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Thermomechanical phenomena during rough rolling of steel slab Chen, Wei Chang 1992

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THERMOMECHANICAL PHENOMENA DURING ROUGH ROLLING OF STEEL SLAB by WEI CHANG CHEN B.A.Sc., Beijing University of Iron and Steel Technology, 1983 M.Sc., University of Science and Technology, Beijing, 1986 A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Applied Science  in THE FACULTY OF GRADUATE STUDIES in the DEPARTMENT OF METALS AND MATERIALS ENGINEERING We accept this thesis as confirming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA November 1991 W.C. Chen  In presenting this thesis in partial fulfillment 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 purpose may be granted by the head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Metals and Materials Engineering The University of British Columbia Vancouver, British Columbia Canada  Date .1 v^( „  Abstract  A mathematical model has been developed to predict the temperature distribution through a slab during rough rolling. The heat transfer model is based on two-dimensional heat flow, and includes the bulk heat flow due to high speed slab motion, but ignores heat conduction in the rolling direction. At high temperature, an oxide scale forms due to exposure to air and must be considered in the heat transfer analysis. The thick scale which forms during reheating helps to insulate the slab during transportation of the slab to the rolling stands. Prior to rolling, the heavy oxide is removed by the descale sprays, and a much thinner oxide layer is formed during the very short exposure of the slab to air during rough rolling. The temperature results predicted by the model have been validated by comparison with another model. The results show that the work roll chilling has a significant effect on the temperature distribution in the slab in the roll gap and approximately 33% of the total heat lost by the slab is extracted by the work rolls; however, the chilling affect is confined to a very thin surface layer on the slab, approximately 2.5% of the slab thickness. To measure the roll chilling effect, pilot mill tests have been conducted at CANMET and UBC. In these tests, the surface and the interior temperatures of specimens during rolling have been recorded using a data acquisition system. The corresponding heat transfer coefficients in the roll bite have been back-calculated by a trial-and-error method using the heat transfer model developed. The heat transfer coefficient has been found to increase along the arc of contact and reaches a maximum and then declines until the exit of the roll bite. It is important to note that the mean heat transfer coefficient in the roll gap is strongly dependent on the mean roll pressure. At  low mean roll pressure, such as in the case of rolling plain carbon steels at elevated temperature, the maximum heat transfer coefficient in the roll bite is in 25-50kW/m 2-°C range. As the roll ■  pressure increases, the maximum heat transfer coefficient also increases to approximately 700 kW/m2-°C. Obviously, the high pressure improves the contact between the roll and the slab surface thereby reducing the resistance to heat flow. The mean roll gap heat transfer coefficient at the interface (HTC) has been shown to be linearly related to the mean roll pressure. These results were employed to calculate the thermal history of the slab during industrial rough rolling; the results are in good agreement with the data in the literature. In addition to the thermal history, the strain and strain rate distribution also affect the evolution of microstructure of rolled steels. In the present project, heat transfer and deformation during rough rolling of a slab have been analyzed with the aid of a coupled finite element model based on the flow formulation approach. In the model, sliding friction is assumed to prevail along the arc of contact and the effect of roll flattening has been incorporated. The model has been validated by comparing the results from the pilot mill tests. It confirms that the deformation of a slab in the roll gap is inhomogeneous and just beneath the surface very high strain rates of approximately 5-10 times the nominal strain rate are reached due to the redundant shearing. The maximum strain rate is attained at the entrance to the roll bite just beneath the rolls. The corresponding strain distribution through the thickness is also non-uniform, being lowest at the center and highest at the surface. The temperature gradient near the surface of the slab is very large due to work roll chilling; this is consistent with results obtained from the finite-difference model. The predicted roll forces are in good agreement with the measured values for the 9-pass schedule currently employed on the roughing mill at Stelco's Lake Erie Works and the pilot mill tests.  iii  Table of Contents  Abstract ^  ii  Table of Contents ^  iv  Table of Tables ^  xi  Table of Figures ^  xiii  Nomenclature ^  xxii  Acknowledgements ^  xxxiii  Chapter 1 INTRODUCTION ^  1  1.1 Rough Rolling ^  2  1.2 Thermomechanical Processing ^  3  1.3 Hot Deformation of Steel Slab during Rough Rolling ^  4  Chapter 2 LITERATURE REVIEW ^ 2.1 Thermal Analysis of the Rough Rolling Process ^ 2.1.1 Heat Transfer during Rough Rolling ^  8 8 8  2.1.1.1 Heat Transfer in the Slab ^  9  2.1.1.2 Heat Losses during Rolling ^  10  2.1.1.2.1 Radiative and Convective Heat Loss ^  iv  10  2.1.1.2.2 High Pressure Water Descaling ^  12  2.1.1.2.3 Roll Chilling ^  13  2.1.1.3 Heat Generation during Rolling ^ 2.1.2 Heat Transfer in the Work Roll ^  13 15  2.1.2.1 Work Roll Cooling System ^  15  2.1.2.2 Heat Transfer in the Work Roll ^  16  2.1.3 Heat Transfer Characterization at the Roll-Slab Interface ^ 19 2.2 Oxidation of Steels at High Temperature ^  27  2.2.1 Oxidation Mechanisms of Metals at High Temperature ^ 27 2.2.2 Iron Oxide Scale Growth in Air ^  28  2.2.3 Effect of Oxide Scale on Heat Transfer ^  33  2.3 Finite Element Analysis on Hot Deformation ^  37  2.3.1 Governing Equations for Finite Element Analysis ^  37  2.3.2 Application of Finite Element Analysis in Metal Forming ^ 41 Chapter 3 SCOPE AND OBJECTIVES ^  44  3.1 Objectives and Scope ^  44  3.2 Methodology ^  45  Chapter 4 EXPERIMENTAL MEASUREMENTS ^ 4.1 Test Design ^  47 47  v  4.1.1 Thermocouple Design and Data Acquisition System ^  48  4.1.2 Preparation of Samples ^  49  4.1.3 Test Facilities ^  51  4.2 Test Procedures ^  53  4.2.1 Test Schedule at CANMET ^  53  4.2.2 Test Schedule at UBC ^  57  4.3 Thermal Responses of Instrumented Specimens ^  59  4.3.1 Thermal Responses ^  59  4.3.2 Surface Temperature ^  64  Chapter 5 HEAT TRANSFER MODEL DEVELOPMENT ^ 5.1 Mathematical Modelling ^  70 70  5.1.1 Heat Conduction in Slab during Rough Rolling ^  70  5.1.2 Boundary Conditions ^  73  5.1.2.1 Initial Condition ^  73  5.1.2.2 Boundary Conditions ^  73  5.1.3 Heat Conduction in the Work Roll ^  76  5.1.4 Numerical Solution ^  78  5.2 Modification of the Model for Roll Gap Heat-Transfer Coefficient Calculation ^ 81 5.2.1 HTC Solution ^  81  vi  5.2.2 Convergence of the Numerical Solution for the Modified Model ^ 82 5.2.3 Verification of the Modified Model ^  86  5.3 Sensitivity Analysis for the Roughing Model ^  87  5.4 Verification of the Roughing Model ^  93  Chapter 6 ROLL GAP HEAT TRANSFER COEFFICIENT ANALYSIS ^ 98 6.1 Roll Gap Heat Transfer Coefficient Analysis ^  98  6.1.1 HTC Variation along the Arc of Contact ^  98  6.1.2 Influences of Rolling Parameters on HTC ^  105  6.1.3 Pressure Dependence of HTC ^  109  6.2 A Preliminary Theoretical Consideration of HTC during Hot Rolling ^ 115 6.2.1 Fenech et al.'s Model ^ 6.2.2 Roll Gap Heat transfer Coefficient (HTC) ^  115 117  6.3 Discussion ^  120  6.4 Summary ^  122  Chapter 7 THERMAL PHENOMENA DURING ROUGH ROLLING ^ 123 7.1 Heat Transfer Characterizations during Rough Rolling ^  123  7.1.1 Heat Transfer Characterizations of a Slab ^  123  7.1.2 Temperature Distribution in the Work Roll ^  135  7.2 Oxide Scale Growth of Steels during Rough Rolling ^  vii  137  7.2.1 Oxide Scale Growth Rate of Steels at High Temperature ^ 137 7.2.2 Assumptions for Oxide Scale Formation on the Steel Slab ^ 140 7.2.3 Oxide Scale Growth of Steels during Rough Rolling ^  143  7.3 Oxidation Effect on Heat Transfer of a Slab ^  144  7.3.1 Effect of Emissivity of Oxide Scale ^  145  7.3.2 Effect of Oxide Scale Thickness ^  146  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling ^ 147 7.3.3.1 Temperature Distributions in the Roll Gap ^  147  7.3.3.2 Thermal Histories of Slab ^  150  7.3.3.3 Effect of Rolled Materials on Thermal History of slab ^ 152 7.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling ^ 156 Chapter 8 DEFORMATION ANALYSIS ON A SLAB DURING ROUGH ROLLING 158  ^ 8.1 Velocity-Pressure Formulation ^  158  8.1.1 Formulation for Deformation ^  158  8.1.2 Thermal Coupling ^  161  8.2 Finite Element Solution ^  161  8.2.1 Finite Element Discretization ^  161  8.2.2 Mechanical and Thermal Boundary Conditions ^  163  viii  8.2.3 Sequence of the Solution ^ 8.3 Verification of the Source-Code of the FEM Program ^ 8.3.1 Heat Transfer Analysis of a Plate ^  164 166 166  8.3.2 Interface Temperature Distribution of a Pilot Mill Test ^ 169 8.4 Rough Rolling Deformation Analysis ^  170  8.4.1 Velocity profiles ^  170  8.4.2 Strain Rate Distribution ^  172  8.4.3 Strain Distribution ^  175  8.4.4 Thermal Field in the Roll Bite ^  178  8.5 Validation of the Model ^  182  8.5.1 Deformation Observation from Pilot Mill Tests ^  182  8.5.2 Roll Force Prediction ^  186  Chapter 9 SUMMARY AND CONCLUSION ^  189  9.1 Summary and Conclusion ^  189  9.2 Future Work ^  193  REFERENCES ^  194  APPENDICES ^  201  A Finite Difference Nodal Equations ^ A.1 Nodes in the Slab ^  ix  201 201  A.2 Nodes in the Rolls ^ B Velocity-Pressure Finite Element Derivation ^  203 205  B.1 Deformation Formulation ^  205  B.2 Thermal Coupling ^  206  B.3 Iso-parametric Element ^  210  B.4 Evaluation of Element Matrix ^  213  x  ^  Table of Tables  Table 4.1 Specifications of the pilot mill at CANMET ^  51  Table 4.2 Specifications of the pilot mill at UBC ^  51  Table 4.3 Conditions employed in rolling tests to determine the influence of successive rolling passes on interface heat transfer coefficient for AISI 304L stainless steel ^  53  Table 4.4 Conditions employed in rolling tests to determine the influence of rolling pressure on interface heat transfer coefficient for a 0.05%C low carbon steel Table 4.5 Tests for microstructural evolution study ^  54 55  Table 4.6 Conditions employed in rolling tests to determine the influence of rolling temperature on interface heat transfer coefficient for AISI 304L stainless steel ^  55  Table 4.7 Tests conducted to determine the influences of rolling speed on interface heat-transfer coefficient ^  56  Table 4.8 Tests for HTC measurements at UBC ^  57  Table 4.9 Regression results for tests at CANMET ^  68  Table 4.10 Regression results for tests at UBC ^  69  xi  Table 5.1 Sensitivity of time step size on the predicted surface temperature ^ 88 Table 5.2 Sensitivity of node size on the predicted surface temperature ^ 90 Table 5.3 Sensitivity of emissivity on the predicted surface temperature ^ 91 Table 5.4 Conditions used in validation of the work roll module ^  96  Table 6.1 Rolling Conditions for material type influence on the HTC ^ 108 Table 6.2 Mean pressure for tests at CANMET ^  113  Table 6.3 Mean pressure for tests at UBC ^  113  Table 7.1 Operating conditions for the 7-pass schedule ^  124  Table 7.2 Operating conditions for the 9-pass schedule ^  132  Table 8.1 Typical deformation parameters for the 9-pass schedule ^ 170 Table 8.2 Comparison of the measured roll force with the FEM prediction for 9-pass rolling ^  187  Table 8.3 Comparison of the measured roll forces with the FEM prediction for the CANMET stainless steel tests ^  188  xi i  Table of Figures  Figure 1.1 Mathematical models in thermomechanical processing ^  6  Figure 1.2 A typical layout of 1/2 continuous hot strip mill ^  7  Figure 2.1 Spray cooling of work roll during rolling process ^  16  Figure 2.2 Contact between real surfaces ^  21  Figure 2.3 Temperature distribution through surface in contact ^  21  Figure 2.4 Fenech et al's heat transfer model for thermal contact ^  22  Figure 2.5 Calculated and measured h e vs. apparent pressure ^  24  Figure 2.6 The relationship between the roll gap heat-transfer coefficient and the mean pressure along the arc of contact for two successive passes on the pilot mill ^ 26 Figure 2.7 Schematic diagram of multi-layer of iron oxide scale ^  28  Figure 2.8 Relative thickness of magnetite (Fe 3 04) and hematite (Fe 2O 3 ) ^ 31 Figure 2.9 Emissivity of oxide scale during metal heating ^  34  Figure 2.10 Intensity of heat transfer through oxide scale during reheating ^ 35 Figure 2.11 Dependence of temperature through thickness on heating time ^ 36  Figure 3.1 Methodology adopted in thermomechanical analysis ^  46  Figure 4.1 Schematic diagram of specimen employed in the thermal response measurements ^  49  Figure 4.2 Schematic layout of the test facilities at UBC ^  52  Figure 4.3 Thermal response of thermocouples for Test RLC12-1 ^ 59 Figure 4.4 Thermal response of thermocouples for Test 1LC-6 ^  60  Figure 4.5 Thermal response of thermocouples during tests at CANMET(SS6P71) ^ 61 Figure 4.6 Thermal response of thermocouples for Test 3 ^  62  Figure 4.7 Thermal response of thermocouples for Test 7-1 ^  63  Figure 4.8 Thermal response of thermocouples for Test 5-1 ^  63  Figure 4.9 Surface temperature in the roll bite for Test SS6P71 with 1.0m/s ^ 64 Figure 4.10 Surface temperature in the roll bite for Test SS-8 with 1.5m/s ^ 65 Figure 4.11 Surface temperature in the roll bite for Test SS-15 with 0.5m/s ^ 65 Figure 4.12 Surface temperature in the roll bite for Test 6 ^  67  Figure 4.13 Surface temperature in the roll bite for Test 8-1 ^  67  Figure 5.1 Schematic diagram of hot rolling ^  71  xiv  Figure 5.2 Discretization of the slice in the slab and the roll for finite difference analysis 79  ^ Figure 5.3 Flow chart of the temperature solution of the model ^  80  Figure 5.4 Flow chart of HTC calculation ^  81  Figure 5.5 Effect of Eps on HTC magnitude ^  82  Figure 5.6 Effect of mesh size and time step on accuracy ^  84  Figure 5.7 Effect of time step on HTC value ^  85  Figure 5.8 Effect of mesh size on HTC value ^  85  Figure 5.9 Comparison of the predicted and the measured temperature ^ 86 Figure 5.10 Effect of time-step sizes on the surface temperature under radiative and natural convective cooling ^  87  Figure 5.11 Effect of node size on the predicted surface temperatures under radiative and natural convective cooling ^  89  Figure 5.12 Effect of emissivity on the surface temperature under radiative and natural convective cooling ^  90  Figure 5.13 Effect of the roll gap heat transfer coefficient on the surface temperature in the roll bite ^  92  Figure 5.14 Comparison of the temperature distribution predicted by the current model with that of Devadas ^  xv  93  Figure 5.15 Comparison of the temperature distribution predicted by the current model under descaling with that of Devadas ^  94  Figure 5.16 Comparison of the temperature distribution predicted by the current model in the roll bite with that of Devadas ^  94  Figure 5.17 Comparison of the thermal history of a strip predicted by the roughing model with the data from Devadas ^  95  Figure 5.18 Comparison of the model results with an analytical solution for the work roll 95  ^ Figure 6.1 HTC variation for Test SS6P71 ^  99  Figure 6.2 HTC variation for Test SS8 ^  100  Figure 6.3 HTC variation for Test SS9 ^  101  Figure 6.4 HTC variation for Test SS18 ^  101  Figure 6.5 HTC variation for Test SS19 ^  102  Figure 6.6 HTC variation for Test 1LC-6 ^  103  Figure 6.7 HTC variation for Test 3LC-1 ^  103  Figure 6.8 HTC variation for Test 8-1 ^  104  Figure 6.9 Effect of roll reduction on HTC for a rolling temperature of 950°C and a roll speed of 1.5m/s ^  105  xvi  Figure 6.10 Effect of roll speed on HTC for a rolling temperature of 1050°C and 38.9% reduction ^  106  Figure 6.11 Influence of rolling temperature on HTC for approximately 35% reduction and 1.5m/s ^  107  Figure 6.12 Influence of material type on the magnitude of HTC ^  108  Figure 6.13 Distribution of HTC and roll pressure in the roll bite for SS-19 ^ 109 Figure 6.14 Distribution of HTC and roll pressure in the roll bite for SS-15 ^ 110 Figure 6.15 Distribution of HTC and roll pressure in the roll bite for SS-8 ^ 110 Figure 6.16 Comparison of surface temperature predicted by mean HTC with the measured one for Test S6P71 ^  112  Figure 6.17 Relation of mean HTC with mean roll pressure ^  114  Figure 6.18 Hardness data for stainless steel-416 ^  118  Figure 6.19 Mean HTC data vs. mean roll pressure for the tests conducted at CANMET and at UBC ^  120  Figure 6.20 Specimen surface profiles before and after rolling ^  121  Figure 7.1 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel ^  125  Figure 7.2 The difference of thermal histories for the head and tail end during 7-pass rolling of 0.05%C plain carbon steel ^  xvii  127  Figure 7.3 The temperature distribution of the slab in the roll bite during 7-pass rolling of a 0.05%C plain carbon steel ^  131  Figure 7.4 Thermal history half-way along the length of the slab during 9-pass rolling of a 0.05%C plain carbon steel ^  133  Figure 7.5 The difference of thermal histories for the head and tail end during 9-pass rolling of a 0.05%C plain carbon steel ^  134  Figure 7.6 Comparison of thermal histories for the 7 and 9-pass rolling of a 0.05%C plain carbon steel ^  135  Figure 7.7 Temperature distribution in the work roll during 7-pass rolling of a 0.05%C plain carbon steel ^ Figure 7.8 Oxide scale growth of iron and some steels at 1200°C ^ Figure 7.9 Distances associated with the interface i ^  136 139 141  Figure 7.10 Oxide scale thickness half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel ^ 143 Figure 7.11 Effects of mesh size on surface temperature ^  144  Figure 7.12 Effects of oxide scale emissivity on surface temperature ^ 145 Figure 7.13 Effects of oxide scale thickness on the surface temperature of the slab ^ 146 Figure 7.14 Effects of oxide scale thickness on the temperature at the scale/steel interface 147  ^  xviii  Figure 7.15 Temperature distribution half-way along the length of slab in the roll bite during 7-pass rolling of a 0.05%C plain carbon steel with oxidation ^ 148 Figure 7.16 Surface and interface temperature half-way along the length of the slab in the roll bite with and without oxidation for Pass 5 and 7 of a 0.05%C plain carbon steel rolling  149  Figure 7.17 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel with oxidation ^  150  Figure 7.18 Thermal history half-way along the length of the slab with and without oxidation during 7-pass rolling of a 0.05%C plain carbon steel ^ 151 Figure 7.19 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.025%Nb bearing steel with oxidation ^  153  Figure 7.20 Thermal history at the head and the tail end of the slab during 7-pass rolling of a 0.025%Nb bearing steel with oxidation ^  153  Figure 7.21 Comparison of thermal history half-way along the length of the slab during 7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation ^ 154 Figure 7.22 Comparison of thermal history half-way along the length of the slab during 7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation ^ 154 Figure 7.23 Comparison of thermal history half-way along the length of the slab during 7-pass rolling with and without heat loss to the work rolls ^ Figure 8.1 Finite element discretization in the roll bite of the slab ^  157 162  Figure 8.2 Sequence of the solution for the finite element analysis ^ 165  xix  Figure 8.3 Verification of the heat transfer module of the finite element program ^ 167 Figure 8.4 Comparison of the numerical upper surface temperature distribution of the plate with an analytical solution ^  168  Figure 8.5 Comparison of the FEM prediction with a pilot mill test result of the interface temperature distribution in the roll bite ^  169  Figure 8.6 Typical velocity profiles during 9-pass rolling ^  171  Figure 8.7 Effective strain rate distribution in the roll bite in the 9-pass rolling ^ 174 Figure 8.8 Effective strain distribution in the roll bite ^  177  Figure 8.9 Temperature distribution in the roll bite ^  180  Figure 8.10 Temperature rise distribution in the roll bite ^  181  Figure 8.11 Illustration of deformation of the embedded pin and the apparent shear angle 183  01 ^  Figure 8.12 Comparison of the effective strain distribution predicted by the model and the pilot mill tests ^ 185 Figure 8.13 Deformation of the pin in a pilot mill test SS-19 ^  186  Figure 8.14 Comparison of the roll forces predicted and measured during 9-pass rolling ^ 187 Figure A.1 Schematic diagram of the control volume for the nodes in the slab ^ 202 Figure A.2 Schematic diagram of the control volume for the nodes in the rolls ^ 204  xx  Figure B.1 Eight-node quadratic rectangular element ^  xxi  211  Nomenclature  {d}^nodal velocities matrix [a")^nodal pressure matrix A^surface area, m2 b^space between nozzle, m Bi^Biot number C, Co^constant Cm^concentration of metal, mol/m 3 Cps^specific heat of slab, J/kg Cp„,^specific heat of water, J/kg Ce^empirical constant d^nozzle hydraulic diameter, m [D]^plasticity matrix D^diameter of work roll, m Eps^matching limit for determination of heat transfer coefficient, °C  f^field variable {f)^thermal force matrix  {f}^velocity force matrix {F}^volumetric force F(x)^modifying weighting function in upwinding technique  thermal force G, H e^linear function in an element Gr^Grashof number  H^hardness of material, N/m 2 H(t),H^slab thickness in the roll bite, m  H0 , H,^slab entry and exit thickness, respectively, m Hn^slab thickness at the neutral point in the roll bite, m HTC^roll gap heat transfer coefficient, W/m 2 -°C h, h'^heat transfer coefficient, W/m 2-°C he^pseudo contact heat transfer coefficient, W/m 2 -°C  convective heat transfer coefficient, W/m 2 -°C hig^latent heat of vaporization of water, J/kg  hgap^roll gap heat transfer coefficient, W/m 2-°C h n^natural convective heat transfer coefficient, W/m 2-°C hrad^pseudo radiative heat transfer coefficient, W/m2-°C h„^roll cooling heat transfer coefficient, W/m 2 -°C  h e^element length, m h(t)^heat transfer coefficient at a particular residence time, W/m 2-°C h(t)^heat transfer coefficient for each cooling zone on the roll surface Jacobian matrix  J^mechanical equivalent of heat, CaVN-m Jo, Jr^Bessel Function of its first kind cationic flux, mol/m2-s  k^thermal conductivity, W/m-°C k^shear strength, N/m 2 thermal conductivity of metal 1, W/m-°C  k2^thermal conductivity of metal 2, W/m-°C combined thermal conductivity of metal 1 and 2, W/m-°C  kf^fluid thermal conductivity, W/m-°C thermal conductivity of roll, W/m-°C  k3^thermal conductivity of slab, W/m-°C  kw^thermal conductivity of water, W/m-°C kinetic constant of scaling, m2/s  kP^parabolic rate constant for the growth of oxide scale, cm 2/s [KT]^stiffness matrix for temperature  xxiv  velocity component of the finite element stiffness matrix [P]^pressure component of the finite element stiffness matrix -4^unit velocity vector at the edge of an element /4, distance between the nozzle and the roll surface, m Lo^Slab length, m  an experimental constant n^number of contact per unit area normal direction of the interface N^rolling speed, rev/min [NJ^shaping function Nu^Nusselt number p^roll pressure, kg/mm2  P.^water jet pressure, kPa Pa^apparent  pressure, kg/mm2  Pe^Peclet number Pr, Pr,^Prantle number P,,,^total load on contacting surface, N  heat flux, W/m 2 qs^heat source, W/m3  X XV  q1  ^  qg  frictional heat, W/m2  ^deformation heat, W/m 2  qconv^convective  heat loss, W  qox^incident heat flow density from steel oxidation, kJ/m 2 -s qr^radiative qs^heat  heat loss, W  flux due to spray cooling, W/m 2  AQ,^total heat loss during rough rolling, J total heat loss except to rolls during rough rolling, J AQ,(%)^percent of heat loss to rolls to total heat loss during rough rolling r^percent reduction R^universal gas constant, 8.31 J/mol-°K R, r^radius, m  Ar^length step in R direction Re^Reynold number Re w^Reynold  number for water at roll surface  SStefan-Boltzmann constant, jim2_0K4 IS) ,S y^deviatoric stress, N/m 2 s^proportion of steam formed  x xvi  t^time, s {t}^traction on stress boundary T, To^temperature, °C T,^initial temperature, °C T„,ax^temperature of critical heat flux, °C  roll temperature, °C TS^slab temperature, °C T,^saturation temperature of water, °C  water temperature, °C ambient temperature, °C Tjt)^ambient temperature along the rougher table, °C T.,(t*)^ambient temperature along the roll surface, °C  A Tdef^temperature rise due to deformation, °C AT ^surface excess temperature, °C AT,^plate temperature drop due to descaling, °C AT),^temperature rise of water, °C -4  horizontal velocity vector at the edge of an element, m/s average velocity along the edge of an element, m/s  xxvii  velocity in horizontal direction, m/s vertical velocity vector at the edge of an element, m/s  velocity, m/s velocity in vertical direction, m/s velocity vector in finite element method velocity, m/s spatial derivative of velocity, s -1 relative velocity between roll and slab surface, m/s water spray velocity, m/s rate of water flow per unit width of plate, m 2/min water flux rate, 1/m 2-s width of water spray, m weighting function initial scale thickness, cm dimension along rolling direction, m length step in X direction length step in Y direction dimension through thickness of slab, m  x xv ii i  z^dimension through width of slab, m  a^thermal diffusivity, m 2/s a'^angle of the velocity direction to the rolling direction, rad a ^parameter used in upwinding technique ;  impingement angle of water spray to the roll surface  13.^parameter used in upwinding technique X^scale thickness, cm 8^skin layer thickness of roll, m 5 1, 5 2^average  void height for two metals respectively, m  Sp^variation of pressure Sy^variation of velocity 43^variation of strain rate c(T)^emissivity as a function of temperature LA^  v^  fraction of contact area  volume strain rate, strain rate, s -1  Ex^strain rate in X direction, s 1  e'y^strain rate in Y direction, s"' es^strain rate in Z direction, s -1 shear strain rate, s -1 e^ xy 6^effective  strain rate, s -1  spatial strain rate, s -1 11  7  ^  heat generation efficiency factor  ^  7  shear strain  ^parameter used in upwinding technique  II  ^viscosity, N/s-m2  1-11  ^  1-Lw  friction coefficient  ^viscosity  of water, N/s-m 2  co^angular velocity, rad/s 9  ^  (P.  P  roll contact angle, rad  ^roll contact angle at neutral point, rad  ^  density, kg/m 3  P S^slab density, kg/m3 Pv  ^density of water vapor, kg/m 3  P.  ^density of water, kg/m 3  xxx  a^flow stress, N/m 2 am^mean flow stress, N/m2 ast^surface tension of water, N/m 2 (7x^flow stress in X direction, N/m2 ay^flow stress in Y direction, N/m2 az^flow  stress in Z direction, N/m2  a^effective flow stress, N/m 2 0^contact angle in the roll bite, rad A0^angle step at 0 direction in cylindric coordinate system, rad tan(0 1)^numerical differential of the deflection of the pin after deformation  ,`1-1^natural coordinates;  Subscripts and Superscripts (e)^element Gaus^Gauss point i, j^finite element node positions n, n+1^current time and time at next time step p^pressure r^roll  s^slab T^transposed matrix or vector  v^velocity w^water x, y^X and Y direction  Acknowledgement  I would like to express my sincere appreciations to Professors I.V. Samarasekera and E.B. Hawbolt, for their guidance, stimulating discussions, immense encouragement and support throughout the project. I would also like to thank Mr. Neil Walker, Serge Milaire and Bernhard Sauter for their assistances in the research work. Financial assistances from NSERC is gratefully acknowledged. I would also like to thank Stelco Inc. for providing the data on rough rolling and CANMET MTL for their assistance to provide the necessary facilities to conduct the pilot mill tests. Thanks are also extended to my fellow graduate students in this department for their interactions and discussions, in particular Kenneth E. Scholey, Sanjay Chandra, Sunil Kumar. I would also like to take this opportunity to express my gratitude to my mother and my wife, Xiaoli, for their understanding and support.  Chapter 1  INTRODUCTION  Steel strip is produced from molten steel by continuous casting, reheating, rough rolling and finish rolling units in sequence. Continuous casting has generally replaced static casting in the steel industry for the production of steel slabs. Although significant improvements in mould design which have been implementing worldwide have resulted in better slab quality, and some success ]  has been realized in the development of a process of continuous casting and direct rolling, a procedure which would obviously save energy and reduce the cost of producing steel, much work remains to be done. Rough rolling, which is conducted between the reheating furnace and the finishing mills, reduces the slab thickness from that produced at the caster to that required for finish rolling. Moreover, for microalloyed steels, rough rolling establishes the final austenite grain size in the transfer bar leaving the roughing mill, together with the size and distribution of precipitates that would influence the evolution of the microstructure . With the development of continuous-casting ]  and direct-rolling, the slab entering the roughing mill will have a thermal field that is somewhat different from slabs that have been traditionally reheated. A thorough understanding of the influence of the thermal field obtained from the reheating furnace on rough rolling will be of value in determining the limits on the through-thickness gradients that can be tolerated in the mill without adverse effects on the shape and properties of the slab. Another practice which is being advocated is low-temperature rolling which can reduce energy cost for reheating, scale loss and improved mechanical properties due to a finer structure. The major  1  1.1 Rough Rolling  problems associated with implementing this innovation is the difficulty of securing high enough finishing temperatures, fluctuations in the strip width and thickness and the roughening of the surface [21131 .  The feasibility of low-temperature rolling for a given steel plant can only be fully assessed ■  with models of the reheat furnace and the roughing mill. In spite of new techniques, such as near-net-shape continuous castingt41 , that are also being developed worldwide, rough rolling is still an important unit in the production of steel strip. Mathematical models of the thermomechanical processing of steel strip have been developed at UBC for all the unit processes except rough rolling, as shown in Fig.1.1 151 . Developing models to quantify the thermal gradient through the thickness and the deformation behavior during its passage through the roughing mill, would permit a coupling of those other models to completely describe the thermomechanical processing of the steel strip in the hot forming operation.  1.1 Rough Rolling There are three types of rolling mills for steel strip production which differ from each other by the layout of the roughing mill(i) 1/2 continuous strip rolling unit, which consists of one reversing roughing mill and a continuous finish rolling unit; (ii) 3/4 continuous strip rolling unit which consists of one reversing mill and two four-high mills in series and a continuous finish rolling unit; (iii) full continuous strip rolling mill which has no reversing mill at all. A typical layout of the 1/2 continuous hot strip mill is schematically shown in Fig.1.2. The roughing unit mainly consists of the reversing mill, the vertical edger, the slab descaler and the coil box.  2  1.2 Thermomechanical Processing At Stelco, there is one reversing roughing mill preceding the 5-stand continuous finishing mill, as in Type (i). 7- and 9-pass schedules are currently employed to reduce the thickness of the slab from 240mm to 21 mm depending on the grade of steel. Before rough rolling, the slab discharged from the reheating furnace goes through a slab-descaler to break the scale formed in the reheating furnace, followed by a vertical edger to adjust the width of slab to final specifications. Breakage of oxide scale in the slab-descaler and in the vertical edger facilitates the removal of scale by high pressure water jets located prior to the reversing mill. Between successive rolling passes, the exposed slab surface is oxidized, due to the high temperature of approximately 1200°C, and scale is reformed. The scale insulates the surface of the slab during the rough rolling process, but the effects may be small depending on the thickness of the scale formed. This secondary scale also must be removed because if it is rolled in, it deterioratively affects the surface quality of the strip. At Stelco, secondary oxide descaling is conducted by high pressure water jets before the second and the sixth passes during reversing rough rolling. After several passes facilitated by reverse rolling, the slab is elongated to a length equal to 10 times its original value and is coiled in the coil box in preparation for finish rolling.  1.2 Thermomechanical Processing Thermomechanical processing refers to technologies aimed at controlling the microstructure and mechanical properties of steel and other metals, during deformation and accelerated cooling. Controlled rolling, controlled cooling and direct quenching are typical examples of such technologies. Such processing reduces the cost of steel manufacturing by minimizing or even eliminating heat treatment after hot deformation, thus increasing the productivity of high-grade steels.  3  1.3 Hot Deformation of Steel Slab during Rough Rolling  Controlled rolling has played an important role in the development of the high-strength low alloy steels (HSLA). It is a technique which produces strong and tough steels by refining the grain size. In controlled rolling, recrystallization during or immediately after hot rolling is controlled by the rolling conditions and by presence of microalloyed elements such as niobium, titanium and vanadium  [61 .  During cooling after controlled rolling, the transformation of the austenite (y) produces  fine ferrite (a) grains. Furthermore, the precipitation of carbides or nitrides or carbonitrides of the microalloying elements also contribute to the strengthening. Controlled cooling process have also been developed to produce small ferrite grain structures [61 . In recent research and development at Hoesch Stahl, Dortmund m , on HS LA steels, researchers claim that the rolling-induced scatter of the mechanical properties of thermomechanically rolled heavy plate and hot-strip are influenced by the rolling temperature ranges of the rough and finish rolling operations together with the amount of reduction and cooling conditions in the finishing mill. Thus, if the final structure and properties of a steel are to be predicted with greater confidence, then it is apparent that attention must be paid to both the roughing and finishing operations.  1.3 Hot Deformation of Steel Slab during Rough Rolling To predict the thermal gradient through the thickness of the slab and the overall evolution of microstructure during the production of strip, the deformation behavior of the slab during rolling needs to be studied. It is characterized by the velocity, strain rate and strain distribution as well as the temperature distribution; it is the combination of these parameters that determines whether dynamic recrystallization is initiated within the roll bite and also controls the rate of static recrystallization following a given pass.  4  1.3 Hot Deformation of Steel Slab during Rough Rolling  During rough rolling, large reduction results in non-uniform deformation in the flow domain. The non-uniform deformation and temperature distribution must lead to a variation in microstructure and mechanical properties. To gain insight into the deformation behavior of the slab, a coupled finite element method is often adopted, and the material is always assumed to behave as a rigid-plastic material during hot rolling. Due to the large ratio of width to thickness during slab rolling, spread in the transverse direction may be assumed to be negligible and conditions of plane strain prevail. This simplifies the analysis to two dimensions. Using the coupled finite element method, the thermal and deformation field including strain rate and strain distribution can be computed. Moreover, an accurate roll force prediction should be possible by means of the finite element analysis. This kind of analysis is also helpful for the design of a hot strip mill, the control of gauge and shape of the product and the final estimation of the mechanical properties.  5  CONTINUOUS CASTING J.K. Brimacombe I.V. Samarasekera o User friendly heat transfer model under development. o Input data - Heat flux data for moulds, Heat-transfer coefficient data for sprays gathered from extensive in-plant measurements over the past 15 years.  REHEAT FURNACE P.V. Barr J.K. Brimacombe A. Burgess o User friendly heat transfer model complete. o Predicts through thickness and transverse temperature distribution of slab including skidmarks for any furnace geometry and slab shape.  cr\  r.. ROUGHING MILL I.V. Samarasekera E.B. Hawbolt o Thermal and deformation models. o Prediction of microstructural evolution In HSLA steels.  FINISHING MILL I.V. Samarasekera E.B. Hawbolt o User friendly model to predict heat-transfer and through thickness temperature distribution, is complete. o Prediction of roll forces and microstructure development for C-Mn steels.  Figure 1.1 Mathematical models in thermomechanical processing  SLAB REHEATING FURNACE Furnace Discharge SLAB DESCALER  Scale Breaking  VERTICAL EDGER REVERSING ROUGHING MILL COIL BOX CROP-SHEAR  Initial Reduction of Slab to Bar  BAR DESCALER FINISHING MILLS COOLING BANK  Removal of Scale from Bar'  0  DOWN COILER Strip Cooling 0  Coil Discharge  0  ri) cr av  Figure 1.2 A typical layout of 1/2 continuous hot strip mill er  2.1.1 Heat Transfer during Rough Rolling  Chapter 2  LITERATURE REVIEW  To enhance the competitiveness of steel in relation to alternate materials, new technologies such as thermomechanical processing have been developed to improve mechanical properties and enhance the quality while reducing costs. To ensure that the specifications of final products are consistently met, operating parameters must be optimized. To increase the reliance on basic knowledge and to reduce dependence on empiricism, fundamental knowledge of the physical metallurgy of steel should be incorporated into mathematical models of the industrial process. The models should be validated through in-plant measurements and be capable of predicting the thermomechanical history and microstructure evolution during processing.  2.1 Thermal Analysis of the Rough Rolling Process 2.1.1 Heat Transfer during Rough Rolling During rolling, the slab loses heat energy by radiation and convection to air, by conduction due to contact with the support rollers, by convection to high pressure water sprays and by conduction to work rolls. Heat is generated due to deformation and friction at the roll/slab interface. The internal temperature distribution of the slab changes in response to these events.  8  2.1.1 Heat Transfer during Rough Rolling  2.1.1.1 Heat Transfer in the Slab Heat conduction in the slab is governed by the following equation: aT , )  VIcs (VT.,)+ 4s = ps Cps i-w  (2.1)  where T, is the temperature of slab; eq. s is the heat generation due to deformation and friction in the roll bite, it is zero outside the roll gap; and k„ p s and C ps are the thermal conductivity, density and specific heat of slab respectively. Miller adopted a hydraulic analogue technique to compute the one-dimensional temperature distribution through the thickness of a steel slab during hot rolling to plate, for both conventional and controlled rolling procedures. The method simulated the surface heat losses by radiation, convection to surroundings and conduction to mill rolls [81 . Several investigators have attempted to obtain an analytical solution by making simplifying as sumptionsE 9H 111 . Seredynski [91 assumed that the heat loss to the rolls is compensated by the heat gain due to deformation and assumed there is no temperature gradient in the slab. Pavlossoglou [w]1111 did not consider the heat loss due to water sprays and the heat generated by deformation in his analysis. In other literature, numerical analysis based on the finite difference method is widely used1121-117] . Hollander developed a mathematical model to predict the temperature distribution in steel slabs from the reheating furnace, during roughing, finish rolling, and during cooling on the run-out table, which was used to optimize the mill performance and to design a new hot strip mill. Yanagi [131 simulated the strip temperature in a strip mill from the reheating furnace to the coiler. Hatta et a1.f 141 utilized a heat transfer model to compare the slab temperature and the length of the rolling mill for different arrangements. They also calculated the steel temperature change from the 9  2.1.1 Heat Transfer during Rough Rolling  reheating furnace in rougher and finish rolling by means of a mathematical model for a full and a three quarters continuous type hot strip mills (see Section 1.1) to predict the rolling productivities t153 . In most of these models, the thermal resistance between the roll and the workpiece was ignored and conduction at the interface was computed by assuming that the roll and the workpiece are semi-infinite bodies. Devadas and Samarasekera r1631173 have developed a model to predict the temperature distribution of the strip in finish rolling taking into account roll chilling due to cold rolls by means of an experimentally determined interface heat transfer coefficient. The model was extended to predict the microstructure evolution of the finished product as a function of mill operating parameters.  2.1.1.2 Heat Losses during Rolling During the rough rolling process, the heat losses from the slab consist of three major components: radiation and convection to the environment, high pressure water descaling and roll chilling due to contact with cold rolls.  2.1.1.2.1 Radiative and Convective Heat Loss The slab loses heat by radiation and convection to the environment in the region between the reheating furnace and the first descaler, between passes, and from the last pass to the coil box. The radiation heat transfer obeys the Stefan-Boltzmann law as follows: q,. = S EA ((T + 273.1)4 — (T + 273.1) 4)  (2.2)  Where S is the Stefan-Boltzmann' s constant; the emissivity, e, may vary greatly depending on such factors as the amount of scale present on the slab surface, the presence of water vapor, smoke and  10  2.1.1 Heat Transfer during Rough Rolling  dust etc., and T s and T., denote the slab surface temperature and the ambient temperature in °C. Based on suitable mathematical analysis, SeredynsId 191 derived an expression of the emissivity of a hot plate as a function of temperature T s :  Ts.^T., e(Ts)= ^ (0.1251000 0.38)+1.1 1000  (2.3)  For the ease of computation, a pseudo heat transfer coefficient is adopted and expressed  as [16) :  kid = Se(T)  ((Ti + 273.1)4 — (T.,+ 273.1)4 ) (Ts—T.)  (2.4)  For convection, the heat transfer is governed by the equation below:  qs,,,,,,= A hs,,,,,,(Ts — T.,)^  (2.5)  In rough rolling, the mode of thermal convection to air is primarily natural convection because the speed of the slab on the transfer tables is relatively low 1131-1151 . Therefore, the influence of forced convection was neglected, since its value is much smaller than that of the natural convection  1151 .  Since heat removal by thermal convection to air is small compared with the total heat loss during rolling, the heat transfer coefficient, h,, for natural convection is fixed at 8.37 W/m 2-°C, as calculated from the equation 1251 :  1 Nu =0.15(Gr • Pr)'^ where Gr • Pr is between 8 x 10 6 and 10' 11 ' 31 for the upper surface of the warm plates.  11  (2.6)  2.1.1 Heat Transfer during Rough Rolling  2.1.1.2.2 High Pressure Water Descaling The high pressure water sprays are used before certain rolling passes to ensure removal of oxide scale which would adversely influence surface quality. The resulting water flow impinges on the slab and extracts heat. The heat transfer caused by water boiling at the surface of the moving slab cannot be easily defined analytica1ly [121 . However, Seredynskim used the heat balance method by examining relevant factors to set up an equation for the temperature change:  AT  '  30Wwp,,,{C„„(AT,, + s(85 AT„,))+sX} —  RNH0Cps/cps  (2.7)  where W is the rate of water flow per unit width of plate (m 2/min); p„, is the density of water (1000 kg/m3 ); Cry is the specific heat of water (4186 J/kg-°C); AT,„ is the temperature rise of cooling water (°C); s is the proportion of steam generated; X is the latent heat of water(J/kg); R is the radius of the rolls (m); Nis the rolling speed (rev/min) and Ho is the plate thickness (m). It is obvious that the inclusion of the proportion of water vapor, s, makes the formula inconvenient to use. Kokado and Hatta assigned a constant value of 1.163 kW/m2 -°C as a water jet descaling heat transfer coefficientf 13],[141, while Hollander 112' has shown that it varies from 12.5-20.9 kW/m 2-°C. Yanagi 1153 measured the surface temperature of undeformed bars being cooled by descaler sprays and found that the heat removed by the descaler spray could be calculated with an empirical equation developed on the basis of laboratory experiments, assuming that the spray was even over an area 1.0 (m) x w (m):  qs = 5.4 x 103 147"( T " r 54^(2.8) -  26  12  2.1.1 Heat Transfer during Rough Rolling  where W is the water flux(1/m2 -s) and qs is found to be independent of the slab surface temperature. Sasaki et a/. (181 developed a correlation based on measurements made at lower water fluxes and lower pressure: h =708W °35T; 11 + 0.116^  (2.9)  where 1.6 < W < 41.71 1/m 2 -s; 700 < T s < 1200 °C; 196 < p„, < 490 kPa. For the present study, the water jet pressure is approximately 13.72 MPa, which is the same as in the case studied by Devadas and S amarasekera 1161 . According to their investigation, the correlation of Eq.(2.9) was deemed suitable.  2.1.1.2.3 Roll Chilling During rolling, the work rolls heat up due to contact with the hot slab. Obviously, as the temperature increases, thermal stresses increase. Thermal cycling of the rolls causes failure of the roll surface by fatigue and thereby shortens the work roll life. To improve work roll life, water sprays are employed to keep the roll temperature low. In addition, lubrication is sometimes employed to reduce friction and to thereby reduce the roll force. Due to the low temperature of the work rolls, the slab surface is chilled during rolling. This effect will be considered in the present study by means of an interface heat transfer coefficient back-calculated from the CANMET pilot mill measurements.  2.1.1.3 Heat Generation during Rolling Apart from the internal heat content of a workpiece prior to rolling, heat may be acquired 1) by plastic deformation of the workpiece, 2) by frictional effects at the interface of the roll and slab, 3) by physical and metallurgical changes occurring in the workpiece during rolling.  13  2.1.1 Heat Transfer during Rough Rolling  Assuming that the plastic deformation in the roll bite is uniform, the temperature rise due to hot deformation can be derived by converting the mechanical work into heat energy, i.e.: ATdef = i  1 a . n( H0 ) 1 vi-^  j ros cps  (2.14)  where 1 is the heat generation efficiency factor from deformation work; for steel, it is in the range 0.80-0.85 E211 ; J is the mechanical equivalent of heat; a is the flow stress of the rolled materials with strong temperature dependence; H o and H 1 are the workpiece thickness at entry and exit respectively. With respect to the frictional heating effect in the roll bite, Hatta et al. [141 derived an expression:  qf = v r i.tfp^  (2.15)  where p is the pressure in the arc of contact and an average value may be calculated from the measured roll force. yr is the relative velocity between the slab and the work roll and varies along the arc of contact. It is expressed as t141 :  v r =R0)  1 H„ cos (i). H cos0  (2.16)  where o) is the roll speed (rpm); Hn and 9„ are the thickness and the contact angle at the neutral point where the roll speed is equal to that of the workpiece, whilst H and 9 are the thickness and the contact angle at some position in the roll gap. Attempts have been made by many workers to calculate the friction coefficient from the mill data. Denton and Crane t271 have predicted a value of 0.25 at 1000°C, increasing to 0.4 at 1100°C, while Roberts  rni  has proposed the following correlation  for unlubricated rolls: i.if = 4.86 x 10 -4 T, — 0.0714^  14  (2.17)  2.1.2 Heat Transfer in the Work Roll  The oxidation of the slab surface as a source of heat during the rolling process does not appear to have been discussed in the literature. However, the reaction is exothermic and the rate of heat generation depends on such parameters as the slab temperature, the type of oxide being formed, the thickness of the existing scale and the rate at which the oxide layer is developing. In the present study, it is assumed that it is small compared to the heat generation due to deformation and friction and is therefore neglected. Of the metallurgical phenomena occurring in the steel, the solid state reaction of most concern in hot rolling is that associated with the decomposition of austenite to ferrite and cementite. During rough rolling, the temperature of the slab is always higher than the transformation temperature except within the very thin surface layer in the roll bite. Thus it is assumed that this component can be ignored. 2.1.2 Heat Transfer in the Work Roll 2.1.2.1 Work Roll Cooling System In the rolling process, the slab is deformed by rotating rolls which are at a lower temperature. As a consequence, a large amount of heat is transferred from the slab to the work rolls during contact. To improve roll life and to achieve high surface quality, the surface finish and profile of the rolls must be closely controlled. Roll wear or spalling, which may occur due to thermal/mechanical fatigue, can be greatly affected by roll cooling. Improper or insufficient cooling causes large thermal gradients near the roll surface resulting in thermal stresses causing surface spalling. Therefore, designing a good roll cooling system has been a concern of mill designers and operators.  15  2.1.2 Heat Transfer in the Work Roll  In the steel industry, water is generally employed as the cooling medium. For cooling and lubrication of rolls, water spray cooling systems are generally used, as shown in Fig.2.1.  cooling spray  vs Figure 2.1 Spray cooling of work roll during rolling process Numerous types and sizes of spray nozzles are commercially available. For roll cooling applications, a flat, fan-shaped spray pattern is usually preferredt 191 .  2.1.2.2 Heat Transfer in the Work Roll Use of sprays or other systems to cool the roll is a complicated process involving jet impingement on a rotating surface. The roll speed often exceeds the coolant velocity and strongly influences the flow pattern developed. The surface temperature when it emerges from the roll bite is always higher than the saturation temperature of the water, and boiling occurs. The boiling may be in nucleate, transition or film boiling regions, depending on the surface temperature" ) . The heat removal from the roll surface can be expressed as:  = hr,(Tr —Toj ni^(2.10)  16  2.1.2 Heat Transfer in the Work Roll  The heat transfer coefficient, h,, and the exponent, n 1 , are dependent on the cooling pattern on the roll surface'', and may be affected by many relevant parameters. These include: 1) the water properties, such as the density (p,), the dynamic viscosity (N) as well as the density and dynamic viscosity of the water vapor, and the surface tension (o w ); 2) the thermal properties of the water and its vapor phase, such as the thermal conductivity (k„,), the specific heat (C PS,), the water temperature (TO and its saturation temperature (Ts .); 3) the geometric properties: the nozzle hydraulic diameter (d), the roll diameter (D), the spacing between nozzles (b), the distance between the nozzle and the  roll surface (1), and the impingement angle ((3); and 4) the operating parameters: the spray velocity (vw), the rotating speed (a)), and the surface temperature, Tr'. Many investigators have attempted to quantify the spray heat transfer coefficient' 221, t231 . In a survey by Tseng et al. [223 the heat transfer coefficient is reported to be in the range 1 to 1 lkW/m 2-°C ,  when the roll surface temperature is below 100°C (without boiling) and 6 to 40 kW/m 2-°C for temperatures above 100°C (with boiling). Hogshead (231 reported an average heat transfer coefficient of 2 to 5 KW/m 2 -°C, for roll surface temperature in the range of 88°C to 115°C; he investigated the effect of rotating speeds (0 to 1800 rpm) and spray rates (0.126 to 1.262 L's). In an attempt to study local heat transfer coefficients, Tseng et a/. 122I reported that the maximum heat transfer coefficient for a jet using city water was found to be 6 kW/m 2-°C directly in the impingement zone and 1 to 2 kW/m2-°C away from the impingement zone. Some heat transfer coefficient correlations have also been developed for the cooling of rolls with spray of water 1241-E261 . In the regions of impingement of spray water on the roll surface, a correlation between the heat transfer coefficient and water flux has been determined by Yamaguchi et al .[221 . el,.  1.291 x 10 5 147 "21^(2.11)  17  2.12 Heat Transfer in the Work Roll  This correlation is based on experimental measurements of the thermal response of a heated plate to spray cooling. The plate temperature was in the range of 100-400°C and the water flux varied from 5 x 10 3 to 5 x 104 1/m2-min In the regions just below the zone of spray water impingement, the surface of the rolls are covered with a film of water streaming down from the above spray zone. Depending on the surface temperature of the rolls, the heat transfer modes fall into three categories: natural convection when the surface temperature is less than the saturation temperature; nucleate boiling when the temperature is between the saturation temperature and the critical heat flux temperature; and unstable film boiling when the temperature is higher than the critical heat flux temperature 1161 . For the natural convection, the following correlation was adopted f253 :  D  f  h =(—)i 0.1 1 [(0.5Re 2 + GrD )Pr] 0.35}  (2.12)  For nucleate boiling, Rohsenow' si 261 correlation was employed: =  w hfg (  S (p w – p ) 0.5 (Y„^  x  C pwAT  x  C ehhP rf`  (2.13)  where n=1 for water. For the heat transfer coefficient of unstable film boiling, Nukiyama' s1261 boiling heat transfer coefficient data were interpolated for the present study.  18  2.13 Heat Transfer Characterization at the Roll/Slab Interface  2.1.3 Heat Transfer Characterization at the Roll/Slab Interface Owing to the paucity of data on the heat transfer coefficient at the interface of the work roll and slab, many investigators have assumed perfect thermal contact 19111311151191 , which is clearly not the case. In Yanagi's [131 model, because the thermal resistance had been calculated for rolling to be of the order of 0.86 x 10 -5m2 °C/W, he treated the thermal resistance between the roll surface and workpiece as negligible and utilizes the semi-infinite model for both the roll and the strip. Hatta et a/." 411151 ignored the heat loss from the slab to the work roll because the region affected by the roll chilling was considered to be confined to the slab surface layer. Stevens et am have estimated that the heat transfer coefficient at the roll-gap interface was 37.6 kW/m 2 -°C during the first t=30 ms in the arc of contact and 18 kW/m2-°C thereafter in the range e= t + 0.094 seconds. Murata et a1. 1211 have back-calculated the heat transfer coefficient that would apply to hot rolling under a variety of conditions by measuring the surface temperatures of a specimen heated to 780°C and brought in contact with a low temperature specimen at 22-30°C during compression with a constant pressure of 5 kg/mm2. The results show that the lubricants affect the value of the heat transfer coefficient and the existence of oxide scale decreases the heat transfer coefficient. In Stevens et al.'s model, they measured the roll sub-surface temperatures and estimated the surface temperature. They found that the equation: T:+1.  h  T: + (T: —T:)( ^ ' cr x {1 — exp(tiOerfc(A 1 4t51 A i k,. -  (2.14)  where: A^(k,.-4cc+ks-4(Tr) k1c, r  19  (2.15)  2.13 Heat Transfer Characterization at the Roll/Slab Interface  gave a good agreement with their experimental data if the resistance of the insulating layer were characterized by two different values of surface heat transfer coefficients, h gap and h'gap , within the roll bite where a change from h g„i, to h ' gap occurred after a given time of contact, 30 ms. They postulated that the cause was the phase transformation that occurs at the surface of the strip as its temperature was depressed due to contact with the roll. This, in turn, increases the surface roughness of the strip with an increase in thermal resistance. Sellars 1291 pointed out that the determination of this heat transfer coefficient between the roll surface and slab varied considerably from slab to slab, presumably because of variations in the surface oxide condition. On the assumption that the oxide scale thickness was reduced during rolling in proportion to slab thickness, the model allowed the heat transfer coefficient to increase proportionally. It was found that a unique value of the coefficient could be applied to the different passes in the multipass experimental rolling schedule. In his model, the value of heat transfer coefficient was determined by trial-and-error by comparing the experimental and predicted temperature at the center of the slab, and a mean heat transfer value of 200 kW/m 2-°C was obtained for rolling a 19 mm thick slab of type 304 stainless steel in three passes. Obviously, thermal contact resistance exists in the heat transfer between the roll surface and slab during hot rolling because a seemingly smooth surface viewed with various degrees of magnification reveals a series of rather randomly spaced hills and valleys, as shown in Fig.2.2 [311 , where the hills and valleys have been exaggerated for clarity. Fenech, Henry and Rohsenow 1301 developed a model with heating one end and cooling the other of the two different contacting materials resulting in nearly unidirectional heat flow in the region away from the contacting surface, as shown in Fig.2.3, with: q , dT v dT — = K l — = K2  A^cbci^cbc2  -  20  (2.16)  2.1.3 Heat Transfer Characterization at the Roll/Slab Interface  Tool  \ \ \ \ \ \ \\\  Workpiece  Figure 2.2 Contact between real surfacesi m i  ATc  M.-  T  Figure 2.3 Temperature distribution through surface in contact t301  21  ^  2.1.3 Heat Transfer Characterization at the Roll/Slab Interface  A pseudo contact heat transfer coefficient he was defined as hc  (qIA)  — AT  (2.17)  There is no localized interface contact resistance but rather a region of influence in the neighborhood of the contact. Clearly, increasing the number of locations of contact per unit area would change the shape of the flow-line significantly, even though the fraction of area in contact, C A , and the roughness of the surface remained the same.  SI 62  n  Cross  t  Section  czy'cj. ;F)^..:, O — 0 .  View  ,:Ds47 ,4 ,  ^ ^(a) ^ Actual Model  0 0 0 0 0 0 0 0 0 0 0  (b)  ^  Button Model  (c)  ^  Heat Channel for Button Model  Figure 2.4 Fenech et al.'s heat transfer model for thermal contact (30I In their idealized mathematical model, shown in Fig.2.4, they considered the steady state heat conduction: V 27' = 0  ^  (2.18)  and proposed that the heat transfer coefficient is of the form: .f(ki ,k2 ,kf,5 1 ,52 , n,eA  22  ^ )  (2.19)  2.1.3 Heat Transfer Characterization at the Roll/Slab Interface  and expressed as: 1  (2.20)  h, ^  d2/k2  The unknown d 1 and d 2 are obtained by solving for the constants with boundary conditions. Therefore, the general heat transfer coefficient was expressed as k  f [ (1 8 +2 ,  h,  _ED  4.26 ,1W  —  eA  + 1^4,26 ,171 =+1  \  1  -  1.16A( IT 1  k,^k2  _ e,2  4)  [ 1 _ kr^  (^  81+82^  )] l  [3°1  4.264W-e-IA-+ 1  . 1)  -  4.26EA4 72  k2  —  4264;1 + 1  (2.21)  eA  For contacts under heavy loading where the asperities would be deformed or contacts in rarified atmospheres, the conductance through the fluid in the void may be neglected, so that the heat transfer coefficient is:  h, — 81  .46(24 )03 + 82 +  (2.22)  7  where lc, can be expressed as: -k- = i +12- J. (  With the application of the model to noncrystalline materials, Moore s] observed that when two surfaces of different metals are pressed together, the irregularities of the softer surface undergo full plastic deformation while the peaks of the harder metal are embedded in the other surface. If p a is the apparent pressure on the contact, the total load, 1 3,, is then given by P.= p aA^  23  (2.23)  ^ ^  2.1.3 Heat Transfer Characterization at the Roll-Slab Interface  The measured values of tic vs. pressure are the circles shown in Fig.2.5, where the vertical lines represent the estimated limits of precision in the data and the various dots represent calculated values of h utilizing Eq. 2.22. ,,  Apparent Pressure Pa (kg/ma x 10) 0.7^7.0^70^700 1^u I I I WI^I I I 1I I L. 567.8 1 r 1 ill il^ 10 5 :^  _^  ....  o Experiment h e • Calculated h e —Best line through the calculated points  10 4  56.78  C)  •  5.678  Armco—iron/ aluminum  io2-  11111111^I^I^111111^  10^102^103  0.5678  Apparent pressure Pa(psi)  Figure 2.5 Calculated and measured h e vs. apparent pressurem The agreement between calculated and measured results in Fig.2.5 is remarkably good and the same excellent agreement has been repeated in numerous additional tests t301 . In the case of surfaces in relative motion, the limited number of reported measurements described above have yielded widely different values and a lack of fundamental understanding about the heat transfer at the roll-slab interface has preluded rationalization of these differences. Samarasekera (mi pointed out that the contact points between the two surface offer a markedly lower resistance path for heat flow in comparison to adjacent regions where heat transfer occurs by conduction through air gaps, the dimensions of which depend on the surface topgraphy of the roll  24  2.1.3 Heat Transfer Characterization at the Roll/Slab Interface  and strip. Therefore it is proposed that a major fraction of the heat is transferred via the contacting points. Unlike the case of two flat surfaces under compression, friction exists at the roll/strip interface due to sliding between the roll and the strip. It has also been proposed that the link between the friction and heat transfer at the interface is the fraction of the total area that is in direct contact. Wanheim and Bay 1321 have postulated that the real contact area depends on both the interfacial pressure and the shear strength in the real contact zone as follows: p  mceAk 1-tf  (2.24)  where me is an experimental constant within the range of 0 to 1; k is the shear strength for the deformed material; 1.4 is the frictional coefficient at the interface. Wilson and co-workers  m i  [33 34  studied the influence of bulk strain and relative velocity between the contacting surfaces on the fractional area of contact and showed that the fractional contact area increases monotonically with increasing bulk strain whilst there is decrease in the tendency for the workpiece and tools to conform with increasing relative speed. S araarasekeran showed that the variation in heat-transfer coefficient with reduction, rolling speed and lubrication observed through pilot mill tests on 316L stainless steel could be explained on the basis of the influence of these rolling parameters on fractional contact area. A linear relationship was shown to exist between the heat-transfer coefficient and mean roll pressure for two passes pointing to a strong analogy with friction.  25  2.1.3 Heat Transfer Characterization at the Roll-Slab Interface 200  U •  160  120  z u. 0 80  LI  1.4J U. (two Th.ckness (mm)  cc 40  — 25 5 --- 12 7  0^ 10  Reduction (•/..) 50 35  20^30^40 MEAN PRESSURE ( kWrnm z )  Figure 2.6 The relationship between the roll gap heat-transfer coefficient and the mean pressure along the arc of contact for two successive passes on the pilot millf 311 The results of this work are shown in Fig.2.6 from which it is evident that for the first pass the heat-transfer coefficient increases linearly with increasing pressure up to 15 kg/mm2, but deviates thereafter and approaches an equilibrium value of 57 kW/m 2-°C. For the second pass, there is a definite linear relationship between the interfacial heat-transfer coefficient and pressure and the slope of the line is remarkably similar to that of the linear portion of the first curve. This indicates that the interfacial heat transfer coefficient is strongly dependent on the real area of contact and consequently that the primary mode of heat transfer must be conduction across contacting asperities. However, further work is necessary to confirm these preliminary findings and to explore the applicability of the relationships determined to other materials such as C-Mn and microalloyed steels, where oxidation is likely to occur.  26  2.2.1 Oxidation Mechanisms of Metals at High Temperature  2.2 Oxidation of Steels at High Temperature 2.2.1 Oxidation Mechanisms of Metals at High Temperature Few metals are stable when exposed to the atmosphere at high temperature and are consequently oxidized. From consideration of the equation M(s)+0 (g)= MO(s)  22  (2.25)  it is obvious that the solid reaction product MO will separate the two reactants as shown below M(Metal)1 MO (Oxide)I 0 2 (Gas)  In order for the reaction to proceed further, one or both reactants must penetrate the scale, i.e. either metal must be transported through the oxide to the oxide-gas interface and react there, or oxygen must be transported to the oxide-metal interface and react there. Therefore, the mechanisms by which the reactants may penetrate the oxide layer are seen to be an important part of the mechanism by which high temperature oxidation occurs t361 . For iron or steel, the consequence of oxidation is the formation of a multi-layer scale at the surface, i.e., Fe/Fe0/Fe 3 04/Fe2 03/02 , as schematically shown in Fig.2.7.  27  ^  2.2.2 Iron Oxide Scale Growth in Air  0  2  Fe  4041 1,^ ^1, ^  44  2  0  3  Fe 3 04 FeO  4  Figure 2.7 Schematic diagram of multi-layer of iron oxide scale  2.2.2 Iron Oxide Scale Growth in Air Wagner 1373 developed a theory (the Wagner Theory) to describe high temperature oxidation of metals. In fact, the theory describes the oxidation under highly idealized conditions. In his theory, the following assumptions were made: The oxide layer is a compact, perfectly adherent scale; Migration of ions or electrons across the scale is the rate controlling process; Thermodynamic equilibrium is established at both the metal-scale and scale-gas interfaces; The oxide scale shows only small deviations from stoichiometry; Thermodynamic equilibrium is established locally throughout the scale;  28  2.2.2 Iron Oxide Scale Growth in Air  (6)  The scale is thick compared with the distance over which space charge effects (electrical double layer) occur;  (7)  Oxygen solubility in the metal may be neglected. Since thermodynamic equilibrium is assumed to be established at the metal-scale and scale-gas  interfaces, it follows that activity gradients of both metals and non metals (such as oxygen, etc) are established across the scale. Consequently, metal ions and oxygen ions will tend to migrate across the scale in opposite directions. Because the ions are charged, this migration will cause an electric field to be set up across the scale resulting in consequent transport of electrons from the metal to the atmosphere. The relative migration rates of cations, anions, and electrons are therefore balanced such that no net charge transfer occurs across the oxide layer as a result of ionic migration. Based on the above assumptions, the parabolic rate law was applied to describe the growth of oxide scale:  dt x  ^  (2.26)  where xis the oxide scale thickness and lc; is the parabolic rate constant and can be derived according to the cationic flux, which may also be expressed by: j,  dx dt  m—  (2.27)  where C. is the concentration of metal in the oxide scale. The simultaneous formation of wustite, magnetite, and hematite during the oxidation of iron in the range of 700 to 1250 °C has been studied extensively by Paidassi [391 . From the study, the  29  2.2.2 Iron Oxide Scale Growth in Air growth of each of the three oxides and the overall scale was found to follow the parabolic rate law. The thickness ratio of the oxides appears to be independent of time. Paidassir  391 deduced  the parabolic  rate constant of the three oxides, kp ' , from metallographic measurements individually as follows growth of wustite: -1 kP '(F e0) = 2.88 e -405®/RT (cm 2s )  (2.28)  growth rate of magnetite: 3e -405MRT(cm 2s-1)  (2.29)  kp Ve20 3 ) = 2.70 x 10-4e -4°5°CiIRT (cm 2s -1 )  (2.30)  ki:(Fe3 04 ) = 5.25 x 10growth rate of hematite:  The total growth rate: ki:(Total)  = 3.05 e -405®/RT (cm 2S -1 )  (2.31)  Fig.2.8 summarizes the experimental results of Paidassi for relative thickness of FeO, Fe 3 04 , and Fe2 0 3 at different temperatures.  30  2.2.2 Iron Oxide Scale Growth in Air  6  202  5  0  700^800^900^1000^1100^1200^1300  TEMPERATURE, ° C  Figure 2.8 Relative thickness of magnetite (Fe 3 04 ) and hematite (Fe2 0 3 ) Zhadan et al.E 381 provided a procedure for calculating decarburization and scaling during hot rolling of carbon steel. The thickness of scale layer, x, grows according to a parabolic law as the time of isothermal holding, t, increases:  x = (20) 2  (2.32)  where le e is the kinetic constant of scaling _6 lc: =7 .1 x10 exp  138.27 x 103) (m2s RT  (2.33)  T is the temperature of the metal in °K and R is the universal gas constant (8.32 J mol l °K-1 ). A theory for the growth of a three-layered scale on a pure metal has been presented by Hsu m ,  together with an application of the theory to the oxidation of iron at 800°C to 1200°C in which Fe i _ x0, Fe3 04 and Fe20 3 are formed simultaneously m . According to the theory, the growth of  31  2.2.2 Iron Oxide Scale Growth in Air  wustite with simultaneous formation of magnetite and hematite on iron is predominantly controlled by bulk diffusion of iron cations in the wustite. The oxygen diffusion in wustite during the oxidation of iron at 800°C to 1200°C is negligible; the growth of magnetite with simultaneous growth of wustite and hematite is predominantly controlled by cation diffusion through vacancies and interstitial sites in magnetite; and the growth of hematite cannot be explained only by bulk diffusivity of cation and anion determined from single-crystal Fe 20 3 . Ionic transport through short-circuit diffusion paths such as grain boundaries, dislocation pipes, and other flaws in hematite is predominantly controlling the growth of Fe2 03 . The oxidation of steels during reheating has been investigated by several researchers 1411-(431 . Obaro 14t ' pointed out that the multi-layer oxide consists of an outer layer of Fe 203 followed by a layer of Fe 3 04 and an inner layer of FeO close to the metal surface. Below 570°C, the scale consists essentially of Fe3 04 covered by a thin layer of a- and y-Fe 2 03 . Above 570 °C, the scale consists essentially of FeO with only a thin layer consisting of Fe 3 04 and Fe 20 3 . Within the scale layer, the concentration of the metal decreases from the inside to outside. The growth of the sublayers follows the parabolic growth law and is proportional to the relative thickness of the three oxides depending on the diffusion rates through the layers, the gradient of the chemical potentials across the layers and the relative porosity of the layers. In Ormerod et al.' sE433 investigation, the influences of process variables (steel chemistry variables) and their interaction on scale formation on steel in reheating furnace systems were examined and illustrated with data drawn from other literature sources. The model employed to provide the simulation results for steels is noted below:  x = lad" + Zoci  +  (2.34)  where x denotes the scale thickness in mm; 8 = 8(T), is a function of temperature; ti = t( Comp.,  t), is a function of composition and time t; I' = F(comp., t), is a function of composition and time  32  2.2.3 Effect of Oxide Scale on Heat Transfer  t, ai and f34  are also functions of composition and time  t, where ai  denotes the influence of single  elements while 13 4 denotes the interactive effects of pairs of elements. With the above equation, it is possible to compute the scale growth in air.  2.2.3 Effect of Oxide Scale on Heat Transfer Although a small amount of scale formation in the vicinity of 0.1% is advantageous in the reheating furnace because such scaling removes minor surface defects like oscillation marks present in continuously cast products m , excessive scale formation represents a yield loss, influences the heat transfer at the surface and has an adverse influence on product quality. Mathematical modelling was used to analyze the scale formation rates for conditions of heating carbon steels in continuous pusher type furnacesm. The rate of growth of scale was determined by solving the differential equation: dx.  ko  B  dt  2x exP  (I')  (2.35)  with initial condition:  x(t) 1 , .0= x0  (2.36)  where t is the heating time, hr; Tc,„ is the mass average temperature of the scale in °K; x o is the initial thickness of the scale in m;  B and ko are empirical coefficients characterizing the kinetics of oxidation  of the steel in 7C and m2 /hr respectively. The change in the radiation characteristics of the scale during heating of the metal is shown in Fig 2.9.  33  2.2.3 Effect of Oxide Scale on Heat Transfer  Figure 2.9 Emissivity of oxide scale during metal heating '1'- <0.4 hr; '2'- 1.2 hr; '3'- 1.4 hr, '4'- 2.1 hr, '5'- > 2.4 hr where eA. denotes the emisivity of the scale and X is the wavelength. From the figure, a fall in the spectral emisivity of the scale in the shortwave range of the spectrum was observed during metal heating. In their study, the incident heat flow on the metal was chosen as the analyzed parameter of external heat transfer, since its formation is determined in many ways by the magnitude and nature of distribution of thermal loads in the furnace working chamber. The incident heat flow density, q.,„ from steel oxidation was defined by the expression: dx  q. = Q.P. d7 where  Q0  (2.37)  is the amount of heat released during the formation of 1 kilogram of oxides in kJ; p ox is  the density of the scale in kg/m3 . For a study of the influence of a reduction in scale emissivity in the course of metal heating on the intensity of heat transfer, dependences q=f(t) were determined for different conditions and the results are shown in Fig.2.10.  34  2.2.3 Effect of Oxide Scale on Heat Transfer  q,kW -  -  415 /  350  /  /  ..q.... ^  ■  ,..........-•■  N,,  ,/  ...  ,  250 150 0  0^2,5^43,0 u l ti  0,5  -  Figure 2.10 Intensity of heat transfer through oxide scale during reheating '1': heating with real values of scale emissivity and thickness; '2': heating with constant values of scale emissivity; '3': heating with constant values of scale emissivity and thickness Curve 1 corresponds to real values of scale emissivity and thickness; curve 2 corresponds to the initial stage of heating of metals in the furnace with an emissivity of 0.90-0.93 in the 1-15 µm wavelength range with constant scale emissivity but increasing scale thickness; and curve 3 corresponds to the constant scale emissivity and constant scale thickness of initial value. The results show that with a constantly high surface emissivity of the heated metal the heat treatment schedule is achieved with lower densities of the incident heat flow. In this case, a reduction in &A, of the scale has the greatest effect on the magnitude of qox with a heating time of 1.3-1.4 hours. According to Fig 2.9, the scale emissivity begins to fall rapidly in the shortwave region of the spectrum. Fig.2.11 shows the dependence of the temperature at different positions through slab thickness on heating time.  35  2.2.3 Effect of Oxide Scale on Heat Transfer  0^1,0^7,0  ^  40 1,h  Figure 2.11 Dependence of temperature through thickness on heating time 1- surface of scale; 2- upper surface of metal; 3- low surface of metal; 4- center of metal The figure shows that, during slab heating in the furnace, the temperature difference through the thickness of the scale layer reaches its maximum magnitude (about 90°C) in the 1.3 hr-1.4 hr time interval, which corresponds to the end of the first heating zone. Subsequently, AT decreases as heating continues, and before discharge from the furnace, it amounts to about 10°C. Very few studies have been conducted on the effect of scale formation on heat transfer of metals during hot rolling. Hollander {121 pointed out that the scale thickness before first descaler was in the range 1.5 - 3.0mm and before the second descaler was 100-150 pm and at the third descaler was 10-15 pm, which are all dependent on surface temperature and time. In the model of hot rolling, Hollander adopted a constant scale thickness, although its effect on the temperature distribution through the slab was not presented.  36  2.3.1 Governing Equations for Finite Element Analysis  2.3 Finite Element Analysis on Hot Deformation As the application of computer-aided techniques (computer-aided design, manufacturing, and engineering, in short for CAD/CAM/CAE) in the metal forming industry increases considerably, process simulation and/or modelling for the investigation and understanding of deformation mechanics has become a major area of research. The finite element method (FEM) is playing an important role in modelling forming processes.  2.3.1 Governing Equations for Finite Element Analysis The governing equation for the finite element analysis can be obtained from the virtual work principle which is simply an alternative statement of the equilibrium conditions. The virtual work principle states: "In a system for which internal forces (stresses) and external applied forces are in equilibrium, the application of any (virtual) system of displacement and corresponding internal strain compatible with it, results in equality of external and internal work". There are two formulations, namely, flow formulation and solid formulation, which are widely used in metal forming. Flow formulation assumes that the deforming material has a negligible elastic response, while solid formulation includes elasticity. Despite the recent advances, the application of the solid formulation to metal forming processes is limited. In the analysis of metal forming, plastic strains are usually much larger than elastic strains and the flow stress is much dependent on the strain rate which characterize the liquid flow. For this reason the assumption of rigid-plastic or rigid-viscoplastic behavior of material for hot deformation is acceptable. As a consequence, the flow formulation is widely used in metal forming process, especially in the hot deformation process. In the flow formulation, the current, real (Cauchy), stress of deformed body a is related to the rate of deformation, or strain rate, i f45j.  37  2.3.1 Governing Equations for Finite Element Analysis  ^G=Gi^ where the matrix G may be dependent on total strain invariants (say  e,  (2.38)  temperature T and indeed  the rate of strain i itself). Thus G =G(i,i,T)^  (2.39)  For incompressible flow, the rate of the volumetric straining is zero, i.e. e t, = ix + i), + iz = [m]T fil = o^(2.40) where  ^[m]T  =  [1,  1,  1,^o, o, o]  ^For such fluids, the mean stress a m , (cy..  ax +Cry + az  3^  ^  (2.41)  — p ), is not defined and has to be sought  from equilibrium relations. The deviatoric stress, a':  101 = {G} -{Gm } = {(5} + [A1]/9  (2.42)  {(0 = [D] fil  (2.43)  For a linear fluid we can write:  where  38  2.3.1 Governing Equations for Finite Element Analysis  2 O O [D] = p, 0 O O  0 2 0 0 0 0  0 0 0 00 0 0 0 2 0 0 0 =1.4D^(2.44) 0 2 0 0 0 0 2 0 0 0 0 2_  where g is viscosity and [D ° ] is a diagonal matrix. For non-Newtonian fluids, this viscosity is variable and dependent on {i} and temperature. The virtual principle can be also applied if we replace displacements by velocities and strain rates, and the equilibrium of a specified mass by an arbitrary surface can be considered at an instant of time. Thus we can write: L{E•i} T {a}dV — {51, } T {F}dV —{8v} T {INS = 0^(2.45) st  where {8v} and {SO are virtual velocity and strain rate changes in flow domain Vin which tractions {t} are specified on the boundary S, and {F) are volumetric force in the flow domain V, and where {8v} is zero on boundary S where velocities are givenE 461 . y  For compatibility, Ey  = (vi +  ^  (2.46)  which can also be expressed as matrix form: {SO = [L] {SO  ^  (2.47)  For any pressure variation Sp, internal work is zero due to incompressibility, so 8pivdV =0^  39  (2.48)  2.3.1 Governing Equations for Finite Element Analysis  Inserting constitutive equations (2.42) to (2.44), we can rewrite equation (2.45) as:  L {450 T  IADIfeldV+ f SivpdV — f {8v } T {F}dV — 1{8v} T {i}dS = 0^(2.49) v^v^s,  Pressure is also a variable as well as velocities in the above formulation. Thus the formulation described above is referred to as the velocity-pressure approach. Besides the velocity-pressure approach in the flow formulation, there is another approach called Direct Penalty Form (or Penalty Function Approach) which is also widely used. The governing equation for this approach is derived according to a variational principle which is equivalent to the virtual work statements subject to constraint: iv =M T  {0 =0  ^  (2.50)  The functional is defined as:  cb=fv {i}T,,,D0, fildV — fv {v} T {F}dV — Is {v } T {7}dS^(2.51) It is possible to modify the functional in a variety of ways to introduce the constraint (2.50). A penalty [471 is imposed on the integral of the square of the error, and this is multiplied by a positive large number K. Thus we can enforce the constraint by minimizing: 43=c1)+K I i,icIV =0+-1 5 iTUMPK[Alf)e.dV^(2.52) ^v v  If we replace [D] by [Dlin Eq. (2.51), where [D1= pt.[D1+[M]2K[M] T then we can easily see that: CD = vf {i} T [D1 {i}dV — fv {v} T {F}dV — sii {v} T {t}CIS^(2.53)  40  2.3.2 Application of Finite Element Analysis in Metal Forming  This functional is similar to the Eq.(2.51) and can be minimized by making its first-order variation be zero. Obviously the variable pressure has been eliminated in the Penalty Function Approach. This makes computation more effective, but the penalty form can become ill-conditioned if K is very large; however, no difficulties have been encountered using values 1451 : K = 107-1°11  (2.54)  The governing equations (2.49) or Eq.(2.53) can be solved by discretization with velocities and pressure as basic variables.  2.3.2 Application of Finite Element Analysis in Metal Forming The application of the finite-element method to metal-forming problems began as an extension of structural analysis technique to the plastic deformation regime. It was first introduced by Zienkiewicz t491 , Dawsonf 50I and Li and Kobayashit511 et al. to the metal forming field. To date, many investigators have analyzed deformation of metals r45H561 by the finite-element method because it can yield a more accurate prediction of strain and strain rate distribution during deformation. The most important development in the application of FEM to the deformation analysis is the inclusion of the effects of strain rate and temperature on the mechanical properties and of thermal coupling in the solution. Li and Kobayashi t511 analyzed metal deformation in the roll bite under plain-strain conditions at room temperature by the rigid-plastic finite element method. They reported that homogeneity of deformation in the roll bite is influenced by the reduction per pass and the roll gap geometry. They observed single and double peaks in the pressure distribution along the arc of contact at high and low reductions respectively. An elasto-viscoplastic approach has been adopted by Grober t521 for hot rolling, and he has shown that strain inhomogeneity increases with increasing reduction and higher coefficients of friction. Zienkiewici 491 and Dawsont 501 on the other hand developed models 41  2.3.2 Application of Finite Element Analysis in Metal Forming  using flow formulation assuming viscoplastic materials behavior in their analysis. Jaint 471 adopted the flow formulation method to analyze several metal deformation processes, such as extrusion and flat rolling, utilizing both the velocity-pressure approach and the penalty-function approach. Compared to the penalty solution, the velocity-pressure solution yields better results for the same mesh. On the other hand, since a larger number of variables (velocities and pressure) are involved, the computational cost is more in the latter than in the former. Furthermore the pressure distribution obtained by the velocity-pressure approach in some cases is poor, so that some sort of upwinding technique has to be adopted to get a smooth variation. For coupled problems, realistic thermal boundary conditions have not been incorporated at the roll/slab interface because of paucity of data describing the heat transfer phenomenon at this interface. There is, in fact, a roll chilling effect during hot rolling" 71 , and about 38% of the total heat energy in the slab is extracted by the work rolls [311 , so that a large temperature gradient is established near the surface due to the chilling affect. The magnitude of the gradient depends on the heat transfer coefficient at the interface, which may be back-calculated from the measured surface temperature. Ignoring this effect, obviously tends to overestimate the surface temperature and underestimate the local deformation resistance. More recently, some investigators have incorporated roll chilling in deformation analysis (5311551 . Beynon [533 presented a coupled Eulerian finite-element method to analyze the deformation and temperature distribution in aluminium specimens during hot rolling. They computed the effective strain from the instantaneous effective strain rate using the Petrov-Galerkin method. They have shown that the strain over a region at the center of the slab approaches the nominal true strain corresponding to the applied reduction and increases significantly above the nominal value near the surface owing to the redundant strain required to force the material to be rolled into the roll gap. Silvonen r541 et al. undertook a similar analysis for laboratory rolling of steel  42  2.3.2 Application of Finite Element Analysis in Metal Forming  and demonstrated that the maximum values of strain rate were obtained near the entry and exit adjacent to the roll, with a rigid zone retained between these two zones. Pietrzky and Lenard[551 's recent analysis of a low speed laboratory rolling operation is also in agreement with the work of Silvonen et al. E541  .  Dawson (561 has taken the finite element analysis a step further and shown that the method can be adopted to predict mechanical property changes, namely hardness, by setting up a constitutive model for plastic deformations of the workpiece, during the flat rolling of aluminum. No similar work has been done on steel rolling and very few models have been directly applied to examine deformation during hot rolling of steel under industrial conditions.  43  3.1 Objectives and Scope  Chapter 3  SCOPE AND OBJECTIVES  In order to improve the final mechanical properties of rolled products, its deformation history has to be controlled. This is difficult at best because the mechanical properties are dependent on many factors, such as temperature, strain, strain rate and composition. To determine the thermomechanical history during rolling mathematical models are useful. 3.1 Objectives and Scope From the literature review in Chapter 2, it appears that there are real merits in modelling the hot rolling process and this study attempts to address this, by undertaking the following: (1)  To characterize the heat transfer at the interface between the work roll and the rolled slab during hot rolling;  (2)  To develop a two-dimensional mathematical model to predict the temperature distribution of a slab during rough rolling;  (3)  To analyze the effect of oxide scale on the heat transfer of slab during rough rolling;  (4)  To modify the existing finishing mill model of deformation based on the finite element method to describe the deformation in the roll bite;  (5) To validate the heat transfer model with the results from other models;  44  3.2 Methodology  (6) To validate the deformation model by comparing the measured roll forces and the strain distribution with model prediction.  3.2 Methodology The thermal history of a slab during rough rolling will be determined by adopting a finite difference method which has been widely used to predict the temperature distribution in steel slabs during rolling. The parabolic law of oxide scale growth has been incorporated to investigate the effect of scale on the thermal history. To validate the model, the results will be compared with those from another model which has been validated f171 . Since it has been estimated that nearly 38% of the total heat in the slab is extracted by the work roll 311 , the heat transfer at the roll/slab interface was characterized by pilot mill tests at CANMET and at UBC. The heat transfer coefficient in the roll bite was back-calculated from the measured surface temperature of the specimen. This data has been employed in the model to calculate the effect of roll chilling. It is necessary to know the deformation behavior of a slab during rough rolling to predict the evolution of microstructure. An existing coupled finite element model for finishing rolling  [461  has  been modified to predict the distribution of velocity, strain rate, strain and temperature in the roll bite. The model has been validated by comparing the model-predicted and the measured roll forces and the strain distribution of the specimen obtained in the pilot mill tests. The overall conception of the project is schematically shown in Fig.3.1.  45  EXPERIMENTAL Thermal Response Measurements  Deformation Observation  FINITE DIFFERENCE MODEL HTC Calculation  Thermal History of a Slab  FINITE ELEMENT MODEL  Oxide Scale Growth  Deformation Analysis of a Slab  Boundary Conditions BCI , BC2, BC3 , BC4  Velocity BC4  Strain Rate  N  Strain  Figure 3.1 Methodology adopted in the thermomechanical analysis during rough rolling  Temperature in the Roll Bite  4.1 Test Design  Chapter 4  EXPERIMENTAL MEASUREMENTS  As reviewed in Section 2.1.3, the characterization of heat transfer at the roll/slab interface is plagued with uncertainty. Therefore, pilot mill tests were conducted at CANMET and at UBC in which the specimen's surface temperature during the rolling process was measured by thermocouples. This chapter presents an investigation of the thermal response that was observed and its impact on the heat transfer at the roll/slab interface. 4.1 Test Design  To characterize the heat-transfer at the roll/slab interface, the thermal response at the roll surface or the slab surface or even both should be measured during hot rolling. However, due to the very short contact time (of the order of 0.05 seconds), surface oxidation, high reduction, it is not easy to make successful measurements. Fortunately, Devadas et a/." 71 have developed a technique that was successfully employed for stainless steels.  47  4.1.1 Thermocouple Design and Data Acquisition System  4.1.1 Thermocouple Design and Data Acquisition System Because of the short contact time and a large deformation during rolling, the system for thermal response measurements must have a fast response. Devadas et al. [173 arrived at an optimum thermocouple diameter that gave a good thermal response. Through hot rolling tests( 900°C, 10% and 20% reduction), using the INCONEL sheathed thermocouples, Devadas et al. established that a thermocouple wire of 0.25 mm diameter gave a satisfactory thermal response consistent with the roll contact time and the thermocouples remained intact after the tests. In addition, they found that an intrinsic thermocouple has less thermal mass than a beaded thermocouple. Therefore a 0.25mm diameter CHROMEL-ALUMEL wire was spot-welded to the surface of a sample to form an intrinsic thermocouple with junctions approximately 0.4mm apart. It is also important that the thermoelectric junctions be located on the sample surface to record the temperature of this surface during rolling. The thermal responses were recorded using a data acquisition system, which consists of a portable microcomputer (COMPAQ), a DT2805 data transmission board, and a DT707T external board. Data acquisition was manually triggered just prior to the entry of a specimen into the roll bite. Four channels with one load cell, one central temperature and two surface temperatures were recorded within a period of 3 seconds at a data acquisition rate of 1500 Hz..  48  4.1.2 Preparation of Samples  4.1.2 Preparation of Samples To determine the heat transfer coefficient at the interface during hot rolling, industrial rolling conditions should be simulated in the mill trials. In industrial rolling, especially in rough rolling, oxidation occurs at the slab surface during transport. If oxidation is allowed to occur on the specimen, detachment of the thermocouples takes place before or during rolling. In order to avoid this kind of problem, specimens were initially fabricated from AISI 304L stainless steels. However since it was unclear whether the heat-transfer coefficient determined for the stainless steels is applicable for carbon steels and microalloyed steels, techniques were devised for testing the latter. These techniques will be described later. Two grooves, 1.5 mm deep, 1 5 mm wide and 22 mm long were milled on the surface of each test specimen, shown schematically in Fig. 4.1. 2.8 /45( 22"7/  (2)  Figure 4.1 Schematic diagram of specimen employed in the thermal response measurements (Dimensions in mm)  49  4.1.2 Preparation of Samples  INCONEL sheathed thermocouples were spot-welded on the surface. The two surface thermocouples (TC1 and TC2) provide a check on reproducibility because the conditions experienced by TC1 and TC2 are almost the same. At half-way between TC1 and TC2, there is a 2 mm diameter hole located at the half thickness of the specimen, as shown in Fig. 4.1. A thermocouple was inserted into the hole to measure the temperature at the center. For ease of handling of the test specimen, an approximately one meter long wire rod was welded to each end of the specimen. The three INCONEL sheathed thermocouples were bunched loosely around the rod and led to the data acquisition board. Before reheating, an electrical check was conducted to make sure that the thermocouples were in working order. The thermocouples were calibrated at two reference temperatures (0°C and 100°C). The voltage signal, compensated using an electronic ice point, was transmitted to the data acquisition board by using fiberglass-insulated CHROMEL-ALUMEL wires. To observe the strain distribution after rolling, a 5mm diameter pin was inserted into a 5mm diameter hole at the center of the working face of the specimen with its axis perpendicular to the rolling plane, as also shown in Fig.4.1. The pin was made of the same material as the sample. This eliminates any differences in deformation behavior between the pin and the specimen. The dimensions of the specimens for tests at CANMET were 100 mm wide, 135 mm long and 50 mm thick for the 0.05%C low carbon steel and 100 mm wide, 150 mm long and 25 mm thick for the stainless steel. The dimensions of the specimens for tests at UBC were 50 8 mm wide, 127 mm long and 12.7 mm thick and were fabricated of 0.05% Carbon plus 0.025% Niobium microalloyed steel.  50  4.1.3 Test Facilities  4.1.3 Test Facilities Some tests were conducted on the rolling facility at CANMET's Metals Technology Laboratories (MTL) and others on a rolling mill at the University of British Columbia. The rolling facility at CANMET consists of a single two-high reversible hot-rolling stand whose specifications are shown in Table 4.1:  Table 4.1 Specifications of the pilot mill at CANMET Motor Drive  225 kW  Motor Speed  150/300/450 rpm  Max. Roll Separating Force  4.5 MN  Roll Diameter  460, 457 mm 0-130 mm  Roll Gap Setting  The heating facility at CANMET MTL consists of a 0.255 m 3 globar element furnace, with digital programmable control, permitting attainment of a desired reheat temperature to within ±5°C. An ingot is placed at the center of the hearth and the test samples are placed against the ingot so that the same average temperature is attained (281 . The pilot mill at UBC is also a two-high reversing mill and its specifications are shown in Table 4.2;  Table 4.2 Specifications of the pilot mill at UBC Rolling Mill  STANAT  Rolling Speed  34.3/67.6 rpm  Roll Diameter  100 mm  Roll Material  AISI 4340  Max. Roll Separating Force  0.2 MN  51  4.1.3 Test Facilities  A tube furnace was used as the reheating facility at UBC with a hearth width of 100mm. To minimize the heat loss from the specimen before rolling, the furnace was placed directly in front of the rolling mill. The specimen was heated at the center of the hearth, and the temperature of the specimen was monitored by the attached thermocouples. In order to control the scale formation on the specimen surface during reheating, the exit and the entry of the hearth was closed using a piece of asbestos plate and Nitrogen (N2 ) was passed through the hearth. The whole apparatus is schematically shown in Fig.4.2. Once the rolling temperature was reached, the specimen was held at that temperature for 15 minutes to improve the temperature homogeneity. Just before the specimen was rolled, the data acquisition system was switched on. After each test, the thermocouples were tested for mechanical and electrical stability. If the thermocouples were still functional, the specimens were rerolled. Rolling Mill  ^  Tube Furnace  Load Cell  Figure 4.2 Schematic layout of the test facilities at UBC  52  4.2.1 Test Schedule at CANMET  4.2 Test Procedures 4.2.1 Test Schedule at CANMET Five trials have been conducted at CANMET to obtain temperature-time data under different conditions. Two kinds of materials, AISI 304L stainless steel and 0.05%C low carbon steel, were employed to investigate the effect of material type on the interface heat transfer coefficient. Trial 1 was conducted to investigate the effect of successive rolling passes on the heat transfer coefficient using AISI 304L stainless steel specimen containing a pin to measure the strain. The test conditions are shown in Table 4.3. Trial 2 was conducted on the 0.05%C low carbon steel samples containing pins to investigate the effect of rolling pressure on the interface heat transfer coefficient; only one pass was carried out for each test. The test conditions are shown in Table 4.4. The specimens were quenched after rolling for microstructure evolution.  Table 4.3 Conditions employed in rolling tests to determine the influence of successive rolling passes on interface heat transfer coefficient for AISI 304L stainless steel Test No. Pass No.  Rolling  Initial Final  Percent  Temperature Ho H 1 Reduction (mm) (mm) (SC) (%) P71 SS6  1250  Rolling Speed (m/s)  25.40 22.86  9.0  1.0  P72  Not recorded 22.86 21.34  6.65  1.0  P73  Not recorded 21.34 19.05  10.73  1.0  P74  Not recorded 19.05 14.73  22.67  1.0  53  4.2.1 Test Schedule at CANMET  Table 4.4 Conditions employed in rolling tests to determine the influence of rolling pressure on interface heat transfer coefficient for a 0.05%C low carbon steel Test  Rolling  No.  Temperature  Initial Final Ho  H1  Percent  Time To Rolling  Reduction  Quench  Speed  (°C)  (ram) (mm)  (%)  (s)  (m/s)  RLC12-1  1250  12.65  8.00  36.7  13.65  1.5  SLC12-5  1250  12.48  8.00  35.9  13.40  1.5  RLC12-2  1250  12.73  /  /  /  /  SLC12-6  1250  12.48  5.99  52.0  6.8  1.5  RLC12-3  1250  6.72  2.69  60.0  5.4  1.5  SLC12-7  1250  7.43  2.95  60.3  4.1  1.5  RLC12-4  1250  6.50  2.388  63.36 *  6.7  1.5  SLC12-8  1250  6.10  2.362  61.28 *  6.0  1.5  Trial 4 was conducted also with the low carbon steels for microstructural evolution studies and a different number of passes were carried out for each test. However only the surface temperatures of the first rolling pass of the tests 1LC-6 and 3LC-1 (with *) were recorded. The conditions for the first pass are stated in Table 4.5. Trial 5 was conducted to determine the dependence of rolling temperature on the heat-transfer coefficient employing AISI 304L. Only one pass was carried out for each test and the test conditions are given in Table 4.6.  54  4.2.1 Test Schedule at CANMET  Table 4.5 Tests for microstructural evolution study Test No.  Total Rolling Initial Final Percent Time To Rolling Temperature Ho Pass H 1 Reduction Quench Speed Number (°C) (mm) (mm) (m/s) (%) (s)  1LC-6 *  1  1250.0  51.8  48.26  6.8  14.0  1.0  3LC- 1 *  3  1250.0  51.8  48.26  6.8  8.0  1.0  3LC-7  5  1250.0  51.8  48.26  6.8  4.5  1.0  3LC-8  7  1250.0  51.8  48.26  6.8  7.4  1.0  Table 4.6 Conditions employed in rolling tests to determine the influence of rolling temperature on interface heat transfer coefficient for AISI 304L stainless steel Test No.  Rolling Initial Final Percent Rolling Temperature Ho H1 Reduction Speed (°C) (mm) (mm) (%) (m/s)  SS-4  850.0  25.8  18.4  28.7  1.5  SS-8  850.0  25.9  16.7  35.5  1.5  SS-13  850.0  26.0  14.48  44.3  1.5  SS-16  950.0  25.9  18.82  27.3  1.5  SS-11  950.0  25.8  16.4  36.2  1.5  SS-18  950.0  25.8  14.12  45.3  1.5  SS-9  1050.0  26.0  18.99  27.0  1.5  SS-19  1050.0  26.0  16.92  34.9  1.5  SS-17  1050.0  25.9  14.4  44.4  1.5  55  4.2.1 Test Schedule at CANMET  Trial 6 was conducted to investigate the dependence of the heat-transfer coefficient on rolling speed; the test conditions are given in Table 4.7 Table 4.7 Tests conducted to determine the influence of rolling speed on interface heat-transfer coefficient Test  Rolling  No.  Temperature  Initial Final Ho  H1  Percent  Rolling  Reduction  Speed  (°C)  (mm) (mm)  (%)  (m/s)  SS-15  950  25.80 15.75  38.9  0.5  SS-20  950  25.80 15.75  38.9  1.0  56  4.2.2 Test Schedule at UBC  4.2.2 Test Schedule at UBC Due to the difficulties encountered at CANMET in measuring the thermal response for the low carbon steel and the stainless steel for large deformation, some supplementary tests have been conducted at UBC using a HSLA steel (0.05%C and 0.025% Nb steel). The test schedule is shown in Table 4.8.  Table 4.8 Tests for HTC measurements at UBC Test  Rolling  Initial  Final  Percent  Contact  No.  Temperature  H1  Reduction  Time  (°C)  Ho (mm)  (mm)  (%)  (s)  Testl  850  11.30  9.55  15.51  0.0265  Test2  950  12.58  9.75  21.27  0.0328  Test3  950  12.60  9.45  25.0  0.0355  Test4  1050  9.45  8.00  15.34  0.0241  Test5  1050  12.55  11.6  7.51  0.0195  Test5-1  1050  11.60  10.03  13.51  0.0251  Test5-2  1050  10.03  8.10  19.27  0.0278  Test6  1050  12.60  12.20  3.17  0.0126  Test?  1050  12.60  11.81  6.27  0.0178  Test?-1  1050  11.81  11.50  2.62  0.0111  Test8  1050  12.55  11.53  8.13  0.0202  Test8-1  850  11.53  10.20  11.54  0.0231  Test9  850  12.60  11.15  11.51  0.0241  57  4.2.2 Test Schedule at UBC  The rolling speed was constant at 0.354 m/s for all of the tests. The contact time in the roll bite was calculated according to the rolling speed.  58  4.3.1 Thermal Response  4.3 Thermal Response of Instrumented Specimens 4.3.1 Thermal Response As the specimen passed through the roll gap, the data acquisition system recorded the three thermocouple mV signals and the rolling mill load cell signal over a three-second period. Temperature conversion were obtained according to the data in the handbook [581 . An equation for thermocouple type K, Chromel-Alumel, was adopted with a 0.7°C error within the temperature range from 0°C to 1370°C. The roll force conversion was obtained through the calibration of the rolling mill load cell prior to testing. For Trial 2 conducted at CANMET using the low carbon steel, all the thermal responses recorded were flawed as shown in Fig.4.3 for Test RLC12-l.  1400 ^  42 —44  1200 -+,  TC1 TC2  1000 -* O  800 O  n.  600 400 200  1  -if^+ ++*+-4- ++  Tcent  —50  Load  —52  V4  -0 0  0  —58  1. *1-  41-+"4+;;4411" 4t t.p41. f  ^+  E  —56  -1+4" • 4 4 1.4t4.#4. t# .444.  -1-  0^ 09  —48  —54  E  -  —46  —60  ^ +*  t ^1^1.05 0.9 5  1.1  —62 1.15  Time (s)  Figure 4.3 Thermal response of thermocouples for Test RLC12-1  59  4.3.1 Thermal Response  The thermocouples detached from the specimen before or during rolling, apparently due to scale formation. For Trial 4, only one thermocouple on the surface remained functional for each test even during small reductions, as shown in Fig.4.4 for Test 1LC-6. The major reasons for the thermocouple problems are: 1) oxide scale formation on the low carbon steel specimen surface because no inert gas was applied in the furnace during heating; 2) large reduction (above 35%) for each test in Trial 2. Trial 1, Trial 5 and Trial 6 were successful as compared to Trial 2 and Trial 4, because stainless steel specimens were utilized in these Trials. However, there were still some thermocouples that detached before or during rolling of the stainless steel samples. For example, Test SS-4, SS-11, SS-13, SS-16, SS-17 completely failed, i.e., neither Thermocouple 1 nor Thermocouple 2 at the two positions on the surface recorded the thermal responses, and only the first pass for Test S S6 ( in short for SS6P71) in Trial 1 was acceptable because of the small reduction employed.  0 —10 —20 —30  u o 1000 -  —40  a) 800 -  n. 600  —50  TC1  E 0 a  -  —60  TC2  —70  Tcent  —80 Load  —90 0.66 0.68^0.7 Time (s)  Figure 4.4 Thermal response of thermocouples for Test 1LC-6  60  0 -J  4.3.1 Thermal Response  Fig.4.5 shows the thermal response of the surface and center thermocouples during the rolling of a stainless steel specimen at CANMET. The response of the surface thermocouple are virtually identical.  1200 1150 1100  towaswolowsaftwoormArisit  — 1050 0 % 1000  -  05  & 950 a.> H 900 -  r: Tsurf-2  850800-  Tcent  7\  r^  ,"^  750^ 0  0.02^0.04^0.06^0.08^0.1^0.12^0.14  ^  Time(s)  Figure 4.5 Thermal response  0.16  of thermocouples during tests at CANMET(SS6P71)  The effect due to contact with the work rolls is evident in the response of Thermocouples, TC1 and TC2. In general, the tests conducted CANMET, although a microalloyed  at UBC were more trouble-free than those conducted at  steel (0.025%Nb) was used as the specimen. Because a tube  furnace was used with N2 flowing through due to the limitation of the motor  its hearth, scale rarely formed during reheating. However,  power and the higher strength of the microalloyed steels, the  reduction for each test was limited to less than 20%, otherwise the specimen jammed in the roll gap. Evidence of this is seen in the results of Test 3 presented in Figure 4.6.  61  4.3.1 Thermal Response  25  1000 900  -  TC1 ° 0  800 -  -20  Load  a) 700 a)  -  10 o  a) 600 -  500  5  0 400 ^ I^ 0^0.1^0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s)  Figure 4.6 Thermal responses of thermocouples for Test 3 It is very interesting to see that the oscillation of the surface temperature corresponds well with that of the roll force. When the roll force increases, the surface temperature decreases, and vice versa. This indicates that the surface temperature is dependent on the roll force. Good reproducibility of the thermocouple response was obtained in many of the tests, as shown in Fig.4.7, whilst in others it was poor, see Fig.4.8. In Fig.4.8, it is evident that the minimum temperature for the second thermocouple (TC2) is about 200°C higher than that for the first thermocouple (TC1). An anomalous response of TC2 was due to the bad installation of the thermocouple prior to rolling. When INCONEL sheathing of the thermocouple protruded above the surface after it was placed in the groove, good contact between specimen and the rolls was not achieved at that location. A sudden increase in roll force at the thermocouple location is evident of this improper placement. Data such as this was disregarded.  62  4.3.1 Thermal Response 1200  Figure 4.7 Thermal responses of themocouples for Test 7-1  1 1 00  14  1000  -12  900  -10  800  -  8  2  0  700 O  600  500  400  0  0.1^0.2  -  4  -  2  0 0.3 0.4 0.5 0 6 T1rne(s)  Figure 4.8 Thermal responses of themocouples for Test -1  63  4.3.2 Surface Temperature  4.3.2 Surface Temperature The change in the surface temperature of the sample in the roll bite depends on the reduction, rolling speed, initial rolling temperature and the material type being rolled. Some typical surface temperature changes are shown in Fig.4.9 to Fig.4.11 for Test SS6P71, SS-8, SS-15 with different rolling speeds and reductions.  1150 1100 1050  -4TC2  1000 -  5th Poly  0  m  950 900 850 800 -  750  0  0.005  0.01^0.015  0.02  Contact Time (s)  Figure 4.9 Surface temperature in the roll bite for Test SS6P71 with 1.0 m/s  64  0.025  43.2 Surface Temperature  850 800 TCI  750  11th Ploy  700 a)  650  a 1,3 600 it  550 500 450 400  0.005^0.01^0.015^0.02^0.025 Contact Time (s)  0  ^  0.03  ^  0.035  Figure 4.10 Surface temperature in the roll bite for Test SS-8 with 1.5 m/s 900 850 800 -  0  al  TC1  750 m 700 -  11th Poly  la  0 650 —  a. 600 -  E  550 500 450 400  0  0.01 0.02 0.03  0.04 0.05 0.06 Contact Time (s)  0.07 0 .08 0.09  Figure 4.11 Surface temperature in the roll bite for Test SS-15 with 0.5 m/s  65  ^  0.1  4.3.2 Surface Temperature  From Fig.4.9, it is apparent that approximately 36 data points were acquired in the 0.025 seconds of contact time; this time depends on the rolling speed and reduction during the test. For Test SS6P71, the contact time calculated from rolling theoryf 211 is approximately 0.0258 seconds, which corresponds well to the time shown in the figure. The maximum rate of change of the surface temperature is present at the early time; the slope -A decreases with increasing time in the roll bite; near the exit of roll bite, the surface temperature approaches a minimum and finally tends to rebound. This phenomenon is more obvious in Fig.4.10 and 4.11 because of larger reductions and/or lower speeds. Obviously, if the reduction is relatively small, the location of the minimum temperature is closer to the exit of the roll bite. On the other hand, if the contact time is long enough as in the case of low speed and large reduction, the surface temperature of the roll and the slab would approach each other and then the surface temperature of the slab could subsequently rebound due to heat conduction from the interior of the slab to the surface. This phenomenon has also been observed in tests at UBC, as shown in Fig.4.12 and Fig.4.13 for thermocouple 1 of Test 6 and Test 8-1 respectively, with different reductions of 3.17% and 11.54% respectively. To produce a function that represents the measured surface temperature, a polynomial regression has been conducted on the data over the range of contact time. However, for some tests such as SS-9, SS-18 and SS-19 at CANMET, and Test 3 and Test 4 at UBC, because the thermocouples failed halfway in the roll bite, only some of the data points were analyzed. The regression line has been shown in the each of the Figures 4.9 through 4.13 with the surface temperature reading.  66  4.3.2 Surface Temperature  1050 ^ 1000  -  950 0 900 0 ........ a) L.. 850 D 15 800 L. E a 4:)^750 )--700 -  650 0.002^0.004 0.006^0.008^0.01 Contact Time (s)  0.012^0.014  Figure 4.12 Surface temperature in the roll bite for Test 6 900  a TC1  ..f.-.). 750 0 s ' 700 a) L. 2 650 a L. 2:', 600 -  6th Poly  .-  E  550 500-  400 ^ 0  0.005 '  0.01^0.015 Contact Time (s)  ^  0.02  Figure 4.13 Surface temperature in the roll bite for Test 8-1  67  0.025  4.3.2 Surface Temperature  The regression results for the successful tests at CANMET and at UBC are listed in Table 4.9 and Table 4.10, respectively,  Table 4.9 Regression results for tests at CANMET R2  S„x  6  0.9978  5.170  SS6P71(TC2)  5  0.9983  4.149  SS8  11  0.9914  8.892  SS-18  8  0.9746  15.29  SS-9  7  0.9978  7.256  SS-19  5  0.9940  10.161  SS-15  11  0.9971  4.668  SS-20  11  0.9973  6.345  3LC-1  4  0.9903  6.680  1LC-6  6  0.9986  3.498  Test  Order of  No.  regression  SS6P71(TC1)  68  4.3.2 Surface Temperature  Table 4.10 Regression results for tests at UBC Test  Order of  No.  regression  Test3(TC1)  R2  Sys  8  0.9990  5.873  Test4(TC1)  6  0.9990  4.595  Test6(TC1)  7  0.9997  3.134  Test6(TC2)  6  0.9952  12.984  Test7-1(TC1)  5  0.9980  7.123  Test7-1(TC2)  6  0.9987  5.007  Test8-1(TC1)  7  0.9899  15.403  where R 2 is the coefficient of determination and  s,  yx  is the standard error of the temperature estimate.  The results show that the regression curve can be used to represent the acquired data. The regression results obtained will be used in Chapter 6 to back-calculate the interface heat-transfer coefficient for different rolling conditions by using the heat-transfer model developed in Chapter 5.  69  Chapter 5  HEAT TRANSFER MODEL DEVELOPMENT  To predict the thermal history of a slab during rough rolling, a two-dimensional mathematical model has been developed using the finite difference technique. The model was also modified to calculate the heat transfer coefficient in the roll bite from the thermal response measurements obtained at CANMET and UBC. The model was verified by comparing the experimental results with an analytical solution.  5.1 Mathematical Modelling 5.1.1 Heat Conduction in the Slab during Rough Rolling As described in the literature review, the heat losses from a slab during rough rolling mainly consist of radiation to the surrounding atmosphere, convection to the air and water spray, and conduction to the work rolls. To compute the temperature distribution in a slab through the thickness as well as along the length, the general heat conduction equation has to be solved subject to the boundary conditions for rough rolling. The governing equation can be written as follows: aT  v(k vT)+4. = PsCpa ats s  (5.1)  To solve the above equation, a completely implicit finite difference method was adopted. The following assumptions were made:  70  5.1.1 Heat Conduction in the Slab during Rough Rolling  Work Roll D  Y Control Volume  vs HO —.Slab  /((A)  Work Roll  Figure 5.1 Schematic diagram of hot rolling 1)  Assume the rolling process was in steady state, i.e., the temperature in the slab was only a function of location, not a function of time;  2)  Heat flow along the transverse direction was ignored because the ratio of width of the slab to the thickness is large during slab rolling ( B 0/110 > 6 );  3)  Owning to the high speed of the motion of the slab in rough rolling (about 3m/s, with the Peclet number of approximately 4 x 10 5), conduction along the length of the slab is negligible in comparison with heat transfer by bulk motion;  4)  For hot rolling problems, the metal flow velocity in the direction of the slab's thickness is much less than that along the rolling direction and it decreases with depth and reaches zero at the center line of the slab, for this reason, the bulk motion heat flow in the depth direction was ignored;  71  5.1.1 Heat Conduction in the Slab during Rough Rolling  5)  The deformation was assumed uniform and the heat generation due to deformation was assumed to increase temperature uniformly at every location in the control volume;  6)  Heat is generated at the interface due to friction; one third was assumed to flow to the slab and the other two thirds to the work roll because of the lower temperature of the work roll, in accordance with other investigators 114m281 ;  7)  Due to symmetry, only half the slab above the center line was considered;  8)  A constant steel density was used in this model, while the other thermal physical properties were temperature dependent 1591 ;  9) Scale formation was ignored in this model, but has been considered in Chapter 7. Based on the above assumptions and by using a Eulerian description, the governing equation of heat transfer in the slab can be simplified as:  aT -a+ aT)^ +qs=psc„v  (5.2)  Employing the transformation: X= X t  ^  (5.3)  the Eq.(5.2) can be converted into a one-dimensional transient heat transfer problem, i.e.,  a(  k  .^aT  ^+qs =p,C s —  ay say^p at  (5.4)  where t is the time taken for an elemental control volume of the slab to travel a distance x measured from a reference point, e.g., at the exit point of the reheating furnace, see Fig. 5.1.  72  5.1.2 Boundary Conditions  Since conduction in the  X direction has been ignored, the governing equation can be applied  to every position along the length of the slab with its specific boundary conditions. In the present study, only the thermal histories at the head end, tail end and the middle point along the length of the slab were computed.  5.1.2 Boundary Conditions To solve the governing equation, corresponding boundary conditions must be satisfied. i.e., the heat losses from a slab during rough rolling must be specified.  5.1.2.1 Initial Condition The exit of reheating furnace was taken as an initial point for the model, i.e.: Ho 2'  Lo , 0..y — Ts =To t =0, 0 .. x^  (5.5)  5.1.2.2 Boundary Conditions The boundary conditions at specific position were expressed as follows: For symmetrical cooling, at the center line: t>0,  aT,  y=0,—ks — ay =0  (5.6)  and at the slab surface:  t > 0, y =  H (t)  aTs  k — = h(t)(T, — T.,(t))  2 ' '  73  an  (5.7)  5.1.2 Boundary Conditions  where H(t) is the slab thickness at each zone; h(t) is the heat transfer coefficient for each zone and determined by the correlations elucidated below; n corresponds to the direction normal to the slab surface; and T(t) is the environmental temperature for each zone during rough rolling. The correlations adopted in this model for specific heat loss modes are described as below.  Radiation and Convection to air Radiative heat loss occurs throughout rough rolling when the slab is exposed to air. In this model, a pseudo radiative heat transfer coefficient has been adopted and is expressed asr 91 :  h rad = S E(T)  ((Ts + 273.15) 4 — (T + 273.15) 4) (Ts — T.„)  (5.8)  where the radiative emissivity of the slab surface, e(T), is as follows 193 : T  E(T)= 1^ 0 00 (0.125 000 0.38) +1.1 1  ^  (5.9)  Actually, natural convection also occurs for the slab during radiative heat transfer. For rough rolling, a convective heat transfer coefficient of 8.37W/m2-°0 131-(151 was employed in the model.  High Pressure Water Descaling At Stelco, a set of water jets ( one above and one below) are located on the each side of the roughing mill. The water jets are operated at a pressure of 13.72 MPa with high water flow rates. Empirical correlations for high pressure sprays are scarce. An empirical correlation obtained by Sasaki et al. 1181 for low pressure sprays is given as:  h =708W °15T: 12 + 0.116^  74  (5.10)  5.1.2 Boundary Conditions  where the water flux varies from 1.6 < W < 41.7(//m 2s),; the surface temperature varies from 700 °C < Ts < 1200 °C and the pressure ranges from 196 < p < 490 kPa. Devadast 281 pointed out that although this correlation was based on the measurements made at lower water fluxes and low pressures, the h compares favorably with the heat transfer data published by Kohringm for high pressure water sprays and Yanagi [133 . Because the parameters of the descalers for the roughing mill and the finishing mill are the same, the above correlation was employed in the current model. Roll Chilling  Nearly 38% of the total heat loss from the slab is extracted by the cold rolls. A thermal contact heat transfer coefficient h g v , back-calculated from the experimental studies, was employed to quantify the heat transfer at the roll slab interface. Heat generation in the Roll Bite  The temperature rise due to deformation can be converted from the mechanical work with an efficiency factor of approximately 11=0.80-0.85 for steels E213 . H0 ) 1 a AT =r1^ln— def JoC^H s ps \ 1  (5.11)  where a is the flow stress of the rolled material, J is the mechanical equivalent of heat, CalIN-m. For carbon steels, a correlation developed by Misaka et a/. [65) was adopted, 2851 + 2968C – 1120C 2 )  a = 9.81 xf x exp (^ 0.126 – 1.75C + 0.594C2+^ T3 + 273  X  0.21 ' 0.13  £ X 6  (5.12)  where f is a factor considering the effects of chemical composition other than carbon content; for plain carbon steel f = 1, but for others it was expressed as: f = 0.916 + 0.18Mn + 0.389V + 0.191Mo + 0.004Ni^(5.13)  75  5.1.3 Heat Conduction in the Work Roll  Temperature rise due to friction was assumed to be confined to the interface [141 :  4f V ?PIP  ^  (5.14)  where v r is a relative velocity between the roll and the slab at the interface and was expressed as: H„cos 1p„ vr = Ro) 1 ^ H cos  (5.15)  and the friction coefficient, li f, is given by t211 : .tf = 4.86 x lei; — 0.074^  (5.16)  5.1.3 Heat Conduction in the Work Roll The heat lost from the slab is gained by the work roll during rolling, so that the slab and the work roll are coupled. The general governing equation for the rolls was expressed as: a er 1 a( aT,J + 1 a r aT, ) 4..k( k aTr ) .,coprcpr aT 71 -57. rk r Dr^7- 2 Y) k r DO ) az r aZ ) --  (5.17)  The following assumptions were made for the work roll model: 1)  The temperature distribution in the rolls was in cyclic steady state during rolling and the temperature change was confined within a thin layer, 8, beneath the roll surface; see also Fig. 5.1. This assumption was confirmed by measuring the surface temperature of the work roll, which was removed after a period of continuous rolling. A constant surface temperature at different position along the circumference was observed" 91-12°3 ;  2)  Heat flow along the axial direction of the rolls was ignored because the length of the rolls is much larger than the layer thickness, 8;  76  5.1.3 Heat Conduction in the Work Roll  3)  Heat conduction along the peripheral direction(9) was negligible compared with the bulk heat flow, because the Peclet number is high, nearly 1.5x10 5 ;  4)  The thermal conditions of the top roll and bottom roll were assumed to be identical;  5) A constant roll density was used in this model, while the other thermo-physical properties are temperature dependent m . Based on the above assumptions, the governing equation for rolls can be simplified as:  ( aT,^ aT =coprCpr6  r ar  rk  ,.ar  (5.18)  Employing the transformation, 0 =coxt^  (5.19)  The Eq.(5.18) can be converted into:  l a^aT  r  rkr1.  =p r Cpr  aT at  r  (5.20)  where t is the time taken for an elemental volume of the rolls to rotate through an angle, 0, measured from a reference point of slab entry at the roll bite. The initial condition for the roll governing equation is:  t = 0, Tr = T„9^  (5.21)  The boundary conditions are as follows:  aTr t >0, r =R —8, -kr -5;- =0^ (5.22) .  and at the surface,  77  5.1.4 Numerical Solution  t >0, r =R, –k r  a , a —=h(t)(7,,--Tjt*))  (5.23)  The layer thickness, 0, within which a cyclic temperature variation occurs during each revolution of the roll, was defined by Tseng 1191 as the depth where the temperature changes cyclically by greater than 1%. It was expressed as: 200Bi(aBil ffi+2) ln ^ R^-N12P e –1^el2(Bi2IP e +4 213i1A  5^2  (5.24)  -  where y is the bite angle. If the heat transfer coefficient, h, varies along the roll surface, the maximum value should be used because a larger value of h would result in a thicker layer, S. The above equation indicates that the thermal layer thickness depends not only on the Peclet number, Pe, but also on the Biot number, Bi, and the bite angle, cp. In some previous studies [221(281 , the layer thickness was assumed to be a function of the Peclet number only. The heat transfer coefficient, h(t*), varied with the roll cooling modes as described in Chapter 2. All the correlations have been verified by Devadas  [281  .  5.1.4 Numerical Solution To solve the governing equations of the slab and the roll with all the boundary conditions, an implicit finite difference technique was employed ( see Appendix A ). A slice of the slab was divided into a number of nodes outside the roll gap. While in the roll gap, the two governing equations had to be solved simultaneously, so a slice of the roll surface layer corresponding to the one in the slab was also divided into a number of nodes, as shown schematically in Fig. 5.2. In the roll bite, the slab thickness decreases step by step according to the continuity principle expressed as Eq.5.25.  78  5.1.4 Numerical Solution  INTERFACE  HO/2  SLAB  H1/2 Vg  X  Figure 5.2 Discretization of the slice in the slab and the roll for finite difference analysis Ho xv0 =H xv =fin xv„.H, xv i  (5.25)  where v is the horizontal velocity in the rolling direction, H the slab thickness, and subscript 0, 1, n stand for entry, exit and neutral point respectively. For the determination of the neutral point at  which the slab velocity is equal to the roll surface velocity in the roll bite, the following expression was adopteei: H„—  H0+3H1 4  (5.26)  From the Eq.(5.25), the slice velocity in the slab increases along the rolling direction, but the speed of the roll does not change; this results in different time steps for the slice in the roll surface layer and the slice in the slab. A flow chart is given in Fig.5.3 showing the steps involved in the model. The thermal history at each position along the length of slab can be computed, but only the head end, the tail end and the middle point were considered in this study to minimize the computational time.  79  5.1.4 Numerical Solution  (start  )  Initial Data  Pass No.=1  Subroutine for radiative and natural convective heat transfer Yes  Subroutine for radiative and natural convective heat transfer Subroutine for boiling Heat transfer ^11.  4  Subroutine roll gap heat transfer  Work roll heat transfer model  Roll temperature cyclic steady ?  Subroutine after rolling for radiation and natural convection heat tansfer No End of Roughing ? Yes CStop  Figure 5.3 Flow chart of the temperature solution of the model  80  5.2.1 HTC Solution  5.2 Modification of the Model for Roll Gap Heat-Transfer Coefficient Calculation 5.2.1 HTC Solution The model was modified to compute the roll gap heat-transfer coefficient (HTC) from the surface temperature. In the model, a trial-and-error method was adopted to determine the magnitude of the HTC for every time step in the roll bite. The flow chart for the calculation is shown in Fig. 5.4. t = 0 ) ^1  "1  initial HTC  ( stop )  Figure 5.4 Flow chart of HTC calculation An initial guess was made for the HTC in the first time step, and then the surface temperature predicted by the model is compared with the measured temperature. If the two temperatures match, which means that the difference is less than a limited value, Eps, the computation proceeds to the  81  5.2.2 Convergence of the Numerical Solution for the Modified Model  next time step until the contact time in the roll bite is over. This calculation produces the HTC distribution as a function of time along the arc of contact. By this procedure, all of the thermal response measurements for different rolling conditions could be converted to heat transfer coefficient variations in the roll bite.  5.2.2 Convergence of the Numerical Solution for the Modified Model Determination of Eps According to the solution technique described above, it is important to establish a reasonable limit, Eps for the allowable difference between the measured and the computed temperature; This value would affect the HTC magnitude calculated, as shown in Fig.5.5.  oC  _  60 )  (NI^50 E 40 U  Q)  30  Eps  0  U  L._  O  1.0 C  20  • 4.0 ° C 0  0  o 15.0 ° C  10  0 4^  0.000  0.005^0.010^0.015^0.020^0.025 Contact Time (s)  Figure 5.5 Effect of Eps on HTC magnitude Theoretically, the smaller the Eps, the better the accuracy of the HTC magnitude. Eventually a limit is reached because of errors in temperature measurements as well as round off errors in the computer. The standard error of estimate for the dependent variable temperature, T, stands for the average deviation of  the experimental data around the regression line. From Table 4.9 and Table 82  5.2.2 Convergence of the Numerical Solution for the Modified Model  4.10, the Sy, value is known for the tests to vary from approximately 2°C to 15°C. In Fig. 5.5, the Eps magnitudes of 1, 4 and 15°C were investigated; The results show that the larger magnitude of Eps, such as 15°C, has a lower maximum HTC, but for the values of 1 and 4, on average, the HTC  distributions do not change significantly. In the present study, a value of Eps equal to 4, was adopted for all of the tests. Effect of Time Step and Mesh Size on HTC Magnitude  The implicit finite difference method does not have the stability problem, but truncation errors can cause the accuracy to suffer slightly. As mentioned before, it is necessary to select a very small time step for back-calculation of the HTC, but how small should it be? As pointed out by Thomas et a1. 1611 , unless stability problems are encountered, accuracy generally improves with the refinement  of both mesh size and time step. However, until the time-step size is reduced below a critical value, mesh refinement does not lead to improvement, as shown in Fig.5.6. Once the time step is smaller than the critical value, finer meshes greatly improve accuracy. This critical time-step size is smaller for finer meshes. Continued refinement of the time-step size results in little further improvement and eventually the accuracy worsens. Thus, every given mesh has an inherent limit with respect to accuracy and an optimum range of time-step size associated with it. Moreover, an optima criteria (k )  At  was proposed which states that the dimensionless number, —1-p^ ,Cp,0.02 , appears to remain a constant of roughly 0.1 at the optimum of each of the three kinds of meshes, coarse mesh, medium mesh and fine mesh, examined by Thomas a a1. 1611 in Fig.5.6. The effect of different time-step sizes on the HTC value was analyzed and the results are shown in Fig.5.7. From Fig.5.7, it is apparent that time-step sizes of 0.0002 and 0.00005 seconds do not change the HTC value significantly, but a small deviation occurs for a time-step size of 0.001 seconds. Therefore, a time-step size of 0.0002 seconds was adopted in the present study.  83  5.2.2 Convergence of the Numerical Solution for the Modified Model 10 — Dupont -Motrix lumped q T. (tz ) -  Dupont - S t (Word 4neor q (t z )  O  0  to  0  0 Coorse Mesh 0 Medium Mesh A Fine Mesh  0 a,  a, 0 0-1  0-43  a, O  .>t  001  2^5^10^20^50^100 200 Number of Time Steps to 600s  Figure 5.6 Effect of mesh size and time step on accuracy  1611  Fig.5.8 shows the effect of the mesh sizes in the specimen and in the surface layer of the roll on the value of the HTC. From Fig.5.8, it is evident that 150 nodes in the roll and in the specimen respectively, are sufficient; further refinement of the mesh size yields little improvement to the result. According to the above analysis, for the present study, a time step of 0.0002 second and a node size of 150 in the roll and in the specimen were adopted. These sizes satisfied the optima criteria for mesh and time step size adopted by Thomas et al.E611 .  84  5.2.2 Convergence of the Numerical Solution for the Modified Model  60 50 40  a) 0  30  05 20 "rn C 0  10  0 ^ 0.020 0.000^0.005^0.010^0.015  0.025  Contact Time (s)  Figure 5.7 Effect of time step on HTC value 60  0  E  50  40 C  0 CD 0  30  20 C  0 F--  10  0  0  0.000  ^  0.005  ^  0.010  ^  0.015  ^  0.020  Contact Time (s)  Figure 5.8 Effect of mesh size on HTC value  85  ^  0.025  5.2.3 Verification of the Modified Model  5.2.3 Verification of the Modified Model The heat transfer coefficient variation with time in the roll bite has been determined using the procedure described earlier. Obviously, the computed heat transfer coefficient for the roll bite should yield temperatures that are good agreement with the measured values. A typical example of the agreement is shown in Fig.5.9 for the Test SS6P71.  1050 0  2 1000 0  950 -  g  -  F"  900 -  0.01^0.015 Contact Time (s)  Figure 5.9 Comparison of the predicted and  86  ^  0.025  the measured temperature  5.3 Sensitivity Analysis for the Roughing Model  5.3 Sensitivity Analysis for the Roughing Model Time Step  During rough rolling, using a very small time step for every zone is impractical due to the computation cost. For this reason, the use of a varying time step was investigated. Fig.5.10 shows the effect of time step size on radiative and natural convective cooling. 1260  Dt=0.005(s)  1240  Dt=0.0 1(s) )4( Dt=0.05(s)  U 1220 tu  L 3 15 1200 L a)  Dt=0.1(s)  x Dt=0.5(s)  a) 1180  1160  1140  0  1  0^20  30  Time(s)  Figure 5.10 Effect of time step size on the surface temperature under radiative and natural convective cooling The figure shows that the time step has little effect on the surface temperature distribution in the interstand region. These results are further illustrated in Table 5.1.  87  5.3 Sensitivity Analysis for the Roughing Model  Table 5.1 Sensitivity of time step size on the predicted surface temperature Time Step of  Surface^Temperature^Ts (°C) at  cooling (s)  0.5 (s)  15 (s)  30 (s)  0.005  1233.89  1170.78  1143.84  0.01  1233.82  1170.78  1143.84  0.05  1233.97  1170.84  1143.84  0.1  1234.15  1170.86  1143.85  0.5  1235.48  1170.93  1143.91  A very small deviation (approximately 1.3°C) occurs for the largest time steps of 0.1 and 0.5 second at the beginning of cooling. This is because the cooling rate at the surface is faster at the beginning due to the uniform temperature distribution through the thickness of the slab. However, a large time step, such as 0.1 and 0.5 second, would cause errors in the computation of distance for each zone which is fixed. Therefore, a time step of 0.01 second was adopted in this model for radiative cooling in the interstand region. In the roll bite, a time step of 0.0002 second was employed and 0.0005 second was deemed necessary for the descaling zone. Node Size  Fig.5.11 shows the effect of node sizes in the slice in the slab on surface temperatures under radiative and convective cooling. From the figure, it can be seen that a further increase of the number of the nodes in the half-thickness of the slab has little effect on the surface temperature.  88  5.3 Sensitivity Analysis for the Roughing Model  1260  300-300  1240 -  250-250 cY  1220 -  200-200  a) — 0 1200 -  100-100 x 50-50  c,)  a 1180 )  1160  -  1140 ^  0  20  10  30  Time (s)  Figure 5.11 Effect of node size on the predicted surface temperatures under radiative and natural convective cooling The exact temperature variations with different node numbers at different time are also shown in Table 5.2. Only a very small deviation from the convergence occurs for node numbers of 50 and 100. Moreover, the deviation is larger at the beginning of cooling than later for all of the node numbers examined. During rough rolling, the thickness of the slab is significantly reduced. Theoretically, the number of nodes in the half-thickness can be reduced as the thickness decreases, but in the computation, it is not worth changing the node size, because it would require interpolation which would introduce further errors. Thus a constant node number of 200 was maintained throughout the rolling process.  89  5.3 Sensitivity Analysis for the Roughing Model  Table 5.2 Sensitivity of node size on the predicted surface temperature No. of  Surface Temperature Ts(°C)  Node  at  through  0.5  15  30  half-  (s)  (s)  (s)  300  1233.76  1170.77  1143.83  250  1233.78  1170.77  1143.83  200  1233.82  1170.78  1143.84  100  1234.2  1170.82  1143.86  50  1236.14  1170.98  1143.95  Thickness  1260 o E=0.6 K E=0.7 )1( E=0.8 x E=0.9  1240  a 1220  o g°  g 0 X  0  23 1200  o  x  X 0 0 g 0 0  I— 1160-  X 0 X gx°00 XX^0 DO 0 E=f(Ts) X X^ XX (:)00 00 00 X% Xg X g _^0000 X X g x X g x sz  1140-  )4( )4( x x 5e^x --x xx x  ..  m Ii Lai 1180 a  -  Ea)  1120  0  x  x  w  10  20 Time (s)  Figure 5.12 Effect of emissivity on the surface temperature under radiative and natural convective cooling  90  30  5.3 Sensitivity Analysis for the Roughing Model  Emissivity Fig.5.12 shows the effects of emissivity on the surface temperature in the zone before the first descaler. Obviously, the emissivity has a large effect on the temperature. The predicted surface temperatures are also shown in Table 5.3.  Table 5.3 Sensitivity of emissivity on the predicted surface temperature Surface^Temperature^Ts(°C) at Emissivity  0.5 (s)  15 (s)  30 (s)  0.6  1237.91  1189.23  1167.70  0.7  1236.06  1180.86  1156.91  0.8  1234.21  1172.78  1146.57  0.9  1232.38  1164.95  1136.66  e(T6)  1233.82  1170.78  1143.84  Eq.(5.9)  Roll Gap Heat Transfer Coefficient The effect of roll gap heat transfer coefficients on the temperature distribution has been examined, and the results are shown in Fig.5.13.  91  5.3 Sensitivity Analysis for the Roughing Model  1150 11001050 ° 0 m 0 ty  1000 950 900 -  a_ a) 850 -  F-  800 -  hgap=30kW/m'—°C  +  0 MK^^  ^  4'  ^  3sE +  hgap=50kW/m'—°C )4E hgap=75kW/m 2 -0 C  ^  Ill  ^ )tE + +  ^  ^  ^ ^ ^ 00 ^ )i( + + + ^ ^ c hgap=100kW/m 2 —°C ^ * .4)* ^ ^ ^ ^ + ^ )4( + ++ —° as NE ^  ^  ^  W  ^  ++  W )4( ^  ^  700  0  +  0 0 NE A )4( ^  750 -  ++  NE )4(^+  ^ ^  ^4. -4+  )k NE )4( 0 )4E W )WE 0 ^  0.01^0.02  Ci 1:3 CO  ^  0.03  Contact Time (s)  Figure 5.13 Effect of the roll gap heat transfer coefficient on the surface temperature of the slab in the roll bite From the figure, it is evident that with the same initial temperature distribution, an increase in the roll gap heat transfer coefficient from 30kW/m2 -°C to 100kW/m2 -°C causes a reduction in slab surface from 882°C to 738 °C at the exit of the roll gap. Although the chilling effect is confined to a thin subsurface layer, it has been shown that it affects the final temperature distribution in a multi-pass rolling schedule 1313 . Furthermore, it affects the material flow stress and flow pattern and therefore influences the structure and properties of the steel. This emphasizes the need to properly characterize the interface heat-transfer coefficient.  92  5.4 Verification of the Roughing Model  5.4 Verification of the Roughing Model To assess the accuracy of the model, each module was tested by comparing the results of this roughing model to the results of other models in the literature s] for the same test conditions. In Fig.5.14 to Fig.5.16, the model-predicted temperature distribution through the half thickness of a strip at a specific position has been compared to the results obtained by Devadas  [28]  ,  for radiative/natural convective cooling and during descaling.  -6, 1040 co 32 1035 a ) 1030  -  11. I— 1025 -  10^15  ^  20  ^  25  Half Thickness (mm)  Figure 5.14 Comparison of the temperature distribution predicted by the current model with that of Devadas (281 Fig.5.16 shows the temperature distribution through the half thickness of the strip at the roll gap exit of the first pass. From the sensitivity analysis, it is known that the roll gap heat transfer coefficient has a very strong effect on the temperature distribution, especially in the layer beneath the surface. Thus, the slight difference between the results is likely due to differences in the roll gap heat-transfer coefficient.  93  5.4 Verification of the Roughing Model  1050  1 000  -  0 Literature Data  V 0  ,0m i-5 .L I-V,  950 -  900  Roughing Model  -  850 -  800  0^5^10^15^20^25  Half Thickness (mm)  Figure 5.15 Comparison of the temperature distribution predicted by the current model during descaling with that of Devadast 281  1— 1 -  4^6^8  ^  10  ^  12  Half Thickness (mm)  Figure 5.16 Comparison of the temperature distribution predicted by the current model in the roll bite with that of Devadas [28)  94  5.4 Verification of the Roughing Model  1100 1000 0 ° N.—, a)  900 -  800 a) a_  500  Tsurf (Roughing Model) )1( Tcent (Literature Data) 0  700  600  Tcent (Roughing MOdel)  Tsurf (Literature Data) -  0  0^2^4^6^8^10  12^14^16  ^  18  Time (s)  Figure 5.17 Comparison of the thermal history of a strip predicted by the roughing model with the data from Devadast 281 500 490 0  480 -  Numerical  -a 470 0 460 11 450  -  E440  -  E  Analytical  i!-I! 430 420 - . .. -  .410 400  0  0.5  ^  1  1.5  2^2.5 Time (s)  ...  3  ^  ... _.. .. . 3.5  4  Figure 5.18 Comparison of the model results with an analytical solution for the work roll  95  5.4 Verification of the Roughing Model  Fig.5.17 shows the thermal history of a strip predicted by the current model under the same condition as those employed by Devadas E281 . The strip experiences a radiative heat loss before two sets of high pressure water sprays, and a backwash spray prior to the first rolling pass. Only one pass was considered since subsequent passes utilize the same module. It is apparent that a good agreement is obtained everywhere although minor difference are evident in the roll gap. The difference in the predicted temperatures for the center line results from a different formula being employed to calculate the strain rate, which determines the flow stress and heat of generation. Fig.5.18 shows the numerical results from the work roll heat transfer module and the results of a corresponding analytical solutiont 251 . The analytical solution is given by the following Fourier Series: T -^- 2^Ji(1-1-k) exp(-14at/R2./0(Mkk r Ti^k = 1 ilk JAN) + JAN)  (5.27)  where Jo and J I are Bessel Function of the first kind andli k.XkR, is the kth root of the transcendental equation:  Bi^.11(1k)  ^  (5.28)  The conditions used in the comparison are shown in Table 5.4. Table 5.4 Conditions used in validation of the work roll module h (W/m2-°C)  (W/m-°C)  ic,  ar (m2/s)  1160.0  29.0  7.0x10-6  Bi  R (m)  10.0 0.50  A very good agreement has been obtained in the figure which indicates that work roll heat transfer module is valid and accurate.  96  5.4 Verification of the Roughing Model  Finally, from all of these analyses, it can be concluded that the roughing model developed in this chapter can be applied to simulate rolling. In Chapter 7, the model will be further extended to incorporate the effects of oxidation.  97  Chapter 6  ROLL GAP HEAT TRANSFER COEFFICIENT ANALYSIS  The thermal response measurements from the pilot mill tests at CANMET and at UBC were described in Chapter 4, and a mathematical model was developed to determine the roll gap heat-transfer-coefficient (HTC) in Chapter 5. The model has been employed to compute the roll gap heat-transfer coefficient for different rolling conditions and these results are presented in this chapter.  6.1 Roll Gap Heat Transfer Coefficient Analysis 6.1.1 HTC Variation along the Arc of Contact Fig.6.1(a) and (b) show the HTC variation along the arc of contact determined from the response of two thermocouples located on the same sample, SS6P71, in Trial 1. The results are very similar to those obtained in an earlier study by Devadas [283 . The HTC is seen to increase and reach an apparent plateau followed by a decrease. The maximum heat-transfer coefficient is in the range of 50-60kW/m2-°C range for both thermocouples. The maximum value of the heat-transfer coefficient appears to be very sensitive to rolling parameters, as is shown in Fig.6.2 in which a maximum value of approximately 600kW/m2 -°C is realized. The reasons for this will be discussed later.  98  ^  6.1.1 HTC Variation along the Arc of Contact  70 ^ o 60i3  E  50  -4-  40 -  '0' 8 30  g 20  7? 10)  0  0  0.005  ^  0.01^0.015  0.02  0.025  Contact Time (s)  (a) Thermocouple 1 60 ^  E  50 -  6 40-06  'a  :4-; 30 0 U  . +i: 20 L  t  o c IL -a 10 to 0  0^ 0.005  0.01^0.015  Contact Time (s)  (b) Thermocouple 2 Figure 6.1 HTC variation for Test SS6P71  99  0.025  -  6.1.1 HTC Variation along the Arc of Contact  700 '-S  0  "E  600 500 400 -  8 300 L  v) 200 -  .+ *  0  L  o  100 -  al  0 ^ ^ 0.01^0.015^0.02^0.025 0 0.005 Contact Time (s)  0.03  0.035  Figure 6.2 HTC variation for Test SS8 100 o 90  E  ^  0  80 70 -  "Ea) 60 -  '6 • O L  co  50  ^  ^  ^  ^  ^  ^  o  ^  0 ^  a  ^  ^  ^^  40 30  -  20  0  o  ^  ^  o^ 0  0.001  0.002^0.003^0.004 Contact Time (s)  0.005  Figure 6.3 HTC variation for Test SS9  100  ^  0.006  ^  -• 6.1.1 HTC  Variation along the Arc of Contact  0^0.002 0.004 0.006 0.008 0.01^0.012 0.014 0.016  Contact Time (s)  Figure 6.4 HTC variation for Test SS 18 ^350 ^ 0  "E 300 —  v 250—— C •z 200 — — )  0 150 —  a) "c7 100o .4  _  0  500^ 0  0.004^0.008^0.012^0.016^0.02  Contact Time (s)  Figure 6.5 HTC variation for Test SS19  101  6.1.1 HTC Variation along the Arc of Contact  40 ^ 35 3 30 .1e  -  50  0.005  0.01^0.015^0.02 Contact Time (s)  ^  0.025  0.03  Figure 6.6 HTC variation for Test 1LC-6 30 0 11111111111 11 11111111111111  E25  -  20  -  c O  6  -  Os 0  0.005  0.01^0.015^0.02 Contact Time (s)  0.025  Figure 6.7 HTC variation for Test 3LC-1  102  0.03  6.1.1 HTC  Variation along the Arc of Contact  In some tests, due to failure of the thermocouples halfway in the arc of contact, only some of the surface temperature data was analyzed. The corresponding HTC for these tests, as shown in Fig.6.3 through 6.5 are seen to be similar to that obtained in Figure 6.1 and 6.2. The two successful tests for the 0.05%C plain carbon steel show the same features as the results obtained for the stainless steel, but the maximum HTC values are approximately 25kW/m 2-°C to 35 kW/m2 -°C, as shown in Fig. 6.6 and 6.7, this value is less than that obtained for the stainless steel. All of these differences are considered to be significant and will be shown to be related to the rolling conditions. The results from the tests conducted at UBC are also similar and an example of the results for Test 8-1 is shown in Fig.6.8. 400 ^  E  350  -  iiii 1111.1.tri,  3 300  E 250 -  4  m  200 0 150 -  500^ 0  0.005  0.01^0.015  0.02  0.025  Contact Time (s)  Figure 6.8 HTC variation for Test 8-1 Although a third type of material (0.05%C plus 0.025%Nb steel) was employed as the specimens in the tests at UBC, the variation of the HTC in the arc of contact has the same features  103  6.1.1 HTC Variation along the Arc of Contact  as observed for the other two materials. This indicates that the general form of the HTC variation in the arc of contact is common no matter what kinds of materials are rolled. In addition, the maximum HTC value for the microalloyed steel varies from about 100kW/m 2 -°C to 500kW/m2 -°C depending on rolling conditions. The above figures show that the HTC values are influenced by the rolling parameters, such as roll percent reduction, roll speed, rolling temperature, and material type. The effects of these variables are presented in the subsequent section.  104  6.1.2 Influences of Rolling Parameters on HTC  6.1.2 Influences of Rolling Parameters on HTC Devadas et al. f173 have shown that the roll-strip HTC is influenced by percent reduction, roll speed, and lubrication. They also demonstrated a basic dependence of the HTC on pressure. They did not however explore the effect of different materials and the influence of rolling temperature; These variables were examined in this study. Percent Reduction  Fig.6.9 shows the influence of percent reduction on the HTC. 600 0  E 500 400 (73  .i 4  0  300 -  412 200 v a L_ I— 100 -  6  0^  0  0.004  0.008^0.012  0.016  0.02  Contact Time (s)  Figure 6.9 Effect of roll reduction on HTC for a rolling temperature of 950°C and a roll speed of 1.5m/s Increasing the reduction from 34.9% to 45.3% increases the maximum HTC from 320kW/m2 -°C to 544kW/m2-°C. All other conditions for the two tests were the same. The lower HTC will be shown to be related to the lower contact pressure between the roll and the specimen  for the lower percent reduction.  105  6.1.2 Influences of Rolling Parameters on HTC  Rolling Speed The effect of rolling speed was investigated using the CANMET test results, as shown in Fig.6.10.  500 ^ 0  450 400  350 -  •tn_ 300 t- 250 o  ° 200 m 150 a 1 100 -  .`_-  5  50 0 ^ 0  0.02  0.04^0.06  0.08  01  Contact Time (s)  Figure 6.10 Effect of rolling speed on HTC for a rolling temperature of 1050°C and 38.9% reduction From the figure, it is apparent that a higher maximum HTC is observed for the higher rolling speed of 1.0m/s. This can be attributed to the fact that the higher speed increases the strain rate and consequently the roll pressure. Furthermore, the contact time is reduced by a factor of two.  106  6.1.2 Influences of Rolling Parameters on HTC  Rolling Temperature  An increase in rolling temperature from 850°C to 1050°C causes a significant reduction in the maximum heat-transfer coefficient, as shown in Fig.6.11. This is attributed.to the fact that at the lower temperature the deformation resistance of the material is significantly higher, which leads to a higher pressure along the arc of contact. Again, pressure emerges as an important variable affecting the interface heat-transfer coefficient. Material Type  The grade of steel also has a strong influence on the magnitude of the heat-transfer coefficient, as illustrated in Fig.6.12. It is evident that the microalloyed steel gives rise to a higher interface HTC, whilst the low carbon steel is associated with the lowest. However, it should be noted that the microalloyed steel was tested at 1050°C and the low carbon steel and the stainless steel at 1250°C as shown in Table 6.1. This lower rolling temperature results in a higher roll pressure and consequently a higher interface heat-transfer coefficient. Table 6.1 Rolling Conditions for material type influence on the HTC  Rolling Initial Final Reduction Rolling Contact Mean H2 Test No. Temperature H 1 Speed Time Pressure (°C) (mm) (min) (%) (m/s) (s) (kg/mm2) 3LC-1  1250.0  51.8 48.26  6.8  1.0  0.0285  5.40  SS6P71  1250.0  25.40 22.86  9.0  1.0  0.0241  10.46  Test?-1  1050.0  11.81 11.50  2.62  0.35  0.0111  20.14  107  6.1.2 Influences of Rolling Parameters on HTC  700 600 500 400 -  0 300 L a)  c o 200 0  0  0.005^0.01^0.015^0.02^0.025^0.03  0  0.035  Contact Time (s)  Figure 6.11 Influence of rolling temperature on HTC for approximately 35% reduction and 1.5m/s 200 0 O  180 160  -  —e— Low Carbon Steel (0.05%C) — 3LC-1  -  \ _Ne 140 -  :(7) 0 (.)  —  A—  Stainless Steel (AISI 3041) — SS6P71 —xMicrodloyed Steel (0.025%1%40 — Test 7-1  120 100 80 -  ta  ta  0  60-  L  F  0  40-  I 20-  o^ 0  Figure  0.005  0.01^0.015^0.02 Contact Time (s)  0.025  0.03  6.12 Influence of material type on the magnitude of HTC  108  ^  6.1.3 Pressure Dependence of HTC  6.1.3 Pressure Dependence of HTC From the above analysis, and from the work of Devadas et a1. f171 it is apparent that the HTC ,  is influenced by roll pressure. For steels, a higher roll pressure can result from a heavier reduction, a lower rolling temperature, a higher roll speed, and a higher strength material. To investigate the dependence of the HTC on roll pressure, the HTC variation in the arc of contact is compared with the roll pressure distribution in Fig.6.13 through Fig.6.15 for Test SS-19, SS-15, SS-8, respectively. The roll pressure distribution in the roll bite was obtained with the aid of a finite element model which will be described in Chapter 8.  35  350 ^ 0  300 -  `■  -  30  -  25  -  20  -Y 250 -5 200  -15  (  8 150-  -  a) 2100a  aa)  -  5  -  0  50-  0^ —1  0^1^2^3^4  10  5  -5 —10  Distance along the rot bite (cm)  Figure 6.13 Distribution of HTC and roll pressure in the roll bite for SS-19  109  Pik  6.1.3 Pressure Dependence of HTC  400  ^  350 3 300250  -  200  -  HTC --Ea— R oft Pressure  L 150  0  50  -40  -  30  -  20  -10  100--  E  -0  13 50 a) -  0  -1  0^1^2^3  4  ^ —10 5  Distance along the roll bite (cm)  Figure 6.14 Distribution of HTC and roll pressure in the roll bite for SS-15  0 700  60  E N 600  50  E 40 E cr, 30  500 0 400  a)  a)  300  20 (0  0 200 0L 100 +C;  1 0 °-  u L  a)  c  0  0  0^11^1^4^1^1 ^10 4 —1^0^1^2^3^5 Distance along the Roll Bite (cm)  Figure 6.15 Distribution of HTC and roll pressure in the roll bite for SS-8  110  6.1.3 Pressure Dependence of HTC  Although the rolling conditions were very different from one another for the above three tests, the HTC variation in the roll bite corresponds well with the roll pressure variation. This indicates that the HTC value is closely related to the pressure in the arc of contact. Moreover, it is interesting to note that the heat-transfer coefficient increases at first and reaches a maximum and decreases, whilst the roll pressure stays high for a much longer period in the roll bite. From the study by Samarasekeraf 311 , it is known that the heat-transfer coefficient in the arc of contact is dependent on the real area of contact. Moreover, WilsonI 34) found that with increasing the relative speed, the fractional area in contact decreases. In the case of rolling, a high relative sliding exists at both the entry and the exit of the roll bite. The speed of the rolled stock at the entry side is always less than the roll speed, so called backward slip, whilst on the exit side, the stock's speed is larger than that of the rolls, so called forward slip; in the region between these two, a neutral region exists where the roll speed is close to the stock's speed, as schematically shown in Fig.6.15. Therefore, the real contact area at the entry and exit would be less than in the middle of the roll bite. Consequently, the heat-transfer coefficient is lower at the entry and exit although the pressure remains high. For practical purposes, it was considered useful to correlate the mean heat-transfer coefficient per pass with the mean roll pressure. This would facilitate use of this data in computer models. Comparison of the surface temperature predicted by the mean heat-transfer coefficient (47.6kW/m2 °C) with the measured one for Test S6P71 (TC2) is shown in Fig.6.16. From the figure, it is apparent that the predicted temperature is lower than the measured one at the beginning of the roll bite due to the higher HTC and then becomes higher, and finally it approaches the measured temperature at the exit of the roll bite. This indicates that the mean HTC can be used to predict the temperature distribution in a slab.  111  6.1.3 Pressure Dependence of HTC  1150 1100  Test S6P71 (TC2)  1050 v °• 1000 m L. 3 1-5 950 i_ a) QE 900 a) f850 -  o Measured Temperature Calculated using Mean HTC  0 0  0 ^ 0  00  0^ 00 0  800 750  0  00  0.d05  ^  0 0 00  0 0 0^ 00  0  0.01^0.015^ 0.02 0.025  Contact Time (s)  Figure 6.16 Comparison of surface temperature predicted by mean HTC with the measured one for Test S6P71 The mean roll pressure was obtained from the measured roll force divided by the contact area; the latter was defined as the product of the contact length in the roll bite and the width of the specimen. The mean HTC value, on the other hand, was defined as the average value in the roll bite. It was obtained by numerical integration of the HTC along the roll bite divided by the length of the contact. The mean roll pressure is listed with the rolling conditions in Table 6.2 and Table 6.3 for each successful test at CANMET and at UBC.  112  6.1.3 Pressure Dependence of HTC  Table 6.2 Mean pressure for the tests at CANMET Test No.  Roll Measured Material Initial Rolling Percent Roll Type Thickness Tempera- Red. Mean Speed Force Pressure ture Ho (m/s) (Tons) (Kg/mm2 ) (mm) (°C) (%)  SS6P71(TC1)  25.4  1250  9.0  1.0  25.17  10.46  SS6P71(TC2)  25.4  1250  9.0  1.0  25.17  10.46  SS8  25.9  850  35.5  1.5  189.56  41.54  SS-18  Stain-  25.8  950  45.3  1.5  204.89  39.81  SS-9  less  26.0  1050  27.0  1.5  101.83  25.54  SS-19  Steel  26.0  1050  34.9  1.5  139.97  30.88  SS-15  25.8  950  38.9  0.5  156.47  32.83  SS-20  25.8  950  38.9  1.0  165.2  34.66  3LC-1  0.05%C  51.8  1270  6.8  1.0  15.47 *  5.4*  1LC-6  Low Carbon Steel  51.8  1270  6.8  1.0  15.91  5.6  Note: the number with * is calculated by Sims equation [571 because the roll force was not recorded. Table 6.3 Mean pressure for the tests at UBC Test No.  Material Initial Rolling Percent Roll Roll Measured Type Thickness Tempera- Red. Speed Force Mean ture Pressure Ho (mm) (°C) (%) (m/s) (Tons) (Kg/mm2)  Test3(TC1)  12.60  950  25.0  0.35  22.21  34.88  Test4(TC1)  9.45  1050  15.34  0.35  16.41  37.21  Test6(TC1)  Micro-  12.60  1050  3.17  0.35  4.19  18.40  Test6(TC2)  alloyed  12.60  1050  3.17  0.35  4.19  18.40  Test7-1(TC1)  Steel  11.81  1050  2.62  0.35  4.27  21.24  Test7-1(TC2)  11.81  1050  2.62  0.35  4.27  21.24  Test8-1(TC1)  11.53  850  11.54  0.35  13.30  31.95  113  •  6.1.3 Pressure Dependence of HTC  700  ^E  col 200 a  L  I-  - 100 o 0  10  15^20^25^30^35  40  45  Mean Pressure (kg/mm')  (a) the tests at CANMET for the AISI 304L stainless steel 500 ^ 0E 450 350 -  m 300 -  .5  250 -  • 200L  17, 1501 L 10015 50 0  18  20^22^24^26^28^30^32^34^36 Mean Pressure (kg/mm 2 )  38  (b) the tests at UBC for the 0.05%C + 0.025%Nb tnicroalloyed steel Figure 6.17 Relation of mean HTC with mean roll pressure  114  6.2.1 Fenech et al.'s Model  The relation of mean HTC with the mean roll pressure is shown in Fig.6.17(a) for the tests at CANMET and (b) for the tests at UBC. The maximum HTC value of each test is also shown in Fig.6.17 for reference. From the figure, it is apparent that there is a linear relationship between the mean HTC and the mean roll pressure for stainless steel (AISI 304L) and for the microalloyed steel(0.05%C with 0.025%Nb). Due to the paucity of data for the low carbon steel (0.05%C), the same kind of relationship has not been established, but is likely to be valid.  6.2 A Preliminary Theoretical Consideration of HTC during Hot Rolling A linear relationship of the mean HTC value with the mean roll pressure has been established according to the pilot mill test results. A theoretical explanation is encouraged based on the postulated mechanism for heat transfer between the strip and rolls given by Samarasekera  t311  . In  this section, a model developed by Fenech et a1. E301 for static surfaces in contact was modified to explain the observed linear dependence.  6.2.1 Fenech et al.'s Model It is known that 'nominally' flat surfaces contact at only a few discrete points, as shown in Fig.2.2. For the heat conduction at the interface, since the thermal conductivity of metals is generally much greater than the thermal conductivity of the fluid filling the interstices, heat flow tends to channel through the points of contact. When the pressure on the contact is increased, the peaks in contact are deformed and the contact points are increased both in size and in number. The heat flow through the interface can be expressed as: q^dT , d7' ^tc2.— A dxi^dx2  —Ki  Thus, the contact heat transfer coefficient, k, can be defined as  115  (6.1)  6.2.1 Fenech  et al.'s  hc= OTc^  Model  (6.2)  For the determination of the heat transfer coefficient between two real surfaces, Fenech et a1. [301 set up a model for the thermal contact problem, as shown in Fig.2.4. For most contacts the height of the void is small compared with its width. Under this condition it is reasonable to neglect both the radial conduction and convection heat transfer in the interstices. Heat transfer by radiation is also neglected. For the purposes of analysis in Fenech et al.'s model, the geometry of the contacting surface is idealized as cylindrical contacts of uniform radius c, equally spaced in a triangular array (Fig.2.4(b)). One such 'button' contact is shown in Fig.2.4(c), where 6 1 , 82 are the average void heights for the two materials. A heat flow channel is assumed to be cylindrical, of radius , a. With these assumptions no heat is transferred between channels, and the heat transfer coefficient derived for one channel is representative of the entire surface. Assuming steady state, the general heat transfer coefficient has been derived by Fenech et a1. (301 as follows: , kf^ 4.26 ,171.—+ 1 4.264; +1 \ ^1 ^ + 1.16A(T± ) s,+s,[ (1 — CA2) ^eA^1 k2 —  ^)  h, — (1_ c2 ) [ — lc/ ^8...L 82 )] 426 -,17t-e5i+ 1 4.26 •  8 1 +82 k  k2  i^ •  4.266A 4/7t  ^+ 1  (6.3)  A  k2  The above expression is the sum of two fractions. The first fraction, with the square brackets, represents the heat flow across the voids, and the second fraction represents the heat flow through the metallic contact. The heat transfer coefficient is obviously of the form: (6.4)  116  6.2.2 Roll Gap Heat transfer Coefficient (HTC)  where cA is the fractional area in contact; n is the number of contacts per unit area, and ko k2 , kf are the thermal conductivities of the contact materials 1, 2 and of the fluid between them, respectively. 6.2.2 Roll Gap Heat transfer Coefficient (HTC) Since contact between the rolls and steel must also occur across asperities, Fenech's model may be applied to this situation, although it must be borne in mind that Fenech's et al.'s model does not account for any influence of the relative motion of the surfaces in contact. For rolling, the contact is always subjected to heavy loading and plastic deformation occurs. Furthenuore, if rolling is carried out without lubrication and no oxidation occurs, a simplification with kf = 0 may be assumed. As a consequence, the expression (Eq. (6.3)) can be simplified as follows: CA (6.5)  H7'C = A 0.47A/(L 2 8 1^82  k2  +^  where lcc is the combined thermal conductivities of ki and k2 , which is defined as  1 _ 1( 1 1 ki +  (6.6)  During hot deformation, the asperities on the roll surface are not deformed and become embedded in the steel resulting in good contact between the rolls and the steel, so that the average void height for the rolls and the steel, S i and 8 2 , approaches zero although it would never be equal to zero. As a consequence, the effect of these two terms in Eq.(6.5) could be ignored and the expression for HTC be further simplified as:  117  6.2.2 Roll Gap Heat transfer Coefficient (HTC)  HTC —  (6.7)  0 .47 -q ( f-2 )  The variable, n, in the above equation, is the number of contacts per unit area which must be determined by experiment. To determine the value of n, a relationship between the hardness of the softer material, and the size of the indentation area(E 2A/n) is assumed according to hardness data obtained Fenech et a1.E 301 , as shown in Fig.6.18. 600  550  500  O _  450  400  350 004^0.1^  1.0  ^  10.0  Indentation area = < 2/n (mit 2 )  Figure 6.18 Hardness data for stainless steel-416 13°1 The relationship is expressed as 1  H = Co ^ E2A  118  (6.8)  6.2.2 Roll Gap Heat transfer Coefficient (HTC)  where C o is a constant determined by a hardness test on the specific material. For the plastic flow of material considered, Pullen and Williamson [62] found that the contact area due to the interaction of microcontacts is not proportional to the normal load( which is always assumed by many other investigatorst" b[63].1641 ). They proposed a good approximation: ,2  °A  Pp -1 + P p  (6.9)  where Pp =--H-, and P. is apparent pressure at the contact surface. Substituting Eqs.(6.8) and (6.9) into Eq.(6.7) gives the following expression for the roll gap heat transfer coefficient, HTC, HTC =C H"^  (6.10)  where C is a constant which can be expressed as: C—  1 0.474C,;  (6.11)  Since the hardness in a hardness test is directly related to the applied stress of the indenter, and the applied stress is approximately equal to three times the flow stress of the material at that specific condition 163m641 , a linear relationship between the roll gap heat transfer HTC and apparent pressure has been established theoretically. Obviously, the definition of the roll gap heat transfer coefficient HTC and the apparent pressure P. are consistent with the mean HTC and mean roll pressure described previously.  119  6.3 Discussion  6.3 Discussion From the studies by Samarasekera t3n and the above analysis, it is known that the roll gap HTC is very dependent on the real area of contact. Therefore, the variation of HTC in the arc of contact should correspond well to that of pressure, which has been confirmed by the pilot mill test (see Fig.6.13 to 6.15). Relative motion between the contacting surfaces decrease the interface heat-transfer coefficient f341 . To explore the relationship between mean HTC value and mean roll pressure, the data obtained from all of the tests at CANMET and at UBC are shown in Fig.6.19. 300  6'  -  "E 250 -  0 Low Carbon Steel (0.05%C) -4-  Stainless Steel (304L) )sE Microalloyed Steel (0.025%Nb)  N(  '200 c co 75 -,,- 150 - Best fitting line -  `  i  4-  -  0 0  -I-  "6 50 0 = 0 ^ ^ ^ 10 5 15 20^25^30^35 Mean Pressure (kg/mm')  40  45  Figure 6.19 Mean HTC data vs. mean roll pressure for tests conducted at CANMET and at UBC The figure indicates a linear relationship between the mean HTC and the mean roll pressure. This can be used to determine a heat-transfer coefficient for industrial rolling from the rolling load.  120  6.3 Discussion  (a) before rolling  (w) after rolling  Surface Profile (x 100)  Figure 6.20 Specimen surface profiles before and after rolling  121  6.4 Summary  It should be borne in mind that roughness of the surfaces in contact have not been accounted for in the current study, which may be the reason for the scatter in the results. Further investigation is needed to fully characterize the effect of the roughness. Fig.6.20 illustrates the change in roughness of the surface after rolling. The surface clearly is much 'flatter' than before rolling due to plastic deformation of the surface. This would result in a higher fractional area in contact and consequently, a higher HTC value. For the effect of lubrication or oxidation during rolling, factors not accounted for in the current study, the effect of thermal conductivity of the lubricant or oxide scale (kf) must be considered as a separate resistance in any computation. The existence of a lubricant or an oxide scale in the roll bite would act as an additional thermal resistance and therefore alter the heat loss to the rolls.  6.4 Summary From the test results and preliminary theoretical considerations, the following conclusions can be reached. The roll gap HTC value can be influenced by many factors, such as roll reduction, rolling temperature, roll speed, roll and rolled material and their roughness. Because all of the factors are related to the roll pressure, a linear function for roll gap heat transfer coefficient versus mean roll pressure has been found; This can be used to determine a heat-transfer coefficient in industrial rolling from the rolling load. The application of lubricants or the existence of oxide scale would be considered as an additional thermal resistance between the roll surface and the rolled material surface.  122  7.1.1 Heat Transfer Characterizations of a Slab  Chapter 7  THERMAL PHENOMENA DURING ROUGH ROLLING  A mathematical model to predict the thermal history of a slab during rough rolling has been developed in Chapter 5, but the oxidation of steel at high temperature was ignored. In order to investigate the oxidation effect on the thermal history of the slab, the mathematical model was supplemented by a module describing the scale formation process.  7.1 Heat Transfer Characterizations during Rough Rolling 7.1.1 Heat Transfer Characterizations of a Slab Prior to rolling, a slab is reheated in a reheating furnace to a preset temperature. The surface temperature should not rise above 1280°0 121 , since above this temperature the oxide scale formed on the slab surface melts; this molten scale is very hard to remove later. Furthermore, the scale build-up in the furnace becomes excessive above this temperature. The minimum temperature in the slab should be high enough to ensure that the steel is in the austenite phase and is homogeneous in temperature. The reheating temperatures currently employed at Stelco's LEW are 1250°C for a 7-pass schedule and 1265°C for a 9-pass schedule for plain carbon steels. The rough rolling operation at Stelco's LEW consists of the following sequence of events. A slab is reheated to 1250°C or 1265 °C in the reheating furnace, and transportated from the exit of the furnace to the edging mill on the roller table. It is then edge-rolled to break the scale on the surface, which is subsequently removed by high pressure water jets. The slab is subsequently rolled on a reversing roughing mill until the final dimensions are reached. During the rolling process, a 123  7.1.1 Heat Transfer Characterizations of a Slab  secondary scale is formed and two descaling operations are conducted once before the second pass and once before the sixth pass, respectively. After rough rolling, the slab which is 10 times its original length is transported to the coil box located between the roughing mill and the finishing mill. Coiling and subsequent uncoiling reverses the head and tail end of the transfer bar with the result that the colder tail end is fed first into the finishing mill.  7-pass Schedule The operating conditions of the 7-pass schedule currently employed at Stelco's LEW for plain carbon steels is shown in Table 7.1. Table 7.1 Operating conditions for the 7-pass schedule Pass No.  Ho (cm)  H1 (cm)  Reduction (%)  Speed (rpm)  Roll Force (MN)  1  24.00  22.61  5.8  56.9  7.07  2  22.61  21.00  7.1  56.9  -6.86  3  21.00  19.01  9.5  56.9  -6.86  4  19.01  15.11  20.5  56.9  11.66  5  15.11  10.00  33.8  56.7  14.20  6  10.00  5.00  50.0  41.0  14.47  7  5.00  2.12  57.6  67.2  14.64  The initial dimension of the slab for 7-pass schedule is 240.0 (mm) x 1130 (mm) x 4800.0 (mm). In the model described in Chapter 5, the temperature distribution through the thickness of the slab at the exit of the reheating furnace was assumed to be uniform, and the scale formation was ignored. Because the heat transfer coefficient distribution in the roll bite varies with the rolling  124  7.1.1 Heat Transfer Characterizations of a Slab  conditions, in the model an average roll gap heat transfer coefficient for pass 1 to pass 7 was estimated according to the mean roll pressure of each pass as being 27.7, 23.0, 18.4, 27.3, 30.5, 32.1, 49.7kW/m2 -°C respectively. The thermal histories of the head end, middle position and the tail end of the slab were calculated. The thermal history half-way along the length of the slab is shown in Fig.7.1. 1300 1200 1100 a)  2  1000  a. c) 900  ---- Center ----- H/40 — Surface -- Average  800 700  0^10^20^30^40^50  ^ ^ ^ ^ 60 70 90 80  Time (s)  Figure 7.1 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel From the Fig.7.1, it is evident that the slab surface temperature changes significantly because of the chilling induced by high pressure water jets used to descale and by contact with the cold rolls in each pass. Each steep surface temperature decrease corresponds to an individual pass, and the smaller temperature reductions just before the 1st, 2nd and 6th pass are due to descaling. The surface temperature of the slab decreases by as much as 250°C to 350°C due to contact with cold rolls, and 100 to 150°C due to descaling water. However, the surface temperature quickly rebounds due to heat conduction from the center of the slab; the temperature decrease due to the radiative  125  7.1.1 Heat Transfer Characterizations of a Slab  and natural convective heat loss in the interpass region is much less. The temperature right beneath the surface, at a depth of H/40, changes relatively smoothly. This means the chilling effect is confined to a thin layer immediately beneath the surface. This is because the time of contact with the work roll is very short, of the order of 0.05 seconds, and during this time the roll surface heats up, reducing the driving force for heat transfer. The temperature at the center line does not decrease until the last two passes, but it increases gradually due to heat generation by plastic deformation. The center temperature decreases after the last two passes because of the significant reduction of slab thickness. From this result, it can be concluded that the heat generation due to plastic deformation in the metal forming process cannot be ignored. However, the mean temperature of the slab throughout the thickness decreases continuously and more sharply as the slab thickness is reduced. This also indicates that roll chilling has a significant effect on the temperature distribution, in spite of the heat generation due to deformation and friction. Moreover, it can also be seen that the cooling rate of a slab before rough rolling is much smaller than that after rolling, and the mean temperature decreases only by as much as 10°C before the last three passes. This results from small reductions of the first four passes. Thus, the rolling schedule affects the thermal history of a slab, even if the total reduction is the same This can also be seen from the thermal histories of the head end and tail end, as shown below.  126  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200 0 0 1100 a) 1000 05 a  900 800 700  Head Center — Head Surface --- Tail Center — Tail Surface 0^10^20^30^40^50^60^70^80^90  Time (s)  Figure 7.2 The difference of thermal histories for the head and tail end during 7-pass rolling of a 0.05%C plain carbon steel Fig.7.2 compares the thermal histories of the head and tail end of the slab. The steep reductions in temperature due to descaling and rolling are off-set by the difference in time corresponding to the length of the slab. After rolling, the through-thickness temperature distribution at both ends is significantly different; the center temperature at the head end is lower than that at the tail end. This is because when the tail end was being rolled, the reduced thickness head end was experiencing radiative and convective heat loss. However, the mean temperature of the head end at the exit of the roughing mill in the last pass is higher than that at the same location for the tail end, as shown in the Fig.7.2; This is consistent with common knowledge' 121 . The local chilling caused by the roll affects the temperatures distribution and potentially the roll forces. Although the details of the chilling cannot be seen very clearly from the earlier figures, it is illustrated in Fig.7.3, which shows the temperature distribution of the slab in the roll bite, for each pass.  127  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200  '(3 1100 0  a)  2 cu E  1--  1000 900 800  - ---- Center ----- H/40 ^ Surface Average  700 0.00^0.01^0.02  0.03  Contact Time (s) (a) Pass 1 1300 1200 1100 N -15 1000  Ew 900 800  ----- Center - ---- H/40 — Surface - - Average  700 0.00^0.01^0.02 Contact Time (s)  (b) Pass 2  0.03  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200 1100 a) D  6 1000 L..  4  a) o_ Ea, 900 1— 800  --- Center ---- H/40 ^ Surface Average  700 ^ 0.01 0.00  0.02  ^  0.03  Contact Time (s)  (c) Pass 3 1300 1200 1100 ----- Center  a)I__  ----- H/40  D  1:; 1000 L.  ^ Surface  .  a) Q E Q) 900 1--  — Average  800 700 ^ 0.00^0.01^0.02^0.03 0.04 Contact Time (s)  (b) Pass 4  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200 1100 ---- Center --- H/40  a)  D 6 1000  ^ Surface  4  a) a E w 900 1--  Average  800 700 ^ 0.00^0.01^0.02^0.03^0.04 0.05 Contact Time (s)  (e) Pass 5 1300 1200 1100 a) D  (53  4  — Center — H/40 — Surface --- Average  1000  a) a E cv 900 1-800  700 0.00^0.01^0.02^0.03^0.04^0.05^0.06^0.07 Contact Time (s)  (f) Pass 6  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200  '(3 0 1100 Center H/40 ^ Surface Average  0 1000 4-  a) a Q,E 900 800  700 ^ 0.03 0.00^0.01^0.02 Contact Time (s)  (g)  Pass 7  Figure 7.3 The temperature distribution of the slab in the roll bite during 7-pass rolling of a 0.05%C plain carbon steel A very large temperature gradient exists at the exit from each pass. The temperature decrease in the roll bite is as much as 250°C to 350°C. For the first four passes, there is little change in the center temperature because the heat generated due to the small reduction balances the heat conduction to the surface caused by the steep gradients in the roll bite. For the last three passes, on the other hand, the heat generated due to plastic deformation is higher than the heat loss by conduction through the thickness, because the reduction are in excess of 30%; as a result, the center temperature increases by 5 to 10°C. Moreover, as the reduction increases, the contact time between the rolls and the slab is extended, and after about 0.03 second contact the surface temperature does not decline further but rebounds slowly. This phenomena has been observed in the pilot mill tests. The reason is that after long contact times, the roll surface temperature is sufficiently high and the  131  7.1.1 Heat Transfer Characterizations of a Slab  driving force for heat transfer is small. In addition, the surface cooling rate at the beginning of contact for Pass 7 is higher than those for the rest of passes because of the higher heat transfer coefficients of 49.7 kW/m 2 -°C. 9-pass Schedule The 9-pass schedule for plain carbon steels employed at Stelco's LEW is shown in Table 7.2. Table 7.2 Operating conditions for the 9-pass schedule Pass No.  Flo (cm)  H1 (cm)  Reduction (%)  Speed (rpm)  Roll Force (MN)  1  24.42  21.00  14.00  47.0  13.04  2  21.00  19.50  7.14  55.5  8.08  3  19.50  17.50  10.26  53.3  11.45  4  17.50  15.50  11.43  56.4  9.84  5  15.50  14.00  9.68  55.0  10.19  6  14.00  11.50  17.86  57.0  11.77  7  11.50  8.50  26.09  56.1  16.19  8  8.50  4.70  44.71  54.7  20.72  9  4.70  2.40  48.94  55.0  20.47  The initial dimension of the slab for the 9-pass schedule is 244.2 (mm) x 1316.0 (mm) x 8290.0 (mm).  132  7.1.1 Heat Transfer Characterizations of a Slab  Fig.7.4 shows the thermal history half-way along the length of a slab for 9-pass rolling for a plahn carbon steel with 0.05%C. The initial temperature was assumed uniform at 1265°C. Average roll gap heat transfer coefficients of 28.2, 25.1, 36.5, 28.9, 40.4, 36.5, 48.6, 58.5, 79.8kW/m 2-°C were employed based on the estimation of the mean roll pressures for Pass 1 to Pass 9 respectively. The features of the thermal history are similar to those of 7-pass rolling. The difference in thermal histories of the head end and the tail end of a slab for 9-pass rolling is shown in Fig.7.5. 1300 1200 - -,  1100 0  a. 0  1000 f; a_  E  GW  0  a  co  0  0  900 800  ----- Center ----- 11/40 Surface Average  0  -  a a^co  "1 0  (0^CO  -  a  o a -  I..)^CA^(0^  -0 Cri^a^-0 (8^(0  700  Co  600  ' 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Time (s)  Figure 7.4 Thermal history half-way along the length of a slab during 9-pass rolling of a 0.05%C steel  133  7.1.1 Heat Transfer Characterizations of a Slab  1300 1200  1100 L  1000 0_ Ea) 900 800 700  0 10 20 30 40 50 60 70 80 90 100 110 120 130  Time (s) Figure 7.5 The difference of thermal histories for the head and tail end during 9-pass rolling of a 0.05%C plain carbon steel Fig.7.6 compares the thermal histories of a slab half-way along the length for 7- and 9-pass rolling. A longer rolling period is required for 9-pass rolling and hence a higher initial temperature of 1265°C was utilized to achieve the desired finishing temperature. Although the total reduction is almost the same as for the 7-pass schedule, there is a significant difference between the two thermal histories due to the different rolling schedules. Therefore, in order to investigate the evolution of microstructure of a slab during the rolling process, the thermal and deformation histories must be examined for a given rolling schedule.  134  7.1.2 Temperature Distribution in the Work Roll  1300 1200 •  /(-3 0 1100  7—pass ^ Center Surface  .6 1000  -  6 0  I"  9—pass - ---- Center — Surface  900 800 700  ' 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Time (s)  Figure 7.6 Comparison of thermal histories for the 7- and 9-pass rolling of a 0.05%C plain carbon steel  7.1.2 Temperature Distribution in the Work Roll During reversing rough rolling, the work roll rotation is reversed for each successive pass, i.e., the entry and exit of the roll bite for a slab is changed alternatively. For this situation, a symmetrical roll cooling system was always adopted in reversing rough rolling, as shown in Fig.2.1. Because each rolling pass was conducted under the same work roll with the same cooling conditions, the temperature distribution in the work roll for each pass is unlikely to be significantly different except perhaps the magnitude of the temperature. This can be seen from Fig. 7.5 for Pass 1 and Pass  2 in 7-pass rolling.  135  7.1.2 Temperature Distribution in the Work Roll  400 — Surface (r=R) -- Subsurface (r=R–Delta/40) Inner Face (r=R–Delta)  300  200 \  E 100  40^80^120 160 200 240 280 320 360 Angular Position (°)  (a) Pass 1 400 — Surface (r=R) -- Subsurface (r=R–Delta/40) — Inner Face (r=R–Delta)  300 -  -g• 200 /A  ,  \  100 _ 40^80^120 160 200 240  280 320  Angular Position (°)  (b) Pass 2 Figure 7.7 Temperature distribution in the work roll during 7-pass rolling of a 0.05%C plain carbon steel  136  360  7.2.1 Oxide Scale Growth Rate of Steels at High Temperature  The figures show the cyclic thermal steady temperature distribution within the surface layer, S. This layer in which the temperature changes by more than 1°C is 13.06 mm thick for Pass 1 and 12.94mm thick for Pass 2. At the work roll surface, the temperature undergoes a very significant change. During contact with a hot slab, the roll surface temperature rises very rapidly to a maximum which is dependent on the slab surface temperature, reduction and rolling speed. Upon exit from the roll bite, the roll surface is cooled by a stream of water flowing from a spray above. When the hot surface comes directly under the water spray the surface cools to below the interior temperature. Beyond the water spray, the surface experiences radiative and convection heat loss, so that the surface temperature rebounds by heat conduction from inside and approaches the interior temperature. Due to the symmetrical placement of sprays, the surface is again cooled by the direct water spray on the entry side. However, at this time, because the surface temperature was low, so the cooling rate at the surface was lower than that in the water spray zone at the exit side. Furthermore, the surface continues to receive heat from the interior causing a slight increase in temperature until the surface makes contact with the hot slab again, and the next revolution starts. Obviously, the work roll surface can fail in fatigue if the thermal cycling is large; so it is very important to design a good work roll cooling system to extend the work roll life. However, this is beyond the scope of the current study.  7.2 Oxide Scale Growth of Steels during Rough Rolling The model adopted above did not consider the oxidation effect on heat transfer during rolling. Before studying the effect, the oxide scale growth at high temperature was investigated.  7.2.1 Oxide Scale Growth Rate of Steels at High Temperature As described in Section 2.2, for iron and steels, the consequence of oxidation is the formation of a multi-layer scale, i.e., Fe/FeOfFe 3 04/Fe2 0 3 . The growth rate of the scale layer generally follows a parabolic law, as stated in Eq.(2.26).  137  7.2.1 Oxide Scale Growth Rate of Steels at High Temperature  The oxidation kinetics of iron during reheating has been studied and an Arrhenius law has been proposed for the scale growth' 381 . The parabolic rate constant has been presented in Eq.(2.31). From Eq.(2.26) and Eq.(2.31), an explicit relationship for the iron oxide scale thickness, x, with time, t, and temperature, T, may be obtained. r  x = 24.7 Nt exp —  ^  10190  T + 273  (7.1)  During oxidation, the scale temperature is not always uniform, so that an average temperature of the scale is generally used. For steel oxidation, Ormerod IV et al. E421 proposed a model to consider the effect of steel composition on the growth of scale using the following equation, (see also Eq.(2.34)):  x = Otr+Zai +E134^(7.2)  where 0 = 0(T), ti = T(Comp., t), F = F(comp., t), and a, and (i ii are also functions of composition and time t, as described in Section 2.2.2. Fig.7.8 shows the oxide scale growth on pure iron and on steels, namely DS0006S, DS 0507L and DS3388A at 1200°C as a function of time; These are the steels currently being rolled at Stelco's LEW. The compositions of the steels has been obtained from Devadas t2s1 .  138  7.2.1 Oxide Scale Growth Rate of Steels at High Temperature  3  DS0006S x DS0507L 0 DS3338A  2.5 -  E  2  PURE IRON  0.5 -  0  0  0.5^ 1 1.5^2 Time (hr)  2.5  ^  3  (a) Within 3 Hours 0.25  x 0.2 -  E E  DS0006S of Stelco Pure Iron  0.15  0 m  0.1  0.05-  0^ x 0  10^20^30  40^50^60 Time (s)  70  ^  80  ^  90^100  (b) Within 100 seconds Figure 7.8 Oxide scale growth of iron and steels at 1200°C  139  7.2.2 Assumptions for Oxide Scale Formation on the Steel Slab  From Fig.7.8(a), it is evident that, the oxide scale growth for the steels currently being rolled at Stelco's LEW does not deviate significantly from that of pure iron, according to Eq.(7.l). To further examine oxide growth within the time period of rough rolling, computations were conducted for 100 seconds and the result is shown in Fig.7.8(b) for steel grade DS0006S. Due to the complexity of Eq.(7.2), Eq.(7.1) was applied to the steel rolling to consider the oxidation effect on heat transfer in a slab.  7.2.2 Assumptions for Oxide Scale Formation on the Steel Slab In order to compute the oxide scale formation, the following simplifications were made: 1) The oxide scale was assumed to be compact and perfectly adherent, as assumed in the Wagner theory 1373 ; 2) Because the primary scale layer thickness formed on the slab in the reheating furnace varies from 1.5mm to 3 0inm 1121 , an initial scale thickness of 2 0 mm was used; 3) It was assumed that this scale was completely removed by the high pressure descaling water, since improper descaling is known to damage the surface quality. After descaling of the primary scale, a secondary oxide scale formed on the slab surface. The growth of the scale was dependent on temperature and time according to the parabolic law; 4) Because the scale growth is dependent on time and temperature which is a coupled problem, the additivity principle was applied to the scale growth calculation, i.e., within a very short time step, the temperature was assumed constant and the scale growth was computed before moving to the next time step at a new temperature; 5) Due to the elongation of the slab, the scale formed on the surface probably breaks into a number of scale islands in the roll bite. For simplification, it has been assumed that the scale remained continuous but that the scale thickness was reduced according to the total reduction of that pass; 140  7.2.2 Assumptions for Oxide Scale Formation on the Steel Slab  6) According to the oxide scale studies [38],[40]-(42], above D e 570°C, the scaling layer consists essentially of FeO with only a thin layer consisting of Fe3 04 and Fe20 3 . For this reason, the thermal physical properties of FeO were chosen to represent the entire oxide scale layer. A constant density of 5800 kg/m3 was applied [661 , while the thermal conductivity and specific heat were assumed to be temperature dependent [661[671 ; 7) The heat of oxidation was ignored; 8) The emissivity of the primary oxide scale was taken as 0.65 before the first descaler and 0.8 for the secondary (much thinner) scale according to the study by Kuznetsova et a1. 1431 ; 9) The overall steel thickness does not change during the formation of the scale. Based on the above assumptions, the formation of an oxide scale layer was simulated during rough rolling.  Oxide Scale  Interface  Steel Y  Figure 7.9 Distance associated with the interface i  141  7.2.2 Assumptions for Oxide Scale Formation on the Steel Slab  Due to different thermal physical properties of the steels and the oxide scale, a combined physical property was employed at the interface, as in the case of a composite material  [681 .  The  expression for the thermal conductivity is shown below:  =  1- ji kar  (7.3)  where fi is the ratio defined in terms of the distances shown in Fig.7.9:  f=  ( 5x), + ( 8x),  The effectiveness of the expression has been validated by Patankert 681 .  142  (7.4)  7.2.3 Oxide Scale Growth of Steels during Rough Rolling  7.2.3 Oxide Scale Growth of Steels during Rough Rolling According to the assumptions made, the roughing model developed in Chapter 5 was supplemented by a module describing the scale formation process. The thickness of the oxide scale is shown in Fig.7.10 assuming an initial scale thickness of 2 0 mm.  2.5  100  90  E  80  2  70  0  2 1.5  60  --x—  Scale Growth  U  15  Taal Scale Thickness  50 40  1  v)  0 15  30 20  0.5  10 0  0^10^20^30^40^50^60  70  80  0 90  Time (s)  Figure 7.10 Oxide scale thickness half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel From the figure, it is apparent that the thickness of the secondary scale formed during rolling was much smaller than the initial scale thickness, due to the much shorter time ( approximately 1 to 2 minutes) of exposure compared to the reheating period (2 to 3 hours). The growth of scale during rough rolling may be seen more clearly in the same figure with the Y-axis on the right side showing relative growth. It can be seen that the initial 2mm scale restricts further scale grow; The increase in scale thickness within the initial 30 seconds was approximately 2gm. Following high pressure descaling, the secondary scale grew faster on the newly exposed surface. However, the  143  7.3 Oxidation Effect on Heat Transfer of a Slab  total growth of secondary scale thickness during roughing was less than 100).tm. Obviously, the slab scale thickness is dependent on the temperature and the exposure time, parameters which are dependent on the operating conditions, rolling temperature, rolling reduction and rolling speed.  7.3 Oxidation Effect on Heat Transfer of a Slab Before considering the oxidation effect on heat transfer, a sensitivity analysis on mesh size in the oxide scale layer has been conducted, as shown in Fig.7.11.  c)  1250  0  1248 m z -16 1246  w Q. cE) 1244 0 0  1242 -  ,.. 4) 1240 -  N=5  a ...  ■ ...„^IN .  N=10 NE N=20  V.'411, N. 4,^+4„.^so . NI IN ., ". dlir ) • +4im . .1 N. 4-4.  SiSisit  Nr^  4-4. +4.  -1-4.....  N=30 1238 -  x  N=40 m 1236 a  A  N=50  1234 ^ 0  2  4  6  8  ^  10  Time (s)  Figure 7.11 Effect of mesh size on surface temperature From the figure, it is evident that for an initial thickness of 2.0mm, 30 nodes through the scale layer was sufficient; the predicted temperature after 10 second exposure differs only by 1°C on increasing the number of nodes to 50. Therefore, a node number of 30 for the initial scale thickness of 2.0mm was adopted in the model.  144  7.3.1 Effect of Emissivity of Oxide Scale  7.3.1 Effect of Emissivity of Oxide Scale To examine the effect of emissivity of oxide scale on the temperature distribution in a slab, the emissivity was varied from 0.5 to 0.9; The effect on the change in surface temperature are shown in Fig.7.12. For this calculation it has been assumed that the slab has a surface scale thickness of 2.0 mm and is losing heat only by radiation and natural convection. 1260 0  1240 1220 1200 O  1180 L  0  1160  L  0  c)  1140  0.5 '; 31  0.6 , q1  0.7  NI  i&P  C't  0.8  :02  '^  ,, ...  0.9 .......................................................................  1120 1100 1080 1060  Time (s)  Figure 7.12 Effect of oxide scale emissivity on surface temperature As expected, the emissivity has a strong effect on the scale surface temperature. After 30 seconds of the radiation heat loss to the air, the surface temperatures was 1122.63°C for an emissivity of 0.5 and 1069°C for 0.9. Thus an accurate determination of the emissivity is required for the analysis of the radiative heat transfer. According to a study of oxide scale emissivity [431 , a value of 0.65 was employed in the model for the primary oxide scale and 0.8 for the secondary scale.  145  7.3.2 Effect of Oxide Scale Thickness  7.3.2 Effect of Oxide Scale Thickness  It is obvious that the scale significantly affects the temperature distribution in a slab, because the thermal conductivity of the oxide scale is about 10 to 15 times less than that of the steel  671  . The  effect of the scale thickness on the surface temperature and the temperature at the interface between the oxide scale and the steel are shown in Fig.7.13 and Fig.7.14, respectively. Fig.7.13 shows that the thicker the scale, the lower the surface temperature, whilst Fig.7.14, shows a corresponding increase in the interface temperature. This indicates that the scale layer prevents heat loss to the surroundings.  0 1200 o ID L. n -8 1150 L o a  0 0.603mm  4:) 1— 1100 ca 0 o 44_ to 1050 -  1000  -I-  1.206mm )4( 1.809mm s 2.412mm 3.015mm  0  4  ^ ^ ^ 6  8  Time (s)  Figure 7.13 Effect of oxide scale thickness on the surface temperature of the slab  146  10  ▪^ ▪  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  1250 2-, 1248 m  15  -  1246 -  15 III 1244  -4o 0 o  *NE -4-+4.  Xxxxxxxx  w 1242 0 11)  ++++++++  1240 -  uu ,  la5 1238 -  )1E)4Ex +++.4.4.4.++4._  www ww  ▪ 1236 m ti) 1234 -°2 0 1232 •  1230 ^ 0  4^6 Time (s)  8  ^  10  Figure 7.14 Effect of oxide scale thickness on the temperature at the scale/steel interface  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling 7.3.3.1 Temperature Distribution in the Roll Gap The temperature distribution of the slab in the roll bite during the third and fourth pass of a 7-pass rolling schedule are shown in Fig.7.15(a) and (b) when scale is present on the surface. From the figures, it is evident that the surface temperature decreases very sharply at the beginning, and then decreases more slowly towards the exit of the roll bite (see Fig.7.15(a)) and may even increase (see Fig.7.15(b)) due to heat conduction from the interior of the slab. The effect of oxidation on the slab temperature distribution in the roll bite is more clearly shown in Fig.7.16(a) and (b) for the fifth and seventh pass during 7-pass rolling.  147  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  1300 1200 1100 a) L._ "  40  1000  a) o_ 0 900 E 1—  800  --- Center —^Scale/Steel Interface — Scale Surface Average —  700 ^ 0.01 0.00  0.02  ^  0.03  Contact Time (s) (a) Pass 3 1300  1200 0(-3 a,  1100  D 0 21 )  E iv 1—  1000  900  ---- Center ---- Scale/Steel Interface ^ Scale Surface — Average  800 0.00^0.01^0.02^0.03^0.04 Contact Time (s)  (b) Pass 4 Figure 7.15 Temperature distribution half-way along the length of the slab in the roll bite during 7-pass rolling of a 0.05%C plain carbon steel with oxidation  148  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  1300 With Oxidation  1200 -3 0  Without Oxidation  --- Scale/Steel Interface  -- Subsurface (H/400)  — Scale Surface  ---- Steel Surface  1100  15' 1000 CD  E  0  900  F-  800 700  0.00^0.01^0.02^0.03^0.04  ^  0.05  Contact Time (s)  (a) Pass 5 1300 1200  With Oxidation  Without Oxidation  ---- Scale/Steel Interface  -- Subsurface (H/400)  — Scale Surface  --- Steel Surface  1100 3  1000  47, 900 (t)^800 700 600  ^ 0.00^0.01^0.02 0.03 Contact Time (s) (b) Pass 7 Figure 7.16 Surface and interface temperature half-way along the length of the slab in the roll bite with and without oxidation for Pass 5 and 7 of a 0.05%C plain carbon steel rolling  149  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling In the figures, the temperature distribution with the scale being present were compared to that with no scale being formed. It is evident that the surface temperature change along the arc of contact when an oxide is present is very different from that of an unoxidized surface. The surface temperature without scale decreases gradually, whereas the surface temperature with oxide scale decreases very sharply at the beginning of the roll bite and then changes more slowly, as described previously. The magnitude of the temperature decrease in the roll gap due to roll chilling is lower in the presence of an oxide scale. Moreover, there is a significant difference in the thermal gradient established in the slab because of the insulating effect of the oxide scale.  7.3.3.2 Thermal History of Slab The thermal history of a slab when the scale is present for the 7-pass schedule is shown in Fig.7.17. 1300 1200 1100  z  0 0  1000  s-  D  4(5'  900 800  700 600 500  0 CD (0 0  FD  -  -0  -U 0 0) CD co  (n  N Center ---- Scale/Steel Interface — Scale Surface — Average  a  co  0^10^20^30^40^50^60  70  80  90  Time (s)  Figure 7.17 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.05%C plain carbon steel with oxidation  150  •  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  The temperature difference between the surface and the scale/steel interface was very large before the first descaler, due to the presence of the relatively thick primary oxide scale. After the first descaler, the difference became smaller because a much thinner secondary oxide scale layer was formed, ( see also Fig.7.10 for the scale growth). This indicates that if the rolling speed is very fast and the interpass time is very short, as in the finishing mill, where temperatures are also lower, the effect of the secondary oxide scale on the heat transfer could be ignored. However, in the case of rough rolling, the primary thick scale formed on the slab surface has a heavy insulating effect on the heat loss from the surface of the slab, and therefore affects the temperature distribution in the slab. 1300 1200  1100 - 1000 •  900  With Oxidation  •  800  Center ^ Scale Surface  700 600 500  Without Oxidation -- Center — Steel Surface  0^10^20^30^40^50^60^70^80^90 Time (s)  Figure 7.18 Thermal history half-way along the length of the slab with and without oxidation during 7-pass rolling of a 0.05%C plain carbon steel  151  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  The thermal histories at the surface of a slab with and without oxidation are compared in Fig.7.18. It is obvious from the figure that although the secondary oxide scale has a relatively small effect on the temperature distribution in the slab, the thermal histories for the two situations (with and without oxidation) are very different; The difference results mostly from the existence of the initial oxide scale. In addition, due to the insulating effect of the oxide scale, the temperature after rough rolling is higher than in the absence of oxidation.  7.3.3.3 Effect of Rolled Materials on Thermal History of slab For comparison of thermal histories between different rolled materials, the 7-pass rolling schedule for the low carbon steel has been applied to the rolling of the microalloyed steel with 0.025%Nb. The roll forces for each pass of microalloyed steel rolling were estimated by Sims equation. Based on the roll forces, the heat transfer coefficients for each pass were estimated by its mean roll pressure from the pilot mill tests as being 63.9, 80.6, 99.4, 81.9, 70.2, 82.1, 394.2kW/m 2 -°C. The thermal history half-way along the length of the slab is shown in Fig.7.19 and the thermal histories at the head and the tail end of the slab is shown in Fig.7.20. From the figures, it is apparent that a heavier roll chilling effect was obtained for each pass due to the higher mean roll gap heat transfer coefficient, especially for the last pass.  152  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  1300 1200 1100 0  1000  0  0  900 -  L  7  800 -  a)  700  CI  Ea)  0 CD^C7  a aT  a (T) a  _u  (-0  a  U3  600 -  -U  o a  a  -  cr)  -o  ----- Center 500 ------- Interface 400 Surface — Average 300 0  0)  0^(.0  a  0'  a  0' (,)  1:1  a  ci) CP  cr) 0  En  10^20^30^40^50^60^70^80^90  Time (s)  Figure 7.19 Thermal history half-way along the length of the slab during 7-pass rolling of a 0.025%Nb bearing steel with oxidation 1300  1  1200 1100 1000 900 800 700 600  ----- Head Center — Head Surface --- Tail Center  500 400 -  --- Tail Surface  300 0  10^20^30^40^50^60^70^80^90  Time (s)  Figure 7.20 Thermal history at the head and the tail end of the slab during 7-pass rolling of a 0.025%Nb bearing steel with oxidation  153  73.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling  1300 1200 1100 1000 900 a) 800 0  700  E  (t)^600 F-  -----^Center (0.025%Nb)  500  ------- Surface (0.025%Nb) Center (Plain Carbon)  400  Surface (Plain Carbon)  300 0^10 20 30 40 50 60^70 80 90 Time (s)  Figure 7.21 Comparison of thermal history half-way along the length of the slab during 7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation 1400 1300 1200 1100 1000  With Oxidation ----- Scale/Steel Interface  -- Subsurface (H/400)  — Scale Surface  -- Steel Surface  900 z t 800 a) o_ 700 600  Without Oxidation  ----------  ------  -------  ---------- ^  500 400 300 0.00  0.01^0.02  ^  0.03  Contact Time (s) Figure 7.22 Comparison of thermal history half-way along the length of the slab  during 7-pass rolling a 0.05c70C and a 0.025%Nb bearing steel with oxidation  154  7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling A comparison of the thermal history between the low carbon steel and the microalloyed steel is shown in Fig.7.21. Due to the heavier roll chilling, the final temperature distribution for the microalloyed steel is lower than that of the low carbon steel. The temperature distribution through the half-thickness of the slab in the roll bite for the 7th pass of the microalloyed steel rolling with and without oxidation is shown in Fig.7.22. The figure shows that a much larger thermal gradient exists between the scale surface and the scale/steel interface. Due to the smaller thermal diffusivity of the scale and the higher mean heat transfer coefficient, the surface temperature decrease very rapidly at the beginning of the roll bite, and the thermal gradient gets larger, as the scale thickness decreases in the roll bite as assumed, the driving force for the heat conduction from the interior of the slab is getting larger, therefore the scale surface temperature rebounds quickly.  155  7.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling  7.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling The heat loss modes of the slab during rough rolling are mainly via radiation and natural convection to the air, convection due to descaling and the conduction to the work rolls. The heat loss to the work rolls can be compared to the heat loss due to radiation and convection with the aid of the roughing model. The model was run by assuming no heat loss to the work rolls during rough rolling. The results were compared to the final temperatures of 7-pass rolling. It is known that the heat content of a slice before and after roughing can be expressed as.  Qo  = f P C T dxdy^ s  Qf =  ps so  ps Cps Tsjdxdy  ^  (7.5)  (7.6)  where Tso , Tsf are the initial and final temperature distribution through thickness before and after rolling, respectively. Thus, the total heat loss for a slice is the difference between the initial and the final heat content.  AQ, = Qo — Qf^  (7.7)  Therefore, the percent heat loss to the work rolls can be obtained by the following expression,  AQ,(%)=  AQ, —  AQ,  where AQ,, denotes the total heat loss due to radiation and convection.  156  (7.8)  7.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling  According to the above equations, the total heat loss to the work rolls is obtained to be approximately 33.2% during 7-pass rough rolling. The heat loss to the work rolls estimated by Hollander" 21 during strip rolling (including roughing and finishing) was about 38%. These results all confirm the importance of the efforts to characterize the heat transfer in the roll bite. The thermal history of a slab without heat loss to the work rolls is compared with that with roll chilling in Fig.7.23. 1300 1200 ----  1  ■•■■J  ...-•■••■■•••-■  1100 0  (1)  1000  Without Oxidation  900  ---- Center Steel Surface  800 700 600  No Heat Loss To Rolls --- Center --- Steel Surface  500  0^10^20^30^40^50^60^70^80^90 Time (a) Figure 7.23 Comparison of thermal history half-way along the length of the slab during 7-pass rolling with and without heat loss to the work rolls From the figure, it is evident that because of no heat loss to the work rolls the final temperature distribution is higher.  157  8.1.1 Formulation for Deformation  Chapter 8  DEFORMATION ANALYSIS ON A SLAB DURING ROUGH ROLLING  In the previous chapters, the thermal field in the slab has been computed using a finite-difference technique. The heat of deformation was independently estimated as described earlier. In reality, the heat transfer and deformation are coupled through this heat generation term, and the equations should be solved simultaneously to obtain an accurate description of the deformation field. This Chapter presents the results of this analysis. 8.1 Velocity-Pressure Formulation 8.1.1 Formulation for Deformation For the velocity-pressure approach, the basic variables are the velocities and pressure. The basic equation can be expressed as: fv {50 T {a}c/V — f {5v } T {F}dV — f {5v } T fildS = 0^(8.1) s,  where V is the flow domain; 5 denotes the differential on the corresponding variable. The volumetric force {F} are specified on flow domain V and {t} on stress boundaries Sr, whilst {v} is the velocity vector, which, in this case, is as, {V}  T  {u,v}  ^  (8.2)  where u and v are the horizontal and vertical velocities in the domain, respectively. The strain rate vector, {0, is expressed as, 158  8.1.1 Formulation for Deformation  foT=te„,y,ky}  ^  (8.3)  Due to the incompressibility in metal deformation, the volumetric straining rate and the internal work associated with any pressure variation should be zero:  = k+ y = [my {0 = 0 where: [M  ] T -=  (8.4)  {110}.  L  (8.5)  'S/iiv = 0  For compatibility, the following equation must be satisfied: {SE} = [L] {6v}  (8.6)  where [L] is:  -a  0 a =^0^ ax  [L]  (8.7)  ay  a _5.7 5T a  .  For the stress vector, it can be divided into two parts, i.e., deviatoric stress {S} and mean stress  {1  {1 {S} = ay — 1}6m a^0 xY  = {G} + [  M]p  because the pressure, p, is related to the mean stress, cv„„ through the following equation:  159  (8.8)  8.1.1 Formulation for Deformation  P =  ^= —(csx + 6y )/2^  (8.9)  Assuming that the material is isotropic, the relation (2.38), now applicable only to the deviatoric stresses, can be written as: {S} =[D]^  (8.10)  where:  2 0 0 ^ [D] =^0 2 0 0 0 1 -  (8.11)  The scalar, is known as the viscosity, which in general is still dependent on the strain, strain rate and temperature of the material. According to the definition of the equivalent stress, cs, and the equivalent strain rate, e, --2^3 = 2- S S Y -  2^  2. .  —EE-. e^=^ 3 -  (8.12)  (8.13)  the Eq.(8.10) can be replaced as: =  (8.14)  so that the viscosity, 1.t, can be expressed from Eq.(8.14): (8.15) Whilst the equivalent stress, 6, is conventionally made equal to the flow stress under uniaxial load.  160  8.2.1 Finite Element Discretization  In Eq(8.15), if the effective strain rate, c, goes to zero, the viscosity, t, may become infinity. As this may lead to numerical problems in subsequent calculations, it is usual to provide a very large cut-off value for the viscosity, to prevent this difficulty.  8.1.2 Thermal Coupling During deformation, the temperature depends not only on the heat lost to the environment and rolls, but also on the heat generated; The latter is determined by the mechanical properties of the metal being deformed. Thus heat transfer and deformation are coupled through the heat generation term, qs , which is expressed as: .  4s=P =i-t  x1±+alay l 3a y^ ax^ 12 4 a1 2 {( 21 2} 4. (ax1^2 ± ay1 )T  (8.16)  where u, v are velocity in X and Y direction, respectively. The governing equation of heat transfer has to be solved simultaneously with the deformation equations for the material flow. In the analysis, an iterative procedure has been used to obtain the best estimate of temperature, strain and strain rate, which will be described in the subsequent section.  8.2 Finite Element Solution 8.2.1 Finite Element Discretization For the solution of the problem, the flow domain must be discretized into a number of elements, as schematically shown in Fig.8.1.  161  8.2.1 Finite Element Discretization  V  ROLL Free Surface  Vs = Vr Frictionalrayer Free Surface  1111111111111  0  xl  v=0  X  Figure 8.1 Finite element discretization in the roll bite of the slab In the solution, an 8-node iso-parametric element was adopted as shown above. It is well established that the finite element interpolation for pressure, p, should be at least one degree less than that for the velocity componentsf 463 , since the continuity equation does not contain a pressure term. As such parabolic interpolation was used for velocities and temperature and linear interpolation for pressure in the present analysis, 3 x 3 Gaussian integration points were employed for numerical integration. From the basic equations of deformation, the following matrix equation can be obtained:  ve l [KI T  [  K P]  { a v} }  0 j fel  { In} 0  0  (8.17)  where t ((i v }, tap } are the nodal velocities and pressure respectively. The corresponding finite element equation for the temperature solution can be expressed as: [K T] {T,} +^= 0^  162  (8.18)  8.2.2 Mechanical and Thermal Boundary Conditions  where (f) is the thermal force which contains the heat generation of deformation. The details of the formulation of the solution can be seen in Appendix B. In the current study, a fixed grid system with 10 elements in the half thickness by 20 elements in the rolling direction was used. The meshes were divided subject to the estimated flow streamlines, which will be described later in detail. Because the deformation is mostly concentrated in the roll bite, the number of elements was 15 in the roll bite, 3 on the entry side and 2 on the exit side, as shown in Fig.8.1. 8.2.2 Mechanical and Thermal Boundary Conditions To solve Eq.(8.17) and Eq.(8.18), specific boundary conditions must be satisfied. Because the flow domain is constrained in the roll bite, all the boundary conditions were specified as: At the entrance side of the roll bite, see also Fig.8.1, 1/0 x=0, 0^  '  v = 0;  T = T0(y)^(8.19)  v = 0;  —IcsTa =  at the exit side,  x =x1 , 0<—Y  aTs  0  (8.20)  along the center line of the slab, due to symmetry, aTs  Y =0,^v =0;^—Ic-53=  (8.21)  whilst along the arc of contact, the slab velocity was assumed to be the same as the roll velocity. To consider the friction effect which actually exists at the roll/slab interface due to the slippage between the two, a very thin layer of elements was inserted along the arc of contact, as shown in  163  8.2.3 Sequence of the Solution  Fig.8.1. The outer nodes of these elements were attached to the roll surface, and the friction effect was accounted for by relating the shear stress, T, associated with flow to the normal pressure, p, by means of a friction coefficient, ;I f, as expressed below: T.pxpf^(8.22) The thermal boundary condition at the interface, was expressed as: = hgap (Ts —TO  ^  (8.23)  where n, stands for the normal direction of the contact arc. For the two top free surfaces at each side of the roll bite, no surface tractions existed. However, for the thermal situation, the boundary conditions were specified as: aTs ad (Ts —T.,) ay^r  —k ---= h  (8.24)  where h„d is the combined heat transfer coefficient for radiation and natural convection. The finite-difference approach was particularly useful in determining hgap . It would have been extremely difficult to use the finite-element model to back-calculate the interface heat-transfer coefficients because the latter is an Eulerian formulation. One would have had to determine the complete variation of hga„ along the arc of contact by trial-and-error, beginning with a trial solution of this variation.  8.2.3 Sequence of the Solution For a thermal coupled plastic deformation problem, an iterative procedure has to be adopted to obtain a solution. The sequence of the solution is shown in Fig.8.2.  164  8.2.3 Sequence of the Solution  (Start Input Initial Data Iter. No. =1 Assumed temperature, Constant viscosity, Constant yield, to solve velocities  4 Modifying viscosity to get new velocities from Eq.(8.17) Calculate q, to get temperature from Eq.(8.18) Modifying the yield, Iter. = Iter. + 1  solve flow Eq.(8.17) No  Velocity Convergent ?  Yes Output strain rate, strain, temperature, roll force. ( Stop ) Figure 8.2 Sequence of the solution for the finite  element analysis  Given an assumed temperature distribution (room temperature, for example) in the domain, to solve the flow problem from Eq.(8.17), the velocities are determined. The thermal energy equation (8.18) was then solved with the trial velocities. The convective term in the thermal energy equation was handled using an upwinding weight function to improve the results. Once the temperature field was known, the flow problem was solved again. The above steps could be repeated until the velocity field was convergent. In the present study, less than 0.1% of velocity variation for each nodal value was taken as a convergence limit. With this restriction, the number of iterations varied from 8 to 35 depending on the percent reduction for the industrial rolling schedule at S telco, On average,  165  8.3.1 Heat Transfer Analysis of a Plate  nearly a half minute of CPU time was needed for one iteration on the SGI at UBC. Since the nature of the matrix obtained in Eq.(8.18) for solution of the temperature is unsymmetrical, an unsymmetric solver, so called Frontal Unsymmetric Matrix Solver t691 , was adopted in the solution. It should be pointed out that since a linear interpolation function was used for the pressure, the accuracy of the pressure solution was less than that of velocities, Therefore, the average pressure for each element of the inserted layer at the interface was employed to consider the friction effect, as seen in Eq.(8.22), rather than the nodal values.  8.3 Verification of the Source-Code of the FEM Program The finite-element program was originally developed by Kumar et al. (463 for finish rolling. It has been modified for application to rough rolling. There are two main modules, with one for the thermal analysis and the other for the deformation analysis. Because of the complexity of the finite-element program, it is necessary to verify the source-code of the program before applying it to the deformation analysis. To verify the program, an analytical solution for a simple heat transfer problem and a pilot  mill  test result was compared  with the model prediction.  8.3.1 Heat Transfer Analysis of a Plate To verify the heat transfer module in the source-code, a rectangular plate with specific thermal boundary conditions and no deformation was considered, as shown in Fig.8.3.  166  8.3.1 Heat Transfer Analysis of a Plate  = -200000.0 WIm2 10^  200  9  199  8  198  7  197  6  196  5  195  4  194  • cm  • cr  193  0  2  192  1  191 1.47 m  ^  x1  TO = 1250 C  Figure 8.3 Verification of the heat transfer module of the finite element program The boundary conditions for the above problem were specified as: H^aTs  x=0,x 1 , 0^—k 2^ax  (8.25)  0 5x 5x 1 , y =0, T =To  (8.26)  aT  s  y =^—k — Const. s ay  (8.27)  For the steady state, it is obvious that the upper surface temperatures of the plate for all of the nodes are the same along the boundary and equal to: Upper Surface T =^+T ks 2^°  167  (8.28)  ^  8.3.1 Heat Transfer Analysis of a Plate  With q = -2.0 x 105 W/m2; To = 1250.0°C, k = 30.0 W/m-°C; and H/2 = 0.15 m, the upper surface temperature can be easily calculated to be equal to 250.0°C. Using these same conditions, the nodal temperature obtained by the finite element program was exactly 250.0°C at each node on the upper surface. In addition, the temperature symmetry at the center line of width of the plate was obtained due to the symmetrical boundary conditions. The temperature distribution through the half thickness is shown in Fig.8.4. ^1300 ^ 1200 1100 1000 0  900  -  L. 800 • a)  700 600  •  500 400 300 200 ^ 0  ^ ^ ^ 2^4^6^8^10 12 14 16 Half Thickness (cm)  Figure 8.4 Comparison of the numerical upper surface temperature distribution of the plate with an analytical solution From the figure, it is evident that an excellent agreement has been obtained between the analytical solution and the results predicted by the FEM model. It indicates that the thermal module of the program is accurate and valid.  168  8.3.2 Interface Temperature Distribution of a Pilot Mill Test  8.3.2 Interface Temperature Distribution of a Pilot Mill Test To verify the deformation module, the conditions of a pilot mill test (Thermocouple 2 (TC2) of Test SS6P71 at CANMET, see Table 6.1) were simulated with the aid of the model. The roll gap interface temperature distribution obtained is compared with experimental measurements in Fig.8.5.  1150 ^ Test S6P71  1100 1050 am L z  Experimental x FEM Prediction  1000 950 -  L  E  900 850 800 750  0  x xxx>c7c,cx  0.5^1^1.5^2^2 5 Distance along the roll bite (cm)  Figure 8.5 Comparison of the FEM prediction with a pilot mill test result of the interface temperature distribution in the roll bite A very good agreement has been obtained in the above figure to indicate that the source-code of the program can be applied to the deformation analysis of industrial rolling. It also affirms that the use of the finite-difference method to back calculate heat-transfer coefficient is valid since the computed values of HTC, when employed in the finite-element code, yield the same interface temperature distribution as seen in Fig.8.5. Deviations between the two solutions will be prevalent in the interior of the slab where deformation heating is significant.  169  8.4.1 Velocity Profiles  8.4 Rough Rolling Deformation Analysis The application of the FEM model can provide insight into metal flow during deformation.  The basic variables employed to describe the material behavior are velocity, strain rate, strain and temperature. The deformation analysis was conducted for a 9-pass rolling schedule. The typical deformation parameters for the 9-pass schedule are listed in Table 8.1. Table 8.1 Typical deformation parameters for the 9-pass schedule Pass  Ho  Reduction  Roll  Nominal  Nominal  Contact  No.  (cm)  (%)  Speed  Effective  Effective  Length  (m/s)  Strain  Strain Rate  (cm)  (s-1 ) 1  24.42  14.00  2.75  0.174  3.49  13.72  2  21.00  7.14  3.25  0.086  3.04  9.12  3  19.50  10.26  3.12  0.125  3.71  10.52  4  17.50  11.43  3.30  0.140  4.40  10.52  5  15.50  9.68  3.22  0.118  4.15  9.12  6  14.00  17.86  3.33  0.227  6.43  11.75  7  11.50  26.09  3.28  0.349  8.90  12.86  8  8.50  44.71  3.20  0.684  15.15  14.45  9  4.70  48.94  3.22  0.776  20.36  11.28  8.4.1 Velocity Profiles In the velocity-pressure approach of the finite-element analysis, velocity is one of the basic variables. It gives the flow pattern of the metal during rolling. Fig.8.6(a) and (b) show the nodal velocity profiles for Pass 1 and Pass 2 during 9-pass rolling at Stelco's LEW.  170  8.4.1 Velocity Profiles  ROLL  E  0  to  cis  a) 0 0)  5  0  0  •  0  10^15  I  20  •  1^•  25  Distance along the roll bite (cm)  (a) Pass 1  E0  ROLL  to  .0 To 0) O  O  C 0  (Ti  0  I  5^  Distance along the  ^ 15 10 ro ll bite (cm)^  20  (b) Pass 2 Figure 8.6 Typical velocity profiles during 9-pass rolling  171  8.4.2 Strain Rate Distribution From the profiles, it is apparent that the velocities increase along the roll bite because of the reduction. The directions of the velocity vectors show that the vertical velocity is the largest at the entry point and decreases along the thickness and the roll bite and finally reaches zero at the center line and at the exit side of the roll bite. Moreover, the velocity at a given depth along the roll gap forms a line which corresponds to the stream line. This indicates that the finite element discretization of the domain is consistent with the stream line, as it is seen in Fig.8.6. The results also show that the direction of the velocity near the center line is almost parallel to the rolling direction due to zero vertical velocity, whilst the velocity near the surface is parallel to the arc of contact. In addition, although the features of the velocity profiles are similar for each pass, the magnitude of the velocity is dependent upon the roll speed and the percent reduction.  8.4.2 Strain Rate Distribution The strain rate distribution in the roll bite can be obtained by the continuity equation: • cif = 5 (vij + vii )  (8.29)  Typical strain rate distributions in the roll bite for Pass 1, Pass 2 and Pass 3 in 9-pass rolling are shown through Fig.8.7(a) to (c).  172  8.4.2 Strain Rate Distribution  Effective Strain Rate in Roll Gap 1st pass ROLL  et 4,  O  5.0 co  3-■ 4.)  C.)  E r0  CL.) N ro 0 ^  °  5^10^15^20  Distance along the rolling direction (cm)  (a) Pass 1 with nominal strain rate of 3.49 s -1  Effective Strain Rate in Roll Gap 2nd pass  4.) N  -  0 C.) ,  0  0  4-■  N-  a 0  5^ 10^ 15  ^  Distance along the rolling direction (cm)  (b) Pass 2 with nominal strain rate of 3.04  20  8.4.2 Strain Rate Distribution  Effective Strain Rate in Roll Gap 3rd pass  ROLL  '0  -0  5^10^15  ^  20  Distance along the rolling direction (cm)  (c) Pass 3 with nominal strain rate of 3.71 s -1 Figure 8.7 Effective strain rate distribution in the roll bite in the 9-pass rolling From the above three figures, it is evident that the effective strain rate distribution is similar from pass to pass. The common features of the distributions are as follows: a very high strain rate just beneath the surface at entry to and exit from the roll bite; the strain rates at these locations are significantly higher than the nominal effective strain rate owing to the high redundant shear associated with constraining the metal to flow into and out of the roll gap. In addition, there is a dead zone beneath the rolls, approximately half way along the arc of contact, in which the strain rate is small. This is consistent with the results obtained from laboratory rolling by Silvonen et Clij 531.(541 .  Moreover, the nominal strain rate for each pass is reached at the entry and exit of the roll  bite. For higher reductions and higher strain rates, ( comparing Pass 3, 10.26% reduction at 3.71s -1 with Pass 2, 7.14% reduction at a nominal effective strain rate of 3.040, the regions of high strain rate extend closer to the center line and are wider. This indicates that a higher percent reduction  174  8.4.3 Strain Distribution  and a higher strain rate produces more intense shearing, which is consistent with the results obtained by Dawsont561 . In addition, the plastic deformation zone in the roll gap is wider near the surface than at the center line. This is also consistent with the result obtained by the slip-line-field method for rolline 01 and by Dawson [561 .  8.4.3 Strain Distribution Strain is one of the major factors influencing microstructure evolution during rolling, therefore the strain distribution during deformation has to be known. However, the velocity-pressure approach does not permit the direct determination of strain from the basic variable, velocity. One way of obtaining the strain is the integration of the effective strain rate along the stream line, i.e.: 6 =^ildt  ^  s J.  (8.30)  Because the velocities are known for the steady state after the solution is convergent, the stream lines can be determined from the direction of the metal flow: = tan(a') = v(x,y) dx^u(x,y) '  (8.31)  and therefore,  stream line: y = .10 tan(a')dx^ (8.32) where a' is the angle of the direction of the velocity vector to the rolling direction. Once the strain is determined it allows the effects of strain-hardening to be accounted for in the finite-element model. Traditionally, with the flow formulation method, the effect of strain hardening is ignored and the flow stress is considered to be a function only of strain rate and temperature.  175  8.43 Strain Distribution  To account for strain hardening, the metal flow pattern is estimated following the first iteration by integrating the effective strain rate along the stream lines. The flow stress at each point is estimated as a function of strain, strain rate and temperature, and the equations are resolved to obtain new estimates of velocity, temperature and pressure in the domain. The strain distribution is recomputed and the procedure is repeated. Fig.8.8 (a) to (c) shows the effective strain distribution determined by this method for Passes 1, 2 and 3 in 9-pass rolling.  Effective Strain in Roll Gap  o  1st pass  ROLL  a  o  CI) 0 4-4 0^.  55  a f•-■  a  •  r.0  0  0 ,  —  0  0  0 0  44-4  a NV a  cn ^  cn Q 0  5^10^15^20  Distance along the rolling direction (cm)  (a) Pass 1 with nominal strain of 0.174  176  8.4.3 Strain Distribution  Effective Strain in Roll Gap 2nd pass  ROLL 0  -.0.15  0, 20  0  0  -^ 5^ 10^ 15  Distance along the rolling direction (cm)  (b) Pass 2 with nominal strain of 0.086  Effective Strain in Roll Gap  0  3rd pass  A 0  0 CO  0 O O  a)  00  O O  O  -•  cq  Ca 0 0  0  ^  5^10^15^20  Distance along the rolling direction (cm)  (c) Pass 3 with nominal strain of 0.125 Figure 8.8 Effective strain distribution in the roll bite  177  20  8.4.4 Thermal Field in the Roll Bite  It is evident that the strain distribution in the roll bite is non-uniform. The strain is higher at the surface than at the center line due to the effects of the redundant shearing brought about by constraining the metal to flow through the roll gap, and the consequent higher local strain rate. Over a region at the center of the slab near the exit of the roll bite, the computed strain approaches the nominal effective strain corresponding to the applied reduction. The figures also show the strong influence of redundant shearing, since the effective strain at the exit of the roll bite is larger than the nominal strain, which is consistent with results obtained by Beynon  et al.E 531 .  8.4.4 Thermal Field in the Roll Bite The thermal field in the roll bite can also be determined with this model. In the present study, the through thickness temperature distribution along the boundary at the entry side and the heat flux at the roll/steel interface were given by the finite difference model developed in Chapter 5. Fig.8.9 (a) to (c) shows the temperature distribution in the roll bite for Passes 1, 2 and 3 in 9-pass rolling.  178  ^  8.4.4 Thermal Field in the Roll Bite  Temperature Distribution in Roll Gap 1st pass  ROLL 1160  -1000.^• ^1240  1240  ------------  0^5^10^15^20^25 Distance along the rolling direction (cm)  (a) Pass 1  Temperature Distribution in Roll Gap 2nd pass  40z^  ROLL ___ _^_ ^ 1160^_______ 1160  0 — C.1  -------------------  ---  -1240---------------  -  0 s. m  -  CU  0 O  0^5^10^15  Distance along the rolling direction (cm)  (b) Pass 2  20  8.4.4 Thermal Field in the Roll Bite  Temperature Distribution in Roll Gap 3rd pass  "ff  1080ROLL  el ieo ------_______-__ ---_-o , ,__ _________ H __,..,, -_-__ ,—, 120 ----1 ^_______________1160:---^-:-  _______________  124o  le° ^______ 1240  to (u ,  L.^  ,1‘..,a1/.  .‘  .*.)  C 2 C  as co c  ,  20  Distance along the rolling direction (cm)  (c) Pass 3 Figure 8.9 Temperature distribution in the roll bite The roll chilling can also be observed from the figures and is confined to a very thin layer, the temperature distribution over two thirds of the half thickness does not change significantly. This is consistent with the prediction of the finite-difference model. From the deformation analysis, it is evident that the deformation is inhomogeneous, which results in a non-uniform generation of heat. The results corresponding to the temperature rise due to deformation along the roll bite are shown in Fig.8. 10 (a) and (b) for two different rolling conditions, Pass 1 and Pass 2 in 9-pass rolling.  180  8.4.4 Thermal Field in the Roll Bite  Temperature Rise in Roll Gap 1st, pass  ROLL  0  25  5^10^15^20  Distance along the rolling direction (cm)  (a) Pass 1  Temperature Rise in Roll Gap 2nd pass  ROLL 0  0.5o ^  3.0 0  5^  - ---------  10^  15  ^  Distance along the rolling direction (cm)  (b) Pass 2 Figure 8.10 Temperature rise distribution in the roll bite  181  20  8.5.1 Deformation Observation from Pilot Tests  The temperature rise due to deformation is much less near the center than near surface. This is consistent with the strain rate and temperature distribution, because the low temperature near the surface results in a high flow stress and the temperature rise is proportional to the product of the flow stress and the strain rate. The maximum of temperature rise is approximately 38°C for the first pass and 14°C for the second pass at the exit of the roll bite; This number is dependent on the roll speed and reduction. Although the highest temperature rise occurs near the surface, the surface temperature of the slab is still a minimum in the roll gap due to roll chilling This also indicates that the roll chilling effect cannot be ignored for the thermomechanical analysis of steels.  8.5 Validation of the Model 8.5.1 Deformation Observation from Pilot Tests Due to the importance of the strain distribution on the evolution of the microstructure, the strain distribution predicted in the roll bite should be verified by experimental measurements. This was accomplished by conducting pilot mill tests at CANMET, SS-8, SS-15 and SS-19 instrumented with pins, as described earlier. The test conditions are the same as listed in Table 4.6 and Table 4.7 in Chapter 4. In the test, a 5mm diameter pin of the same material as the specimen was inserted into a 5 mm diameter hole located at the center of the specimen. The deflection profile of the pin subsequent to rolling was employed to determine the effective strain distribution through the thickness, as described by Sakai et al.r713 . In their study, the effective strain and redundant shear strain (y) were estimated on the basis of three assumptions: 1) The ratio of incremental shear strain to incremental compressive strain is constant during rolling; 2)  The incremental compressive strain is uniform through the thickness;  182  8.5.1 Deformation Observation from Pilot Tests  3) Plane strain conditions prevail in the deformation zone, - 2  1 -r  6.^ n^ —  = 24412- 1111  where  1  1^^  (8.33)  tan%  (8.34)  1-r  .,\ 1 + f (1 - r) 2 r(2 -r)  (8.32)  j  where r is the percent reduction in thickness; tan% is the slope of the interface between the pin and sample, as shown in Fig.8.11.  x HO Y  Embedded pin Before rolling  ^  ^  y=(y1 + y2)/2 After rolling  Figure 8.11 Illustration of deformation of the embedded pin and the apparent shear angle 0 1 After the test, the values of deflection, (x, y), at each position along the interface was measured for each side of the pin, and an average was obtained for the two sides of the pin. A polynomial regression was performed on the data (x, y), and numerical differentiation (dy/dx) was employed to obtain the tan0 1 . Fig.8.12(a) to (c) shows the distribution of effective strain through the half thickness of a specimen predicted by the FEM model and measured from the pilot mill tests. 183  8.5.1 Deformation Observation from Pilot Tests  0.8 0.750.7j  0.650.6  U Z.: 0.55 0.50.45-  0.4  0  1^2^3^4^5^6^';  9  Half thickness (mm)  (a) Test SS-8 with nominal strain of 0.5063 1^ 0.95 0.9 g  o  0.85  S-  In  0.8 -  g3 0.75  0.7 0.65 0.6 0.55 0.5  0  1  3^4^5^6 Half thickness (mm)  7  (b) Test SS-15 with nominal strain of 0.5416  184  8  9  ^  8.5.1 Deformation Observation from Pilot Tests  0.8 ^ 0.75 0.7 -  f 0.65).  E  IT)  0.6 -  Ua)  :7- 0.55 w -  0.5 0.45 0.4  0  1  2^3^4^5^6 Half thickness (mm)  8  ^  9  (c) Test SS-19 with nominal strain of 0.4681 Figure 8.12 Comparison of the effective strain distribution predicted by the model and the pilot mill tests There is a good agreement between measurements and predictions for the strain over the region near the center. The deviation near the surface is probably due to separation between the pin and the specimen, as seen in Fig. 8.13. The test conditions are shown in Table 8.3.  185  8.5.2 Roll Force Prediction  3^9^1 0 5.0-7E19 4  I  1 1^1 2^1' 3  234667897 '12  VIII^III^III^IIII,IIIIII^III^III^III^III 1111111H111111111111111111111111111  Figure 8.13 Deformation of the pin in a pilot mill test SS-19 8.5.2 Roll Force Prediction The roll force can be calculated from Eq.(8.17) for each pass of the 9-pass rolling by summing the normal component of nodal pressure along the arc of contact. The comparison between the predicted and measured roll force for a 9-pass rolling schedule at Stelco is shown in Fig.8.14 and Table 8.2. The rolling conditions were given in Table 7.2.  186  8.5.2 Roll Force Prediction  25 20  0 -  E 0  Measured  z  15  FEM -  0  8  5-  0  0  3^4^5^6^7 Pass No. for 9-pass Schedule  8  ^  9  Figure 8.14 Comparison of the roll forces predicted and measured during 9-pass rolling Table 8.2 Comparison of the measured roll force with the FEM prediction for 9-pass rolling Pass No.  Friction Reduction Coefficient (%) 010  Roll Force (Tons/cm) (Measured)  Roll Force (Tons/cm) (FEM)  Relative Error (%)  1  14.0  0.45  10.10  9.19  -8.98  2  7.14  0.45  6.29  5.77  -8.26  3  10.26  0.45  8.92  6.93  -22.26  4  11.43  0.45  7.84  7.17  -8.55  5  9.68  0.45  8.12  6.07  -25.20  6  17.86  0.45  10.08  9.29  -7.82  7  26.09  0.45  13.20  12.53  -5.07  8  44.71  0.45  17.28  20.01  +15.80  9  48.94  0.45  16.87  20.90  +23.89  187  8.5.2 Roll Force Prediction  It is evident that reasonable agreement has been obtained for the 9-pass rolling schedule with the exception of some passes where deviations between measured and predicted were significant. The deviations could be attributed to the error in the measured data for those passes because some of the data does not appear reasonable; for example, the percent reduction for Pass 4 is larger than that of Pass 3 and the rolling temperature is lower, but the measured roll force for Pass 4 is smaller than Pass 3. The situation is similar for Pass 5, Pass 8 and Pass 9, each of which results in a large deviation between the measured and the predicted roll force. The friction coefficient for each pass was assumed to be 0.45 due to the high rolling temperature r211 . The roll forces obtained for the pilot mill tests are also compared in Table 8.3.  Table 8.3 Comparison of the measured roll force with the FEM prediction for the CANMET stainless steel tests Test No.  Reduction Rolling (%)  Temp. (°C)  Roll  Friction  Speed Coefficient (m/s) (11)  Roll  Roll  Relative  Force  Force  Error  (Tons)  (Tons)  (%)  (measured)  (FEM)  SS-8  35.5  850  1.5  0.30  200.80  232.00  +15.54  SS-15  38.9  950  0.5  0.30  161.0  181.00  +12.42  SS-19  34.9  1050  1.5  0.30  140.00  132.86  -5.28  Because the rolling temperature was relative low, a friction coefficient of 0.3 was employed for each test. The deviation in the roll forces is probably a result of the equation for flow stress for the AISI 304L steel which is more suitable for high strain rate conditions f723 .  188  9.1 Summary and Conclusion  Chapter 9  SUMMARY AND CONCLUSION  In this study, the thermal and mechanical behavior of a slab during rough rolling has been investigated, and some conclusions can be made. Recommendations for the future work have also been made.  9.1 Summary and Conclusion A mathematical model has been developed to predict the thermal history of a slab during rough rolling. The model takes into consideration cooling due to high pressure water, roll chilling, radiation with natural convection before and after rolling, and heat generation due to friction and plastic deformation in the roll bite. The model is also able to calculate the thermal history at each position along the length of the slab and therefore able to determine the difference in temperature distribution at the head and tail end of the slab. Due to the coupling of the slab and the work roll in the roll bite, a module for heat transfer calculation has been developed for the roll bite which includes the work roll. It accounts for the heat gain in the roll due to the contact with the slab and the subsequent cooling of the roll due to water spray out of the roll bite. It is assumed that the rolls reach a cyclic steady state. The model is based on one dimensional transient heat flow by transforming the rolling direction coordinate into the time coordinate. A completely implicit finite difference technique is adopted to solve the governing equations. To characterize the heat transfer coefficient at the roll/slab interface, pilot mill tests were conducted at CANMET and at UBC using specimens instrumented with thermocouples which were  189  9.1 Summary and Conclusion  spot welded on the surface. Different rolling conditions were examined to measure their effect on the thermal response of the sample surface. The roughing model was modified to back-calculate a heat transfer coefficient to yield temperatures which match the measured surface temperature. A trial-and-error method was adopted in the calculation. The following are the main findings from the analysis of the pilot mill test results. (a)  The heat transfer coefficient(HTC) increases gradually at the beginning of the roll bite, and then more rapidly until a maximum value is reached; finally the heat transfer coefficient decreases until the exit of the roll bite;  (b)  The maximum value of heat transfer coefficient varies from 25kW/m 2 -°C to approximately 700kW/m2 -°C. This values increases with percent reduction, increasing roll speed, decreasing temperature and increasing strength of material;  (c)  The variation of the heat transfer coefficient in the roll bite corresponds well with the distribution of roll pressure; a relationship between the two has been proposed;  (d)  In order to apply the pilot mill test results to the industrial rolling, a mean heat transfer coefficient is obtained from the heat transfer coefficient variation in the roll bite. It was found that the mean value is linearly dependent on the mean roll pressure; the influence of the rolling parameters, temperature, roll speed, percent reduction on the mean heat-transfer coefficient, can be attributed to the influence of roll pressure;  (e) A preliminary theoretical consideration also reveals a linear relationship between the mean heat transfer coefficient and the mean roll pressure for each of materials examined (0.05%C plain carbon steel, 0.05%C plus 0.025%Nb steel, AISI 3041 stainless steel) for conditions of no lubrication and no oxidation.  190  9.1 Summary and Conclusion  The roughing model has been applied to predict the thermal history of a plain carbon steel slab for a 7- and 9-pass rolling schedule at Stelco's LEW. The roll gap heat transfer coefficient for each pass is estimated from the mean roll pressure. The model is validated by comparing the predicted temperature with results from the literature. To consider the effect of oxide scale on heat transfer, an oxide scale growth module is included in the roughing model; a parabolic law is used to account for the oxide scale growth during rough rolling. The important results are described below. (a)  The model predicted surface temperature reflects the cooling experienced by the slab during rough rolling. For instance contact with the rolls depresses the surface temperature by approximately 250°C to 350°C in the roll bite. On exit from the roll bite, the surface temperature rebounds very quickly due to the heat conduction from the interior. After reaching a maximum temperature, it decreases more gradually due to radiative heat loss. The surface temperature also undergoes a very steep change due to water spray descaling after which it also rebounds quickly.  (b)  The roll chilling is confined only to a very thin layer of the slab thickness with a depth of approximately H/40 during rough rolling. The center temperature increases in the roll bite due to the heat of deformation, which indicates that the heat generation due to deformation must not be ignored for the heat transfer analysis;  (c)  The fact that the temperature distribution at the tail end is lower than that at the head end at the same position is confirmed by the model prediction;  (d)  The thermal history is specific for each rolling schedule. Therefore, for the prediction of the evolution of the microstructure during rolling, the corresponding thermal history must be determined by the specific rolling schedule;  191  9.1 Summary and Conclusion  (e) The secondary scale growth is limited to about 100i.tm during 7-pass rough rolling. It is mainly dependent on the rolling temperature and rolling speed; the oxide scale has an insulating function and its effect on heat transfer of the slab is dependent on the scale thickness. In the case of rough rolling, a much larger effect on heat transfer from the initial scale has been obtained, in comparison to effect of the secondary oxide scale, because the thickness of the initial scale formed in the reheating furnace is from 1.5mm to 3.0mm This indicates that the oxide scale effect cannot be ignored during rough rolling.  To gain insight on the deformation behavior of steel during rolling, a 2-D finite element program was modified and applied to the roughing process. It has been verified by an analytical solution and a pilot mill test. In the program, a thermally coupled velocity-pressure approach was adopted. The solution was iteratively obtained by solving the energy equation for temperature and the basic deformation equation for velocities. The model has been applied to the industrial rolling process at Stelco's LEW for 9-pass schedule. The strain rate distribution has been obtained directly from the velocity solution, whilst the strain distribution is obtained by the integration of the effective strain rate along the stream lines in the roll bite. The deformation behaviors for several typical rolling passes, including the temperature distribution, have been presented and the measured roll forces are compared with the model prediction for a 9-pass rolling schedule. To validate the model, three pilot mill tests were conducted at CANMET using stainless steel specimens. In the test, the effective strain distribution through the thickness at the exit of the roll bite was investigated by inserting a pin of the same material into the specimen. The measured effective strain has been compared with model prediction. The conclusions for the deformation analysis are as follow. (a) The model predicted effective strain distribution and the roll forces for industrial rolling and laboratory rolling compare favorably with the measured values. Although differences in  192  9.2 Future Work  effective strain arise at the surface, this is probably due to the separation between the pin and the specimen at this location. The difference in roll forces for some passes may arise from measurement error; (b)  Model predictions reveal that deformation within the slab is non-uniform during rolling. The highest strain rate distribution is concentrated just beneath the surface around the entry point and exit point, whilst between these two regions is a dead zone with a low strain rate. The nominal effective strain is only approached over the central region at the exit of the roll bite; the strain increases considerably as the surface is approached due to the redundant shearing;  (c)  The temperature rise due to heat of deformation is higher near the surface than near the center line, but is mashed by roll chilling.  9.2 Future Work Thermomechanical analysis of the slab has been conducted, paving the way for future work which should focus on the prediction of evolution of microstructure and the mechanical properties during rolling. For the prediction of evolution of microstructure of HS LA steels, considerable work is still required to develop the equations describing the precipitation kinetics and its effect on recrystallization. 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Sims, "The Calculation of Roll Force and Torque in Hot Rolling Mills", Proc. Instn. Mech. Engrs., 168,(1954), pp. 191-200.  [58]  An OMEGA Group Company, 'OMEGA Complete Temperature Measurement Handbook and Encyclopedia', OMEGA Engineering, Inc., Vol. 26, T-12, 1988.  [59]  B.I.S.R.A., Ed., 'Physical Constants of some Commercial Steels at Elevated Temperatures', Butterworths, London, 1953, pp. 1-38.  [60]  F.C. Kohring, Iron and Steel Eng., Vol.62, 1985, (6), pp. 30-36.  [61]  B.G. Thomas, I.V. Samarasekera, J.K. Brimacombe, "Comparison of Numerical Modeling Techniques for Complex, Dimensional, Transient Heat-Conduction Problems", Metallurgical Transactions B, Vol. 15B, June, 1984, pp. 307-318.  [62]  J. Pullen and J.B.P. Williamson, Proc. R. Soc. Lond., 327A, 159-173(1973).  [63]  B.B. Mikic, "Thermal Contact Conductance; Theoretical Considerations", Int. J. Heat Mass Transfer, Vol.17, pp. 205-214.  [64]  M.G. Cooper, B.B. Mikic, and M.M. Yovanovich, "Thermal Contact Conductance", Int. J. Heat Mass Transfer, Vol.12, pp. 279-300.  [65]  Y.Misaka, T.yokoi, R. Takahashi and H. Nagai, J. Iron and Steel Inst. Japan, 67, 1981, A53.  [66]  'The Oxide Handbook', Translated from Russian by C. Nigel Turton and Tatiana I. Turton, IFI/PLENUM, 1973.  [67]  G.I. Kolchenko and N.P. Kuznetsova, "Coefficient of heat conductivity of scale at high temperature", Izvestiga Vysshikh Uchebnykh Zavedenii, Chemaya Metallurgiya, 1984, (11), 141.  199  REFERENCE  [68]  S.V. Patanker, ' Numerical Heat Transfer and Fluid Flow', Hemisphere Publication Corp., New York, 1980.  [69]  P.Hood, " Frontal solution program for unsymmetric matrices", Int. J. For Numerical Methods in Engineering, Vol. 10, pp. 379-399 (1976).  [70]  L.R. Underwood, 'The Rolling of Metals, Theory and Experiment', John Wiley & Sons Inc., New York, 1950, pp. 57-93.  [71]  Tetsuo Sakai, Yoshihiro Saito, Kenji Hirano, "Deformation and recrystallization of low carbon steel in high speed hot rolling", Transactions ISIJ, Vol.28, 1988, pp. 1028-1035.  [72]  S.L. Semiatin and J.H. Holbrook, " Plastic Flow Phenomenology of 304L Stainless Steel", Metallurgical Trans. A, Vol. 14A, Aug. 1983, pp. 1681-1694  [73]  O.C. Ziewkiewicz, K. Morgan, ' Finite Element and Approximation', New York, Wiley, 1983.  200  A.1 Nodes in the S lab  APPENDICES  Appendix A Finite Difference Nodal Equations A completely implicit finite difference technique has been used in solving the one dimensional transient heat transfer problem in this study. By the technique, the domain of heat transfer could be divided into a number of time steps, at each time step, a slice in the domain could also be divided into a number of nodes with either equal spacing or unequal spacing. In the slice, there are three kinds of nodes for the slab and the work roll which should be specified, namely: i) Nodes at the surface which is in exposure to environment; ii) Interior nodes; iii) Adiabatic nodes at the center line of the slab or at the inner side surface of the layer in the rolls. Because different coordinate systems ware adopted for the slab and the rolls, i.e, rectangular system and circular system, different nodal equations were specified as below.  A.1 Nodes in the Slab The slice in the domain at each time step experienced different environment inside the roll gap and outside the roll gap, but a general heat transfer coefficient h(t) would represent the different situations, therefore, general equations for each of the three nodal equations can be obtained as follows by the control volume method. Surface Nodes  For the surface nodes, a heat balance can be set up for the control volume as shown in Fig. A.1.  201  A.1 Nodes in the Slab  Figure A.1 Schematic diagram of the control volume for the nodes in the slab The roll chilling was considered by coupling the roll surface node, Nr, by means of the roll gap heat transfer coefficient.  n'  —Tk.,  l +1 s+1 1 Ics ^ Ax  + h(t)(T: TNsn +1 ) Ax + Ay^ cos(9) .  f  1^n  TV At TN, A2y 4, ) ay^ps cps^ (A.1) g  2  where q f, 4g are the heat generation due to friction and deformation, respectively; Oi is the contact angle in the roll bite; they all become zero outside the roll bite.  202  A.2 Nodes in the Rolls  Interior Nodes  A similar heat balance can be also set up for the interior nodes in the slab as shown in Fig.A.1. 1  — T n^T:t++11^+ 1^Tin + ks ^ + g Ay Ax = p,,C i„ ^ Ay Ax^(A.2) Ay^Ay^ At  i 7 +1 1 kr ^  Center Line Nodes  For the nodes at the center line, owning to the assumption of an adiabatic boundary, the heat balance can be given as below, see also in Fig.A.1. Tr —Tr'^ Tr1 ks ^ Ax + AyAx = p,Cp, ^ Ay Ax Ay^ At  (A .3)  A.2 Nodes in the Rolls The slice in the surface layer of rolls is in circular coordinate system. The nodal equations for the three kinds of nodes can be expressed as follows. Surface Nodes  The heat balance for the roll surface nodes can be set up as below, see Fig.A.2 kr  nr+11 —nr+1^Arr+1 —^Ar^Ar Ar^R^AO + h O A ) (TZ 1 1^r+ 1 )R AO = p r Cpr At^R — AO ^(A.4) 4 )  2  where h(t*) is heat transfer coefficient along the roll surface including the roll gap heat transfer coefficient.  203  A.2 Nodes in the Rolls  Figure A.2 Schematic diagram of the control volume for the nodes in the rolls Interior Nodes  The heat balance for the interior nodes is as below: Tnil _ Tin +1 kr  Ar^  iv  (  r  2  Tn.:11_ Tn. + 1  AO + kr  Ar ■ r+  Ar  2 J AO  PrC Pr  Ti  n At  rAeAr (A .5)  Inside Surface Nodes  An adiabatic boundary condition was assumed to the inside surface, so the heat balance is set up as below: Tri  —77 +1^Ar^Trl^Ar —5+ AO = prCPr ^ R —5+Ar^2 At 4  kr ^ R  204  (A .6)  B.1 Deformation Formulation  Appendix B Velocity-Pressure Finite Element Derivation B.1 Deformation Formulation The governing equation for the deformation analysis was expressed as below which is the same as Eq.(8.1):  fv feil T falc1V  —  {8v} T {F}dV  —  s,  {SO T {t}dS = 0^(B .1)  Inserting the constitutive equation (8.10), we can write,  fv feil TPIDI {i}dV +  8evpdV —  {8v } T {F}dV —I {45v T T { -INS = 0^(B.2) s,  By using trial shaping functions as: {v} = ENT^=^{ay}  (B.3)  NiP a iP = [V] {aP {P}=^ r=1  (B.4)  i =1  }  where m, n are the total number of nodes per element for pressure and velocities; {ay}, aP} are the nodal velocities and pressure respectively, and substituting the Eqs.(8.4), (8.6) into the Eq.(B.2), we can get: [LW,][Arlf 1.1[D 0la][Incllifal +[1 . [(Mi r' [L][Nif [N P jdV] — f {1■11 T {F}dV  —j  [N1 T fildS = 0^  (B.5)  The terms [N7 and [N"] are referred to as shaping functions or interpolation functions which will be described later. 205  B.2 Thermal Coupling  From Eq.(8.5), we can write:  [DI Vi T [M] T^=  0^(B .6)  Combining the equations (B.5) and (B.6), a simple symmetric set of equations can be obtained as the same as Eq.(8.17): [Kulifal^{{i}}^0^(B .7) j[ail^o 0^t T^ L[KP]^ or [IC] {a} + {f} = 0^  (B .7(a))  where =  [[1][111 li ] T g[D °1[[L,][N]JcIV^  (B .8(a))  [M T [1 ]W v l] T  (B .8(b))  t  = — [All T IF^— s [N1 T {i}dS^  (B .8(c))  B.2 Thermal Coupling  For the thermal effect of the two-dimensional steady state problems, the governing equation (energy equation) in Cartesian co-ordinate system, Eq.(8.16), can be written as:  'a2-r ^a2r: + s  kS  ax 2  ay /  + 43^PsCps  206  aTs^aT's  U^± V  ax^ay  (B .9)  B.2 Thermal Coupling  where u and v are velocities along the two coordinate axes, X and Y. Using the trial shaping function: nt  = E Ni Ti = [N] {Ts } 1=1  (B.10)  where n e is the total number of nodes per element and [T s } is the nodal temperature value; Ni or [NJ is also referred to as the shaping function which has the same properties as described in the previous section; and applying Galerkin method of weighted residuals, and using integration by parts for second-order derivatives, the equation (B.9) reduces to:  ^aN.^aN. ^i E Ics K,;Ti + y, p c „Kw {F1 }+ s Ni ks (--e^ly){T}edS J.0^J.0 ;  s  (B.11)  where  aN, aN aN, aN `dxdy  K K`'  Kvu  and  ;  ;  (B.12(a))  ax ax^ay ay  aN ^aN ;  ;  =^f(uNi --a7+vNi 7-y-  =  y  ,.Ni dxdy  (B.12(b))  (B.12(c))  The last term on the right-hand side of Eq.(B.11) is not effective inside the domain and on the boundaries, where prescribed values give the necessary conditions. It vanishes if homogeneous natural boundary conditions are assumed in those sections of the boundary where T s is not specified, such as, in rolling case, at the roll/slab interface, the center line and the boundary of the exit side. But only the upper surface boundaries are non-homogeneous.  207  B.2 Thermal Coupling  Eq.(B.9) contains convective terms on the right hand side of the equation. If the mesh size exceeds a certain critical value, the solution are oscillatory, and at high velocities acceptable answers can only be obtained by an excessive reduction in the element size' s] . This difficulty can be overcome by introducing an upwinding technique which was originally used in the finite element methods in terms of 'upwinding difference' (backward differences with respect to the velocity direction) for the convective terms. The most important feature of the numerical schemes proposed relies in the choice of the weighting functions. In conventional Galerkin formulationsi n] , these are chosen equal to the shape functions, but it is clear that other choices are possible, and schemes where the weighting functions are not equal to the shaping function have been used [461 . For one dimensional problems, the weighting function were taken as: Wi =Wi (x,a)=Ni (x)—aF(x)^  (B.13)  for the corner nodes; and: VVi = Wi (x, f3) = Ni (x) +413F(x)  ^  (B.14)  for the middle nodes in eight-noded element. The function F(x) was given by: F(x) =  r I)2)7,1 1—k/T)+1  5 x  - ix 2 (x)  4/t e  0..x << — h e^(B.15)  where an element of span was assumed as (0, h e) and a and 13 are parameters. For two dimensional problems with an 8-noded element, 8 parameters are defined along the four sides of the element, i.e., a, and 0,, i =1, 4. They were expressed as below:  208  ^ ^  B.2 Thermal Coupling  Pi 12^12 o p ^(B + 2^y^y  0c, = 2(tanh ) 1 + 3  —  i  -  1 ^4 13,^  tanh(!) Y  .16)  (B.17)  ye  where y=-7-; u i; was calculated as below:  :7 7  2-1. (u->i -> uj  )  (B.18)  ' lij  where ui and u; are the velocity vectors at the node i and j; and /i; is a unit vector in the positive direction of the line through i and j. The sign of kJ also determines the sign of the parameters oc i and P i . An easy way of obtaining weights is by means of linear blending of the one-dimensional  weights. The weighting functions adopted in this study for each node are as follows: W1(x)Y, a(x)Y)) =^Y Ge2 a4F Wer^aiF(G e2)^(B .19(a)) W2 (x , y , 0(x , y )) = N2 (x , y) + 4He2 P I F (Ge, )^  (B .19(b))  W3 (X , y , cc(x , y )) = N3 (x, y) — Gei cc2F (I/ el ) — H e2 a 1 F (Gel )^(B .19(c)) .  Y ,(3 (x Y)) = 1\14(x ,Y) -1- 4 G eiP2F (HeI)^ W5 (x, y, ct(x, y))  (B .19(d))  = N5 (x, y)-- Gei oc2F^— -I (x2F (Gel )^(B .19(e))  Wo(x, Y 13 (x, Y )) = N6 (X, Y)+ 414/ f32F(Gei )^ -  W7 (x,y,a(x,y)) = 1V7 (x ,y)— Ge2 a4F (HeI) —  209  el  (B .19(f))  oc2F (Gel )^(B .19(g))  B .3 Isoparametric Element  14i8(x ,P(x,Y)) = Njx,y)  + 4Ge, 134F (Hei )^  (B .19(h))  where NA, y) are shaping functions; and Gel =  (1 +x G .) 2  (B .20(a ))  G e2  (1 —xG ) 2  (B .20(b))  Het  Het  -  (1 + ycaus)  (B .20(c))  2 (1 —xGaus ) 2  (B .20(d))  where xG„,„ and yGaus are the coordinate value at Gauss point in each element.  B.3 Isoparametric Element In the finite element method, interpolation of a field property f(x, y) defined over an element is introduced in a form: f(x,y)=  I, N;(x, y)f  ^  (B.21)  where f is a function value associated with ith node, and Ni (x, y) is the shaping function. It approximates the behavior of a field variable, for example velocity and pressure, over the domain of an element. Further, it is a function of position with the property that N, is unity at node i and zero at all other nodes and this produces the correct velocity and pressure at each node. There are various types of elements, depending upon the shape of the element and the polynomial order of shape functions. In this study, eight-noded isoparametric elements were used  210  B.3 Isoparametric Element  to better estimate the curve boundaries in the flow domain. Fig.B.1 shows a quadratic rectangular element with 8 nodes in the natural coordinate and global coordinate systems. The element defined in the natural coordinate system is sometimes called the parent element. 2  7  4  3  3  1  6  8 -1  4  1  -1  ^• 5^2  1  x  (b)  (a)  Figure B.1 Eight-node quadratic rectangular element (a) parent element in natural coordinate system and (b) isoparametric element after Cartesian mapping in global coordinate system The shaping functions were defined by: NA,I1)= 4 +4,4)( 1^+r1;T1-— 1)  (B.22(a))  for corner nodes; 1 Ni(4, 11) = —2 ( 1 — 42) + Ni (4, 11) = 2 ( 1 + 4,4) (1, -TC)  211  =0  (B.22(b))  11 1 = 0  (B .22(c))  B.3 Isoparametric Element  for mid-side nodes. where („ti t ) are the natural coordinates of a node at one of its corners. These functions have the properties that when =  4 and Il i = the value is unity but at any  other nodes E i # 4 and rh # rj the value is zero.^Therefore, using a simplified notation, nodal properties are defined as: 8  x =^NA,i)xi^(B 1=1  .23(a))  Y =1=E1 N,r1)y1  (B .23 (b))  {v} =^Ni (toi){vi } 1=1  (B .23(c))  T =^N A,i)T  (B .23(d))  But for the variable, pressure, due to the restriction of one degree of interpolation less than that of velocities, a 4-node isoparametric element was used with linear interpolation of shaping function as below: 1 NATI) = -4 ( 1 + U) +ill)  (B .24)  The nodal property for pressure is defined as below: 4  P = N,(4, 11)P, =1  212  (B .25)  B.4 Evaluation of Element Matrix  B.4 Evaluation of Element Matrix For the derivative of the field property, f, it is easy to be obtained according to the Eq.(B.21), i.e.:  of^aN, ax =  (B .26(a ))  of = aN.If 8  (B .26(b))  ay^i=i ay  The derivative of shaping functions to natural coordinates can be expressed as:  aN,^aN, axwa y ax^ay  (B .27(a))  aN,^aN, ax aN, a y  (B .27(b ))  ax an + ay an  or in matrix form:  aN,  aN, arl  - ax ay - {aN, a a a ax ay aN  _ an  an_ ay  ;  {aN, a } (B .28)  = Vl  ay  The matrix [J] is termed as the Jacobian matrix of the coordinate transformation. To convert the results to the global coordinate system the Jacobian matrix in inverted to change the Eq.(B.28) to:  {aN, ax aN, ay  =  213  aN,} a4 aN, an  (B .29)  B.4 Evaluation of Element Matrix  The derivatives of the field viable f ( which are velocity, temperature in the current study) can also be defined as:  {of ax of ay  =  am aN2 ax ax am aN2 ay ay  aN8 ax aN8  f2 fli  ay _  = [fl-1  aN, aN2  aN8  aN, am  aN8  a a4 an an  f8  a  1 f2  f8  (B.29)  The above transformation makes the Gauss-Legendre integration over the domain of each element possible noting that dxdy =Id] dEATI. 3 x 3 integration points have been used in the current study.  214  BIOGRAPHICAL INFORMATION  NAME:^CH^.^WEI  c)-0-,Q(.4-  MAILING ADDRESS: , of HF V LS Al-di) MA 14A (4L5 Ei/4 4 -  -  Q 4 C^v6 T PLACE AND DATE OF BIRTH: Z.H (AVG—  ,^4NA , juLY  5",  I 16 3  EDUCATION (Colleges and Universities attended, dates, and degrees):  SE!) /NIT Uivt vER,Si T\^4f41)^ -  U Ntv^ScA C^ALC) Toe,4-40 Lo cry  f  BEI^  POSITIONS HELD:  PUBLICATIONS (if necessary, use a second sheet):  AWARDS:  Complete one biographical form for each copy of a thesis presented to the Special Collections Division, University Library. DE-5  Lo Cr Y,  133 13 1,5c  forj, ,^Sc  

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