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The behaviour of copper on gallium arsenide Macquistan, David Alexander 1989

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I  C H A R A C T E R I Z A T I O N OF T H E C O O L I N G AND T R A N S F O R M A T I O N OF STEELS ON A R U N - O U T 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 . S c , The University of British Columbia,  1986  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES METALS AND MATERIALS  ENGINEERING  We accept this thesis as confonning to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A August 1988 ©Craig Allen McCulloch, 1988  In  presenting  degree at the  this  thesis  in partial  fulfilment  University  of British  Columbia,  of  the  I agree  requirements  for  an  advanced  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  department  or  understood  by  his  or  her  representatives.  It  is  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of  M  e  t  a  l  s  a  n  d  Materials  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  Enqineering  head of copying my  my or  written  ABSTRACT 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 k W / m C . Isothermal 2,  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°C 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  L I T E R A T U R E 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 2.3 Review of Related Models  10 ,  2.4 Figures  13 16  3.0 SCOPE A N D OBJECTTVES  17  3.1 Scope  17  3.2 Objectives  18  4.0 P R O C E D U R E  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 4.4 Tables and Figures  37 39  5.0 R E S U L T S 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 5.2 Phase transformation  62 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 5.3 Mathematical Model  69 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  ,  8.3 Hatta et al. Thermal Boundary Layer Calculations  vii  147 147  LIST O F T A B L E S 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  76  Table IV  Industrial plant cooling conditions 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 Table V  77  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 O F 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  Figure 3  Schematic of the S T E L C O Lake Erie Works Hot Strip Mill  Run-out Table Figure 4  41  Specific Heat as a Function of Temperature for a 0.34 % carbon  steel, BISRA, w/o phase transformation Figure 5  42  Thermal Conductivity as a Function of Temperature for a 0.06 %  plain carbon steel, BISRA Figure 6  43  Thermal Conductivity as a Function of Temperature for a 0.08 %  plain carbon steel, BISRA Figure 7  44  Thermal Conductivity as a Function of Temperature for a 0.23 %  plain carbon steel, BISRA Figure 8  40  45  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  50  Experimental verification of T A C 3 and T A C 1  ix  Figure 13  A typical dilation versus time plot for an isothermal dilatometer  test Figure 14  51 A typical dilation and temperature versus time plot showing  transformation start and finish times Figure 15  52  Experimental dilation and thermal dilation plots, used with  divergence method (Campbell[27]) for calculation of transformation start Figure 16  Flow sheet for the iterative solution of the Avrami fraction  transformed equation as a function of temperature Figure 17  81 Hatta laminar water bar heat transfer coefficient as a function of  contact radius Figure 19  82  Thermal profile model sensitivity to changes in the water  temperature for the Kokada film boiling heat transfer coefficient Figure 20  84  Thermal profile model literature heat transfer coefficients 0.05%  carbon, 3.89 mm gauge, target coiling temperature 720*C Figure 22  85  Thermal profile model literature heat transfer coefficients, 0.05%  carbon, 2.62 mm gauge, target coiling temperature 720*C Figure 23  83  Thermal profile model sensitivity to changes in the support roller  conduction cooling Figure 21  54  Black zone radius as a function of a constant steel surface  temperature Figure 18  53  86  Thermal profile model literature heat transfer coefficients, 0.07%  carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720° C  x  87  Figure 24  Thermal profile model literature heat transfer coefficients, 0.07%  carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 7 2 0 ° C Figure 25  88  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 Figure 28  91  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 Figure 29  92  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 Figure 30  93  Thermal profile model, overall heat transfer coefficient calculated  from plant pyrometer measurements, 0.05% carbon, 2.62 mm gauge, target coiling temperature 720*C Figure 31  94  Thermal profile model, overall heat transfer coefficient calculated  from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 620"C Figure 32  95  Thermal profile model, overall heat transfer coefficient calculated  from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target  xi  coiling temperature 540°C Figure 33  96  Thermal profile model sensitivity, overall heat transfer coefficient  calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling temperature 720°C Figure 34  97  Thermal profile model sensitivity, individual laminar water bar  heat transfer coefficient, 10 kW/m C with 20 kW/m C and 5 kW/m C 2,  2,  2o  deviations,target coiling temperature 720*C Figure 35  98  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 Figure 38  101  ln(b) Avrami coefficient for the isothermal formation of ferrite in  the 0.34 carbon steel Figure 39  102  ln(b) Avrami coefficient for the isothermal formation of pearlite in  the 0.34 carbon steel Figure 40  103  Avrami coefficient, n , for the austenite-to-ferrite transformation in f  the 0.34 % C, plain carbon steel Figure 41  104  Avrami coefficient, n,,, for the austenite-to-pearlite transformation  in the 0.34 % C, plain carbon steel Figure 42  105  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 Figure 44  107  Average Avrami coefficient V for 0.34% carbon compared to  other experimental values (Campbell[27]) Figure 45  108  Comparison of the ln(b) Avrami coefficient for the  austenite-ferrite transformation in several plain-carbon steels(Campbell[27]) Figure 46  Comparison of the ln(b) Avrami coefficient for the  austenite-pearlite transformation in several plain-carbon steels(Campbell[27])... Figure 47  109  110  Temperature as a function of time for a continuous cooling rate of  27'C/s  Ill  Figure 48  Thermal and Experimental dilatometer values as a function of  time for a cooling rate of 27*C/s Figure 49  112  The undercooling for the austenite-to-ferrite start temperature as a  function of cooUng rate Figure 50  113  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 k W / m C , Table rn run one cooling conditions, and a transformation start 2,  temperature of 732'C (dT/dt = 7°C/s)  118  xiii  Figure 55  Surface thermal profile, using the overall heat transfer coefficient,  1 kW/m C, Table in run two cooling conditions, and a transformation start 2o  temperature of 732'C (dT/dt = TCJs) Figure 56  119  Surface thermal profile, using the overall heat transfer coefficient,  1 kW/m °C, Table LTI run three cooling conditions, and a transformation start 2  temperature of 732'C (dT/dt = 7'C/s) Figure 57  120  Surface thermal profile, using the overall heat transfer coefficient,  1 kW/m *C, Table DI run one cooling conditions, and a transformation start 2  temperature of 688'C (dT/dt = 45'C/s) Figure 58  121  Surface thermal profile, using the overall heat transfer coefficient,  1 kW/m *C, Table in run two cooling conditions, and a transformation start 2  temperature of 688'C (dT/dt = 45'C/s) Figure 59  122  Surface thermal profile, using the overall heat transfer coefficient,  1 kW/m "C, Table DI run three cooling conditions, and a transformation start 2  temperature of 688'C (dT/dt = 45'C/s) Figure 60  123  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) Figure 61  124  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) Figure 63  126  Effect on predicted center line temperature of changes in the  number of nodes through thickness Figure 64  127  Effect on predicted center line temperature of changes in the step  size, where the step size equals the strip velocity times the time increment Figure 65  Effect of the ±0.3m/s deviation of the strip velocity on the  predicted temperature profile Figure 66  129  Industrial cooling profile simulated on the Gleeble high  temperature testing machine Figure 67  130  Industrial cooling conditions, simulated on a Gleeble high  temperature testing machine Figure 68  128  131  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 experience. 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 variables 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 experimental(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 thermomechanical 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 transfer 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  L r T E R A T U R E REVIEW  2  2.1  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  L I T E R A T U R E 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 C ± 2 0 0 W / m C for the bank of water curtain cooling, 1300 W/m *C ± 2 0 0 2,  2,  2  W / m C for the bank of laminar water bar cooling, and 900 W / m C ± 1 5 0 W / m C for the 2o  2o  2,  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°C/s up to 200°C/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  L I T E R A T U R E 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°C 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 k W / m C for an individual water curtain; they 2,  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/m "C for the region outside the contact zone which compares 2  well with the Farber and Scorah[3] value of between 150 W / m C and 170 W / m C . 2,  2.1.2  2o  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  L I T E R A T U R E 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 "C was adopted for natural convection in air in the Hatta model. 2  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  and  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  Pr  is the Prandd number  The heat transfer coefficient for the film boiling region is,  6  L I T E R A T U R E REVIEW  a™ = 200*  where  and  2420-21.70V)  2.1.2  ...2  ,•8  T  w  is the average water temperature, in Kelvin  T  s  is the steel temperature, in Kelvin  T  S A X  is the saturation temperature of the water, in Kelvin.  The water saturation temperature under one atmosphere pressure is 100 °C. The water temperature at which the transition from water contact cooling to film boiling cooling occurs, Tom-, is described by the equation, -(7^-1150)  cm ~ The value of  ...3  8  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 °C. 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, 100 C+68°C o  2  = 84°C  which is used as T for Eq.2. w  7  ...4  L I T E R A T U R E REVIEW  2.1.2  A horizontal water velocity is needed to derive a water film thickness as well as for 1  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 zone , a boundary layer phenomena, is the thickness of 2  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. E q . l is insensitive to the water flow rate but relatively sensitive to the area of contact under the laminar water bar . 3  1  Equation 1 in the appendix  2  Water film thickness, described in the appendix  3  From E q . A l in the appendix  8  L I T E R A T U R E 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 rolls and continuously 4  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 C can 2o  be calculated.  4  On the inside of the curve above the slab  9  L I T E R A T U R E REVIEW  2.2  2.2  Phase Transformation  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 °C 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")  •••  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 Scheil  5  in 1935. A n additive system is one in which the transformation is only a function of the  5  For incubation not for phase transformation as such  10  5  L I T E R A T U R E 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 > 0.28^ 20  ...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  L I T E R A T U R E 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  L I T E R A T U R E 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  L I T E R A T U R E 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  L I T E R A T U R E 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±, B , and Q used in, k  dt  where  'az^  = A (Z + e)%> k  k  is the rate of transformation for phase k,  Jk  4  is the fraction of phase k transformed, is the fraction of y phase transformed,  e  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  L I T E R A T U R E REVIEW  2.4 Figures  >«r>+  c o  "9 v $ v O v g v O v.e( w  o  5  co CM o a d ci d ci a  O) CD  -Mac  5 -K> O + +  <  •  -max •MIX  1  (fl m  1  I  -  CO  CM  T-  i-  1  1  1  O) oo s  1  1—  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  2.4  SCOPE A N D OBJECTIVES  3  3.1  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 prediction for the coiled steel from dilatometer data and 1  the integrated model.  1  ferrite-pearlite ratios  18  PROCEDURE  4  4.1  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  'la7 a7 37j= '3T +  +  P C  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. q is calculated by taking the fraction transformed for a given time step (which is f  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 large , a zero temperature gradient is 1  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  ,  ^fdr)  ,1ft?  4.1.1  ...10  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. x>0,  Boundary Conditions,  -k^  z = 0,  z = d  = h(x)(T-T ) A  "  A  l  Initial Conditions  x=0,0£z£d ...12  T = Tj 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 independent , all three were included as variables for each grade of 2  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 0.34 weight percent carbon steel. 3  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 ±0.0105 between 700 °C 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 E q . 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 E q . 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 F O R T R A N and run on the University of British Columbia Amdahl V8 mainframe computer with approximately 250 seconds C P U 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 C P U 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 ± 0 . 3 kW/m K is reported for an overall heat transfer coefficient 2  for a water curtain cooling bank and a value of 1.3 ± 0 . 2 5 kW/m K is given for an overall 2  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] E q . l . a,•WB= 0.063* -  ,8)  *Re**Pr  ... i  While the film boiling heat transfer coefficient is calculated using the Kokada et al.[7] relationship Eq.2. a™ = 200*  2420-21.7(7V)  (Ts ~ TSAT)*  28  ...2  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 -1150 =—  ...3  5  T 1  _g  cm  The Reynolds number, Prandd number, and k are temperature dependent and calculated internally in the model. T , the temperature of the water in the film boiling w  section, is greater than or equal to 68"C by the definition of Tcxst- At one atmosphere pressure T can be assumed to have a maximum value of 100'C. Therefore, T must w  w  always have a value between 68*C and 100 C. A n 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°C; 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 °C. A water curtain cooling heat transfer value of 17 kW/m °C has been reported by 2  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 °C. This value is less than 2  the Kokada et al.[7] value for a water temperature of 100 °C, 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 °C for a steel temperature of 2  1000 ' C which increases to 990 W / m C for a steel temperature of 500 ' C . The water 2o  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 roller , a heat flow of 75 kW/m with an average roll/slab 4  temperature difference of 900 *C was given resulting in an 83 W / m ' C heat transfer 2  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 F E X T , ROT, and D C . 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, F E X T , ROT, and D C which are shown schematically in Figure 3. The plant pyrometers were IRCON 2000 series with a 700°C to 1100'C range for the F E X T pyrometer and a 500°C 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  Dilatometer  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°C 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°C 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. It is recognized that these 5  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 t » as shown in Figure 13. AV  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°C is 0.45, ( A D / A D = X  5  30 minutes at TAc3 + 50°C, 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: ...13  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  36  PROCEDURE  4.3.4  temperature values. Thus at some time, t,  (AD/AT), = | - r „ , r „ r , y 2  1 +  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, t , the time that would be required v  to produce Fx(k-l) at temperature T(k). Eq.5 is rearranged to deterrnine the virtual time,  , [  .1  v -  . . . 15  . ~  V  )  b  The time step, dt, is added to t and a new fraction transformed, Fx(k), is calculated v  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 ' is calculated and in a similar v  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°C, T(k)' becomes T(k)" and the process repeats until the 0.05°C 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 Tables and Figures Low carbon  Low carbon-  Medium carbon  niobium 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  Al  0.030  0.047  0.043  Table I  Composition of the three steel chemistries intended for use in this study.  39  4.4  40  PROCEDURE  o  o O c 5 o Q  8 ©-  03  CD k_  3 CO 03  c  'E  (5-V4  Q_  03  p—  o  CO Q_ DC CM CL CL  ©-fc v  ®4  ©4  CO  m  CD CO  ej  -•—»  —LU -eeoo oo olo  Figure 3  c  'c  Schematic of the S T E L C O Lake Erie Works Hot Strip Mill Run-out Table  41  4.4  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  o o  Thermal Conductivity W/mC  Figure 5  Thermal Conductivity as a Function of Temperature for a 0.06 % plain  carbon steel, BISRA  43  4.4  P R O C E D U R E 4.4  44  PROCEDURE  4.4  o o  o o CM  o o o  o CD  o o co  CO  E  o o  o o CM  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  Water Curtain  Film Boiling Laminar: Water Bar  y Roller  Film Boiling  Roller  Film Boiling Roller  Figure 10  The six types of cooling regime experienced by the steel strip  48  4.4  PROCEDURE  i  i  i  i  i  i  i  i  i  i  i  i  i  i  r  4.4  Q  Heat Transfer Coefficient  Kw/mC Figure 11  2  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  P R O C E D U R E 4.4  P R O C E D U R E 4.4  •^1  Figure 13  Dilation  A typical dilation versus time plot for an isothermal dilatometer test.  51  PROCEDURE  4.4  §  Dilation ^  Temperature ( Q  Figure 14....A typical dilation and temperature versus time plot showing transformation start and finish times.  52  P R O C E D U R E 4.4  53  PROCEDURE  T(k)  Fx(k-1)  4.4  START  t .Fx(k) v  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  5.1.1  RESULTS & DISCUSSION 5.1  Heat Transfer Coefficient 5.1.1  Literature  5.1.1.1  Laminar Water  Bar  Cooling  The Hatta et al.[5] laminar water bar heat transfer coefficient calculated using E q . 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. T o 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 temperature. Figure 17 shows the results of this model calculation for steel surface temperatures in the range from 400°C to 1100'C. For steel surface temperatures greater than 600°C the 'black zone' radius changes slowly with temperature. A n average value, 33.7 mm, was chosen for the temperature range of 700°C 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 Figure 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 k W / m C calculated for a contact radius of 33.7 mm. The thermal profile model 2o  55  RESULTS & DISCUSSION  5.1.1  combines the Colas and Sellars[2] water curtain heat transfer coefficient of 17 k W / m C , the laminar water bar heat transfer coefficient calculated with E q . l , and a 2o  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 Figure 11, to be sensitive to the cooling water temperature, T . T o assess the sensitivity w  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 . The results of this model are shown in Figure 19, it is 1  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°C, and 778*C for the respective water film temperatures, T , of w  68°C, 84°C, and 100'C, while the measured pyrometer value at that location is 718*C ±12'C.  5.1.1.3  Support Roller Contact Cooling  The Diener and Drastik[7] support roller, conduction cooling, heat-transfer coefficient is an approximation for a different physical system; therefore to evaluate the sensitivity to the 83 W/m *C value the model was employed. A ±50% change in the 2  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 °C. 2  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 E q . 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 ±12*C and 717"C ±8"C respectively. For the 0.07 carbon with niobium grade with gauges of 3.89 mm and 2.69 mm, the model predicted surface temperatures at the down-coiler are 776*C and 758°C, the results are shown in Figures 22 and 24. The pyrometer readings at the corresponding positions in-plant are 710*C ±15°C and 714*C ±8"C. In all four cases the in-plant pyrometer measured values are 40° 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 plaincarbon 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 predicted 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 temperature 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 ±14*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 carbon 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°C. Basic run-out table parameters for the trial coil runs, along with average temperature values and standard deviations are also listed in Table II. A n 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 D C 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 sample . 2  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 conditions experienced by the 0.05 carbon, 3.89 mm gauge strip, coiled at 718°C ±18*C various values for an overall laminar water bank heat transfer coefficient were evaluated. A value of l k W / m C gave the best fit with a predicted temperature of 716"C for the 2e  strip surface at the down-coiler pyrometer location. The plant pyrometer value of 7 1 8 * C ± 1 8 * C i s shown with the model predictions in Figure 27. To evaluate the effect of using the lkW/m *C overall effective heat transfer coef2  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 headers that were turned on during that particular run. Model predicted top surface temperatures have been compared with in-plant temperature 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°C for the 2.62 mm gauge. This is close agreement with the respective pyrometer values of 710°C ±15*C and 714'C ±8*C. The 0.05 carbon, 2.62mm gauge prediction of 716*C is also in close agreement with the pyrometer value of 718*C ± 8 ' C . The l k W / m ° C heat transfer coef2  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 coefficient 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°C and 550°C, in which phase transformation should have some effect. Figure 31 shows the predicted surface value at the down-coiler position of 620° C as well as the pyrometer value of 620° C ±14°C for the 630°C 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°C ±16"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 prediction is shown in Figure 33, with a ± 0 . 2 k W / m C change resulting in predicted surface 2o  temperature values of 736'C and 700°C for a respective decrease and increase of 0.2 kW/m *C in the overall heat transfer coefficient 2  Both values are within the ±18*C  deviation of the 718°C pyrometer value. For all six figures one trend emerges; the model consistently predicts a value 3  slightly less than the measured PI pyrometer temperature, slighdy greater than than the measured P2 pyrometer value, and a much higher value than the P3 and P4 measured 4  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°C to 200*C  61  RESULTS & DISCUSSION  5.1.2.2  5.1.2  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 C, approximately the same as that calculated by 2,  Hatta et al.[5] for an average contact radius. This value, along with the Colds and Sellars[2] water curtain heat transfer coefficient of 17kW/m *C, and the Kokada et al.[7] 2  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/m *C heat transfer coefficient resulted in 2  a predicted down-coiler position surface temperature of 741°C. The same model, under similar conditions, with a 20kW/m *C laminar water bar heat transfer coefficient, yields 2  a prediction of 730*C while a 5kW/m °C laminar water bar heat transfer coefficient 2  results in a prediction of 746'C as compared to the pyrometer reading of 718*C±12"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 dilatometric 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 measuring 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 second 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 H S L A steel can have a 50 percent recrystallization time two orders of magnitude 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 H S L A steel. With the incorporation 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°C 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(equilibrium) 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 intercept 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 typical of the isothermal 5  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 experimental variation, an average, n, value for ferrite, n , and pearlite, fi , is shown in Figures 40 f  p  and 41 respectively. Using the average values, n and fip, new ln(b ) and ln(b ) values f  f  p  were calculated for each isothermal test Figure 42 shows the temperature dependent values of ln(b ) for fi equal to 1.25. Figure 43 shows the calculated values for ln(b ) f  f  p  withfipequal to 1.14. The best fit equations for this new data are shown in the respective 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 and fip values with various f  grades of steel, while the ln(b ), and ln(b ) values in Figures 45 and 46 indicate that a f  p  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°C/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 equilibrium 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 microstructural 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 anneal during dilatometer sample preparation was insufficient 6  to produce a homogenous product An overall average austenite grain size of 29 um ± 8 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 pearlite outlined by a small fraction ferrite. At slow cooling rates it was difficult to determine 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 furnace 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 ( composition 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 M n then does the transfer bar sample. This would encourage more pearlite formation for a given cooling rate, consistent with the microstructure observed in Figure 53. The percentage ferrite value of .16 ±.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 ± 9 um is in good agreement with the continuous cooling sample value of 29 um ± 8 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  Coiling 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 temperature 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 temperature.  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 coefficient 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 ±8*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 ± 7 ° C and 676'C ± 1 1 *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 ±8*C. The value for run two, shown in Figure 58, of 686*C predicted versus 680'C ± 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 ±12*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 literature derived heat transfer coefficients rather than the overall heat transfer coefficient value of 1 kW/m 'C. Using the Hatta et al.[6] laminar water bar heat transfer 2  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 downcoiler position of the run-out table of 711'C and the pyrometer reading is 686*C ±8*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 ±7*C, as shown in Figure 61. For the run three cooling conditions the model top surface predicted temperature is 709*C with an equivalent pyrometer location value of 676*C ±12*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 thickness 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 variation in the model temperature prediction. The industrial data from Table D l shows that the strip velocity deviation is ± 0 . 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 ± 0 . 3 m/s variation in strip velocity results in a ± 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 Figures 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°C to 930*C, holding at 930°C for 2 seconds, cooling to 900"C in 3 seconds (10°C/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 transformation start temperature for a cooling rate of 7°C/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 ± 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 Strip Velocity Strip Gauge Strip Width  720"C target coiling temperature 359.6 m/min 3.89 mm 1.056 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Tempera ture(C) Deviation ±  910 6  770 6  856 12  856 17  759 11  786 5  730 11  0.054 Carbon  630*C target coiling temperature 3.89 mm 1.052 m  Strip Width Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Tempera ture(C) Deviation ±  882 26  671 49  620 14  825 14  700 1  680 9  639 12  0.054 Carbon Strip Velocity Strip Gauge Strip Width  550*C target coiling temperature 375.3 m/min 3.89 mm 1.052 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Tempera ture(C) Deviation ±  893 7  662 10  559 16  802 8  700 0  660 10  625 0  0.07 Carbon w/niobium Strip Velocity Strip Width  550°C target coiling temperature 377.9 m/min 3.89 mm 1.053 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  895 74  637 81  539 36  847 28  700 0  645 12  625 0  Table Ha Plant Conditions for Four Runs  73  RESULTS & DISCUSSION  0.07 Carbon w/niobium Strip Velocity Strip Gauge Strip Width  5.4  720*C target coiling temperature 329.9 nVmin 3.89 mm 1.053 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  917 36  757 5  710 15  863 16  716 7  773 9  716 13  0.07 Carbon w/niobium Strip Velocity Strip Gauge Strip Width  630"C target coiling temperature 348.0 m/min 3.89 mm 1.053 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  921 29  721 10  629 10  819 20  700 0  704 25  637 9  0.07 Carbon w/niobium Strip Velocity Strip Gauge Strip Width  720*C target coiling temperature 328.1 m/min 3.89 mm 1.056 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  924 12  752 7  711 26  865 21  704 7  767 10  723 23  0.07 Carbon w/niobium  550*C target coiling temperature 2.62 mm 1.049 m  Strip Gauge Strip Width Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Tempera ture(C) Deviation ±  922 8  685 11  564 32  788 40  700 0  681 24  626 3  Table LTb Plant Conditions for Four Runs  74  RESULTS & DISCUSSION 5.4  0.07 Carbon w/niobium Strip Velocity Strip Gauge Strip Width  720°C target coiling temperature 414.3 m/min 2.62 mm 1.050 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  917 7  753 4  714 8  831 18  711 7  769 6  724 17  0.054 Cartaan Strip Veloc ity Strip Gaugej Strip Width  630°C target c oiling temperature 45'3.8 rn/min 2. 62 mm 1.1353 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  895 7  717 8  629 14  835 4  701 3  732 15  638 11  0.054 Carbon Strip Velocity  550*C target coiling temperature 457.9 m/min  Strip Width  ...1.053 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Tempera ture(C) Deviation ±  897 6  667 8  564 12  838 5  707 11  662 9  625 0  0.054 Carbon Strip Velocity Strip Gauge Strip Width  720°C target coiling temperature 458.2 rn/min 2.62 mm 1.050 m  Pyrometer  FEXT  ROT  DC  PI  P2  P3  P4  Temperature(C) Deviation ±  897 6  779 4  718 8  864 15  784 6  795 11  735 9  Table lie Plant Conditions for Four Runs  75  RESULTS & DISCUSSION  Plant cooling conditions, 0.34 weight percent carbon. Run  Number of  FEXT  DC  Strip  Number  Sprays On  Temperature  Temperature  Velocity  CC)  CC)  (m/s)  1  10 ± 2  914 ± 9  686 ± 8  5.24 ± . 3  2  9 ±2  909 ± 8  680 ± 7  4.89 ± . 3  3  11 ± 3  924 ± 10  676 ± 1 1  5.20 ± 3  Table m Industrial Plant Cooling Conditions.  76  5.4  RESULTS & DISCUSSION  5.4  Metallographic Data Fraction Ferrite  Undercooling CC)  Gamma Grain Size (um)  5'Os  0.53  42  19 ± 2 *  C C T @ 10 °C/s  0.51  63  28 ± 7 *  C C T @ 15 'as  0.56  66  22 ± 3 *  C C T @ 27 'as  0.65  81  18 ± 2 *  °c/s  0.15  90  C C T @ 51 *C/s  0.41  95  C C T @ 55 °C/s  0.38  C C T @ 65 'as  0.40  101  32 ± 6*  C C T @ 78 'as  0.07  114  37 ± 5 *  C C T @ 103 *C/s  0.13  118  36 ± 4 *  Sample CCT@  C C T @ 40  C C T average  37 ± 7 * 28 ± 3 *  29 ± 9 *  0.34 C down-coiler  0.16  35 ± 9  0.34 C Gleeble  0.39  18 ± 3  •....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. A n overall average prior austenite grain diameter of 29 um ± 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 tabulated values for, cooling rate, fraction ferrite, undercooling, and average austenite grain size.  77  RESULTS & DISCUSSION  5.4  Composition  Table V  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  Al  0.043  0.023  Comparison of the composition of the down-coiler and transfer bar medium  carbon test samples.  78  RESULTS & DISCUSSION  Grain size as a function of coiling temperature for a low carbon steel 0.054 % carbon, 3.89 mm gauge, 720 °C  17 ± 2 um  0.054 % carbon, 3.89 mm gauge, 630 ' C  14 ± 1 um  0.054 % carbon, 2.69 mm gauge, 720 "C  16 ± 2 um  0.054 % carbon, 2.69 mm gauge, 630 *C  16 ± 2 um  Table VI  Grain size versus coiling temperature for a 0.054 weight percent carbon  grade steel.  79  5.4  RESULTS & DISCUSSION  Model temperature predictions for various cooling conditions Model Temperature  Pyrometer  Values  Reading  Figure 54  706*C  686'C ± 8 ' C  7'C/s = 732'C  Figure 55  699'C  680'C ± 7 ' C  7'C/s = 732'C  Figure 56  700'C  676'C ± 1 2 ' C  7'C/s = 732'C  Figure 57  688'C  686'C ± 8 ' C  45'C/s = 688'C  Figure 58  685'C  680'C ± 7 ' C  45'C/s = 688'C  Figure 59  676'C  676°C ±12*C  45'C/s = 688'C  Figure 60  711'C  686*C ±8*C  26'C/s = 710'C  Figure 61  705'C  680'C ±7*C  26°as = 710'C  Figure 62  709'C  676'C ± 1 2 ' C  26'C/s = 710'C  Cooling rate = T  san  Table VII.„..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  5.4  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  Heat Transfer Coefficient Kw/m C  Figure 18  Hatta laminar water bar heat transfer coefficient as a function of contact  radius  82  5.4  O  §  i c ?  Distance from Finish Stand F4 meters  2: ?  RESULTS & DISCUSSION  o o  o m  O  J  o o  C  D  o m  c  o  o o  o  o  o w  r  ^  o o  r  ^  o m  c  o  o 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 m O  >  o o o  >  o w c o  o o o 5  f  o w  »  *  o o r  ^  o m c  o  o 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°C.  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 « » 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 i C  n O  o O  o J  o O  w O  o C  o  o  O  h  o o  m -  r  o m ^  C  o o 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  Temperature (C)  Figure 26  A sample temperature profile from the plant data.  90  5.4  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 m  o o  o m  o o  o m  o o  o m  o o  o w  o 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°C  94  RESULTS & DISCUSSION  o o  o o L no oo m o o om o mo oo L o o oo 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 oo mo oo m o o o m oo om oo L o c 7 ) c o a o a o r « 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°C  96  RESULTS & DISCUSSION  5.4  o o  o o o o o o o o fflcooiooooooooeoSsSNN  o  o  o  o  o  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° 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*»S-r^r»-r*»-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 °C and 2  showing the effect of using 20 kW/m *C and 5 k W / m C on the target coiling 2  2o  temperature of 720*C.  98  RESULTS & DISCUSSION  Fraction Transformed  Figure 35  Isothermal dilatometer results for 673*C test  99  5.4  RESULTS & DISCUSSION  Fraction Transformed  Figure 36  Fraction transformed as a function of time for constant temperature =  673 C. #  100  5.4  RESULTS & DISCUSSION  •  OJ  \ • \ •\ h  CNJ  CO  CM CO CD  \  CO  >  if) C\J  \  CO ©  «CD —« Q CO  TT  d  CO  •\  II r-  • \ CVJ  d  in  d  T -  IT)  CM i  If)  d  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»i-^CMcoTrir)cDr^coo>c\i-i-CMoo CM o o o o o o I I I I I i CiM iCM i i i 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 , for the austenite-to-ferrite transformation in the 0.34 f  % 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-  CO  ^  CM  ^  to o  T-  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  ln(b), ferrite  Figure 42  Calculated ln(b) values for the ferrite transformation assuming ii = 1.25, f  for the 0.34% carbon steel.  106  5.4  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  LOTTOOCNJT—cxjo^cqr^couo  CNJ C\i C\j C\i C\j  T^T-^T-^T—i-^T-^-r^T-^-r^  n Figure 44  o o o o o  value  Average Avrami coefficient V for 0.34% carbon compared to other  experimental values (Campbell[27])  108  5.4  RESULTS & DISCUSSION  5.4  O 00  o  r^. r-. r*. r^ CM CM CM C\l C\J J  CD CD  <  To  I  I  )  CD CD CD CD CD CD  GO  "55  CO <  CD CD  CO GO  Q  o Tt  rr co C M T - 3 o o o o o . n  T-  y-  •  -r- T -  + O < X  O  o  «  CM  o o  +• o  o oo  • x x  o CO  +  o  + +  CO  I CM  i-  I  1  O  i-  I  I  CM  CO  rr  CM  rf i  If)  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  <  h-  o  s o  CD "O  o  •  ~  CO  c 3  RESULTS & DISCUSSION  •  9  © <  o oo  o  CO  _ o  CD CD © CD CD CD CD CD CD CD CD CD CM,  O co d  CM  T-  O  T-  CM  o  wcyScyjcy) c/5co "55 com coo o o .O T - C M C O T T cor*, Q. oo oo o o E  CM  CO  X  CO i  <o+ • > O  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  5.4  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  112  5.4  RESULTS & DISCUSSION  Undercooling ( C ) Figure 49  5.4  (785 - Tst)  The undercooling for the austenite-to-ferrite start temperature as a function  of the cooling rate.  113  RESULTS & DISCUSSION  I  1  1  1  1  1  1  i  C O r ^ C O U O T T C O C N J - r - O o  o  o  o  o  o  o  h  o  Percentage Ferrite  Figure 50....Fraction ferrite as a function of cooling rate, from metallographic examination.  114  o  5.4  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 *C, Table DT-Run #1 cooling conditions, 3.94mm gauge, and a transformation 2  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 *C, Table DJ-Run #2 cooling conditions, 3.94mm gauge, and a transformation 2  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 *C, Table DI-Run #3 cooling conditions, 3.94mm gauge, and a transformation 2  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 *C, Table HI-Run #1 cooling conditions, 3.94mm gauge, and a transformation 2  start temperature of 688°C ( 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 « 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 °C, Table Hi-Run #2 cooling conditions, 3.94mm gauge, and a transformation 2  start temperature of 688°C ( 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 cooJOTOioocooococoi^r^.r^r^r^cococD  corfojoajcDTrcNiococoTrcMOcocorr  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°C ( dt/dt = 26'C/s).  126  RESULTS & DISCUSSION  to in in in m U D m m m <<* -<t -t  TJ-  TT  CO CO CO  Temperature (C)  Figure 63  Effect on predicted center line temperature of changes in the number of  nodes through thickness.  127  5.4  RESULTS & DISCUSSION  5.4  E E  co N CO Q. ©  CO  Temperature (C)  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  RESULTS & DISCUSSION  o o  o o o o o o o o o o o o o o o  T T C N O C O C O r f C N J O C O C O T f C N O C O C O  C D O C n o O O O C O C O C O N N N N S C O C O  Temperature (C)  Figure 65  Effect of the ± 0 . 3 m/s deviation of the strip velocity on the predicted  temperature profile.  129  5.4  RESULTS & DISCUSSION  5.4  Temperature (C)  Figure 66....Industrial cooling profile simulated on the Gleeble high temperature testing machine.  130  RESULTS & DISCUSSION  Temperature (C)  Figure 67  Industrial cooling conditions, simulated on a Gleeble high temperature  testing machine.  131  RESULTS & DISCUSSION  Figure 68  5.4  Microstructure of the Gleeble cooling simulation sample of the Table UJ  run one cooling conditions.  132  CONCLUSIONS  6  6.1  CONCLUSIONS 6.1 Summary A mathematical model has been formulated to predict the through thickness  temperature distribution of strip as it cools on the run-out table of a hot-strip mill. The model incorporates the heat of transformation associated with the austenite to ferrite and ferrite plus pearlite phase change. The phase transformation has been characterized by an Avrami equation and the coefficients, n and b, have been determined from isothermal diametral dilatometer tests. The transformation start temperature is a function of the cooling rate experienced by the steel and a relationship between the cooling rate and the temperature at which the phase transformation starts has been determined from continuous cooling diametral dilatometer tests. The total fraction ferrite that will form is also a function of the cooling rate and the fraction ferrite has been characterized as a function of cooling rate using the quantitative analysis capability of the Wild-Leitz image analyzer with polished and etched continuous cooling test samples. In-plant temperature measurements for various strip cooling regimes were obtained and used with the thermal profile model to calculate an overall effective heat transfer coefficient for a laminar water bar cooling bank. Back calculation of a heat transfer coefficient for individual laminar water bar headers using the plant temperature data was unsuccessful due to the insensitivity of the strip temperature to individual  133  CONCLUSIONS  6.1  laminar water bars due to the short strip residence rime under any individual header. Metallographic examination of strip samples from the in-plant temperature data acquisition tests to determine a relationship between coiling temperature and grain size was inconclusive as no obvious relationship was seen. Heat transfer coefficients for various cooling regimes; laminar water bar, water curtain, film boiling, support roll contact, and air cooling by convection/radiation, were found in the literature. These values were used in the overall model incorporating phase transformation along with industrial cooling conditions to predict strip thermal profiles and the predicted temperatures were compared with plant pyrometer values. The effect of variations in the various heat transfer coefficients on the model thermal profile predictions was examined and the model was shown to be relatively insensitive to small variations in the heat transfer coefficients obtained from literature or calculated. The model did, however, show some sensitivity to variations in strip velocity. The cooling regime predicted by the overall model, incorporating phase transformation, was simulated on a Gleeble 1500 testing machine. The resulting transformation start temperature of 730*C is similar to that predicted by the emperical transformation start versus constant cooling rate data for an average cooling rate of 7'C/s. The dilatometer test samples were from transfer bar stock, due to specimen size constraints, and had not received the finish stand rolling that a down_coiled sample would. To determine the validity of using transfer bar stock to predict down-coiled  134  CONCLUSIONS  6.2  reactions a down-coiled sample was acquired and metallographic examination of the transfer bar, down-coiled, and Gleeble samples gave reasonable agreement in prior austenite grain size for the samples tested.  6.2 Conclusions (i)  For a 0.34 carbon plain-carbon steel the  Avrami equation coefficients were determined as: n (ferrite) =  1.25  . . . 16  ln(b) (ferrite) =  23.86 - 0.038559 * ( T )  . . . 17  fi (pearlite) =  1.14  . . . 18  ln(b) (pearlite) =  25 - 0.040 * ( T )  . . . 19  (ii)  The transformation start temperature as a  function of the cooling rate (dT/dt) is: T (start) =  785 - [ 38.0933 + 2.1164(dT/dt)  - 0.0242(dT/dt) + 0.00011043(dT/dt) ] 2  (iii)  3  . . . 20  The total percentage ferrite formed as a function of  cooling rate (dT/dt) is: Fraction ferrite = (iv)  0.59317 - 0.0048177(dT/dt)  A n overall average effective heat transfer coefficient for a  laminar water bar cooling bank of 1 kW/m2'C has been calculated and shown to be effective for describing the plant data.  135  . . . 21  CONCLUSIONS  (v)  A n individual laminar water bar heat transfer coefficient  cannot be calculated with the present data due to the short residence time of the strip under any one individual spray. (vi)  While no correlation between grain size and coiling  temperature was observed, a larger number of tests might separate trends from experimental scatter. (vii)  The overall model, incorporating phase transformation,  calculated a strip thermal profile for various industrial cooling conditions, for a 0.34 carbon plain-carbon steel. For the calculated overall heat transfer coefficient of 1 k W / m C , using a transformation start temperature of 732"C 2,  calculated for the cooling rate of 7*C/s experienced in the air cooling section of the run-out table, the model predicted temperatures 19*C to 24*C greater than the measured pyrometer values. (viii)  For the same conditions as in (vii) with a transformation start  temperature of 688'C calculated for the 45*C/s cooling rate experienced in the water cooling section of the run-out table, the model predicted temperatures of l ' C to 5*C greater than the pyrometer readings. (ix)  Using the literature derived heat transfer coefficient values  instead of the calculated overall heat transfer coefficient, with the same cooling conditions as in (vii) and (viii), and a cooling rate of 26°C/s, which is the average of the 45*C/s and 7*C/s values, model predictions were compared with pyrometer values. For this cooling rate the transformation  136  6.2  CONCLUSIONS  start temperature is 710*C and the model predictions are 25*C to 33'C greater than the plant pyrometer readings. These literature derived heat transfer coefficients are useful but require fine tuning for the effects of strip velocity which was not present in the initial experiments used to produce these relationships. (x)  The cooling regime used for the model predictions was  simulated on a Gleeble testing machine and the model predictions were within XX* C of the simulation temperatures. The simulation sample started phase transformation at approximately 730" C which is also the experimentally produced transformation start temperature for a cooling rate of 7'C/s. (xi)  The continuous cooling and Gleeble test samples, made from  transfer bar stock, and a down-coiler sample, where examined to compare the prior austenite grain sizes. The average prior austenite grain size for the continuous cooling dilatometer samples of 29 um ± 8 um compares well with the down-coiler value of 35 um ± 9 um. The Gleeble sample shows some local microstructural variation due to banding, as shown in Figure 68, and an average austenite grain size of 18 um ± 3 urn. This grain size is finer than that obtained in the continuous cooling transformation samples but is similar in magnitude. The closer austenite grain size similarity between the continous cooling samples and the down-coiler sample supports the use of transfer bar to measure the transformation kinetics.  137  6.2  CONCLUSIONS  6.3  6.3 Future Considerations Modification of the existing hardware to allow characterization of other low carbon and H S L A grades of steel as well as duplication of the prior thermo-mechanical history is needed to increase the scope of the work. The addition of the Gleeble high temperature testing machine means that the deformation due to the finishing stands may be duplicated in the future or that a method of using test samples of smaller dimensions will allow the use of down-coiler samples rather than the transfer bar samples currently employed. Modifications to the model to include prior thermo-mechanical history and grain growth effects prior to the run-out table should be added to allow grain growth prediction on the run-out table as well as prediction of final grain size and thus some mechanical properties prediction. As the grades of steel that can be characterized are increased the bainite and martensite phase transformation kinetics should be characterized to increase the scope of the model. Finally the run-out table model could be integrated into an overall rolling mill model for use in production forecasting.  138  BIBLIOGRAPHY  7  7  BIBLIOGRAPHY  1 G . Tacke, H . Litzke and E . Raquet, "Investigations into the Efficiency of cooling systems for Wide-Strip Hot Rolling Mills and Computer-aided Control of Strip Cooling", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of A I M E , Pittsburgh, Pennsylvania, August 19-21, 1985, pp 35-54. 2 R. Colas and C M . Sellars, "Computed Temperature Profiles of Hot Rolled Plate and Strip During Accelerated Cooling", (CTM convention , Winnipeg, Manitoba 1987) 3 E . A . Farber and R.L. Scorah, "Heat Transfer to Water Boiling Under Pressure", Transactions of the A S M E , 70 (1948), pp.369-384. 4 Natsuo Hatta, Jun-ichi Kokada and Koichi Hanasaki, "Numerical Analysis of Cooling Characteristics for Water Bar", Transactions of ISU, Volume 23, 1983, pp 555-564. 5 Natsuo Hatta, Jun-ichi Kokada, Hirohiko Takuda, Jun Harada and Keizo Hiraku, "Predictable Modeling for Cooling Process of a Hot Steel Plate by Laminar Water Bar", Archiv fur das Eisenhuttenwesen, 55(1984) Nr. 4 April, pp.143-148. 6 Jun-ichi Kokada, Natsuo Hatta, Hirohiko Takuda, Jun Harada and Nobuo Yasuhira, "An Analysis of Film Boiling Phenomena of Subcooled Water Spreading Radially on a Hot Steel Plate", Archiv fur das Eisenhiittenwessen, Nr. 55(1984) March 7 Arnulf Diener and Alfons Drastik, "Heat Exchange Between Strands and Guide Rollers in the Secondary Cooling Zone of a Slab Continuous Casting Machine", Fourth Japan-Germany Seminar, Nov. 1980, Tokyo, The Iron And Steel Institute of Japan 8 "Physical Constants of Some Commercial Steels at Elevated Temperatures", Edited by the British Iron and Steel Research Association, Metallurgy (General) Division Thermal Treatment Sub-Committee, London, 1953 9 M . A^rami: "Kinetics of Phase Change. I, General Theory", Journal of Chemical Physics, December 1939, pp.1103-12. 10 M . Avrami: "Kinetics of Phase Change. LI, Transformation-Time Relations for Random Distribution of Nuclei", Journal of Chemical Physics, February 1940, pp.212-224.  139  BIBLIOGRAPHY  7  11 M . Avrami: "Granulation, Phase Change, and Microstructure. Kinetics of Phase Change HI", Journal of Chemical Physics, February 1941, pp. 177-84. 12 J.W. Cahn, "The Kinetics of Grain Boundary Nucleated Reactions", Acta Metallurgies Vol. 4, September 1956, pp.449-459. 13 Prakash K . Agarwal and J.K. Brimacombe, "Mathematical Model of Heat Flow and Austerute-Pearhte Transformation in Eutectoid carbon Steel Rods for Wire", Metallurgical Transactions B, Vol 12B, March 1981, pp.121-133. 14 M . B . Kuban, R. Jayaraman, E.B. Hawbolt, and J.K. Brimacombe: "An Assessment of the Additivity Principle in Predicting Continuous-Cooling Austenite-to-Pearlite Transformation Kinetics Using Isothermal Transformation Data", Metallurgical Transactions A , Vol. 17A, September 1986, pp. 1493-1503. 15 E . B . Hawbolt, B. Chau, and J.K. Brimacombe, "Kinetics of Austenite-Pearlite Transformation in Eutectiod Carbon Steel", Metallurgical Transactions A , Vol. 14A, September 1983, pp.1803-15. 16 E . B . Hawbolt, B. Chau, and J.K. Brimacombe: "Kinetics of Austenite-Ferrite and Austenite-Pearlite Transformations in a 1025 Carbon Steel", Metallurgical Transactions A , Vol. 16A, April 1985, pp.565-77. 17 M . Morita, K . Hashiguchi, O. Hashimoto, M . Nishida, and S. Okano, "On-Line Transformation Detector for Property Control of Hot Rolled Steel", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of A J M E , Pittsburgh, Pennsylvania, August 19-21, 1985, pp.449-461 18 Eric N . Hinrichsen, "Hot Strip Mill Runout Table Cooling - A System View of Control, Operation and Equipment", Yearly Proceedings of the AISE, 1976, pp.403-408. 19 A . Hurkmans, G.A. Duit, Th.M. Hoogendoom and F. Hollander, "Accelerated cooling and the Transformation of Steel", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of A I M E , Pittsburgh, Pennsylvania, August 19-21, 1985, pp.481-499 20 Edward B. Longenberger, "Computer Modeling of a Hot Strip Mill Run out Table", Proceedings of the XXVIII Conference on Mechanical Working and Steel Processing, The Iron and Steel Society of the AIME, Pittsburgh, PA, October 26-28, 1986, pp. 169-172  140  BIBLIOGRAPHY  7  21 Yushi Miyake, Teruyuki Nishide and Shoichi Moriya, "Device and System for Controlled Cooling for Hot Strip Mill", Transactions of the ISIJ, vol 20, 1980 pp.496-503 22 H . Yada, "Prediction of Microstructural Changes and Mechanical Properties in Hot Strip Rolling", Presented at the Accelerated Cooling of Steel Symposium at the Conference of the Canadian Institute of Mining and Metallurgy, Winnipeg, Manitoba, August 24, 1987. 23 Zacay and AAronson editors, "Decomposition of Austenite by Diffusional Processes", 1962, pp. 336 24 P J . Berensen, "Film-Boiling Heat Transfer From a Horizontal Surface", Transactions of the A S M E , Journal of Heat Transfer, vol 83, August 1961, pp.351-358. 25 K . W . Andrews, "Empirical Formula for the Calculation of Some Transformation Temperatures", Journal of the Iron and Steel Institute, July 1965, pp. 721-727 26 A . le Bon, J. Rofes-Vemis, and C. Rossard, "Recrystallization and Precipitation during Hot Working of a Nb-Bearing H S L A Steel", Metal Science, volume 9, 1975, pp36-40. 27  P. Campbell, Private communication of unpublished work.  28  C . Devadas, Private communication of unpublished work.  29 Yukihisa Kuriyama, Matsuo Ataka, Masayuki Nakanishi, Masayuki Miyatake, Kouichiro Goto and Shuichi Hamauzu, "On-line Model for Prediction of Strip Temperature in Hot Rolling", Transactions of the ISIJ, vol. 23 number 9, 1983, p.B338. 30 J.S. Kirkaldy and E . A . Baganis, "Thermodynamic Prediction of the Ae3 Temperature of Steels with Additions of Mn, Si, Ni, Cr, Mo, Cu", Metallurgical Transactions A , Vol. 9A, April 1978, pp.495-501.  141  APPENDIX  APPENDIX 8.1 N O M E N C L A T U R E  */ K c.  Avrami coefficient, 'b' for ferrite  D  Laminar water bar nozzle internal diameter, (m)  Fx(k  -1)  Fx(k)  Avrami coefficient, 'b' for pearlite Specific heat, (W/kg*C)  Fraction transformed in the previous time step Fraction transformed in the current time step  8  Acceleration due to gravity, (m/s )  h  Heat transfer coefficient, (W/m *C)  H  Laminar water bar header height over run-out table, (m)  k  Thermal conductivity, (W/m'C)  n  f  P  N  "/ "P  2  2  Avrami coefficient, V for ferrite Avrami coefficient, V for pearlite average n(ferrite) average n(pearlite)  Pr  Prandtl number  4.  Heat flux generated due to phase transformation, (W)  Q  Water flow rate, (m /s)  r  Laminar water bar contact radius, (m)  Re  3  Reynolds number  142  APPENDIX  t  Time, (s)  t  Virtual time, (s)  v  /•  Time for completion of 20% of the phase transformation, (s)  r  Time for completion of 90% of the phase transformation, (s)  T  CRTT  Critical water temperature for film boiling to water contact cooling transition, (*C)  T(k)  Node temperature at this time step, (°C)  7(j^y  Internally calculated model temperature values, (*C)  T(k)  Internally calcualted model temperature values, (*C)  T  s  TSAJ.  T  Steel surface temperature, (°C) Water saturation temperature, (*C) Transformation start temperature, (*C)  T  w  x,y,z  Water temperature, (*C) Spatial coordinates, (m)  X  Fraction transformed  a  Ferrite phases  a  Film boiling heat transfer coefficient, (W/m C) 2,  n  ot^j  Laminar water bar heat transfer coefficient, (W/m C) 2#  y  Austenite phase  5  Water film thickness, (m)  143  8.1  APPENDIX  6V AFx{k),dFx(k)  Thermally affected water film thickness, (m) Incremental fraction transformed for this time step  At,dt  Model time step, (s)  Ax,etc  Model step size, (m)  Az  Model node size, (m)  e  Emissivity  v  Strip velocity, (m/s)  p  Density, (kg/m )  ej  Stefan Boltzmann constant, (W/m K )  3  2  144  4  8.1  APPENDIX  8.2.1  8.2 Finite Difference Equation Derivation Based on the assumptions in section 4 and using 200 nodes through the thickness of the strip, the basic finite difference equations are;  8.2.1  The top surface node, dUt dt ~  p*C*A*bc^dT 2  x  * dt  .Al  where dUi -jfi- = <?2-1 + 4lT- 1 + Qrad + 4  ....A2  ....A3 4 - =h *A*<T' "-T '+") Fr  l  c  Fr  2  ....A4  ....AS Qnai  = the heat generatedby the phase transformation  ....A6  Due to the small size of the time step and the relativly large, compared to the step size, fluid volume it will be assumed that the surrounding fluid ( air or water) is of  145  APPENDIX  8.2.2  constant temperature. To facilitate the solution of the radiation calculations the node temperature from the current time step is used for the future time step calculations. Which results in the equation :  2A/  1  1  2  .A7  fp*C,*A^ 1  2A/  8.2.2 Interior nodes The interior nodes were less difficult to describe due to the absence of radiation or convection effects in Eq.A.7. Through a similar set of assumptions to those in the top surface node, Eq.A.8 was produced. o*C *Ar ^  = T'* * a  ....A8  2*Ar  +4 rtcai  8.2.3 The bottom node As with the formulation of the top surface node (1) equations and with similar fluid temperature and radiation cooling simplifications the bottom node, node 'n' in this example, could be described, after manipulation by Eq.A.9.  146  APPENDIX  8.3  -k*Ax*T.• - I ....A9  „JP*C,*Ax  + h *T'„ c  +  a*f^(T'J-(T:))  + <i rtcai  8.2.4 Solution The equations for the top and bottom surfacve nodes along with the interior nodes can be gatherd together as a sparse tridiagonal matrix. A number of simple subroutines exist to make this problem readily solvable by computer.  8.3 Hatta et al. Model Thermal Boundary Layer Calculation The thermal boundary layer concept used by Hatta et al. is effected by the water velocity, which is calculated with ....A10  where Q is the water flow rate in m / min, D is the diameter of the laminar water bar 3  nozzle in meters, H is the height above the strip in meters, and g is the acceleration due to gravity. The thickness of the water film layer is then calculated by E q . A . l 1.  ...All k*dr 2KV  W  147  APPENDIX  8.3  For E q . A . l 1 k is the step number and dr is the incremental distance. V is the w  water velocity from Eq.A.10. The over prediction of cocooling by the model caused Hatta et al.[5] to postulate that the entire water film thickness was not affected by the heat flow. They hypothesize a thermally affected zone, a type of boundry layer in the water film and calculate the thickness with Eq.A.12. ....A12  In Eq.A.l2 Re is the Reynolds number and Pr is the Prandd number. If the water film from the laminar water bar was thought of as radially symmetrical and divided into k rings of thickness r and area, ....A13  dS = (7C*(2* - l)*dr ) 2  The calculation of the water temperature of any one cell, using the thermal zone concept could then be formulated as, dS(k-l)  oy*-l)-cW*-l)  , dS(k)  dTik)  r  T(k) =  + 8 (*-l)' wr  148  Mi  ....A14  

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