)  Cooling Rate CCIs)  0-1  78.1  49.2  25.4  1-2  104.2  70.2  9.45  2-3  164.1  74.3  8.07  3-4  167.2  72.2  8.74  4-5  175.0  85.9  5.40  5-7  181.3  93.4  4.28  8.4 Solute Redistribution after 8 - y Transformation The solute concentrations in the interdendric area with a fixed size (20 |i) after solidification for Sample IC1 are shown in Figure 8.12 and those after the 8 - y transformation (at 1350 °C) are presented in Figure 8.13. The latter shows that the maximum concentration of all the solute increases in the region corresponding to the segregation line. As manganese hardly diffuses at lower temperature, these results imply that manganese and sulphur (probably associated together with as MnS) cause the segregation lines. However, the solute increase (especially manganese) 3 - 4 mm below  140  the surface can be seen also in the solute concentration distribution after solidification (in Figure 8.13), which means that solute does not redistribute significantiy during 5 - y transformation. During and after solidification solute hardly diffuses in the enlarged primary dendrite arm spacing (diffusion length is enlarged) due to the mild cooling and the solute remaining in the interdendric region after solidification causes the segregation. For Sample IC7, the solute concentrations after 5 - y transformation are shown in Figure 8.14. Except for the region near the surface, there is no significant change in the solute distributions. Not only in Sample IC7 but also in Sample IC1, large changes in the solute distribution are calculated at the surface. These indicate that segregation should exist just below the surface. In other words, a negative segregation should be observed at the surface. From the photographs in this study, negative segregation (i.e. white bands) was not found at the surface. Perhaps, surface segregation caused by carbon sintering from the mold flux [16] [17] and positive segregation at the surface reported by Takeuchi and Brimacombe [35] may compensate for the white bands. Although the existence of negative segregation near the surface is not clear, the segregation line in Figures 5.8 or 5.9 can be explained by interdendric segregation. In the etched photograph, these interdendric regions with rich solute form a band and they are observed as a segregation line. However, contrary to the initial hypothesis, the interdendric segregation is not strongly affected by solute redistribution due to the 8 - y transformation.  141  Distance from the Surface (mm) Figure 8.12  Distance from the Surface (mm)  Solute Concentrations after Solidification for  Sample IC1  142  Distance from the Surface (mm) Figure 8.13  Distance from the Surface (mm)  Solute Concentrations after 6-y Transformation for  Sample IC1  (at 1350  °C)  143  Figure 8.14 Solute Concentrations after 8 - y Transformation for Sample IC7 (at 1350 °C)  144  8.5 Segregation below the Surface in the Sticking Shell From the preceding sections, it seems clear that segregation lines below the surface in the sticking shell are related to redistribution of solute due to solidification, and it appears that enlarged primary dendrite arm spacing enhance the solute segregation. This tendency is expected from Equations (2.4) and (2.5) of Brody and Hearings [53]. The equations show that a low cooling rate reduces interdendric segregation, if the primary dendrite arm spacing is constant. However, explicitly, the effect of the primary arm spacing on the interdendric segregation is larger than that of the cooling rate. Although the segregation lines are formed by interdendric segregation, not due to the cooling rate change, but due to the primary arm spacing change, it is also true that the primary arm spacing is strongly related to the cooling rate. From the discussion in Section 8.1, the cooling rate change in the sticking shell is not caused by mold flux remelting phenomena. The discontinuous cooling rate change must be caused by a mechanical change of the boundary conditions of the sticking shell, perhaps due to the appearance of an air gap between the sticking shell and the solid mold flux. If the transverse section of the sticking shell in the mold is considered, except for the vicinity of the initial sticking area at the meniscus, there is a thin shell in the ruptured area surrounded by a thicker sound shell, as shown in Figure 8.15. The sound shells are cooler and shrink faster than the ruptured part. Therefore, the ruptured part is put into tension and a depression occurs, similar to the mechanism of the depression with longitudinal surface crack which is a result of "necking" [73]. In contrast, during the casting of the shell corresponding to Sample IC7 which experienced the discontinuous change in the cooling curve only at the surface (as shown in Figure 8.8), perhaps an air gap appeared independently of the shell shrinkage because of the mechanical disturbance of the other sticking A-4 which created the breakout, as shown in Figure 5.3.  145  Tension  Ruptured Shell  Air Gap  ///////////// Figure 8.15 Mechanism for Appearance of Air Gap during the Sticking If this mechanism for the appearance of an air gap is true, all sticking-type breakout shells would have a segregation line below the surface. However, these segregation lines have never been reported in the previous studies of sticking-type breakouts. Probably because, in steels in which the 5 - y transformation does not occur immediately after solidification, the deformation of the sticking shell is not large enough to cause an air gap. It can be said that these segregation lines may appear only in the steel with 8 - y transformation at a higher temperature because of the shell shrinkage enhanced by the transformation but not solute redistribution.  146  8.6 Non-equilibrium Phase Diagram Experimentally, it has already been confirmed that the 8 - y transformation of steel is not based on the equilibrium diagram [74] [75]. Therefore, previous work on crack formation accompanied by the 8 - y transformation should be reconsidered from a viewpoint based on the non-equilibrium diagram. As a result of the mass transfer model, the non-equilibrium solidus temperature, the initial temperature of the 8 - y transformation (T ) and the completed temperature of the Mi  8 - y transformation (T ) can be predicted. Figure 8.16 shows the calculated A4c  non-equilibrium temperatures for Sample IC1. The solidus temperature changes from 1498 °C (the equilibrium temperature) to 1460 °C. The initial temperature and the completed temperature also shift from the equilibrium temperature. It should be noted, therefore, that the cooling curves, as shown in Figure 8.7 and 8.8, will be slightly different from the cooling curves based on the heat transfer calculation with the equilibrium temperatures. Figure 8.17 demonstrates non-equilibrium temperatures in the Fe-C phase diagram with different primary arm spacing and different cooling rate. Apparently, an increase in cooling rate and an increase in primary arm spacing enhance the deviation from the equilibrium temperature. However, because the primary arm spacing itself is affected by the cooling rate in reality, the deviation is not always increased with an increase in the cooling rate. Although some theoretical or empirical correlations between the primary arm spacing and the cooling rate have been proposed, they cannot be easily applied for the case of continuously cast steel, because the effect of convection in the liquid on the primary arm spacing cannot be ignored. Therefore, for the prediction of the  147  non-equilibrium transformation in continuously cast steel, it will be necessary to obtain the correlation between the primary arm spacing and the thermal conditions accompanying liquid flow.  Time (s) Figure 8.16 Non-equilibrium Transformation Temperature Calculated for IC1  148  --equilibrium  1240  1  0  1  0.04  1  X 10C/S  1  0.08  1  1  1  A  1  1  450  1  0.12 0.16 0.20 Carbon wt %  1  C/s  1  0.24  1  1—  0.28  (a) Effect of Cooling Rate ( X = 50 \im ) x  0  0.04  0.08  0.12  0.16  Carbon wt %  0.20  0.24  0.28  (b) Effect of Primary Arm Spacing ( Cooling Rate = 10 °Cls )  Figure 8.17 Equilibrium and Non-equilibrium Phase Diagram (a) with Different Cooling Rate (b) Different Primary Arm Spacing  149  9 Summary and Conclusions Sticking of the shell in the mold, which often occurs in a high speed continuous slab casting machine, can be detected with thermocouples in the mold copper plate and prevented from developing to create a breakout by reduction of the casting speed; However previous work has not revealed why and how the sticking initiated at the meniscus. The objectives of this study were to identify the causes of sticking by examining the dendrite structure in a sticking-type breakout shell and to elucidate the mechanism of sticking with mathematical models of fluid flow in the mold flux channel and heat transfer at the meniscus. First of all, to link information from dendrite structure at a slab surface to casting conditions, the application of correlation in the literature between secondary dendrite arm spacing and local cooling rate has been examined. The secondary dendrite arm spacing in the subsurface of a slab has been measured and correlated against local cooling rate calculated with measured mold heat-flux. Based on this correlation, the following can be stated:  (1) The experimental correlation of Suzuki between secondary dendrite arm spacing and cooling rate can be applied for high cooling rates experienced during continuous casting. (2) For the application of this experimental correlation to a continuous casting slab, the definition of cooling rate should be the average cooling rate between the liquidus and solidus temperature.  However, the accuracy of the cooling rate at the slab surface calculated with a computer model depends on the finite-difference mesh size. Therefore, an analytical solution has been introduced to correlate the secondary dendrite arm spacing to the heat flux at the slab  150  surface.  A sticking breakout slab exhibiting five sticking events was studied. Small holes were observed at the surface in the sticking shells and were most likely the site of entrapment of solid mold flux which subsequently was scoured out during sand blasting.  From a metallographic examination of the sticking-type breakout shell, it has been found that:  (1) A shell which initially sticks has a coarse dendrite structure. (2) The geometrical shape of the initial sticking shell in a longitudinal cross section is parabolic and resembles the meniscus shape. (3) With one exception, segregation lines are found in the sticking shell typically 1 ~ 3 mm below and almost parallel to the surface.  From secondary ion mass spectroscope analysis, the solutes concentrating in these segregation lines were determined to be Mn and S.  For the analysis of meniscus phenomena such as the initiation of sticking, a mathematical model based on heat transfer in the mold flux was formulated. The following results were obtained:  (1) The average heat flux from the slab calculated with this model is almost the same as the heat flux predicted from the dendrite arm spacing with the analytical solution. (2) In the region near the meniscus, the heat flux from the slab is different from the heat  151  flux to the mold, because an amount of heat from the slab is carried by mold flux in the casting direction. (3) Convection in the liquid mold flux has a significant influence on the heat flux from the slab, but it has little influence on the heat flux to the mold. (4) The heat flux to the mold is strongly related to the solid mold flux thickness.  With respect to oscillation mark formation:  (1) The mathematical model, based on the mechanism of oscillation mark formation due to the movement of the mold flux rim proposed by Kawakami et al. can explain the effects of the mold flux properties and the operating conditions on the oscillation mark depth reported in previous work. (2) If the solid mold flux film thickness is reduced, the mechanism of the oscillation mark formation is better explained by Emi's model, concerning the change of the pressure in the mold flux channel.  To analyze the initial sticking event, the meniscus level has been changed in the computer simulation and it has been found that:  (1) A rapid rise in the meniscus level enhances the interaction between the shell and the solid mold flux rim and creates a hot spot below the interaction point. (2) The solid flux rim profile calculated after a rise of the meniscus agrees with the measured profile of the initial sticking shell in a longitudinal cross section. (3) A fall in the meniscus level does not have a significant effect on the interaction between the shell and the solid mold flux rim.  152  Therefore, the mechanism of the initiation of a sticking-type breakout can be proposed as follows:  If the meniscus level rises, the shell has a deep notch due to the interaction between the mold flux rim and the shell. When a thicker mold flux rim moves downward, it contacts the shell above the notch and the shell sticks to the mold flux rim. During the upstroke motion of the mold, tensile forces on the shell cause a rupture at the deep notch which is the hottest and weakest.  Since the hot spot is the most likely point to be ruptured, conditions which minimize temperature of the hot spot were sought with the model. Most of the conditions required to reduce hot-spot formation are exactly opposite to those required to minimize oscillation mark depth. Notwithstanding this, there are a few technique to reduce the occurrence of sticking and to improve the surface quality and they are to use a low melting point powder and, probably, to maintain a deep mold flux pool.  To analyze the segregation line, a mass transfer model has been formulated based on a consideration of 8 - y transformation. From this analysis:  (1) The segregation observed in the sticking shell appears to be a band of interdendric segregation enhanced by enlarged primary dendrite arm spacings. 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