{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Materials Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"McCulloch, Craig Allen","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-08-30T02:35:12Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1988","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Applied Science - MASc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"A mathematical model has been developed to predict the thermal history of strip during cooling on the run-out table of a hot strip mill. The model incorporates phase transformation kinetics and accounts for the heat of transformation. To characterize the cooling by laminar water sprays, in-plant trials were conducted at the Stelco Lake Erie Works hot strip mill. The temperature data was used in the thermal model to calculate an overall heat transfer coefficient for a laminar water bank of 1 kW\/m\u00b2\u00b0C. Isothermal diametral dilatometer testing was used to generate phase transformation kinetics for a 0.34 weight percent plain carbon steel. Continuous cooling dilatometer testing was used to calculate the transformation start time as a function of the cooling rate. The high cooling rates of 40 \u00b0C\/s to 50\u00b0C\/s, experienced on the run-out table had the effect of depressing the transformation start temperature by over 100\u00b0C.\r\nThe phase transformation kinetics were incorporated in a phase transformation model and employed to predict thermal profiles for a 0.34 carbon plain-carbon steel. The temperature predictions were within 25\"C of the plant pyrometer readings using the calculated overall heat transfer coefficient and within 35\u00b0C of the plant pyrometer values using literature derived heat transfer coefficients.\r\nA simulation of the model predicted cooling conditions on a Gleeble high temperature testing machine showed that the transformation was occurring at approximately 730\u00b0C. The empirical transformation start time, obtained from cooling rate versus transformation start time tests, which was used in the phase transformation portion of the model, and the Gleeble simulation gave excellent agreement with the model thermal profile predictions.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/27900?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"I CHARACTERIZATION OF T H E COOLING AND TRANSFORMATION OF STEELS O N A RUN-OUT T A B L E OF A HOT-STRIP M I L L By CRAIG A L L E N M C C U L L O C H B.A.Sc, The University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E O F MASTER OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES M E T A L S A N D MATERIALS ENGINEERING We accept this thesis as confonning to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 1988 \u00a9Craig Allen McCulloch, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M e t a l s a n d M a t e r i a l s E n q i n e e r i n g The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3\/81) A B S T R A C T A mathematical model has been developed to predict the thermal history of strip during cooling on the run-out table of a hot strip mill. The model incorporates phase transformation kinetics and accounts for the heat of transformation. To characterize the cooling by laminar water sprays, in-plant trials were conducted at the Stelco Lake Erie Works hot strip mill. The temperature data was used in the thermal model to calculate an overall heat transfer coefficient for a laminar water bank of 1 kW\/m 2 , C. Isothermal diametral dilatometer testing was used to generate phase transformation kinetics for a 0.34 weight percent plain carbon steel. Continuous cooling dilatometer testing was used to calculate the transformation start time as a function of the cooling rate. The high cooling rates of 40 *C\/s to 50*C\/s, experienced on the run-out table had the effect of depressing the transformation start temperature by over 100'C. The phase transformation kinetics were incorporated in a phase transformation model and employed to predict thermal profiles for a 0.34 carbon plain-carbon steel. The temperature predictions were within 25\"C of the plant pyrometer readings using the calculated overall heat transfer coefficient and within 35\u00b0C of the plant pyrometer values using literature derived heat transfer coefficients. A simulation of the model predicted cooling conditions on a Gleeble high temperature testing machine showed that the transformation was occurring at approximately 730*C. The empirical transformation start time, obtained from cooling ii rate versus transformation start time tests, which was used in the phase transformation portion of the model, and the Gleeble simulation gave excellent agreement with the model thermal profile predictions. iii T A B L E OF CONTENTS Abstract ii Table of Contents iv List of Tables viii List of Figures ix Acknowledgment xvi 1.0 INTRODUCTION 1 2.0 LITERATURE REVIEW 3 2.1 Heat Transfer on the Run-out Table 3 2.1.1 Heat Transfer Coefficients for Water Bar and Water Curtain Cooling from Plant Data 4 2.1.2 Heat Transfer Coefficients for Water Bar Cooling from Experimental Measurements 5 2.1.3 Heat Transfer Coefficient for Roll Contact Cooling from Experimental Measurements 9 2.2 Phase Transformation Kinetics 10 2.3 Review of Related Models , 13 2.4 Figures 16 3.0 SCOPE A N D OBJECTTVES 17 3.1 Scope 17 3.2 Objectives 18 4.0 PROCEDURE 19 iv 4.1 Mathematical Model 19 4.1.1 Formulation 20 4.1.2 Numerical Methods 24 4.2 Heat Transfer Coefficient 27 4.2.1 Calculation from Literature 28 4.2.2 Calculation from Plant Data 30 4.3 Phase Transformation Characterization 33 4.3.1 Material 33 4.3.2 Metallography 33 4.3.3 Dilatometer 34 4.3.3.1 Isothermal Tests 35 4.3.3.2 Continuous Cooling Tests 36 4.3.4 Phase Transformation Model Calculations 37 4.4 Tables and Figures 39 5.0 RESULTS A N D DISCUSSION 55 5.1 Heat Transfer Coefficient 55 5.1.1 Literature 55 5.1.1.1 Laminar Water Bar Cooling 55 5.1.1.2 Film Boiling Cooling 56 5.1.1.3 Support Roller Contact Cooling 56 5.1.1.4 Combined Cooling 57 5.1.2 Plant Trials 58 v 5.1.2.1 Overall Heat Transfer Coefficient 59 5.1.2.1.1 Calculation 59 5.1.2.1.2 Sensitivity 60 5.1.2.2 Individual Heat Transfer Coefficient 62 5.2 Phase transformation 63 5.2.1 Material 63 5.2.2 Isothermal Cooling Tests 64 5.2.3 Continuous Cooling Tests 65 5.2.3.1 Metallography 66 5.2.3.2 Coiling Temperature 67 5.2.4 Model Phase Transformation Calculations 69 5.3 Mathematical Model 71 5.3.1 Sensitivity 71 5.3.2 Validation 72 5.4 Tables and Figures 73 6.0 CONCLUSIONS 133 6.1 Summary 133 6.2 Conclusions 135 6.3 Future Considerations 138 7.0 BIBLIOGRAPHY 139 8.0 APPENDIX 142 8.1 Nomenclature 142 vi 8.2 Derivation of Finite Difference Equations 145 8.2.1 Top Surface Node 145 8.2.2 Interior Nodes 146 8.2.3 Bottom Surface Node 146 8.2.4 Solution , 147 8.3 Hatta et al. Thermal Boundary Layer Calculations 147 vii LIST OF TABLES Table I Composition for the three steel chemistries used. 39 Table IIa....Plant conditions for four runs 73 Table Ho....Plant conditions for four runs 74 Table IIc....Plant conditions for four runs 75 Table IJJ Industrial plant cooling conditions 76 Table IV Metaliographic data for the 0.34 carbon samples, for the down-coiler sample.the continuous cooling samples, and the Gleeble simulation sample; with tabulated values for, cooling rate, fraction ferrite, undercooling, and average austenite grain size 77 Table V Comparison of the composition of the down-coiler and transfer bar medium carbon samples 78 Table VI....Grain size versus coiling temperature for 0.054 weight percent carbon grade steel 79 Table VH....Tabulated model predictions, for low (7'C\/s) and high (45'C\/s) cooling rates, and for the literature heat transfer coefficients at an average cooling rate, (26'C\/s) 80 viii LIST OF FIGURES Figure 1 Specific Heat as a Function of Temperature for five carbon levels, BISRA 16 Figure 2 Hot-strip geometry used for the model 40 Figure 3 Schematic of the STELCO Lake Erie Works Hot Strip Mill Run-out Table 41 Figure 4 Specific Heat as a Function of Temperature for a 0.34 % carbon steel, BISRA, w\/o phase transformation 42 Figure 5 Thermal Conductivity as a Function of Temperature for a 0.06 % plain carbon steel, BISRA 43 Figure 6 Thermal Conductivity as a Function of Temperature for a 0.08 % plain carbon steel, BISRA 44 Figure 7 Thermal Conductivity as a Function of Temperature for a 0.23 % plain carbon steel, BISRA 45 Figure 8 Thermal Conductivity as a Function of Temperature for a 0.34 % plain carbon steel, BISRA 46 Figure 9 Flow chart for the basic program 47 Figure 10 The six types of cooling regime experienced by the steel strip 48 Figure 11 The various film boiling heat transfer coefficients from Kokada et al.[6] for three cooling water temperatures with two values from the Berensen[24] horizontal surface boiling equation 49 Figure 12 Experimental verification of TAC3 and TAC1 50 ix Figure 13 A typical dilation versus time plot for an isothermal dilatometer test 51 Figure 14 A typical dilation and temperature versus time plot showing transformation start and finish times 52 Figure 15 Experimental dilation and thermal dilation plots, used with divergence method (Campbell[27]) for calculation of transformation start 53 Figure 16 Flow sheet for the iterative solution of the Avrami fraction transformed equation as a function of temperature 54 Figure 17 Black zone radius as a function of a constant steel surface temperature 81 Figure 18 Hatta laminar water bar heat transfer coefficient as a function of contact radius 82 Figure 19 Thermal profile model sensitivity to changes in the water temperature for the Kokada film boiling heat transfer coefficient 83 Figure 20 Thermal profile model sensitivity to changes in the support roller conduction cooling 84 Figure 21 Thermal profile model literature heat transfer coefficients 0.05% carbon, 3.89 mm gauge, target coiling temperature 720*C 85 Figure 22 Thermal profile model literature heat transfer coefficients, 0.05% carbon, 2.62 mm gauge, target coiling temperature 720*C 86 Figure 23 Thermal profile model literature heat transfer coefficients, 0.07% carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720\u00b0 C 87 x Figure 24 Thermal profile model literature heat transfer coefficients, 0.07% carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 720\u00b0C 88 Figure 25 Thermal profile model literature heat transfer coefficients, 0.05% carbon, 2.62 mm gauge, target coiling temperature 620\"C 89 Figure 26 A sample temperature profile from the plant data. 90 Figure 27 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 720\"C 91 Figure 28 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720'C 92 Figure 29 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 720*C 93 Figure 30 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 2.62 mm gauge, target coiling temperature 720*C 94 Figure 31 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 620\"C 95 Figure 32 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target xi coiling temperature 540\u00b0C 96 Figure 33 Thermal profile model sensitivity, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 720\u00b0C 97 Figure 34 Thermal profile model sensitivity, individual laminar water bar heat transfer coefficient, 10 kW\/m 2 ,C with 20 kW\/m2 ,C and 5 kW\/m2 oC deviations,target coiling temperature 720*C 98 Figure 35 Isothermal dilatometer results for 673*C test, dilation-time and temperature-time 99 Figure 36 AD\/AT as a function of time for the 673'C isothermal test 100 Figure 37 Isothermal dilatometer test sample plot lnln(l\/(l-FX)) vs ln(t) for 673\"C fraction ferrite transformed 101 Figure 38 ln(b) Avrami coefficient for the isothermal formation of ferrite in the 0.34 carbon steel 102 Figure 39 ln(b) Avrami coefficient for the isothermal formation of pearlite in the 0.34 carbon steel 103 Figure 40 Avrami coefficient, nf, for the austenite-to-ferrite transformation in the 0.34 % C, plain carbon steel 104 Figure 41 Avrami coefficient, n,,, for the austenite-to-pearlite transformation in the 0.34 % C, plain carbon steel 105 Figure 42 Calculated ln(b) values for the ferrite transformation assuming n<= 1.25, for 0.34% carbon steel 106 xii Figure 43 Calculated ln(b) values for the pearlite transformation assuming Ap = 1.14, for 0.34% carbon steel 107 Figure 44 Average Avrami coefficient V for 0.34% carbon compared to other experimental values (Campbell[27]) 108 Figure 45 Comparison of the ln(b) Avrami coefficient for the austenite-ferrite transformation in several plain-carbon steels(Campbell[27]) 109 Figure 46 Comparison of the ln(b) Avrami coefficient for the austenite-pearlite transformation in several plain-carbon steels(Campbell[27])... 110 Figure 47 Temperature as a function of time for a continuous cooling rate of 27'C\/s I l l Figure 48 Thermal and Experimental dilatometer values as a function of time for a cooling rate of 27*C\/s 112 Figure 49 The undercooling for the austenite-to-ferrite start temperature as a function of cooUng rate 113 Figure 50 Fraction ferrite as a function of cooling rate, from metallographic examination 114 Figure 51 Continuously cooled dilatometer sample 115 Figure 52 Continuously cooled dilatometer sample showing banding 116 Figure 53 Medium carbon down-coiler sample 117 Figure 54 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m 2 , C, Table rn run one cooling conditions, and a transformation start temperature of 732'C (dT\/dt = 7\u00b0C\/s) 118 xiii Figure 55 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m 2 o C, Table in run two cooling conditions, and a transformation start temperature of 732'C (dT\/dt = TCJs) 119 Figure 56 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m 2 \u00b0C, Table LTI run three cooling conditions, and a transformation start temperature of 732'C (dT\/dt = 7'C\/s) 120 Figure 57 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m 2*C, Table DI run one cooling conditions, and a transformation start temperature of 688'C (dT\/dt = 45'C\/s) 121 Figure 58 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m 2*C, Table in run two cooling conditions, and a transformation start temperature of 688'C (dT\/dt = 45'C\/s) 122 Figure 59 Surface thermal profile, using the overall heat transfer coefficient, 1 kW\/m2\"C, Table DI run three cooling conditions, and a transformation start temperature of 688'C (dT\/dt = 45'C\/s) 123 Figure 60 Surface thermal profile, using the literature heat transfer coefficients, Table DI run one cooling conditions, and a transformation start temperature of 710\"C (dT\/dt = 26'C\/s) 124 Figure 61 Surface thermal profile, using the literature heat transfer coefficients, Table DI run two cooling conditions, and a transformation start temperature of 710'C (dT\/dt = 26'C\/s) 125 Figure 62.....Surface thermal profile, using the literature heat transfer xiv coefficients, Table DJ run three cooling conditions, and a transformation start temperature of 710*C (dT\/dt = 26'C\/s) 126 Figure 63 Effect on predicted center line temperature of changes in the number of nodes through thickness 127 Figure 64 Effect on predicted center line temperature of changes in the step size, where the step size equals the strip velocity times the time increment 128 Figure 65 Effect of the \u00b10.3m\/s deviation of the strip velocity on the predicted temperature profile 129 Figure 66 Industrial cooling profile simulated on the Gleeble high temperature testing machine 130 Figure 67 Industrial cooling conditions, simulated on a Gleeble high temperature testing machine 131 Figure 68 Microstructure of the Gleeble cooling simulation sample for the Table HI run one cooling conditions 132 xv Acknowledgment I would like to acknowledge the support provided for this project by NSERC and Stelco Inc. The guidance of my thesis supervisors I.V. Samarasekera and E.B. Hawbolt is also much appreciated. On the experimental side, Keith Barnes, Henk Averink, and Barbara Zbinden were of major assistance in securing the industrial temperature data as was Bihn Chau with the dilatometer phase transformation kinetics. xvi INTRODUCTION 1 1 INTRODUCTION Historically the production of flat rolled product has been based on previous experi-ence. New materials or new physical properties for existing materials, were produced by a trial and error process by examining the effects of minor modifications to the rolhng\/ccoling conditions. The introduction of the computer to industrial applications initially changed the experience reservoir from the on-line staff to the machine, with little modification of the trial and error methods historically used. Approximately one half of the finished steel in North America is in the form of sheet or strip, with hot rolled strip being used in a wide variety of applications ranging from auto body to the shells of stoves and refrigerators. Furthermore, there has been a proliferation of non ferrous products in the market place of lightweight materials with mechanical properties equivalent to those of steel. The steel industry has responded by developing lighter gauge steel to lower the weight without an attendant loss of strength. To produce a strip steel of improved physical properties with decreased weight the vari-ables in the process must be well controlled; the final overall physical properties are affected by the chemical composition and thermo-mechanical history of the steel. Thus, the initial composition of the steel, the casting and reheating processes, together with the thermo-mechanical history of the steel during rolling and subsequent cooling profoundly influences the final mechanical properties of the strip. The economic down-tum at the start of this decade emphasized the need for tighter control of the hot rolling process to minimize costs. The use of niobium-titanium-1 INTRODUCTION 1 vanadium additives for increased strength became routine, although mill scheduling was accomplished by trial and error. Hot strip mills world wide are now controlled by empirical or semi-empirical computer models with some operator input, but, a better thenno-mechanical understanding of the entire rolling mill is needed in order to produce control models that decrease deviations from target physical properties. The rolling mill and the run-out table have been modeled from both an experimen-tal(empirical) and a theoretical(mathematical) point of view in an effort to provide better mill operational control. Mathematical models of these processes are also being developed in order to provide a better understanding of the theoretical aspects of thermo-mechanical processing of steel. One area that is not well understood is the relationship between the process variables and the transformation from austenite to ferrite\/pearlite during run-out table cooling. The phase transformation effects have historically been included in models by incorporating a specific heat value that includes the recalescence due to phase transformation, however, the cooling rate strongly influences the start of transformation and thus kinetics must be considered in any modeling effort. The object of this thesis was to model the temperature and microstructure of the hot-strip as it passes along the run-out table from exit of the finishing stands until entrance to the pinch roller of the down-coiler. A better understanding of the heat trans-fer to the cooling water sprays and of the kinetics of the phase transformation was sought in an attempt to provide a realistic model of the process. 2 LrTERATURE REVIEW 2.1 2 LITERATURE REVIEW The current literature on hot strip run-out table cooling covers all aspects of the process. These range from a description of new cooling techniques to composition control for the production of steels with improved mechanical properties. They also include reviews of models for process control. Of particular interest to this thesis are measurements of or expressions quantifying the heat transfer between the strip and cooling water on the run-out table, phase transformation kinetics under non-equilibrium conditions and mathematical models coupling the two phenomena. 2.1 Heat Transfer on the Run-Out Table On the run-out table the strip is cooled by a laminar water bar or water curtain. The former technique has received its name because of the 'glassy' or 'rod-like' non-turbulent appearance rather than due to a strict Reynolds number definition of laminar flow. A water curtain has been described as a continuous water bar in that it resembles a laminar sheet or 'curtain' of water. Cooling occurs by forced convection in the zone of direct contact and by film boiling across a vapour barrier in the region surrounding the impact zone. The heat transfer coefficient for laminar water bar, water curtain, and film boiling may be determined empirically from in-plant temperature measurements or from laboratory experiments. Heat is also transferred to the support rolls by conduction and to the surrounding air by radiation. 3 LITERATURE REVIEW 2.1.1 2.1.1 Heat Transfer Coefficients for Water Bar and Water Curtain Cooling from Plant Data In this approach heat transfer coefficients are back calculated from in-plant strip surface temperature measurements. Tacke et al.[l] studied the two types of water cooling as well as the spray nozzle cooling previously employed on run-out tables. To a run-out table with a standard spray nozzle header configuration, they added a bank of laminar water bar headers and a bank of water curtain headers . Each cooling bank had a blow-off and a pyrometer mounted before and after the bank to give an accurate reading of surface temperature. Using the measured temperature changes across a cooling bank in a finite element model they calculated an overall heat transfer coefficient for each type of cooling bank as a function of the water application rate. From this they derived values of 1800 W \/ m 2 , C \u00b1 2 0 0 W \/ m 2 , C for the bank of water curtain cooling, 1300 W\/m2*C \u00b1200 W\/m 2 o C for the bank of laminar water bar cooling, and 900 W\/m 2 o C \u00b1150 W \/ m 2 , C for the bank of spray nozzle cooling. While the water curtain cooling had the highest heat transfer coefficient, the water application was uneven over the strip. The laminar water bar gave an even water application over the full strip width and so was chosen for plant use even thoygh the resultant heat transfer coefficient was less than that for water curtain cooling. They report cooling rates of 50\u00b0C\/s up to 200\u00b0C\/s depending on strip gauge with laminar water bar cooling. They also report that front end loading the cooling, that is using water sprays at the finish mill end of the run-out table resulted in an increase in yield strength over tail end loading, cooling with the sprays at the down-coiler end of the 4 LITERATURE REVIEW 2.1.2 run-out table. They note that in a series of over seven hundred strips using the laminar water bar cooling coupled with the calculated heat transfer coefficients in their plant control model the two sigma deviation was reduced from an average value of 45\u00b0C down to 20'C. Colds and Sellars[2] have calculated a heat transfer coefficient for an individual water curtain by using a finite difference heat transfer model with various values included for water curtain and film boiling cooling. The resulting thermal profile is compared to observed values to arrive at a result of 17 kW\/m 2 , C for an individual water curtain; they comment on the difficulty in producing an exact value due to the short residence time of the strip under the water curtain contact area which they assume to have a diameter of two to three times the water curtain diameter. They use a film boiling cooling heat transfer value of 150 W\/m2\"C for the region outside the contact zone which compares well with the Farber and Scorah[3] value of between 150 W \/ m 2 , C and 170 W\/m 2 o C. 2.1.2 Heat Transfer Coefficients for Water Bar Cooling from Experimental Measurements Individual laminar water bar heat transfer coefficients and values for the associated film boiling cooling in the surrounding region were presented in three articles by Hatta[5,6], Kokada[7], et al. The Hatta et al. results were based on an examination of the cooling associated with a single laminar water bar over a stationary 10 mm thick stainless steel plate with low water flow rates. The plate was instrumented with five thermocouples inserted from the back of the plate at a depth of 8 mm from the top surface 5 LITERATURE REVIEW 2.1.2 at 20 mm increments from the water bar contact center line. The plate was heated in a reducing furnace to hot-strip temperatures, it was then removed and placed under a laminar water bar header. The temperature change over time for various water temperature and flow rates was recorded. This data was then used in a finite difference model to derive an equation for a heat transfer coefficient. A heat transfer coefficient of 15.93 W\/m 2\"C was adopted for natural convection in air in the Hatta model. The Hatta et al. experiments produced a number of general observations about the water cooling under a laminar water bar. First, that there is a 'black zone' around the area under the water bar which did not show any boiling phenomena. Second, around this 'black zone' was an area of film boiling. Third, the transition between the boiling and non-boiling areas appeared to be instantaneous; that is, there did not appear to be a visible transition cooling regime. Using these observations and the data produced from the thermocouples, a heat transfer coefficient equation was obtained, . . . 1 where 0.063 is an experimentally derived constant k is the thermal conductivity of the water, in W\/mK r is the laminar water bar contact radius, in meters Re is the Reynolds number and Pr is the Prandd number The heat transfer coefficient for the film boiling region is, 6 LITERATURE REVIEW 2.1.2 a\u2122 = 200* 2420-21.70V) . . . 2 ,\u20228 where T w is the average water temperature, in Kelvin T s is the steel temperature, in Kelvin and T S A X is the saturation temperature of the water, in Kelvin. The water saturation temperature under one atmosphere pressure is 100 \u00b0C. The water temperature at which the transition from water contact cooling to film boiling cooling occurs, Tom-, is described by the equation, The value of ranges from 18.75 \"C for a steel temperature of 1000 *C to 100'C for a steel temperature of 350 *C. Hatta, Kokada, et al. noted that film boiling was not observed for a water temperature lower than 68 \u00b0C. The cooling water temperature, T w , must therefore be between the minimum critical transition temperature, Tdm- = 68 *C, and the saturation temperature, T S A X = 100 *C. The water temperature used in the model is a simple average of these two values, -(7^-1150) 8 . . . 3 cm ~ 1 0 0 o C + 6 8 \u00b0 C 2 = 84\u00b0C . . . 4 which is used as T w for Eq.2. 7 LITERATURE REVIEW 2.1.2 A horizontal water velocity1 is needed to derive a water film thickness as well as for computation of the Reynolds number used in Eq.l; this is calculated based on the assumption that the horizontal water velocity is equal to the vertical water velocity which is determined by the water flow rate. Hatta et al. used the heat transfer coefficients calculated in Eq. 1 and Eq.2 to calculate a thermal profile for the plate. The calculated profiles were then compared to the thermocouple data and it was found that greater cooling was predicted by the calculated heat transfer coefficient than was observed experimentally. To compensate for this over cooling Hatta et al.[6] postulated that there is a 'thermal zone' in the water film layer, not all of the water film thickness was affected by the heat flow. The thermal zone2, a boundary layer phenomena, is the thickness of water above the plate that is heated in a finite time period. This was used in Hatta's model with Eq. 1 for the area under the laminar water bar and Eq.2 for the film boiling zone to calculate a new thermal profile which gave good agreement with the experimental results. Eq. l is insensitive to the water flow rate but relatively sensitive to the area of contact under the laminar water bar3. 1 Equation 1 in the appendix 2 Water film thickness, described in the appendix 3 From Eq.Al in the appendix 8 LITERATURE REVIEW 2.1.3 2.1.3 Heat Transfer Coefficient for Roll Contact Cooling from Experimental Measurements Diener and Drastik[7] examined heat flow between guide rolls4 and continuously cast slab using instrumented rolls and developed heat flux profiles for various cooling types. Using their 'quasi-stationary' heat flux value of 75 kW\/m with an average temperature difference of 900 *C, an average heat transfer coefficient of 83 W\/m 2 o C can be calculated. 4 On the inside of the curve above the slab 9 LITERATURE REVIEW 2.2 2.2 Phase T rans fo rmat ion The phase transformation and its associated heat of transformation has been characterized with a variety of methods, the predominant one being the use of a modified specific heat value. The specific heat values tabulated by the British Iron and Steel Research Association, BISRA[8], include the effects of the phase change by incorporating the heat of transformation in the specific heat value to give a greatly increased value at the phase transformation temperature, as can be seen in Figure 1. If the specific heat is taken as a temperature dependent value which includes the heat of transformation, then the thermal effects of the phase transformation can be accounted for in this way. The BISRA specific heat values were obtained from plant measurements made in the early 1950's and do not include the modem alloys. The data range is based only on the weight percent carbon and a variation in carbon content of 0.06 to 0.40 weight percent; individual values are an average for a 50 \u00b0C temperature range. The BISRA values are for equilibrium and any effects of cooling rate are ignored. The use of the isothermal kinetics to describe the continuous cooling transformation is based on the Avrami [10] formula, X = l-expH>r\") \u2022 \u2022 \u2022 5 where X is the fraction transformed, t is time, and b and n are two coefficients called the 'Avrami' coefficients. This is based on the additivity concept first postulated by Scheil5 in 1935. An additive system is one in which the transformation is only a function of the 5 For incubation not for phase transformation as such 10 LITERATURE REVIEW 2.2 temperature and the fraction previously transformed. In an additive system a continuous process can be approximated as the sum of a series of discrete steps; this is very useful in mathematical modeling. The Avrami formula presents the fraction transformed ( X ) as a function of time (t) and the two 'Avrami' coefficients 'b' and V . Avrami[9,10,ll], and later Cahn[12] postulated separate criterion for determining if a system is additive. Avrami described an ' isokinetic' condition in which the ratio of nucleation and growth rate is constant. Cahn described a site saturation criterion based on preferential nucleation sites. Agarwal and Brimacombe[13] used the additivity concept in their model of rod cooling, noting that while the system being examined did not satisfy either criterion, the results from the model based on the assumption of an additive system agreed with experimental observations. Kuban et al.[14] examined the additivity of the austenite to pearlite transformation to determine conditions over which additivity applied. They postulated a criterion of 'effective site saturation' based on the concept that most of the growth of the new phase, pearlite, is growth at the initially nucleated sites with the sites nucleated near the end of the transformation contributing very little to the overall volume change. The effective site saturation criterion was found to be valid if the time for twenty percent of the transformation was experimentally greater than 0.28 times the time for ninety percent of the transformation, r 2 0> 0.28^ . . . 6 Hawbolt et al.[15,16] examined the austenite-to-pearlite transformation for a eutectoid steel and the austenite-to-ferrite and pearlite transformations for a 1025 steel using a dilatometer to determine phase transformation kinetics and start temperatures. 11 LITERATURE REVIEW 2.2 The Avrami coefficients, n, and, b, were determined from isothermal tests. The transformation start time (or temperature) and the total fraction ferrite formed as a function of cooling rate were determined using continuous cooling tests. A different method of dealing with the phase transformations occurring on a run-out table was examined by Morita et al.[17]. They used an on-line transformation detector measuring the change in magnetic resistance of the strip to determine the fraction transformed on the run-out table. The concept of an on-line transformation detector under the strip is potentially very desirable. However, machine calibration and data interpretation seem dependent on trial and error. Until a theoretical model capable of interpreting the change in magnetic resistance in terms of the kinetics of the austenite to ferrite and ferrite plus pearlite transformation is available, the on-line transformation detector, while sophisticated, requires substantial experimental data to describe the transformation behavior. 12 LITERATURE REVIEW 2.3 2.3 Review of Related Models Of the various published models pertaining to run-out table cooling of hot strip, most are intended for use in mill control. They range from the Hinrichsen[18] dynamic systems approach to the Hurkmans et al.[19] experimentally produced deformation transformation model. Hinrichsen[18] modeled the run-out table as a dynamic simulation via a systems control approach used widely in chemical engineering applications. He formulated a dynamic model of the run-out table including the gain and dead time of each component, from the spray water valves to the run-out table pyrometer. He then used experimental data to tune the response of the dynamic model. The entire process is controlled with a proportional-integral controller using modified feedback compensated feed-forward control to prevent cumulative errors from inducing increasing oscillation. This is a widely used control system in areas where there is small variability in the desired output product. With hot-strip, current production requires output of many products with different properties from the same production line, which makes this type of model of limited utility. A basic model used in a wide variety of plants is the basic heat transfer model as exemplified by Tacke et al.[l]. This model is described in the preceding section A. and is an excellent example of the use of empirical data and mathematical modeling to control run-out table output. The heat transfer models found in the literature vary in their levels of sophistication. These range from the simple Longenberger[20] model to the sophisticated Tacke et al.[l] model. The Longenberger[20] one dimensional model 13 LITERATURE REVIEW 2.3 discretizes the strip through thickness into three nodes and the resulting model is tuned through statistical regression. Miyake[21] has produced a more sophisticated model which mathematically characterizes the losses due to radiation and water cooling but is still fine tuned with empirical data. The Tacke et al.[l] model, previously described, uses a finite element approach and back calculated heat transfer coefficients to produce an on-line control model and represents the most sophisticated of the purely heat transfer models. More complex still are the models that add microstructural considerations to the basic thermal model. Yada[20] uses additivity and the assumption of a transformation rate independent of time. The entire rolling mill is approximated as a series of independent models; one for hot deformation, resistance to hot deformation, a temperature profile model, a transformation model based on nucleation and growth, and a structure versus properties model. The model outputs are combined to produce a prediction of the final microstructure and the physical properties and are used as an on-line mill control model. The model is used on-line to compensate strip cooling for variations in strip velocity to maintain the consistency of the strip properties. Yada notes that some form of on-line microstructural information would be useful during processing to eliminate cumulative errors and to this end he suggests the use of the magnetic transformation detector described by Morita et al.[17]. The most comprehensive approach is that of Hurkmans et al.[19] in which dilatometry is used to characterize the phase transformation kinetics for a given chemistry. The dilatometric data for a given test is reduced to a group of between thirty 14 LITERATURE REVIEW 2.3 to sixty points which are then fitted to a cubic spline interpolation. From the interpolated data a set of thirty data values are produced and used for all future calculations. From the fitted data set the rate of diametral change over time and the rate of temperature change over time is calculated. This is similar to the method used by Hawbolt et al.[15,16] but, with the interpolation of the raw data, variations due to experimental differences between individual data runs should be minimized. The diametral change with time data is integrated to produce fraction transformed data. This data is then fitted to an equation by a least squares approximation to produce values for the constants A\u00b1, B k , and Q used in, dt where 'az^ Jk 4 e = Ak(Zk + e)%> is the rate of transformation for phase k, is the fraction of phase k transformed, is the fraction of y phase transformed, is a small number needed in integration of the differential equation. The constants are derived for a given phase, composition, and austenitizing condition. Hurkmans et al.[19] have used the model for ferrite, pearlite, bainite, and martensite transformations. This data is used in an in-plant control model and has resulted in a reduction of overall water consumption while maintaining the desired microstructure. 15 LITERATURE REVIEW 2.4 2.4 Figures c o \"9 v $ v O v g v O v.e( w co CM o a o 5 d ci d ci a O) CD 5 >\u00abr>+ -Mac -K> O + < + \u2022 - m a x \u2022 M I X 1 1 -(fl m CO CM T -I 1 1 1 1 1\u2014 i - O) oo s to in ^ d d d d d ci Specific Heat W\/kg C (X 1000) Figure 1 Specific Heat as a Function of Temperature for five carbon levels, BISRA 16 SCOPE A N D OBJECTIVES 3.1 3 SCOPE A N D OBJECTIVES 3.1 Scope The impetus for this work lies in the need to link the microstructure and properties of hot band to processing parameters in the hot strip mill. This requires the integration of effects of composition, casting, reheat, rough and finish rolling, run-out table cooling, and down-coiler cooling on the microstructure. This may be best accomplished by developing mathematical models of the individual processes and linking them up to trace the changes in the microstructure due to processing. This project focuses on the cooling and phase transformations on the run-out table of a hot strip mill subject to certain limitations. This examination is limited to the run-out table without regard to the prior thermo-mechanical history, even though this is accepted as having an effect on the microstructure. The model incorporates heat transfer and phase transformation kinetics associated with the cooling and any thermally generated run-out table stresses or strains are ignored. The model will examine only a medium carbon (< 0.40% ) steel and the resulting austenite to ferrite and ferrite plus pearlite phase transformations. The bainite and martensite transformation kinetics will be left to future workers. Transformation and cooling in the down-coiler is also outside the scope of the model. 17 SCOPE A N D OBJECTIVES 3.2 3.2 Objectives (i) Production of a heat transfer model of the hot strip on the run-out table, from exit from the final stand of the finish mill until entrance into the pinch roller of the down-coiler. (ii) Determination of phase transformation kinetics for a medium carbon, plain carbon steel of 0.34 % C. (iii) Determination of individual and overall heat transfer coefficients for laminar water bar spray banks. (iv) Integration of the heat transfer coefficients and phase transformation kinetics in an overall heat transfer model to predict coiling temperatures. (v) Microstructure prediction1 for the coiled steel from dilatometer data and the integrated model. 1 ferrite-pearlite ratios 18 PROCEDURE 4.1 4 PROCEDURE 4.1 Mathematical Model The strip geometry assumed for this model is shown in Figure 2. The model has been formulated for the Stelco Lake Erie Works Hot Strip Mill Run-Out Table, which is shown schematically in Figure 3. The cooling water for this run-out table is delivered by laminar water bar sprays over the top of the strip and water curtain spray for the bottom of the strip. The cooling system consists of five banks of sprays with six headers in each bank. The five banks cover the first half of the run-out table with banks one, two, and three used as the main cooling banks and the fifth bank used to trim the strip temperature to the desired down-coiler temperature. Bank four was being installed and was not in use for the duration of this work. On the run-out table hot steel strip moves at high speed and undergoes rapid cooling. The significant phenomena that occur as a result are internal heat flow, variable external heat transfer, phase transformation and associated heat generation. To mathematically model the hot-strip on the run-out table, the following is required: (i) basic physical description of the strip and the layout of the run-out table, (ii) ....heat flow equations, (iii) ...boundary conditions, (iv) ....phase transformation and recalesence equations. The heat flow equations are well understood and will be described in section 4.1 along with the basics of the mathematical model. The external environment the strip sees 19 PROCEDURE 4.1.1 varies down the length of the run-out table. Heat transfer occurs by convection and radiation to the air as well as by convection and film boiling to the cooling water. The various heat transfer regimes, the resulting heat transfer coefficients, and the theoretical and empirical formulae for their calculation will be examined in section 4.2. The phase transformations and recalescence as well as the methods for their characterization will be exarnined in section 4.3 with the figures for sections 4.1,4.2, and 4.3 following in section 4.4. 4.1.1 Formulation The basic unsteady state equation for a three dimensional control volume is, k{s?+B? +^ r- + v p c ' l a 7 + a7 + 3 7 j = P C ' 3 T where the first three terms account for internal heat conduction, qg is the heat generated by the phase change, and the last three terms involving the velocity, v, of the strip are the heat flow due to bulk motion; the right hand side is the energy change in the volume as a function of time. qfis calculated by taking the fraction transformed for a given time step (which is detailed in 4.3.4) and multiplying by the volume of one node. The calculated volume transformed is used with the Zacay and AAronson[23] values for the heat generated by phase transformation per mole along with a density value to produce a heat flux for a 20 PROCEDURE 4.1.1 given fraction transformed. In order to simplify Eq.8 a number of assumptions about the physical geometry of the hot strip as it travels on the run-out table were made: (i) The strip is continuous and no distinction is made between the head end, tail end, or central portion of the strip. (ii) The process is operating at steady state and the temperature profile at a fixed location is invariant with time. (iii) Since the width to thickness ratio is large1, a zero temperature gradient is assumed across the strip width perpendicular to the direction of travel. (iv) Although a Biot number calculation based on an overall heat transfer coefficient indicates that there should be no gradient in the z direction through the strip thickness, the local heat transfer coefficient beneath a water spray is sufficiendy high to produce internal gradients. Therefore, (v) The rate of heat transferred into a stationary control volume due to bulk motion of the strip is much greater than the rate of heat transfer by conduction so the latter term in the x direction will be assumed to be negligible, Thus the governing equation simplifies to 1 1 meter wide to 0.004 meters thick 21 PROCEDURE 4.1.1 , ,1ft? ^fdr) . . . 1 0 Therefore while the through thickness nodes must be solved simultaneously, the steps along the axis of travel may be solved sequentially, which greatly simplifies the model calculations. The boundary and initial conditions for Eq.8 for a strip of thickness'd' are given below. Boundary Conditions, x > 0 , z = 0, z = d -k^ = h(x)(T-TA) \" A l Initial Conditions x = 0 , 0 \u00a3 z \u00a3 d T = Tj . . . 1 2 As can be seen in Eq.l 1 the heat removed from the surface is a product of the temperature difference between the strip and cooling medium and a heat transfer coefficient, h(x); the heat transfer coefficient is a function of the type of cooling at the particular location which will be examined in section B. The basic physical properties for steel were derived from the British Iron and Steel Research Association data tables[8]. BISRA compiled values for specific heat, thermal conductivity, density, thermal expansion, thermal diffusivity, and resistivity. The data for density, specific heat, and thermal conductivity were examined for temperature 22 PROCEDURE 4.1.1 dependence over the conditions of the run-out table and while the density was found to be relatively temperature independent2, all three were included as variables for each grade of steel. The BISRA specific heat and thermal conductivity are strongly temperature dependent A cubic spline interpolation of the BISRA[8] specific heat and density data was used to provide equations for the model. The temperature dependence of the specific heat data can be seen in Figure 1. This specific heat data includes the effects of the heat generated by the phase transformation; for this model a specific heat value that is independent of the heat of transformation is required since the latter has been incorporated separately. As the variation of specific heat with temperature for non-equilibrium conditions is not known a simple linear approximation of the austenite and ferrite regions in Figure 1 was used. Figure 4 shows the linear extrapolations of the specific heat of the gamma and alpha regions for 0.343 weight percent carbon steel. Initially a weighted average of the specific heat values was to have been used with the proportion of the gamma and alpha phases determining the proportion of the austenite and ferrite specific heats used. As the specific heat values are only linear approximations of discrete data points, a weighted average was viewed as having greater precision than 2 7.615 gm\/cm \u00b10.0105 between 700 \u00b0C and 950 ' C 3 The BISRA data is an interpolation of the 0.23 weight percent carbon value and the 0.40 weight percent carbon values for plain carbon steel. An interpolation of the low alloy values gave similar results. 23 PROCEDURE 4.1.2 the data would allow. The model, therefore, uses the austenite specific heat value at temperatures greater than the transformation start temperature and the ferrite value for temperatures at or below this temperature. The thermal conductivity data from the BISRA tables is described as a pair of linear equations with an inflection point at a temperature that varies according to the carbon content. The values for 0.06,0.08,0.23, and 0.34 weight percent carbon are shown in Figures 5, 6,7, and 8 respectively. The values for the heat generated by the phase transformation were taken as 776 cal\/mole[23] for the austenite\/fenite transformation and 1000 cal\/mole[23] for the austenite\/pearlite. 4.1.2 Numerical methods Equation 10, subject to the boundary conditions given in Eq. l 1, was solved numerically by an implicit finite difference method. The strip thickness was discretized into a series of nodes and finite difference equations were formulated for each node; the equations are derived in Appendix Eq.A. l to Eq.A.8. Figure 9 shows the flow chart of the computation scheme. The physical data, such as strip gauge and speed, cooling water flow rates, spray position, run-out table length, and steel composition are inputs to the model together with an initial steel temperature. The program computes the position along the run-out table and the heat transfer conditions for that location are determined. The coefficients for the tridiagonal matrix are calculated, the matrix is then solved and the node temperatures are altered. The data is then output and the position counter is 24 PROCEDURE 4.1.2 incremented; if the down-coiler position has not been reached the process starts over with a new position calculation. For the second and subsequent calculations the temperature of all the nodes at that location are examined to determine if any are less than the transformation start temperature. When the node temperature is below the transformation start temperature the model becomes slightly more complicated. The fraction transformed, and subsequently the amount of heat generated, and the resulting temperature increase are a function of the temperature at which the transformation takes place. The calculation of recalescence is therefore an iterative process which is repeated until the difference in two succeeding temperatures is below an error value. This process is exarnined in greater detail in section 4.3. The choice of a time step and through thickness node size for the model was based on the diameter of the laminar water bar. The time step for this model is a distance along the strip divided by the strip velocity. The laminar water bar diameter at the header nozzle is slighdy less than 40 mm and so to ensure that the step size is capable of resolving an individual laminar water bar, the step size had to be at least less than half the laminar water bar diameter or just under 20 mm. A step size of 10 mm was chosen so that each laminar water bar would be represented by at least three steps. 200 nodes through thickness were chosen after running various values for the number of nodes through thickness with the model and a 10 mm step size. The results of the model tests with various combinations of step size and through thickness nodes will be shown in section 5.3. 25 PROCEDURE 4.1.2 The model testing and validation is obtained through comparison of predicted thermal history and microstructure with plant data and the microstructure in down-coiler samples and will be exarnined in Chapter 5 sections 5.2 and 5.3. The model was written in FORTRAN and run on the University of British Columbia Amdahl V8 mainframe computer with approximately 250 seconds CPU time in an elapsed time of one-half hour. The model was also run on a C O M P A Q portable Ll personal computer with a 80286 CPU and an 80287 math co-processor, with an approximate running time of 3 1\/2 hours for a 200 node by 10mm step size configuration. Due to the different floating point representations of the two machines, double precision was necessary for the Amdahl while only single precision was needed for the Personal Computer. 26 PROCEDURE 4.2 4.2 Heat Transfer Coefficient The magnitude of the heat flow from the steel surface to the surrounding fluid, which consists of air, water, or some combination of the two, is deterrnined by the local heat transfer coefficient. The hot steel strip experiences six different cooling regimes as it proceeds along the run-out table, as shown schematically in Figure 10 and described as, (1) air cooling on the top and bottom of the strip, (2) air cooling on the top of the strip with roller contact below, (3) cooling by film boiling on top and air cooling below, (4) cooling by film boiling on top and roller contact below, (5) laminar water bar cooling on top and roller contact below, (6) cooling by film boiling on top and water curtain cooling below. To describe the six cooling regimes, the following five heat transfer coefficients are needed, (a) convection and radiation cooling to air, (b) conduction to the water cooled support rollers, (c) convection to the vapour film surrounding the laminar water bar, (d) convection to the laminar water bar, (e) convection to the water curtain. 27 PROCEDURE 4.2.1 The five heat transfer coefficients are developed from theoretical relationships found in the accelerated water cooling Uterature, examined in section 4.2.1, and by back calculation from plant temperature measurements, examined in section 4.2.2. 4.2.1 Calculation from Literature From an examination of the literature on accelerated cooling of hot steel strip it is clear that relatively few studies have been performed for the determination of heat transfer coefficients between the moving strip and the cooling water, either in plant or by laboratory simulation. The plant trial-derived values are best illustrated by the Tacke et al.fl] paper in which 1.8 \u00b10.3 kW\/m 2 K is reported for an overall heat transfer coefficient for a water curtain cooling bank and a value of 1.3 \u00b10.25 kW\/m 2 K is given for an overall heat transfer coefficient for a laminar water bar. These two values are for an entire bank of water sprays and include convective cooling in the contact zone beneath a water curtain or water bar and cooling by film boiling in the surrounding region. The laminar water bar heat transfer coefficient can be calculated using the Hatta et al.[5] Eq. l . While the film boiling heat transfer coefficient is calculated using the Kokada et al.[7] relationship Eq.2. a, = 0.063* - *Re**Pr ,8) ... i \u2022WB 2420-21.7(7V) (Ts ~ TSAT)* . . . 2 a\u2122 = 200* 28 PROCEDURE 4.2.1 The temperature at which a transition from Eq. 1 type cooling to Eq.2 type cooling takes place is calculated with Eq.3. r 5 -1150 . . . 3 T = \u2014 1 cm _g The Reynolds number, Prandd number, and k are temperature dependent and calculated internally in the model. T w , the temperature of the water in the film boiling section, is greater than or equal to 68\"C by the definition of Tcxst- At one atmosphere pressure T w can be assumed to have a maximum value of 100'C. Therefore, T w must always have a value between 68*C and 100#C. An average of these two values was used in Eq.2. Figure 11 plots the film boiling heat transfer coefficient as a function of the difference in temperature between the water and the steel surface. The values are plotted for 68*C, 100'C, and the average value 84*C. The two Berensen[24] values are for film boiling on a horizontal surface for a water film-steel surface temperature difference of 816*C and 636\u00b0C; these represent the average temperature difference realized just before and after the water cooling zones on the run-out table. The Berensen values agree with the Kokada et al. Eq.2 values calculated with a water temperature of 68 \u00b0C. A water curtain cooling heat transfer value of 17 kW\/m 2 \u00b0C has been reported by Colds and Sellars[2], assuming the existence of a surface oxide layer in order to produce a 'black zone' that will appear black at the calculated temperatures. They have employed a heat transfer coefficient for film boiling cooling of 150 W\/m 2 \u00b0C. This value is less than the Kokada et al.[7] value for a water temperature of 100 \u00b0C, is much less than the Berensen[24] values, but, agrees quite well with the Farber and Scorah[3] values for 29 PROCEDURE 4.2.2 small diameter wires. Eq.2 predicts a value of 520 W\/m 2 \u00b0C for a steel temperature of 1000 ' C which increases to 990 W\/m 2 o C for a steel temperature of 500 'C. The water curtain cooling for the hot strip is only on the underside of the strip; film boiling cooling does not occur as the water immediately falls off of the strip. For this reason, the Colds and Sellers film boiling heat transfer coefficient was ignored and the Kokada et al. film boiling heat transfer coefficient was used for the top surface along with Eq. 1 for the laminar water bar in this model. There does not appear to be any literature describing heat transfer at the support roller in a hot strip mill run-out table. However, Diener and Drastik[7] reported some data on heat transfer to support rollers in the secondary cooling zone of a continuous slab caster. For a water spray cooled roller4, a heat flow of 75 kW\/m with an average roll\/slab temperature difference of 900 *C was given resulting in an 83 W\/m 2 'C heat transfer coefficient As there is no available data on the size of the roller\/strip contact area, a value of one model step size has been used. 4.2.2 Calculation from Plant Data An alternative method of generating heat transfer coefficients is to use the mathematical model to back calculate specific machine dependent values from in-plant surface temperature measurements. To gather this data, a C O M P A Q portable computer with a Data Translation DT2805\/DT707T data acquisition board was connected to four pyrometers positioned along the run-out table of the Stelco L E W Hot-Strip Mill. All the 4 a 0.3 meter diameter roll of 1.75 meters length, 16 Cr and 44 Mo. 30 PROCEDURE 4.2.2 pyrometer and plant engineering log data was stored on 5 1\/4 inch, high density, floppy diskettes. The four pyrometers, PI, P2, P3, and P4, are shown schematically in Figure 3 and were mounted to the hand rail of the walkway over the run-out table water cooling bank section. The four walkway pyrometers were supplied and installed by Stelco Research and Development specifically for trial data acquisition; calibration for these units was done by Stelco with an IRCON portable black body. This device was also used to calibrate the three plant pyrometers FEXT, ROT, and DC. Units PI and P2 were IRCON R series two colour units with a range of 700*C to 1400'C, while units P3 and P4 are single colour IRCON 6000 units with a 500*C to 1500*C range. As these units were only in place for the twelve runs during the trials, air and water blow-offs were not in place at the strip locations measured. Additional data in the form of the engineering logs for the trials was available from the rolling mill computer. This data listed the speed, gauge, average number of cooling sprays, finish mill exit temperature, and down-coiler temperature of the strip along with the standard deviations of these values for one run. Temperature data was also available, in the plant engineering log for the three permanently installed plant pyrometers, FEXT, ROT, and D C which are shown schematically in Figure 3. The plant pyrometers were IRCON 2000 series with a 700\u00b0C to 1100'C range for the F E X T pyrometer and a 500\u00b0C to 800'C range for the ROT and D C pyrometers. The plant pyrometers were aimed at areas with water and air blow-offs and were recorded in the engineering logs with all water, speed, and physical data taken at one second intervals. 31 PROCEDURE 4.2.2 The heat transfer coefficients were calculated by using the plant pyrometer temperature data for strips that were coiled at a high enough temperature that recalescence effects did not occur until the down-coiler. A value for a heat transfer coefficient was input into the model and the resulting thermal profile compared to the pyrometer data. The best fit with the pyrometer data will be taken as the heat transfer coefficient for that set of conditions. An overall heat transfer coefficient for an entire cooling bank of six headers was calculated as was a value for individual header laminar water bars. 32 PROCEDURE 4.3.1 4.3 Phase Transformation Characterization 4.3.1 Material Three grades of steel were chosen for the phase transformation kinetics characterization due to availability of test samples and plant temperature data. These were a 0.054 weight percent carbon, a 0.074 weight percent carbon with 0.024 weight percent niobium, and a 0.343 weight percent carbon. These steels will be referred to as the 0.05 carbon, 0.07 carbon with niobium, and 0.34 carbon steels respectively for the rest of this thesis. The chemical composition for all three steels is listed in Table L 4.3.2 Metallography Down-coiler samples were obtained from Stelco for the various steel chemistries examined. These were transversely sectioned, polished to a five micrometer diamond surface, etched with 5% Picral etch and photographed on Polaroid type 55 positive negative film. The percentage ferrite for the down-coiler was determined with a Wild-Leitz Image Analyzer, using five randomly selected sample areas per specimen. It was necessary to determine the percentage ferrite and percentage pearlite in the continuously cooled dilatometer test samples from metallographic studies; a visible transition from ferrite to pearlite in the dilation-time plots was not observable at the high cooling rates used. 33 PROCEDURE 4.3.3 4.3.3 D i la tometer The diametral dilatometer, which measures the change in diameter of a tubular sample divring isothermal or continuous cooling conditions has been previously described by Hawbolt et al.[4,5] In this device, a thin walled tube is used as a specimen and the diametral dilation is measured. A thin walled tube is used to minimize internal temperature gradients and to provide the same cooling rate around the periphery of the specimen. A control thermocouple is attached to the outside of the tube at the plane of the dilation measurement The diameter change, as a function of time and temperature, is recorded and is used to provide phase transformation kinetics and transformation start times or temperatures as a function of time, temperature, and cooling rate. The AC3 temperature of 785*C and the AC1 temperature of 723*C, were calculated using the Andrews[25] formula and checked using a very slow heating rate for the 0.34 weight percent carbon sample. The experimental values of 800\u00b0C for AC3 and 733*C for AC1 are shown in the temperature-time plot of Figure 12. All samples were heated to 850\" C and held for 3 minutes. The samples were then air cooled to 820\u00b0C and held for 1 minute. The isothermal test samples were then rapidly cooled to the test temperature while the continuous cooling samples were cooled at a constant rate for the duration of the test. As the down-coiler strip was too thin for preparation of dilatometer samples, the tubular samples were machined from transfer bar taken at the end of the rougher rolling stage, which precedes the finish rolling stage. The transfer bar samples were cut to 34 PROCEDURE 4.3.3 approximate sample dimensions and then fully annealed.5 It is recognized that these samples do not duplicate the grain size and thermal history of the steel as it arrives at the run-out table. However, the transfer bar does have the same chemistry. The isothermal transformation kinetics obtained from the annealed transfer bar samples are characteristic of a given austenitizing condition (grain size). 4.3.3.1 Isothermal Dilatometer Tests The Avrami coefficients, b, and, n, are determined from data generated during isothermal diametral dilation tests. The isothermal dilatometer tests measure diametral dilation versus time at a constant temperature. From the dilation-time data the onset of dilation change is taken as the transformation start time, or tA V\u00bb as shown in Figure 13. The fraction transformed, for ferrite or pearlite, which is proportional to the diametral dilation, is calculated by dividing the dilation value at time, t, by the dilation value associated with completion of each transformation. The equilibrium fraction ferrite that will form at a given temperature is calculated from the Fe-C phase diagram using a lever law and an extrapolation of the y and lines to temperatures below the TAC1 using the Kirkaldy et al.[30] equations. The fraction transformed that corresponds to this equilibrium fraction ferrite (AD(ferrite) in Figure 13) is used as the ferrite stop, pearlite start point. Thus, the total fraction pearlite that will form is one minus the total fraction ferrite. For example, if the total fraction ferrite that forms at 680\u00b0C is 0.45, (AD\/AD X = 5 30 minutes at TAc3 + 50\u00b0C, followed by furnace cooling. 35 PROCEDURE 4.3.3 0.45 ), then the fraction ferrite transformed at a given time, t, is the measured ferrite dilation divided by 0.45 which gives the ferrite fraction transformed. The fraction transformed for pearlite is obtained by dividing the measured pearlite dilation by 0.55. The transformations can be described using the Avrami equation, Eq.5, in the form: The Avrami coefficients, n, and, b, are calculated from the graph of lnln(l\/(l-X)) versus ln(t); with, n, as the slope and ln(b) as the intercept, where ln(t) = 0. This assumes that n is a constant value during the isothermal test, as is indicated by the experimental data. 4.3.3.2 Continuous Cooling Tests Continuous cooling tests were performed by passing a controlled flow of cooling gas over the interior and exterior surface of the hollow tubular sample while measuring dilation and temperature versus time. Typical data, shown in Figure 14, is used to calculate the transformation start temperature as a function of cooling rate. The transformation start temperature (or time) for each cooling rate was determined for a range of cooling rates equivalent to those obtained on the run-out table. This temperature was calculated using the diametral dilation versus time and the temperature versus time data shown in Figure 14. AD\/AT is calculated using six dilation values and the corresponding six temperature values. The difference between the average of the first three dilation values and the second three dilation values is divided by the difference between the average of the first three temperature values and the second three . . .13 36 PROCEDURE 4.3.4 temperature values. Thus at some time, t, (AD\/AT), = | - r \u201e 2 , r \u201e 1 + r , y d ^ J ...14 the point at which AD\/AT changes slope is taken as the transformation start temperature, as shown in Figure 15. This is an effective procedure for determining the transformation start temperature (or time) because both dilation and temperature are affected by the onset of transformation; the heat of transformation causes recalescence in the temperature-time response. 4.3.4 Phase Transformation Model Calculations The model incorporates relationships describing the calculated transformation start temperature and the experimental percentage ferrite formed as a function of the cooling rate. The phase transformation rate at any time step in the model is assumed to be a function of the fraction transformed and the temperature at which the transformation takes place; this assumes that the phase transformation is additive. The fraction that undergoes transformation during one time increment generates a finite amount of heat which in turn raises the temperature of the node. This requires an iterative solution to determine the temperature and amount transformed; this is detailed in the flow chart shown in Figure 16. 37 PROCEDURE 4.3.4 The fraction transformed in the previous time step, Fx(k-1), and the current node temperature, T(k), are used to calculate a virtual time, tv, the time that would be required to produce Fx(k-l) at temperature T(k). Eq.5 is rearranged to deterrnine the virtual time, , .1 . . . 15 [v - . V ~ b ) The time step, dt, is added to t v and a new fraction transformed, Fx(k), is calculated for temperature T(k). The difference between Fx(k-1) and Fx(k) is the fraction transformed, dFx(k), for the time increment, dt, and is used to calculate a new temperature T(k)' based on T(k) and the heat generated by the new fraction transformed, dFx(k). Using T(k)' and Fx(k) a new virtual time t v' is calculated and in a similar manner a new temperature T(k)\". T(k)\" - T(k)' is compared to an acceptable error value (0.05 *C). If the difference is lower than O.OS'C the loop is exited. If the temperature difference is greater than 0.05\u00b0C, T(k)' becomes T(k)\" and the process repeats until the 0.05\u00b0C limit is satisfied. It should be noted that the model uses through-thickness nodes to model an observed rebound of surface temperature after a cooling spray. At strip velocities ranging from 5 m\/s to 7 m\/s the re-heating times are too short for a reverse transformation to austenite and the temperature is usually too low. For this reason, the model assumes that no transformation will take place if the temperature difference between the current step and the previous step is positive, that is if the node is increasing in temperature there is no reverse transformation. 38 PROCEDURE 4.4 4.4 Tables and Figures Low carbon Low carbon-niobium Medium carbon c 0.054 0.074 0.343 Mn 0.270 0.540 0.700 P 0.006 0.005 0.008 S 0.011 0.008 0.009 Si 0.020 0.017 0.009 Cu 0.044 0.021 0.021 Ni 0.007 0.008 0.006 Cr 0.062 0.012 0.023 Mo 0.002 0.003 0.003 V 0.000 0.000 0.000 Nb 0.000 0.024 0.000 A l 0.030 0.047 0.043 Table I Composition of the three steel chemistries intended for use in this study. 39 40 PROCEDURE 4.4 03 CD k_ 3 CO 03 c 'E 03 8 \u00a9-Q_ (5-V4 p\u2014 o DC \u00ae4 CO Q_ \u00a94 CM CL CL ej \u00a9-fc v \u2014 L U o o O c 5 o Q c CO m CD -\u2022\u2014\u00bb CO -ee-oo oo olo 'c Figure 3 Schematic of the STELCO Lake Erie Works Hot Strip Mill Run-out Table 41 PROCEDURE 4.4 Specific Heat W\/kg C Figure 4 Specific Heat as a Function of Temperature for a 0.34 % carbon steel, BISRA, with out phase transformation. 42 PROCEDURE 4.4 o o Thermal Conductivity W\/mC Figure 5 Thermal Conductivity as a Function of Temperature for a 0.06 % plain carbon steel, BISRA 43 PROCEDURE 4.4 44 PROCEDURE 4.4 o o o o CM o o o o CD o o co o o o o CM CO E Thermal Conductivity W\/mC Figure 7 Thermal Conductivity as a Function of Temperature for a 0.23 % plain carbon steel, BISRA 45 PROCEDURE 4.4 o Thermal Conductivity W\/mC Figure 8 Thermal Conductivity as a Function of Temperature for a 0.34 % plain carbon steel, BISRA 46 PROCEDURE 4.4 47 PROCEDURE 4.4 Film Boiling Laminar: y Water Bar Film Boiling Film Boiling Water Curtain Roller Roller Roller Figure 10 The six types of cooling regime experienced by the steel strip 48 PROCEDURE 4.4 i i i i i i i i i i i i i i r Q Heat Transfer Coefficient Kw\/mC 2 Figure 11 The various Film boiling heat transfer coefficients from Kokada et al.[6] for three cooling water temperatures with two values from the Berensen[24] horizontal surface boiling equation. 49 PROCEDURE 4.4 PROCEDURE 4.4 \u2022^ 1 Dilation Figure 13 A typical dilation versus time plot for an isothermal dilatometer test. 51 PROCEDURE 4.4 \u00a7 Dilation ^ Temperature ( Q Figure 14....A typical dilation and temperature versus time plot showing transformation start and finish times. 52 PROCEDURE 4.4 53 PROCEDURE 4.4 T(k) Fx(k-1) tv.Fx(k) START CALCULAT6 T(k)' t' ,Fx(k) v CALCULATE T(k)\" STOP Figure 16 Flow sheet for the iterative solution of the fraction transformed as a function of temperature. 54 RESULTS & DISCUSSION 5.1.1 5 R E S U L T S & D I S C U S S I O N 5.1 Heat Transfer Coefficient 5.1.1 Literature 5.1.1.1 Laminar Water Bar Cool ing The Hatta et al.[5] laminar water bar heat transfer coefficient calculated using Eq. l is sensitive to the value of the contact radius, r. Colds and Sellars[2] in their water curtain heat transfer calculation have noted that a value of two to three times the water curtain width seemed reasonable for a contact diameter. To examine the effect of steel surface temperature on the contact radius or 'black zone' diameter a simple one dimensional model of laminar water bar cooling, using Equations 1 and 3 and the Hatta et al.[5] heat flow and thermal layer calculations (in the appendix section 8.3) was used to calculate the radius of the 'black zone' as a function of a constant steel surface tem-perature. Figure 17 shows the results of this model calculation for steel surface tem-peratures in the range from 400\u00b0C to 1100'C. For steel surface temperatures greater than 600\u00b0C the 'black zone' radius changes slowly with temperature. An average value, 33.7 mm, was chosen for the temperature range of 700\u00b0C to 900'C; this is the range of interest on the run-out table. The heat transfer coefficient for various contact radii between 0.1 mm and 100 mm was calculated and the results are presented in Fig-ure 18. The heat transfer coefficient values seen in Figure 18 are stable for any contact radius greater than 20 mm with an average heat transfer coefficient value of 11 kW\/m 2 o C calculated for a contact radius of 33.7 mm. The thermal profile model 55 RESULTS & DISCUSSION 5.1.1 combines the Colas and Sellars[2] water curtain heat transfer coefficient of 17 kW\/m 2 o C, the laminar water bar heat transfer coefficient calculated with Eq. l , and a film boiling heat transfer coefficient calculated with Eq.2. 5.1.1.2 Film B oiling Cooling The film boiling heat transfer coefficient calculated with Eq.2 was shown, in Fig-ure 11, to be sensitive to the cooling water temperature, T w . To assess the sensitivity of the thermal profile model predictions to this parameter, the through strip thermal profile was modeled using film boiling heat transfer coefficients calculated with Eq.2. The minimum, average, and maximum values for the water film temperature of 68'C, 84*C, and lOO'C respectively were used along with the laminar water bar heat transfer coefficient calculated in E q . l 1 . The results of this model are shown in Figure 19, it is evident that the model predictions of strip surface temperature are only mildly sensitive to the water film temperature. The predicted surface temperatures at the down-coiler location are 769*C, 774\u00b0C, and 778*C for the respective water film temperatures, T w , of 68\u00b0C, 84\u00b0C, and 100'C, while the measured pyrometer value at that location is 718*C \u00b112 'C. 5.1.1.3 Support Roller Contact Cooling The Diener and Drastik[7] support roller, conduction cooling, heat-transfer coeffi-cient is an approximation for a different physical system; therefore to evaluate the sen-sitivity to the 83 W\/m 2*C value the model was employed. A \u00b150% change in the 1 33.7 millimeter contact radius 56 RESULTS & DISCUSSION 5.1.1 support roller conduction heat transfer coefficient results in a top surface, down-coiler location, temperature prediction of 772\"C and 775*C respectively. The temperature predictions are shown in Figure 20, as is the predicted value of 773*C for the Diener and Drastik[7] heat transfer coefficient of 83 W\/m 2 \u00b0C. 5.1.1.4 Combined Cooling The literature derived heat transfer coefficient values were combined into one model to test the predictions in relation to the plant pyrometer data from section 5.1.2. The laminar water bar heat transfer coefficient from Eq. l , the film boiling heat transfer coefficient from Eq.2, and the Diener and Drastik[7] support roller contact heat transfer coefficient were employed as input to the thermal profile model and used to predict top surface strip temperatures under a variety of cooling conditions from the plant data. For the 0.05 carbon, 720*C target coiling temperature, the model predictions for the strip surface are 774\"C and 765\"C at the down-coiler position for the 3.89 mm and 2.69 mm gauges respectively. These results are shown in Figures 21 and 23 and it is seen that the corresponding in-plant temperature measurements at the same location are considerably lower with values of 718*C \u00b112*C and 717\"C \u00b18\"C respectively. For the 0.07 carbon with niobium grade with gauges of 3.89 mm and 2.69 mm, the model pre-dicted surface temperatures at the down-coiler are 776*C and 758\u00b0C, the results are shown in Figures 22 and 24. The pyrometer readings at the corresponding positions in-plant are 710*C \u00b115\u00b0C and 714*C \u00b18\"C. In all four cases the in-plant pyrometer measured values are 40\u00b0 C to 55\" C lower than the model predicted values. 57 RESULTS & DISCUSSION 5.1.2 The Hatta et al.[5] laminar water bar and the Kokada et al.[6] film boiling heat transfer coefficients were experimentally derived using a stationary stainless steel plate under a water nozzle as opposed to the plant cooling conditions of a moving plain-carbon strip under a water nozzle. The composition of the plate should have little or no effect, while strip movement will elongate the water contact area and this may increase cooling. The overall result would be lower pyrometer temperatures than pre-dicted by the model and these lower predictions are what has been observed with the model calculations based on literature derived heat transfer coefficients. The model predictions for the cooling conditions experienced by a 3.89 mm 0.05 carbon steel with a 630*C target coiling temperature are shown in Figure 25. These cooling conditions were modeled to determine what effect the lower coiling tempera-ture would have on the model prediction-pyrometer temperature difference and the model prediction of 725*C is much higher than the pyrometer value of 629*C \u00b114*C. 5.1.2 Plant Trials The run-out table history for a total of twelve strips is listed in Table Ua, lib, and lie. The variables for the twelve runs were; 0.05 percent carbon and 0.07 percent car-bon with 0.024 percent niobium, 3.89 millimeter and 2.69 millimeter gauges, and three aim coiling temperatures of 720'C, 630*C, and 550\u00b0C. Basic run-out table parameters for the trial coil runs, along with average temperature values and standard deviations are also listed in Table II. An example set of temperature readings for each pyrometer, for one trial, taken at one second intervals, are plotted in Figure 26. The readings for 58 RESULTS & DISCUSSION 5.1.2 the three permanent plant pyrometers, F E X T , ROT, and DC have been compensated at the pyrometer for emissivity. PI and P2 being two colour units need no emissivity compensation. P3 and P4 are recorded at an emissivity of 1.00 and therefore must be compensated for a strip emissivity of 0.80. 5.1.2.1 Overall Heat Transfer Coefficient 5.1.2.1.1 Calculation An overall heat transfer coefficient was calculated using the plant temperature data. At the cooling rates experienced on the run-out table, between 40*C\/s and 50*C\/s, the phase transformation starts at a lower temperature than in an air cooled sample2. To avoid including phase transformation effects in the calculation of an overall heat transfer coefficient, only those coils with an aim coiling temperature greater than 700*C were used for the calculations. Using the thermal profile model with the cooling condi-tions experienced by the 0.05 carbon, 3.89 mm gauge strip, coiled at 718\u00b0C \u00b118*C vari-ous values for an overall laminar water bank heat transfer coefficient were evaluated. A value of lkW\/m 2 e C gave the best fit with a predicted temperature of 716\"C for the strip surface at the down-coiler pyrometer location. The plant pyrometer value of 7 1 8 * C \u00b1 1 8 * C i s shown with the model predictions in Figure 27. To evaluate the effect of using the lkW\/m 2*C overall effective heat transfer coef-ficient with the cooling conditions experienced by other strips, the thermal profile 2 This is examined in the continuous cooling section of section 5.2.4 59 RESULTS & DISCUSSION 5.1.2 model was run for a number of other grade\/gauge combinations. For a given strip, depending on the finish mill exit temperature, gauge, strip speed, and desired coiling temperature; the number of headers turned on in each bank is varied. In the model the overall effective heat transfer coefficient is applied over the region spanning the head-ers that were turned on during that particular run. Model predicted top surface temperatures have been compared with in-plant tem-perature measurements for a 3.89 mm gauge strip of 0.07 carbon with niobium, a 2.62 mm gauge strip of the same composition, and a 2.62 mm gauge strip with 0.05 carbon, and the results are shown in Figures 28, 29 and 30. The 0.07 carbon with niobium gives predicted values of 715'C for the 3.89 mm gauge and 715\u00b0C for the 2.62 mm gauge. This is close agreement with the respective pyrometer values of 710\u00b0C \u00b115*C and 714'C \u00b18*C. The 0.05 carbon, 2.62mm gauge prediction of 716*C is also in close agreement with the pyrometer value of 718*C \u00b18'C. The lkW\/m 2 \u00b0C heat transfer coef-ficient appears to be valid for the plant cooling data. 5.1.2.1.2 Sensitivity To check the sensitivity of model predictions with the overall heat transfer coeffi-cient the thermal profile model was run using the 0.05 carbon, 3.89 mm gauge cooling conditions for the two lower coiling temperatures of 630\u00b0C and 550\u00b0C, in which phase transformation should have some effect. Figure 31 shows the predicted surface value at the down-coiler position of 620\u00b0 C as well as the pyrometer value of 620\u00b0 C \u00b114\u00b0C for the 630\u00b0C target coiling temperature sample. Figure 32 displays the model down-coiler 60 RESULTS & DISCUSSION 5.1.2 position temperature prediction of 656\"C for the 550*C target coiling cooling conditions as well as the measured pyrometer value of 559\u00b0C \u00b116\"C. These two figures show no discernible trend for the relation of the overall heat transfer coefficient predictions and the coiling temperature. The effect of changing the overall heat transfer coefficient on the model predic-tion is shown in Figure 33, with a \u00b10.2 kW\/m 2 o C change resulting in predicted surface temperature values of 736'C and 700\u00b0C for a respective decrease and increase of 0.2 kW\/m 2*C in the overall heat transfer coefficient Both values are within the \u00b118*C deviation of the 718\u00b0C pyrometer value. For all six figures3 one trend emerges; the model consistently predicts a value slightly less than the measured PI pyrometer temperature, slighdy greater than than the measured P2 pyrometer value, and a much higher4 value than the P3 and P4 measured pyrometer readings. The model predictions agree quite well, however, with the readings for the three permanently installed plant pyrometers. The consistent deviation of the readings from the pyrometers installed for the trials makes the data of doubtful value. This deviation is possibly due to the lack of air and water blow-offs, which are employed at the permanent pyrometer locations, resulting in an unclear view of the strip surface for the temporary pyrometers. 3 Figures 27 through 32. 4 100\u00b0C to 200*C 61 RESULTS & DISCUSSION 5.1.2 5.1.2.2 Individual Heat Transfer Coefficient for Laminar Water Cooling Individual heat transfer coefficients for laminar water bar sprays were determined based on an initial value of 10kW\/m 2 ,C, approximately the same as that calculated by Hatta et al.[5] for an average contact radius. This value, along with the Colds and Sel-lars[2] water curtain heat transfer coefficient of 17kW\/m2*C, and the Kokada et al.[7] film boiling heat transfer calculation from Eq.2, was input to the thermal profile model. The temperature distribution through the strip was predicted for a 0.05 weight percent carbon steel of 3.89mm gauge and with a 720*C target coiling temperature. The results are presented in Figure 34. The 10kW\/m2*C heat transfer coefficient resulted in a predicted down-coiler position surface temperature of 741\u00b0C. The same model, under similar conditions, with a 20kW\/m2*C laminar water bar heat transfer coefficient, yields a prediction of 730*C while a 5kW\/m 2 \u00b0C laminar water bar heat transfer coefficient results in a prediction of 746'C as compared to the pyrometer reading of 718*C\u00b112\"C. The large changes in the individual laminar water bar heat transfer coefficient result in temperature prediction changes that are less than the deviation of the pyrometer reading. Under these conditions an accurate individual heat transfer coefficient value cannot be calculated. 62 RESULTS & DISCUSSION 5.2.1 5.2 Phase Transformation 5.2.1 Material Of the three steels intended for use in this study, two were not amenable to dilato-metric characterization with the current machine configuration. To date, the kinetics of the isothermal and continuous cooling austenite decomposition to ferrite and ferrite plus pearlite have been characterized only for the plain-carbon grade. Thus, all the model phase transformation calculations pertain to the 0.34 carbon steel. The in-plant cooling conditions available for this steel are listed in Table LTf. Modifications are currently being made to incorporate the diametral dilation mea-suring capability on a Gleeble high temperature testing machine. The isothermal and continuous cooling decomposition kinetics of the low carbon steel will be measured by other workers when these modifications have been completed. The 0.07 carbon with niobium grade presents a problem. Le Bon et al.[26], in a study of the recrystallization of niobium and plain carbon steels present data showing a 50 percent static recrystallization time for a niobium-free grade of approximately 1 sec-ond with 100 percent recrystallization by 2 seconds at a temperature of 900\" C. In the temperature range of interest for run-out table cooling, 700*C to 900*C, a niobium bearing HSLA steel can have a 50 percent recrystallization time two orders of magni-tude higher, or approximately 100 seconds. The strip velocity on the run-out table is such that the total transit time is approximately 20 seconds. Thus, the 0.07 carbon with niobium grade is entering the down-coiler in an unrecrystallized condition. The present 63 RESULTS & DISCUSSION 5.2.2 dilatometer setup cannot duplicate this unrecrystallized condition and therefore cannot simulate the appropriate run-out table conditions for the HSLA steel. With the incorpo-ration of the dilatometer into the Gleeble system it is hoped that this problem can be resolved. 5.2.2 Isothermal Cooling Tests Figure 35 shows isothermal dilatometer results for a 673\u00b0C test The horizontal line indicates the fraction of ferrite formed, as deterrnined from the extrapolated lines of the phase diagram {Kirkaldy et al.[30}. The fraction of the equilibrium amount of each phase formed is obtained by dividing the measured dilation by the maximum(equi-librium) dilation for each product phase. The results are shown in Figure 36 for both the ferrite and pearlite transformations. Figure 37 shows the ferrite data for a typical ln ln ( 1 \/ (1-FX)) as a function of ln( t) plot for an isothermal temperature test of 673*C. The slope of a best fit line is the Avrami coefficient value, n, while the inter-cept at ln(t) = 0 is the natural logarithm of the Avrami coefficient, b. Figure 37 is for the austenite to ferrite transformation at 673*C and is typical5 of the isothermal transformation data for the formation of ferrite in plain-carbon steels. The ln(b) values for the ferrite and pearlite transformation, for a range of isothermal temperatures are plotted in Figures 38 and 39 respectively. Figures 40 and 41, show the Avrami time exponent, n, for the ferrite and pearlite transformations respectively. 5 See Hawbolt et al.[16] 64 RESULTS & DISCUSSION 5.2.3 The additivity concept requires, n, to be a constant, independent of temperature. Assuming that, n, is a constant and that the scatter in the test data is due to experimen-tal variation, an average, n, value for ferrite, n f, and pearlite, fip, is shown in Figures 40 and 41 respectively. Using the average values, n f and fip, new ln(bf) and ln(bp) values were calculated for each isothermal test Figure 42 shows the temperature dependent values of ln(bf) for fif equal to 1.25. Figure 43 shows the calculated values for ln(bp) with fip equal to 1.14. The best fit equations for this new data are shown in the respec-tive figures and will be used in the model to characterize the Avrami coefficient, b. The Avrami time exponent, n, and the ln b parameter for the ferrite and pearlite transformations are in good agreement with data reported for other steel grades by Campbell[27]. Figure 44 shows good agreement for the n f and fip values with various grades of steel, while the ln(bf), and ln(bp) values in Figures 45 and 46 indicate that a linear fit best describes the parameters. 5.2.3 Continuous Cooling Tests Figure 47 shows the dilation-time and temperature-time data for a continuous cooling dilatometer trial, with a cooling rate of 27\u00b0C\/s. The AD\/AT values produced from this data are shown as a function of time in Figure 48. The temperature-time data in Figure 47 gives a transformation start temperature of 704\"C for the transformation start time deterrnined in the AD\/AT plot, Figure 48. The undercooling, the difference between the continuous cooling rate transformation start temperature and the equilib-rium value of 785*C; is listed in Table l Y and shown in Figure 49 as a function of the 65 RESULTS & DISCUSSION 5.2.3 cooling rate. 5.2.3.1 Metallography Figure 50 shows the average fraction ferrite as a function of cooling rate for the sectioned and etched continuous cooling dilatometer samples; the linear best fit line is also shown. The best fit equation for fraction ferrite was used in the model to calculate the total fraction ferrite formed as a function of the cooling rate, which was input as an initial model parameter. The fraction ferrite was determined at five random locations on the cross section of each tubular sample using the quantitative analysis capability of the Wild-Leitz image analyzer. A typical microstructure is shown in Figure 51. The wide scatter is due to the inhomogenous nature of the samples which exhibit a micro-structural banding, as shown in Figure 52. This is attributed to segregation associated with solidification during continuous casting. It is apparent that rough rolling in the plant and a full furnace anneal6 during dilatometer sample preparation was insufficient to produce a homogenous product An overall average austenite grain size of 29 um \u00b18 urn was calculated from the individual values listed in Table IV. Counting the grains was very difficult as only in the high cooling rate samples were the grains easily recognizable as large areas of pear-lite outlined by a small fraction ferrite. At slow cooling rates it was difficult to deter-mine if each pearlite cluster represented one prior austenite grain or if multiple clusters 6 30 minutes at a temperature 50'C above the A3 temperature followed by a fur-nace cool 66 RESULTS & DISCUSSION 5.2.3 were the result of a single austenite grain. For this reason while the local deviation from the average ( 18um to 37um) is greater than expected for the one austenitizing condition experienced by all of the samples, an average prior austenite grain size value was calculated. A sample of down-coiler material with the same nominal composition ( compo-sition listed in Table V ) as the transfer bar samples used in the dilatometer tests was polished and etched for Wild-Leitz image analysis of percentage ferrite and prior austenite grain size, with the results listed in Table V and the structure shown in Figure 53. However, the chemical analysis does indicate that the down-coiler sample contains more C and Mn then does the transfer bar sample. This would encourage more pearlite formation for a given cooling rate, consistent with the microstructure observed in Fig-ure 53. The percentage ferrite value of .16 \u00b1.03 compares with the continuous cooling test samples for a cooling rate of 90*C\/s which is higher than that experienced by the strip. The prior austenite grain size of 35 um \u00b19 um is in good agreement with the continuous cooling sample value of 29 um \u00b18 um and supports the use of transfer bar samples to characterize the conditions in a down-coiler sample for these experiments. 5.2.3.2 Coil ing Temperature Samples from the 720*C and the 630*C target coiling temperature, 0.05 carbon continuous cooling dilatometer tests were examined to determine if the coiling tempera-ture had an observable effect on the grain size. The four samples were polished to 5 67 RESULTS & DISCUSSION 5.2.3 (im diamond and etched for 20 seconds with 2 percent Nital. The results are listed in Table VI and do not show an overall effect on grain size attributable to coiling temper-ature. 68 RESULTS & DISCUSSION 5.2.4 5.2.4 Model Phase Transformation Calculations Using the industrial data, listed in Table HI, obtained from Devadas[28] for the 0.34 weight percent carbon grade, the austenite decomposition kinetics from section 5.2.3, the phase transformation model, and the calculated overall heat transfer coeffi-cient from section 5.1.2, a series of thermal profile predictions were made. Run one, shown in Figure 54, predicted a down-coiler position, (DC), strip top surface temperature of 706\"C at the run-out table location that has a pyrometer reading of 686*C \u00b18*C. Figures 55 and 56, for runs two and three respectively, result in similar agreement with predictions of 699'C and 700\"C versus pyrometer readings of 680\"C \u00b17\u00b0C and 676'C \u00b111 *C respectively. The percentage ferrite and transformation start temperature for these model predictions were calculated for a cooling rate of 7\"C\/s, which is the strip surface cooling rate prior to the transformation of the strip. From Figures 54, 55, and 56 it was seen that the strip undergoes two basic cooling regimes on the run-out table, one under the laminar water bar cooling banks with an average cooling rate of 45\"C\/s, and a second of air cooling on the balance of the run-out table at 7\"C\/s. Figures 57, 58, and 59 show the results of repeating the cooling conditions used in Figures 54, 55, and 56 for a cooling rate of 45*C\/s instead of 7*C\/s. In Figure 57 the model prediction of 688*C is very close to the pyrometer temperature of 686*C \u00b18*C. The value for run two, shown in Figure 58, of 686*C predicted versus 680'C \u00b1 7 ' C pyrometer reading, and the run three values shown in Figure 59 of a 676*C model prediction for a pyrometer reading of 676'C \u00b112*C, show similar agreement. The 69 RESULTS & DISCUSSION 5.2.4 phase transformation model results are also listed in Table VLT. The model incorporating phase transformation was also run with the industrial cooling conditions from Table IV, those used for Figures 54 to 59, but, using the litera-ture derived heat transfer coefficients rather than the overall heat transfer coefficient value of 1 kW\/m 2 'C. Using the Hatta et al.[6] laminar water bar heat transfer coefficient, the Kokada et al.[7] Film boiling heat transfer coefficient, the Colas and Sellars[2] water curtain values, and the 0.34 phase transformation kinetics as input to the phase transformation model, a thermal profile prediction was made. The industrial cooling conditions as presented in Table DI, with a cooling rate input of 26*C\/s, which is the average cooling rate of the combined 45*C\/s and 7*C\/s sections were used. For run one cooling conditions the model predicted a top surface temperature for the down-coiler position of the run-out table of 711'C and the pyrometer reading is 686*C \u00b18*C, these values are shown in Figure 60. The model predicted temperature for run two cooling conditions was 705\"C with a pyrometer reading of 680*C \u00b17*C, as shown in Figure 61. For the run three cooling conditions the model top surface predicted tem-perature is 709*C with an equivalent pyrometer location value of 676*C \u00b112*C. The results are listed in Table V H with the other phase transformation model results, all of which show reasonable agreement with the pyrometer values. 70 RESULTS & DISCUSSION 5.3.1 5.3 Mathematical Model 5.3.1 Sensitivity The time step for the model is the step size along the direction of strip travel divided by the strip velocity. A 10 mm step size, which results in a time step of 1.3 to 2.0 milliseconds, was chosen based on the size of the laminar water bar, as previously stated in section 4.1.2. The thermal profile model was run with various numbers of nodes through the thickness of the strip and the results are plotted in Figure 63. The predicted final temperature value increases as the number of nodes through the thick-ness of the strip increases or decreases towards the value of 200 nodes, so this value was used in the model which results in an average node thickness of 15 urn to 20 urn. The step size was varied from 10 mm to determine the effect on model predictions. Using 200 nodes through thickness, the results are shown in Figure 64 and it can be seen that varying the step size from 5mm to 40mm results in less than 10\"C of varia-tion in the model temperature prediction. The industrial data from Table Dl shows that the strip velocity deviation is \u00b10.3 m\/s. Using the overall model incorporating phase transformation, cooling conditions from run one Table Dl, with a 45 'C\/s cooling rate input; the model was run three times with the only variable being the strip velocity, with values of 5.54 m\/s, 5.24 m\/s, and 4.94 m\/s. Figure 65 shows that the \u00b10.3 m\/s variation in strip velocity results in a \u00b1 H \" C deviation in the predicted temperature, showing that the model is sensitive to the strip velocity. 71 RESULTS & DISCUSSION 5.3.2 5.3.2 Validation The overall model predicted temperatures for the down-coiler position on the run-out table gave reasonable agreement with the pyrometer readings as shown in Fig-ures 54 to 62, and listed in Table VLL To provide some external validation of the overall model incorporating phase transformation, the cooling conditions experienced by the 0.34 carbon steel in run one Table DI, were duplicated on a Gleeble high temperature testing machine, recently installed in the Metals and Materials Engineering department. The test conditions, shown in Figure 66, consisted of heating from room temperature to 800*C at 400*C\/s, heating at 65*C\/s from 800\u00b0C to 930*C, holding at 930\u00b0C for 2 seconds, cooling to 900\"C in 3 seconds (10\u00b0C\/s), cooling to 770*C in 3 seconds (43\"C\/s), and then cooling to room temperature at 7*C\/s. The results of this simulation are shown in Figure 67 with some model predicted temperature values included to show the accuracy of the duplication of the model predicted cooling rates. Examination of the test data showed that transformation started at approximately 732*C, which is approximately the transfor-mation start temperature for a cooling rate of 7\u00b0C\/s. The simulation test sample was polished and etched in 5 percent Picral and is shown in the photomicrograph, Figure 68, with the percentage ferrite of 0.39 and a prior austenite grain size of 18 um \u00b1 6 |im in reasonable agreement with the transfer bar and down-coiler samples listed in Table rv. 72 RESULTS & DISCUSSION 5.4 5.4 Tables and Figures 0.054 Carbon 720\"C target coiling temperature Strip Velocity 359.6 m\/min Strip Gauge 3.89 mm Strip Width 1.056 m Pyrometer F E X T ROT DC PI P2 P3 P4 Tempera ture(C) 910 770 856 856 759 786 730 Deviation \u00b1 6 6 12 17 11 5 11 0.054 Carbon 630*C target coiling temperature 3.89 mm Strip Width 1.052 m Pyrometer F E X T ROT DC PI P2 P3 P4 Tempera ture(C) 882 671 620 825 700 680 639 Deviation \u00b1 26 49 14 14 1 9 12 0.054 Carbon 550*C target coiling temperature Strip Velocity 375.3 m\/min Strip Gauge 3.89 mm Strip Width 1.052 m Pyrometer F E X T ROT D C PI P2 P3 P4 Tempera ture(C) 893 662 559 802 700 660 625 Deviation \u00b1 7 10 16 8 0 10 0 0.07 Carbon w\/niobium 550\u00b0C target coiling temperature Strip Velocity 377.9 m\/min 3.89 mm Strip Width 1.053 m Pyrometer F E X T ROT D C PI P2 P3 P4 Temperature(C) 895 637 539 847 700 645 625 Deviation \u00b1 74 81 36 28 0 12 0 Table Ha Plant Conditions for Four Runs 73 RESULTS & DISCUSSION 5.4 0.07 Carbon w\/niobium 720*C target coiling temperature Strip Velocity 329.9 nVmin Strip Gauge 3.89 mm Strip Width 1.053 m Pyrometer F E X T ROT D C PI P2 P3 P4 Temperature(C) Deviation \u00b1 917 36 757 5 710 15 863 16 716 7 773 9 716 13 0.07 Carbon w\/niobium 630\"C target coiling temperature Strip Velocity 348.0 m\/min Strip Gauge 3.89 mm Strip Width 1.053 m Pyrometer F E X T ROT DC PI P2 P3 Temperature(C) Deviation \u00b1 921 29 721 10 629 10 819 20 700 0 704 25 P4 637 9 0.07 Carbon w\/niobium 720*C target coiling temperature Strip Velocity 328.1 m\/min Strip Gauge 3.89 mm Strip Width 1.056 m Pyrometer F E X T ROT DC PI P2 P3 P4 Temperature(C) 924 752 711 865 704 767 723 Deviation \u00b1 12 7 26 21 7 10 23 0.07 Carbon w\/niobium 550*C target coiling temperature Strip Gauge 2.62 mm Strip Width 1.049 m Pyrometer F E X T ROT D C PI P2 P3 P4 Tempera ture(C) 922 685 564 788 700 681 626 Deviation \u00b1 8 11 32 40 0 24 3 Table LTb Plant Conditions for Four Runs 74 RESULTS & DISCUSSION 5.4 0.07 Carbon w\/niobium 720\u00b0C target coiling temperature Strip Velocity 414.3 m\/min Strip Gauge 2.62 mm Strip Width 1.050 m Pyrometer F E X T ROT DC PI P2 P3 P4 Temperature(C) Deviation \u00b1 917 7 753 4 714 8 831 18 711 7 769 6 724 17 0.054 Carta Strip Veloc Strip Gauge Strip Width an 630\u00b0C target c ity 45' j 2. 1.1 oiling temperature 3.8 rn\/min 62 mm 353 m Pyrometer F E X T ROT D C PI P2 P3 P4 Temperature(C) Deviation \u00b1 895 7 717 8 629 14 835 4 701 3 732 15 638 11 0.054 Carbon 550*C target coiling temperature Strip Velocity 457.9 m\/min Strip Width ...1.053 m Pyrometer F E X T ROT D C PI P2 P3 P4 Tempera ture(C) 897 667 564 838 707 662 625 Deviation \u00b1 6 8 12 5 11 9 0 0.054 Carbon 720\u00b0C target coiling temperature Strip Velocity 458.2 rn\/min Strip Gauge 2.62 mm Strip Width 1.050 m Pyrometer F E X T ROT DC PI P2 P3 P4 Tempera ture(C) Deviation \u00b1 897 6 779 4 718 8 864 15 784 6 795 11 735 9 Table lie Plant Conditions for Four Runs 75 RESULTS & DISCUSSION 5.4 Plant cooling conditions, 0.34 weight percent carbon. Run Number Number of Sprays On F E X T Temperature C C ) D C Temperature C C ) Strip Velocity (m\/s) 1 10 \u00b12 914 \u00b1 9 686 \u00b18 5.24 \u00b1.3 2 9 \u00b12 909 \u00b18 680 \u00b17 4.89 \u00b1.3 3 11 \u00b13 924 \u00b1 10 676 \u00b111 5.20 \u00b1 3 Table m Industrial Plant Cooling Conditions. 76 RESULTS & DISCUSSION 5.4 Metallographic Data Sample Fraction Ferrite Undercooling C C ) Gamma Grain Size (um) C C T @ 5'Os 0.53 42 19 \u00b1 2 * C C T @ 10 \u00b0C\/s 0.51 63 28 \u00b1 7 * C C T @ 15 'as 0.56 66 22 \u00b1 3 * C C T @ 27 'as 0.65 81 18 \u00b1 2 * C C T @ 40 \u00b0c\/s 0.15 90 C C T @ 51 *C\/s 0.41 95 37 \u00b1 7 * C C T @ 55 \u00b0C\/s 0.38 28 \u00b1 3 * C C T @ 65 'as 0.40 101 32 \u00b1 6* C C T @ 78 'as 0.07 114 37 \u00b1 5 * C C T @ 103 *C\/s 0.13 118 36 \u00b1 4 * C C T average 29 \u00b1 9 * 0.34 C down-coiler 0.16 35 \u00b1 9 0.34 C Gleeble 0.39 18 \u00b1 3 \u2022....variation due to difficulties in counting technique as all C C T samples were the result of the same austenitizing conditions of 3 minutes at 850*C, 1 minute at 820*C, followed by continuous cooling rate. An overall average prior austenite grain diameter of 29 um \u00b1 9 um will be used. Table IV Metallographic data for the 0.34 carbon samples, for the down-coiler sam-ple, the continuous cooling test samples, and the Gleeble simulation sample; with tabu-lated values for, cooling rate, fraction ferrite, undercooling, and average austenite grain size. 77 RESULTS & DISCUSSION 5.4 Composition Transfer bar sample Down-coiler sample C 0.343 0.370 Mn 0.700 0.840 P 0.008 0.018 S 0.009 0.010 Si 0.009 0.150 Cu 0.021 <0.02 Ni 0.006 <0.08 Cr 0.023 < 0.025 Mo 0.003 0.000 V 0.000 0.000 Nb 0.000 0.000 A l 0.043 0.023 Table V Comparison of the composition of the down-coiler and transfer bar medium carbon test samples. 78 RESULTS & DISCUSSION 5.4 Grain size as a function of coiling temperature for a low carbon steel 0.054 % carbon, 3.89 mm gauge, 720 \u00b0C 17 \u00b1 2 um 0.054 % carbon, 3.89 mm gauge, 630 ' C 14 \u00b1 1 um 0.054 % carbon, 2.69 mm gauge, 720 \"C 16 \u00b1 2 um 0.054 % carbon, 2.69 mm gauge, 630 *C 16 \u00b1 2 um Table VI Grain size versus coiling temperature for a 0.054 weight percent carbon grade steel. 79 RESULTS & DISCUSSION 5.4 Model temperature predictions for various cooling conditions Model Temperature Values Pyrometer Reading Cooling rate = Tsan Figure 54 706*C 686'C \u00b1 8 ' C 7'C\/s = 732'C Figure 55 699'C 680'C \u00b1 7 ' C 7'C\/s = 732'C Figure 56 700'C 676'C \u00b112 'C 7'C\/s = 732'C Figure 57 688'C 686'C \u00b1 8 ' C 45'C\/s = 688'C Figure 58 685'C 680'C \u00b17 'C 45'C\/s = 688'C Figure 59 676'C 676\u00b0C \u00b112*C 45'C\/s = 688'C Figure 60 711'C 686*C \u00b18*C 26'C\/s = 710'C Figure 61 705'C 680'C \u00b17*C 26\u00b0as = 710'C Figure 62 709'C 676'C \u00b112 'C 26'C\/s = 710'C Table VII.\u201e..Tabulated model predictions, for low (7*C\/s) and high (45*C\/s) cooling rates, and for the literature heat transfer coefficients at an average cooling rate, (26'C\/s). 80 RESULTS & DISCUSSION 5.4 O C O C D t ^ O C O ^ T f C M O C O t O ^ C M O ^ Black Zone Radius ( mm) Figure 17 Black zone radius as a function of a constant steel surface temperature, per Hatta et al.[4] 81 RESULTS & DISCUSSION 5.4 Heat Transfer Coefficient Kw\/m C Figure 18 Hatta laminar water bar heat transfer coefficient as a function of contact radius 82 O \u00a7 i Distance from Finish Stand F4 2: c ? meters ? RESULTS & DISCUSSION o o o o o o o o o o m o m o w o m o O J C D c o o o r ^ r ^ c o c o Temperature (C) Figure 21 Thermal profile model literature heat transfer coefficients 0.05% carbon, 3.89 mm gauge, target coiling temperature 720*C. 85 RESULTS & DISCUSSION 5.4 o o o o o o o o o o m o w o w o m o O > o > c o o 5 f \u00bb * r ^ c o c o Temperature (C) Figure 22 Thermal profile model literature heat transfer coefficients, 0.07% carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720\u00b0C. 86 RESULTS & DISCUSSION 5.4 o o c n o ) c o o o r ^ r > . c o c D Temperature (C) Figure 23 Thermal profile model literature heat transfer coefficients, 0.05% carbon, 2.62 mm gauge, target coiling temperature 720'C. 87 RESULTS & DISCUSSION 5.4 o o o o o o o o o o m o m o m o u D O c o c n c o c o r \u00ab \u00bb r ^ c o c o Temperature (C) Figure 24 Thermal profile model literature heat transfer coefficients, 0.07% carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 720 *C. 88 RESULTS & DISCUSSION 5.4 o o o o o o o o o o i n o w o m o m o C O O J O O C O h - r ^ C O C O Temperature (C) Figure 25 Thermal profile model literature heat transfer coefficients, 0.05% carbon, 3.89 mm gauge, target coiling temperature 630*C. 89 RESULTS & DISCUSSION 5.4 Temperature (C) Figure 26 A sample temperature profile from the plant data. 90 RESULTS & DISCUSSION Temperature (C) Figure 27 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 720\"C 91 RESULTS & DISCUSSION Temperature (C) Figure 28 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720*C 92 RESULTS & DISCUSSION o o o o o o o o o o o o m o m o m o m o w o Temperature (C) Figure 29 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 720'C 93 RESULTS & DISCUSSION o o o o o o o o o o o o i n o m o m o L o o m o o i c o o o o o r ^ r x - c o c o m i o Temperature (C) Figure 30 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 2.62 mm gauge, target coiling temperature 720\u00b0C 94 RESULTS & DISCUSSION o o o o o o o o o o o o L n o m o m o m o L o o c o c o o o o o r ^ r - c o c o m m Temperature (C) Figure 31 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 630* C 95 RESULTS & DISCUSSION o o o o o o o o o o o o L o o m o m o m o m o c 7 ) c o a o a o r \u00ab r ^ c o c o m m Temperature (C) Figure 32 Thermal profile model, overall heat transfer coefficient calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 550\u00b0C 96 RESULTS & DISCUSSION 5.4 o o o o o o o o o o o o o o o fflcooiooooooooeoSsSNN Temperature (C) Figure 33 Thermal profile model sensitivity, Overall Heat Transfer Coefficient Calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 720\u00b0 C 97 RESULTS & DISCUSSION o o o o o o o o o o o o o o o o o o o o o o eo CM o cn oo h- co m ^ co CM t- o o> oo r*. co i n -^ r co cvj cjiO)0>oioococooocooococococor*\u00bbS-r^r\u00bb-r*\u00bb-S'r^r ,>. Temperature (C) Figure 34 Thermal profile model results for a 0.05 carbon, 3.89 mm gauge steel using an individual laminar water bar heat transfer coefficient of 10 kW\/m 2 \u00b0C and showing the effect of using 20 kW\/m2*C and 5 kW\/m 2 o C on the target coiling temperature of 720*C. 98 RESULTS & DISCUSSION 5.4 Fraction Transformed Figure 35 Isothermal dilatometer results for 673*C test 99 RESULTS & DISCUSSION 5.4 Fraction Transformed Figure 36 Fraction transformed as a function of time for constant temperature = 673#C. 100 RESULTS & DISCUSSION \u2022 \\ \u2022 \\ \u2022 \\ CO CD \\ if) CO \\ C\\J > CD \u00ab\u2014\u00ab Q CO CO II \u2022 \\ r -\u2022 \\ OJ h CNJ CO CM CO \u00a9 T T d d CVJ d in d T - IT) CM i If) c\\i In(ln(1\/(1-Fx))) Figure 37 Isothermal 673*C austenite to ferrite kinetics plotted as lnln(l\/(l-FX)) versus ln(t). 101 RESULTS & DISCUSSION 5.4 ln(b), ferrite Figure 38 ln(b) Avrami coefficient for the isothermal formation of ferrite in the 0.34 carbon steel. 102 RESULTS & DISCUSSION i i i i i i I i i i i i i i i i i r Tru3cor^coc\u00bbi-^CMcoTrir)cDr^coo>c\\i-i-CMoo o o o o o o i i i i i i I I I I I CM CM CM i i i ln(b), pearlite Figure 39 ln(b) Avrami coefficient for the isothermal formation of pearlite in the 0.34 carbon steel. 103 RESULTS & DISCUSSION 5.4 n (ferrite) Figure 40 Avrami coefficient, n f, for the austenite-to-ferrite transformation in the 0.34 % C, plain carbon steel. 104 RESULTS & DISCUSSION o -O i _ C3 O o> CO ' c i I I i r m co m CM m T -C O ^ C M ^ to T -o cn cn in oo d n (pearlite) Figure 41 Avrami coefficient, n,,, for the austenite-to-pearlite transformation in the 0.34 % C, plain carbon steel. 105 RESULTS & DISCUSSION 5.4 ln(b), ferrite Figure 42 Calculated ln(b) values for the ferrite transformation assuming ii f = 1.25, for the 0.34% carbon steel. 106 RESULTS & DISCUSSION 5.4 ln(b), pearlite Figure 43 Calculated ln(b) values for the pearlite transformation assuming ftp = 1.14, for the 0.34% carbon steel. 107 RESULTS & DISCUSSION 5.4 LOTTOOCNJT\u2014cxjo^cqr^couo CNJ C\\i C\\j C\\i C\\j T ^ T - ^ T - ^ T \u2014 i - ^ T - ^ - r ^ T - ^ - r ^ o o o o o n v a l u e Figure 44 Average Avrami coefficient V for 0.34% carbon compared to other experimental values (Campbell[27]) 108 RESULTS & DISCUSSION 5.4 +\u2022 < \u2022 x x r^ . r-. r*. r^ CM J CM I CM I C\\l ) C\\J < CD CD CD CD CD CD CD CD CD CD To GO \"55 CO GO Q rr co CM T - 3 o o o o o n . T- y- -r- T - O \u2022 + O < X \u00ab o o \u2022 + + + CO ~ I I 1 I I CM i - O i - CM CO r f i If) O 00 o CO o T t o CM o o o oo o CO o rr CM CO < h-o s o CD \" O c 3 CO ln(b), ferrite Figure 45 Comparison of the ln(b) Avrami coefficient for the austenite-ferrite transformation is several plain-carbon steels(Campbell[27]). 109 RESULTS & DISCUSSION 5.4 \u2022 9 \u00a9 < O co d CD CD \u00a9 CD CD CD CD CD CD CD CD CD wcyScyjcy) c\/5co com coo o o T -CMCOTT cor*, o o o o o o X < o + \u2022 > CM, \"55 .O Q. E CO O o oo o CO _ o o CM CM T - O T - CM CO i i co 00 ln(b), pearlite Figure 46 Comparison of the ln(b) Avrami coefficient for the austenite-ferrite transformation is several plain-carbon steels(Campbell[27]). 110 RESULTS & DISCUSSION 5.4 Dilation - Temperature Figure 47 Dilatometer-time and temperature-time data for a continuous cooling rate of 27'C\/s* 111 RESULTS & DISCUSSION 5.4 112 RESULTS & DISCUSSION 5.4 Undercooling ( C ) (785 - Tst) Figure 49 The undercooling for the austenite-to-ferrite start temperature as a function of the cooling rate. 113 RESULTS & DISCUSSION 5.4 I 1 1 1 1 1 1 i h o C O r ^ C O U O T T C O C N J - r - O o o o o o o o o Percentage Ferrite Figure 50....Fraction ferrite as a function of cooling rate, from metallographic examination. 114 RESULTS & DISCUSSION Figure 51 Continuously cooled dilatometer sample. 115 RESULTS & DISCUSSION RESULTS & DISCUSSION RESULTS & DISCUSSION Temperature (C) Figure 54 Surface thermal profile, using the overall heat transfer coefficient = 1 kW\/m 2*C, Table DT-Run #1 cooling conditions, 3.94mm gauge, and a transformation start temperature of 732'C ( dT\/dt = 7'C\/s). 118 RESULTS & DISCUSSION o o o o o o o o o o o o o o o o o o o C O r r c \\ J O C O C D r f C \\ J O O O C D T t C \\ J O C O C O T t CnO5O)O)COCOCO00 0 0 S N N N N ( D t O ( O Temperature (C) Figure 55 Surface thermal profile, using the overall heat transfer coefficient = 1 kW\/m 2*C, Table DJ-Run #2 cooling conditions, 3.94mm gauge, and a transformation start temperature of 732'C ( dT\/dt = 7'C\/s). 119 RESULTS & DISCUSSION o o o o o o o o o o o o o o o o o o o C D ^ C N J O C O C D T f C N J O C O t O T r C N J O C O C O T f O l C I C n c n c O C O O O C O C O N N N N N O l D C D Temperature (C) Figure 5 6 Surface thermal profile, using the overall heat transfer coefficient = 1 kW\/m 2*C, Table DI-Run #3 cooling conditions, 3.94mm gauge, and a transformation start temperature of 732'C ( dt\/dt = 7'C\/s). 120 RESULTS & DISCUSSION o o O O O O Q O O O O O O O O O O J O ) 0 > 0 0 0 0 0 0 0 0 0 3 N N N S N ( 0 Temperature (C) Figure 57 Surface thermal profile, using the overall heat transfer coefficient = 1 kW\/m 2*C, Table HI-Run #1 cooling conditions, 3.94mm gauge, and a transformation start temperature of 688\u00b0C ( dt\/dt = 45*C\/s). 121 RESULTS & DISCUSSION o o o o o o o o o o o o o o o o t ( v J O 0 0 \u00ab 0 t C v | O 0 0 ( f i t C M O 0 0 O ) O ) O ) C O C O C O 0 0 0 0 N N N N N C O Temperature (C) Figure 58 Surface thermal profile, using the overall heat transfer coefficient = 1 kW\/m 2 \u00b0C, Table Hi-Run #2 cooling conditions, 3.94mm gauge, and a transformation start temperature of 688\u00b0C ( dt\/dt = 45'C\/s). 122 meters RESULTS & DISCUSSION o o o o o o o o o o o o o o o o o o o c o r f o j o a j c D T r c N i o c o c o T r c M O c o c o r r cooJOTOioocooococo i^r^ .r^r^r^cococD Temperature (C) Figure 60 Surface thermal profile, using the literature heat transfer coefficients , Table DI-Run #1 cooling conditions, 3.94mm gauge, and a transformation start temperature of 710*C ( dt\/dt = 26*C\/s). 124 RESULTS & DISCUSSION Temperature (C) Figure 61 Surface thermal profile, using the literature heat transfer coefficients , Table LTI-Run #2 cooling conditions, 3.94mm gauge, and a transformation start temperature of 710\"C ( dt\/dt = 26'C\/s). 125 RESULTS & DISCUSSION Temperature (C) Figure 62 Surface thermal profile, using the literature heat transfer coefficients , Table HI-Run #3 cooling conditions, 3.94mm gauge, and a transformation start temperature of 710\u00b0C ( dt\/dt = 26'C\/s). 126 RESULTS & DISCUSSION 5.4 to in in in m U D m m m <<* -