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

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THERMOMECHANICAL PHENOMENA DURING ROUGH ROLLINGOFSTEEL SLABbyWEI CHANG CHENB.A.Sc., Beijing University of Iron and Steel Technology, 1983M.Sc., University of Science and Technology, Beijing, 1986A Thesis submitted in partial fulfillment ofthe requirements for the degree ofMaster of Applied ScienceinTHE FACULTY OF GRADUATE STUDIESin theDEPARTMENT OF METALS AND MATERIALS ENGINEERINGWe accept this thesis as confirmingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIANovember 1991W.C. ChenIn presenting this thesis in partial fulfillment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available for referenceand study. I further agree that permission for extensive copying of this thesis for scholarly purposemay be granted by the head of my Department or by his or her representatives. It is understoodthat copying or publication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Metals and Materials EngineeringThe University of British ColumbiaVancouver, British ColumbiaCanada„Date .1 v^(AbstractA mathematical model has been developed to predict the temperature distribution through aslab during rough rolling. The heat transfer model is based on two-dimensional heat flow, andincludes the bulk heat flow due to high speed slab motion, but ignores heat conduction in the rollingdirection. At high temperature, an oxide scale forms due to exposure to air and must be consideredin the heat transfer analysis. The thick scale which forms during reheating helps to insulate the slabduring transportation of the slab to the rolling stands. Prior to rolling, the heavy oxide is removedby the descale sprays, and a much thinner oxide layer is formed during the very short exposure ofthe slab to air during rough rolling. The temperature results predicted by the model have beenvalidated by comparison with another model. The results show that the work roll chilling has asignificant 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 confinedto 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 andUBC. In these tests, the surface and the interior temperatures of specimens during rolling havebeen recorded using a data acquisition system. The corresponding heat transfer coefficients in theroll bite have been back-calculated by a trial-and-error method using the heat transfer modeldeveloped. The heat transfer coefficient has been found to increase along the arc of contact andreaches a maximum and then declines until the exit of the roll bite. It is important to note that themean heat transfer coefficient in the roll gap is strongly dependent on the mean roll pressure. Atlow 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 700kW/m2-°C. Obviously, the high pressure improves the contact between the roll and the slab surfacethereby reducing the resistance to heat flow. The mean roll gap heat transfer coefficient at theinterface (HTC) has been shown to be linearly related to the mean roll pressure. These results wereemployed to calculate the thermal history of the slab during industrial rough rolling; the results arein good agreement with the data in the literature.In addition to the thermal history, the strain and strain rate distribution also affect the evolutionof microstructure of rolled steels. In the present project, heat transfer and deformation during roughrolling of a slab have been analyzed with the aid of a coupled finite element model based on theflow formulation approach. In the model, sliding friction is assumed to prevail along the arc ofcontact and the effect of roll flattening has been incorporated. The model has been validated bycomparing the results from the pilot mill tests. It confirms that the deformation of a slab in the rollgap is inhomogeneous and just beneath the surface very high strain rates of approximately 5-10times the nominal strain rate are reached due to the redundant shearing. The maximum strain rateis attained at the entrance to the roll bite just beneath the rolls. The corresponding strain distributionthrough the thickness is also non-uniform, being lowest at the center and highest at the surface. Thetemperature gradient near the surface of the slab is very large due to work roll chilling; this isconsistent with results obtained from the finite-difference model. The predicted roll forces are ingood agreement with the measured values for the 9-pass schedule currently employed on theroughing mill at Stelco's Lake Erie Works and the pilot mill tests.iiiTable of ContentsAbstract ^  iiTable of Contents ^  ivTable of Tables  xiTable of Figures ^  xiiiNomenclature  xxiiAcknowledgements ^  xxxiiiChapter 1 INTRODUCTION ^  11.1 Rough Rolling ^  21.2 Thermomechanical Processing ^  31.3 Hot Deformation of Steel Slab during Rough Rolling ^  4Chapter 2 LITERATURE REVIEW ^ 82.1 Thermal Analysis of the Rough Rolling Process ^  82.1.1 Heat Transfer during Rough Rolling ^  82.1.1.1 Heat Transfer in the Slab  92.1.1.2 Heat Losses during Rolling ^  102.1.1.2.1 Radiative and Convective Heat Loss ^  10iv2.1.1.2.2 High Pressure Water Descaling ^  122.1.1.2.3 Roll Chilling ^  132.1.1.3 Heat Generation during Rolling ^  132.1.2 Heat Transfer in the Work Roll  152.1.2.1 Work Roll Cooling System ^  152.1.2.2 Heat Transfer in the Work Roll  162.1.3 Heat Transfer Characterization at the Roll-Slab Interface ^ 192.2 Oxidation of Steels at High Temperature ^  272.2.1 Oxidation Mechanisms of Metals at High Temperature ^ 272.2.2 Iron Oxide Scale Growth in Air ^  282.2.3 Effect of Oxide Scale on Heat Transfer ^  332.3 Finite Element Analysis on Hot Deformation  372.3.1 Governing Equations for Finite Element Analysis ^  372.3.2 Application of Finite Element Analysis in Metal Forming ^ 41Chapter 3 SCOPE AND OBJECTIVES ^ 443.1 Objectives and Scope ^  443.2 Methodology ^  45Chapter 4 EXPERIMENTAL MEASUREMENTS ^ 474.1 Test Design ^  47v4.1.1 Thermocouple Design and Data Acquisition System ^  484.1.2 Preparation of Samples ^  494.1.3 Test Facilities ^  514.2 Test Procedures  534.2.1 Test Schedule at CANMET^ 534.2.2 Test Schedule at UBC  574.3 Thermal Responses of Instrumented Specimens ^  594.3.1 Thermal Responses ^  594.3.2 Surface Temperature  64Chapter 5 HEAT TRANSFER MODEL DEVELOPMENT ^ 705.1 Mathematical Modelling ^  705.1.1 Heat Conduction in Slab during Rough Rolling ^  705.1.2 Boundary Conditions ^  735.1.2.1 Initial Condition  735.1.2.2 Boundary Conditions ^  735.1.3 Heat Conduction in the Work Roll ^  765.1.4 Numerical Solution ^  785.2 Modification of the Model for Roll Gap Heat-Transfer Coefficient Calculation ^ 815.2.1 HTC Solution ^  81vi5.2.2 Convergence of the Numerical Solution for the Modified Model ^ 825.2.3 Verification of the Modified Model ^  865.3 Sensitivity Analysis for the Roughing Model  875.4 Verification of the Roughing Model ^  93Chapter 6 ROLL GAP HEAT TRANSFER COEFFICIENT ANALYSIS ^ 986.1 Roll Gap Heat Transfer Coefficient Analysis ^  986.1.1 HTC Variation along the Arc of Contact  986.1.2 Influences of Rolling Parameters on HTC ^  1056.1.3 Pressure Dependence of HTC ^  1096.2 A Preliminary Theoretical Consideration of HTC during Hot Rolling ^ 1156.2.1 Fenech et al.'s Model ^  1156.2.2 Roll Gap Heat transfer Coefficient (HTC) ^  1176.3 Discussion ^  1206.4 Summary  122Chapter 7 THERMAL PHENOMENA DURING ROUGH ROLLING ^ 1237.1 Heat Transfer Characterizations during Rough Rolling ^  1237.1.1 Heat Transfer Characterizations of a Slab ^  1237.1.2 Temperature Distribution in the Work Roll  1357.2 Oxide Scale Growth of Steels during Rough Rolling ^  137vii7.2.1 Oxide Scale Growth Rate of Steels at High Temperature ^ 1377.2.2 Assumptions for Oxide Scale Formation on the Steel Slab  1407.2.3 Oxide Scale Growth of Steels during Rough Rolling ^  1437.3 Oxidation Effect on Heat Transfer of a Slab ^  1447.3.1 Effect of Emissivity of Oxide Scale  1457.3.2 Effect of Oxide Scale Thickness ^  1467.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling ^ 1477.3.3.1 Temperature Distributions in the Roll Gap ^  1477.3.3.2 Thermal Histories of Slab ^  1507.3.3.3 Effect of Rolled Materials on Thermal History of slab ^ 1527.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling  156Chapter 8 DEFORMATION ANALYSIS ON A SLAB DURING ROUGH ROLLING^  1588.1 Velocity-Pressure Formulation ^  1588.1.1 Formulation for Deformation  1588.1.2 Thermal Coupling ^  1618.2 Finite Element Solution  1618.2.1 Finite Element Discretization ^  1618.2.2 Mechanical and Thermal Boundary Conditions ^  163viii8.2.3 Sequence of the Solution ^  1648.3 Verification of the Source-Code of the FEM Program ^  1668.3.1 Heat Transfer Analysis of a Plate ^  1668.3.2 Interface Temperature Distribution of a Pilot Mill Test ^ 1698.4 Rough Rolling Deformation Analysis ^  1708.4.1 Velocity profiles ^  1708.4.2 Strain Rate Distribution  1728.4.3 Strain Distribution ^  1758.4.4 Thermal Field in the Roll Bite ^  1788.5 Validation of the Model ^  1828.5.1 Deformation Observation from Pilot Mill Tests ^  1828.5.2 Roll Force Prediction ^  186Chapter 9 SUMMARY AND CONCLUSION ^  1899.1 Summary and Conclusion ^  1899.2 Future Work ^  193REFERENCES  194APPENDICES ^ 201A Finite Difference Nodal Equations ^  201A.1 Nodes in the Slab ^  201ixA.2 Nodes in the Rolls ^  203B Velocity-Pressure Finite Element Derivation ^  205B.1 Deformation Formulation ^  205B.2 Thermal Coupling ^  206B.3 Iso-parametric Element  210B.4 Evaluation of Element Matrix ^  213xTable of TablesTable 4.1 Specifications of the pilot mill at CANMET ^ 51Table 4.2 Specifications of the pilot mill at UBC ^  51Table 4.3 Conditions employed in rolling tests to determine the influenceof successive rolling passes on interface heat transfer coefficientfor AISI 304L stainless steel ^  53Table 4.4 Conditions employed in rolling tests to determine the influenceof rolling pressure on interface heat transfer coefficient^for a 0.05%C low carbon steel    54Table 4.5 Tests for microstructural evolution study ^  55Table 4.6 Conditions employed in rolling tests to determine the influenceof rolling temperature on interface heat transfer coefficientfor AISI 304L stainless steel ^  55Table 4.7 Tests conducted to determine the influences of rolling speed on interfaceheat-transfer coefficient ^  56Table 4.8 Tests for HTC measurements at UBC ^  57Table 4.9 Regression results for tests at CANMET ^ 68Table 4.10 Regression results for tests at UBC ^  69xiTable 5.1 Sensitivity of time step size on the predicted surface temperature ^ 88Table 5.2 Sensitivity of node size on the predicted surface temperature ^ 90Table 5.3 Sensitivity of emissivity on the predicted surface temperature ^ 91Table 5.4 Conditions used in validation of the work roll module ^  96Table 6.1 Rolling Conditions for material type influence on the HTC ^ 108Table 6.2 Mean pressure for tests at CANMET ^  113Table 6.3 Mean pressure for tests at UBC ^  113Table 7.1 Operating conditions for the 7-pass schedule ^  124Table 7.2 Operating conditions for the 9-pass schedule  132Table 8.1 Typical deformation parameters for the 9-pass schedule ^  170Table 8.2 Comparison of the measured roll force with the FEM prediction for 9-passrolling ^  187Table 8.3 Comparison of the measured roll forces with the FEM prediction for theCANMET stainless steel tests ^  188xiiTable of FiguresFigure 1.1 Mathematical models in thermomechanical processing ^  6Figure 1.2 A typical layout of 1/2 continuous hot strip mill ^  7Figure 2.1 Spray cooling of work roll during rolling process ^  16Figure 2.2 Contact between real surfaces ^  21Figure 2.3 Temperature distribution through surface in contact ^  21Figure 2.4 Fenech et al's heat transfer model for thermal contact ^  22Figure 2.5 Calculated and measured h e vs. apparent pressure ^  24Figure 2.6 The relationship between the roll gap heat-transfer coefficient and the meanpressure along the arc of contact for two successive passes on the pilot mill ^ 26Figure 2.7 Schematic diagram of multi-layer of iron oxide scale ^  28Figure 2.8 Relative thickness of magnetite (Fe 304) and hematite (Fe 2O3) ^ 31Figure 2.9 Emissivity of oxide scale during metal heating ^  34Figure 2.10 Intensity of heat transfer through oxide scale during reheating ^ 35Figure 2.11 Dependence of temperature through thickness on heating time ^ 36Figure 3.1 Methodology adopted in thermomechanical analysis ^  46Figure 4.1 Schematic diagram of specimen employed in the thermal responsemeasurements ^  49Figure 4.2 Schematic layout of the test facilities at UBC ^  52Figure 4.3 Thermal response of thermocouples for Test RLC12-1 ^  59Figure 4.4 Thermal response of thermocouples for Test 1LC-6  60Figure 4.5 Thermal response of thermocouples during tests at CANMET(SS6P71) ^ 61Figure 4.6 Thermal response of thermocouples for Test 3 ^  62Figure 4.7 Thermal response of thermocouples for Test 7-1  63Figure 4.8 Thermal response of thermocouples for Test 5-1 ^  63Figure 4.9 Surface temperature in the roll bite for Test SS6P71 with 1.0m/s ^ 64Figure 4.10 Surface temperature in the roll bite for Test SS-8 with 1.5m/s ^ 65Figure 4.11 Surface temperature in the roll bite for Test SS-15 with 0.5m/s ^ 65Figure 4.12 Surface temperature in the roll bite for Test 6 ^  67Figure 4.13 Surface temperature in the roll bite for Test 8-1 ^  67Figure 5.1 Schematic diagram of hot rolling ^  71xivFigure 5.2 Discretization of the slice in the slab and the roll for finite difference analysis^  79Figure 5.3 Flow chart of the temperature solution of the model ^  80Figure 5.4 Flow chart of HTC calculation ^  81Figure 5.5 Effect of Eps on HTC magnitude ^  82Figure 5.6 Effect of mesh size and time step on accuracy ^  84Figure 5.7 Effect of time step on HTC value ^  85Figure 5.8 Effect of mesh size on HTC value ^  85Figure 5.9 Comparison of the predicted and the measured temperature ^ 86Figure 5.10 Effect of time-step sizes on the surface temperature under radiative andnatural convective cooling ^  87Figure 5.11 Effect of node size on the predicted surface temperatures under radiative andnatural convective cooling ^  89Figure 5.12 Effect of emissivity on the surface temperature under radiative and naturalconvective cooling ^  90Figure 5.13 Effect of the roll gap heat transfer coefficient on the surface temperature in theroll bite ^  92Figure 5.14 Comparison of the temperature distribution predicted by the current modelwith that of Devadas ^  93xvFigure 5.15 Comparison of the temperature distribution predicted by the current modelunder descaling with that of Devadas ^  94Figure 5.16 Comparison of the temperature distribution predicted by the current model inthe roll bite with that of Devadas ^  94Figure 5.17 Comparison of the thermal history of a strip predicted by the roughing modelwith the data from Devadas ^  95Figure 5.18 Comparison of the model results with an analytical solution for the work roll^  95Figure 6.1 HTC variation for Test SS6P71 ^  99Figure 6.2 HTC variation for Test SS8  100Figure 6.3 HTC variation for Test SS9 ^  101Figure 6.4 HTC variation for Test SS18  101Figure 6.5 HTC variation for Test SS19 ^  102Figure 6.6 HTC variation for Test 1LC-6  103Figure 6.7 HTC variation for Test 3LC-1 ^  103Figure 6.8 HTC variation for Test 8-1  104Figure 6.9 Effect of roll reduction on HTC for a rolling temperature of 950°C and a rollspeed of 1.5m/s ^  105xviFigure 6.10 Effect of roll speed on HTC for a rolling temperature of 1050°C and 38.9%reduction ^  106Figure 6.11 Influence of rolling temperature on HTC for approximately 35% reductionand 1.5m/s ^  107Figure 6.12 Influence of material type on the magnitude of HTC ^  108Figure 6.13 Distribution of HTC and roll pressure in the roll bite for SS-19 ^ 109Figure 6.14 Distribution of HTC and roll pressure in the roll bite for SS-15 ^ 110Figure 6.15 Distribution of HTC and roll pressure in the roll bite for SS-8 ^ 110Figure 6.16 Comparison of surface temperature predicted by mean HTC with themeasured one for Test S6P71 ^  112Figure 6.17 Relation of mean HTC with mean roll pressure ^  114Figure 6.18 Hardness data for stainless steel-416 ^  118Figure 6.19 Mean HTC data vs. mean roll pressure for the tests conducted at CANMETand at UBC^  120Figure 6.20 Specimen surface profiles before and after rolling ^  121Figure 7.1 Thermal history half-way along the length of the slab during 7-pass rolling of a0.05%C plain carbon steel ^  125Figure 7.2 The difference of thermal histories for the head and tail end during 7-passrolling of 0.05%C plain carbon steel ^  127xviiFigure 7.3 The temperature distribution of the slab in the roll bite during 7-pass rolling ofa 0.05%C plain carbon steel ^  131Figure 7.4 Thermal history half-way along the length of the slab during 9-pass rolling of a0.05%C plain carbon steel ^  133Figure 7.5 The difference of thermal histories for the head and tail end during 9-passrolling of a 0.05%C plain carbon steel ^  134Figure 7.6 Comparison of thermal histories for the 7 and 9-pass rolling of a 0.05%C plaincarbon steel ^  135Figure 7.7 Temperature distribution in the work roll during 7-pass rolling of a 0.05%Cplain carbon steel ^  136Figure 7.8 Oxide scale growth of iron and some steels at 1200°C ^  139Figure 7.9 Distances associated with the interface i ^  141Figure 7.10 Oxide scale thickness half-way along the length ofthe slab during 7-pass rolling of a 0.05%C plain carbon steel ^  143Figure 7.11 Effects of mesh size on surface temperature ^  144Figure 7.12 Effects of oxide scale emissivity on surface temperature ^ 145Figure 7.13 Effects of oxide scale thickness on the surface temperature of the slab ^ 146Figure 7.14 Effects of oxide scale thickness on the temperature at the scale/steel interface^  147xviiiFigure 7.15 Temperature distribution half-way along the length of slab in the roll biteduring 7-pass rolling of a 0.05%C plain carbon steel with oxidation ^ 148Figure 7.16 Surface and interface temperature half-way along the length of the slab in theroll bite with and without oxidation for Pass 5 and 7 of a 0.05%C plain carbon steelrolling   149Figure 7.17 Thermal history half-way along the length of the slab during 7-pass rolling ofa 0.05%C plain carbon steel with oxidation ^  150Figure 7.18 Thermal history half-way along the length of the slab with and withoutoxidation during 7-pass rolling of a 0.05%C plain carbon steel ^ 151Figure 7.19 Thermal history half-way along the length of the slab during 7-pass rolling ofa 0.025%Nb bearing steel with oxidation ^  153Figure 7.20 Thermal history at the head and the tail end of the slab during 7-pass rolling ofa 0.025%Nb bearing steel with oxidation ^  153Figure 7.21 Comparison of thermal history half-way along the length of the slab during7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation ^ 154Figure 7.22 Comparison of thermal history half-way along the length of the slab during7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation ^ 154Figure 7.23 Comparison of thermal history half-way along the length of the slab during7-pass rolling with and without heat loss to the work rolls ^  157Figure 8.1 Finite element discretization in the roll bite of the slab ^  162Figure 8.2 Sequence of the solution for the finite element analysis ^  165xixFigure 8.3 Verification of the heat transfer module of the finite element program ^ 167Figure 8.4 Comparison of the numerical upper surface temperature distribution of theplate with an analytical solution ^  168Figure 8.5 Comparison of the FEM prediction with a pilot mill test result of the interfacetemperature distribution in the roll bite ^  169Figure 8.6 Typical velocity profiles during 9-pass rolling ^  171Figure 8.7 Effective strain rate distribution in the roll bite in the 9-pass rolling ^ 174Figure 8.8 Effective strain distribution in the roll bite ^  177Figure 8.9 Temperature distribution in the roll bite  180Figure 8.10 Temperature rise distribution in the roll bite ^  181Figure 8.11 Illustration of deformation of the embedded pin and the apparent shear angle01 ^  183Figure 8.12 Comparison of the effective strain distribution predicted by the model and thepilot mill tests ^  185Figure 8.13 Deformation of the pin in a pilot mill test SS-19 ^  186Figure 8.14 Comparison of the roll forces predicted and measured during 9-pass rolling ^ 187Figure A.1 Schematic diagram of the control volume for the nodes in the slab ^ 202Figure A.2 Schematic diagram of the control volume for the nodes in the rolls ^ 204xxFigure B.1 Eight-node quadratic rectangular element ^  211xxiNomenclature{d}^nodal velocities matrix[a")^nodal pressure matrixA^surface area, m2b^space between nozzle, mBi^Biot numberC, Co^constantCm^concentration of metal, mol/m 3Cps^specific heat of slab, J/kgCp„,^specific heat of water, J/kgCe^empirical constantd^nozzle hydraulic diameter, m[D]^plasticity matrixD^diameter of work roll, mEps^matching limit for determination of heat transfer coefficient, °Cf^field variable{f)^thermal force matrix{f}^velocity force matrix{F}^volumetric forceF(x)^modifying weighting function in upwinding techniquethermal forceG, He^linear function in an elementGr^Grashof numberH^hardness of material, N/m 2H(t),H^slab thickness in the roll bite, mH0, H,^slab entry and exit thickness, respectively, mHn^slab thickness at the neutral point in the roll bite, mHTC^roll gap heat transfer coefficient, W/m 2-°Ch, h'^heat transfer coefficient, W/m2-°Che^pseudo contact heat transfer coefficient, W/m 2-°Cconvective heat transfer coefficient, W/m 2-°Chig^latent heat of vaporization of water, J/kghgap^roll gap heat transfer coefficient, W/m 2-°Chn^natural convective heat transfer coefficient, W/m 2-°Chrad^pseudo radiative heat transfer coefficient, W/m2-°Ch„^roll cooling heat transfer coefficient, W/m2-°Che^element length, mh(t)^heat transfer coefficient at a particular residence time, W/m 2-°Ch(t)^heat transfer coefficient for each cooling zone on the roll surfaceJacobian matrixJ^mechanical equivalent of heat, CaVN-mJo, Jr^Bessel Function of its first kindcationic flux, mol/m2-sk^thermal conductivity, W/m-°Ck^shear strength, N/m2thermal conductivity of metal 1, W/m-°Ck2^thermal conductivity of metal 2, W/m-°Ccombined thermal conductivity of metal 1 and 2, W/m-°Ckf^fluid thermal conductivity, W/m-°Cthermal conductivity of roll, W/m-°Ck3^thermal conductivity of slab, W/m-°Ckw^thermal conductivity of water, W/m-°Ckinetic constant of scaling, m2/skP^parabolic rate constant for the growth of oxide scale, cm 2/s[KT]^stiffness matrix for temperaturexxivvelocity 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, mLo^Slab length, man experimental constantn^number of contact per unit areanormal direction of the interfaceN^rolling speed, rev/min[NJ^shaping functionNu^Nusselt numberp^roll pressure, kg/mm2P.^water jet pressure, kPaPa^apparent pressure, kg/mm2Pe^Peclet numberPr, Pr,^Prantle numberP,,,^total load on contacting surface, Nheat flux, W/m2qs^heat source, W/m3X XVq1^frictional heat, W/m2qg^deformation heat, W/m2qconv^convective heat loss, Wqox^incident heat flow density from steel oxidation, kJ/m 2-sqr^radiative heat loss, Wqs^heat flux due to spray cooling, W/m 2AQ,^total heat loss during rough rolling, Jtotal heat loss except to rolls during rough rolling, JAQ,(%)^percent of heat loss to rolls to total heat loss during rough rollingr^percent reductionR^universal gas constant, 8.31 J/mol-°KR, r^radius, mAr^length step in R directionRe^Reynold numberRew^Reynold number for water at roll surfaceSStefan-Boltzmann constant, jim2_0K4IS) ,S y^deviatoric stress, N/m2s^proportion of steam formedxxvit^time, s{t}^traction on stress boundaryT, To^temperature, °CT,^initial temperature, °CT„,ax^temperature of critical heat flux, °Croll temperature, °CTS^slab temperature, °CT,^saturation temperature of water, °Cwater temperature, °Cambient temperature, °CTjt)^ambient temperature along the rougher table, °CT.,(t*)^ambient temperature along the roll surface, °CA Tdef^temperature rise due to deformation, °CAT^surface excess temperature, °CAT,^plate temperature drop due to descaling, °CAT),^temperature rise of water, °C-4horizontal velocity vector at the edge of an element, m/saverage velocity along the edge of an element, m/sxxviivelocity in horizontal direction, m/svertical velocity vector at the edge of an element, m/svelocity, m/svelocity in vertical direction, m/svelocity vector in finite element methodvelocity, m/sspatial derivative of velocity, s -1relative velocity between roll and slab surface, m/swater spray velocity, m/srate of water flow per unit width of plate, m 2/minwater flux rate, 1/m2-swidth of water spray, mweighting functioninitial scale thickness, cmdimension along rolling direction, mlength step in X directionlength step in Y directiondimension through thickness of slab, mxxviiiz^dimension through width of slab, ma^thermal diffusivity, m2/sa'^angle of the velocity direction to the rolling direction, rada ;^parameter used in upwinding techniqueimpingement angle of water spray to the roll surface13.^parameter used in upwinding techniqueX^scale thickness, cm8^skin layer thickness of roll, m51, 52^average void height for two metals respectively, mSp^variation of pressureSy^variation of velocity43^variation of strain ratec(T)^emissivity as a function of temperatureLA^fraction of contact areav^volume strain rate,strain rate, s -1Ex^strain rate in X direction, s1e'y^strain rate in Y direction, s"'es^strain rate in Z direction, s -1e^shear strain rate, s -1xy6^effective strain rate, s -1spatial strain rate, s -111^heat generation efficiency factor7^shear strain7^parameter used in upwinding techniqueII^viscosity, N/s-m21-11^friction coefficient1-Lw^viscosity of water, N/s-m2co^angular velocity, rad/s9^roll contact angle, rad(P.^roll contact angle at neutral point, radP^density, kg/m3PS^slab density, kg/m3Pv^density of water vapor, kg/m3P.^density of water, kg/m3xxxa^flow stress, N/m2am^mean flow stress, N/m2ast^surface tension of water, N/m 2(7x^flow stress in X direction, N/m2ay^flow stress in Y direction, N/m2az^flow stress in Z direction, N/m2a^effective flow stress, N/m20^contact angle in the roll bite, radA0^angle step at 0 direction in cylindric coordinate system, radtan(0 1)^numerical differential of the deflection of the pin after deformation,`1-1^natural coordinates;Subscripts and Superscripts(e)^elementGaus^Gauss pointi, j^finite element node positionsn, n+1^current time and time at next time stepp^pressurer^rolls^slabT^transposed matrix or vectorv^velocityw^waterx, y^X and Y directionAcknowledgementI would like to express my sincere appreciations to Professors I.V. Samarasekera and E.B.Hawbolt, for their guidance, stimulating discussions, immense encouragement and supportthroughout the project. I would also like to thank Mr. Neil Walker, Serge Milaire and BernhardSauter for their assistances in the research work.Financial assistances from NSERC is gratefully acknowledged. I would also like to thankStelco Inc. for providing the data on rough rolling and CANMET MTL for their assistance to providethe necessary facilities to conduct the pilot mill tests.Thanks are also extended to my fellow graduate students in this department for theirinteractions 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 1INTRODUCTIONSteel strip is produced from molten steel by continuous casting, reheating, rough rolling andfinish rolling units in sequence. Continuous casting has generally replaced static casting in the steelindustry for the production of steel slabs. Although significant improvements in mould designwhich have been implementing worldwide ] have resulted in better slab quality, and some successhas been realized in the development of a process of continuous casting and direct rolling, a procedurewhich would obviously save energy and reduce the cost of producing steel, much work remains tobe 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 thetransfer bar leaving the roughing mill, together with the size and distribution of precipitates thatwould influence the evolution of the microstructure ] . With the development of continuous-castingand direct-rolling, the slab entering the roughing mill will have a thermal field that is somewhatdifferent from slabs that have been traditionally reheated. A thorough understanding of the influenceof the thermal field obtained from the reheating furnace on rough rolling will be of value indetermining the limits on the through-thickness gradients that can be tolerated in the mill withoutadverse effects on the shape and properties of the slab.Another practice which is being advocated is low-temperature rolling which can reduce energycost for reheating, scale loss and improved mechanical properties due to a finer structure. The major11.1 Rough Rollingproblems associated with implementing this innovation is the difficulty of securing high enoughfinishing 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 asnear-net-shape continuous castingt41 , that are also being developed worldwide, rough rolling is stillan important unit in the production of steel strip. Mathematical models of the thermomechanicalprocessing 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 thicknessand the deformation behavior during its passage through the roughing mill, would permit a couplingof those other models to completely describe the thermomechanical processing of the steel strip inthe hot forming operation.1.1 Rough RollingThere are three types of rolling mills for steel strip production which differ from each otherby the layout of the roughing mill-(i) 1/2 continuous strip rolling unit, which consists of one reversing roughing mill and acontinuous finish rolling unit;(ii) 3/4 continuous strip rolling unit which consists of one reversing mill and two four-highmills 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. Theroughing unit mainly consists of the reversing mill, the vertical edger, the slab descaler and the coilbox.21.2 Thermomechanical ProcessingAt Stelco, there is one reversing roughing mill preceding the 5-stand continuous finishingmill, as in Type (i). 7- and 9-pass schedules are currently employed to reduce the thickness of theslab from 240mm to 21 mm depending on the grade of steel. Before rough rolling, the slab dischargedfrom the reheating furnace goes through a slab-descaler to break the scale formed in the reheatingfurnace, followed by a vertical edger to adjust the width of slab to final specifications. Breakageof oxide scale in the slab-descaler and in the vertical edger facilitates the removal of scale by highpressure water jets located prior to the reversing mill. Between successive rolling passes, the exposedslab 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 besmall depending on the thickness of the scale formed. This secondary scale also must be removedbecause if it is rolled in, it deterioratively affects the surface quality of the strip. At Stelco, secondaryoxide descaling is conducted by high pressure water jets before the second and the sixth passesduring reversing rough rolling.After several passes facilitated by reverse rolling, the slab is elongated to a length equal to10 times its original value and is coiled in the coil box in preparation for finish rolling.1.2 Thermomechanical ProcessingThermomechanical processing refers to technologies aimed at controlling the microstructureand mechanical properties of steel and other metals, during deformation and accelerated cooling.Controlled rolling, controlled cooling and direct quenching are typical examples of suchtechnologies. Such processing reduces the cost of steel manufacturing by minimizing or eveneliminating heat treatment after hot deformation, thus increasing the productivity of high-gradesteels.31.3 Hot Deformation of Steel Slab during Rough RollingControlled rolling has played an important role in the development of the high-strength lowalloy steels (HSLA). It is a technique which produces strong and tough steels by refining the grainsize. In controlled rolling, recrystallization during or immediately after hot rolling is controlled bythe rolling conditions and by presence of microalloyed elements such as niobium, titanium andvanadium [61 . During cooling after controlled rolling, the transformation of the austenite (y) producesfine ferrite (a) grains. Furthermore, the precipitation of carbides or nitrides or carbonitrides of themicroalloying elements also contribute to the strengthening. Controlled cooling process have alsobeen developed to produce small ferrite grain structures [61 .In recent research and development at Hoesch Stahl, Dortmund m , on HS LA steels, researchersclaim that the rolling-induced scatter of the mechanical properties of thermomechanically rolledheavy plate and hot-strip are influenced by the rolling temperature ranges of the rough and finishrolling operations together with the amount of reduction and cooling conditions in the finishingmill. 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 RollingTo predict the thermal gradient through the thickness of the slab and the overall evolution ofmicrostructure during the production of strip, the deformation behavior of the slab during rollingneeds to be studied. It is characterized by the velocity, strain rate and strain distribution as well asthe temperature distribution; it is the combination of these parameters that determines whetherdynamic recrystallization is initiated within the roll bite and also controls the rate of staticrecrystallization following a given pass.41.3 Hot Deformation of Steel Slab during Rough RollingDuring 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 microstructureand mechanical properties.To gain insight into the deformation behavior of the slab, a coupled finite element method isoften adopted, and the material is always assumed to behave as a rigid-plastic material during hotrolling. Due to the large ratio of width to thickness during slab rolling, spread in the transversedirection may be assumed to be negligible and conditions of plane strain prevail. This simplifiesthe analysis to two dimensions. Using the coupled finite element method, the thermal anddeformation field including strain rate and strain distribution can be computed. Moreover, anaccurate roll force prediction should be possible by means of the finite element analysis. This kindof analysis is also helpful for the design of a hot strip mill, the control of gauge and shape of theproduct and the final estimation of the mechanical properties.5FINISHING MILLI.V. SamarasekeraE.B. Hawbolto User friendly modelto predict heat-transferand through thicknesstemperature distribution,is complete.o Prediction of rollforces and microstructuredevelopment for C-Mnsteels.CONTINUOUS CASTINGJ.K. BrimacombeI.V. Samarasekerao User friendly heat transfermodel under development.o Input data - Heatflux data for moulds,Heat-transfer coefficientdata for sprays gatheredfrom extensive in-plantmeasurements over thepast 15 years.REHEAT FURNACEP.V. BarrJ.K. BrimacombeA. Burgesso User friendly heat transfermodel complete.o Predicts throughthickness andtransverse temperaturedistribution ofslab includingskidmarks for anyfurnace geometryand slab shape.ROUGHING MILLI.V. SamarasekeraE.B. Hawbolto Thermal and deformationmodels.o Prediction of micro-structural evolutionIn HSLA steels.r..cr\Figure 1.1 Mathematical models in thermomechanical processingSLAB REHEATINGFURNACEFurnaceDischargeSLAB DESCALERVERTICAL EDGERScale BreakingREVERSINGROUGHING MILLCOIL BOXCROP-SHEARInitial Reductionof Slab to Bar BAR DESCALERFINISHING MILLSCOOLING BANKRemoval of Scale from Bar'DOWN COILERStrip CoolingCoil DischargeFigure 1.2 A typical layout of 1/2 continuous hot strip mill000ri)craver2.1.1 Heat Transfer during Rough RollingChapter 2LITERATURE REVIEWTo enhance the competitiveness of steel in relation to alternate materials, new technologiessuch as thermomechanical processing have been developed to improve mechanical properties andenhance the quality while reducing costs. To ensure that the specifications of final products areconsistently 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 mathematicalmodels of the industrial process. The models should be validated through in-plant measurementsand be capable of predicting the thermomechanical history and microstructure evolution duringprocessing.2.1 Thermal Analysis of the Rough Rolling Process2.1.1 Heat Transfer during Rough RollingDuring rolling, the slab loses heat energy by radiation and convection to air, by conductiondue to contact with the support rollers, by convection to high pressure water sprays and by conductionto work rolls. Heat is generated due to deformation and friction at the roll/slab interface. Theinternal temperature distribution of the slab changes in response to these events.82.1.1 Heat Transfer during Rough Rolling2.1.1.1 Heat Transfer in the SlabHeat conduction in the slab is governed by the following equation:aT , )VIcs (VT.,)+ 4s = psCpsi-w (2.1)where T, is the temperature of slab; eq. s is the heat generation due to deformation and friction in theroll bite, it is zero outside the roll gap; and k„ p s and Cps are the thermal conductivity, density andspecific heat of slab respectively.Miller adopted a hydraulic analogue technique to compute the one-dimensional temperaturedistribution through the thickness of a steel slab during hot rolling to plate, for both conventionaland 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 simplifyingas sumptionsE9H111 . Seredynski [91 assumed that the heat loss to the rolls is compensated by the heatgain due to deformation and assumed there is no temperature gradient in the slab. Pavlossoglou [w]1111did not consider the heat loss due to water sprays and the heat generated by deformation in hisanalysis.In other literature, numerical analysis based on the finite difference method is widelyused1121-117] . Hollander developed a mathematical model to predict the temperature distribution insteel slabs from the reheating furnace, during roughing, finish rolling, and during cooling on therun-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.f141 utilized a heat transfer model to compare the slab temperature and the length of therolling mill for different arrangements. They also calculated the steel temperature change from the92.1.1 Heat Transfer during Rough Rollingreheating furnace in rougher and finish rolling by means of a mathematical model for a full and athree 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 andconduction at the interface was computed by assuming that the roll and the workpiece aresemi-infinite bodies. Devadas and Samarasekera r1631173 have developed a model to predict thetemperature distribution of the strip in finish rolling taking into account roll chilling due to coldrolls by means of an experimentally determined interface heat transfer coefficient. The model wasextended to predict the microstructure evolution of the finished product as a function of milloperating parameters.2.1.1.2 Heat Losses during RollingDuring the rough rolling process, the heat losses from the slab consist of three majorcomponents: radiation and convection to the environment, high pressure water descaling and rollchilling due to contact with cold rolls.2.1.1.2.1 Radiative and Convective Heat LossThe slab loses heat by radiation and convection to the environment in the region between thereheating furnace and the first descaler, between passes, and from the last pass to the coil box. Theradiation 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 suchfactors as the amount of scale present on the slab surface, the presence of water vapor, smoke and10kid = Se(T) (Ts—T.)((Ti + 273.1)4 — (T.,+ 273.1)4)(2.4)2.1.1 Heat Transfer during Rough Rollingdust etc., and Ts 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 ofa hot plate as a function of temperature Ts :s.^T.,e(Ts)=^ (0.1251000 0.38)+1.110T0 (2.3)For the ease of computation, a pseudo heat transfer coefficient is adopted and expressed as [16) :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 thespeed of the slab on the transfer tables is relatively low 1131-1151 . Therefore, the influence of forcedconvection 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 duringrolling, the heat transfer coefficient, h,, for natural convection is fixed at 8.37 W/m 2-°C, as calculatedfrom the equation 1251 :1Nu =0.15(Gr • Pr)'^ (2.6)where Gr • Pr is between 8 x 106 and 10' 11 '31 for the upper surface of the warm plates.11AT' RNH0Cps/cps30Wwp,,,{C„„(AT,, + s(85 — AT„,))+sX}(2.7)2.1.1 Heat Transfer during Rough Rolling2.1.1.2.2 High Pressure Water DescalingThe high pressure water sprays are used before certain rolling passes to ensure removal ofoxide scale which would adversely influence surface quality. The resulting water flow impingeson the slab and extracts heat. The heat transfer caused by water boiling at the surface of the movingslab cannot be easily defined analytica1ly [121 . However, Seredynskim used the heat balance methodby examining relevant factors to set up an equation for the temperature change:where W is the rate of water flow per unit width of plate (m2/min); p„, is the density of water (1000kg/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 ofthe rolls (m); Nis the rolling speed (rev/min) and Ho is the plate thickness (m). It is obvious thatthe inclusion of the proportion of water vapor, s, makes the formula inconvenient to use. Kokadoand Hatta assigned a constant value of 1.163 kW/m2-°C as a water jet descaling heat transfercoefficientf13],[141, while Hollander112' has shown that it varies from 12.5-20.9 kW/m 2-°C. Yanagi 1153measured the surface temperature of undeformed bars being cooled by descaler sprays and foundthat the heat removed by the descaler spray could be calculated with an empirical equation developedon 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" 26( -"T r54^(2.8)122.1.1 Heat Transfer during Rough Rollingwhere 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 andlower pressure:h =708W°35T;11 + 0.116^ (2.9)where 1.6 < W < 41.71 1/m2-s; 700 < Ts < 1200 °C; 196 < p„, < 490 kPa. For the present study, thewater jet pressure is approximately 13.72 MPa, which is the same as in the case studied by Devadasand S amarasekera 1161 . According to their investigation, the correlation of Eq.(2.9) was deemedsuitable.2.1.1.2.3 Roll ChillingDuring rolling, the work rolls heat up due to contact with the hot slab. Obviously, as thetemperature increases, thermal stresses increase. Thermal cycling of the rolls causes failure of theroll surface by fatigue and thereby shortens the work roll life. To improve work roll life, watersprays are employed to keep the roll temperature low. In addition, lubrication is sometimes employedto 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 bymeans of an interface heat transfer coefficient back-calculated from the CANMET pilot millmeasurements.2.1.1.3 Heat Generation during RollingApart 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.131 H„ cos (i).H cos0vr =R0) (2.16)2.1.1 Heat Transfer during Rough RollingAssuming that the plastic deformation in the roll bite is uniform, the temperature rise due tohot deformation can be derived by converting the mechanical work into heat energy, i.e.:H1 a  . n( 0 )ATdef = i j ros cps 1 vi-^ (2.14)where 1 is the heat generation efficiency factor from deformation work; for steel, it is in the range0.80-0.85E211 ; J is the mechanical equivalent of heat; a is the flow stress of the rolled materials withstrong temperature dependence; Ho and H1 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 = vri.tfp^ (2.15)where p is the pressure in the arc of contact and an average value may be calculated from themeasured roll force. yr is the relative velocity between the slab and the work roll and varies alongthe arc of contact. It is expressed as t141 :where o) is the roll speed (rpm); Hn and 9„ are the thickness and the contact angle at the neutralpoint where the roll speed is equal to that of the workpiece, whilst H and 9 are the thickness andthe contact angle at some position in the roll gap. Attempts have been made by many workers tocalculate the friction coefficient from the mill data. Denton and Crane t271 have predicted a value of0.25 at 1000°C, increasing to 0.4 at 1100°C, while Roberts rni has proposed the following correlationfor unlubricated rolls:i.if = 4.86 x 10-4T, — 0.0714^ (2.17)142.1.2 Heat Transfer in the Work RollThe oxidation of the slab surface as a source of heat during the rolling process does not appearto have been discussed in the literature. However, the reaction is exothermic and the rate of heatgeneration depends on such parameters as the slab temperature, the type of oxide being formed, thethickness of the existing scale and the rate at which the oxide layer is developing. In the presentstudy, it is assumed that it is small compared to the heat generation due to deformation and frictionand is therefore neglected.Of the metallurgical phenomena occurring in the steel, the solid state reaction of most concernin hot rolling is that associated with the decomposition of austenite to ferrite and cementite. Duringrough rolling, the temperature of the slab is always higher than the transformation temperatureexcept within the very thin surface layer in the roll bite. Thus it is assumed that this componentcan be ignored.2.1.2 Heat Transfer in the Work Roll2.1.2.1 Work Roll Cooling SystemIn 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 rollsmust be closely controlled.Roll wear or spalling, which may occur due to thermal/mechanical fatigue, can be greatlyaffected by roll cooling. Improper or insufficient cooling causes large thermal gradients near theroll surface resulting in thermal stresses causing surface spalling. Therefore, designing a good rollcooling system has been a concern of mill designers and operators.15cooling spray2.1.2 Heat Transfer in the Work RollIn the steel industry, water is generally employed as the cooling medium. For cooling andlubrication of rolls, water spray cooling systems are generally used, as shown in Fig.2.1.vsFigure 2.1 Spray cooling of work roll during rolling processNumerous 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 RollUse of sprays or other systems to cool the roll is a complicated process involving jetimpingement on a rotating surface. The roll speed often exceeds the coolant velocity and stronglyinfluences the flow pattern developed. The surface temperature when it emerges from the roll biteis always higher than the saturation temperature of the water, and boiling occurs. The boiling maybe in nucleate, transition or film boiling regions, depending on the surface temperature" ) . The heatremoval from the roll surface can be expressed as:= hr,(Tr —Tojni^(2.10)162.1.2 Heat Transfer in the Work RollThe heat transfer coefficient, h,, and the exponent, n 1 , are dependent on the cooling pattern on theroll surface'', and may be affected by many relevant parameters. These include: 1) the waterproperties, such as the density (p,), the dynamic viscosity (N) as well as the density and dynamicviscosity of the water vapor, and the surface tension (o w); 2) the thermal properties of the water andits 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 theroll 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 asurvey by Tseng et al. [223 , the heat transfer coefficient is reported to be in the range 1 to 1 lkW/m 2-°Cwhen the roll surface temperature is below 100°C (without boiling) and 6 to 40 kW/m 2-°C fortemperatures above 100°C (with boiling). Hogshead (231 reported an average heat transfer coefficientof 2 to 5 KW/m2-°C, for roll surface temperature in the range of 88°C to 115°C; he investigated theeffect of rotating speeds (0 to 1800 rpm) and spray rates (0.126 to 1.262 L's). In an attempt to studylocal heat transfer coefficients, Tseng et a/. 122I reported that the maximum heat transfer coefficientfor a jet using city water was found to be 6 kW/m 2-°C directly in the impingement zone and 1 to 2kW/m2-°C away from the impingement zone.Some heat transfer coefficient correlations have also been developed for the cooling of rollswith spray of water1241-E261 . In the regions of impingement of spray water on the roll surface, acorrelation between the heat transfer coefficient and water flux has been determined by Yamaguchiet al [221 .. el,. 1.291 x 105147"21^(2.11)172.12 Heat Transfer in the Work RollThis correlation is based on experimental measurements of the thermal response of a heated plateto spray cooling. The plate temperature was in the range of 100-400°C and the water flux variedfrom 5 x 103 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; nucleateboiling when the temperature is between the saturation temperature and the critical heat fluxtemperature; and unstable film boiling when the temperature is higher than the critical heat fluxtemperature1161 . For the natural convection, the following correlation was adopted f253 :D fh =(—)i 0.1 1 [(0.5Re 2 + GrD )Pr]0.35}For nucleate boiling, Rohsenow' si 261 correlation was employed:S (pw – p ) 0.5 C pwATx= whfg((Y„^x C ehhP rf`(2.12)(2.13)where n=1 for water. For the heat transfer coefficient of unstable film boiling, Nukiyama' s1261boiling heat transfer coefficient data were interpolated for the present study.182.13 Heat Transfer Characterization at the Roll/Slab Interface2.1.3 Heat Transfer Characterization at the Roll/Slab InterfaceOwing to the paucity of data on the heat transfer coefficient at the interface of the work rolland slab, many investigators have assumed perfect thermal contact 19111311151191 , which is clearly notthe case.In Yanagi's [131 model, because the thermal resistance had been calculated for rolling to be ofthe order of 0.86 x 10 -5m2°C/W, he treated the thermal resistance between the roll surface andworkpiece as negligible and utilizes the semi-infinite model for both the roll and the strip. Hattaet a/."411151 ignored the heat loss from the slab to the work roll because the region affected by theroll chilling was considered to be confined to the slab surface layer. Stevens et am have estimatedthat the heat transfer coefficient at the roll-gap interface was 37.6 kW/m2-°C during the first t=30ms in the arc of contact and 18 kW/m2-°C thereafter in the range e= t + 0.094 seconds. Murata eta1. 1211 have back-calculated the heat transfer coefficient that would apply to hot rolling under avariety of conditions by measuring the surface temperatures of a specimen heated to 780°C andbrought in contact with a low temperature specimen at 22-30°C during compression with a constantpressure of 5 kg/mm2. The results show that the lubricants affect the value of the heat transfercoefficient 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 thesurface temperature. They found that the equation:(^T:+1. T: + (T: —T:) h ' cr x {1 — exp(tiOerfc(A 1 -4t51A ik,.(2.14)where:A^(k,.-4cc+ks-4(Tr)rk1c, (2.15)192.13 Heat Transfer Characterization at the Roll/Slab Interfacegave a good agreement with their experimental data if the resistance of the insulating layer werecharacterized by two different values of surface heat transfer coefficients, h gap and h'gap , within theroll bite where a change from h g„i, to h ' gap occurred after a given time of contact, 30 ms. Theypostulated that the cause was the phase transformation that occurs at the surface of the strip as itstemperature was depressed due to contact with the roll. This, in turn, increases the surface roughnessof the strip with an increase in thermal resistance.Sellars 1291 pointed out that the determination of this heat transfer coefficient between the rollsurface and slab varied considerably from slab to slab, presumably because of variations in thesurface oxide condition. On the assumption that the oxide scale thickness was reduced duringrolling in proportion to slab thickness, the model allowed the heat transfer coefficient to increaseproportionally. It was found that a unique value of the coefficient could be applied to the differentpasses in the multipass experimental rolling schedule. In his model, the value of heat transfercoefficient was determined by trial-and-error by comparing the experimental and predictedtemperature at the center of the slab, and a mean heat transfer value of 200 kW/m 2-°C was obtainedfor 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 andslab during hot rolling because a seemingly smooth surface viewed with various degrees ofmagnification 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 1301developed a model with heating one end and cooling the other of the two different contactingmaterials 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(2.16)20\ \ \ \ \ \ \\\ATcM.-2.1.3 Heat Transfer Characterization at the Roll/Slab InterfaceToolWorkpieceFigure 2.2 Contact between real surfacesi m iTFigure 2.3 Temperature distribution through surface in contact t30121SI62CrossSectionViewt ,:Ds47 ,4 nO, czy'cj.;F)^..:,— 0 .0 0 0 00 0 00 0 0 02.1.3 Heat Transfer Characterization at the Roll/Slab InterfaceA pseudo contact heat transfer coefficient he was defined as(qIA)hc — AT (2.17)There is no localized interface contact resistance but rather a region of influence in the neighborhoodof the contact. Clearly, increasing the number of locations of contact per unit area would changethe shape of the flow-line significantly, even though the fraction of area in contact, C A, and theroughness of the surface remained the same.^(a)^(b)^(c)^Actual Model Button Model Heat Channelfor Button ModelFigure 2.4 Fenech et al.'s heat transfer model for thermal contact (30IIn their idealized mathematical model, shown in Fig.2.4, they considered the steady state heatconduction:V27' = 0^ (2.18)and proposed that the heat transfer coefficient is of the form:.f(ki ,k2,kf,51 ,52, n,eA )^ (2.19)221h, ^d2/k2(2.20)2.1.3 Heat Transfer Characterization at the Roll/Slab Interfaceand expressed as:The unknown d 1 and d2 are obtained by solving for the constants with boundary conditions.Therefore, the general heat transfer coefficient was expressed as [3°1 .\ -kf4.26,1W— + 1^4,26,171 -=+1eA11.16A(1 )4.26EA4 72[ (1 _ED IT —k2h,8 +2, k,^k2 1 (2.21)kr^(^)]_ e,24) [ 1 _ 4.264W-e-IA-+ 1 4264;1 + 1eA81+82^ lFor contacts under heavy loading where the asperities would be deformed or contacts inrarified atmospheres, the conductance through the fluid in the void may be neglected, so that theheat transfer coefficient is:h, —.46(247)0381 + 82 +(2.22)(where lc, can be expressed as: -k- = i +12- J.With the application of the model to noncrystalline materials, Moore s] observed that whentwo surfaces of different metals are pressed together, the irregularities of the softer surface undergofull plastic deformation while the peaks of the harder metal are embedded in the other surface. Ifpa is the apparent pressure on the contact, the total load, 1 3,, is then given byP.= paA^ (2.23)232.1.3 Heat Transfer Characterization at the Roll-Slab InterfaceThe measured values of tic vs. pressure are the circles shown in Fig.2.5, where the vertical linesrepresent the estimated limits of precision in the data and the various dots represent calculatedvalues of h,, utilizing Eq. 2.22.Apparent Pressure Pa (kg/ma x 10)^0.7^7.0^70^7001 r 1 ill il I I 1I I L. 567.810 5 :^ 1^u I I I WI I_ ....o Experiment he• Calculated he—Best line through thecalculated points56.78C)5.678•Armco—iron/ aluminum11111111^I^I^111111^10^102^103Apparent pressure Pa(psi)Figure 2.5 Calculated and measured h e vs. apparent pressuremThe agreement between calculated and measured results in Fig.2.5 is remarkably good and the sameexcellent agreement has been repeated in numerous additional tests t301 .In the case of surfaces in relative motion, the limited number of reported measurementsdescribed above have yielded widely different values and a lack of fundamental understandingabout 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 lowerresistance path for heat flow in comparison to adjacent regions where heat transfer occurs byconduction through air gaps, the dimensions of which depend on the surface topgraphy of the rollio2- 0.5678104242.1.3 Heat Transfer Characterization at the Roll/Slab Interfaceand strip. Therefore it is proposed that a major fraction of the heat is transferred via the contactingpoints. Unlike the case of two flat surfaces under compression, friction exists at the roll/stripinterface due to sliding between the roll and the strip. It has also been proposed that the link betweenthe 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 interfacialpressure and the shear strength in the real contact zone as follows:pmceAk1-tf(2.24)where me is an experimental constant within the range of 0 to 1; k is the shear strength for thedeformed material; 1.4 is the frictional coefficient at the interface. Wilson and co-workers [33m34 istudied the influence of bulk strain and relative velocity between the contacting surfaces on thefractional area of contact and showed that the fractional contact area increases monotonically withincreasing bulk strain whilst there is decrease in the tendency for the workpiece and tools to conformwith increasing relative speed. S araarasekeran showed that the variation in heat-transfer coefficientwith reduction, rolling speed and lubrication observed through pilot mill tests on 316L stainlesssteel could be explained on the basis of the influence of these rolling parameters on fractional contactarea. A linear relationship was shown to exist between the heat-transfer coefficient and mean rollpressure for two passes pointing to a strong analogy with friction.25200U•1602.1.3 Heat Transfer Characterization at the Roll-Slab Interface120(twoTh.ckness(mm)Reduction(•/..)— 25---  1257503520^30^40MEAN PRESSURE ( kWrnmz )Figure 2.6 The relationship between the roll gap heat-transfercoefficient and the mean pressure along the arc of contactfor two successive passes on the pilot millf 311The results of this work are shown in Fig.2.6 from which it is evident that for the first passthe heat-transfer coefficient increases linearly with increasing pressure up to 15 kg/mm2, butdeviates thereafter and approaches an equilibrium value of 57 kW/m 2-°C. For the second pass, thereis a definite linear relationship between the interfacial heat-transfer coefficient and pressure andthe slope of the line is remarkably similar to that of the linear portion of the first curve. This indicatesthat the interfacial heat transfer coefficient is strongly dependent on the real area of contact andconsequently 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 theapplicability of the relationships determined to other materials such as C-Mn and microalloyedsteels, where oxidation is likely to occur.zu.0 80LI1.4JU.cc 400 ^10262.2.1 Oxidation Mechanisms of Metals at High Temperature2.2 Oxidation of Steels at High Temperature2.2.1 Oxidation Mechanisms of Metals at High TemperatureFew metals are stable when exposed to the atmosphere at high temperature and areconsequently oxidized. From consideration of the equationM(s)+02 (g)= MO(s)2 (2.25)it is obvious that the solid reaction product MO will separate the two reactants as shown belowM(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. eithermetal must be transported through the oxide to the oxide-gas interface and react there, or oxygenmust be transported to the oxide-metal interface and react there. Therefore, the mechanisms bywhich the reactants may penetrate the oxide layer are seen to be an important part of the mechanismby which high temperature oxidation occurs t361 .For iron or steel, the consequence of oxidation is the formation of a multi-layer scale at thesurface, i.e., Fe/Fe0/Fe 304/Fe203/02, as schematically shown in Fig.2.7.27024041^1,^44^1, 4Fe 2 03Fe 3 04FeO2.2.2 Iron Oxide Scale Growth in AirFigure 2.7 Schematic diagram of multi-layer of iron oxide scale2.2.2 Iron Oxide Scale Growth in AirWagner1373 developed a theory (the Wagner Theory) to describe high temperature oxidationof metals. In fact, the theory describes the oxidation under highly idealized conditions. In histheory, 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;282.2.2 Iron Oxide Scale Growth in Air(6) The scale is thick compared with the distance over which space charge effects (electricaldouble 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-gasinterfaces, it follows that activity gradients of both metals and non metals (such as oxygen, etc) areestablished across the scale. Consequently, metal ions and oxygen ions will tend to migrate acrossthe scale in opposite directions. Because the ions are charged, this migration will cause an electricfield to be set up across the scale resulting in consequent transport of electrons from the metal tothe atmosphere. The relative migration rates of cations, anions, and electrons are therefore balancedsuch 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 growthof oxide scale:dt x^ (2.26)where xis the oxide scale thickness and lc; is the parabolic rate constant and can be derived accordingto the cationic flux, which may also be expressed by:dxj, m —dt(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 ironin the range of 700 to 1250 °C has been studied extensively by Paidassi [391 . From the study, the292.2.2 Iron Oxide Scale Growth in Airgrowth 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 parabolicrate constant of the three oxides, kp ' , from metallographic measurements individually as followsgrowth of wustite:growth rate of magnetite:growth rate of hematite:The total growth rate:kP '(F e0) = 2.88 e -405®/RT (cm 2s-1)ki:(Fe304) = 5.25 x 10-3e -405MRT(cm 2s-1)kpVe203) = 2.70 x 10-4e -4°5°CiIRT (cm 2s -1)ki:(Total) = 3.05 e -405®/RT (cm 2S -1 )(2.28)(2.29)(2.30)(2.31)Fig.2.8 summarizes the experimental results of Paidassi for relative thickness of FeO, Fe 304 , andFe203 at different temperatures.302025602.2.2 Iron Oxide Scale Growth in Air700^800^900^1000^1100^1200^1300TEMPERATURE, ° CFigure 2.8 Relative thickness of magnetite (Fe304) and hematite (Fe203)Zhadan et al.E381 provided a procedure for calculating decarburization and scaling during hotrolling of carbon steel. The thickness of scale layer, x, grows according to a parabolic law as thetime of isothermal holding, t, increases:x = (20)2where lee is the kinetic constant of scalinglc: =7 .1 x10_6 exp 138.27 x 103) (m2sRT(2.32)(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 Hsum ,together with an application of the theory to the oxidation of iron at 800°C to 1200°C in whichFe i _ x0, Fe304 and Fe203 are formed simultaneouslym . According to the theory, the growth of312.2.2 Iron Oxide Scale Growth in Airwustite with simultaneous formation of magnetite and hematite on iron is predominantly controlledby bulk diffusion of iron cations in the wustite. The oxygen diffusion in wustite during the oxidationof iron at 800°C to 1200°C is negligible; the growth of magnetite with simultaneous growth ofwustite and hematite is predominantly controlled by cation diffusion through vacancies andinterstitial sites in magnetite; and the growth of hematite cannot be explained only by bulk diffusivityof cation and anion determined from single-crystal Fe 203. Ionic transport through short-circuitdiffusion paths such as grain boundaries, dislocation pipes, and other flaws in hematite ispredominantly controlling the growth of Fe203 .The oxidation of steels during reheating has been investigated by several researchers 1411-(431 .Obaro14t ' pointed out that the multi-layer oxide consists of an outer layer of Fe203 followed by alayer of Fe304 and an inner layer of FeO close to the metal surface. Below 570°C, the scale consistsessentially of Fe304 covered by a thin layer of a- and y-Fe 203. Above 570 °C, the scale consistsessentially of FeO with only a thin layer consisting of Fe 304 and Fe203 . Within the scale layer, theconcentration of the metal decreases from the inside to outside. The growth of the sublayers followsthe parabolic growth law and is proportional to the relative thickness of the three oxides dependingon the diffusion rates through the layers, the gradient of the chemical potentials across the layersand the relative porosity of the layers. In Ormerod et al.' sE433 investigation, the influences of processvariables (steel chemistry variables) and their interaction on scale formation on steel in reheatingfurnace systems were examined and illustrated with data drawn from other literature sources. Themodel 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 time322.2.3 Effect of Oxide Scale on Heat Transfert, ai and f34 are also functions of composition and time t, where ai denotes the influence of singleelements while 134 denotes the interactive effects of pairs of elements. With the above equation, itis possible to compute the scale growth in air.2.2.3 Effect of Oxide Scale on Heat TransferAlthough a small amount of scale formation in the vicinity of 0.1% is advantageous in thereheating furnace because such scaling removes minor surface defects like oscillation marks presentin continuously cast productsm , excessive scale formation represents a yield loss, influences theheat 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 heatingcarbon steels in continuous pusher type furnacesm. The rate of growth of scale was determinedby solving the differential equation:dx. ko Bdt 2x exP(2.35)(I')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 initialthickness of the scale in m; B and ko are empirical coefficients characterizing the kinetics of oxidationof the steel in 7C and m2 /hr respectively. The change in the radiation characteristics of the scaleduring heating of the metal is shown in Fig 2.9.332.2.3 Effect of Oxide Scale on Heat TransferFigure 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 hrwhere eA. denotes the emisivity of the scale and X is the wavelength. From the figure, a fall in thespectral emisivity of the scale in the shortwave range of the spectrum was observed during metalheating. In their study, the incident heat flow on the metal was chosen as the analyzed parameterof external heat transfer, since its formation is determined in many ways by the magnitude andnature of distribution of thermal loads in the furnace working chamber. The incident heat flowdensity, q.,„ from steel oxidation was defined by the expression:dxq. = Q.P. d7 (2.37)where Q0  is the amount of heat released during the formation of 1 kilogram of oxides in kJ; pox isthe density of the scale in kg/m3 .For a study of the influence of a reduction in scale emissivity in the course of metal heatingon the intensity of heat transfer, dependences q=f(t) were determined for different conditions andthe results are shown in Fig.2.10.342.2.3 Effect of Oxide Scale on Heat Transferq,kW-//// ^■..q....,..........-•■-N,,,,...0 0,5 0^2,5^43,0 -u l tiFigure 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 thicknessCurve 1 corresponds to real values of scale emissivity and thickness; curve 2 corresponds tothe initial stage of heating of metals in the furnace with an emissivity of 0.90-0.93 in the 1-15 µmwavelength range with constant scale emissivity but increasing scale thickness; and curve 3corresponds to the constant scale emissivity and constant scale thickness of initial value. The resultsshow that with a constantly high surface emissivity of the heated metal the heat treatment scheduleis achieved with lower densities of the incident heat flow. In this case, a reduction in &A, of the scalehas the greatest effect on the magnitude of qox with a heating time of 1.3-1.4 hours. According toFig 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 thicknesson heating time.415350250150352.2.3 Effect of Oxide Scale on Heat Transfer0^1,0^7,0^40 1,hFigure 2.11 Dependence of temperature through thickness on heating time1- surface of scale; 2- upper surface of metal; 3- low surface of metal; 4- center of metalThe figure shows that, during slab heating in the furnace, the temperature difference through thethickness of the scale layer reaches its maximum magnitude (about 90°C) in the 1.3 hr-1.4 hr timeinterval, which corresponds to the end of the first heating zone. Subsequently, AT decreases asheating 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 ofmetals during hot rolling. Hollander {121 pointed out that the scale thickness before first descaler wasin the range 1.5 - 3.0mm and before the second descaler was 100-150 pm and at the third descalerwas 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 distributionthrough the slab was not presented.362.3.1 Governing Equations for Finite Element Analysis2.3 Finite Element Analysis on Hot DeformationAs the application of computer-aided techniques (computer-aided design, manufacturing, andengineering, in short for CAD/CAM/CAE) in the metal forming industry increases considerably,process simulation and/or modelling for the investigation and understanding of deformationmechanics has become a major area of research. The finite element method (FEM) is playing animportant role in modelling forming processes.2.3.1 Governing Equations for Finite Element AnalysisThe governing equation for the finite element analysis can be obtained from the virtual workprinciple which is simply an alternative statement of the equilibrium conditions. The virtual workprinciple states: "In a system for which internal forces (stresses) and external applied forces are inequilibrium, the application of any (virtual) system of displacement and corresponding internalstrain compatible with it, results in equality of external and internal work".There are two formulations, namely, flow formulation and solid formulation, which are widelyused in metal forming. Flow formulation assumes that the deforming material has a negligibleelastic response, while solid formulation includes elasticity. Despite the recent advances, theapplication of the solid formulation to metal forming processes is limited. In the analysis of metalforming, plastic strains are usually much larger than elastic strains and the flow stress is muchdependent on the strain rate which characterize the liquid flow. For this reason the assumption ofrigid-plastic or rigid-viscoplastic behavior of material for hot deformation is acceptable. As aconsequence, the flow formulation is widely used in metal forming process, especially in the hotdeformation process.In the flow formulation, the current, real (Cauchy), stress of deformed body a is related tothe rate of deformation, or strain rate, i f45j.372.3.1 Governing Equations for Finite Element Analysis^G=Gi^ (2.38)where the matrix G may be dependent on total strain invariants (say e, temperature T and indeedthe rate of strain i itself). ThusG =G(i,i,T)^ (2.39)For incompressible flow, the rate of the volumetric straining is zero, i.e.et, = ix + i), + iz = [m]T fil = o^(2.40)where^[m]T = [1, 1, 1,^o, o, o] (2.41)ax +Cry + az^For such fluids, the mean stress am, (cy..  3^— p ), is not defined and has to be soughtfrom equilibrium relations.The deviatoric stress, a':For a linear fluid we can write:where101 = {G} -{Gm } = {(5} + [A1]/9{(0 = [D] fil(2.42)(2.43)382.3.1 Governing Equations for Finite Element Analysis 2 0 0 0 0 0-O 2 0 0 0 0O 0 2 0 0 00 0 0 2 0 0O 0 0 0 2 0O 0 0 0 0 2_[D] = p, =1.4D^(2.44)where g is viscosity and [D°] is a diagonal matrix. For non-Newtonian fluids, this viscosity isvariable and dependent on {i} and temperature.The virtual principle can be also applied if we replace displacements by velocities and strainrates, and the equilibrium of a specified mass by an arbitrary surface can be considered at an instantof time. Thus we can write:L{E•i}T {a}dV — {51, }T {F}dV —{8v} T {INS = 0^(2.45)stwhere {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 Sy where velocities are givenE461 .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, so8pivdV =0^ (2.48)392.3.1 Governing Equations for Finite Element AnalysisInserting constitutive equations (2.42) to (2.44), we can rewrite equation (2.45) as:L {450TIADIfeldV+ 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 formulationdescribed above is referred to as the velocity-pressure approach.Besides the velocity-pressure approach in the flow formulation, there is another approachcalled Direct Penalty Form (or Penalty Function Approach) which is also widely used. Thegoverning equation for this approach is derived according to a variational principle which isequivalent to the virtual work statements subject to constraint:iv =MT {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 positivelarge 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 ^vIf we replace [D] by [Dlin Eq. (2.51), where [D1= pt.[D1+[M]2K[M]T then we can easily see that:CD = fv{i}T [D1 {i}dV — fv{v}T {F}dV — isi{v}T {t}CIS^(2.53)402.3.2 Application of Finite Element Analysis in Metal FormingThis functional is similar to the Eq.(2.51) and can be minimized by making its first-order variationbe 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 isvery large; however, no difficulties have been encountered using values 1451 :K = 107-1° 1 (2.54)The governing equations (2.49) or Eq.(2.53) can be solved by discretization with velocitiesand pressure as basic variables.2.3.2 Application of Finite Element Analysis in Metal FormingThe application of the finite-element method to metal-forming problems began as an extensionof structural analysis technique to the plastic deformation regime. It was first introduced byZienkiewiczt491 , Dawsonf50I and Li and Kobayashit511 et al. to the metal forming field. To date, manyinvestigators have analyzed deformation of metals r45H561 by the finite-element method because itcan yield a more accurate prediction of strain and strain rate distribution during deformation. Themost important development in the application of FEM to the deformation analysis is the inclusionof the effects of strain rate and temperature on the mechanical properties and of thermal couplingin the solution.Li and Kobayashi t511 analyzed metal deformation in the roll bite under plain-strain conditionsat room temperature by the rigid-plastic finite element method. They reported that homogeneity ofdeformation in the roll bite is influenced by the reduction per pass and the roll gap geometry. Theyobserved single and double peaks in the pressure distribution along the arc of contact at high andlow reductions respectively. An elasto-viscoplastic approach has been adopted by Grober t521 forhot rolling, and he has shown that strain inhomogeneity increases with increasing reduction andhigher coefficients of friction. Zienkiewici 491 and Dawsont501 on the other hand developed models412.3.2 Application of Finite Element Analysis in Metal Formingusing flow formulation assuming viscoplastic materials behavior in their analysis. Jaint 471 adoptedthe flow formulation method to analyze several metal deformation processes, such as extrusion andflat 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 samemesh. 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 distributionobtained by the velocity-pressure approach in some cases is poor, so that some sort of upwindingtechnique has to be adopted to get a smooth variation.For coupled problems, realistic thermal boundary conditions have not been incorporated atthe roll/slab interface because of paucity of data describing the heat transfer phenomenon at thisinterface. There is, in fact, a roll chilling effect during hot rolling"71 , and about 38% of the totalheat energy in the slab is extracted by the work rolls [311 , so that a large temperature gradient isestablished near the surface due to the chilling affect. The magnitude of the gradient depends onthe heat transfer coefficient at the interface, which may be back-calculated from the measuredsurface temperature. Ignoring this effect, obviously tends to overestimate the surface temperatureand underestimate the local deformation resistance.More recently, some investigators have incorporated roll chilling in deformationanalysis (5311551 . Beynon [533 presented a coupled Eulerian finite-element method to analyze thedeformation and temperature distribution in aluminium specimens during hot rolling. Theycomputed the effective strain from the instantaneous effective strain rate using the Petrov-Galerkinmethod. They have shown that the strain over a region at the center of the slab approaches thenominal true strain corresponding to the applied reduction and increases significantly above thenominal value near the surface owing to the redundant strain required to force the material to berolled into the roll gap. Silvonen r541 et al. undertook a similar analysis for laboratory rolling of steel422.3.2 Application of Finite Element Analysis in Metal Formingand demonstrated that the maximum values of strain rate were obtained near the entry and exitadjacent to the roll, with a rigid zone retained between these two zones. Pietrzky and Lenard[551 'srecent analysis of a low speed laboratory rolling operation is also in agreement with the work ofSilvonen et al. E541 .Dawson(561 has taken the finite element analysis a step further and shown that the method canbe adopted to predict mechanical property changes, namely hardness, by setting up a constitutivemodel for plastic deformations of the workpiece, during the flat rolling of aluminum. No similarwork has been done on steel rolling and very few models have been directly applied to examinedeformation during hot rolling of steel under industrial conditions.433.1 Objectives and ScopeChapter 3SCOPE AND OBJECTIVESIn order to improve the final mechanical properties of rolled products, its deformation historyhas to be controlled. This is difficult at best because the mechanical properties are dependent onmany factors, such as temperature, strain, strain rate and composition. To determine thethermomechanical history during rolling mathematical models are useful.3.1 Objectives and ScopeFrom the literature review in Chapter 2, it appears that there are real merits in modelling thehot 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 slabduring hot rolling;(2) To develop a two-dimensional mathematical model to predict the temperature distribution ofa 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 methodto describe the deformation in the roll bite;(5) To validate the heat transfer model with the results from other models;443.2 Methodology(6) To validate the deformation model by comparing the measured roll forces and the straindistribution with model prediction.3.2 MethodologyThe thermal history of a slab during rough rolling will be determined by adopting a finitedifference method which has been widely used to predict the temperature distribution in steel slabsduring rolling. The parabolic law of oxide scale growth has been incorporated to investigate theeffect of scale on the thermal history. To validate the model, the results will be compared withthose 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 thework roll 311 , the heat transfer at the roll/slab interface was characterized by pilot mill tests atCANMET and at UBC. The heat transfer coefficient in the roll bite was back-calculated from themeasured surface temperature of the specimen. This data has been employed in the model tocalculate the effect of roll chilling.It is necessary to know the deformation behavior of a slab during rough rolling to predict theevolution of microstructure. An existing coupled finite element model for finishing rolling [461 hasbeen modified to predict the distribution of velocity, strain rate, strain and temperature in the rollbite. The model has been validated by comparing the model-predicted and the measured roll forcesand the strain distribution of the specimen obtained in the pilot mill tests.The overall conception of the project is schematically shown in Fig.3.1.45ThermalHistoryof a SlabHTCCalculation DeformationAnalysisof a SlabTemperaturein theRoll BiteVelocity StrainRate StrainFINITE ELEMENT MODELNBoundary ConditionsBCI , BC2, BC3 , BC4BC4OxideScaleGrowthFINITE DIFFERENCE MODELEXPERIMENTALThermalResponseMeasurementsDeformationObservationFigure 3.1 Methodology adopted in the thermomechanical analysisduring rough rolling4.1 Test DesignChapter 4EXPERIMENTAL MEASUREMENTSAs reviewed in Section 2.1.3, the characterization of heat transfer at the roll/slab interface isplagued with uncertainty. Therefore, pilot mill tests were conducted at CANMET and at UBC inwhich the specimen's surface temperature during the rolling process was measured bythermocouples. This chapter presents an investigation of the thermal response that was observedand its impact on the heat transfer at the roll/slab interface.4.1 Test DesignTo characterize the heat-transfer at the roll/slab interface, the thermal response at the rollsurface or the slab surface or even both should be measured during hot rolling. However, due tothe very short contact time (of the order of 0.05 seconds), surface oxidation, high reduction, it isnot easy to make successful measurements. Fortunately, Devadas et a/."71 have developed atechnique that was successfully employed for stainless steels.474.1.1 Thermocouple Design and Data Acquisition System4.1.1 Thermocouple Design and Data Acquisition SystemBecause of the short contact time and a large deformation during rolling, the system forthermal response measurements must have a fast response. Devadas et al. [173 arrived at an optimumthermocouple 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 thata thermocouple wire of 0.25 mm diameter gave a satisfactory thermal response consistent with theroll contact time and the thermocouples remained intact after the tests. In addition, they found thatan intrinsic thermocouple has less thermal mass than a beaded thermocouple. Therefore a 0.25mmdiameter CHROMEL-ALUMEL wire was spot-welded to the surface of a sample to form an intrinsicthermocouple with junctions approximately 0.4mm apart. It is also important that the thermoelectricjunctions 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 aportable microcomputer (COMPAQ), a DT2805 data transmission board, and a DT707T externalboard. Data acquisition was manually triggered just prior to the entry of a specimen into the rollbite. Four channels with one load cell, one central temperature and two surface temperatures wererecorded within a period of 3 seconds at a data acquisition rate of 1500 Hz..484.1.2 Preparation of Samples4.1.2 Preparation of SamplesTo determine the heat transfer coefficient at the interface during hot rolling, industrial rollingconditions 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 kindof problem, specimens were initially fabricated from AISI 304L stainless steels. However since itwas unclear whether the heat-transfer coefficient determined for the stainless steels is applicablefor carbon steels and microalloyed steels, techniques were devised for testing the latter. Thesetechniques will be described later.Two grooves, 1.5 mm deep, 1 5 mm wide and 22 mm long were milled on the surface of eachtest specimen, shown schematically in Fig. 4.1./45(2.822"7/(2)Figure 4.1 Schematic diagram of specimen employedin the thermal response measurements (Dimensions in mm)494.1.2 Preparation of SamplesINCONEL sheathed thermocouples were spot-welded on the surface. The two surfacethermocouples (TC1 and TC2) provide a check on reproducibility because the conditionsexperienced by TC1 and TC2 are almost the same. At half-way between TC1 and TC2, there is a2 mm diameter hole located at the half thickness of the specimen, as shown in Fig. 4.1. Athermocouple was inserted into the hole to measure the temperature at the center. For ease ofhandling of the test specimen, an approximately one meter long wire rod was welded to each endof the specimen. The three INCONEL sheathed thermocouples were bunched loosely around therod and led to the data acquisition board. Before reheating, an electrical check was conducted tomake sure that the thermocouples were in working order. The thermocouples were calibrated attwo reference temperatures (0°C and 100°C). The voltage signal, compensated using an electronicice point, was transmitted to the data acquisition board by using fiberglass-insulatedCHROMEL-ALUMEL wires.To observe the strain distribution after rolling, a 5mm diameter pin was inserted into a 5mmdiameter hole at the center of the working face of the specimen with its axis perpendicular to therolling plane, as also shown in Fig.4.1. The pin was made of the same material as the sample. Thiseliminates 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 longand 50 mm thick for the 0.05%C low carbon steel and 100 mm wide, 150 mm long and 25 mmthick 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% Niobiummicroalloyed steel.504.1.3 Test Facilities4.1.3 Test FacilitiesSome tests were conducted on the rolling facility at CANMET's Metals TechnologyLaboratories (MTL) and others on a rolling mill at the University of British Columbia. The rollingfacility at CANMET consists of a single two-high reversible hot-rolling stand whose specificationsare shown in Table 4.1:Table 4.1 Specifications of the pilot mill at CANMETMotor Drive 225 kWMotor Speed 150/300/450 rpmMax. Roll Separating Force 4.5 MNRoll Diameter 460, 457 mmRoll Gap Setting  0-130 mmThe heating facility at CANMET MTL consists of a 0.255 m 3 globar element furnace, withdigital 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 sothat the same average temperature is attained (281 .The pilot mill at UBC is also a two-high reversing mill and its specifications are shown inTable 4.2;Table 4.2 Specifications of the pilot mill at UBCRolling Mill STANATRolling Speed 34.3/67.6 rpmRoll Diameter 100 mmRoll Material AISI 4340Max. Roll Separating Force 0.2 MN514.1.3 Test FacilitiesA tube furnace was used as the reheating facility at UBC with a hearth width of 100mm. Tominimize the heat loss from the specimen before rolling, the furnace was placed directly in frontof the rolling mill. The specimen was heated at the center of the hearth, and the temperature of thespecimen was monitored by the attached thermocouples. In order to control the scale formation onthe specimen surface during reheating, the exit and the entry of the hearth was closed using a pieceof asbestos plate and Nitrogen (N2) was passed through the hearth. The whole apparatus isschematically shown in Fig.4.2. Once the rolling temperature was reached, the specimen was heldat that temperature for 15 minutes to improve the temperature homogeneity. Just before the specimenwas rolled, the data acquisition system was switched on. After each test, the thermocouples weretested for mechanical and electrical stability. If the thermocouples were still functional, thespecimens were rerolled.Rolling Mill^Tube FurnaceLoad CellFigure 4.2 Schematic layout of the test facilities at UBC524.2.1 Test Schedule at CANMET4.2 Test Procedures4.2.1 Test Schedule at CANMETFive trials have been conducted at CANMET to obtain temperature-time data under differentconditions. Two kinds of materials, AISI 304L stainless steel and 0.05%C low carbon steel, wereemployed to investigate the effect of material type on the interface heat transfer coefficient. Trial1 was conducted to investigate the effect of successive rolling passes on the heat transfer coefficientusing AISI 304L stainless steel specimen containing a pin to measure the strain. The test conditionsare shown in Table 4.3.Trial 2 was conducted on the 0.05%C low carbon steel samples containing pins to investigatethe effect of rolling pressure on the interface heat transfer coefficient; only one pass was carriedout for each test. The test conditions are shown in Table 4.4. The specimens were quenched afterrolling for microstructure evolution.Table 4.3 Conditions employed in rolling tests to determine the influenceof successive rolling passes on interface heat transfer coefficientfor AISI 304L stainless steelTest No. Pass No. RollingTemperature(SC)InitialHo(mm)FinalH1(mm)PercentReduction(%)RollingSpeed(m/s)SS6P71 1250 25.40 22.86 9.0 1.0P72 Not recorded 22.86 21.34 6.65 1.0P73 Not recorded 21.34 19.05 10.73 1.0P74 Not recorded 19.05 14.73 22.67 1.0534.2.1 Test Schedule at CANMETTable 4.4 Conditions employed in rolling tests to determine the influenceof rolling pressure on interface heat transfer coefficientfor a 0.05%C low carbon steelTestNo.RollingTemperature(°C)InitialHo(ram)FinalH1(mm)PercentReduction(%)Time ToQuench(s)RollingSpeed(m/s)RLC12-1 1250 12.65 8.00 36.7 13.65 1.5SLC12-5 1250 12.48 8.00 35.9 13.40 1.5RLC12-2 1250 12.73 / / / /SLC12-6 1250 12.48 5.99 52.0 6.8 1.5RLC12-3 1250 6.72 2.69 60.0 5.4 1.5SLC12-7 1250 7.43 2.95 60.3 4.1 1.5RLC12-4 1250 6.50 2.388 63.36* 6.7 1.5SLC12-8 1250 6.10 2.362 61.28* 6.0 1.5Trial 4 was conducted also with the low carbon steels for microstructural evolution studiesand a different number of passes were carried out for each test. However only the surfacetemperatures of the first rolling pass of the tests 1LC-6 and 3LC-1 (with *) were recorded. Theconditions for the first pass are stated in Table 4.5.Trial 5 was conducted to determine the dependence of rolling temperature on the heat-transfercoefficient employing AISI 304L. Only one pass was carried out for each test and the test conditionsare given in Table 4.6.544.2.1 Test Schedule at CANMETTable 4.5 Tests for microstructural evolution studyTestNo.TotalPassRollingTemperatureInitialHoFinalH1PercentReductionTime ToQuenchRollingSpeedNumber (°C) (mm) (mm) (%) (s) (m/s)1LC-6* 1 1250.0 51.8 48.26 6.8 14.0 1.03LC- 1 * 3 1250.0 51.8 48.26 6.8 8.0 1.03LC-7 5 1250.0 51.8 48.26 6.8 4.5 1.03LC-8 7 1250.0 51.8 48.26 6.8 7.4 1.0Table 4.6 Conditions employed in rolling tests to determine the influenceof rolling temperature on interface heat transfer coefficientfor AISI 304L stainless steelTestNo.RollingTemperature(°C)InitialHo(mm)FinalH1(mm)PercentReduction(%)RollingSpeed(m/s)SS-4 850.0 25.8 18.4 28.7 1.5SS-8 850.0 25.9 16.7 35.5 1.5SS-13 850.0 26.0 14.48 44.3 1.5SS-16 950.0 25.9 18.82 27.3 1.5SS-11 950.0 25.8 16.4 36.2 1.5SS-18 950.0 25.8 14.12 45.3 1.5SS-9 1050.0 26.0 18.99 27.0 1.5SS-19 1050.0 26.0 16.92 34.9 1.5SS-17 1050.0 25.9 14.4 44.4 1.5554.2.1 Test Schedule at CANMETTrial 6 was conducted to investigate the dependence of the heat-transfer coefficient on rollingspeed; the test conditions are given in Table 4.7Table 4.7 Tests conducted to determine the influence of rolling speedon interface heat-transfer coefficientTestNo.RollingTemperatureInitialHoFinalH1PercentReductionRollingSpeed(°C) (mm) (mm) (%) (m/s)SS-15 950 25.80 15.75 38.9 0.5SS-20 950 25.80 15.75 38.9 1.0564.2.2 Test Schedule at UBC4.2.2 Test Schedule at UBCDue to the difficulties encountered at CANMET in measuring the thermal response for thelow carbon steel and the stainless steel for large deformation, some supplementary tests have beenconducted at UBC using a HSLA steel (0.05%C and 0.025% Nb steel). The test schedule is shownin Table 4.8.Table 4.8 Tests for HTC measurements at UBCTestNo.RollingTemperature(°C)InitialHo(mm)FinalH1(mm)PercentReduction(%)ContactTime(s)Testl 850 11.30 9.55 15.51 0.0265Test2 950 12.58 9.75 21.27 0.0328Test3 950 12.60 9.45 25.0 0.0355Test4 1050 9.45 8.00 15.34 0.0241Test5 1050 12.55 11.6 7.51 0.0195Test5-1 1050 11.60 10.03 13.51 0.0251Test5-2 1050 10.03 8.10 19.27 0.0278Test6 1050 12.60 12.20 3.17 0.0126Test? 1050 12.60 11.81 6.27 0.0178Test?-1 1050 11.81 11.50 2.62 0.0111Test8 1050 12.55 11.53 8.13 0.0202Test8-1 850 11.53 10.20 11.54 0.0231Test9 850 12.60 11.15 11.51 0.0241574.2.2 Test Schedule at UBCThe rolling speed was constant at 0.354 m/s for all of the tests. The contact time in the roll bitewas calculated according to the rolling speed.584.3.1 Thermal Response4.3 Thermal Response of Instrumented Specimens4.3.1 Thermal ResponseAs the specimen passed through the roll gap, the data acquisition system recorded the threethermocouple 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 equationfor thermocouple type K, Chromel-Alumel, was adopted with a 0.7°C error within the temperaturerange from 0°C to 1370°C. The roll force conversion was obtained through the calibration of therolling mill load cell prior to testing.For Trial 2 conducted at CANMET using the low carbon steel, all the thermal responsesrecorded were flawed as shown in Fig.4.3 for Test RLC12-l.1000 -*O800 -O600 -n.E- 400 -1400^1200 -+,200 10 ^09TC1TC2TcentLoad42—44—46—48—50—52—54—56—58—60—621.15E-000-if^+ ++*+-4- ++-1+4" •t# .44.  4 1.4t4.#4.V441-+"4+; 4411"4tpf. 41.1. *1--1- ^+ +^* t0.95 ^1^1.05(s)Time1.1Figure 4.3 Thermal response of thermocouples for Test RLC12-159u 1000 -oa)800 -n.6000.66 0.68^0.7Time (s)0—10—20—30—40—50—60—70—80—90E-0a0- JTC1TC2TcentLoad4.3.1 Thermal ResponseThe thermocouples detached from the specimen before or during rolling, apparently due toscale formation. For Trial 4, only one thermocouple on the surface remained functional for eachtest even during small reductions, as shown in Fig.4.4 for Test 1LC-6. The major reasons for thethermocouple problems are: 1) oxide scale formation on the low carbon steel specimen surfacebecause no inert gas was applied in the furnace during heating; 2) large reduction (above 35%) foreach 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 somethermocouples 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 norThermocouple 2 at the two positions on the surface recorded the thermal responses, and only thefirst pass for Test S S6 ( in short for SS6P71) in Trial 1 was acceptable because of the small reductionemployed.Figure 4.4 Thermal response of thermocouples for Test 1LC-6604.3.1 Thermal ResponseFig.4.5 shows the thermal response of the surface and center thermocouples during the rollingof a stainless steel specimen at CANMET. The response of the surface thermocouple are virtuallyidentical.750 0^12001150 -1100towaswolowsaftwoormArisitr:Tsurf-27\,"^ r^Tcent 0— 1050 -% 1000 -05& 950-a.>H 900-850-800-0.02^0.04^0.06^0.08^0.1^0.12^0.14^0.16Time(s)Figure 4.5 Thermal response 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 at UBC were more trouble-free than those conducted atCANMET, although a microalloyed steel (0.025%Nb) was used as the specimen. Because a tubefurnace was used with N2 flowing through its hearth, scale rarely formed during reheating. However,due to the limitation of the motor power and the higher strength of the microalloyed steels, thereduction for each test was limited to less than 20%, otherwise the specimen jammed in the rollgap. Evidence of this is seen in the results of Test 3 presented in Figure 4.6.61251000TC1Load900 - -200° 800 -a)700 -a)a) 600 -500- 10 o- 54.3.1 Thermal Response400^ I^ 00^0.1^0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Time (s)Figure 4.6 Thermal responses of thermocouples for Test 3It is very interesting to see that the oscillation of the surface temperature corresponds wellwith that of the roll force. When the roll force increases, the surface temperature decreases, andvice 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, asshown in Fig.4.7, whilst in others it was poor, see Fig.4.8. In Fig.4.8, it is evident that the minimumtemperature for the second thermocouple (TC2) is about 200°C higher than that for the firstthermocouple (TC1). An anomalous response of TC2 was due to the bad installation of thethermocouple prior to rolling. When INCONEL sheathing of the thermocouple protruded abovethe surface after it was placed in the groove, good contact between specimen and the rolls was notachieved at that location. A sudden increase in roll force at the thermocouple location is evidentof this improper placement. Data such as this was disregarded.62-10-8 20-4-20.1^0.200.3 0.4 0.5 0 6T1rne(s)400014-121 1 001000900800700O6005004.3.1 Thermal Response1200Figure 4.7 Thermal responses of themocouples for Test 7-1Figure 4.8 Thermal responses of themocouples for Test -1631100 --4-TC25th Poly105001000 -m950 -900 -850 -800 -11500.01^0.015Contact Time (s)750 0 0.005 0.02 0.0254.3.2 Surface Temperature4.3.2 Surface TemperatureThe 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 surfacetemperature changes are shown in Fig.4.9 to Fig.4.11 for Test SS6P71, SS-8, SS-15 with differentrolling speeds and reductions.Figure 4.9 Surface temperature in the roll bitefor Test SS6P71 with 1.0 m/s64TCI11th Ploy43.2 Surface Temperature850800750700a) 650a1,3 600it 5505004504000 0.005^0.01^0.015^0.02^0.025^0.03^0.035Contact Time (s)Figure 4.10 Surface temperature in the roll bitefor Test SS-8 with 1.5 m/s9008500TC111th Poly800 - al750 -m 700 -—0 650 -a. 600 -E550la500 -450 -4000Contact0.06Time (s)0.04 0.05 0.07 0 .08 0.09^0.10.01 0.02 0.03Figure 4.11 Surface temperature in the roll bitefor Test SS-15 with 0.5 m/s654.3.2 Surface TemperatureFrom Fig.4.9, it is apparent that approximately 36 data points were acquired in the 0.025seconds of contact time; this time depends on the rolling speed and reduction during the test. ForTest 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 surfacetemperature 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 lowerspeeds. Obviously, if the reduction is relatively small, the location of the minimum temperature iscloser to the exit of the roll bite. On the other hand, if the contact time is long enough as in the caseof low speed and large reduction, the surface temperature of the roll and the slab would approacheach other and then the surface temperature of the slab could subsequently rebound due to heatconduction from the interior of the slab to the surface. This phenomenon has also been observedin tests at UBC, as shown in Fig.4.12 and Fig.4.13 for thermocouple 1 of Test 6 and Test 8-1respectively, with different reductions of 3.17% and 11.54% respectively.To produce a function that represents the measured surface temperature, a polynomialregression has been conducted on the data over the range of contact time. However, for some testssuch as SS-9, SS-18 and SS-19 at CANMET, and Test 3 and Test 4 at UBC, because thethermocouples failed halfway in the roll bite, only some of the data points were analyzed. Theregression line has been shown in the each of the Figures 4.9 through 4.13 with the surfacetemperature reading.661050 ^1000 -950 -0 900 -0........a) 850 -L..D15L. 800 -Ea4:)^750 -)---700650..f.-. .) 750 -0s.-' 700 -a)L.2 650 -aL.2:', 600 -E550 -500-aTC16th Poly4.3.2 Surface Temperature0.002^0.004 0.006^0.008^0.01Contact Time (s)0.012^0.014Figure 4.12 Surface temperature in the roll bite for Test 6900400 ^0 '0.005 0.0250.01^0.015^0.02Contact Time (s)Figure 4.13 Surface temperature in the roll bite for Test 8-1674.3.2 Surface TemperatureThe regression results for the successful tests at CANMET and at UBC are listed in Table4.9 and Table 4.10, respectively,Table 4.9 Regression results for tests at CANMETTestNo.Order ofregressionR2 S„xSS6P71(TC1) 6 0.9978 5.170SS6P71(TC2) 5 0.9983 4.149SS8 11  0.9914 8.892SS-18 8 0.9746 15.29SS-9 7 0.9978 7.256SS-19 5 0.9940 10.161SS-15 11 0.9971 4.668SS-20 11 0.9973 6.3453LC-1 4 0.9903 6.6801LC-6 6 0.9986 3.498684.3.2 Surface TemperatureTable 4.10 Regression results for tests at UBCTestNo.Order ofregressionR2 SysTest3(TC1) 8 0.9990 5.873Test4(TC1) 6 0.9990 4.595Test6(TC1) 7 0.9997 3.134Test6(TC2) 6 0.9952 12.984Test7-1(TC1) 5 0.9980 7.123Test7-1(TC2) 6 0.9987 5.007Test8-1(TC1) 7 0.9899 15.403where R2 is the coefficient of determination and sy,x is the standard error of the temperature estimate.The results show that the regression curve can be used to represent the acquired data. The regressionresults obtained will be used in Chapter 6 to back-calculate the interface heat-transfer coefficientfor different rolling conditions by using the heat-transfer model developed in Chapter 5.69Chapter 5HEAT TRANSFER MODEL DEVELOPMENTTo predict the thermal history of a slab during rough rolling, a two-dimensional mathematicalmodel has been developed using the finite difference technique. The model was also modified tocalculate the heat transfer coefficient in the roll bite from the thermal response measurementsobtained at CANMET and UBC. The model was verified by comparing the experimental resultswith an analytical solution.5.1 Mathematical Modelling5.1.1 Heat Conduction in the Slab during Rough RollingAs described in the literature review, the heat losses from a slab during rough rolling mainlyconsist of radiation to the surrounding atmosphere, convection to the air and water spray, andconduction to the work rolls. To compute the temperature distribution in a slab through the thicknessas well as along the length, the general heat conduction equation has to be solved subject to theboundary conditions for rough rolling. The governing equation can be written as follows:aTv(ksvT)+4. = PsCpa a st (5.1)To solve the above equation, a completely implicit finite difference method was adopted. Thefollowing assumptions were made:70Work RollDControl Volume YvsHO —.-Slab5.1.1 Heat Conduction in the Slab during Rough Rolling/((A)Work RollFigure 5.1 Schematic diagram of hot rolling1) Assume the rolling process was in steady state, i.e., the temperature in the slab was only afunction of location, not a function of time;2) Heat flow along the transverse direction was ignored because the ratio of width of the slabto 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 thePeclet number of approximately 4 x 105), conduction along the length of the slab is negligiblein comparison with heat transfer by bulk motion;4) For hot rolling problems, the metal flow velocity in the direction of the slab's thickness ismuch less than that along the rolling direction and it decreases with depth and reaches zeroat the center line of the slab, for this reason, the bulk motion heat flow in the depth directionwas ignored;715.1.1 Heat Conduction in the Slab during Rough Rolling5) The deformation was assumed uniform and the heat generation due to deformation wasassumed 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 slaband the other two thirds to the work roll because of the lower temperature of the work roll,in accordance with other investigators114m281 ;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 propertieswere 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 equationof 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( .^aTk ^+qs =p,C s —ay say p atwhere t is the time taken for an elemental control volume of the slab to travel a distance x measuredfrom a reference point, e.g., at the exit point of the reheating furnace, see Fig. 5.1.(5.4)725.1.2 Boundary ConditionsSince conduction in the X direction has been ignored, the governing equation can be appliedto every position along the length of the slab with its specific boundary conditions. In the presentstudy, only the thermal histories at the head end, tail end and the middle point along the length ofthe slab were computed.5.1.2 Boundary ConditionsTo 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 ConditionThe exit of reheating furnace was taken as an initial point for the model, i.e.:Ho't =0, 0 .. x^2Lo, 0..y — Ts =To5.1.2.2 Boundary ConditionsThe boundary conditions at specific position were expressed as follows:For symmetrical cooling, at the center line:aT,t>0, y=0,—ks —ay =0and at the slab surface:t > 0, y = H (t)  ' k' an—aTs = h(t)(T, — T.,(t))2(5.5)(5.6)(5.7)73hrad = SE(T) (Ts — T.„)((Ts + 273.15)4 — (T + 273.15)4)(5.8)5.1.2 Boundary Conditionswhere H(t) is the slab thickness at each zone; h(t) is the heat transfer coefficient for each zone anddetermined by the correlations elucidated below; n corresponds to the direction normal to the slabsurface; 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 airRadiative heat loss occurs throughout rough rolling when the slab is exposed to air. In thismodel, a pseudo radiative heat transfer coefficient has been adopted and is expressed asr 91 :where the radiative emissivity of the slab surface, e(T), is as follows 193 :E(T)= 1 0^00 (0.1251T000 0.38) +1.1^(5.9)Actually, natural convection also occurs for the slab during radiative heat transfer. For roughrolling, a convective heat transfer coefficient of 8.37W/m2-°0131-(151 was employed in the model.High Pressure Water DescalingAt Stelco, a set of water jets ( one above and one below) are located on the each side of theroughing 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 bySasaki et al. 1181 for low pressure sprays is given as:h =708W °15T:12 + 0.116^ (5.10)745.1.2 Boundary Conditionswhere 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. Devadast281 pointed out thatalthough this correlation was based on the measurements made at lower water fluxes and lowpressures, the h compares favorably with the heat transfer data published by Kohringm for highpressure water sprays and Yanagi [133 . Because the parameters of the descalers for the roughing milland the finishing mill are the same, the above correlation was employed in the current model.Roll ChillingNearly 38% of the total heat loss from the slab is extracted by the cold rolls. A thermal contactheat transfer coefficient hg v, back-calculated from the experimental studies, was employed toquantify the heat transfer at the roll slab interface.Heat generation in the Roll BiteThe temperature rise due to deformation can be converted from the mechanical work with anefficiency factor of approximately 11=0.80-0.85 for steels E213 .1  a H0 )AT =r1^l —def JoC^Hs psn \ 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 )0.126 – 1.75C + 0.594C2+^ 0.21 ' 0.13a = 9.81 xf x exp X £ X 6T3 + 273 (5.12)where f is a factor considering the effects of chemical composition other than carbon content; forplain carbon steel f = 1, but for others it was expressed as:f = 0.916 + 0.18Mn + 0.389V + 0.191Mo + 0.004Ni^(5.13)755.1.3 Heat Conduction in the Work RollTemperature rise due to friction was assumed to be confined to the interface [141 :4f V ?PIP^ (5.14)where vr is a relative velocity between the roll and the slab at the interface and was expressed as:vr = Ro)H„cos 1p„1 H^ cos (5.15) and the friction coefficient, lif, is given by t211 :.tf = 4.86 x lei; — 0.074^ (5.16)5.1.3 Heat Conduction in the Work RollThe heat lost from the slab is gained by the work roll during rolling, so that the slab and thework roll are coupled. The general governing equation for the rolls was expressed as:1 a( aT,J + 1 a r aT, ) 4..k(k aTr ) .,coprcpr aTer17-57. 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 thetemperature 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 atdifferent 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 ismuch larger than the layer thickness, 8;765.1.3 Heat Conduction in the Work Roll3) Heat conduction along the peripheral direction(9) was negligible compared with the bulk heatflow, 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 propertiesare temperature dependentm .Based on the above assumptions, the governing equation for rolls can be simplified as:Employing the transformation,( aT,^aTr ar rk,.ar =coprCpr6 (5.18)0 =coxt^ (5.19)The Eq.(5.18) can be converted into:l a^aTr aTrrkr1. =prCpr at (5.20)where t is the time taken for an elemental volume of the rolls to rotate through an angle, 0, measuredfrom 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:aTrt >0, r =R —8, -kr -5;-. =0^ (5.22)and at the surface,775.1.4 Numerical Solution,t >0, r =R, –kraa —=h(t)(7,,--Tjt*)) (5.23)The layer thickness, 0, within which a cyclic temperature variation occurs during eachrevolution of the roll, was defined by Tseng 1191 as the depth where the temperature changes cyclicallyby greater than 1%. It was expressed as:5^2ln ^200Bi(aBil ffi+2) R^-N12P e –1^el2(Bi2IP e +4-213i1A(5.24)where y is the bite angle. If the heat transfer coefficient, h, varies along the roll surface, the maximumvalue should be used because a larger value of h would result in a thicker layer, S. The aboveequation indicates that the thermal layer thickness depends not only on the Peclet number, Pe, butalso on the Biot number, Bi, and the bite angle, cp. In some previous studies [221(281 , the layer thicknesswas 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 Chapter2. All the correlations have been verified by Devadas [281 .5.1.4 Numerical SolutionTo solve the governing equations of the slab and the roll with all the boundary conditions, animplicit finite difference technique was employed ( see Appendix A ). A slice of the slab wasdivided into a number of nodes outside the roll gap. While in the roll gap, the two governingequations had to be solved simultaneously, so a slice of the roll surface layer corresponding to theone 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 principleexpressed as Eq.5.25.78H1/2SLABHO/2XINTERFACEVg5.1.4 Numerical SolutionFigure 5.2 Discretization of the slice in the slab and the rollfor finite difference analysisHo 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 atwhich the slab velocity is equal to the roll surface velocity in the roll bite, the following expressionwas adopteei:H0+3H1H„— 4 (5.26)From the Eq.(5.25), the slice velocity in the slab increases along the rolling direction, but thespeed of the roll does not change; this results in different time steps for the slice in the roll surfacelayer and the slice in the slab.A flow chart is given in Fig.5.3 showing the steps involved in the model. The thermal historyat each position along the length of slab can be computed, but only the head end, the tail end andthe middle point were considered in this study to minimize the computational time.79^11.Initial DataPass No.=1Subroutine for radiative andnatural convective heat transferYesSubroutine for radiative andnatural convective heat transferSubroutine for boiling Heat transferSubroutine roll gap heat transfer4Work roll heat transfer modelRoll temperature cyclic steady ?Subroutine after rollingfor radiation and natural convection heat tansferNoEnd of Roughing ?YesCStop5.1.4 Numerical Solution(start )Figure 5.3 Flow chart of the temperature solution of the model805.2.1 HTC Solution5.2 Modification of the Model for Roll Gap Heat-Transfer Coefficient Calculation5.2.1 HTC SolutionThe model was modified to compute the roll gap heat-transfer coefficient (HTC) from thesurface temperature. In the model, a trial-and-error method was adopted to determine the magnitudeof 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"1initial HTC( stop )Figure 5.4 Flow chart of HTC calculationAn initial guess was made for the HTC in the first time step, and then the surface temperaturepredicted 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 the81UQ) 300UL._200100EpsO1.0 C• 4.0 °Co 15.0 °CoC_)(NI^50 -E405.2.2 Convergence of the Numerical Solution for the Modified Modelnext time step until the contact time in the roll bite is over. This calculation produces the HTCdistribution as a function of time along the arc of contact. By this procedure, all of the thermalresponse measurements for different rolling conditions could be converted to heat transfercoefficient variations in the roll bite.5.2.2 Convergence of the Numerical Solution for the Modified ModelDetermination of EpsAccording to the solution technique described above, it is important to establish a reasonablelimit, Eps for the allowable difference between the measured and the computed temperature; Thisvalue would affect the HTC magnitude calculated, as shown in Fig.5.5.600 4 0.000^0.005^0.010^0.015^0.020^0.025Contact Time (s)Figure 5.5 Effect of Eps on HTC magnitudeTheoretically, the smaller the Eps, the better the accuracy of the HTC magnitude. Eventually alimit is reached because of errors in temperature measurements as well as round off errors in thecomputer. The standard error of estimate for the dependent variable temperature, T, stands for theaverage deviation of the experimental data around the regression line. From Table 4.9 and Table825.2.2 Convergence of the Numerical Solution for the Modified Model4.10, the Sy, value is known for the tests to vary from approximately 2°C to 15°C. In Fig. 5.5, theEps magnitudes of 1, 4 and 15°C were investigated; The results show that the larger magnitude ofEps, such as 15°C, has a lower maximum HTC, but for the values of 1 and 4, on average, the HTCdistributions do not change significantly. In the present study, a value of Eps equal to 4, was adoptedfor all of the tests.Effect of Time Step and Mesh Size on HTC MagnitudeThe implicit finite difference method does not have the stability problem, but truncation errorscan cause the accuracy to suffer slightly. As mentioned before, it is necessary to select a very smalltime step for back-calculation of the HTC, but how small should it be? As pointed out by Thomaset a1. 1611 , unless stability problems are encountered, accuracy generally improves with the refinementof 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 smallerthan the critical value, finer meshes greatly improve accuracy. This critical time-step size is smallerfor finer meshes. Continued refinement of the time-step size results in little further improvementand eventually the accuracy worsens. Thus, every given mesh has an inherent limit with respectto accuracy and an optimum range of time-step size associated with it. Moreover, an optima criteria(k ) Atwas proposed which states that the dimensionless number, —1--p^0.02 , appears to remain a constant,Cp, of roughly 0.1 at the optimum of each of the three kinds of meshes, coarse mesh, medium meshand 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 areshown in Fig.5.7. From Fig.5.7, it is apparent that time-step sizes of 0.0002 and 0.00005 secondsdo not change the HTC value significantly, but a small deviation occurs for a time-step size of 0.001seconds. Therefore, a time-step size of 0.0002 seconds was adopted in the present study.83— Dupont -Motrix lumped qT. (tz)- Dupont - S t (Word 4neor q(tz )0 Coorse Mesh0 Medium MeshA Fine Mesh00-435.2.2 Convergence of the Numerical Solution for the Modified Model10O0to0a,a,0-10a,O.>t2^5^10^20^50^100 200Number of Time Steps to 600sFigure 5.6 Effect of mesh size and time step on accuracy 1611Fig.5.8 shows the effect of the mesh sizes in the specimen and in the surface layer of the rollon the value of the HTC.From Fig.5.8, it is evident that 150 nodes in the roll and in the specimen respectively, aresufficient; 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 anode size of 150 in the roll and in the specimen were adopted. These sizes satisfied the optimacriteria for mesh and time step size adopted by Thomas et al.E611 .001845.2.2 Convergence of the Numerical Solution for the Modified Model60504030a)005 20"rnC01000.000^0.005^0.010^0.015^0.020Contact Time (s)Figure 5.7 Effect of time step on HTC value0.025600E 5040C-030CD020C0F--10000.000^0.005^0.010^0.015^0.020^0.025Contact Time (s)Figure 5.8 Effect of mesh size on HTC value851050 -02 1000 -0950 -g-F" 900 -5.2.3 Verification of the Modified Model5.2.3 Verification of the Modified ModelThe heat transfer coefficient variation with time in the roll bite has been determined usingthe procedure described earlier. Obviously, the computed heat transfer coefficient for the roll biteshould yield temperatures that are good agreement with the measured values. A typical exampleof the agreement is shown in Fig.5.9 for the Test SS6P71.0.01^0.015^0.025Contact Time (s)Figure 5.9 Comparison of the predicted and the measured temperature8612601240U 1220tuL315 1200La)a) 118011601 0^2011400 30Dt=0.005(s)Dt=0.0 1(s))4(Dt=0.05(s)Dt=0.1(s)xDt=0.5(s)5.3 Sensitivity Analysis for the Roughing Model5.3 Sensitivity Analysis for the Roughing ModelTime StepDuring rough rolling, using a very small time step for every zone is impractical due to thecomputation cost. For this reason, the use of a varying time step was investigated. Fig.5.10 showsthe effect of time step size on radiative and natural convective cooling.Time(s)Figure 5.10 Effect of time step size on the surface temperatureunder radiative and natural convective coolingThe figure shows that the time step has little effect on the surface temperature distribution inthe interstand region. These results are further illustrated in Table 5.1.875.3 Sensitivity Analysis for the Roughing ModelTable 5.1 Sensitivity of time step size on the predicted surface temperatureTime Step ofcooling(s)Surface^Temperature^Ts(°C)at0.5(s)15(s)30(s)0.005 1233.89 1170.78 1143.840.01 1233.82 1170.78 1143.840.05 1233.97 1170.84 1143.840.1 1234.15 1170.86 1143.850.5 1235.48 1170.93 1143.91A very small deviation (approximately 1.3°C) occurs for the largest time steps of 0.1 and 0.5second at the beginning of cooling. This is because the cooling rate at the surface is faster at thebeginning 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 foreach zone which is fixed. Therefore, a time step of 0.01 second was adopted in this model forradiative cooling in the interstand region.In the roll bite, a time step of 0.0002 second was employed and 0.0005 second was deemednecessary for the descaling zone.Node SizeFig.5.11 shows the effect of node sizes in the slice in the slab on surface temperatures underradiative and convective cooling. From the figure, it can be seen that a further increase of thenumber of the nodes in the half-thickness of the slab has little effect on the surface temperature.88300-300250-250200-200100-100x50-505.3 Sensitivity Analysis for the Roughing Model12601240 -cY 1220 -a)—0 1200 -c,)a) 11801160 -1140 ^0 10 20Time (s)30Figure 5.11 Effect of node size on the predictedsurface temperatures under radiative and natural convective coolingThe exact temperature variations with different node numbers at different time are also shownin 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 numbersexamined.During rough rolling, the thickness of the slab is significantly reduced. Theoretically, thenumber of nodes in the half-thickness can be reduced as the thickness decreases, but in thecomputation, it is not worth changing the node size, because it would require interpolation whichwould introduce further errors. Thus a constant node number of 200 was maintained throughoutthe rolling process.89oE=0.6KE=0.7)1(E=0.8xE=0.9E=f(Ts)12601240a 1220023.. 1200mIiLai 1180 -aEa)I— 1160-1140-og°oX X 0 0g 0 0x x xX 0X gx°00X DOX^0gX^0XXXX% (:)00 00 00X X g x szx x 5e^xw--x xx x)4( )4(g 0X g _^0000X X g x11200 10 20 30Time (s)5.3 Sensitivity Analysis for the Roughing ModelTable 5.2 Sensitivity of node size on the predicted surface temperatureNo. ofNodeSurface Temperature Ts(°C)atthrough 0.5 15 30half- (s) (s) (s)Thickness300 1233.76 1170.77 1143.83250 1233.78 1170.77 1143.83200 1233.82 1170.78 1143.84100 1234.2 1170.82 1143.8650 1236.14 1170.98 1143.95Figure 5.12 Effect of emissivity on the surface temperatureunder radiative and natural convective cooling905.3 Sensitivity Analysis for the Roughing ModelEmissivityFig.5.12 shows the effects of emissivity on the surface temperature in the zone before thefirst descaler.Obviously, the emissivity has a large effect on the temperature. The predicted surfacetemperatures are also shown in Table 5.3.Table 5.3 Sensitivity of emissivity on the predicted surface temperatureEmissivitySurface^Temperature^Ts(°C)at0.5(s)15(s)30(s)0.6 1237.91 1189.23 1167.700.7 1236.06 1180.86 1156.910.8 1234.21 1172.78 1146.570.9 1232.38 1164.95 1136.66e(T6)Eq.(5.9)1233.82 1170.78 1143.84Roll Gap Heat Transfer CoefficientThe effect of roll gap heat transfer coefficients on the temperature distribution has beenexamined, and the results are shown in Fig.5.13.9111501100-1050 -0° 1000 -950 -m0900 -tya_a) 850 -F-800 -750 -7000+ 0MK^^^ 4' ^3sE +^ +)tEIll+^^ ^ ^hgap=30kW/m'—°Chgap=50kW/m'—°C)4Ehgap=75kW/m 2-0C^ )i(^ *++ +^ 00^ ^ c^hgap=100kW/m 2 —°C^ )* )4(NE.4-++ + +^ ^ ^ ^—° as^ W + +^ W )4( + +^^ 4++NE )4(^ ^4. -NE^ A )4(0 0^ NE )4(^ ^ 0)k)4E W )WE0 ^ Ci 1:3 CO0.01^0.02 0.03Contact Time (s)5.3 Sensitivity Analysis for the Roughing ModelFigure 5.13 Effect of the roll gap heat transfer coefficienton the surface temperature of the slab in the roll biteFrom the figure, it is evident that with the same initial temperature distribution, an increasein the roll gap heat transfer coefficient from 30kW/m2-°C to 100kW/m2-°C causes a reduction inslab surface from 882°C to 738 °C at the exit of the roll gap. Although the chilling effect is confinedto a thin subsurface layer, it has been shown that it affects the final temperature distribution in amulti-pass rolling schedule 1313 . Furthermore, it affects the material flow stress and flow pattern andtherefore influences the structure and properties of the steel. This emphasizes the need to properlycharacterize the interface heat-transfer coefficient.92-6, 1040 -co32 1035 -a) 1030 -11.I— 1025 -5.4 Verification of the Roughing Model5.4 Verification of the Roughing ModelTo assess the accuracy of the model, each module was tested by comparing the results of thisroughing model to the results of other models in the literatures] for the same test conditions.In Fig.5.14 to Fig.5.16, the model-predicted temperature distribution through the halfthickness 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.10^15^20^25Half Thickness (mm)Figure 5.14 Comparison of the temperature distribution predictedby the current model with that of Devadas (281Fig.5.16 shows the temperature distribution through the half thickness of the strip at the roll gapexit of the first pass. From the sensitivity analysis, it is known that the roll gap heat transfercoefficient has a very strong effect on the temperature distribution, especially in the layer beneaththe surface. Thus, the slight difference between the results is likely due to differences in the rollgap heat-transfer coefficient.931050 1 000 -0Literature DataRoughing Model900 -850 -800950 -5.4 Verification of the Roughing ModelV00,mi - 5L.V,I-0^5^10^15^20^25Half Thickness (mm)Figure 5.15 Comparison of the temperature distribution predictedby the current model during descaling with that of Devadast 2811-—4^6^8^1 0^12Half Thickness (mm)Figure 5.16 Comparison of the temperature distribution predictedby the current model in the roll bite with that of Devadas [28)94500490 -480 --a 470 -0460 -11 450 -E440 -Ei!-I! 430 -420 -.410 -4000 0.5^1 1.50NumericalAnalytical- . ..- ...... _.. .. .2^2.5Time (s)3^3.5 45.4 Verification of the Roughing Model11001000 -0° 900 -N.—,a)a)a_800 -700600 -500 Tcent (Roughing MOdel)Tsurf (Roughing Model))1(Tcent (Literature Data)0Tsurf (Literature Data)012^14^16^180^2^4^6^8^10Time (s)Figure 5.17 Comparison of the thermal history of a strip predicted bythe roughing model with the data from Devadast281Figure 5.18 Comparison of the model results with an analytical solutionfor the work roll955.4 Verification of the Roughing ModelFig.5.17 shows the thermal history of a strip predicted by the current model under the samecondition as those employed by Devadas E281 . The strip experiences a radiative heat loss before twosets of high pressure water sprays, and a backwash spray prior to the first rolling pass. Only onepass was considered since subsequent passes utilize the same module. It is apparent that a goodagreement is obtained everywhere although minor difference are evident in the roll gap. Thedifference in the predicted temperatures for the center line results from a different formula beingemployed 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 ofa corresponding analytical solutiont 251 . The analytical solution is given by the following FourierSeries:T -^- 2^Ji(1-1-k) Ti^k = 1 ilk JAN) + JAN) exp(-14at/R2./0(Mkkr (5.27)where Jo and J I are Bessel Function of the first kind andli k.XkR, is the kth root of the transcendentalequation: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 moduleh(W/m2-°C)ic,(W/m-°C)ar(m2/s)Bi R(m)1160.0 29.0 7.0x10-6 10.0 0.50A very good agreement has been obtained in the figure which indicates that work roll heattransfer module is valid and accurate.965.4 Verification of the Roughing ModelFinally, from all of these analyses, it can be concluded that the roughing model developed inthis chapter can be applied to simulate rolling. In Chapter 7, the model will be further extended toincorporate the effects of oxidation.97Chapter 6ROLL GAP HEAT TRANSFER COEFFICIENT ANALYSISThe thermal response measurements from the pilot mill tests at CANMET and at UBC weredescribed in Chapter 4, and a mathematical model was developed to determine the roll gapheat-transfer-coefficient (HTC) in Chapter 5. The model has been employed to compute the rollgap heat-transfer coefficient for different rolling conditions and these results are presented in thischapter.6.1 Roll Gap Heat Transfer Coefficient Analysis6.1.1 HTC Variation along the Arc of ContactFig.6.1(a) and (b) show the HTC variation along the arc of contact determined from theresponse of two thermocouples located on the same sample, SS6P71, in Trial 1. The results arevery similar to those obtained in an earlier study by Devadas [283 . The HTC is seen to increase andreach an apparent plateau followed by a decrease. The maximum heat-transfer coefficient is in therange of 50-60kW/m2-°C range for both thermocouples. The maximum value of the heat-transfercoefficient appears to be very sensitive to rolling parameters, as is shown in Fig.6.2 in which amaximum value of approximately 600kW/m2-°C is realized. The reasons for this will be discussedlater.986.1.1 HTC Variation along the Arc of Contact^70 ^i3o 60-E5040 --4-'0'8 30g 207?) 10-0060 ^50 -E6 40--60'a:4-; 30 -0UL.+it: 20 -ocIL-a 10to00.005^0.01^0.015Contact Time (s)(a) Thermocouple 10.02 0.0250^ 0.01^0.015Contact Time (s)(b) Thermocouple 2Figure 6.1 HTC variation for Test SS6P710.005 0.02599^0E 80 -70 - ^^ ^o^^ ^^"Ea) 60 -'6• 50 -O40 -^^Lco- 30 -^o020100o 90 -0 ^a ^^^o0^ 0.0016.1.1 HTC Variation along the Arc of Contact700'-S0"E 600500 -400 -8 300 -L.+v)* 200 -0L100 -oal00^0.005^0.01^0.015^0.02^0.025Contact Time (s)Figure 6.2 HTC variation for Test SS80.002^0.003^0.004Contact Time (s)Figure 6.3 HTC variation for Test SS90.03 0.0350.005^0.0061006.1.1 HTC Variation along the Arc of Contact0^0.002 0.004 0.006 0.008 0.01^0.012 0.014 0.016Contact Time (s)Figure 6.4 HTC variation for Test SS 18^350^0"E 300 —v 250——C•z—)• 200 —0 150 —a)"c7 100-o.4_-50-0^0 ^0 0.004^0.008^0.012^0.016^0.02Contact Time (s)Figure 6.5 HTC variation for Test SS191010.005 0.01^0.015^0.02^0.025Contact Time (s)300E25 -20 -cO-60.01^0.015^0.02Contact Time (s)Os0 0.0250.005 0.0311111111111 11 1111111111111140 ^35 -3 30 -.1e5-Figure 6.6 HTC variation for Test 1LC-600.036.1.1 HTC Variation along the Arc of ContactFigure 6.7 HTC variation for Test 3LC-110250-iiii 1111.1.tri,0.01^0.015Contact Time (s)0 ^0 0.005 0.02 0.025400 ^E 350 -3 3004E 250 -m200 -0150 -6.1.1 HTC Variation along the Arc of ContactIn some tests, due to failure of the thermocouples halfway in the arc of contact, only some ofthe surface temperature data was analyzed. The corresponding HTC for these tests, as shown inFig.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 theresults obtained for the stainless steel, but the maximum HTC values are approximately 25kW/m2-°Cto 35 kW/m2-°C, as shown in Fig. 6.6 and 6.7, this value is less than that obtained for the stainlesssteel. All of these differences are considered to be significant and will be shown to be related tothe rolling conditions.The results from the tests conducted at UBC are also similar and an example of the resultsfor Test 8-1 is shown in Fig.6.8.Figure 6.8 HTC variation for Test 8-1Although a third type of material (0.05%C plus 0.025%Nb steel) was employed as thespecimens in the tests at UBC, the variation of the HTC in the arc of contact has the same features1036.1.1 HTC Variation along the Arc of Contactas observed for the other two materials. This indicates that the general form of the HTC variationin the arc of contact is common no matter what kinds of materials are rolled. In addition, themaximum HTC value for the microalloyed steel varies from about 100kW/m 2-°C to 500kW/m2-°Cdepending on rolling conditions.The above figures show that the HTC values are influenced by the rolling parameters, suchas roll percent reduction, roll speed, rolling temperature, and material type. The effects of thesevariables are presented in the subsequent section.1040.0160.004 0.026000E 500 -400(734.i 300 -0412 200 -vaL_I— 100 -60 ^0 0.008^0.012Contact Time (s)6.1.2 Influences of Rolling Parameters on HTC6.1.2 Influences of Rolling Parameters on HTCDevadas et al. f173 have shown that the roll-strip HTC is influenced by percent reduction, rollspeed, and lubrication. They also demonstrated a basic dependence of the HTC on pressure. Theydid not however explore the effect of different materials and the influence of rolling temperature;These variables were examined in this study.Percent ReductionFig.6.9 shows the influence of percent reduction on the HTC.Figure 6.9 Effect of roll reduction on HTC for a rolling temperatureof 950°C and a roll speed of 1.5m/sIncreasing the reduction from 34.9% to 45.3% increases the maximum HTC from320kW/m2-°C to 544kW/m2-°C. All other conditions for the two tests were the same. The lowerHTC will be shown to be related to the lower contact pressure between the roll and the specimenfor the lower percent reduction.105500 ^0 450 -400350 -tn_ 300 -•t- 250 -o° 200 -m150 -a1.`_- 100 -5 50 -0.04^0.06Contact Time (s)0 ^0 0.02 010.086.1.2 Influences of Rolling Parameters on HTCRolling SpeedThe effect of rolling speed was investigated using the CANMET test results, as shown inFig.6.10.Figure 6.10 Effect of rolling speed on HTC for a rolling temperatureof 1050°C and 38.9% reductionFrom the figure, it is apparent that a higher maximum HTC is observed for the higher rollingspeed of 1.0m/s. This can be attributed to the fact that the higher speed increases the strain rate andconsequently the roll pressure. Furthermore, the contact time is reduced by a factor of two.1066.1.2 Influences of Rolling Parameters on HTCRolling TemperatureAn increase in rolling temperature from 850°C to 1050°C causes a significant reduction inthe maximum heat-transfer coefficient, as shown in Fig.6.11. This is attributed.to the fact that atthe lower temperature the deformation resistance of the material is significantly higher, which leadsto a higher pressure along the arc of contact. Again, pressure emerges as an important variableaffecting the interface heat-transfer coefficient.Material TypeThe 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 lowcarbon steel is associated with the lowest. However, it should be noted that the microalloyed steelwas tested at 1050°C and the low carbon steel and the stainless steel at 1250°C as shown in Table6.1. This lower rolling temperature results in a higher roll pressure and consequently a higherinterface heat-transfer coefficient.Table 6.1 Rolling Conditions for material type influence on the HTCRolling Initial Final Reduction Rolling Contact MeanTest No. Temperature H 1 H2 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 5.40SS6P71 1250.0 25.40 22.86 9.0 1.0 0.0241 10.46Test?-1 1050.0 11.81 11.50 2.62 0.35 0.0111 20.14107700600 -500 -400 -0 300 -La)co 200 -00.005^0.01^0.015^0.02^0.025^0.03Contact Time (s)00 0.035200—e—Low Carbon Steel (0.05%C) — 3LC-1—A—Stainless Steel (AISI 3041) — SS6P71—x-Microdloyed Steel (0.025%1%40 — Test 7-1:(7)0(.)120 -100 -80 -tata0LF060-40-I 20-0.01^0.015^0.02Contact Time (s)Figure 6.12 Influence of material type on the magnitude of HTCo0^ 0.005 0.030.025180 -0O160 -\_Ne 140 -6.1.2 Influences of Rolling Parameters on HTCFigure 6.11 Influence of rolling temperature on HTCfor approximately 35% reduction and 1.5m/s1086.1.3 Pressure Dependence of HTC6.1.3 Pressure Dependence of HTCFrom the above analysis, and from the work of Devadas et a1. f171 , it is apparent that the HTCis 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 ofcontact 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 aidof a finite element model which will be described in Chapter 8.35- 30- 25- 20-15- 10- 5- 0--5—105^350 ^0300 -`■-Y 250 --(5 2008 150-a)2100-a50-aa)0 ^—1 0^1^2^3^4Distance along the rot bite (cm)Figure 6.13 Distribution of HTC and roll pressure in the roll bite for SS-19109400^503 300-L 1500 100-E-200 -350 -250 -Pik1-3 50 -a)-40-30-20-10-00^1^2^3Distance along the roll bite (cm)0-1 4^5—10HTC--Ea—R oft Pressure6.1.3 Pressure Dependence of HTCFigure 6.14 Distribution of HTC and roll pressure in the roll bite for SS-150 700EN 6005000 400a)u 300La)0 200c0L100+C;0^11^1^4^1^1 ^10—1 0^1^2^3 5Distance along the Roll Bite (cm)Figure 6.15 Distribution of HTC and roll pressure in the roll bite for SS-86050E40 Ecr,30a)20 (01 0 °-0041106.1.3 Pressure Dependence of HTCAlthough 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 indicatesthat the HTC value is closely related to the pressure in the arc of contact. Moreover, it is interestingto 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 bySamarasekeraf311 , it is known that the heat-transfer coefficient in the arc of contact is dependent onthe real area of contact. Moreover, WilsonI 34) found that with increasing the relative speed, thefractional area in contact decreases. In the case of rolling, a high relative sliding exists at both theentry and the exit of the roll bite. The speed of the rolled stock at the entry side is always less thanthe roll speed, so called backward slip, whilst on the exit side, the stock's speed is larger than thatof the rolls, so called forward slip; in the region between these two, a neutral region exists wherethe roll speed is close to the stock's speed, as schematically shown in Fig.6.15. Therefore, the realcontact 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 coefficientper 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 theroll bite due to the higher HTC and then becomes higher, and finally it approaches the measuredtemperature at the exit of the roll bite. This indicates that the mean HTC can be used to predict thetemperature distribution in a slab.111115011001050 -v°• 1000 -mL.31-5 950 -i_a)Q-E 900a)f-000 0 ^ 0000 000 0 00^00 000 ^0^ 00Test S6P71 (TC2)oMeasured TemperatureCalculated using Mean HTC6.1.3 Pressure Dependence of HTC750 0 0.d05 0.02850 -800 -0.01^0.015^ 0.025Contact Time (s)Figure 6.16 Comparison of surface temperature predictedby mean HTC with the measured one for Test S6P71The 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 thespecimen. The mean HTC value, on the other hand, was defined as the average value in the rollbite. It was obtained by numerical integration of the HTC along the roll bite divided by the lengthof the contact. The mean roll pressure is listed with the rolling conditions in Table 6.2 and Table6.3 for each successful test at CANMET and at UBC.1126.1.3 Pressure Dependence of HTCTable 6.2 Mean pressure for the tests at CANMETTestNo.MaterialTypeInitialThicknessHo(mm)RollingTempera-ture(°C)PercentRed.(%)RollSpeed(m/s)RollForce(Tons)MeasuredMeanPressure(Kg/mm2)SS6P71(TC1)Stain-lessSteel25.4 1250 9.0 1.0 25.17 10.46SS6P71(TC2) 25.4 1250 9.0 1.0 25.17 10.46S S 8 25.9 850 35.5 1.5 189.56 41.54SS-18 25.8 950 45.3 1.5 204.89 39.81SS-9 26.0 1050 27.0 1.5 101.83 25.54SS-19 26.0 1050 34.9 1.5 139.97 30.88SS-15 25.8 950 38.9 0.5 156.47 32.83SS-20 25.8 950 38.9 1.0 165.2 34.663LC-1 0.05%CLowCarbonSteel51.8 1270 6.8 1.0 15.47 * 5.4*1LC-6 51.8 1270 6.8 1.0 15.91 5.6Note: 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 UBCTestNo.MaterialTypeInitialThicknessHo(mm)RollingTempera-ture(°C)PercentRed.(%)RollSpeed(m/s)RollForce(Tons)MeasuredMeanPressure(Kg/mm2)Test3(TC1)Micro-alloyedSteel12.60 950 25.0 0.35 22.21 34.88Test4(TC1) 9.45 1050 15.34 0.35 16.41 37.21Test6(TC1) 12.60 1050 3.17 0.35 4.19 18.40Test6(TC2) 12.60 1050 3.17 0.35 4.19 18.40Test7-1(TC1) 11.81 1050 2.62 0.35 4.27 21.24Test7-1(TC2) 11.81 1050 2.62 0.35 4.27 21.24Test8-1(TC1) 11.53 850 11.54 0.35 13.30 31.9511340 4515^20^25^30^35Mean Pressure (kg/mm')700E^col 200 -aLI-- 100 -o010386.1.3 Pressure Dependence of HTC(a) the tests at CANMET for the AISI 304L stainless steel500 ^0 450 -E-350 -m 300 -.5250 -• 200-L17, 150-1L• 100-15 50 -018 20^22^24^26^28^30^32^34^36Mean Pressure (kg/mm 2)(b) the tests at UBC for the 0.05%C + 0.025%Nb tnicroalloyed steelFigure 6.17 Relation of mean HTC with mean roll pressure1146.2.1 Fenech et al.'s ModelThe relation of mean HTC with the mean roll pressure is shown in Fig.6.17(a) for the tests atCANMET and (b) for the tests at UBC. The maximum HTC value of each test is also shown inFig.6.17 for reference. From the figure, it is apparent that there is a linear relationship between themean HTC and the mean roll pressure for stainless steel (AISI 304L) and for the microalloyedsteel(0.05%C with 0.025%Nb). Due to the paucity of data for the low carbon steel (0.05%C), thesame kind of relationship has not been established, but is likely to be valid.6.2 A Preliminary Theoretical Consideration of HTC during Hot RollingA linear relationship of the mean HTC value with the mean roll pressure has been establishedaccording to the pilot mill test results. A theoretical explanation is encouraged based on thepostulated mechanism for heat transfer between the strip and rolls given by Samarasekera t311 . Inthis section, a model developed by Fenech et a1. E301 for static surfaces in contact was modified toexplain the observed linear dependence.6.2.1 Fenech et al.'s ModelIt is known that 'nominally' flat surfaces contact at only a few discrete points, as shown inFig.2.2. For the heat conduction at the interface, since the thermal conductivity of metals is generallymuch greater than the thermal conductivity of the fluid filling the interstices, heat flow tends tochannel through the points of contact. When the pressure on the contact is increased, the peaks incontact are deformed and the contact points are increased both in size and in number. The heatflow through the interface can be expressed as:q^dT , d7'— ^tc2.—A Ki dxi^dx2(6.1)Thus, the contact heat transfer coefficient, k, can be defined as1156.2.1 Fenech et al.'s Modelhc= OTc^ (6.2)For the determination of the heat transfer coefficient between two real surfaces, Fenech eta1. [301 set up a model for the thermal contact problem, as shown in Fig.2.4. For most contacts theheight of the void is small compared with its width. Under this condition it is reasonable to neglectboth the radial conduction and convection heat transfer in the interstices. Heat transfer by radiationis also neglected.For the purposes of analysis in Fenech et al.'s model, the geometry of the contacting surfaceis 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 voidheights for the two materials. A heat flow channel is assumed to be cylindrical, of radius , a. Withthese assumptions no heat is transferred between channels, and the heat transfer coefficient derivedfor one channel is representative of the entire surface.Assuming steady state, the general heat transfer coefficient has been derived by Fenech eta1. (301 as follows:h, — ^)s,+s,[ (1 — CA2) ^eA^1^1kf^,(1_ c2 ) [ —  lc/ ^8...L 82 )] 426 -,17t-e5i+ 1 4.26 ^+ 14.26,171.—+ 1 4.264; +1 \81 +82 k i A • k2^ + 1.16A(T± —) 4.266A4/7t• k2k2(6.3)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 throughthe metallic contact. The heat transfer coefficient is obviously of the form:(6.4)1166.2.2 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 arethe 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 modelmay be applied to this situation, although it must be borne in mind that Fenech's et al.'s model doesnot 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 simplificationwith kf = 0 may be assumed. As a consequence, the expression (Eq. (6.3)) can be simplified asfollows:CAH7'C = (6.5)0.47A/(L2A81^82 k2+^where lcc is the combined thermal conductivities of ki and k2 , which is defined as1 _ 1( 1 1ki + (6.6)During hot deformation, the asperities on the roll surface are not deformed and becomeembedded in the steel resulting in good contact between the rolls and the steel, so that the averagevoid height for the rolls and the steel, S i and 82, approaches zero although it would never be equalto zero. As a consequence, the effect of these two terms in Eq.(6.5) could be ignored and theexpression for HTC be further simplified as:1176.2.2 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 bedetermined by experiment. To determine the value of n, a relationship between the hardness of thesofter material, and the size of the indentation area(E2A/n) is assumed according to hardness dataobtained Fenech et a1.E301 , as shown in Fig.6.18.O_600550500450400350004^0.1^ 1.0^10.0Indentation area = < 2/n (mit2)Figure 6.18 Hardness data for stainless steel-416 13°1The relationship is expressed as1H = Co^E2A(6.8)1181C —0.474C,;(6.11)6.2.2 Roll Gap Heat transfer Coefficient (HTC)where Co 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 contactarea due to the interaction of microcontacts is not proportional to the normal load( which is alwaysassumed 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 heattransfer coefficient, HTC,HTC =C H"^ (6.10)where C is a constant which can be expressed as: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 thatspecific condition 163m641 , a linear relationship between the roll gap heat transfer HTC and apparentpressure has been established theoretically. Obviously, the definition of the roll gap heat transfercoefficient HTC and the apparent pressure P. are consistent with the mean HTC and mean rollpressure described previously.119300-6'"E 250 -'200 -c`co75i-,-, 150 -000Low Carbon Steel (0.05%C)-4-Stainless Steel (304L))sEMicroalloyed Steel (0.025%Nb)Best fitting lineN(4--I-6.3 Discussion6.3 DiscussionFrom the studies by Samarasekera t3n and the above analysis, it is known that the roll gap HTCis very dependent on the real area of contact. Therefore, the variation of HTC in the arc of contactshould correspond well to that of pressure, which has been confirmed by the pilot mill test (seeFig.6.13 to 6.15). Relative motion between the contacting surfaces decrease the interfaceheat-transfer coefficient f341 .To explore the relationship between mean HTC value and mean roll pressure, the data obtainedfrom all of the tests at CANMET and at UBC are shown in Fig.6.19. "6 50 -0=5^10^15^20^25^30^35Mean Pressure (kg/mm')Figure 6.19 Mean HTC data vs. mean roll pressurefor tests conducted at CANMET and at UBCThe 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.040 45120(w) after rollingSurface Profile(x 100)6.3 Discussion(a) before rollingFigure 6.20 Specimen surface profiles before and after rolling1216.4 SummaryIt should be borne in mind that roughness of the surfaces in contact have not been accountedfor in the current study, which may be the reason for the scatter in the results. Further investigationis needed to fully characterize the effect of the roughness.Fig.6.20 illustrates the change in roughness of the surface after rolling. The surface clearlyis much 'flatter' than before rolling due to plastic deformation of the surface. This would result ina 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 currentstudy, the effect of thermal conductivity of the lubricant or oxide scale (kf) must be considered asa separate resistance in any computation. The existence of a lubricant or an oxide scale in the rollbite would act as an additional thermal resistance and therefore alter the heat loss to the rolls.6.4 SummaryFrom the test results and preliminary theoretical considerations, the following conclusionscan be reached.The roll gap HTC value can be influenced by many factors, such as roll reduction, rollingtemperature, roll speed, roll and rolled material and their roughness. Because all of the factors arerelated to the roll pressure, a linear function for roll gap heat transfer coefficient versus mean rollpressure has been found; This can be used to determine a heat-transfer coefficient in industrialrolling from the rolling load.The application of lubricants or the existence of oxide scale would be considered as anadditional thermal resistance between the roll surface and the rolled material surface.1227.1.1 Heat Transfer Characterizations of a SlabChapter 7THERMAL PHENOMENA DURING ROUGH ROLLINGA mathematical model to predict the thermal history of a slab during rough rolling has beendeveloped in Chapter 5, but the oxidation of steel at high temperature was ignored. In order toinvestigate the oxidation effect on the thermal history of the slab, the mathematical model wassupplemented by a module describing the scale formation process.7.1 Heat Transfer Characterizations during Rough Rolling7.1.1 Heat Transfer Characterizations of a SlabPrior to rolling, a slab is reheated in a reheating furnace to a preset temperature. The surfacetemperature should not rise above 1280°0 121 , since above this temperature the oxide scale formedon the slab surface melts; this molten scale is very hard to remove later. Furthermore, the scalebuild-up in the furnace becomes excessive above this temperature. The minimum temperature inthe slab should be high enough to ensure that the steel is in the austenite phase and is homogeneousin temperature. The reheating temperatures currently employed at Stelco's LEW are 1250°C for a7-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 exitof the furnace to the edging mill on the roller table. It is then edge-rolled to break the scale on thesurface, which is subsequently removed by high pressure water jets. The slab is subsequently rolledon a reversing roughing mill until the final dimensions are reached. During the rolling process, a1237.1.1 Heat Transfer Characterizations of a Slabsecondary scale is formed and two descaling operations are conducted once before the second passand once before the sixth pass, respectively. After rough rolling, the slab which is 10 times itsoriginal length is transported to the coil box located between the roughing mill and the finishingmill. Coiling and subsequent uncoiling reverses the head and tail end of the transfer bar with theresult that the colder tail end is fed first into the finishing mill.7-pass ScheduleThe operating conditions of the 7-pass schedule currently employed at Stelco's LEW forplain carbon steels is shown in Table 7.1.Table 7.1 Operating conditions for the 7-pass schedulePass No. Ho(cm)H1(cm)Reduction(%)Speed(rpm)Roll Force(MN)1 24.00 22.61 5.8 56.9 7.072 22.61 21.00 7.1 56.9 -6.863 21.00 19.01 9.5 56.9 -6.864 19.01 15.11 20.5 56.9 11.665 15.11 10.00 33.8 56.7 14.206 10.00 5.00 50.0 41.0 14.477 5.00 2.12 57.6 67.2 14.64The 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 theslab at the exit of the reheating furnace was assumed to be uniform, and the scale formation wasignored. Because the heat transfer coefficient distribution in the roll bite varies with the rolling124130012001100a)10002a.c) 900800700---- Center----- H/40— Surface-- Average7.1.1 Heat Transfer Characterizations of a Slabconditions, in the model an average roll gap heat transfer coefficient for pass 1 to pass 7 was estimatedaccording 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.7.1.0^10^20^30^40^50^60^70^80^90Time (s)Figure 7.1 Thermal history half-way along the length of the slabduring 7-pass rolling of a 0.05%C plain carbon steelFrom the Fig.7.1, it is evident that the slab surface temperature changes significantly becauseof the chilling induced by high pressure water jets used to descale and by contact with the cold rollsin each pass. Each steep surface temperature decrease corresponds to an individual pass, and thesmaller temperature reductions just before the 1st, 2nd and 6th pass are due to descaling. Thesurface temperature of the slab decreases by as much as 250°C to 350°C due to contact with coldrolls, and 100 to 150°C due to descaling water. However, the surface temperature quickly reboundsdue to heat conduction from the center of the slab; the temperature decrease due to the radiative1257.1.1 Heat Transfer Characterizations of a Slaband natural convective heat loss in the interpass region is much less. The temperature right beneaththe surface, at a depth of H/40, changes relatively smoothly. This means the chilling effect isconfined to a thin layer immediately beneath the surface. This is because the time of contact withthe work roll is very short, of the order of 0.05 seconds, and during this time the roll surface heatsup, reducing the driving force for heat transfer. The temperature at the center line does not decreaseuntil 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 ofslab thickness. From this result, it can be concluded that the heat generation due to plasticdeformation in the metal forming process cannot be ignored. However, the mean temperature ofthe slab throughout the thickness decreases continuously and more sharply as the slab thickness isreduced. 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 smallerthan that after rolling, and the mean temperature decreases only by as much as 10°C before the lastthree passes. This results from small reductions of the first four passes. Thus, the rolling scheduleaffects the thermal history of a slab, even if the total reduction is the same This can also be seenfrom the thermal histories of the head end and tail end, as shown below.126Head Center— Head Surface--- Tail Center— Tail Surface7.1.1 Heat Transfer Characterizations of a Slab1300120000 1100a)100005a9008007000^10^20^30^40^50^60^70^80^90Time (s)Figure 7.2 The difference of thermal histories for the head and tail endduring 7-pass rolling of a 0.05%C plain carbon steelFig.7.2 compares the thermal histories of the head and tail end of the slab. The steep reductionsin temperature due to descaling and rolling are off-set by the difference in time corresponding tothe length of the slab. After rolling, the through-thickness temperature distribution at both ends issignificantly different; the center temperature at the head end is lower than that at the tail end. Thisis because when the tail end was being rolled, the reduced thickness head end was experiencingradiative and convective heat loss. However, the mean temperature of the head end at the exit ofthe roughing mill in the last pass is higher than that at the same location for the tail end, as shownin 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 rollforces. Although the details of the chilling cannot be seen very clearly from the earlier figures, itis illustrated in Fig.7.3, which shows the temperature distribution of the slab in the roll bite, foreach pass.127- ---- Center----- H/40^ Surface- Average----- Center- ---- H/40— Surface- - Average7.1.1 Heat Transfer Characterizations of a Slab13001200'0(3 1100a)10002cuE 9001--8000.037000.00^0.01^0.02Contact Time (s)(a) Pass 1130012001100N-15 1000Ew 9008000.037000.00^0.01^0.02Contact Time (s)(b) Pass 2--- Center---- H/40^ Surface Average7.1.1 Heat Transfer Characterizations of a Slab130012001100a)D46 1000L..a)o_Ea, 9001—800700130012001100a)I__D1:.; 1000L.a)QEQ) 9001--8007000.02^0.03Contact Time (s)(c) Pass 3----- Center----- H/40^ Surface— Average0.00^0.010.00^0.01^0.02^0.03^0.04Contact Time (s)(b) Pass 4---- Center--- H/40^ Surface Average— Center— H/40— Surface--- Average7.1.1 Heat Transfer Characterizations of a Slab130012001100a)D46 1000a)awE 9001--8007000.00^0.01^0.02^0.03^0.04^0.05Contact Time (s)(e) Pass 5130012001100a)D4(53 1000a)acvE 9001--8007000.00^0.01^0.02^0.03^0.04^0.05^0.06^0.07Contact Time (s)(f) Pass 67.1.1 Heat Transfer Characterizations of a Slab13001200'(30 11004-0  1000a)aQ,E 900 - Center- H/40^ Surface Average8007000.00^0.01^0.02^0.03Contact Time (s)(g) Pass 7Figure 7.3 The temperature distribution of the slab in the roll biteduring 7-pass rolling of a 0.05%C plain carbon steelA very large temperature gradient exists at the exit from each pass. The temperature decreasein the roll bite is as much as 250°C to 350°C. For the first four passes, there is little change in thecenter temperature because the heat generated due to the small reduction balances the heatconduction 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 byconduction through the thickness, because the reduction are in excess of 30%; as a result, the centertemperature increases by 5 to 10°C. Moreover, as the reduction increases, the contact time betweenthe rolls and the slab is extended, and after about 0.03 second contact the surface temperature doesnot 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 the1317.1.1 Heat Transfer Characterizations of a Slabdriving force for heat transfer is small. In addition, the surface cooling rate at the beginning ofcontact for Pass 7 is higher than those for the rest of passes because of the higher heat transfercoefficients of 49.7 kW/m2-°C.9-pass ScheduleThe 9-pass schedule for plain carbon steels employed at Stelco's LEW is shown in Table7.2.Table 7.2 Operating conditions for the 9-pass schedulePass No. Flo(cm)H1(cm)Reduction(%)Speed(rpm)Roll Force(MN)1 24.42 21.00 14.00 47.0 13.042 21.00 19.50 7.14 55.5 8.083 19.50 17.50 10.26 53.3 11.454 17.50 15.50 11.43 56.4 9.845 15.50 14.00 9.68 55.0 10.196 14.00 11.50 17.86 57.0 11.777 11.50 8.50 26.09 56.1 16.198 8.50 4.70 44.71 54.7 20.729 4.70 2.40 48.94 55.0 20.47The initial dimension of the slab for the 9-pass schedule is 244.2 (mm) x 1316.0 (mm) x 8290.0(mm).1327.1.1 Heat Transfer Characterizations of a SlabFig.7.4 shows the thermal history half-way along the length of a slab for 9-pass rolling for aplahn carbon steel with 0.05%C. The initial temperature was assumed uniform at 1265°C. Averageroll 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-°Cwere 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 rollingis shown in Fig.7.5.13001200110001000f;900a_E800700a.00a0a^co-0 "1a0(0^CO--,GWco0-a -oaI..)^CA (0^-0Cri a^-0(8 (0----- Center----- 11/40  Surface AverageCo600 '0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (s)Figure 7.4 Thermal history half-way along the length of a slabduring 9-pass rolling of a 0.05%C steel1337.1.1 Heat Transfer Characterizations of a Slab130012001100L10000_Ea) 9008007000 10 20 30 40 50 60 70 80 90 100 110 120 130Time (s)Figure 7.5 The difference of thermal histories for the head and tail endduring 9-pass rolling of a 0.05%C plain carbon steelFig.7.6 compares the thermal histories of a slab half-way along the length for 7- and 9-passrolling. A longer rolling period is required for 9-pass rolling and hence a higher initial temperatureof 1265°C was utilized to achieve the desired finishing temperature. Although the total reductionis almost the same as for the 7-pass schedule, there is a significant difference between the twothermal histories due to the different rolling schedules. Therefore, in order to investigate theevolution of microstructure of a slab during the rolling process, the thermal and deformation historiesmust be examined for a given rolling schedule.1347—pass^ Center- Surface9—pass- ---- Center— Surface13001200/(-30 1100-.6 100006 900800700 '0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (s)Figure 7.6 Comparison of thermal histories for the 7- and9-pass rolling of a 0.05%C plain carbon steelI"7.1.2 Temperature Distribution in the Work Roll•7.1.2 Temperature Distribution in the Work RollDuring 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, asymmetrical 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 differentexcept perhaps the magnitude of the temperature. This can be seen from Fig. 7.5 for Pass 1 andPass 2 in 7-pass rolling.135400300200 \E100 — Surface (r=R)-- Subsurface (r=R–Delta/40)Inner Face (r=R–Delta)300 --g• 200 /A\,1007.1.2 Temperature Distribution in the Work Roll40^80^120 160 200 240 280 320 360Angular Position (°)(a) Pass 1400— Surface (r=R)-- Subsurface (r=R–Delta/40)— Inner Face (r=R–Delta)40^80^120 160 200 240Angular Position (°)(b) Pass 2Figure 7.7 Temperature distribution in the work rollduring 7-pass rolling of a 0.05%C plain carbon steel_280 320 3601367.2.1 Oxide Scale Growth Rate of Steels at High TemperatureThe 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 and12.94mm thick for Pass 2. At the work roll surface, the temperature undergoes a very significantchange. During contact with a hot slab, the roll surface temperature rises very rapidly to a maximumwhich is dependent on the slab surface temperature, reduction and rolling speed. Upon exit fromthe roll bite, the roll surface is cooled by a stream of water flowing from a spray above. When thehot 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 thesurface temperature rebounds by heat conduction from inside and approaches the interiortemperature. Due to the symmetrical placement of sprays, the surface is again cooled by the directwater spray on the entry side. However, at this time, because the surface temperature was low, sothe 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 untilthe surface makes contact with the hot slab again, and the next revolution starts. Obviously, thework roll surface can fail in fatigue if the thermal cycling is large; so it is very important to designa good work roll cooling system to extend the work roll life. However, this is beyond the scope ofthe current study.7.2 Oxide Scale Growth of Steels during Rough RollingThe model adopted above did not consider the oxidation effect on heat transfer duringrolling. Before studying the effect, the oxide scale growth at high temperature was investigated.7.2.1 Oxide Scale Growth Rate of Steels at High TemperatureAs described in Section 2.2, for iron and steels, the consequence of oxidation is the formationof a multi-layer scale, i.e., Fe/FeOfFe 304/Fe203 . The growth rate of the scale layer generally followsa parabolic law, as stated in Eq.(2.26).1377.2.1 Oxide Scale Growth Rate of Steels at High TemperatureThe oxidation kinetics of iron during reheating has been studied and an Arrhenius law hasbeen 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, withtime, t, and temperature, T, may be obtained.r— ^10190 x = 24.7 Nt exp T + 273 (7.1)During oxidation, the scale temperature is not always uniform, so that an average temperature ofthe scale is generally used.For steel oxidation, Ormerod IV et al. E421 proposed a model to consider the effect of steelcomposition 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 compositionand 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 0507Land DS3388A at 1200°C as a function of time; These are the steels currently being rolled at Stelco'sLEW. The compositions of the steels has been obtained from Devadas t2s1 .1380.250.2 -xDS0006S of StelcoPure Iron7.2.1 Oxide Scale Growth Rate of Steels at High Temperature3E 22.5 - DS0006SxDS0507L0DS3338APURE IRON0.5 -00.5^ 1.5^21Time (hr)(a) Within 3 Hours0 2.5^3EE0.1500.1m0.05-0 x^0 10^20^30 40^50^60Time (s)70^80^90^100(b) Within 100 secondsFigure 7.8 Oxide scale growth of iron and steels at 1200°C1397.2.2 Assumptions for Oxide Scale Formation on the Steel SlabFrom Fig.7.8(a), it is evident that, the oxide scale growth for the steels currently beingrolled 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 wereconducted for 100 seconds and the result is shown in Fig.7.8(b) for steel grade DS0006S. Due tothe complexity of Eq.(7.2), Eq.(7.1) was applied to the steel rolling to consider the oxidation effecton heat transfer in a slab.7.2.2 Assumptions for Oxide Scale Formation on the Steel SlabIn 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 Wagnertheory1373 ;2) Because the primary scale layer thickness formed on the slab in the reheating furnace variesfrom 1.5mm to 3 0inm1121 , 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 primaryscale, a secondary oxide scale formed on the slab surface. The growth of the scale was dependenton 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 timestep, the temperature was assumed constant and the scale growth was computed before moving tothe 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 numberof scale islands in the roll bite. For simplification, it has been assumed that the scale remainedcontinuous but that the scale thickness was reduced according to the total reduction of that pass;1407.2.2 Assumptions for Oxide Scale Formation on the Steel Slab6) According to the oxide scale studies [38],[40]-(42], aboveD e 570°C, the scaling layer consists essentiallyof FeO with only a thin layer consisting of Fe304 and Fe203 . For this reason, the thermal physicalproperties of FeO were chosen to represent the entire oxide scale layer. A constant density of 5800kg/m3 was applied [661 , while the thermal conductivity and specific heat were assumed to betemperature 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 0.8for 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 duringrough rolling. Oxide ScaleInterfaceSteelYFigure 7.9 Distance associated with the interface i1417.2.2 Assumptions for Oxide Scale Formation on the Steel SlabDue to different thermal physical properties of the steels and the oxide scale, a combinedphysical property was employed at the interface, as in the case of a composite material [681 . Theexpression for the thermal conductivity is shown below:ji=1-karwhere fi is the ratio defined in terms of the distances shown in Fig.7.9:f = (5x), +(8x),(7.3)(7.4)The effectiveness of the expression has been validated by Patankert 681 .14210090800Taal Scale Thickness--x—Scale Growth504030201 00^10^20^30^40^50^60Time (s)09070 8070607.2.3 Oxide Scale Growth of Steels during Rough Rolling7.2.3 Oxide Scale Growth of Steels during Rough RollingAccording to the assumptions made, the roughing model developed in Chapter 5 wassupplemented by a module describing the scale formation process. The thickness of the oxide scaleis shown in Fig.7.10 assuming an initial scale thickness of 2 0 mm.2.5E 22 1.5U115v)015 0.50Figure 7.10 Oxide scale thickness half-way along the length ofthe slab during 7-pass rolling of a 0.05%C plain carbon steelFrom the figure, it is apparent that the thickness of the secondary scale formed duringrolling was much smaller than the initial scale thickness, due to the much shorter time ( approximately1 to 2 minutes) of exposure compared to the reheating period (2 to 3 hours). The growth of scaleduring rough rolling may be seen more clearly in the same figure with the Y-axis on the right sideshowing relative growth. It can be seen that the initial 2mm scale restricts further scale grow; Theincrease in scale thickness within the initial 30 seconds was approximately 2gm. Following highpressure descaling, the secondary scale grew faster on the newly exposed surface. However, the1438^100^21250c)01248 -mz-16 1246wQ.Ec) 1244 -0 1242 -,..4) 1240 -1238 -m 1236 -aN=5N=10NEN=20N=30xN=40AN=50a ...■...„^IN .V.'411,N. 4,^+4„.^so . NI IN .,". dlir )SiSisit • +4-4-4.N. Nr im . .14-4.+4.-1-4.....64Time (s)12347.3 Oxidation Effect on Heat Transfer of a Slabtotal growth of secondary scale thickness during roughing was less than 100).tm. Obviously, theslab scale thickness is dependent on the temperature and the exposure time, parameters which aredependent on the operating conditions, rolling temperature, rolling reduction and rolling speed.7.3 Oxidation Effect on Heat Transfer of a SlabBefore considering the oxidation effect on heat transfer, a sensitivity analysis on mesh sizein the oxide scale layer has been conducted, as shown in Fig.7.11.Figure 7.11 Effect of mesh size on surface temperatureFrom the figure, it is evident that for an initial thickness of 2.0mm, 30 nodes through thescale layer was sufficient; the predicted temperature after 10 second exposure differs only by 1°Con increasing the number of nodes to 50. Therefore, a node number of 30 for the initial scalethickness of 2.0mm was adopted in the model.1447.3.1 Effect of Emissivity of Oxide Scale7.3.1 Effect of Emissivity of Oxide ScaleTo 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 areshown in Fig.7.12. For this calculation it has been assumed that the slab has a surface scale thicknessof 2.0 mm and is losing heat only by radiation and natural convection.';31,q1NIi&P'^:02C't,, ...0.9.......................................................................OL0L0c)1260124012201200118011601140112011001080106000.50.60.70.8Time (s)Figure 7.12 Effect of oxide scale emissivity on surface temperatureAs expected, the emissivity has a strong effect on the scale surface temperature. After 30seconds of the radiation heat loss to the air, the surface temperatures was 1122.63°C for an emissivityof 0.5 and 1069°C for 0.9. Thus an accurate determination of the emissivity is required for theanalysis of the radiative heat transfer. According to a study of oxide scale emissivity [431 , a value of0.65 was employed in the model for the primary oxide scale and 0.8 for the secondary scale.1454^6^8^10Time (s)0 1200 -oIDL.n-8 1150Loa4:)1— 1100 -ca0o4-4_to 1050 -00.603mm-I-1.206mm)4(1.809mms2.412mm3.015mm10000Figure 7.13 Effect of oxide scale thicknesson the surface temperature of the slab7.3.2 Effect of Oxide Scale Thickness7.3.2 Effect of Oxide Scale ThicknessIt is obvious that the scale significantly affects the temperature distribution in a slab, becausethe thermal conductivity of the oxide scale is about 10 to 15 times less than that of the steel 671 . Theeffect of the scale thickness on the surface temperature and the temperature at the interface betweenthe 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 layerprevents heat loss to the surroundings.1467.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling12502-, 1248 -m15 1246 -15III 1244w 1242011) 1240 --la5 1238 -▪ 1236 -mti)1234-°20 1232 -• 1230 ^0-4-*▪^NEo0o ▪ -4-+4.Xxxxxxxx)1E)4Ex+++++++++++.4.4.4.++4._u u ,www ww4^6Time (s)8^10Figure 7.14 Effect of oxide scale thickness on the temperatureat the scale/steel interface7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling7.3.3.1 Temperature Distribution in the Roll GapThe temperature distribution of the slab in the roll bite during the third and fourth pass ofa 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 thebeginning, and then decreases more slowly towards the exit of the roll bite (see Fig.7.15(a)) andmay 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 clearlyshown in Fig.7.16(a) and (b) for the fifth and seventh pass during 7-pass rolling.147--- Center—^Scale/Steel Interface— Scale Surface— Average---- Center---- Scale/Steel Interface^ Scale Surface— Average7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling130012001100a)L._4- "0 1000a)o_0E 9001—80070013001200(0-3a, 1100D02)1 1000Eiv1—9008000.02^0.03Contact Time (s)(a) Pass 30.00^0.010.00^0.01^0.02^0.03^0.04Contact Time (s)(b) Pass 4Figure 7.15 Temperature distribution half-way along the length of the slabin the roll bite during 7-pass rolling of a 0.05%C plain carbon steel with oxidation148With Oxidation Without Oxidation--- Scale/Steel InterfaceScale Surface-- Subsurface (H/400)---- Steel Surface—13001200-30 110015' 1000CD0E 900F-8007007.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough RollingWith Oxidation Without Oxidation---- Scale/Steel InterfaceScale Surface—-- Subsurface (H/400)--- Steel Surface0.00^0.01^0.02^0.03^0.04^0.05Contact Time (s)(a) Pass 51300120011001000347, 900(t)^8007006000.00^0.01^0.02^0.03Contact Time (s)(b) Pass 7Figure 7.16 Surface and interface temperature half-way along the length ofthe slab in the roll bite with and without oxidation for Pass 5 and 7 of a 0.05%C plain carbonsteel rolling1497.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough RollingIn the figures, the temperature distribution with the scale being present were compared tothat with no scale being formed. It is evident that the surface temperature change along the arc ofcontact when an oxide is present is very different from that of an unoxidized surface. The surfacetemperature without scale decreases gradually, whereas the surface temperature with oxide scaledecreases very sharply at the beginning of the roll bite and then changes more slowly, as describedpreviously. The magnitude of the temperature decrease in the roll gap due to roll chilling is lowerin the presence of an oxide scale. Moreover, there is a significant difference in the thermal gradientestablished in the slab because of the insulating effect of the oxide scale.7.3.3.2 Thermal History of SlabThe thermal history of a slab when the scale is present for the 7-pass schedule is shown inFig.7.17.13001200110001000s-D4(5' 900800Center---- Scale/Steel Interface— Scale Surface— Average5000^10^20^30^40^50^60Time (s)Figure 7.17 Thermal history half-way along the length of the slabduring 7-pass rolling of a 0.05%C plain carbon steel with oxidation700600aco70 80 90z00CD(00FD- -0-U0 0)CD co(nN150With OxidationCenter^ Scale SurfaceWithout Oxidation-- Center— Steel Surface7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough RollingThe temperature difference between the surface and the scale/steel interface was very largebefore the first descaler, due to the presence of the relatively thick primary oxide scale. After thefirst descaler, the difference became smaller because a much thinner secondary oxide scale layerwas formed, ( see also Fig.7.10 for the scale growth). This indicates that if the rolling speed is veryfast 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 caseof rough rolling, the primary thick scale formed on the slab surface has a heavy insulating effecton the heat loss from the surface of the slab, and therefore affects the temperature distribution inthe slab.130012001100-• 1000• 900• 8007006005000^10^20^30^40^50^60^70^80^90Time (s)Figure 7.18 Thermal history half-way along the length of the slabwith and without oxidation during 7-pass rolling of a 0.05%C plain carbon steel1517.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough RollingThe thermal histories at the surface of a slab with and without oxidation are compared inFig.7.18. It is obvious from the figure that although the secondary oxide scale has a relatively smalleffect on the temperature distribution in the slab, the thermal histories for the two situations (withand without oxidation) are very different; The difference results mostly from the existence of theinitial oxide scale. In addition, due to the insulating effect of the oxide scale, the temperature afterrough rolling is higher than in the absence of oxidation.7.3.3.3 Effect of Rolled Materials on Thermal History of slabFor comparison of thermal histories between different rolled materials, the 7-pass rollingschedule for the low carbon steel has been applied to the rolling of the microalloyed steel with0.025%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 rollpressure from the pilot mill tests as being 63.9, 80.6, 99.4, 81.9, 70.2, 82.1, 394.2kW/m 2-°C. Thethermal history half-way along the length of the slab is shown in Fig.7.19 and the thermal historiesat 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 dueto the higher mean roll gap heat transfer coefficient, especially for the last pass.152----- Head Center— Head Surface--- Tail Center--- Tail Surface00L7a)CIEa)7.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling 0CD^C70^(.0a(T) a(-0_ua -UU3 acr)00)aaT-oa0'----- Center------- Interface  Surface— Averagea0'(,)1:1aci)CP10^20^30^40^50^60^70^80^90Time (s)Figure 7.19 Thermal history half-way along the length of the slabduring 7-pass rolling of a 0.025%Nb bearing steel with oxidation1130012001100 -1000900 -800 -700600 -500400 -300 -0-oacr)0En130012001100 -1000900 -800700 -600500400 -300 -0 10^20^30^40^50^60^70^80^90Time (s)Figure 7.20 Thermal history at the head and the tail end of the slabduring 7-pass rolling of a 0.025%Nb bearing steel with oxidation153With Oxidation Without Oxidation----- Scale/Steel Interface -- Subsurface (H/400)-- Steel Surface— Scale Surface----------^---------- ------ -------73.3 Effect of Oxide Scale on Thermal History of a Slab during Rough Rolling1300120011001000900a)8000700E(t)^600F------^Center (0.025%Nb)------- Surface (0.025%Nb)Center (Plain Carbon)Surface (Plain Carbon)300 -5004000^10 20 30 40 50 60^70 80 90Time (s)Figure 7.21 Comparison of thermal history half-way along the length of the slabduring 7-pass rolling a 0.05%C and a 0.025%Nb bearing steel with oxidation0.01^0.02^0.03Contact Time (s)Figure 7.22 Comparison of thermal history half-way along the length of the slabduring 7-pass rolling a 0.05c70C and a 0.025%Nb bearing steel with oxidation14001300120011001000900zt 800a)o_ 7006005004003000.001547.3.3 Effect of Oxide Scale on Thermal History of a Slab during Rough RollingA comparison of the thermal history between the low carbon steel and the microalloyed steelis shown in Fig.7.21.Due to the heavier roll chilling, the final temperature distribution for the microalloyed steel islower than that of the low carbon steel. The temperature distribution through the half-thickness ofthe slab in the roll bite for the 7th pass of the microalloyed steel rolling with and without oxidationis shown in Fig.7.22.The figure shows that a much larger thermal gradient exists between the scale surface and thescale/steel interface. Due to the smaller thermal diffusivity of the scale and the higher mean heattransfer 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, thedriving force for the heat conduction from the interior of the slab is getting larger, therefore thescale surface temperature rebounds quickly.1557.4 Estimate of Heat Loss to the Work Rolls during Rough Rolling7.4 Estimate of Heat Loss to the Work Rolls during Rough RollingThe heat loss modes of the slab during rough rolling are mainly via radiation and naturalconvection to the air, convection due to descaling and the conduction to the work rolls. The heatloss to the work rolls can be compared to the heat loss due to radiation and convection with the aidof the roughing model. The model was run by assuming no heat loss to the work rolls during roughrolling. The results were compared to the final temperatures of 7-pass rolling. It is known that theheat content of a slice before and after roughing can be expressed as.Qo = f PsCpsTsodxdy^ (7.5)Qf = psCpsTsjdxdy^ (7.6)where Tso, Tsf are the initial and final temperature distribution through thickness before and afterrolling, respectively. Thus, the total heat loss for a slice is the difference between the initial andthe 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, —(7.8)where AQ,, denotes the total heat loss due to radiation and convection.156---- ■•■■J1 ...-•■••■■•••-■Without Oxidation---- CenterSteel SurfaceNo Heat LossTo Rolls--- Center--- Steel Surface7.4 Estimate of Heat Loss to the Work Rolls during Rough RollingAccording to the above equations, the total heat loss to the work rolls is obtained to beapproximately 33.2% during 7-pass rough rolling. The heat loss to the work rolls estimated byHollander" 21 during strip rolling (including roughing and finishing) was about 38%. These resultsall 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 withroll chilling in Fig.7.23.13001200110001000900(1)8007006005000^10^20^30^40^50^60^70^80^90Time (a)Figure 7.23 Comparison of thermal history half-way along the length of the slabduring 7-pass rolling with and without heat loss to the work rollsFrom the figure, it is evident that because of no heat loss to the work rolls the finaltemperature distribution is higher.1578.1.1 Formulation for DeformationChapter 8DEFORMATION ANALYSIS ON A SLAB DURING ROUGH ROLLINGIn the previous chapters, the thermal field in the slab has been computed using afinite-difference technique. The heat of deformation was independently estimated as describedearlier. 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 thedeformation field. This Chapter presents the results of this analysis.8.1 Velocity-Pressure Formulation8.1.1 Formulation for DeformationFor the velocity-pressure approach, the basic variables are the velocities and pressure. Thebasic equation can be expressed as:fv{50T {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 volumetricforce {F} are specified on flow domain V and {t} on stress boundaries Sr, whilst {v} is the velocityvector, 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 ratevector, {0, is expressed as,1588.1.1 Formulation for DeformationfoT=te„,y,ky}^ (8.3)Due to the incompressibility in metal deformation, the volumetric straining rate and theinternal work associated with any pressure variation should be zero:= k+ y = [my {0 = 0where: [M ]T -= {110}.L 'S/iiv = 0For compatibility, the following equation must be satisfied:{SE} = [L] {6v}where [L] is:0[L] =^0^aaya a_5.7 5T.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 = {G} + [M]pa^0xY(8.8)-aax(8.4)(8.5)(8.6)(8.7)because the pressure, p, is related to the mean stress, cv„„ through the following equation:1598.1.1 Formulation for DeformationP =^= —(csx + 6y )/2^ (8.9)Assuming that the material is isotropic, the relation (2.38), now applicable only to thedeviatoric stresses, can be written as:{S} =[D]^ (8.10)where:2 0 0-[D] =^0 2 0^(8.11)0 0 1The scalar, is known as the viscosity, which in general is still dependent on the strain, strain rateand temperature of the material. According to the definition of the equivalent stress, cs, and theequivalent strain rate, e,the Eq.(8.10) can be replaced as:--2^3Y= 2- S S-2^2e^=^. -.—EE-.3=(8.12)(8.13)(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.1608.2.1 Finite Element DiscretizationIn 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 verylarge cut-off value for the viscosity, to prevent this difficulty.8.1.2 Thermal CouplingDuring deformation, the temperature depends not only on the heat lost to the environmentand rolls, but also on the heat generated; The latter is determined by the mechanical properties ofthe metal being deformed. Thus heat transfer and deformation are coupled through the heatgeneration term, q.s , which is expressed as:4s=P =i-t ax2 a{( 21 2 4 a12} 4. ( 1^2± ay1 )Tx1±+al lx^y^3 a ay(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 deformationequations for the material flow. In the analysis, an iterative procedure has been used to obtain thebest estimate of temperature, strain and strain rate, which will be described in the subsequent section.8.2 Finite Element Solution8.2.1 Finite Element DiscretizationFor the solution of the problem, the flow domain must be discretized into a number of elements,as schematically shown in Fig.8.1.161ROLLFrictionalrayerVs = Vr8.2.1 Finite Element DiscretizationVFree Surface1111111111111Free Surface0 v=0 xl XFigure 8.1 Finite element discretization in the roll bite of the slabIn the solution, an 8-node iso-parametric element was adopted as shown above. It is wellestablished that the finite element interpolation for pressure, p, should be at least one degree lessthan that for the velocity componentsf 463 , since the continuity equation does not contain a pressureterm. As such parabolic interpolation was used for velocities and temperature and linearinterpolation for pressure in the present analysis, 3 x 3 Gaussian integration points were employedfor numerical integration. From the basic equations of deformation, the following matrix equationcan be obtained:vel[KIT[KP]0 j{ a v} }fel{ In}00 (8.17)where t ((iv}, tap} are the nodal velocities and pressure respectively. The corresponding finiteelement equation for the temperature solution can be expressed as:[KT] {T,} +^= 0^ (8.18)1628.2.2 Mechanical and Thermal Boundary Conditionswhere (f) is the thermal force which contains the heat generation of deformation. The details ofthe 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 elementsin 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 rollbite, the number of elements was 15 in the roll bite, 3 on the entry side and 2 on the exit side, asshown in Fig.8.1.8.2.2 Mechanical and Thermal Boundary ConditionsTo solve Eq.(8.17) and Eq.(8.18), specific boundary conditions must be satisfied. Becausethe 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.8.1,1/0x=0, 0^'v = 0; T = T0(y)^(8.19)at the exit side,x =x1 , 0<—Y v = 0;aTs—IcsTa =0(8.20)along the center line of the slab, due to symmetry,aTsY =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 slippagebetween the two, a very thin layer of elements was inserted along the arc of contact, as shown in1638.2.3 Sequence of the SolutionFig.8.1. The outer nodes of these elements were attached to the roll surface, and the friction effectwas accounted for by relating the shear stress, T, associated with flow to the normal pressure, p, bymeans of a friction coefficient, ;If, 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 eachside of the roll bite, no surface tractions existed. However, for the thermal situation, the boundaryconditions were specified as:aTs—k ---= h ad(Ts —T.,)ay^r (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 havebeen extremely difficult to use the finite-element model to back-calculate the interface heat-transfercoefficients because the latter is an Eulerian formulation. One would have had to determine thecomplete variation of hga„ along the arc of contact by trial-and-error, beginning with a trial solutionof this variation.8.2.3 Sequence of the SolutionFor a thermal coupled plastic deformation problem, an iterative procedure has to be adoptedto obtain a solution. The sequence of the solution is shown in Fig.8.2.1648.2.3 Sequence of the Solution(Start  Input Initial DataIter. No. =1Assumed temperature, Constant viscosity,Constant yield, to solve velocities4Modifying viscosity to get new velocitiesfrom Eq.(8.17)Calculate q, to get temperaturefrom Eq.(8.18)Modifying the yield,solve flow Eq.(8.17) Iter. = Iter. + 1Velocity Convergent ?YesNoOutput strain rate, strain,temperature, roll force.( Stop )Figure 8.2 Sequence of the solution for the finite element analysisGiven 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 equationwas handled using an upwinding weight function to improve the results. Once the temperature fieldwas known, the flow problem was solved again. The above steps could be repeated until the velocityfield was convergent. In the present study, less than 0.1% of velocity variation for each nodal valuewas taken as a convergence limit. With this restriction, the number of iterations varied from 8 to35 depending on the percent reduction for the industrial rolling schedule at S telco, On average,1658.3.1 Heat Transfer Analysis of a Platenearly 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 isunsymmetrical, an unsymmetric solver, so called Frontal Unsymmetric Matrix Solver t691, wasadopted 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 pressurefor 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 ProgramThe finite-element program was originally developed by Kumar et al. (463 for finish rolling. Ithas been modified for application to rough rolling. There are two main modules, with one for thethermal analysis and the other for the deformation analysis.Because of the complexity of the finite-element program, it is necessary to verify thesource-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 comparedwith the model prediction.8.3.1 Heat Transfer Analysis of a PlateTo verify the heat transfer module in the source-code, a rectangular plate with specific thermalboundary conditions and no deformation was considered, as shown in Fig.8.3.1669 1998 1987 1976 1965 1954 19421911192= -200000.0 WIm210^ 200• cm1938.3.1 Heat Transfer Analysis of a Plate• cr0 1.47 m^ x1TO = 1250 CFigure 8.3 Verification of the heat transfer moduleof the finite element programThe boundary conditions for the above problem were specified as:H^aTsx=0,x1, 0^—k2^ax0 5x 5x1, y =0, T =ToaTsy =^—ks—ay Const.(8.25)(8.26)(8.27)For the steady state, it is obvious that the upper surface temperatures of the plate for all ofthe nodes are the same along the boundary and equal to:Upper Surface T =^+Tk 2^°s (8.28)1678.3.1 Heat Transfer Analysis of a PlateWith q = -2.0 x 105 W/m2; To = 1250.0°C, k = 30.0 W/m-°C; and H/2 = 0.15 m, the uppersurface temperature can be easily calculated to be equal to 250.0°C.Using these same conditions, the nodal temperature obtained by the finite element programwas exactly 250.0°C at each node on the upper surface. In addition, the temperature symmetry atthe center line of width of the plate was obtained due to the symmetrical boundary conditions. Thetemperature distribution through the half thickness is shown in Fig.8.4.^1300^1200 -1100 -1000 -0900 -L. 800 -• 700 -a)600• 500 -400 -300 -^200 ^0 2^4^6^8^10^12^14^16Half Thickness (cm)Figure 8.4 Comparison of the numerical upper surface temperaturedistribution of the plate with an analytical solutionFrom the figure, it is evident that an excellent agreement has been obtained between theanalytical solution and the results predicted by the FEM model. It indicates that the thermal moduleof the program is accurate and valid.1688.3.2 Interface Temperature Distribution of a Pilot Mill Test8.3.2 Interface Temperature Distribution of a Pilot Mill TestTo 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 rollgap interface temperature distribution obtained is compared with experimental measurements inFig.8.5.Test S6P71ExperimentalxFEM Predictionx xxx>c7c,cx0.5^1^1.5^2^2 5Distance along the roll bite (cm)Figure 8.5 Comparison of the FEM prediction with a pilot mill testresult of the interface temperature distribution in the roll biteA very good agreement has been obtained in the above figure to indicate that the source-codeof the program can be applied to the deformation analysis of industrial rolling. It also affirms thatthe use of the finite-difference method to back calculate heat-transfer coefficient is valid since thecomputed values of HTC, when employed in the finite-element code, yield the same interfacetemperature distribution as seen in Fig.8.5. Deviations between the two solutions will be prevalentin the interior of the slab where deformation heating is significant.1150 ^1100 -10501000 -amLz950 -LE 900850 -800 -750 01698.4.1 Velocity Profiles8.4 Rough Rolling Deformation AnalysisThe 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 andtemperature. The deformation analysis was conducted for a 9-pass rolling schedule. The typicaldeformation parameters for the 9-pass schedule are listed in Table 8.1.Table 8.1 Typical deformation parameters for the 9-pass schedulePassNo.Ho(cm)Reduction(%)RollSpeed(m/s)NominalEffectiveStrainNominalEffectiveStrain RateContactLength(cm)(s-1)1 24.42 14.00 2.75 0.174 3.49 13.722 21.00 7.14 3.25 0.086 3.04 9.123 19.50 10.26 3.12 0.125 3.71 10.524 17.50 11.43 3.30 0.140 4.40 10.525 15.50 9.68 3.22 0.118 4.15 9.126 14.00 17.86 3.33 0.227 6.43 11.757 11.50 26.09 3.28 0.349 8.90 12.868 8.50 44.71 3.20 0.684 15.15 14.459 4.70 48.94 3.22 0.776 20.36 11.288.4.1 Velocity ProfilesIn the velocity-pressure approach of the finite-element analysis, velocity is one of the basicvariables. It gives the flow pattern of the metal during rolling. Fig.8.6(a) and (b) show the nodalvelocity profiles for Pass 1 and Pass 2 during 9-pass rolling at Stelco's LEW.170• I20ROLL10^15Distance along the roll bite (cm)• 1^•25t o5E0cisa)00)000(a) Pass 18.4.1 Velocity ProfilesROLLtoE0.0To0)OOC0(TiIDistance along the roll bite (cm)^155^ 10^ 20(b) Pass 2Figure 8.6 Typical velocity profiles during 9-pass rolling01718.4.2 Strain Rate DistributionFrom the profiles, it is apparent that the velocities increase along the roll bite because of thereduction. The directions of the velocity vectors show that the vertical velocity is the largest at theentry point and decreases along the thickness and the roll bite and finally reaches zero at the centerline and at the exit side of the roll bite. Moreover, the velocity at a given depth along the roll gapforms a line which corresponds to the stream line. This indicates that the finite element discretizationof the domain is consistent with the stream line, as it is seen in Fig.8.6. The results also show thatthe direction of the velocity near the center line is almost parallel to the rolling direction due to zerovertical 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 velocityis dependent upon the roll speed and the percent reduction.8.4.2 Strain Rate DistributionThe 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 rollingare shown through Fig.8.7(a) to (c).1725.05^10^15^20Distance along the rolling direction (cm)Effective Strain Rate in Roll Gap2nd pass8.4.2 Strain Rate DistributionEffective Strain Rate in Roll Gap1st passROLLet4,Oco3-■4.)C.)E r-0CL.) Nro0 ^°4.)-N0C.) ,004-■ N-a0 5^ 10^ 15^ 20Distance along the rolling direction (cm)(a) Pass 1 with nominal strain rate of 3.49 s -1(b) Pass 2 with nominal strain rate of 3.04Effective Strain Rate in Roll Gap3rd passROLL'0 -05^10^15^20Distance along the rolling direction (cm)8.4.2 Strain Rate Distribution(c) Pass 3 with nominal strain rate of 3.71 s -1Figure 8.7 Effective strain rate distribution in the roll bitein the 9-pass rollingFrom the above three figures, it is evident that the effective strain rate distribution is similarfrom pass to pass. The common features of the distributions are as follows: a very high strain ratejust beneath the surface at entry to and exit from the roll bite; the strain rates at these locations aresignificantly higher than the nominal effective strain rate owing to the high redundant shearassociated with constraining the metal to flow into and out of the roll gap. In addition, there is adead zone beneath the rolls, approximately half way along the arc of contact, in which the strainrate is small. This is consistent with the results obtained from laboratory rolling by Silvonen etClij531.(541 . Moreover, the nominal strain rate for each pass is reached at the entry and exit of the rollbite. For higher reductions and higher strain rates, ( comparing Pass 3, 10.26% reduction at 3.71s -1with Pass 2, 7.14% reduction at a nominal effective strain rate of 3.040, the regions of high strainrate extend closer to the center line and are wider. This indicates that a higher percent reduction1748.4.3 Strain Distributionand a higher strain rate produces more intense shearing, which is consistent with the results obtainedby Dawsont561 . In addition, the plastic deformation zone in the roll gap is wider near the surfacethan at the center line. This is also consistent with the result obtained by the slip-line-field methodfor rolline 01 and by Dawson [561 .8.4.3 Strain DistributionStrain is one of the major factors influencing microstructure evolution during rolling, thereforethe strain distribution during deformation has to be known. However, the velocity-pressure approachdoes not permit the direct determination of strain from the basic variable, velocity. One way ofobtaining the strain is the integration of the effective strain rate along the stream line, i.e.:6 =^ildt^ (8.30)s J.Because the velocities are known for the steady state after the solution is convergent, the streamlines can be determined from the direction of the metal flow:' v(x,y)= tan(a') =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 inthe finite-element model. Traditionally, with the flow formulation method, the effect of strainhardening is ignored and the flow stress is considered to be a function only of strain rate andtemperature.1758.43 Strain DistributionTo account for strain hardening, the metal flow pattern is estimated following the first iterationby integrating the effective strain rate along the stream lines. The flow stress at each point isestimated as a function of strain, strain rate and temperature, and the equations are resolved to obtainnew estimates of velocity, temperature and pressure in the domain. The strain distribution isrecomputed and the procedure is repeated.Fig.8.8 (a) to (c) shows the effective strain distribution determined by this method for Passes1, 2 and 3 in 9-pass rolling.oaCI) 04-40^.af•-■a• r.0,0044-4a N-Vacn ^cnQ 0Effective Strain in Roll Gap1st passROLL55005^10^15^20Distance along the rolling direction (cm)o0—0(a) Pass 1 with nominal strain of 0.17417620-^ -5 10^ 15Distance along the rolling direction (cm)Effective Strain in Roll Gap2nd passROLL0-.0.150,20008.4.3 Strain Distribution(b) Pass 2 with nominal strain of 0.086Effective Strain in Roll Gap3rd passOO0^5^10^15^20Distance along the rolling direction (cm)(c) Pass 3 with nominal strain of 0.125Figure 8.8 Effective strain distribution in the roll bite0A00CO0a)00 cqCa0 0OOO-•1778.4.4 Thermal Field in the Roll BiteIt is evident that the strain distribution in the roll bite is non-uniform. The strain is higher atthe surface than at the center line due to the effects of the redundant shearing brought about byconstraining the metal to flow through the roll gap, and the consequent higher local strain rate. Overa region at the center of the slab near the exit of the roll bite, the computed strain approaches thenominal effective strain corresponding to the applied reduction. The figures also show the stronginfluence of redundant shearing, since the effective strain at the exit of the roll bite is larger thanthe nominal strain, which is consistent with results obtained by Beynon et al.E531 .8.4.4 Thermal Field in the Roll BiteThe 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 heatflux 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 in9-pass rolling.178ROLL1160-1000.^•1240 ^1240------------8.4.4 Thermal Field in the Roll BiteTemperature Distribution in Roll Gap1st pass0^5^10^15^20^25Distance along the rolling direction (cm)(a) Pass 1Temperature Distribution in Roll Gap2nd passROLL40z^ ___^ 1160^_______^_^_ 1160---------------------- -1240----------------0 —C.10s. m -CU0O200^5^10^15Distance along the rolling direction (cm)(b) Pass 28.4.4 Thermal Field in the Roll Bite"ff,—,elo , __ _________ ------_______-__,_______________124oTemperature Distribution in Roll Gapieo- _  3rd passle° ^______ 12401080ROLL__,..,,H120 ----1,1‘..,a1/.--- _--^_______ _ ____1160:---^-:-L.^ .‘to.*.)(u ,C2Casco c,Distance along the rolling direction (cm)(c) Pass 3Figure 8.9 Temperature distribution in the roll biteThe 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. Thisis consistent with the prediction of the finite-difference model.From the deformation analysis, it is evident that the deformation is inhomogeneous, whichresults in a non-uniform generation of heat. The results corresponding to the temperature rise dueto 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.20180Temperature Rise in Roll Gap1st, passROLL5^10^15^20Distance along the rolling direction (cm)0 25(a) Pass 18.4.4 Thermal Field in the Roll BiteTemperature Rise in Roll Gap2nd pass0.5o^- --------- ROLL03.005^ 10^ 15^20Distance along the rolling direction (cm)(b) Pass 2Figure 8.10 Temperature rise distribution in the roll bite1818.5.1 Deformation Observation from Pilot TestsThe temperature rise due to deformation is much less near the center than near surface. Thisis consistent with the strain rate and temperature distribution, because the low temperature near thesurface results in a high flow stress and the temperature rise is proportional to the product of theflow stress and the strain rate. The maximum of temperature rise is approximately 38°C for thefirst pass and 14°C for the second pass at the exit of the roll bite; This number is dependent on theroll speed and reduction. Although the highest temperature rise occurs near the surface, the surfacetemperature of the slab is still a minimum in the roll gap due to roll chilling This also indicatesthat the roll chilling effect cannot be ignored for the thermomechanical analysis of steels.8.5 Validation of the Model8.5.1 Deformation Observation from Pilot TestsDue to the importance of the strain distribution on the evolution of the microstructure, thestrain distribution predicted in the roll bite should be verified by experimental measurements. Thiswas accomplished by conducting pilot mill tests at CANMET, SS-8, SS-15 and SS-19 instrumentedwith pins, as described earlier. The test conditions are the same as listed in Table 4.6 and Table4.7 in Chapter 4.In the test, a 5mm diameter pin of the same material as the specimen was inserted into a 5mm diameter hole located at the center of the specimen. The deflection profile of the pin subsequentto rolling was employed to determine the effective strain distribution through the thickness, asdescribed by Sakai et al.r713 . In their study, the effective strain and redundant shear strain (y) wereestimated on the basis of three assumptions:1) The ratio of incremental shear strain to incremental compressive strain is constant duringrolling;2) The incremental compressive strain is uniform through the thickness;1828.5.1 Deformation Observation from Pilot Tests3) Plane strain conditions prevail in the deformation zone,- 26.^1— n (8.32)- r= 24412- 1111 1 -1^r (8.33)where.,\1 1 + f (1 - r)2r(2 -r) tan% j(8.34)where r is the percent reduction in thickness; tan% is the slope of the interface between the pin andsample, as shown in Fig.8.11. HOxYEmbedded pin^ y=(y1 + y2)/2Before rolling After rollingFigure 8.11 Illustration of deformation of the embedded pinand the apparent shear angle 0 1After the test, the values of deflection, (x, y), at each position along the interface was measuredfor each side of the pin, and an average was obtained for the two sides of the pin. A polynomialregression was performed on the data (x, y), and numerical differentiation (dy/dx) was employedto obtain the tan0 1 .Fig.8.12(a) to (c) shows the distribution of effective strain through the half thickness of aspecimen predicted by the FEM model and measured from the pilot mill tests.1830.80.75-0.7-j 0.65-0.6UZ.: 0.550.5-0.45-1^ 2^3^4^5^6^';Half thickness (mm)0.40 90.85ogS- 0.8 -Ing3 0.750.70.65 -0.6 -0.55 -0.50 1 3^4^5^6Half thickness (mm)7 8 91 ^0.95 -0.9 -8.5.1 Deformation Observation from Pilot Tests(a) Test SS-8 with nominal strain of 0.5063(b) Test SS-15 with nominal strain of 0.54161848.5.1 Deformation Observation from Pilot Tests1^0.8 ^0.75 -0.7 -f). 0.65-EIT)0.6 -Ua):7- 0.55 -w0.5 -0.45 -0.4 0 2^3^4^5^6Half thickness (mm)8^9(c) Test SS-19 with nominal strain of 0.4681Figure 8.12 Comparison of the effective strain distribution predictedby the model and the pilot mill testsThere is a good agreement between measurements and predictions for the strain over theregion near the center. The deviation near the surface is probably due to separation between thepin and the specimen, as seen in Fig. 8.13. The test conditions are shown in Table 8.3.185I3^9^1 05.0-7E19 4VIII^III^III^IIII,IIIIII^III^III^III^III1 1^1 2^1' 3234667897 '121111111H1111111111111111111111111118.5.2 Roll Force PredictionFigure 8.13 Deformation of the pin in a pilot mill test SS-198.5.2 Roll Force PredictionThe roll force can be calculated from Eq.(8.17) for each pass of the 9-pass rolling by summingthe normal component of nodal pressure along the arc of contact. The comparison between thepredicted and measured roll force for a 9-pass rolling schedule at Stelco is shown in Fig.8.14 andTable 8.2. The rolling conditions were given in Table 7.2.1860z8.5.2 Roll Force Prediction25MeasuredFEME15 -020 -08^908 05-3^4^5^6^7Pass No. for 9-pass ScheduleFigure 8.14 Comparison of the roll forces predicted and measuredduring 9-pass rollingTable 8.2 Comparison of the measured roll force with the FEM prediction for 9-passrollingPassNo.Reduction(%)FrictionCoefficient010RollForce(Tons/cm)(Measured)RollForce(Tons/cm)(FEM)RelativeError(%)1 14.0 0.45 10.10 9.19 -8.982 7.14 0.45 6.29 5.77 -8.263 10.26 0.45 8.92 6.93 -22.264 11.43 0.45 7.84 7.17 -8.555 9.68 0.45 8.12 6.07 -25.206 17.86 0.45 10.08 9.29 -7.827 26.09 0.45 13.20 12.53 -5.078 44.71 0.45 17.28 20.01 +15.809 48.94 0.45 16.87 20.90 +23.891878.5.2 Roll Force PredictionIt is evident that reasonable agreement has been obtained for the 9-pass rolling schedule withthe 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 someof the data does not appear reasonable; for example, the percent reduction for Pass 4 is larger thanthat of Pass 3 and the rolling temperature is lower, but the measured roll force for Pass 4 is smallerthan Pass 3. The situation is similar for Pass 5, Pass 8 and Pass 9, each of which results in a largedeviation between the measured and the predicted roll force. The friction coefficient for each passwas assumed to be 0.45 due to the high rolling temperaturer211 .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 predictionfor the CANMET stainless steel testsTestNo.Reduction(%)RollingTemp.RollSpeedFrictionCoefficientRollForceRollForceRelativeError(°C) (m/s) (11) (Tons) (Tons) (%)(measured) (FEM)SS-8 35.5 850 1.5 0.30 200.80 232.00 +15.54SS-15 38.9 950 0.5 0.30 161.0 181.00 +12.42SS-19 34.9 1050 1.5 0.30 140.00 132.86 -5.28Because the rolling temperature was relative low, a friction coefficient of 0.3 was employedfor each test. The deviation in the roll forces is probably a result of the equation for flow stress forthe AISI 304L steel which is more suitable for high strain rate conditions f723 .1889.1 Summary and ConclusionChapter 9SUMMARY AND CONCLUSIONIn this study, the thermal and mechanical behavior of a slab during rough rolling has beeninvestigated, and some conclusions can be made. Recommendations for the future work have alsobeen made.9.1 Summary and ConclusionA mathematical model has been developed to predict the thermal history of a slab duringrough 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 andplastic deformation in the roll bite. The model is also able to calculate the thermal history at eachposition along the length of the slab and therefore able to determine the difference in temperaturedistribution at the head and tail end of the slab. Due to the coupling of the slab and the work rollin the roll bite, a module for heat transfer calculation has been developed for the roll bite whichincludes the work roll. It accounts for the heat gain in the roll due to the contact with the slab andthe subsequent cooling of the roll due to water spray out of the roll bite. It is assumed that the rollsreach a cyclic steady state. The model is based on one dimensional transient heat flow bytransforming the rolling direction coordinate into the time coordinate. A completely implicit finitedifference technique is adopted to solve the governing equations.To characterize the heat transfer coefficient at the roll/slab interface, pilot mill tests wereconducted at CANMET and at UBC using specimens instrumented with thermocouples which were1899.1 Summary and Conclusionspot welded on the surface. Different rolling conditions were examined to measure their effect onthe thermal response of the sample surface. The roughing model was modified to back-calculatea heat transfer coefficient to yield temperatures which match the measured surface temperature. Atrial-and-error method was adopted in the calculation. The following are the main findings fromthe analysis of the pilot mill test results.(a) The heat transfer coefficient(HTC) increases gradually at the beginning of the roll bite, andthen more rapidly until a maximum value is reached; finally the heat transfer coefficientdecreases until the exit of the roll bite;(b) The maximum value of heat transfer coefficient varies from 25kW/m 2-°C to approximately700kW/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 thedistribution 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 transfercoefficient is obtained from the heat transfer coefficient variation in the roll bite. It wasfound that the mean value is linearly dependent on the mean roll pressure; the influence ofthe rolling parameters, temperature, roll speed, percent reduction on the mean heat-transfercoefficient, can be attributed to the influence of roll pressure;(e) A preliminary theoretical consideration also reveals a linear relationship between the meanheat transfer coefficient and the mean roll pressure for each of materials examined (0.05%Cplain carbon steel, 0.05%C plus 0.025%Nb steel, AISI 3041 stainless steel) for conditionsof no lubrication and no oxidation.1909.1 Summary and ConclusionThe roughing model has been applied to predict the thermal history of a plain carbon steelslab for a 7- and 9-pass rolling schedule at Stelco's LEW. The roll gap heat transfer coefficient foreach pass is estimated from the mean roll pressure. The model is validated by comparing thepredicted temperature with results from the literature. To consider the effect of oxide scale on heattransfer, an oxide scale growth module is included in the roughing model; a parabolic law is usedto account for the oxide scale growth during rough rolling. The important results are describedbelow.(a) The model predicted surface temperature reflects the cooling experienced by the slab duringrough rolling. For instance contact with the rolls depresses the surface temperature byapproximately 250°C to 350°C in the roll bite. On exit from the roll bite, the surfacetemperature rebounds very quickly due to the heat conduction from the interior. Afterreaching 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 descalingafter which it also rebounds quickly.(b) The roll chilling is confined only to a very thin layer of the slab thickness with a depth ofapproximately H/40 during rough rolling. The center temperature increases in the roll bitedue to the heat of deformation, which indicates that the heat generation due to deformationmust 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 atthe same position is confirmed by the model prediction;(d) The thermal history is specific for each rolling schedule. Therefore, for the prediction ofthe evolution of the microstructure during rolling, the corresponding thermal history mustbe determined by the specific rolling schedule;1919.1 Summary and Conclusion(e) The secondary scale growth is limited to about 100i.tm during 7-pass rough rolling. It ismainly dependent on the rolling temperature and rolling speed; the oxide scale has aninsulating function and its effect on heat transfer of the slab is dependent on the scalethickness. In the case of rough rolling, a much larger effect on heat transfer from the initialscale has been obtained, in comparison to effect of the secondary oxide scale, because thethickness of the initial scale formed in the reheating furnace is from 1.5mm to 3.0mm Thisindicates 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 elementprogram was modified and applied to the roughing process. It has been verified by an analyticalsolution and a pilot mill test. In the program, a thermally coupled velocity-pressure approach wasadopted. The solution was iteratively obtained by solving the energy equation for temperature andthe basic deformation equation for velocities. The model has been applied to the industrial rollingprocess at Stelco's LEW for 9-pass schedule. The strain rate distribution has been obtained directlyfrom the velocity solution, whilst the strain distribution is obtained by the integration of the effectivestrain rate along the stream lines in the roll bite. The deformation behaviors for several typicalrolling passes, including the temperature distribution, have been presented and the measured rollforces 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, theeffective strain distribution through the thickness at the exit of the roll bite was investigated byinserting a pin of the same material into the specimen. The measured effective strain has beencompared 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 andlaboratory rolling compare favorably with the measured values. Although differences in1929.2 Future Workeffective strain arise at the surface, this is probably due to the separation between the pinand the specimen at this location. The difference in roll forces for some passes may arisefrom 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 entrypoint 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 rollbite; the strain increases considerably as the surface is approached due to the redundantshearing;(c) The temperature rise due to heat of deformation is higher near the surface than near thecenter line, but is mashed by roll chilling.9.2 Future WorkThermomechanical analysis of the slab has been conducted, paving the way for future workwhich should focus on the prediction of evolution of microstructure and the mechanical propertiesduring rolling. For the prediction of evolution of microstructure of HS LA steels, considerable workis still required to develop the equations describing the precipitation kinetics and its effect onrecrystallization.In addition, for a complete study of the heat transfer coefficient at the roll/slab interface, otherconditions such as lubrication and the roughness of the rolls are important and must be investigated.193REFERENCEREFERENCE[1] J.K. Brimacombe, F. Weinberg, and E.B. Hawbolt, "Formulation of Longitudinal, MidfaceCracks in Continuously-Cast Slabs", Metallurgical Transaction B, Vol. 10B, June, 1979,pp. 279-292.[2] I. Tamura, H. Sekine, T. Tanaka and C. Ouchi, 'Thermomechanical Processing ofHigh-strengh Low-alloy Steels', Butterworths & Co. (Publishers) Ltd., 1988.[3] A. Suzuki, "Recent Progress in Rolling Mills - Part II", Trans. ISU, Vol. 24, 1984, pp.309-329.[4] J.K. Brimacombe and I.V. Samarasekera, "Fundamental Aspects of Continuous Castingof Near Net Shape Steel Products", Int. Symp. on Casting of Near Net Shape Products,Honolulu, Hawaii, TMS, 1988.[5] I.V. Samarasekera, E.B. Hawbolt, Private Communication, 1989.[6] J.K. Brimacombe, E.B. Hawbolt, I.V. Samarasekera, P.C. Campbell and C. Devadas, "Microstructural Engineering: The Prediction of Microstructure Evolution and MechanicalProperties Based on Steel and Process Characteristics", Proc. of Int. Conf. on PhysicalMetallurgy of Thermomechanical Processing of Steels and Other Metals, (THERMEC-88),Tokyo, Japan, 1988.[7] U. Tenhaven, "Research and Development at Hoesch Stahl AG, Dot tnient, Part 4, HotRolled Products Department", Steel Research, 57, No.12, 1986, pp. 609-612.[8] J.S. Miller, "Design and Use of a Hydraulic Analogue to Determine Temperature Gradientsin a Steel Slab During Various Hot Rolling Procedures", J. Iron and Steel Institute,November 1969,pp. 1444-1453.194REFERENCE[9] F. Seredynski, "Prediction of Plate Cooling During Rolling-Mill Operation", J. Iron andSteel Institute, Vol. 211, 1973, pp. 197-203.[10] J. Pavlossoglou, Arch. Eisenhuttenwes, ibid, pp. 275-279.[11] J. Pavlossoglou, Arch. Eisenhuttenwes, Vol. 52, 1981, Nr.4, pp. 153-158.[12] F. Hollander, " A Model to Calculate the Complete Temperature Distribution in SteelDuring Hot Rolling", in 'Mathematical Models for Metallurgical Process Development',46-78, 1970, London, The lion and Steel Institute.[13] Ken-ichi Yanagi, "Prediction of Strip Temperature for Hot Strip Mills", Trans ISIJ, Vol.16, 1976, pp. 11-19.[14] Natsuo Hatta, Jun-ichi Kokado et al., "Analysis of Slab Temperature Change and rollingMill Line Length in Quasi Continuous Hot Strip Mill Equipped with Two Roughing Millsand Six Finishing Mills", Trans. ISU, Vol. 21, 1981, pp. 270-277.[15] Jun-ichi Kokado and N. Hatta, " Mathematical Model of Full and Three QuartersContinuous Type Hot S trip Mills for Prediction of Temperature Change of Steel and RollingProductivity", J. The Japan Society for Tech, of Plasticity, 19(1978), 213.[16] C. Devadas and I.V. Samarasekera, "Heat Transfer during Hot Rolling of Steel Strip",Ironmaking and Steelmaking, Vol. 13, No.6, 1986, pp. 311-321.[17] C. Devadas, I.V. Samarasekera and E.B. Hawbolt, "The Thermal and Metallurgical Stateof Steel Strip during Hot Rolling: Part I. Characterization of Heat Transfer", MetallurgicalTransaction B, 1990.[18] K. Sasaki, Y. Sugitani and M. Kawasaki, Tetsu-to-Hagane (J. of Iron and Steel Japan),65(1974), pp. 90-96.[19]^A.A. Tseng, F.G. Lin et al., "Roll Cooling and Its Relationship to the Roll life", MetallurgicalTransactions A, Vol. 20A, Nov., pp. 1938-2305.195REFERENCE[20] P.G. Stevens, K.P. Ivens and P. Harper, "Increasing Work-Roll Life by ImprovedRoll-Cooling Practice", J. of the Iron and Steel Institute, Jan. 1971, pp. 1-11.[21] W.L. Robert, " Hot Rolling of Steel", Manufacturing Engineering and Materials Processing,1983, Marcel Dekker, Inc., New York.[22] A.A. Tseng, S.J. Chen, and C.R. Westgate: "Modeling of Materials Processing", ASME.New York, NY, 1987, MD-Vol.3, pp. 51-63.[23] T.H. Hogshead, Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA, 1967.[24] Y. Yamaguchi, M. Nakao, K. Takatsuka, S. Marakami and K. Hirata, Nippon Steel Tech.Rep., 1985, 33, (4).[25] F. Kreith and W.L. Black, 'Basic Heat Transfer', 237-269, 1980, New York, Harper andRow.[26] S. Nukiyama, J Soc. Mech. Eng. Jpn., 1934., 37, 367-394.[27] B.K. Denton and F.A.A. Crane, J. Iron Steel Institute, 30-36.[28] C. Devadas, Ph.D Thesis, The University of British Columbia, 1989.[29] C.M. Sellars, "Computer Modelling of Hot-Working Process", Materials ScienceTechnology, Vol. 1, April 1985, pp. 325-32.[30] H. Fenech, J.J. Henry and W. M. Rohsenow, "Thermal Contact Resistance", in'Developments in Heat Transfer', Editor: W.M. Rohsenow, The M.I.T. Press, Cambridge,Massachusetts, 1964.[31] I.V. Samarasekera, 'Proc. Int. Symp. on the Mathematical Modelling of the Hot Rollingof Steel', 29th Annual Conf. of Metallurgists, CIMM, Hamilton, ON, Canada, Aug., 27-29,1990, Pergamon Press, New York, N.Y.[32] T. Wanheim and N. Bay, Annals of CIRP, 1976, 27-1, pp. 189-193.[33] W.R.D. Wilson and S. Sheu, Int. J. Mech. Sci., 1988, Vol. 30, No. 7, pp. 475-490.196REFERENCE[34] W.R.D. Wilson, "Friction Models for Metal Forming in the Boundary Lubrication Regime",ASME Winter Annual Meeting, December 1988, Chicago, Illinois.[35] A.J.W. Moore, 'Proc. Roy. Soc. (London), A195, 231,(1948).[36] N. Birks and G.H. Meier, 'Introduction to High Temperature Oxidation of Metals', EdwardArnold Ltd., 1983.[37] C.Z. Wagner, Phys. Chem., 21, 25, 1933.[38] V.T. Zhadan, V.D. Breigin, V.A. Trusov, I.E. Oratovskaya, and A.N. Chichaev, " Procedurefor calculating decarburization and scaling during hot rolling of carbon steel", IzvestiyaVUZ Chernaya Metallurgiya, 1987, (11), pp. 77-81.[39] J. Paidassi, "Oxidation of Iron", Acta. Metallurg. 6, 184(1958).[40] H.S. Hsu, "The Formation of Multi-layer Scale on Pure Metals", Oxidation of Metals, Vol.26, Nos. 5/6,1986.[41] Daniel M. Obaro, "Concepts for Minimization of Scale Formation during Reheating ofCarbon Steels", 'Procs. of the International Symposium on Steel Reheat FurnaceTechnology', Edited by F. Mucciardi, CIM, Hamilton, 1990, pp. 268-276.[42] H. Abuluwefa, G. Carayannis, F. Dallaire, R.I.L. Guthrie, J.A. Kozinski, V. Lee, F.Mucciardi, "Oxidation and Decarburization in the Reheating of Steel Slabs", 'Procs. of theInternational Symposium on Steel Reheat Furnace Technology', Edited by F. Mucciardi,CIM, Hamilton, 1990, pp. 243-267.[43] R.C. Ormerod IV, H.A. Becker, E.W. Grandmaison, A. Pollard, P. Rubini, and A. Sobiesiak,"Multifactor Process Analysis with Application to Scale Formation in Steel ReheatSystem", 'Procs. of the International Symposium on Steel Reheat Furnace Technology',Edited by F. Mucciardi, CIM, Hamilton, 1990, pp. 227-242.[44]^N.P. Kuznetsova and G.I. Kolchenko, "Influence of Scale Formation on Heat TransferIntensity in Continuous Furnace", Steel in the USSR, Vol. 18, 1988, pp. 332-333.197REFERENCE[45] O.C. Zienkiewicz, "Flow Formulation for Numerical Solution of Forming Process", in 'Numerical Analysis of Forming Process', Edited by J.F.T. Pittman, O.C. Zienkiewicz, R.D.Wood, and J.M. Alexander, 1984 John Wiley & Sons Ltd.[46] Ashok Kumar, I.V. Samarasekera and E.B. Hawbolt; " Roll Bite Deformation during theHot Rolling of Sheet Strip", private communication, 1990.[47] P.C. Jain, 'Plastic Flow in Solids', Ph.D. Thesis, 1976, Department of Civil Engineering,University of Walse, Swansea.[48] P. Hartley, C.E.N. Sturgess, C. Liu and G.W. Rowe: International Metals Review, 1989,Vol. 34, No.1, pp. 19-34.[49] O.C. Zienkiewicz, P.C. Jain and E. Onate, Int. J. Solids and Structures, 1978, Vol. 14, pp.15-38.[50] P.R. Dawson, 'Application of Numerical Methods of Forming Processes', Winter Meetingof AS ME, San Francisco, 1978, pp. 55-60.[51] G.J. Li and S. Kobayashi, "Rigid-Plastic Finite-Element Analysis of Plane Strain Rolling",Transactions of ASME, Vol. 104, Feb., 1982, pp. 55-64.[52] H. Grolier, proc. of the NUMIFORM-86 Conference, 1986, Gothenburg, August 25-29,pp. 225-229.[53] J.H. Beynon, P.R. Brown, S.I. Mizban, A.R.S. Ponter and C.M. Sellars, 'Proc.Computational Methods for Predicting Materials Processing Defects', Ed. M. Predeleanu,Elsevier, Amsterdam, Holland, 1987, pp. 19-28.[54] A. Silvonen, M. Malinen and A.S. Korhonen, Scandinavian J. Of Metallurgy, 1987, Vol.16, pp. 103-108.[55]^M. Pietrzyk and J.G. Lenard, J. Materials Shaping Technology, 1989, Vol. 7, No.2, pp.117-125.198REE-ERENCE[56] P.R. Dawson, "On Modelling of Mechanical Property Changes during Flat Rolling ofAluminum", IN. J. Solids and Structures, Vol. 23, No. 7, 1987, pp. 947-968.[57] R.B. 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 Handbookand 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 ModelingTechniques 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 MassTransfer, 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 hightemperature", Izvestiga Vysshikh Uchebnykh Zavedenii, Chemaya Metallurgiya, 1984,(11), 141.199REFERENCE[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 NumericalMethods in Engineering, Vol. 10, pp. 379-399 (1976).[70] L.R. Underwood, 'The Rolling of Metals, Theory and Experiment', John Wiley & SonsInc., New York, 1950, pp. 57-93.[71] Tetsuo Sakai, Yoshihiro Saito, Kenji Hirano, "Deformation and recrystallization of lowcarbon 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,1983.200A.1 Nodes in the S labAPPENDICESAppendix A Finite Difference Nodal EquationsA completely implicit finite difference technique has been used in solving the one dimensionaltransient heat transfer problem in this study. By the technique, the domain of heat transfer couldbe divided into a number of time steps, at each time step, a slice in the domain could also be dividedinto a number of nodes with either equal spacing or unequal spacing. In the slice, there are threekinds 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 therolls.Because different coordinate systems ware adopted for the slab and the rolls, i.e, rectangularsystem and circular system, different nodal equations were specified as below.A.1 Nodes in the SlabThe slice in the domain at each time step experienced different environment inside the rollgap and outside the roll gap, but a general heat transfer coefficient h(t) would represent the differentsituations, therefore, general equations for each of the three nodal equations can be obtained asfollows by the control volume method.Surface NodesFor the surface nodes, a heat balance can be set up for the control volume as shown in Fig.A.1.201A.1 Nodes in the SlabFigure A.1 Schematic diagram of the control volumefor the nodes in the slabThe roll chilling was considered by coupling the roll surface node, Nr, by means of the roll gapheat transfer coefficient.1^nIcsn's+11 —Tkl.,+1 Ax ^Ax . + h(t)(T: TNsn +1 ) cos(9)+ f 4, ) ay^ps cps (A.1)TV At TN, A2yAy g 2where qf, 4g are the heat generation due to friction and deformation, respectively; Oi is the contactangle in the roll bite; they all become zero outside the roll bite.202A.2 Nodes in the RollsInterior NodesA similar heat balance can be also set up for the interior nodes in the slab as shown in Fig.A.1.17 +1 1 — Tin^T:t++11^+ 1^Tinkr^ + ks^ + gAy Ax = p,,C i„^Ay Ax^(A.2)Ay Ay AtCenter Line NodesFor the nodes at the center line, owning to the assumption of an adiabatic boundary, the heatbalance can be given as below, see also in Fig.A.1.Tr —Tr'^ Tr1Axks^ + AyAx = p,C ^Ay AxAy p, At (A .3)A.2 Nodes in the RollsThe slice in the surface layer of rolls is in circular coordinate system. The nodal equationsfor the three kinds of nodes can be expressed as follows.Surface NodesThe heat balance for the roll surface nodes can be set up as below, see Fig.A.2nr+11 —nr+1^Arr+1 —^Ar^Arkr Ar^R^AO + h O A) (T 1Z 1^r+ 1 )R AO = p rCpr At^R —4 ) 2AO ^(A.4)where h(t*) is heat transfer coefficient along the roll surface including the roll gap heat transfercoefficient.203A.2 Nodes in the RollsFigure A.2 Schematic diagram of the control volumefor the nodes in the rollsInterior NodesThe heat balance for the interior nodes is as below:Tnil _ Tin +1rAr^( iv Tn.:11_ Tn. + 1AOAr ■Ti n2 r + 2 JAOkr + kr Ar PrC PrInside Surface NodesAt rAeAr (A .5)An adiabatic boundary condition was assumed to the inside surface, so the heat balance is setup as below:Tri —77 +1^Ar^Trl^ArAr 2kr^ R —5+ AO = prCPr^ R —5+-4 At (A .6)204B.1 Deformation FormulationAppendix B Velocity-Pressure Finite Element DerivationB.1 Deformation FormulationThe governing equation for the deformation analysis was expressed as below which is thesame as Eq.(8.1):fvfeilT falc1V — {8v}T {F}dV — {SOT {t}dS = 0^(B .1)s,Inserting the constitutive equation (8.10), we can write,fvfeilTPIDI {i}dV + 8evpdV — {8v } T {F}dV —I {45v TT {-INS = 0^(B.2)s,By using trial shaping functions as:{v} = ENT^=^{ay}i =1 (B.3){P}=^NiPaiP = [V] {aP } (B.4)r=1where m, n are the total number of nodes per element for pressure and velocities; {ay}, aP} are thenodal 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 . [(Mir' [L][Nif [N P jdV]— f {1■11 T {F}dV —j [N1T fildS = 0^ (B.5)The terms [N7 and [N"] are referred to as shaping functions or interpolation functions which willbe described later.205B.2 Thermal CouplingFrom Eq.(8.5), we can write:[DI ViT [M]T^= 0^(B .6)Combining the equations (B.5) and (B.6), a simple symmetric set of equations can be obtainedas the same as Eq.(8.17):L[KP]^0^tT j[ail^o[Kulifal^{{i}}^0^(B .7)or[IC] {a} + {f} = 0^ (B .7(a))where[[1][111 il]T g[D °1[[L,][N]JcIV^ (B .8(a))[MT [1 ]Wtvl]T (B .8(b))= — [AllT IF^— s [N1T {i}dS^ (B .8(c))B.2 Thermal CouplingFor 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:kS'a2-rs^a2r:+ 43^PsCpsaTs^aT's (B .9)ax 2 + ay / U^± Vax^ay=206B.2 Thermal Couplingwhere u and v are velocities along the two coordinate axes, X and Y.Using the trial shaping function:nt= E NiTi = [N] {Ts }1=1 (B.10)where ne is the total number of nodes per element and [T s} is the nodal temperature value; Ni or [NJis also referred to as the shaping function which has the same properties as described in the previoussection; and applying Galerkin method of weighted residuals, and using integration by parts forsecond-order derivatives, the equation (B.9) reduces to:E IcsK,;Ti + y, psc „Kw;J.0^J.0^aN.^aN.i s Niks(--e ly){T}edS (B.11){F1 }+whereaN, aN; aN, aN; `KK`'dxdy (B.12(a))ax ax^ay ayaN;^aN;Kvu =^f(uNi --a7+vNi 7-y- y (B.12(b))and = ,.Nidxdy (B.12(c))The last term on the right-hand side of Eq.(B.11) is not effective inside the domain and on theboundaries, where prescribed values give the necessary conditions. It vanishes if homogeneousnatural 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.207B.2 Thermal CouplingEq.(B.9) contains convective terms on the right hand side of the equation. If the mesh sizeexceeds a certain critical value, the solution are oscillatory, and at high velocities acceptable answerscan only be obtained by an excessive reduction in the element size' s] . This difficulty can be overcomeby introducing an upwinding technique which was originally used in the finite element methods interms of 'upwinding difference' (backward differences with respect to the velocity direction) forthe convective terms.The most important feature of the numerical schemes proposed relies in the choice of theweighting functions. In conventional Galerkin formulationsi n] , these are chosen equal to the shapefunctions, but it is clear that other choices are possible, and schemes where the weighting functionsare 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) = 5 r x I- ix 12 (x)4/te )2)7,1 —k/T)+1 0..x <<— he^(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 thefour sides of the element, i.e., a, and 0,, i =1, 4. They were expressed as below:208B.2 Thermal Coupling^0c, = 2(tanh 2^y^y) 1 + 3 —Pi + -12^12 po i^(B .16)1 ^413,^ (B.17)tanh(!) Yyewhere y=-7-; ui; was calculated as below::7-7 -1.2 (u->i ->uj) ' lij (B.18)where ui and u; are the velocity vectors at the node i and j; and /i; is a unit vector in the positivedirection of the line through i and j. The sign of kJ also determines the sign of the parameters oc iand Pi.An easy way of obtaining weights is by means of linear blending of the one-dimensionalweights. The weighting functions adopted in this study for each node are as follows:W1(x)Y, a(x)Y)) =^Y Ge2 a4F Wer^aiF(Ge2)^(B .19(a))W2(x , y , 0(x , y )) = N2(x , y) + 4He2 P IF (Ge, ) (B .19(b))W3 (X. , y , cc(x , y )) = N3 (x, y) — Gei cc2F (I/ el ) — H e2a1F (Gel )^(B .19(c))Y ,(3(x Y)) = 1\14(x ,Y) -1- 4G eiP2F (HeI)^ (B .19(d))W5(x, y, ct(x, y)) = N5(x, y)-- Gei oc2F^— -I (x2F (Gel )^(B .19(e))Wo(x, Y 13(x, Y )) = N6(X, Y)+ 41-4/ f32F(Gei ) (B .19(f))W7(x,y,a(x,y)) = 1V7(x ,y)— Ge2 a4F (HeI) — el oc2F (Gel )^(B .19(g))209(1 +x-G.)Gel = 2Ge2(B .20(a ))(B .20(b))(1 —xG )22(1 + ycaus)Het (B .20(c))B .3 Isoparametric Element14i8(x ,P(x,Y)) = Njx,y) + 4Ge, 134F (Hei )^ (B .19(h))where NA, y) are shaping functions; andHet(1 —xGaus)2(B .20(d))where xG„,„ and yGaus are the coordinate value at Gauss point in each element.B.3 Isoparametric ElementIn the finite element method, interpolation of a field property f(x, y) defined over an elementis 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. Itapproximates the behavior of a field variable, for example velocity and pressure, over the domainof an element. Further, it is a function of position with the property that N, is unity at node i andzero 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 thepolynomial order of shape functions. In this study, eight-noded isoparametric elements were used210x32(b)(a)4 3176814-1-1 ^•1 5^2B.3 Isoparametric Elementto better estimate the curve boundaries in the flow domain. Fig.B.1 shows a quadratic rectangularelement with 8 nodes in the natural coordinate and global coordinate systems. The element definedin the natural coordinate system is sometimes called the parent element.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 systemThe shaping functions were defined by:NA,I1)= 4 +4,4)( 1^+r1;T1- 1)— (B.22(a))for corner nodes;1Ni(4, 11) = —2 ( 1 — 42) +Ni (4, 11) = 2 ( 1 + 4,4) (1, -TC)= 011 1 = 0(B.22(b))(B .22(c))211B.3 Isoparametric Elementfor 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 anyother nodes E i # 4 and rh # rj the value is zero.^Therefore, using a simplified notation, nodalproperties are defined as:8x =^NA,i)xi^(B .23(a))1=1Y = E N,r1)y1 (B .23 (b))1= 1{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 thanthat of velocities, a 4-node isoparametric element was used with linear interpolation of shapingfunction as below:1NATI) = -4( 1 + U) +ill) (B .24)The nodal property for pressure is defined as below:4P = N,(4, 11)P,=1(B .25)212B.4 Evaluation of Element MatrixB.4 Evaluation of Element MatrixFor the derivative of the field property, f, it is easy to be obtained according to the Eq.(B.21),i.e.:of^aN,ax =of = 8 aN.Ifay^i=i ayThe derivative of shaping functions to natural coordinates can be expressed as:aN,^aN, axwayax^ayaN,^aN, ax aN, ayax an + ay an(B .26(a ))(B .26(b))(B .27(a))(B .27(b ))or in matrix form:aN,aN,arl{- ax ay- aN,a a aax ay aN;_ an an_ ay{aN,a }= Vlay (B .28)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 tochange the Eq.(B.28) to:{aN,axaN,ayaN,}a4aN,an= (B .29)213aN8 aN8= axaN8= [fl-1 aaN8flif2f8{ofaxofay 1f2f8ay _can also be defined as:am aN2ax axam aN2ay ayaN, aN2a a4aN, aman anB.4 Evaluation of Element MatrixThe derivatives of the field viable f ( which are velocity, temperature in the current study)(B.29)The above transformation makes the Gauss-Legendre integration over the domain of eachelement possible noting that dxdy =Id] dEATI. 3 x 3 integration points have been used in the currentstudy.214BIOGRAPHICAL INFORMATION NAME:^CH^.^WEI c)-0-,Q(.4-MAILING ADDRESS:, of HF V LS Al-di) MA-14A (4L5 Ei/4- 4Q 4 C^v6 TPLACE AND DATE OF BIRTH:Z.H (AVG— ,^4NA , juLY 5", I 16 3EDUCATION (Colleges and Universities attended, dates, and degrees):SE!) /NIT Uivt vER,Si -T\^4f41)^ Lo Cr Y, 133 13 1,5cUNtv^ScA C^ALC) Toe,4-40 Lo cry f BEI^ forj, ,^ScPOSITIONS HELD:PUBLICATIONS (if necessary, use a second sheet):AWARDS:Complete one biographical form for each copy of a thesis presentedto the Special Collections Division, University Library.DE-5

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