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Heat transfer and stress generation during forced convective quenching of steel bars Hernández-Morales, José Bernardo 1996

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H E A T T R A N S F E R A N D STRESS G E N E R A T I O N D U R I N G F O R C E D C O N V E C T I V E Q U E N C H I N G OF STEEL B A R S By Jose Bernardo Hernandez-Morales B. Sc. (Metallurgical Eng.) University of Mexico, Mexico City, 1983 M . A . Sc. (Metals and Materials Eng.) University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY IN T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF METALS AND MATERIALS ENGINEERING W E A C C E P T THIS THESIS AS CONFORMING TO T H E REQUIRED STANDARD T H E UNIVERSITY OF BRITISH COLUMBIA M A Y 1996 © J O S E B E R N A R D O H E R N A N D E Z - M O R A L E S ; 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Metals and Materials Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract A n industrial heat treatment based on forced convective quenching of alloyed eutectoid steel bars has been studied within the framework of microstructural engineering. The pro-prietary process is used to produce grinding rods with improved abrasion resistance and toughness. Mathematical models of heat transfer, microstructural evolution and stress generation have been applied to predict the final microstructure and residual stress dis-tribution in 38.1 mm-dia. bars, quenched in a laboratory facility. The models are based on the finite-element method (FEM) and assume axisymmetric geometry. The principle of additivity was invoked to predict the evolution of the microstructure under continuous cooling conditions. The kinetics of diffusional and martensitic phase transformations were characterized using the J M A K and Koistinen-Marburger equations, respectively. The ki-netic parameters were determined through continuous cooling and isothermal tests using the Gleeble 1500 thermomechanical simulator. The existing elasto-plastic stress model incorporates thermal- and transformation-related strains. The mechanical properties were obtained from the literature. The effect of stresses on transformation kinetics was not considered; thus, the thermal-microstructural and mechanical models were effectively uncoupled. The heat-transfer boundary condition has been characterized by acquiring the thermal response in instrumented interstitial-free (IF) steel bars quenched from 1000 °C in flowing water. Three water velocities and water temperatures were investigated. Tests with clean and oxidized bars were also conducted. The surface heat flux, as a function of surface temperature, was estimated by solving the inverse heat conduction problem (IHCP). The solution algorithm was based on the sequential function specification technique. The water temperature was found to have a significant effect on the boiling curves. For n the highest water temperature, 75 °C, a film boiling stage was identified, while only transition, nucleate and convective boiling were observed when using water at 50 and 25 °C. No significant differences in heat extraction were observed between clean and oxidized quenched bars, within the parameter range investigated. The residual stress distribution in selected forced convective quenched IF, 1045 carbon and alloyed steel bars has been determined by means of neutron diffraction. The axial and circumferential residual stresses in the IF and 1045 carbon steel quenched bars were compressive at the surface and tensile at the centre, while the radial component was always tensile. In contrast, the alloyed eutectoid steel bar showed compressive axial, circumferential and radial residual stresses at the centre. The final microstructural and hardness distributions have also been determined in the three bars. The IF steel bar transformed completely to ferrite, while the alloyed eutectoid steel specimen showed an essentially martensitic structure. On the other hand, quenching the 1045 carbon steel bar resulted in an outer ring of martensite and a mixture of diffusional and martenistic products in the core. Comparisons between measured and model-predicted thermal response, final mi-crostructure and residual stress distribution have been made. Fair agreement between measured and predicted values was observed. It was found that the position of the 'nose' of the continuous cooling diagram, when a mixture of diffusional and martensitic prod-ucts was produced, has a significant influence on the predicted final microstructure and, therefore, on the predicted residual stress distribution. The difference in the measured residual stress distributions obtained in the alloyed eutectoid steel specimen, when com-pared with those found in the IF and 1045 carbon steel quenched bars, has been explained based on the sequence of transformations that took place during the quench. The quench of a bar under industrial conditions was also simulated. i n Table of Contents Abstract ii Table of Contents iv List of Tables ix List of Figures xii List of Symbols xxxiii Acknowledgements xxxix 1 Introduction 1 1.1 Microstructural Engineering 1 1.2 Heat Treatment of Grinding Media 2 1.3 Statement of the Problem 3 2 Literature Review 6 2.1 Heat Treatment Design 6 2.1.1 Empirical Approaches 6 2.1.2 Characterization of Quench-Bath Quality 7 2.1.3 Process Modeling 14 2.1.4 Computerized Information Systems 15 2.2 Heat Transfer in Forced Convective Quenching 16 2.2.1 General 16 2.2.2 Surface Heat Flux Characterization 17 iv 2.2.3 Boiling Heat Transfer 19 2.2.4 Forced Convective Quenching 23 2.3 Microstructural Evolution 25 2.3.1 General 26 2.3.2 Isothermal Phase Transformation Kinetics 26 2.3.3 Continuous Cooling Phase Transformation Kinetics 32 2.3.4 Fe-C-X Phase Boundaries 37 2.3.5 Effect of Stresses on the Kinetics of Phase Transformations . . . . 38 2.4 Residual Stresses in Heat Treatments 40 2.4.1 General 40 2.4.2 Residual Stress Measurement Techniques 41 2.5 Mathematical Models of Microstructural Evolution and Stress Generation in Heat Treating Operations 43 2.5.1 Modeling of Heat Transfer and Microstructural Evolution 43 2.5.2 Modeling of Stress Generation 45 3 Scope and Objectives 61 3.1 Scope of the Research Programme 61 3.2 Objectives of the Research Programme 63 4 Heat Transfer Model 64 4.1 Governing Equation 64 4.1.1 Rate of Heat Evolved 65 4.2 Finite-Element Formulation 67 4.2.1 Finite-Element Equations 68 4.2.2 Solution Algorithm 70 4.3 Verification of Mathematical Model of Heat Flow 72 4.3.1 Infinite Solid Cylinder : q = 0 73 v 4.3.2 Infinite Solid Cylinder : q = f(T) 75 4.3.3 Infinite Solid Cylinder : Air Cooling of Eutectoid Steel 76 4.4 Summary 76 5 The Inverse Heat Conduction Problem 88 5.1 Solution Algorithm 89 5.2 Modification for Cylindrical Coordinates 92 5.3 Application to Controlled Air Cooling of Rods 95 5.4 Application to Forced Convective Boiling 95 5.5 Sensitivity coefficients and experimental design 98 6 Stress Model 112 6.1 Stress-Strain Relations 112 6.1.1 Elastic Stress-Strain Relations 114 6.1.2 Plastic Stress-Strain Relations 115 6.1.3 Finite-Element Formulation 120 6.2 Verification of Mathematical Model of Stress Generation . . . . . . . . . 125 6.2.1 Infinite Solid Cylinder : Elastic Thermal Stresses 125 6.2.2 Infinite Solid Cylinder : Elastoplastic Thermal Stresses 127 6.3 Summary 128 7 Laboratory Experiments : Quenching Tests 134 7.1 Quenching Apparatus 135 7.2 Procedure 141 7.3 Metallographic Characterization 143 8 Laboratory Experiments : Transformation Kinetics 161 8.1 Material and Sample Preparation 161 8.2 Procedure 162 vi 8.2.1 Continuous Cooling Tests 162 8.2.2 Isothermal Tests 162 8.2.3 Temperature Gradient in a Gleeble Specimen 163 8.3 Microstructural Characterization 164 9 Laboratory Results and Discussion : Quenching Tests 174 9.1 Measured Temperature Response 174 9.2 Estimated Surface Heat Flux 178 9.3 Metallographic Characterization 185 10 Laboratory Results and Discussion : Transformation Kinetics 215 10.1 Phase Boundaries 215 10.2 Continuous Cooling Tests 216 10.2.1 Alloyed Eutectoid Steels 216 10.2.2 1045 Carbon Steel 218 10.3 Isothermal Tests 219 10.4 Prior-Austenite Grain Size 221 10.5 Ferrite Fraction 222 11 Residual Stress Measurement 239 11.1 Experimental Procedure 239 11.1.1 Campaign 1 . . 239 11.1.2 Campaign 2 241 11.2 Results 2 4 2 12 Mathematical Analysis of Forced Convective Quenching 260 12.1 Verification of the Inverse Analysis 260 12.2 Temperature Response and Microstructural Evolution 264 12.2.1 IF Steel 264 vii 12.2.2 1045 Carbon Steel 266 12.2.3 Alloyed Eutectoid Steel 267 12.2.4 Sensitivity Analysis 269 12.3 Stress Generation 271 12.3.1 Residual Stresses 272 12.3.2 Transient Stresses 275 12.4 Application to Industrial Conditions 277 13 Summary and Conclusions 313 Bibliography 318 A Thermo-Microstructural Model : Finite Element Equations 338 A . l Isoparametric Elements and Numerical Integration 340 B Thermo-Microstructural Model : Semi-Analytical Solution 345 C Stress Model : Finite Element Equations 347 C . l Isoparametric Elements and Numerical Integration 351 D Stress Model : Nonmechanical Strains 354 E Recalescence During Quenching of IF Steel Bars 356 F Isothermal Tests : Data Reduction 359 F . l Linear Regression 359 F.2 Non-Linear Regression 360 vm List of Tables 2.1 Classification of techniques for measuring residual stresses (modified from [130]) 53 4.1 Decision tree to determine the sequence of calculations during phase transformations in a eutectoid steel 77 4.2 Input data used for the comparison of finite-element and analytical so-lutions under Newtonian cooling conditions 78 4.3 Input data used for the comparison of finite-element and analytical so-lutions under non-Newtonian cooling conditions 78 4.4 Input data adopted for the comparison of finite-element and finite-difference simulations of air cooling of eutectoid steel rod 79 5.1 Parameters used for Case 1 : constant surface heat flux 100 5.2 Parameters used for Case 2 : constant heat-transfer coefficient 100 5.3 Parameters used for the simulation of a forced convective quenching ex-periment 101 5.4 IHCP algorithm : summary of sensitivity runs 101 6.1 Input data used for the comparison of finite-element and analytical so-lutions for elastic stresses generated in an infinite solid cylinder by a temperature gradient 129 7.1 Chemical composition of the IF steel used in the forced convective quench-ing experiments (in weight percent) 144 ix 7.2 Chemical composition of the alloyed steels used in the forced convective quenching experiments (in weight percent) 144 7.3 Chemical composition of the 1045 steel used in the forced convective quenching experiments (in weight percent) 145 7.4 Typical operational plant data 145 7.5 Test matrix used for the boiling heat transfer experiments 146 7.6 Quenching conditions for specimens produced for residual stress mea-surements and metallographic analysis 146 7.7 Quenching conditions to study the effect of surface oxidation 147 8.1 Summary of continuous cooling tests for the alloyed eutectoid steel A . . 166 8.2 Summary of continuous cooling tests for the alloyed eutectoid steel B . . 166 8.3 Summary of continuous cooling tests for the alloyed eutectoid steel C. . 167 8.4 Summary of continuous cooling tests for the 1045 carbon steel 167 8.5 Summary of isothermal tests for the alloyed eutectoid steel A 168 8.6 Summary of isothermal tests for the alloyed eutectoid steel A (bainite reaction) 169 8.7 Summary of isothermal tests for the alloyed eutectoid steel B 170 8.8 Summary of isothermal tests for the alloyed eutectoid steel C 171 9.1 Estimated critical heat flux (CHF) using raw and pre-filtered data for a 38.1 mm-dia. IF steel bar quenched in water flowing at 2.8 m s _ 1 at 25 °C189 9.2 Estimated critical heat flux (CHF) as a function of water velocity dur-ing forced convective quenching of 38.1 mm-dia. IF steel bars in water flowing at 3 water temperatures 189 9.3 Average heat flux as a function of water velocity during forced convective quenching of 38.1 mm-dia. IF steel bars in water flowing at 3 water temperatures 189 x 10.1 Phase boundaries calculated from the equations by Kirkaldy et al. and the empirical formula given by Andrews 223 10.2 Measured expansion coefficients for the alloyed eutectoid and the 1045 carbon steel 223 12.1 Quenching conditions simulated in the verification runs of the thermal model 279 12.2 Thermophysical properties of 1045 carbon steel [188] 280 12.3 Thermophysical properties of alloyed eutectoid steel [188] 281 12.4 Thermomechanical properties of IF steel [205] 282 12.5 Thermomechanical properties of 1045 carbon steel [47] 283 F . l Initial values for n and In 6 used to test the robustness of the weighted non-linear regression analysis 363 xi List of Figures 1.1 Thermal, microstructural and stress field interactions during heat treat-ments [7] 5 2.1 Methods adopted to predict the performance of quenching processes. . . 54 2.2 Schematic representation of a typical cooling curve showing the different stages of boiling 55 2.3 Schematic representation of a typical boiling curve for saturated pool boiling [75] 56 2.4 Correspondence between cooling, boiling, and evaporation curves [56]. . 57 2.5 Regimes in boiling heat transfer [87] 58 2.6 Schematic representation of fraction transformed as a function of time, for a typical nucleation and growth phase transformation 59 2.7 Schematic representation of fraction transformed as a function of tem-perature, for a typical martensitic phase transformation 59 2.8 Schematic diagram illustrating the application of the additivity principle. 60 4.1 Typical cooling curve showing recalescence (schematic) 80 4.2 Flowchart for the solution of the thermal-microstructural problem. . . . 81 4.3 Details of the model sequence for evaluating the heat released during austenite-to-pearlite decomposition 82 4.4 Comparison between the finite-element and the analytical solution for the newtonian cooling (Bi= 0.01) of an infinitely long cylinder 83 xii 4.5 Comparison between the finite-element and the analytical solution at the centreline for the Non-newtonian cooling (Bi= 0.5) of an infinitely long cylinder 83 4.6 Function Q0 used in the semi-analytical solution of Newtonian cooling of a long rod with a uniformly distributed heat source 84 4.7 Semi-analytical solution, at the centreline of a 10 mm-dia rod, for Q0,m = 27.5 and 60.0 M W m~ 3 r" 1 84 4.8 Effect of At on the semi-analytical solution, at the centreline of a 10 mm-dia. rod, for Q0,m = 600.0 M W n T 3 s"1 85 4.9 Comparison between the semi-analytical and the numerical (FEM) solu-tion, during cooling of a 10 mm-dia. rod, for Q0,m = 27.5 M W m - 3 s _ 1 . 85 4.10 Comparison between the semi-analytical and the numerical (FEM) solu-tion, during cooling of a 10 mm-dia. rod, for Q0,m = 60.0 M W m - 3 s _ 1 . 86 4.11 Finite-difference [188] and finite-element solutions for the thermal re-sponse at the centreline and surface of a 10 mm-dia. eutectoid carbon steel 86 4.12 Finite-difference [188] and finite-element solutions for fraction trans-formed to pearlite at the centreline and surface of a 10 mm-dia. eutectoid carbon steel 87 5.1 (a) Schematic representation of a one-dimensional, single-sensor IHCP in a flat plate, (b) Discrete temperature measurements 102 5.2 Piecewise aproximation of the surface heat flux as a function of time. . 103 5.3 Flow chart of the sequential function specification algorithm adopted for the solution of the IHCP 104 5.4 Comparison of estimated and input surface heat flux for Case 1 : constant surface heat flux 105 xiii 5.5 Comparison of estimated, using the sequential function specification tech-nique, and analytical thermal response at the surface of the cylinder for Case 1 : constant heat flux 105 5.6 Comparison of estimated and input heat-transfer coefficient for Case 2 : constant heat-transfer coefficient 106 5.7 Comparison of estimated, using the sequential function specification tech-nique, and analytical thermal response at the surface of the cylinder for Case 2 : constant heat-transfer coefficient 106 5.8 Estimated heat-transfer coefficient for 8 mm-dia. steel rods air cooled at 22 m/s; calculated by using a sequential matching approach and using "'• the sequential function specification algorithm 107 5.9 The functional q = f(Ts) adopted for the finite-element simulation of the thermal response in a 38.1 mm-dia. cylinder subjected to forced convective quenching 107 5.10 Calculated thermal response at the centreline, surface, and simulated thermocouple position (r* = 17.75 mm), obtained when the heat flux distribution shown in Figure 5.9 was applied at the surface of a 38.1 mm-dia cylinder 108 5.11 Effect of varying the parameter r adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder 108 5.12 Effect of varying the parameter r on the estimated surface temperature response during the simulated forced convective quenching of a 38.1 mm-dia. cylinder 109 5.13 Effect of varying the value of thermal conductivity adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder 109 xiv 5.14 Effect of varying the value of the thermocouple position adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder 110 5.15 Effect of the thermocouple position on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder 110 5.16 Sensitivity coefficients at several radial positions in a solid cylinder sub-jected to a medium of constant heat-transfer coefficient and fluid tem-perature I l l 6.1 Schematic of loading and unloading paths for a work-hardening material under a uniaxial load 129 6.2 Schematic representation of the scaling factor, r 130 6.3 Flow chart of the stress solution algorithm 131. 6.4 Comparison between analytical and numerical (FEM) predictions of ther-mal stress distributions in a 100 mm-dia. cylinder 132 6.5 Comparison between analytical and numerical (FEM) predictions of ra-dial displacements in a 100 mm-dia. cylinder 132 6.6 Comparison between measurements made by Buhler [204] and Mitter et al. [205] and numerical (FEM) predictions of residual stress distribu-tions in a 50 mm-dia. pure-iron bar quenched in ice water from 850 °C [47]. 133 7.1 Schematic diagram showing the quenching test section 148 7.2 Design of the specimen used for IF steel tests 149 7.3 Design of the specimen used for the alloyed eutectoid and 1045 carbon steel tests 150 7.4 Design of the specimen adaptor 151 7.5 Design of the tapered extension 152 xv 7.6 Calculated velocity and corresponding Reynolds number as a function of water flow rate for an annular region with Di = 46 mm 153 7.7 Calculated hydrodynamic entry length as a function of Reynolds number. 154 7.8 Calculated thermal entry length as a function of Reynolds number for various Prandtl numbers [209] 154 7.9 Flow loop of the quenching apparatus 155 7.10 Design of the aluminum cap 156 7.11 Schematic diagram of the data acquisition configuration . 157 7.12 Magnetic flowmeter calibration : water flow rate and current as a func-tion of valve setting 158 7.13 Temperature response at the centre of the specimen during heating to (a) 1000 °C and (b) 850 °C. 159 7.14 Difference between centre and subsurface temperature prior to the start of the quench for the IF steel tests 160 8.1 Geometry of the Gleeble specimen used for characterizing the phase transformation kinetics 172 8.2 Location of the sample blanks taken from the 100 mm-dia. rods 172 8.3 Schematic representation of continuous cooling tests 173 8.4 Schematic representation of isothermal tests 173 9.1 Measured temperature response at the centre and subsurface of two 38.1 mm-dia. IF steel bars quenched with water flowing at 4.8 m s _ 1 at 50 °C.190 9.2 Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 2.8 m s"1 for 3 values of water tem-perature 190 xv i 9.3 Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 for 3 values of water tem-perature 191 9.4 Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 6.9 m s _ 1 for. 3 values of water tem-perature 191 9.5 Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 25 °C for 3 values of water velocity. 192 9.6 Measured temperature response at the centre of a 38.1 mm-dia. bar quenched with water flowing at 50 °C for 3 values of water velocity. . . 192 9.7 Measured temperature response at the centre of a 38.1 mm-dia. bar quenched with water flowing at 75 °C for 3 values of water velocity. . . 193 9.8 Measured temperature difference between the centre and the subsurface as a function of subsurface temperature during forced convective quench-ing of a 38.1 mm-dia. IF steel bar in water flowing at 25 °C for 3 values of water velocity 193 9.9 Measured temperature difference between the centre and the subsurface as a function of subsurface temperature, during forced convective quench-ing of a 38.1 mm-dia. IF steel bar in water flowing at 50 °C for 3 values of water velocity ' 194 9.10 Measured temperature difference between the centre and the subsurface as a function of subsurface temperature during forced convective quench-ing of a 38.1 mm-dia. IF steel bar in water flowing at 75 °C for 3 values of water velocity 194 9.11 Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 25 °C for 3 values of water velocity 195 xvii 9.12 Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 50 °C for 3 values of water velocity 195 9.13 Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 75 °C for 3 values of water velocity 196 9.14 Maximum cooling rate at the centre as a function of water velocity during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature 196 9.15 Maximum cooling rate at the subsurface as a function of water velocity during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. . 197 9.16 Effect of surface condition on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 4.8 m s _ 1 at 25 °C 197 9.17 Effect of surface condition on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. alloyed steel bar in water flowing at 4.8 m s _ 1 at 32 °C 198 9.18 Effect of initial test temperature on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 4.8 m s _ 1 at 32 °C 198 9.19 Raw and filtered temperature response at the subsurface during forced convective quenching of a 38.1 mm-dia. IF in water flowing at 2.8 m s _ 1 at 25 °C 199 9.20 Experimentally determined temperature response at the subsurface and centreline during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 at 25 °C 200 xvm 9.21 Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar. The raw data of Figure 9.20 were filtered before being input to the computer program. 200 9.22 Comparison between estimated surface heat flux as a function of sur-face temperature, using raw and filtered data, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 at 25 °C 201 9.23 Effect of the parameter r on the residuals obtained during the estimation of surface heat flux during forced convective quenching of a 38.1 mm-dia. IF steel bar when raw data was used 201 9.24 Effect of the parameter r on the residuals obtained during the estimation of surface heat flux during forced convective quenching of a 38.1 mm-dia. IF steel bar when filtered data was used 202 9.25 Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 for 3 values of water temperature 202 9.26 Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 for 3 values of water temperature 203 9.27 Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 6.9 m s _ 1 for 3 values of water temperature 203 9.28 Estimated surface heat flux as a function of time, during forced con-vective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 for 3 values of water temperature 204 9.29 Surface heat flux as a function of wall superheat, during forced convective quenching of thin platinum wires [218] 204 xix 9.30 Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 for 3 values of water temperature 205 9.31 Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 for 3 values of water temperature 205 9.32 Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 6.9 m s _ 1 for 3 values of water temperature 206 9.33 Estimated and calculated (film boiling) surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s - 1 for 3 values of water temperature. 206 9.34 Estimated critical heat flux (CHF) as a function of water velocity, during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature 207 9.35 Average heat flux as a function of water velocity, during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. 207 9.36 Estimated heat-transfer coefficient as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water at 25 °C for 3 values of water velocity 208 9.37 Photomicrographs of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C, at (a) ~ 3 mm from the surface and (b) centre. Magnification : 500 X 209 9.38 Photomicrographs of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C, at (a) ~ 3 mm from the surface and (b) centre. Magnification : 400 X 210 xx 9.39 Photograph of the macrostructure of the 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C 211 9.40 Photomicrographs of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s - 1 at 50 °C. (a) within the martensitic ring, (b) in the transition zone and (c) at the centre. Magnification : 800 X 211 9.40 Photomicrographs of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. (a) within the martensitic ring, (b) in the transition zone and (c) at the centre. Magnification : 800 X 212 9.41 As-quenched hardness distribution in 38.1 mm-dia. IF, alloyed and 1045 carbon steel bars. Use the right axis to read hardness in the IF steel bar. 213 9.42 Hardness distribution near the ends and at mid-length in a 100 mm-.dia. alloyed eutectoid steel bar quenched and tempered under industrial conditions 213 9.43 Outline of prior-austenite grain boundaries in a 38.1 mm-dia. alloyed eutectoid steel quenched bar, etched in a boiling alkaline sodium picrate solution. Magnification : 1000 X 214 9.44 Measured prior-austenite grain area distribution in a 38.1 mm-dia. al-loyed eutectoid steel quenched bar 214 10.1 Experimentally determined cooling curves for the alloyed eutectoid steel B showing the cooling rate for each test 224 10.2 Experimentally determined dilation-temperature curves obtained for con-tinuous cooling of the alloyed eutectoid steel B showing the cooling rate for each test 224 x x i 10.3 Continuous cooling test showing the thermal contraction of the austenite (high temperature) and the low temperature product phase obtained for the 17.5 °C s _ 1 cooling of the alloyed eutectoid steel B 225 10.4 Continuous cooling test showing the procedure to determine the start of transformation for the 17.5 °C s _ 1 cooling of the alloyed eutectoid steel B.225 10.5 Continuous cooling test showing the procedure to determine the end of transformation for the 17.5 °C s~x cooling of the alloyed eutectoid steel B.226 10.6 Continuous cooling diagram for the alloyed eutectoid steel A 226 10.7 Continuous cooling diagram for the alloyed eutectoid steel B 227 10.8 Continuous cooling diagram for the alloyed eutectoid steel C 227 10.9 Transformation start temperature as a function of cooling rate for the alloyed eutectoid steels A , B and C 228 10.10 Transformation start temperature plotted as a function of cooling rate for the alloyed eutectoid Steel A 228 10.11 Measured and predicted C C T transformation start times for the alloyed eutectoid Steel A 229 10.12 Continuous cooling diagram for the 1045 carbon steel 229 10.13 Transformation start temperature plotted as a function of cooling rate for the 1045 carbon steel 230 10.14 Measured and predicted C C T ferrite start time for the 1045 carbon steel. 230 10.15 Measured and predicted C C T pearlite start time for the 1045 carbon steel.231 10.16 Experimentally determined isothermal dilation-time curves for the al-loyed eutectoid steel A 231 10.17 IT diagram for the alloyed eutectoid steel A 232 10.18 IT diagram for the alloyed eutectoid steel B 232 10.19 IT diagram for the alloyed eutectoid steel C 233 xxii 10.20 Kinetic parameters for the pearlitic transformation in the 3 alloyed eutec-toid steels estimated with a 5-parameter, non-linear regression analysis (weighted and non-weighted), (a) n; (b) In b 234 10.21 Kinetic parameter 6, as a function of undercooling, for the pearlitic trans-formation in the 3 alloyed eutectoid steels estimated with a weighted, 5-parameter non-linear regression analysis, for a constant value of n = h. 235 10.22 Kinetic parameter b, as a function of undercooling, for the pearlitic trans-formation in 3 alloyed eutectoid steels 235 10.23 Kinetic parameter n for the bainitic transformation in the alloyed eutec-toid steel A , estimated with a 5-parameter, weighted, non-linear regres-sion analysis 236 10.24 Kinetic parameter b for the bainitic transformation in the alloyed eutec-toid steel A as a function of undercooling, estimated with a weighted, 5-parameter non-linear regression analysis, for a constant value of n = h. 236 10.25 Microstructure of etched Gleeble specimen (alloyed eutectoid steel A cooled at 58 °C s _ 1 ) showing the outline of prior austenite grains. . . . 237 10.26 Measured prior-austenite mean chord length distribution obtained in the alloyed eutectoid steel A Gleeble specimen cooled at 58 °C s _ 1 237 10.27 Ferrite fraction as a function of cooling rate for the 1045 (open circles) and 1025 (filled circles) [98] carbon steel samples 238 11.1 Specimen orientations with respect to neutron beams to measure (a) radial, (b) circumferential, and (c) axial strain components in force con-vective quenched steel bars 249 11.1 Specimen orientations with respect to neutron beams to measure (a) radial, (b) circumferential, and (c) axial strain components in force con-vective quenched steel bars 250 xxin 11.2 Schematic diagram showing the cutting sequence used to extract refer-ence slices for neutron diffraction measurements 251 11.3 Measured diffraction peak profiles in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C 252 11.4 Measured diffraction peak profiles in a 38.1 mm-dia. 1045 steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C 252 11.5 Measured diffraction peak profiles in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C showing the position of the F C C austenite (A( i n ) ) and B C T martensite (M( U 0 ) ) peaks. Data collected during Campaign 1 253 11.6 Measured diffraction peak profiles in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2 253 11.7 Measured and fitted diffraction peak profile in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C 254 11.8 Measured and fitted diffraction peak profile in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. . . . 254 11.9 Measured and fitted diffraction peak profile in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2 255 11.10 Mean scattering angle as a function of radial position obtained from the strain-free reference slices extracted from the 1045 carbon steel specimen. 255 11.11 Control volumes adopted for force balance calculations : (a) axial com-ponent (plan view) and (b) circumferential component 256 11.12 Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C 257 xxiv 11.13 Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. 257 11.14 Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2.258 11.15 Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C 258 11.16 Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C 259 11.17 Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2.259 12.1 Computational domain adopted for the finite-element simulation of ther-mal/mi crostructural evolution during forced convective quenching. . . . 284 12.2 Schematic representation of the boundary conditions adopted for the thermal/mi crostructural simulations 284 12.3 Finite-element mesh adopted for the thermal/mi crostructural simulations. 285 12.4 Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. The boiling curve shown as a broken line in Figure 12.5 was adopted as boundary condition 286 12.5 Boiling curves adopted as boundary condition to simulate forced con-vective quenching of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 50 °C 286 XXV 12.6 Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. The boiling curve shown as a solid line in Figure 12.5 was adopted as boundary condition 287 12.7 Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 6.9 m s _ 1 at 25 °C 287 12.8 Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 2.8 m s"1 at 75 °C 288 12.9 Effect of varying the value of thermal conductivity by ± 10 % adopted in the finite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s"1 at 50 °C 288 12.10 Effect of varying the surface heat flux by ± 10 % adopted in the finite-element simulation on the temperature response at the centre and sub-surface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s"1 at 50 °C 289 12.11 Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 25 °C. . . . 289 12.12 Comparison between predicted and measured temperature responses at the subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 25 °C 290 12.13 Comparison between predicted and measured cooling rates at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C 290 xxvi 12.14 Comparison between predicted and measured cooling rates at the sub-surface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 25 °C 291 12.15 Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. IF steel with water flowing at 4.8 m s"1 at 25 °C 291 12.16 Calculated microstructural evolution at the centre during forced convec-tive quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s"1 at 25 °C 292 12.17 Calculated microstructural evolution at the surface during forced con-vective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s- 1 at 25 °C 292 12.18 Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C 293 12.19 Comparison between predicted and measured temperature responses at mid-radius of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C 293 12.20 Comparison between predicted and measured cooling rates at the centre of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s- 1 at 50 °C 294 12.21 Comparison between predicted and measured cooling rates at mid-radius of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s- 1 at 50 °C 294 12.22 Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s _ 1 at 50 °C 295 XXVll 12.23 Calculated (with modified kinetics) microstructural evolution at the cen-tre during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s _ 1 at 50 °C 295 12.24 Calculated (with modified kinetics) microstructural evolution at the sur-face during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 4.8 m s _ 1 at 50 °C 296 12.25 Calculated (with modified kinetics) final microstructural distribution in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 4.8 m s"1 at 50 °C 296 12.26 Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s"1 at 75 °C 297 12.27 Comparison between predicted and measured temperature responses at the subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s"1 at 75 °C 297 12.28 Comparison between predicted and measured cooling rates at the centre ' of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s"1 at 75 °C 298 12.29 Comparison between predicted and measured cooling rates at the subsur-face of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 298 12.30 Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s"1 at 75 °C 299 12.31 Calculated microstructural evolution at the centre during forced convec-tive quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s _ 1 at 75 °C 299 xxvm 12.32 Calculated microstructural evolution at the surface during forced convec-tive quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s"1 at 75 °C 300 12.33 Calculated final microstructural distribution in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s - 1 at 75 °C. . 300 12.34 Effect of varying the value of Ms adopted in the finite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s - 1 at 75 °C 301 12.35 Effect of varying the value of M s adopted in the finite-element simulation on the microstructural evolution at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 301 12.36 Effect of varying the value of the kinetic constant in the Koistinen-Marburger equation adopted in the finite-element simulation on the tem-perature response at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. . 302 12.37 Effect of varying the value of the kinetic constant in the Koistinen-Marburger equation adopted in the finite-element simulation on the mi-crostructural evolution at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 302 12.38 Finite-element mesh adopted to simulate the thermal/mi crostructural evolution and stress generation during quenching 303 12.39 Computational domain adopted for the finite-element simulation of stress generation during forced convective quenching 304 xxix 12.40 Schematic representation of the boundary conditions applied for the ther-mal/mi crostructural simulations 304 12.41 Schematic representation of the boundary conditions applied for the stress simulations 305 12.42 Comparison between predicted and measured residual stress distribution (radial component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s"1 at 25 °C. 305 12.43 Comparison between predicted and measured residual stress distribu-tion (circumferential (hoop) component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C 306 12.44 Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s"1 at 25 °C 306 12.45 Comparison between predicted and measured residual stress distribution (radial component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. Modified kinetics 307 12.46 Comparison between predicted and measured residual stress distribution (circumferential (hoop) component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. Modified kinetics.307 12.47 Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. Modified kinetics 308 12.48 Comparison between predicted and measured residual stress distribu-tion (radial component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 308 xxx 12.49 Comparison between predicted and measured residual stress distribution (circumferential (hoop) component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 309 12.50 Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C 309 12.51 Evolution of radial, circumferential (hoop) and axial stresses at the sur-face during forced convective quenching of a 38.1 mm-dia. alloyed eutec-toid steel bar 310 12.52 Evolution of radial, circumferential (hoop) and axial stresses at the centre during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar 310 12.53 Evolution of radial, circumferential (hoop) and axial stresses at the sur-face during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar 311 12.54 Evolution of radial, circumferential (hoop) and axial stresses at the centre during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar 311 12.55 Calculated final microstructural distribution in a 100 mm-dia. alloyed eutectoid steel bar quenched from 850 °C with water flowing at 4.8 m s- 1 at 32 °C 312 E . l Start (closed circles) and finish (open circles) temperatures during con-tinuous cooling of tubular IF steel specimens 358 E.2 Predicted temperature response at the centre and mid-radius of a 38.1 mm-dia. IF steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. The start and end of the transformation are shown as broken lines. . . . 358 xxxi F . l (a) Exact data corresponding to n = 2, In 6 = —11 and tAv — 0.5 s. The error bars correspond to 0.05 units and are equal at all levels of t. (b) Exact data linearized using Eq. (F.l) 364 F.2 Non-linear regression applied to exact isothermal kinetic data 365 F.3 Residuals plot against estimated values corresponding to the non-linear regression of Figure F.2 365 F.4 Comparison of weighted and non weighted multiple non-linear regression applied to exact data : (a) estimated and observed values, (b) residuals plot against estimated values 366 F.5 Multiple (3 parameter) weighted non-linear regression applied to exper-imental data : estimated and observed values 367 F.6 Multiple (5 parameter) weighted non-linear regression applied to exper-imental data : estimated and observed values 367 xxxn L i s t o f S y m b o l s w e -- nodal displacement vector, m A -- area, m 2 4 -- partition coefficient, dimensionless AL -- cross-sectional area, m 2 AF -- frame area, / im 2 Aj ~ - Jeffries number, /mi 2 b -- kinetic parameter in Avrami equation, dimensionless [B] -- strain-nodal displacement matrix, m _ 1 Bi -- Biot number, dimensionless [C] -- capacitance matrix, J ° C _ 1 C R -- cooling rate, °C s _ 1 Cp - heat capacity, J k g - 1 K _ 1 D -- diameter, mm D0 -- initial Gleeble specimen diameter, mm D^ki -- material elastic constant tensor, M P a De -- equivalent diameter, m D • -m i n - dilation at the start of the transformation, mm ^max - dilation at the end of the transformation, mm D0 -- initial gleeble specimen diameter, mm E -- Young's modulus, M P a xxxm E Q A D - .equivalent area grain diameter pim {/} - load vector (heat transfer), W {F} - internal force vector, N F - yield function, M P a Fo - Fourier number, dimensionless g - plastic potential function, M P a G - mass flux, kg m - 2 s _ 1 h - heat-transfer coefficient, W m - 2 ° C _ 1 Hp - plastic modulus, M P a A i J t r a n s f - heat of transformation, J k g - 1 i - current, mA J 2 - second invariant of the deviatoric stress tensor, (MPa) 2 J0 - Bessel function of the first kind of order zero, dimensionless Ji - Bessel function of the first kind of order one, dimensionless [K] - stiffness matrix, k - thermal conductivity, J m _ 1 K _ 1 s _ 1 ki - ternary distribution coefficient, dimensionless L - characteristic length, m L h y d - hydrodynamic entry length, mm Lth - thermal entry length, mm Ms - martensite start temperature, °C M f - martensite finish temperature, °C n - kinetic parameter in Avrami equation, dimensionless [N] - shape function matrix, dimensionless NQ - number of grains, dimensionless x x x i v p - pressure, M P a Pw - wetted perimeter, m Pr - Prandtl number, dimensionless q - heat flux, M W m - 2 qavg - average heat flux, M W m~ 2 c7chf - critical heat flux, M W m - 2 g m a x - maximum heat flux, M W m - 2 <?rms - root mean square of the error in q, dimensionless Q - water flow rate, 1 s _ 1 r - number of future time steps, dimensionless r min ~ minimum value of the scaling factor, dimensionless R - radius, mm {R} - load vector (stress), N Re - Reynolds number, dimensionless 5* - least squares function, ° C 2 t - time, s A£ - time increment, s t* - virtual time, s tAVCOT - transformation start time (CCT), s IAV ~ transformation start time (IT), s T - temperature, °C T A e 3 - Ae3 temperature, °C T A c m - Acm temperature, °C Tc - centre temperature, °C xxxv Tf - fluid temperature, °C T0 - initial temperature, °C T s u r f - surface temperature, °C Ts - transformation start temperature, °C T g a t - saturation temperature, °C TAL - °C TA, - °C A T s a t - wall superheat, °C A r s u b - subcooling, °C {u} - displacement vector, m u\ - undercooling below T ^ , °C U3 - undercooling below TA3, °C v - water velocity, m s _ 1 Wi - weight in regression analysis, dimensionless X - fraction transformed, dimensionless Xj - sensitivity coefficient at sensor j, °C W _ 1 m 2 Yj - measured temperature at sensor j, °C a - linear thermal expansion coefficient, ° C _ 1 (3 - '• linear expansion due to transformation, m m" 5ij - Kronecker delta, dimensionless e - strain, m m _ 1 ee - elastic strain, m m _ 1 e° - ' initial ' strain, m m _ 1 £ p - plastic strain, m m _ 1 x x x v i 6th -- thermal strain, m m _ 1 £tr _ - transformation strain, m m _ 1 £tp _ - transformation plasticity strain, m m _ 1 £AE _ - strain due to variation of elastic properties with temperature, m m eAF -- strain due to variation of flow stress with temperature, m m _ 1 Sp - effective plastic strain, m m _ 1 K -- hardening parameter, A -- wavelength, A A m - eigenvalue, m- 1 dA -- loading parameter, dimensionless ^ -- viscosity, kg m- 1 s _ 1 v -- poisson's ratio, dimensionless # -- residual force vector, M P a p -- density, kg m- 3 a - stress, M P a o-e ~ - effective stress, M P a a' -- deviatoric component of stress, M P a T -- isothermal incubation time, s 6 -- mean scattering angle, degrees 0o -- reference mean scattering angle, degrees u -- relaxation parameter, dimensionless xxxvn Coordinates r z 9 Superindices e m m + 1 radial, m axial, m circumferential (hoop), m element level previous time step current time step x x x v n i A c k n o w l e d g e m e n t s I would like to thank Dr. E .B. Hawbolt and Dr. J .K. Brimacombe for their interest and suggestions throughout the course of this work. The assistance of the technical staff at U B C , in particular Mr. B . Chau, Mr. P. Musil and Mr. R. Cardeno is greatly appreci-ated. Thanks are also extended to Mr. J . Lapointe for his help during the construction of the laboratory quenching facility. The residual stress measurements could not have been made without the guidance and technical expertise of Dr J . Root and the person-nel at A E C L (Chalk River Laboratories). The assistance of Prof. J .V. Beck (Michigan State University) in providing the original code CONTA is gratefully acknowledged. Discussions with Dr. S. Cockroft and Dr. Y . Nagasaka on the implementation of their respective computer programs have been also invaluable. Thanks are also extended to my fellow graduate students for their voluntary assistance and to my friends who made this a very enjoyable stay. I would like to express my gratitude to the Universidad Nacional Autonoma de Mexico (Departamento de Ingenieria Quimica Metalurgica) and the Department of Metals and Materials Engineering of the University of British Columbia for the financial support given during my professional studies. Finally, I would like to thank the members of my family for their understanding and encouragement. xxxix Chapter 1 Introduction 1.1 Mic ros t ruc tu ra l Engineer ing Heat treatment processes constitute a very important operation in the processing of engi-neering components. Through a combination of heating and cooling cycles, the optimum microstructure required to obtain the desired mechanical properties can be produced. At the same time, however, residual stresses may be introduced into the component due to volume changes associated with thermal contraction and phase transformations which may generate distortion and even cracking. However, residual stresses are not always detrimental. For example, surface compressive residual stresses contribute to the frac-ture resistance of case-hardened components [1]. It follows that while distortion must be minimized, the residual stress pattern produced by a given heat treatment must be known and optimized. The need for predicting the mechanical properties and residual stress distribution resulting from a given heat treatment has long been recognized. More stringent require-ments of high productivity, performance and low energy consumption have favoured re-search towards the linking of mechanical properties and process variables. Consequently, the trial-and-error procedures that have been followed in the past are being replaced by more fundamental methodologies which incorporate concepts of transport phenomena, mechanical behaviour, solid-solid phase transformations, and microstructure-properties relationships, within a paradigm known as 'microstructural engineering'. 1 Introduction 2 1.2 Heat Treatment of Grinding Media Grinding using either steel rods or balls is the greatest operating cost in mineral process-ing [2]. Most of this cost stems from rod/ball and energy consumption [3]; thus, the wear rate of grinding media in rod mills becomes a factor of economic significance, and meth-ods to reduce grinding media wear, and steel consumption in general, are desirable [4]. By using grinding media of higher quality and, thus, reducing breakage, the grinding rods can be worn to smaller sizes which can reduce operating costs significantly [5]. Abra-sion, impact, erosion, and corrosion, contribute to the wear of grinding media under wet grinding [3]. Typically, grinding media are fabricated of high carbon steel in order to maximize attainable hardness. The addition of alloying elements improves hardenability and hardness and, therefore, abrasion resistance, but it may also result in a more brittle product. A combination of alloying additions and heat treatment is, therefore, required to accomplish optimum mechanical behaviour during service of grinding media in rod mills. Pugh and Ma [6] reported a reduction in grinding rod consumption of up to 25 % when heat treated grinding rods of 100 mm (4 in) dia. were used in production trials at the Endako Mines Division of Placer Dome Inc. The proprietary heat treatment process developed at AltaSteel is based on forced convective quenching of steel rods, followed by self-tempering, and produces a material with a tempered martensitic case and a bainitic/pearlitic core. The hardness profile ranges from Rc 55 at the surface to Rc 40 at the core. The mill tests [6] showed a low initial wear rate; the high surface hardness resulted in an almost constant wear rate until the first 30 mm were worn. By comparison, the initial wear rate in AISI 1090 rods, tested under the same conditions, was twice as high as that of the heat treated rods and decreased as the rod wore down. Clearly, the microstructural distribution plays an important role on wear rate behaviour of grinding rods, as it affects both the abrasion resistance and the toughness. Despite improved rod performance, there are production problems that need to be Introduction 3 addressed. Large transient stresses, responsible for distortion in the final product, are caused by non uniform cooling and/or phase evolution. In the long (1.8 m) rods required for grinding mills, end effects produce significant distortion during quenching; the ends having to be straightened using induction" coils. Variability in the "equalization" tem-perature, i.e., the maximum temperature reached during self-tempering has also been observed. Another area of interest for rod producers is that of quality control; at the moment only crude impact tests are used to asses product quality. Thus, in order to optimize the process, as well as to assist in future developments, it is desirable to gen-erate a predictive tool that would obviate the need for expensive and time-consuming trial-and-error operations. In consequence, fundamental knowledge related to the process needs to be generated. 1.3 Statement of the Problem The thermal, microstructural and stress fields obtained during heat treatment interact in a complex manner as shown in Figure 1.1 [7]. In general, the temperature distribution upon cooling is nonhomogeneous, which generates thermal stresses (interaction 1). The microstructural evolution depends, basically, on the cooling history of the component (interaction 2) and, at the same time, modifies the temperature response through the latent heat evolved during the transformations (interaction 6). An unequal volume change distribution accompanying the phase transformations sets up transformation stresses (interaction 3) which, in turn, may affect the transformation kinetics (interaction 5). Finally, the internal stresses generated upon cooling do mechanical work, part of which is converted into heat (interaction 4); however, the amount of heat generated is negligible and, therefore, this interaction is usually ignored. In addition to these effects, material-related properties such as temperature-dependent thermophysical and thermomechanical properties, prior-austenite grain size distribution Introduction 4 and carbon concentration gradients (in case hardening) play an important role in deter-mining the final microstructure and residual stress distributions as well as distortion. In this study, the concept of microstructural engineering has been applied to study forced convective quenching of steel bars. The approach taken involved the development and application of mathematical models of heat transfer, microstructural evolution and stress generation, coupled with the experimental determination of the active heat-transfer boundary condition in a laboratory facility designed and built for that purpose. Addition-ally, phase transformation kinetics were measured in tubular specimens, and the residual stress distribution and final microstructure and hardness in forced convective quenched steel bars were determined. Introduction 5 \ thermal s t r e s s 4. heat generated by mechanical w o r k 5. stress-induced transformation. 2 t e m p e r a t u r e \ \ 6 . latent dependent p h a s « \ V e a t t ransformat ion metallic s t ruc tures 3. transformation s tress Figure 1.1: Thermal, microstructural and stress field interactions during heat treatments [TI-Chapter 2 L i te ra ture Review In this chapter, the literature relevant to the problem is reviewed. First, a general overview of the methods available to design a heat treatment is presented. Then, mate-rial related to the kinetics of phase transformations in steels, heat transfer under forced convective boiling conditions, and methods for residual stress measurement is reviewed. The final section is dedicated to a discussion of published mathematical models of mi-crostructural evolution and stress generation during heat treatments. 2.1 Heat Treatment Design Several approaches have been developed to predict the performance of a given quenching process. They can be classified as : i) trial-and-error, ii) characterization of quench bath quality, and iii) process modeling. A diagram showing the conceptual differences among these techniques is given in Figure 2.1. As can be seen from this figure, the complexity increases as one moves from the trial-and-error approach to the process modeling method-ology as does the insight gained on the effect of process parameters. In the following, a review of these methodologies is presented. 2.1.1 Empirical Approaches In the past, heat treatment design was conducted exclusively via trial-and-error proce-dures which are clearly expensive as well as time consuming. They effectively act as a 6 Literature Review 7 'black box' and, therefore, the results obtained can only be applied under a restricted set of conditions. Factorial design provides a systematic method for conducting experi-ments, but still fails to give information on the processes that take place during a heat treatment. After selecting the factors that are considered to have an impact on the final result, any one of the standard factorial design plans can be applied. In the case of factorial experimentation (the most commonly used of these standard plans) all levels of a given factor are combined with all levels of each of the other factors [8]. Two examples where factorial design has been successfully applied are the heat treatment of nickel-iron-base superalloys (to optimize for room-temperature tensile properties) [9], and the heat treatment of gears (to optimize for hardness and distortion) [10]. 2.1.2 Characterization of Quench-Bath Quality The resulting mechanical properties after a heat treatment are linked to the microstruc-ture which, in turn, is governed by the heat extraction characteristics of the quench bath. Over the years, a number of tests have been designed to measure the quality of the quench bath, based on either its hardening power (hardenability, Jominy end quench) or its cooling power ( G M quenchometer, hot wire test, interval five-second test, and cooling curve) [11]. 2.1.2.1 Quench-Bath Quality Based on Hardening Power Early attempts to predict mechanical properties were concentrated on the prediction of the hardness that can be obtained from a quenching operation. Thus, the concept of hardenability, 'the depth of useful hardness which can be produced for given quenching conditions' [12] evolved. The hardenability of a particular steel can be predicted, based on its chemistry, using the Grossman method [13]. This method characterizes the effect of the amount of each Literature Review 8 alloying element in a steel on the hardenability as a multiplying factor which is used to obtain an ideal critical diameter (50 % martensite at the centre of a bar quenched in a medium of infinite severity); it should be noted that only a linear effect of composition on hardenability is considered. This method serves only as a tool for comparing steels of different compositions since the heat transfer characteristics of the cooling media are described through an average quantity called 'quench severity'. Kirkaldy [14] pointed out the lack of generality of this method (as evidenced by the number of sets of multiplying factors that have been reported) and attributed it to the linearity of the function adopted for the calculation of the ideal diameter. A critical review of hardenability predictors based on chemical composition alone can be found in [15]. The concept of hardenability is qualitative in nature and, therefore, methods for describing a quantitative relationship between cooling rates and hardness are required. The Jominy end-quench test provides such a relationship. In order to be able to predict the behaviour of a given part, a Jominy test should be conducted under standardized austenitizing and quenching conditions and criteria to correlate experimental and indus-trial results should be proposed. Such empirical relationships between end-quench bar positions and positions in the actual piece are available in the literature for plates [16] and bars [17]. Hardenability predictors can be very useful in the heat treatment industry. Kirkaldy et al. [18] reviewed the state of the art in hardenability prediction and concluded that the Jominy test can be economically replaced by properly calibrated predictors. A n additional advantage is the possibility of on-line control of the process. One such hard-enability predictor was developed by Kirkaldy and Venugopalan [14]. Umemoto et al. [19] used isothermal transformation kinetics information to predict martensite fraction as a function of distance from the quenched end in a Jominy bar and obtained very good agreement between predicted and experimental values. Continuous cooling transformation (CCT) diagrams are also used for designing heat Literature Review 9 treatments [20]. Again, a criterion is needed to correlate the C C T diagrams obtained in a laboratory with the response in actual practice. Rose et al. [21] proposed the use of the time to cool from AC3 to 500 °C as the criterion for an equivalence between C C T diagrams and end-quench tests. Cias [22] found a good correlation when the half-cooling time (the time to reach the mean temperature between AC3 and room temperature) concept was used. It should be noted that Jominy curves as well as C C T diagrams are strictly valid for the cooling and austenitizing conditions under which they were derived. Also, published results may contain empirical errors - as pointed out in [18]. For automation reasons it would be convenient to create a description of C C T dia-grams that would lend itself to automatic retrieving. One possibility (suggested in [23]) would be to save a compilation of C C T curves arranged by composition and grain size such that upon request the closest C C T curve for a given chemistry and grain size could be produced. Alternatively, a set of linear regression formulas calibrated to the property of interest could be deduced. The latter approach has been adopted to generate formu-lae for the prediction of microstructure and hardness of low-alloy steels in the form of critical cooling velocities at 700 °C necessary to attain a given microstructure [24]. The austenitizing conditions have been characterized by a thermally activated grain growth parameter called the austenitizing parameter. 2.1.2.2 Quench-Bath Qual i ty Based on Coo l ing Power While methods based on hardening power are concerned with the ability of a quench bath to produce a given hardness distribution, methods that measure the cooling power are directed towards evaluating the amount of heat that can be extracted by a quench bath. In the Quenchometer test [25], a 22 mm-dia. nickel sphere is heated to 885 °C and subsequently dropped into a wire basket which is suspended in 200 ml of quench oil kept Literature Review 10 at a temperature between 20 and 30 °C. The time the sphere spends between 885 and 354 °C is recorded by first activating a switch when the sphere passes a photoelectric sensor and falls into the quenching bath and then deactivating it when the nickel becomes magnetic (at its Curie temperature, i.e., 354 °C). Typical Quenchometer values for slow, medium and fast quenching oils are 15 to 20, 11 to 14, and 8 to 10 seconds, respectively. Quenchometer results are a good indicator for monitoring oil degradation but it has not been possible to correlate them to final properties in the quenched part. A major drawback of this technique is that it covers only a portion of the cooling path. Con-versely, cooling curves covering the entire range of temperatures can be obtained from instrumented cylindrical metal probes [11,26]. The probes are commonly made of either 304 stainless steel or Inconel 600 and are instrumented with a single thermocouple lo-cated in the geometric centre (probe diameter ranges from 12.7 to 50 mm). From the experimentally obtained temperature vs time curve a cooling rate curve is computed di-rectly as the first derivative of temperature with respect to time. Bates and Totten [26] obtained cooling curves and cooling rate curves for a 25 mm-dia., 304 stainless steel probe quenched in agitated slow, medium and fast quenching oil (with Quenchometer speeds of 21, 14 and 8.5 s). The computed maximum cooling rate of the slow oil was higher than for medium speed oil, which could not have been detected by a Quenchome-ter test alone. Although more informative than the Quenchometer test, the cooling curve method suffers from the fact that a thermocouple located at the geometrical centre of any specimen does not reflect accurately the heat transfer at the surface. Tamura [27] recog-nized this shortcoming and conducted a study involving both the commonly used probe (made of Inconel 600 and instrumented with a thermocouple in the geometrical centre) and a Japanese Industrial Standard (JIS) specimen (made of silver and instrumented with a thermocouple at its surface at mid-length). Both probes are solid cylinders. He compared the response of probes quenched in oil with varying amounts of additives and Literature Review 11 showed that the JIS specimen is more sensitive to changes in heat transfer characteris-tics of the quench bath. An important question regarding the measurement of a cooling curve is that of standardisation, as pointed out by Bodin and Segerberg [28]. Several national standards have been used and a draft proposal for an international standard for oil quenching is under review; a proposal for a similar test for water-based polymer quenchants is being formulated. Once a cooling curve and the corresponding cooling rate curve have been obtained they can be analyzed in a number of ways varying from visual comparison (i.e., plotting several cooling curves side by side in order to compare quench baths) to attempts to correlate cooling curve data to mechanical properties (by including phase transformation characteristics of the specimen) [11,28]. Among the latter, the quench factor analysis (QFA) has received a great deal of attention in the aluminum industry due to its ability to predict the performance of quenching media [29,30]. This method requires a cooling curve and a T T P C-curve1 (iso-strength contours plotted in a temperature vs time graph) and is similar to the additivity rule [31]. The quench factor is essentially the summation, over a temperature range, of fractional times (time spent at a given temperature divided by the time required to reach the T T P curve isothermally at that particular temperature) for a given cooling curve. Both, type and depth of corrosion as well as maximum yield strength have been correlated to quench factor [29]. In general terms, low values of quench factor are associated with high quench rates and high strengths [11]. Bates [32] applied the concept of quench factor analysis to the selection of quench baths for aluminum parts considering attainable tensile properties as well as distortion. In order to minimize distortion, a quenchant that will produce a heat-transfer coefficient slightly higher than the equivalent quench factor in the critical section thickness should be selected [32]. Bates and Totten [33] successfully predicted the as-quenched hardness of 4130, 4140 and 1045 1Even though a T T P curve follows the typical s-shape associated with T T T and C C T curves, it differs in that it is calibrated with respect to the property of interest for a given chemistry Literature Review 12 steels using QFA. Reti et al. [34] have taken this concept one step further and developed the so-called Cooling/Transformation Analysis (CTA). The CTA is a computerized technique devel-oped to quantitatively characterize the quenching performance of a given quenching medium. Like QFA, it provides a single figure (the hardening power) that relates the cooling condition to the resulting microstructure. The starting point for this technique is a measured, directly or indirectly, surface heat flux history in a series of standard probes quenched in several media. The heat flux function is characterized by the maximum heat flux as well as the average heat flux values for the 300 - 400 °C, 400 - 500 °C, 500 -600 °C, 600 - 700 °C and 700 - 850 °C ranges. From this information the corresponding cooling curves and cooling rate curves are obtained. The parameters that characterize the cooling rate curves are the maximum cooling rate, the temperature corresponding to this figure, and the mean cooling rates over the 300 - 400 °C, 400 - 500 °C, 500 - 600 °C, and 600 - 700 °C ranges. Experiments based on computer simulations [34] showed that C T A results are compatible with results produced through cooling curve analysis and QFA. It is interesting to note that studies on cooling curves have shown differences between hardness predictions based on C C T diagrams and those based on cooling curve analysis [35-37]. The reason for this discrepancy is that C C T diagrams are commonly constructed from 'linear' cooling conditions with cooling rates that do not vary significantly and, therefore, the heat extraction history is quite different from one where boiling is present. Gupta [38] proposed a method for selecting quenching conditions based on the com-putation of the heat-transfer coefficient for a number of quench baths. The heat-transfer coefficients were calculated from the solution of the inverse heat conduction problem for instrumented disks (200 mm-dia. x 20 mm thick) made of 304 stainless steel. To select the quench bath that produces a part with a specific hardness distribution, the required thermal history was first obtained from either a C C T diagram or a Jominy curve. This Literature Review 13 thermal response was imposed at the desired location and a mathematical model to solve the inverse heat conduction problem was used to compute the required heat-transfer coefficient vs surface temperature curve. This curve was then compared to the curves obtained with the instrumented disks to select the appropriate quench bath. The method was successfully applied to the quenching of a 100 mm-dia. steel grinding ball. Liscic and Filetin [39] developed software for predicting the hardness distribution in quenched bars from measured cooling curves. The technique is based on measuring the temperature at the surface and at an internal point near the surface of cylindrical specimen instrumented with an specially designed probe. The specimen (50 mm-dia. x 200 mm long) is made of AISI 304 stainless steel to ensure that no heat will be generated due to phase change. The measured temperature responses are used to compute the surface heat flux by directly applying Fourier's law, with the conductivity taken as a constant. The quenching conditions are then described by three functions : T s u r f = f(t), q = F(t) and q = f(Tsurt). A test specimen made of the steel of interest needs then to be quenched and its hardness distribution determined; this data is transformed into the equivalent Jominy curve. With this database, the hardness distribution in a bar of the diameter of interest can be computed by using regression equations from a series of Craft-Lamont diagrams. Liscic et al. [40] suggested a qualitative description of depth of hardening and risk of cracking based on the method described above. For this purpose the maximum flux density (qmax) and the heat extracted (as calculated by integrating // q dt) must be calculated from the q = f{TsuT^) and q = F(t) functions, respectively. For two given quench baths, the one that results in a higher value of the integral of surface heat flux with respect to time will produce a greater depth of hardening (since it corresponds to a higher amount of heat extracted). On the other hand, the quench bath with the lower value of surface temperature corresponding to a maximum heat flux at later times will have a higher risk of cracking (since the largest amount of heat would be extracted at temperatures near or below M J . Literature Review 14 Wetting kinematics measurements have also been proposed as the basis for hardness predictions [41]. Wetting time is the time required for the wetting front to arrive at a given location in the specimen. Since the wetting time irepresents the heat extraction at the surface, the same hardness is expected in locations having the same wetting time. The method requires a calibration curve of hardness versus wetting time obtained under laboratory conditions. After measuring the wetting time under plant conditions the hardness can be predicted based on the calibration curve. A shortcoming of this method is the experimental difficulties associated with measuring the wetting time under plant conditions. Also, the calibration curves are unique to the geometry used to determine them and cannot be easily extended to other geometries. 2.1.3 Process M o d e l i n g None of the predictors previously mentioned includes a careful simulation of thermal histories but rely on either average values of heat transfer conditions or correlations based on cooling rates at particular stages. Some other simplified assumptions need to be made in order to keep the computational requirements to a minimum. In contrast, a process modeling approach can provide detailed thermal, microstructural and mechanical responses obtained during a particular heat treatment. The methodology involves the so-lution (through numerical methods) of the partial differential equations that describe the evolution of the thermal and stress fields. A relatively large data bank containing ther-mophysical and thermomechanical properties, as well as transformation kinetics, must be available. Laboratory-scale simulations are usually required to verify the different mathematical models involved. A vast amount of information regarding the flow of heat as well as microstructural and mechanical information at all stages of the heat treatment is generated but, due to the high non-linearities involved in real problems, extensive computational resources are required. In particularly complex cases, efforts should be Literature Review 15 devoted towards implementing a practical way to visualize the numerical results. A crit-ical component of the models is a detailed heat transfer study to characterize the active boundary condition. Once the microstructural distribution is computed, the mechanical properties can be predicted from empirical correlations. In this fashion, mathematical models can be applied to the prediction of mechanical properties from information of process variables. Crack formation and residual stress distribution can also be predicted if a stress model is available. The process modeling approach was applied by Campbell et al. [42-44] to the simulation of controlled cooling of steel rods in a Stelmor line. A similar approach was adopted by Wallis et al. [45] to optimize the heat treatment of superalloy forgings and by Persampieri et al. [46] to simulate water spray quenching of a flat disc and a hollow cylinder of AISI 4335V steel. Another example of the use of this methodology, as applied to water spray quenching of steel bars, has been presented by Nagasaka et al. [47]. The commercial package HEARTS is also built on the microstruc-tural engineering concept and has been used to model carburized quenching of a gear tooth [48]. A detailed discussion of published mathematical models developed to simulate mi-crostructural evolution and stress generation during heat treatments is presented in a later section. 2.1.4 Computerized Information Systems The simulation models developed, based on first principles, can be complicated to run and may require expert knowledge. They can also be expensive. For these reasons, there has been a strong interest in developing PC-based software for material selection, property prediction, and design of heat treating operations. A series of computerized information systems for steel selection and prediction of mechanical properties have been reported in the literature [49-52]. One of these systems [52] also includes access to data bases of several steel standards and methods for heat treatments. A comprehensive review Literature Review 16 of the application of computerized information systems can be found in [53]. Bodin and Segerberg [54] developed a benchmark testing procedure to evaluate commercially available, PC-based programs for heat treatment of steel components. They tested three programs against each other and experimental measurements of quenched cylindrical specimens of steels of low and high hardenability and found some discrepancies, which they attributed to differences in the characterization of heat extraction. 2.2 Heat Transfer in Forced Convective Quenching The resulting hardness profile and residual stress distribution in a heat treated part depends primarily on cooling rate history. Therefore, characterizing the rate of heat removed from the surface by the quenching medium is of vital importance. But heat extraction during quenching is a complicated process due to the presence of boiling. In this section, the general characteristics of heat transfer, including surface boiling, are presented, followed by a review of techniques for characterizing surface heat flow. Then, specific studies of forced convective boiling are reviewed. 2.2.1 General By recording the temperature response within instrumented specimens and simultane-ously filming the reactions at the surface, a number of researchers [55-57] have established the different modes of heat transfer during quenching of metallic parts in typical quench-ing media. When a component at high temperature is put in contact with a vaporisable liquid, three distinct stages of cooling can be distinguished from the surface temperature response (see Figure 2.2). After a short period characterized by intense boiling, the first stage, known as film boiling, starts. During film boiling a blanket of vapour surrounding the entire piece is formed, thus creating a barrier for heat transfer; the corresponding low rate of heat extraction is reflected in a small rate of change in the temperature response Literature Review 17 in the cooling curve. As the surface temperature drops, the amount of heat reaching the liquid decreases resulting in a partial collapse of the vapour blanket. During this transition boiling regime, the breaking up of the vapour blanket provides fresh sites for metal/liquid contact and, therefore, the heat extraction rate increases. With a further reduction of temperature, smaller bubbles and an increased metal/liquid contact occur, which are characteristic of the nucleate boiling stage. This is the most efficient mode of heat extraction during quenching and results in the maximum surface heat flux. For surface temperatures near the boiling point of the quenchant and below, boiling stops and heat extraction is entirely controlled by either natural or forced convection. In selecting a quenchant for a given application, the complete history of rate of heat extraction at the surface needs to be considered. Ideally, to minimize thermal stress and optimize the microstructure required, a quenching medium would produce low cooling rates at high temperatures, high cooling rates at temperatures where phase transfor-mations occur (i.e., between A3 and Ms), and low cooling rates at the final stages of cooling [58]. Generally speaking, water and brine (aqueous solutions of sodium or cal-cium chloride) produce the highest cooling rates at virtually all stages of cooling; while this may result in higher hardnesses, it is also a potential source of severe distortion and even crack generation. On the other hand, oils produce more uniform, albeit slower, heat extraction rates as compared to water or brine and are favoured in a large number of applications. Recently, polymeric aqueous solutions have received increasing attention due to the variety of heat extraction rates that are possible to obtain with them. Obvi-ously, the selection of the most appropriate quench bath should also include factors such as material hardenability, component shape and surface condition, among others. 2.2.2 Surface Heat F lux Characterization A cooling curve provides valuable information on the different stages of cooling dur-ing quenching. However, computer codes designed for the prediction of microstructural Literature Review 18 and thermomechanical responses in a metallic part require values of either surface heat-transfer coefficient or surface heat flux, as a function of surface temperature, in order to characterize the active boundary condition. There are three possible ways to accomplish this task : 1. Indirect measurement of the surface heat flux. 2. Direct measurement of the surface heat flux. 3. Prediction of the surface heat flux. Diller [59] has reviewed techniques developed for heat flux measurements and sub-classified the category of indirect measurement of surface heat flux according to : 1. Measurements based on spatial temperature difference. A temperature difference in the specimen is measured across a known thermal resistance gauge. The major advantage in using these gauges resides in the fact that the output signals are proportional to the surface heat flux. 2. Measurements based on temperature change with time. The thermal response in the specimen is measured over time with a known thermal capacitance gauge. Given that only a single temperature measurement is required in this case, these gauges are the simplest ones to fabricate. Most of the effort is concentrated on recovering the heat flux - usually through the solution of the inverse heat conduction problem 2. 3. Measurements based on spatial temperature difference in the fluid. The temper-ature gradient in the fluid next to the surface is measured. This approach also involves the accurate knowledge of fluid properties as well as fluid flow characteri-zation and, therefore, it has not been widely used. 2 A detailed discussion on the inverse heat conduction problem is presented in Chapter 5. Literature Review 19 On the other hand, a direct measurement of the energy leaving or reaching a surface at steady or quasi-steady state conditions can provide a very accurate measure of the sur-face heat flux, as long as thermal equilibrium at all temperatures exists. Bamberger and Prinz [60] have pointed out that the rapid change of heat flux with surface temperature in transition boiling makes it impossible to attain thermal equilibrium and, therefore, pre-cludes the direct measurement of heat flux during this stage of boiling. Given that energy exchange is usually achieved through electrical heating, which has power limitations, this technique is not normally used in high-heat flux or high-temperature applications. For either direct or indirect measurements of the surface heat flux, special attention should be given to the design of the gauge. The goals of any heat flux gauge are twofold : i) the flow of fluid over the surface should not be affected by the presence of the gauge, and ii) the conduction pattern in the solid should not be disrupted. Unfortunately, a gauge always alters the local distribution of both temperature and heat flux, but this disruption should be kept to a minimum. Another issue in heat flux measurement is calibration. Although any mode of heat transfer can be applied to calibrate a given gauge, radiation is used in most of the cases. Two limitations of the calibration procedures are : 1) Most gauges are calibrated at room temperature, and 2) Calibration of the transient response is rarely done. An alternative to measure the surface heat flux is to predict it theoretically. How-ever, this is not an easy task, as it involves a coupled heat and fluid flow analysis of the quenching bath. Moreover, the boiling that accompanies quenching involves several modes of heat transfer between the surface of the test piece, the bath and bubbles, as well as kinetics of bubble formation and growth [61-63]. 2.2.3 B o i l i n g Heat Transfer. The different modes of boiling are classified according to the hydrodynamics of the bath and the operational temperature of the fluid with respect to its saturation point. If the Literature Review 20 liquid is quiescent, the boiling mode is termed pool boiling, whereas in forced convective boiling, the liquid is set in motion by external forces. It should be noted that bubble dynamics induce mixing near the surface of the test piece in both modes. When the operational temperature of the liquid is kept below its saturation point subcooled boil-ing occurs; on the other hand, in the case of saturated boiling, the liquid is kept at a temperature slightly above saturation. The bulk of investigations on boiling have been concentrated on saturated pool boil-ing. In an early study, Nukiyama [64] devised a power-controlled experiment to study the boiling behaviour of saturated water at atmospheric pressure. He identified the different stages of boiling on heating and summarized them in a boiling curve : this is a plot of surface heat flux vs excess temperature (the difference between surface and saturation temperature), as shown in Figure 2.3. Two points of particular interest in a boiling curve are the critical heat flux (maximum heat flux and lower temperature bound of the transition region) and the Leidenfrost point (minimum heat flux and upper temperature bound of the transition region). The correspondence between the surface temperature response upon cooling in subcooled pool boiling and the boiling curve is schematically shown in Figure 2.4 [56]. When boiling is present, Newton's law of cooling takes the following form : 1 = HTsuri-TsJ (2.1) where q is the heat flux, T g a t is the saturation temperature of the fluid and h is the heat transfer coefficient which is a strong function of the surface temperature, the geometrical configuration of the test piece, fluid properties and the fluid flow conditions. Clearly, by inspecting the boiling curve, it is not possible to obtain a single correlation to encompass all modes of heat transfer with boiling. Instead, most of the studies have dealt with separate sections of the boiling curve. For example, the heat flux for nucleate pool boiling on clean surfaces can be predicted using an empirical correlation developed Literature Review 21 by Rohsenow [65], which includes the effect of surface tension as well as the particular combination of fluid-surface. On the other hand, film pool boiling correlations have been developed based on experiments on laminar film condensation. Due to the higher temperatures associated with film boiling, radiation effects need to be incorporated. The correlations developed for boiling heat transfer can be classified according to [66] : 1. Correlations of an empirical nature, which make no assumptions about the mech-anisms of heat transfer, but solely attempt a functional relation between the heat-transfer coefficient and independent variables. 2. Correlations that recognize that departure from a thermodynamic equilibrium con-dition can occur and attempt to calculate the actual vapor quality and vapor tem-perature. 3. Semi-theoretical models, where attempts are made to write equations for the indi-vidual hydrodynamic and heat transfer processes in the heated channel and relate them to the heated wall temperature. Hewitt [67] noted that most correlations developed for boiling are of an empirical nature, although correlations obtained for pool boiling have, in general, a sounder physical basis; but even in that number of empirical factors have to be introduced. An issue that has received a good deal of attention is the applicability of correlations obtained under steady-state conditions to applications where transient behaviour occurs [68-72]. While steady-state boiling curves are commonly obtained by controlled heating of the test piece, transient measurements are usually conducted by quenching and lend themselves to obtaining a complete boiling curve - which is not always the case with steady-state measurements involving fluids having relatively high boiling points (like water). Bergles and Thompson [68] performed steady-state and transient experiments with saturated Freon-113, saturated water, and liquid nitrogen, to establish possible discrepancies between steady-state pool boiling data and quench data. For the water Literature Review 22 tests, they found several differences between the two kinds of experiments. In particular, the peak heat flux was characterized by lower values and higher wall superheats in the transient experiments. The minimum heat flux was also shifted to considerably lower values. Similar observations were made with Freon and nitrogen. Unfortunately, the steady-state measurements were conducted with stainless steel, whereas copper was used for the transient ones. Peyayopanakul and Westwater [69] conducted similar experiments, but removed the uncertainty caused by using different metals by testing the circular end of a 5.08 cm copper rod in liquid nitrogen at atmospheric pressure in both, steady-state and transient experiments. Several quenching times were obtained by varying the thickness of the specimen. For the fastest quenches (thinner specimens) a departure from steady-state conditions was observed. A minimum thickness of the copper specimen (in this case 25 mm) was needed to obtain a complete boiling curve equivalent to the steady-state one. A n interesting observation was that different minimum thicknesses were required to obtain quasi-steady-state responses for the different regions of the boiling curve. Huang et al. [70] obtained transient and steady-state boiling curves under forced upflow of water in a hollow copper cylinder using water as the fluid. To minimize the possible influence of external factors, they used the same test section and surface condition for all tests, and devised an in-situ transient calibration of the instrumented specimen. A single data reduction method was used in all cases. In the range of pressure < 7 bar, mass flow rate < 300 kg m - 2 s _ 1 and subcooling < 15 K , good agreement between steady-state and transient boiling curves was observed. Again, steady-state and transient experiments affected each stage of the boiling curve differently. The system response in film boiling was very similar for both modes throughout the parameter range investigated, possibly due to the low cooling rates experienced in that stage. For the high end of the parameter range investigated, the minimum film boiling temperature was lower for the transient experiments. The quench data was consistently lower in the transition region of the boiling curve, while no difference was found in the nucleate stage. Ishigai Literature Review 23 et al. [73] measured the surface heat flux, as a function of surface temperature, near the stagnation point of a plane water jet impinging on a flat surface using steady-state and transient measurements. The samples were made of stainless steel and instrumented with a thermocouple welded to the center of the back face. Their results showed slightly lower values of heat flux in the film boiling region when steady-state measurements were performed. In the nucleate boiling stage, the boiling curve determined by the transient experiment was shifted to higher temperatures. It should be pointed out that the results in the nucleate boiling region of the boiling curve deduced from the steady-state experiments lay on the extension of the nucleate pool boiling curve. Other characteristics of the experimental set-ups that may affect the shape of the boiling curve are [61,62,74] : surface finish, surface contamination, presence of noncon-densible gases, heater thickness, heater material, method of heating, and flow velocity. 2.2.4 Forced Convective Quenching The need for higher cooling rates than those provided by pool boiling has prompted the development of more efficient cooling schemes using forced convective boiling. Spray cooling is used in 'press quenching' of aluminum alloys because it enhances the heat transfer rate in all stages of cooling and increases the Leidenfrost temperature (which could possibly eliminate the film boiling regime entirely) [75]. Impinging liquid jets have been widely used either in submerged or free surface mode; Wolf et al. [74] have recently reviewed the literature available on jet impingement boiling. A number of studies have been conducted on forced flow of boiling helium, which is commonly used for cooling superconducting magnets [76]. Water jet cooling is applied in run-out tables to obtain high heat extraction rates; this accelerated cooling refines the ferrite grain size, producing steels of higher strength [77]. During continuous casting of steel, heat exchange between the copper mould and the cooling water takes place by forced convection. Under ideal operating conditions, the temperature of the flowing water is kept well below its boiling Literature Review 24 point; but, if the mould is hot enough, boiling may occur which can lead to defects in the casting. Samarasekera and Brimacombe [78] computed boiling behaviour diagrams (plots of water velocity vs water pressure, showing areas of boiling and no boiling) based on published correlations for fully developed pool boiling and incipient boiling, and used the results to show that intermittent boiling could occur during the casting of high-carbon steel grades. Several studies have been conducted on the heat transfer during a hypothet-ical loss-of-coolant accident (LOCA) in nuclear power plants [79-85]. Both, reflooding (low pressure) and recovery (high pressure) stages of LOCAs have been considered. The hydrodynamic conditions investigated are falling film of water on the outside surface (relevant to B W R and S G H W R spray cooling) and upflow of water (relevant to B W R and P W R core reflooding). It should be noted that most of the studies related to heat transfer in LOCAs have been conducted at low mass fluxes (< 2000 kg m - 2 s _ 1 ) and relatively low surface temperatures (< 500 °C) While fluid flow in pool boiling is primarily driven by bubble motion, bulk motion as well as buoyancy effects are responsible for fluid flow in forced convective boiling [86]. This results in higher rates of heat extraction (that increase with fluid velocity and subcooling) as shown, schematically, in Figure 2.5 [87]. Near the critical heat flux the forced convective boiling curves for various fluid velocities and subcoolings merge into a single curve known as the fully developed boiling curve. In some systems, this curve lies on an extension of the corresponding nucleate boiling curve for pool boiling. During forced convective boiling a fraction of energy is transported from the heated wall to the liquid through turbulent forced convection, as in single-phase convection. In addition, mechanisms are available that increase heat transport by : 1) creating addi-tional near-wall convection (therefore, augmenting single-phase forced convection) or 2) providing additional heat transfer routes. Vandervort et al. [88] studied the mechanisms for heat transfer in very high heat flux ( > 107 W m - 2 ) subcooled boiling that can be achieved through subcooled forced Literature Review 25 convective boiling of water. They focused on phase distribution measurements at high mass fluxes (> 5000 kg m - 2 s _ 1 ) inside small diameter tubes (0.3 - 2.5 mm-dia.) and its relation to heat transfer, in particular CHF. It was observed that the maximum heat flux attainable increases with mass flux rate and decreases with tube diameter, with q > 108 W m - 2 for a flow of 40,000 kg m - 2 s _ 1 in a tube of 0.3 mm diameter. The accompanying pressure drop suggested that the vapor fraction is low compared with typical values at lower velocities and subcoolings and this was confirmed by visual inspection. They also presented an analysis of forces acting on bubbles at the wall and on departed bubbles. At high heat fluxes, Marangoni forces are likely to dominate, since very large thermal gradients exists near the wall. These large gradients are also responsible for bubble condensation occurring near the wall since the superheated layer extends only a few microns from the wall. For their particular configuration, Vandervort et al. [88] found the bubbles to grow to diameters up to approximately 3 microns and form a bubble boundary layer of approximately 10 microns, i.e., only a few bubble diameters. A number of mechanisms for heat transport were also discussed : when boiling occurs, additional near-wall turbulent motion is created by bubble growth and detachment, and new mechanisms for heat transfer are provided by superheated liquid vaporization, liquid quenching of exposed surface, bubble-induced microconvection and latent heat transport. Recently, Celata et al. [89] assessed the applicability of existing correlations and mod-els to predict the critical heat flux in subcooled forced convective boiling in the range of interest of fusion reactors (0.1 < p < 8.4 MPa, 0.3 < D < 25.4 mm, 0.1 < L < 0.61 m, 2 < G < 90 Mg m - 2 s _ 1 , 90 < A T s u b < 230 K) . They noted that a major obstacle in establishing the applicability of these kinds of correlations is the scarcity of experimental data available. Literature Review 26 2.3 Microstructural Evolution The prediction of microstructural evolution constitutes the essence of any effort to model heat treatments because the final microstructure will define the mechanical properties in the heat treated material. A vast amount of information regarding isothermal transfor-mations in steels, both empirical and fundamental, has been generated over the years; however, for practical industrial heat treatment applications, the kinetics of the trans-formations under continuous cooling/heating conditions are required. In this section, general characteristics of phase transformations occurring during heat treatment of steels are first presented. Then, the kinetic equations describing isothermal transformations in metals are discussed, followed by their implementation to continuous cooling processes. Finally, the effect of stresses on transformation kinetics is presented. 2.3.1 General In general terms, solid-solid phase transformations in alloys can be classified, based on nucleation and growth mechanisms, as 1) thermally activated and 2) athermal. The former are also known as nucleation and growth reactions and are controlled by diffusional mechanisms in which atoms are added to an original embryo via random atom motion. The transformation rate is dependent on temperature (with the rate increasing with undercooling below the equilibrium temperature for the transformation); and the fraction transformed varies with time spent at a given temperature. On the other hand, athermal transformations are characterized by a coordinated atom movement. The transformation interface moves at the order of the speed of sound in solids and, consequently, the fraction transformed does not depend on time spent at a temperature, but solely on temperature. Literature Review 27 2.3.2 Isothermal Phase Transformation Kinetics A schematic plot of fraction transformed as a function of time spent at constant temper-ature for a nucleation and growth transformation is shown in Figure 2.6. The transfor-mation curve takes the typical sigmoidal shape. It can be readily appreciated that the transformation does not start immediately, but that there exists a finite incubation pe-riod. Also, the rate of transformation (the derivative of fraction transformed with time) increases rapidly in the early stages of transformation, then reaches a nearly constant value and, finally, decreases to zero. There are two factors contributing to the reduction of the growth rate : 1) at the late stages of the transformation the diffusion fields overlap causing a reduction in the carbon gradient at the interface, diminishing the driving force for diffusion (this phenomenom is known as soft impingement and is characteristic of long range diffusion-controlled transformations) and 2) the growth of a region is stopped when it reaches the growing interface of another region (i.e., hard impingement). Because of these characteristics of thermally activated transformations, their reaction kinetics at constant temperature cannot be described by simple first-order reaction models. Johnson and Mehl [90] incorporated hard impingement into a kinetic equation by defining the so-called extended volume. Following this assumption, the transformation is allowed to occur as if nucleation and growth could take place in the untransformed as well as in the already transformed regions. With this treatment it is possible to separate the geometrical aspect of hard impingement. By making the following assumptions : • nucleation occurs randomly; • the transformation is isotropic and, therefore, the product has a perfectly spherical shape; • nucleation rate as well as growth rate are constant; and • an increment of extended volume is formed by totally random nucleation, Literature Review the Johnson-Mehl equation is obtained through integration of a differential increment in volume transformed : X = 1 — exp (2.2) 3 Avrami [91-93] extended this treatment to include a variable nucleation rate based on the number of nucleation sites available. For the case of most of the nucleation sites being consumed in the early stages of the transformation (early site saturation), the following equation was obtained :-X = 1 - exp ( - y N0Gst^j (2.3) This equation can be generalized to : X = 1 - exp(-6i n ) (2.4) where b is related to the magnitude of the nucleation and growth rates and is a func-tion of temperature, and n is nearly independent of temperature and represents geom-etry (dimensionality) and growth mode (linear, etc.). This expression is known as the Johnson-Mehl-Avrami-Kolmogorov ( J M A K ) [90-94] equation and is extensively used for predicting isothermal phase transformation kinetics in steels. The parameters n and b are deduced from experimental measurements of dilation as a function of time at constant temperature. Cahn [95] derived rate laws describing transformations in which nucleation takes place on grain boundary surfaces, grain edges, or grain corners. He assumed steady-state conditions, i.e., growth rate and nucleation rate were independent of time, and considered the case where only one kind of site was active. Furthermore, the new phase was assumed to grow with constant radial velocity until it impinged on another growing area. Cahn adopted the extended volume fraction concept proposed by Johnson and Mehl [90], and defined an extended volume fraction (for grain corner nucleation), an extended area fraction (for grain boundary nucleation) and an extended length fraction (for grain edge Literature Review 29 nucleation). The results were presented in the form of curves of log log( l / l — X) vs logi . At low nucleation rates, the rate laws approached the form of Eq. (2.2). In early studies, published isothermal transformation kinetics were obtained under conditions that did not warrant a careful characterization of the heat exchange between the sample and the cooling medium. When this kind of information was applied in a mathematical model of quenching of eutectoid steel rods, poor agreement between measured and predicted centerline temperatures was found [96]. Recognizing that a more careful characterization of the kinetics of the transformations was needed, Hawbolt et al. performed detailed studies on the austenite-to-pearlite transformation in eutectoid carbon steel [97] and the austenite-to-ferrite and austenite-to-pearlite transformations in 1025 carbon steel [98]. To accomplish this task, an apparatus based on a diametral dilatometer was designed to measure temperature and dilation simultaneously on resistively-heated tubular samples. For the eutectoid steel, the parameter n was found to be nearly constant when the experimental data was fitted to the J M A K equation obtained by setting the transformation start time, t = 0, at the start of the transformation (tAv)\ but n varied widely when t = 0 was set to the time to reach TAl • The reason for this discrepancy is that the J M A K equation was developed without any consideration of the incubation period and, therefore, the correct procedure to determine the kinetic constants n and b should only consider the transformation event. The effect of varying the cooling rate to reach the isothermal transformation temperature on the transformation start time was also studied and found to be small : an increase of only 7.5 s (from 14 to 21.5 s) was obtained for a change in cooling rate from 108 °C s _ 1 to 14.65 °C s _ 1 . Similar conclusions regarding the calculation of n and b were drawn from the study of transformations in the 1025 carbon steel. By comparing the thermal response of a thin wafer specimen heat treated in a molten salt bath with the results obtained from the resistance-heated dilatometer, these investigators found that, at the desired test temperatures, isothermal conditions were not attained quickly in the salt bath. Literature Review 30 The prior-austenite grain size plays an important role in the kinetics of phase trans-formations. In general, the rate of transformation is inversely proportional to the grain size. This effect can be modelled using a modified form of the Avrami equation proposed by Umemoto et al. [19] : (-btn\ X(0 = l - e x p ^ J (2.5) where d is the prior-austenite grain diameter and the exponent m takes values from 0 to 3 depending on the geometry of the nucleation site. In contrast to nucleation and growth reactions, the kinetics of athermal transforma-tions do not depend on time spent at a given temperature, but on temperature alone. The evolution of martensite fraction, as a function of temperature, is schematically shown in Figure 2.7. For the case of martensitic transformations in steels, an empirical equation » obtained by Koistinen and Marburger [99] has been found to predict the kinetics of the transformation adequately : X = exp[A(M s - r)] (2.6) where A = 0.011 ° C _ 1 for carbon steels and Ms is the composition sensitive martensite start temperature. Information relating the temperature and time required for the start and end of vari-ous phase transformations for a particular steel is conveniently summarized in isothermal transformation (IT) diagrams. Unfortunately, the incorrect assumption of instantaneous attainment of bath temperature in salt bath tests and the fact that the vast majority of the heat treatments involve continuous heating and cooling limit the usefulness of IT diagrams to such heat treatments as isothermal annealing, austempering and martem-pering [22] (which usually involve salt bath processing). A characteristic feature of IT diagrams is the ' C shape of the iso-transformation contours for reactions controlled by nucleation and growth. Assuming that the velocity Literature Review 31 of phase-boundary propagation for nucleation and growth reactions in steels can be de-scribed based on the diffusion of carbon atoms across the boundary, Zener [100] derived the following relationship between the rate of advance of the boundary and the diffusivity and diffusion distance : f^th ° b a ^ d 1 1 0 6 ^ (^^ff118^011 distance)(diffusivity) (2.7) where the diffusivity is inversely related to temperature through an Arrhenius-type equa-tion and the diffusion distance is inversely proportional to undercooling. This relation-ship provides a basis for a qualitative explanation of the characteristic C-shape of IT diagrams. Slow kinetics are obtained at high temperatures (low undercoolings) where diffusivity is high but the diffusion distance is large; fast kinetics occur at intermediate temperatures; and slow kinetics appear again at low temperatures (high undercoolings) where the diffusion distance is small but the diffusivity is low. Based on classical nucleation theory concepts, Russell [101] obtained a dependence of the rate of establishing steady-state nucleation on driving force and mobility : rate of establishing (driving force) (mobility) . oc — ( 2-o) steady-state nucleation T The driving force increases with undercooling (decreases with increasing temperature) while the mobility exhibits the opposite behaviour. Assuming that the rate of estab-lishing steady-state nucleation is inversely proportional to the incubation time, Eq. (2.8) predicts large incubation times at high and low temperatures and short incubation times at intermediate temperatures, i.e., a C-shape for the transformation start time with a diffusion controlled transformation. Empirical relationships to compute IT diagrams have been reported by Sakamoto Literature Review 32 et al. [102]. A more fundamental approach has been adopted by Kirkaldy and co-workers [14,103,104] to derive semi-empirical formulas for predicting IT curves based on Zener-type equations. 2.3.3 Continuous Cooling Phase Transformation Kinetics Phase transformations during most heat treatments occur under continuous cooling con-ditions. Experimentally determined times to reach various stages of transformation under non-isothermal conditions are presented in continuous cooling transformation (CCT) dia-grams. Typically, they can be obtained from dilatometer records of length vs temperature that are verified through metallographic analysis [20]. The hardness value at the end of the cooling path is also reported. A major drawback of these diagrams is that they are strictly valid only for the specific cooling conditions and the austenite grain size used for their experimental determination. Even though they provide a means of visualiz-ing a great deal of information at a glance, their graphical nature makes it difficult to incorporate the data into a computerized system. The fact that phase transformation kinetics under continuous cooling conditions de-pend not only on steel composition and prior-austenite grain size but also on the cooling path, makes the task of predicting the microstructural evolution during heat treatments a difficult one. Since the cooling rate in a part varies from point to point, deriving the necessary number of empirical relationships between phase distribution and cooling path would require a very large number of measurements, specially for a part of complex geometry. Thus, a more general approach is desired. The first step in modeling microstructural evolution involves the prediction of the incubation time. A common practice consists of obtaining empirical correlations between incubation time and cooling rate [43]. However, this method cannot be generalized easily and, therefore, several researchers have attempted to develop methods to predict the start of austenite decomposition for any arbitrary cooling path. Literature Review 33 Scheil [31] proposed that the amount of undercooling and the incubation time are related. By assuming-that the continuous cooling path to a given transformation tem-perature can be discretized, Scheil postulated that a fraction of the isothermal incubation time is consumed at each 'isothermal' stage and that when the sum of all these fractions reaches unity, the transformation begins. Thus, if is the time spent at a given 'isother-mal' stage and Tj is the isothermal incubation time corresponding to that temperature, the fraction consumed isothermally is given by : and, upon summation, : which is known as the additivity rule or Scheil equation. In their work described previously, Hawbolt et al. [97, 98] computed the incuba-tion time for transformations in both hypoeutectoid and eutectoid carbon steels using Eq. (2.11) and compared it with the experimentally determined start times. They found that by using the additivity rule the start times were consistently overestimated. In an attempt to improve upon the ability to predict the transformation start time, Pham [105] derived 'ideal' T T T curves3 based on measured transformation kinetics under isothermal and non-isothermal conditions. In the latter case, the inverse of the Scheil's additivity rule was applied to back-calculate the 'ideal' isothermal start times. Predicted 3The 'ideal' T T T curve corresponds to a hypothetical infinitely rapid cooling rate to each of the test temperatures and, therefore, can be defined in terms of the steel composition and austenite grain size alone. (2.10) In the limit where A T —> 0 and assuming a constant cooling rate : Literature Review 34 start times, based on the 'ideal' IT curve, for the austenite-to-pearlite transformation in a eutectoid carbon steel subjected to continuous and semi-continuous (Stelmor) cooling were in good agreement with measured values. For non-continuous cooling a reasonable agreement was found, with the predicted times being consistently early. The most plausible reason for the observed discrepancy between theoretical and mea-sured incubation times is that real systems do not comply with the implicit assumption of 'proportional consumption' [106], which requires that a given fraction of the incubation time at a given temperature always corresponds to the same fraction of the total incuba-tion time under non-isothermal conditions for all temperatures involved. Experimental evidence supporting this hypothesis has been reported by Moore [107] As mentioned earlier, transformation kinetics under continuous cooling conditions cannot be characterized from first principles. A methodology to apply semi-empirical de-scriptions of isothermal transformations kinetics to a transformation event under continu-ous cooling conditions has been successfully applied by, among others, the microstructural engineering research group at U B C [96,108,109]. The calculations during the transfor-mation stage are based on the assumption that the rate at which the transformation develops is a function only of the current temperature and fraction transformed, and not of the history of the microstructural evolution; in other words, this assumption, known as the additivity principle, proposes that the material does not have structural 'mem-ory' and thus the continuous transformation can be approximated by a series of additive fractional transformations. The application of the additivity principle is illustrated in Figure 2.8. If X\ is the fraction transformed at (T\,ti), then an additional increment of fraction transformed, AX, after a time increment, At, (At — £2 — ^ 1 ) spent at temperature T2, can be computed by using the kinetic information corresponding to T^. The methodology includes the evaluation of a virtual (fictitious) time that corresponds to the time required to attain X\, at T2, then adding the incremental growth at T2 associated with additional incremental Literature Review 35 time at T 2 . A n expression for this virtual time can be obtained from Eq. (2.4) : where X\ is the fraction transformed at time ti and denotes the inverse of the function X(t). Thus, the effective time increment adopted for the calculation is A t = t 2 — t\ (instead of t2 — ti). It is readily seen that the additivity principle is similar to the additivity rule except that it is applied to the transformation, as opposed to the incubation period. The additivity principle not only provides a tool to apply isothermal data to contin-uous cooling processes, but it also matches the numerical techniques used to solve the energy equation that describes the thermal field evolution. Irrespective of the details of the particular technique employed, the continuous cooling curve is always replaced by a series of small isothermal steps; a large number of such steps results in a more accu-rate simulation, but at the cost of larger memory requirements and a greater number of mathematical operations. Because of the enormous potential of the additivity principle as a tool for applying isothermal data to the prediction of continuous cooling events, it is very important to determine the limits of its applicability. Avrami [91] proposed that the ratio of nucle-ation rate to growth rate (N/G) should be used as a parameter to predict whether the additivity principle can be applied. The criterion he proposed, called the isokinetic con-dition, requires that the ratio remains constant over the temperature range of interest. Cahn [95,110] suggested an expanded set of conditions based on the assumption that the kinetics of the transformation are controlled by the growth rate. Christian [111] pro-posed a somewhat different approach considering that the rate of transformation must be described by the ratio of two independent functions, one dependent on temperature (2.12) Literature Review 36 and the other on fraction transformed (i.e. R = J^Q)- More recently, Kuban et al. [112] conducted an experimental program that included the measurement of nucleation and growth rates of pearlite in a eutectoid, plain-carbon steel and suggested an 'effective site saturation' criterion, where early growth is again the controlling mechanism. There have been few studies concerned with mathematical formalisms of the additivity principle. Verdi et al. [113] introduced the concept of degree of advancement of the transformation, [ix-, and showed that the concept of additivity, as applied to the evolution of phase transformations, holds if the rate of degree of advancement of the transformation depends on X and T alone. Due to the nature of the research on both the additivity rule and the additivity principle, some confusion exists in the literature regarding the nomenclature as well as the equivalence between different descriptions and conditions for additivity. Recently, Mukunthan [106] has presented an excellent review aimed at clarifying these points. Experimental success at using the Avrami equation and applying the additivity prin-ciple to the austenite-to-ferrite transformation in 1010 and 1020 plain carbon steel, was reported by Kamat et al. [114]. By conducting stepped isothermal transformation tests, these researchers clearly showed that the simulated continuous cooling transformation to ferrite could be adequately described using the additivity principle. Using a fundamental, carbon diffusion-controlled mathematical model for ferrite growth kinetics, they showed that the duration of the unsteady-state effects caused by the rapid change of bound-ary conditions and diffusion coefficient is very short, which would explain the observed experimental additivity of the proeutectoid ferrite. Since the Avrami equation was not developed to describe the 'soft impingement' conditions associated with the austenite-to-ferrite transformation, it is used here as an empirical equation, having the correct form. Finally, the bulk of the work that has been done regarding transformation kinetics has been directed towards the cooling of steels. Since heat treatment involves the initial Literature Review 37 heating and austenitizing of the steel, a systematic study of the applicability of the addi-tivity principle to the kinetics of the heating reversion to austenite should be conducted in order to be able to model a full heat treatment. This has been done by Riehm for a eutectoid carbon steel and confirms that the reversion can be adequately described using equations having a similar form [115]. 2.3.4 F e - C - X Phase Boundaries The need to accurately model the multicomponent equilibrium Fe-C-X diagram in hard-enable steels is twofold : 1) typically, parameters in empirical and semi-empirical kinetic equations are evaluated based on the degree of undercooling, and 2) regression equations for calculating equilibrium fractions transformed from the lever rule are required in the case of hypo- and hypereutectoid steels. Empirical formulae to compute TA3, TACI, and TAC3 have been obtained by Andrews [116] using a database comprised of published data on 150 steels. It is important to note that all the formulae show a linear dependence of the respective temperature with the percentage of alloying elements. Theoretically-based formulae have been given by Kirkaldy and co-workers [14,104,117] to predict phase boundaries of the Fe(X,-)-(Fe, XJ3C multicomponent phase diagram for carbon contents up to ~ 2 w/o C. The thermodynamic formalism to describe the 7-a equilibrium is a generalization of the classical 'depression of the freezing point' relation, applied to the Fe-Fe 3C diagram. The temperature deviation is derived, from the equality of chemical potentials. The calculations for the carbide equilibria followed a somewhat different approach due to the fact that there is insufficient knowledge of the solution thermodynamics of the alloy-enriched cementite. Starting with the equality of chem-ical potentials, as given by the Gibbs-Duhem equations, the isothermal concentration deviation of the 7 —»• 7+ FeaC boundary is obtained. Literature Review 38 Once the equilibrium lines have been calculated, the eutectoid composition and tem-perature in an alloyed steel can be computed from the intersection of the two equilibria. By comparing predicted and measured equilibria in low-alloy steels [104], the limits of applicability of this formulation were found as follows : %C < 2 w/o, % M n < 3 w/o, %Si < 1 w/o, %'Ni < 3 w/o, %Cr < 2.5 w/o, %Mo < 2 w/o, %Cu < 3 w/o, others < 1 w/o. For the prediction of IT curves in hardenable steels, the values of interest are the ferrite, pearlite, and bainite asymptotes, i.e., the temperatures to which the start curve of the respective reaction extrapolates at very long times. The ferrite asymptote always corresponds to the A e 3 temperature [104]. In contrast, the pearlite asymptote depends on the partitioning behaviour of alloying elements. Kirkaldy and Venugopalan [14] have proposed an interpolation formula to compute the pearlite asymptote based on the cal-culation of equilibrium (full partitioning) and paraequilibrium (no-partitioning) phase boundaries. Rigorous thermodynamic formulae for equilibrium, paraequilibrium, and no-partition local equilibrium have been given by Hashiguchi et al. [118]. No theoret-ical description of the bainite asymptote has been developed and, therefore, empirical equations are commonly adopted, as they are for the composition invariant martensite transformation [116,119]. 2.3.5 Effect of Stresses on the Kinetics of Phase Transformations The influence of external stresses applied prior to phase transformations is well known. In the case of the martensitic transformation, strain-induced nucleation has been recognized as an important factor in increasing the M g temperature [120-122]. An increase in the nucleation rate has also been reported for the austenite-to-pearlite transformation when the austenite has been previously deformed [123]. In contrast, very few studies on the effect of stresses during the transformation are available. Dubrov [124] conducted a high-temperature metallographic analysis of the Literature Review 39 bainitic transformation under applied tensile stresses in a 40Kh2N3SGV steel. He moni-tored the growth of a needles and observed an increased number of nucleation sites when the transformation developed under stress. The influence of stresses on the kinetics of the transformation varied with the stage at which they were applied. Violle [125] studied the effect of a uniaxial compressive stress on the kinetics of the transformation of /3 to a in U-Cr alloys. For the isothermal case he reported shorter incubation periods and higher rates of transformation at all temperatures; in the case of continuous cooling, the applied stress (up to 25 kg m m - 2 ) resulted in a shift in the actual transformation temper-atures toward higher values. Gautier [126] studied the influence of applied tensile stresses during the pearlitic transformation in eutectoid carbon steels by measuring dilation and electrical resistivity. She reported shorter times for the beginning and end of the isother-mal transformation. This shift in the kinetic curves was a function of both the level of applied stress and the test temperature. It was concluded that the growth rate was not significantly modified and, therefore, the transformation kinetics were accelerated through a change in the nucleation rate alone. It was observed that the nucleation sites remained unchanged but they were saturated in shorter times. Gautier also investigated the influence of applied tensile stresses on the kinetics of the martensitic transformation in a 60 N C D 11 steel. The applied stresses resulted in an increase in the Ms tem-perature. Fernandez et al. [127] reported a similar shift in the transformation kinetic curves of eutectoid steels subjected to an applied stress during the austenite-to-pearlite transformation. Andre et al. [128] measured the hardness distribution in massive and implanted cylin-ders (34 mm-dia. by 102 mm length) of 60SC7 spring steel quenched in water and oil. The implanted samples were fabricated by drilling an axial hole in the specimen and inserting a cylinder made of the same steel in the shaft. Good thermal contact between the two cylinders was ensured by pouring liquid tin between them. Due to the presence of liquid tin, the stresses generated in the outer shell were not transmitted to the inner Literature Review 40 cylinder; also, negligible thermal gradients were generated within the inner cylinder. The authors concluded that internal stresses generated by quenching resulted in an acceler-ation in the pearlite and bainite transformations. A review of the influence of internal stresses on the kinetics of the pearlite transformation during continuous cooling has been presented by Denis et al. [129]. 2.4 Residual Stresses in Heat Treatments The residual stress distribution generated during the manufacture of engineered compo-nents, in combination with the loads experienced in service, determines the components' life. Additionally, mathematical models of stress generation during heat treatments can only be verified against measured residual stresses. Thus, techniques for measuring resid-ual stress distributions are a very important tool in structural analysis. In this section, the concept of residual stress is defined together with current classifications of residual stresses. Then, the generation of residual stresses during quenching is briefly discussed, and the techniques for their measurement are reviewed. 2.4.1 General Residual stresses are those stresses that would exist in a body in the absence of external loads [130]. They are also referred to as internal stresses, initial stresses, inherent stresses, reaction stresses, and locked-in stresses. Residual stresses are elastic only and the residual stress system in a body must be in static equilibrium, i.e., the resultant force and the resultant moment produced by the system of stresses must be zero. According to the characteristic dimension involved, residual stresses can be classified into three subgroups [131] : 1) macrostresses, which vary continuously over macroscopic dimensions larger than the average grain size, 2) microstresses, which act over dimensions of the order of a grain dimension and are distributed non-homogeneously on a macro Literature Review 41 scale, and 3) pseudo-macrostresses, which are the average of microstresses taken over a representative volume. K i m [132] noted that the difference in residual stress among phases in a microstructure would be a more meaningful characteristic of residual stresses. Accordingly, he proposed the following classification : 1) macrostresses, same as above, 2) 'phase macrostresses', which are the average residual stress in a particular phase in a microstructure4, and 3) microstresses, defined as variations of macrostresses and 'phase macrostresses' over small distances. Residual stresses may be generated during manufacturing or during use of parts, and are caused by nonhomogeneous partitioning of inelastic strains both at the micro- and macroscopic scale. The residual stress distribution generated during quenching is due to thermal and transformation volume changes that interact in a complex manner as the part is being cooled from the austenitizing temperature. Thermal contraction occurs at all times, while volume expansion associated with phase transformations occurs only during the transformation. Clearly, the hardenability of the steel and the heat extraction characteristics of the quenching medium, as well as the thermophysical properties of the part, will dictate the final stress profiles. Residual stresses cause distortion and affect the performance of parts. Distortion is caused by the non-uniform nature of the residual stress field inside the body. Two examples where residual stresses can affect a structure adversely are fracture and stress-corrosion cracking. Residual stresses however can be beneficial as in the case of compressive residual stresses near the surface of shot-peened parts. The effect of residual stresses on the behaviour of parts is more important at low levels of applied stress. 4The weighted average of the 'phase macrostresses' would be identical to the macrostresses. Literature Review 42 2.4.2 Res idua l Stress Measurement Techniques Several techniques can be used to measure residual stresses. They are classified, according to the principle used [130], in : 1) stress-relaxation techniques, 2) diffraction techniques, 3) techniques using stress sensitive properties, and 4) cracking techniques. Some examples are given in Table 2.1. In the stress-relaxation techniques the residual stresses are computed by directly measuring the strain released by either cutting the specimen into pieces or removing a piece from the specimen. It is assumed that no further plastic damage is caused during the stress relaxation (cutting) operation. Despite the fact that stress-relaxation methods are destructive, they are widely used [133-137] due to the relatively simple equipment involved. The accuracy and reliability of the measurements require that the data reduction assumptions are valid. The most commonly used method is the hole-drilling strain-gauge method described in the A S T M standard E837 [137]. In all cases, the method is calibrated in a uniform uniaxial stress field of known magnitude. In contrast, diffraction techniques for residual stress measurements are non-destructive. The elastic strains associated with the residual stress field are obtained by measuring the scattering angle for a given plane (or set of planes) when the specimen is irradiated with X-rays [1,138,139] or neutrons [140-143] and comparing it to a suitable reference angle -commonly measured in a stress-free specimen. The measurements can be made in a small, precisely located area, but the techniques are relatively slow. X-rays can only penetrate small distances in metallic specimens and, therefore, only surface residual stresses can be determined using X-ray diffraction. On the other hand, neutrons are approximately a thousand times more penetrating than X-rays [141]; thus, neutron diffraction can be applied to determine residual stresses throughout the entire part. However, the data acquisition time for neutron diffraction is longer, due to the loss of beam intensity with depth of beam penetration. Rather than measuring strains directly, some methods have been developed based Literature Review 43 on the measurement of stress-sensitive properties, such as ultrasonic velocity [144] and Barkhausen noise [145]. The complexity of the interaction of the propagating ultrasonic wave and a lack of consistency in the performance of transducers are important problems related to ultrasound measurements [146]. Barkhausen noise analysis can only be applied to ferromagnetic materials. In addition, it has a narrow range of stress sensitivity, and is restricted to near-surface probing. None of these techniques are used in field work. Residual stress distributions can also be characterized through hydrogen-induced and corrosion cracking [147]; the results are, however, qualitative in nature. 2.5 Mathematical Models of Microstructural Evolution and Stress Genera-tion in Heat Treating Operations Mathematical models of heat treatments are made up of two components : 1) a heat transfer and microstructural evolution module, and 2) a stress generation module. The majority of the modeling efforts produced so far have treated them independently, solving for the heat transfer and microstructural evolution first and then using those results as input to stress generation models. In this section, mathematical models developed to predict microstructural evolution and stress generation in heat treating operations are described. 2.5.1 Modeling of Heat Transfer and Microstructural Evolution The thermal response and microstructural evolution during cooling are linked through transformation kinetics and the heat evolved during the phase change. Early models [7,148] did not include the heat of transformation and assumed constant thermophysical properties. More recent work was based on adopting a modified value of specific heat in the temperature range where phase transformations take place (see, for example, Literature Review 44 [149]). This approach-eliminates the need for calculating the microstructural evolution to compute the rate of heat evolved and, therefore, the calculations are simplified. Agarwal and Brimacombe [96] modelled the microstructural evolution during water-quenching of 8.5 mm-dia. eutectoid steel rods. They computed the pearlite fraction transformed, as a function of time, by adopting an empirical two-parameter equation (similar to the J M A K equation) and invoking the additivity principle. The parame-ters in the empirical kinetic equation were estimated from the corresponding isothermal transformation (IT) diagrams; the transformation was assumed to commence as soon as the local temperature dropped below T ^ . The numerical procedure adopted to solve the heat balance equation was based on the finite-difference method. Iyer et al. [108] improved upon that model by separating the incubation period (as characterized by an experimentally measured start time) from the transformation event. The kinetic pa-rameters in the J M A K equation were experimentally determined rather than calculated from published T T T diagrams. Campbell et al. [43] extended the phase transformation kinetics database to hypoeutectoid and eutectoid steels and incorporated the results in a mathematical model of the Stelmor process. Kumar Singh and Mazumdar [150] compared three methods to model the heat source upon heating and cooling : 1) the fraction transformed vs time relationship was derived from IT diagrams, 2) the fraction transformed vs time relationship was derived from the iron-carbon equilibrium diagram, and 3) the heat source term was assigned a value of zero and a modified heat capacity was adopted. A control volume-based finite-difference procedure was implemented for the numerical solution. The computed results were com-pared against analytical solutions, as well as measured temperature response during heating, air-cooling of hypoeutectoid steel, and water-quenching of eutectoid steel. The only transformation considered, even for the hypoeutectoid grade, was from austenite to pearlite (or vice-versa). For the calculations, a constant density and specific heat were assumed together with a temperature-dependent thermal conductivity. The boundary Literature Review 45 condition was characterized by either a combined radiative plus convective heat flux (for heating and air-cooling) or a constant heat-transfer coefficient (for water-quenching). For the experimental conditions investigated, the authors found that the procedure adopted to model the source term was not critical; they attributed this observation to the rel-atively small contribution of the heat source. However, due to the simplicity of the computational algorithms used in methods 2 and 3, they recommended using them over method 1. The effect of prior-austenite grain size has been included in a recent model of induction hardening [151] by shifting the time scale of the IT diagram. The incubation time, as well as the kinetic parameters in the J M A K equation, were calculated by modifying their base value through a grain size-dependent parameter. For the martensitic transformation, the Ms temperature was explicitly given as a function of prior-austenite grain size. Few mathematical models have included the effect of internal stresses on the kinetics of the transformation. Fernandes et al. [127] recognized the importance of including this effect and modelled it by using a mean value of the equivalent stress between the surface and the centre of an infinitely long cylinder. Experimental values of IT curves for various stress levels were then used to obtain the modified kinetics. A good agreement between calculated and experimental values was found. Inoue et al. [152] modelled the kinetics through a Johnson-Mehl type equation modified for the presence of stresses. Sjostrom [153] used a linear relationship suggested by Denis [154] to account for the effect of stress on the martensite start temperature. Regarding the active boundary condition, a significant number of investigators have adopted a measured surface or near-surface temperature response, while fewer researchers have estimated the heat-transfer coefficient from either available correlations or through the solution of the inverse heat transfer problem. Literature Review 46 2.5.2 M o d e l i n g of Stress Generat ion It has already been stated that volume changes due to thermal and microstructural responses interact in a complex manner to give rise to transient and residual stresses in quenched parts. In addition, the level of stresses generated upon cooling are such that plastic flow always occurs. In those instances where low cooling rates are encountered (e.g. oil quenching of large specimens), viscous effects may also be important. The mechanical behaviour of engineered components is commonly defined through a constitutive material law. This equation describes the relation between stresses and strains [155] and, together with the equilibrium equations and the compatibility equa-tions, define the problem. A mechanistic approach, based on crystal plasticity, including all the interacting micro-mechanisms responsible for plastic deformation of real materials, is desirable. However, such a formulation is extremely complicated and difficult to apply in real situations [156]. Instead, virtually all stress analyses are carried out by adopt-ing the so-called phenomenological (classical) theory of time-independent plasticity [157]. The basic elements of the theory include the assumption of a mathematical continuum, the existence of a plastic potential, the change in the yield surface with plastic flow as described by a hardening rule, and the associated flow rule. A relationship between plastic strain and deviatoric stress has been proposed, both on a total plastic strain (Hencky's equations) and an incremental plastic strain (Prandtl-Reuss equations) basis [158]. However, the total plastic strain theory gives inconsistent results [159] (except in the case of 'proportional loading') and, therefore, all mathematical models of stress generation are based on the Prandtl-Reuss equations of incremental plastic strain. Furthermore, it is possible to separate the incremental strain into its elastic and plastic components. Although strictly valid only for infinitesimal strain increments, this assumption holds true for finite strain increments as well. When the nonmechanical strain increments (such as thermal, transformation, etc.) are explicitly considered, their Literature Review 47 values can also be computed independently and then added to obtain the total elastic and plastic strain increments. The general form of the strain increment, de, then becomes : d £ = ^ elastic + Aplastic + ^thermal + detransf. + • • • (2-13) Early work was devoted to analytical solutions of structural problems and, there-fore, only the simplest material behaviour, i.e., perfectly elastic-perfectly plastic, was considered. With the advances in computer technology and the development of suit-able numerical techniques, constitutive equations that reflect the behaviour of actual components can now be adopted. Among those numerical techniques, the finite-element method has gained popularity as a tool for solving structural problems, due to its ability to handle complex geome-tries. Two commonly adopted methods to include material nonlinearities are [160] : 1) the tangent stiffness approach (where the nonlinearities are included in the equilibrium equations in terms of a 'tangent stiffness'), and 2) the pseudo-forces approach (where the nonlinearities are included in the equilibrium equations as 'pseudo-forces'). One of the first efforts to model stress generation during quenching is due to Weiner and Huddleston [161]. They assumed an elastic-perfectly plastic material obeying the Tresca yield criterion, to compute the transient and residual stresses in solid and hollow cylinders. An analytical solution was obtained for a cylinder cooling with a negligible thermal gradient and a phase transformation that occurred instantly when a critical temperature was reached. Early models obviated the need for a detailed description of the microstructural evo-lution during quenching by using experimentally measured 'effective' thermal expansion coefficients to characterize a combined thermal-transformation strain. In this formula-tion, the 'effective' thermal expansion coefficient was assumed to be an explicit function of either temperature alone [161,162], or temperature and a parameter that character-izes the cooling rate [7]. Inoue and Raniecki [163] have pointed out that the former Literature Review 48 approximation would only hold for through-hardened specimens, given that the rate of transformation in diffusional reactions does not depend on temperature alone. Fujio et al. [148] modelled the generation of stresses in a 50 mm-dia. 1045 carbon steel cylinder quenched in water by assuming a constant heat-transfer coefficient at the surface of the piece. Calculations performed with the chosen value of 5800 W m~ 2 K _ 1 showed large differences with respect to the measured thermal response 1 mm below the surface at early times, but a reasonable agreement otherwise. The authors justified their selection of a constant heat-transfer coefficient by suggesting that the maximum cooling rate is the most important parameter in describing the austenite-to-martensite transformation. They did not include the latent heat of transformation in the analysis. The results of the model were compared with experimentally determined residual stresses obtained using Sachs' technique. Despite the simplifications, the computed residual stresses agreed well with the experimentally measured values. Inoue and Tanaka [7] studied a similar case, i.e., quenching of a 60 mm-dia. bar made of 0.43 % carbon steel in water. In this investigation, the 'effective' coefficient of thermal expansion was assumed to vary with temperature and cooling rate, and the microstructural evolution was estimated from the maximum cooling rate in the part. The latter assumption resulted in a non-continuous approximation of the phase distribution during the quenching process. The latent heat of transformation was not included in the model and the thermophysical properties were assumed to be independent of temperature. In contrast with the work of Fujio and co-workers, the rate of heat extraction at the surface was assumed to vary with temperature following the boiling curve. The residual stress distribution was measured using the Sachs' boring-out technique. Despite having used a more complex model than Fujio et al. [148], good agreement between predicted and measured values was again observed. Inoue and Raniecki [163] presented a more realistic model where the kinetics of the transformations were taken into account via empirical relationships obtained from IT Literature Review 49 diagrams. Since the microstructural evolution could be tracked as the quench progressed, an explicit relationship for the volumetric change accompanying the transformation could be developed. The model was applied to the quenching of a semi-infinite body; the results were presented as residual stress and martensite fraction, as a function of distance from the quenched end, for various cooling rates characterized by a Biot number. Based on their results, the authors established that the residual stresses would vanish on a plane containing 30 to 35 % martensite. A topic that has received a good deal of attention is the modeling of strain hardening during quenching. Denis et al. [164] simulated the water quenching of a 35 mm-dia., 60NCD11 steel bar in water at 20 °C. For this cooling regime, the only reaction taking place was the austenite-to-martensite transformation. The authors assumed a perfectly plastic behaviour within the martensitic transformation interval and an isotropic linear hardening for the austenitic phase. Isotropic hardening was also assumed by Nagasaka et al. [47] in their simulation of water spray quenching of 75 mm-dia., 1035 carbon steel and Ni-Cr alloy steel bars, Buchmayr and Kirkaldy [165] to model the Jominy end-quench test, and Habraken [166] to simulate a combination of air cooling of beams followed by water quenching. The case of a 50 mm-dia. 11.6 % Ni alloy steel cylinder cooled from 900 °C in ice-water was studied by Rammerstorfer et al. [149]. The model allowed for either pure isotropic or pure kinematic strain hardening; the case of perfectly plastic behaviour was also included for comparison. The computed results showed a better agreement with measured residual stresses at the centre of the cylinder when kinematic hardening was adopted to model strain hardening; the calculated values at the surface were very similar in both cases. Sjostrom [153] modelled the quenching of a 60 mm-dia. 1050 steel cylinder in water at 20 °C and a 50 mm-dia. 11% Ni alloy steel cylinder in ice-water. He found that the choice between isotropic and kinematic hardening was not critical for the 1050 steel, whereas kinematic hardening gave better agreement with measured residual stresses for the alloyed steel. Other models [167-169] have included a Literature Review 50 mixed (isotropic-kinematic) linear hardening rule. The related topic of loss of memory of prior deformation through phase transformation was also investigated by Sjostrom in the above-mentioned work [153]. Again, the results were material-dependent, with the refinement being relevant only in the case of the through-hardened specimen. The stress and microstructural fields interact in two ways : 1) the stress field inside a component modifies the transformation kinetics, and 2) the transformation induces the so-called transformation plasticity strain. Reference has previously been made to the treatment of stress-modified kinetics of diffusional and martensitic transformations and their inclusion in mathematical models (see subsection 2.3.5). A description of transformation plasticity and its numerical implementation follows. It has been experimentally observed that macroscopic plastic flow can occur during solid-solid phase transformations (of both pure metals and alloys) for externally applied stresses well below the yield stress of the phases involved when an internal stress state exists in the material. This macroscopic behaviour is the result of localized microscopic plastic flow and is known as transformation plasticity. Early researchers had suggested that a transient loss of cohesion between atoms while they move to new positions in the lattice during the transformation was responsible for this loss of strength, but more modern investigations suggest two basic mechanisms : 1) the weaker of the two phases flows plastically in order to accommodate the strain gen-erated by the transformation front [170], and 2) the product phase orients preferentially in order to minimize the total energy of the system [171]. The first mechanism was pro-posed for transformations that involve a significant change in density while the latter was introduced to explain microplasticity results observed in martensitic transformations. Several quantitative models have been proposed to incorporate transformation plas-ticity in calculations of internal stresses during heat treatments. In general terms, they fall into two categories : 1) adopting an artificially lowered elastic limit during the time Literature Review 51 the transformation takes place, and 2) introducing an additional strain increment term which needs to be determined experimentally. However, it should be noted that a lower-ing of the yield strength implies that the plastic deformation can increase indefinitely and independently of the phase distribution; also, there is not enough experimental evidence to support this treatment. The majority of the expressions to evaluate transformation plasticity in mathematical models have been developed based on the Greenwood and Johnson mechanism [170]. Their original equation suggested a linear relationship existed between the transformation plasticity strain, e t p , and the applied stress : Abrassart [172] modified this equation to include the amount of phase transformed : Note that for /,• = 1 which is very similar to Eq. (2.14). From dilatometric experiments under uniaxial constant stress, Desalos and Guinsberg [173] have proposed the following model : (2-14) (2.15) 3K{l-fz)s dfi (2.17) dt where / ; is fraction transformed and Sij is the deviatoric component of the stress tensor. Giusti [174] and Leblond [175] derived the following theoretical expression for the time Literature Review 52 derivative of the transformation plasticity strain under a multiaxial state of stresses : # = K'sn gUi) f (2-18) where g(fi) is a normalized function of the proportion of phase transformed. Leblond et al. [176] have given the following heuristic arguments that justify the previous relationship : • the equation for etp will be incremental, i.e., it will give ktp. • etp should be proportional to /,• and must be zero for /,• = 0 (since transformation plasticity occurs only if there is a transformation). • etp should be proportional to a. • i ip should not involve volume variation, and should thus be related to the stress deviatoric component. In contrast, very little attention has been given to the mechanism put forward by Magge, whose equation is now given : etp i sin 20 + ^e 0 ( l + cos 26) iJ°{0)dO Je de — (2.19) The vast majority of the work on modeling the mechanical response during cooling of metallic components has been based on elasto-plastic formulations. However, few studies have been conducted using an elasto-viscoplastic approach to model stress generation in casting [177-179] and quenching [180]. One advantage of elasto-viscoplastic models is that they can be used to describe rate-dependent phenomena, such as creep. Also, the steady-state solution of the viscoplastic problem is identical to the corresponding static elasto-plastic solution [181] and, therefore, efficient codes may be written for certain ap-plications. A general formulation for the solution of the elasto-viscoplastic problem can Literature Review 53 be found in Zienkiewicz and Cormeau [178]. The main justification given by the authors for adopting this methodology is that, experimentally, creep and plasticity cannot be ef-fectively separated. In a similar fashion to that adopted in the elasto-plastic formulation, the total strain is separated into an elastic strain, an 'initial' strain, and a viscoplastic strain. The latter is characterized by an strain rate which is nonzero only for stresses above a certain yield value, and is defined in terms of a plastic potential and a fluidity parameter [178]. Literature Review 54 Table 2.1: Classification of techniques for measuring residual stresses (modified from [130]). Stress-relaxation techniques using electric and mechanical strain gauges • Sachs boring-out technique. • Centre-hole (blind-hole) drilling technique. • Rosenthal-Norton sectioning technique. Stress-relaxation techniques using apparatus other than electric and mechanical gauges • Brittle coating-drilling technique. • Photoelastic coating-drilling technique. • Grid system-dividing technique. Diffraction techniques • X-ray diffraction. • Neutron diffraction. Techniques using stress-sensitive properties • Ultrasonic techniques. • Hardness techniques. • Magnetic techniques. Cracking techniques • Hydrogen-induced cracking techniques. • Stress corrosion cracking techniques. Literature Review Tr ia l -and-Er ror Processing Conditions Properties Q u e n c h Bath Qua l i t y Processing Conditions Structure Properties P r o c e s s M o d e l l i n g Processing Conditions Structure Transport Phenomena Properties Materials Science Figure 2.1: Methods adopted to predict the performance of quenching processes. Literature Review 56 Figure 2.2: Schematic representation of a typical cooling curve showing the different stages of boiling. Literature Review Natural Convection Regime Nucleate Boiling Regime Isolated J Bubbles L Transition Boiling Regime Jets and Columns vaoor Blanket Film Boiling Regime O x 3 a X a r— Critical Heat Flux (CHF) *— Leidenfrost Point L Onset of Nucleate Boiling (ONB Wall Superheat log A T s a t Figure 2.3: Schematic representation of a typical boiling curve for saturated pool boi [75]. Literature Review 58 — S u r f a c e t e m p e r a t u r e (8,) L P — s . / / / / / \ b p / /© © \ 1 / / (c) V ® _ / > / / / — S u r f a c e t e m p e r a t u r e ( 9 S ) ® Film boiling (2) Transition boiling (3) Nucleate boiling @ Convection (non-boiling) Figure 2.4: Correspondence between (a) cooling, (b) boiling, and (c) evaporation curves [56]. Literature Review 59 log (Tw-Tszt) Figure 2.5: Regimes in boiling heat transfer [87]. Literature Review 60 Time Figure 2.6: Schematic representation of fraction transformed as a function of time, for a typical nucleation and growth phase transformation. Figure 2.7: Schematic representation of fraction transformed as a function of tempera-ture, for a typical martensitic phase transformation. Literature Review 61 Figure 2.8: Schematic diagram illustrating the application of the additivity principle. The non-isothermal reaction follows the path marked by the arrows. The isothermal transformation kinetic curves at two temperatures are labeled Ti and T 2 . Chapter 3 Scope and Objectives 3.1 Scope of the Research Programme The goal of the present study was to apply the microstructural engineering approach to generate a basis for understanding the microstructural and mechanical response during forced convective quenching (heat treatment) of steel rods. It is part of an ongoing generic project aimed at the prediction of microstructure and mechanical properties obtained from thermal and thermo-mechanical processing operations. To achieve this objective, mathematical models of thermal/microstructural evolution and stress generation were developed. Once debugged and tested, the mathematical models were validated by comparing calculated results with values measured in the lab-oratory. The effect of internal stresses on the transformation kinetics is relatively small; however, the computing costs increase dramatically when this interaction is considered. Also, there is uncertainty as to how to best apply the results of simple mechanical tests to a complex stress state. Thus, in this work, the effect of internal stresses on transfor-mation kinetics was not considered. As a consequence of this assumption, the models could be effectively uncoupled, i.e., the results of the thermal/microstructural model were adopted as input to the stress generation model. The problem at hand is highly nonlinear and, therefore, a numerical solution needs to be implemented. In structural analysis, the finite-element method (FEM) has be-come a de facto standard (mainly due to its ability to handle components of complex 62 Scope and Objectives 63 geometry) and was adopted for the analysis. Based on geometric considerations, an ax-isymmetric formulation was deemed adequate. A finite-element program developed to simulate thermal stresses in fused-cast monofrax-s refractories [182] was used as a basis for the transient thermal/microstructural model while a computer program for the time-independent, elasto-plastic analysis of stress evolution in water spray-quenching of steel bars [47] was adopted to model internal stress generation. Due to difficulties in instrumenting samples under industrial conditions, a laboratory quenching facility was designed and built. Measurements made with this apparatus were used to characterize heat transfer in forced convective quenching as well as provide samples for residual stress measurements (by neutron diffraction) and microstructural characterization to validate the mathematical models. A critical component of the models is the heat transfer boundary condition. Due to a lack of data in the literature for the conditions of interest, the surface heat flux, as a function of surface temperature, was estimated from the measured temperature response of instrumented samples through the solution of the inverse heat conduction problem (IHCP). An existing computer program that implements the sequential function specification technique was modified for this purpose [183]. A characteristic of mathematical models of heat treatments is the large database that is required. For this study, the thermophysical and thermomechanical properties were taken from the literature while the transformation kinetics under isothermal and continuous cooling regimes were measured in the laboratory using a Gleeble 1500 ther-momechanical simulator. Scope and Objectives 64 3.2 Objectives of the Research Programme The objective of the present study was : To formulate, develop and verify mathematical models capable of solving the transient thermal, microstructural and stress fields ob-tained during forced convective quenching of steel bars. To accomplish this goal, the following subtasks were undertaken : • To characterize the surface heat flux, as a function of surface temperature, in forced convective quenching. • To formulate, develop and verify a computer program to solve the inverse heat conduction problem. • To measure transformation kinetics under isothermal and continuous cooling con-ditions. • To characterize the final microstructure and hardness in the quenched samples, for comparison to model predictions. • To measure the residual stress distribution in the quenched samples via neutron diffraction, for comparison to model predictions. C h a p t e r 4 H e a t T r a n s f e r M o d e l To model the evolution of the thermal and microstructural fields, a continuum mechanics approach was adopted. As discussed earlier, the thermal and microstructural fields need to be solved simultaneously, while ignoring both the effect of elasto-plastic deformation on phase transformation kinetics and the heat generated by the deformation. In this chapter, the equation governing heat transfer during quenching of steel, to-gether with the relationships between the kinetics of phase transformation and heat evolved, are presented. The principle of additivity was invoked to compute the advance of the transformations taking place. Then, a solution to this problem, based on the finite-element method, is presented. 4.1 Governing Equation The governing equation for the heat transfer phenomena in an isotropic body, including heat generated due to phase evolution, is as follows where, in general, T = T(x{,t) and the rate of heat generation, q = q(xi,t). For an axisymetric problem, this equation reduces to : V-fc VT + q = pCp — (4.1) (4.2) 65 Heat Transfer Model 66 Ignoring the end effects, the above equation reduces to : For an infinitely long, solid cylinder cooling in water, one can express the boundary conditions (B.C.'s) for the heat transfer problem by the following dT B.C . 1: ^ = 0 at r = 0, f > 0 (4.4) or and dT B . C . 2 : -k — =-h(T-Tf) at r = R, t > 0 (4.5) or i.e., symmetry at the centreline and heat transfer by convection at the surface, respec-tively. The bar is assumed to be at a uniform temperature before the start of quenching. Then, the initial condition (I.C) is I . C . : T(r,t)= T0 0 < r < R, t = 0 (4.6) 4.1.1 Rate of Heat Evolved The rate of heat evolved during a transformation constitutes the link between the thermal and microstructural fields. At each time increment, At, it is directly proportional to the rate of transformation : 9 = ^ ^ ^ (4-7) In order to compute q, the new fraction transformed (which is a function of temper-ature and cooling path) must be known. It follows that an iterative scheme must be Heat Transfer Model 67 implemented. For diffusional transformations, the fraction transformed during continuous cooling can be computed using kinetic information obtained under isothermal conditions by in-voking the additivity principle as follows1: The 'virtual time', the time required to reach the old fraction transformed, mX, at the current iterated temperature, m + 1 T*, is obtained by solving the J M A K equation for time, i.e., obtaining the inverse of the J M A K equation : l+16* = ( r o + 1 X ( T , * ) ) In' 1 •x (4.8) where n and b are kinetic parameters in the J M A K equation. The new fraction trans-formed is then calculated by adding At to m+10* and substituting in the Eq. (2.4) : m+lX* = 1 - exp [-&*( m + 1 0* + At)n"] (4.9) The kinetic parameters n, b and £ A „ c c t are computed via empirical equations that are developed based on isothermal (n and b) and continuous cooling (transformation start, ^AVCCT) tests. It should be noted that, in the model, tAvCCT is calculated at each node ignoring recalescence, i.e., it is assumed that each node cools independently of the rest2. The kinetics of the martensitic transformation are described by the Koistinen and Marburger [99] equation (Eq. (2.6)). In this case, the fraction transformed is only de-pendent on the current temperature. Given that the kinetic relationships are usually expressed in terms of undercooling below the thermodynamic transformation temperature, the relevant sections of the phase •"Til the following equations m + 1 () and m () denote values at the end and beginning of the current time interval, respectively. An asterisk, ()*, denotes a quantity evaluated at the guessed temperature m+1T*. 2Based on the fact that the experimental values of tAvccr a r e obtained from a set of independent experiments conducted at several cooling rates. Heat Transfer Model 68 diagram need to be calculated. To this end, a code based on the algorithm proposed by Kirkaldy et al. [104] was developed. Once the Ae3 and Acm lines were computed, the eutectoid temperature and composition were calculated as their intersection; instead of solving the corresponding equations simultaneously a trial and error procedure was implemented, with a convergence criterion given by e = T A e S ~ T A c m < 0.05 (4.10) -L Acm The code calculates the Acm and Ae3 lines as 2nd and 3rd order polynomials as a function of % C , respectively. For hypo- and hyper-eutectoid steels, the %C vs T functions for temperatures below the eutectoid can be obtained by inverting the polynomials, which was accomplished by using reversion formulae [184] based on the Taylor series expansion. It was found that 2nd and 3rd order reversion formulae are adequate to invert the 7—Fe 3C+7 and 7 — a + 7 lines, respectively. 4.2 Finite-Element Formulation The heat evolved during the transformation, the boundary conditions and the variation of thermophysical properties with temperature, make this a highly non-linear problem and, therefore, an analytical solution is not an option. Instead, one has to resort to a numerical solution. In particular, the finite-element method was chosen to solve Eq. (4.3). The basic components of the finite-element formulation are : 1) the finite-element equations and 2) the solution algorithm for the nonlinear equations. Heat Transfer Model 69 4.2.1 Finite-Element Equations r In the finite-element method the thermal field at the element level is approximated by the following relationship : {T} » {f} = [N]{aY (4.11) where {T} = [T]T represents the thermal field, [N] is the shape function matrix 3, and {a} e is the nodal temperature vector. There is one degree of freedom at each nodal point. The gradient of the thermal field is expressed as V T = [L]{T] = [L][N]{aY (4.12) where [L] is the following linear operator (for an axisymmetric problem) : ^-{!•£} w The [B] matrix is defined by [B] = [L][N] (4.14) then VT = [B]{a}e (4.15) and the heat flux by conduction is computed from the thermal gradient as follows : {q}T = - f c V T = -k[B]{a}e (4.16) By adopting the Galerkin method, a weak form of Eq. (4.3) can be derived, from 3Linear-order (4-node) and quadratic order (8-node) shape functions applicable to two-dimensional isoparametric elements can be found in Ref. [185]. Heat Transfer Model 70 which the following equivalent expression of the governing equation is obtained4 [C]e{aY + [K]e{aY = {RY (4.17) where, for the axisymmetric case, the capacitance matrix [C]e and the conduction matrix [K]e at the element level are given by 5 [C]e = 2TT / [N}Tp Cp r[N] dr dz « 2wr f [N]Tp CP[N] dr (4.18) and [K]e = [Krr]e + [Kzz]e + [Kcv]e (4.19) [Krr]e = 2n f -^-[N]Tkr-^-[N]drdz (4.20) J A* or or w 2nf f — [N]Tkr—[N]drdz JAE or or [Kzz]e = 2*1 ^-[N]Tkr^-[N]drdz (4.21) JA" OZ OZ « 2nf ( —\NYk-\N\drdz JA' oz dz [Kcv]e = 2TT / [N]hrdrdz (4.22) Jce w 27rf / [N]hdrdz Jce respectively. Considering boundary conditions of convection and/or specified surface heat flux, and 4Details of the derivation are given in Appendix A. 5 To facilitate the integration, the quantities that depend on the coordinates are evaluated at the centroidal point (r = J2i •^ rt7*t) a n d a r e denoted by Q. Heat Transfer Model 71 a spatially varying volumetric heat source, the elemental load vector has three compo-nents : {/}e = iUY + {hvY + {fqY (4.23) where {/•} e = 2TT / [N]q r drdz Pa 2?rf / [N}qdrdz (4.24) JAe {/,}" = 2;r / qsr dC « 2?rr / ? s dC7 (4.25) if*,}* = 2TT / /*7> dC7 » 2?rf / hTt dC (4.26) The system of equations given by Eq. (4.18) represents the equivalent finite-element form of the governing differential equation and must be assembled to give a system of equations at the global level before solving. These equations are then solved by adopting the following three-point recurrence scheme [186], as described in [182] : . 4 • + A ^ - J ( 4 - ^ J with a Crank-Nicholson approximation adopted to start the algorithm. 4.2.2 Solut ion A l g o r i t h m Due to kinetic factors, phase transformations in actual processes do not start at the tem-peratures indicated in a phase diagram. Also, the rate of heat released by the austenite decomposition reactions may be larger than the rate of heat removed by the cooling medium, producing recalescence. These two factors combine to produce cooling curves Heat Transfer Model 72 like the one shown, schematically, in Figure 4.1. For each of the sections shown in the figure the heat released was evaluated in a different way : • For conditions under which the transformation either has not yet begun or has already been completed, a value of zero was assigned to the heat generation term. • When the transformation is taking place, a definite amount of heat (related to the current fraction transformed) was released. An iterative scheme was necessary in order to evaluate this contribution. A first approximation for the thermal field was obtained assuming that no heat was released. With this information, the additivity principle was then used to evaluate the fraction transformed and the total heat released, q, was then calculated from Eq. (4.7). The thermal field was re-calculated using this value of q, and the process was repeated until the following convergence criterion was met : where JYL{mJrlT* — m + 1 T)2 is the Euclidean Norm [187] and e is a predetermined convergence criterion. Depending on the cooling rate, the parent phase may transform to a mixture of equi-librium (pearlite, ferrite) and non-equilibrium (bainite, martensite) phases and, there-fore, several possible scenarios needed to be incorporated in the computer code. This was achieved through decision trees, as illustrated in Table 4.1. The progress of the transformations was monitored at all times, and an increment of fraction transformed was computed at every time step. The start of the austenite-to-ferrite, -pearlite, or -bainite transformation was determined by either calculating the time spent below the corresponding transformation temperature and comparing it to the measured incuba-tion time, or by using empirical correlations of start time as a function of cooling rate. v / E ( m + i r ) 2 (4.28) Heat Transfer Model 73 The austenite-to-martensite transformation was assumed to start as soon as the local temperature dropped below M g . A flow diagram of the numerical procedure is shown in Figure 4.2. As an example, details of the sequence for evaluating the heat evolved during the austenite-to-pearlite transformation are shown in Figure 4.3. In order to speed up the solution, a relaxation technique was implemented. The temperature field was under-relaxed after every iteration following : T = toT* + (1 - u)T (4.29) where T* is the current iterated thermal field and T is the thermal field that will be used for the next iteration. A value of 0.5 was used for the under-relaxation parameter, u>. Realizing that the very first iteration under-estimates the actual thermal field, the relaxation factor was set to u < 0.5 for that particular iteration. Typically, less than 10 iterations were needed to achieve convergence. 4.3 Verification of Mathematical Model of Heat Flow The finite-element code developed by Cockroft [182] to simulate heat transfer in the Epic-3 Monofrax-S casting process was modified to model microstructural and thermal evolution during heat treatment. The verification of the computer code was carried out in three stages. First, numerical results for heat conduction, without considering heat generation, were compared against analytical solutions. Secondly, a semi-analytical solution was carried out for the cooling of a cylinder with a uniformly distributed heat source of known strength in order to verify the solution algorithm (without including the kinetics of the phase transformation). Lastly, a finite-difference simulation of thermal and microstructural responses for controlled air cooling of eutectoid steel rod [188] was used to test the complete finite-element code. In all uniform initial temperature Heat Transfer Model 74 distribution was assumed. 4.3.1 Infinite Solid Cylinder : q — 0 The governing equation for the case of no generation nor consumption of energy is 1 & (n dT\ „ dT . . where T = T(r,f). For constant thermophysical properties, Eq. (4.30) reduces to d2T IdT IdT . Or2 r or a ot 4.3.1.1 Newtonian Cooling. An extreme case of Eq. (4.31) is that of Newtonian cooling (Bi < 0.1), where it is assumed that no thermal gradients exist inside the solid cylinder, i.e., the thermal resistance due to convection is much higher than that due to conduction. After performing a heat balance, the ordinary differential equation that describes the problem is [189] - Vp Cp^ - hA(T - Tf) (4.32) with T = T(i) only. After integration, the thermal response, for Tf = 0, in the cylinder is given by, T=T°^(~MV) (433) The input to the analytical and F E solutions is given in Table 4.2. Figure 4.4 compares the F E M and the analytical solution for a 1 mm-dia. rod cooled under Newtonian conditions. Since the numerical solution reflects the fact that a thermal gradient does Heat Transfer Model 75 exist, the temperature plotted corresponds to values at r/R = 0.5. Excellent agreement can be observed. 4.3.1.2 Non-Newton ian Cool ing . The more general case for non-Newtonian cooling of an infinite solid cylinder by con-vection to a medium of constant temperature with a constant heat-transfer coefficient, was also considered. In this case, Eq. (4.31) applies, subject to the following boundary conditions B .C . 1: T= 0 at r = 0, i > 0 (4.34) B .C . 2 : -k^j- = -h(Tf -T) at r = R, t > 0 (4.35) where T = T(r, t). The analytical solution to Eqs. (4.31), (4.34) and (4.35) (for 7> = 0) is, using the method of separation of variables [190], r ( r , t) = 2-^t n i+ffr], m exp( -aA m 2 *) (4.36) K m=l( A™ + H2)J0(\mR) where H = \ ( 4 ' 3 7 ) and the eigenvalues, A m , are the positive roots of either \mJ'0{\mR) + HJ0(XmR) = 0 (4.38) or A m Ji{XmR) - HJ0{\mR) = 0 (4.39) with J0{x), J'0(x) and J\{x) being the Bessel function of the first kind of order zero, its Heat Transfer Model 76 first derivative, and the Bessel function of the first kind of order one, respectively. Eqs. (4.36) and (4.39) can be rewritten in non-dimensional terms (Fo = at/R2,Bi = HR) as : T = 4 (WWXM t , j ( - ( U ) ! F o ) ( 4' 4 0 ) and f(XmR) = XmRJi(XmR) - B\J0{XmR) = 0 (4.41) The data used as input are given in Table 4.3. A comparison between the numerical and the analytical solution for cooling at the centreline of a rod in which non-Newtonian conditions apply (Bi = 0.5) is shown in Figure 4.5. Again, very good agreement can be seen. 4.3.2 Infinite Sol id Cy l inde r : q = f(T) A semi-analytical solution for the Newtonian cooling of an infinitely long cylinder with a uniformly distributed heat source, q = Q0(T), was carried out (see Appendix B). As is the case with virtually all analytical solutions, the thermophysical properties were assumed to be independent of temperature. The specific function, Q0(T), used to test the numerical scheme is shown in Figure 4.6; it was chosen because it resembles the shape of the heat source history in a typical quench. Figure 4.7 shows the thermal response at the centreline for a 10 mm-dia. rod for values of Q0,m of 2.75 x l O 7 and 6.0 x l O 8 W m~ 3 s - 1 (typical values for heat evolved during the austenite-to-pearlite transformation range from 5 x 107 to 3 x 108 W m~ 3 s _ 1 ) . The curve for Qo,m = 0, i.e., no heat evolved, is included for comparison. Note the recalescence produced by the heat generation term. As described in Appendix B , the semi-analytical solution was achieved in a step-wise fashion. Results for a large value of Q0,m obtained using different time intervals, At, showed instabilities when a large value of At was chosen (Figure 4.8). This observation Heat Transfer Model 77 points out the importance of a proper selection of the time interval, A t . The numerical and semi-analytical solutions were compared for values of Q0ym = 2.75 x 107 and 6.0 x l O 8 W m~ 3 s _ 1 (see Figures 4.9 and 4.10, respectively). Excellent agreement was found for both cases6. 4.3.3 Infinite Solid Cylinder : Air Cooling of Eutectoid Steel The predicted thermal response of an infinitely long, 10 mm-dia. rod of eutectoid carbon steel obtained with a finite difference scheme [188], assuming constant thermophysical properties, was also used to verify the mathematical model. The properties adopted are given in Table 4.4. Figure 4.11 shows the results of both the finite difference and the finite element analyses for the 10 mm-dia. rod; very good agreement is evident between comparing the predicted temperatures at both the centre and the surface of the rod. Figure 4.12 shows the corresponding results for microstructural evolution. 4.4 Summary A computer program based on the finite-element method has been developed to simulate heat transfer and microstructural evolution during forced convective quenching of steel bars. Given that the solutions of the thermal and microstructural fields are coupled, through the heat of transformation, an iterative procedure was implemented. To accel-erate the convergence, a relaxation technique was applied. The principle of additivity was invoked to compute the progress of the transformation under continuous cooling conditions. The J M A K equation was used to describe the kinetics of diffusional trans-formations, while the Koistinen and Marburger equation was adopted to compute the martensitic transformation. The code has been verified by comparing results of numerical and analytical solutions. 6No relaxation was applied for these calculations Heat Transfer Model 78 Table 4.1: Decision tree to determine the sequence of calculations during phase transfor-mations in a eutectoid steel. Product Condition Action P /.'.OLD > 0 A / * p = 0 /B,OLD > 0 A / p = 0 /P.OLD > 0-995 A / p = 0 /p,OLD = 0 Check start of transformation 0 < /P.OLD < °-995 Calculate A / p B /«',OLD > 0 A / * B = 0 /P.OLD > 0-995 A / * B = 0 /B.OLD > 0-995 A / * B = 0 /B,OLD = 0 Check start of transformation 0 < /B.OLD < 0-995 Calculate A / B a' /P,OLD + /B,OLD > 0-995 A / * A , = 0 /a',OLD > 0-995 A / ; < = o /<*',OLD = 0 Check start of transformation 0 < / . ' .OLD < 0-995 Calculate A / * , Note : fi = normalized fraction transformed Note : /* = true fraction transformed. Heat Transfer Model 79 Table 4.2: Input data used for the comparison of finite-element and analytical solutions under Newtonian cooling conditions. The corresponding Biot number was Bi= 0.01. Variable Value R 0.001 m k 20 W / ( m K) P 10000 kg /m 3 Cp 200 J/(kg K) h 200 W / ( m 2 K) T0 1000 °C Tf 0 °C At 1.0 s Table 4.3: Input data used for the comparison of finite-element and analytical solutions under non-Newtonian cooling conditions. The corresponding Biot number was Bi= 0.5. Variable Value R 0.050 m k 20 W / ( m K) P 10000 kg /m3 Cp 200 J/(kg K) h 200 W / ( m 2 K) To 1000 ° c Tf 0 ° c At 250.0 s Heat Transfer Model 80 Table 4.4: Input data adopted for the comparison of finite-element and finite-difference simulations of air cooling of eutectoid steel rod [188]. Variable Value k 25 W / ( m K) P 7650 kg /m3 625 J/(kg K) h 160 W / ( m 2 K) (simulates a cooling rate of 9.4 °C s _ 1 ) T0 850 °C Tj 20 °C A # 7 _ P 88200 J kg" 1 In np 2.2 In bp -41.49(7^ - r ) ° - O 7 2 3 e x p [ - 0 . 0 3 6 4 3 ( ( r A l - T)] In tAvccr 29.73 + 0.05816(rA l - T) - 8.622 log(TAl - T) + 6.27 (XC + XMJG.O) Note : The thermophysical properties : k,p and Cp, of both austenite and pearlite were assumed to be equal. Heat Transfer Model Heat Transfer Model ( START ) Read input data Initialize variables { W <- { T m } g = 0 Calculate {T*} {Tprop}^Lo{T:} + (l-u){Tprop} Calculate {T*} Convergence ? y QTRNSF-MICRO n * <- t + A t t>tFl • ( E N D n { T m + 1 } <- {T*} { r } <- {/m+1> Figure 4.2: Flowchart for the solution of the thermal-microstructural probL Heat Transfer Model 83 ( QTRNSFJVIICRO ) T > ? R E T U R N n f m > 0.995 ? — ^ R E T U R N n t* = t — tAl t* > tAVCCT ? >( R E T U R N j n Compute f m+1 (nodal) A / «- / m + 1 - / m Compute <j (nodal) R E T U R N Figure 4.3: Details of the model sequence for evaluating the heat released during austen-ite-to-pearlite decomposition. Heat Transfer Model 84 - 1 0 0 1 • 1 • 1 1 1 • 1 0 10 20 30 40 Time, s Figure 4.4: Comparison between the finite-element and the analytical solution for the newtonian cooling (Bi= 0.01) of an infinitely long cylinder. 1100 4> H Analy t ica l O FEM Center l ine - e — e — e — e — o _i i i— 500 1000 1500 2000 2500 3000 Time, s Figure 4.5: Comparison between the finite-element and the analytical solution at the centreline for the Non-newtonian cooling (Bi= 0.5) of an infinitely long cylinder Heat Transfer Model 85 100 200 300 400 500 600 700 800 900 o Temperature, C Figure 4.6: Function Q0 used in the semi-analytical solution of Newtonian cooling of a long rod with a uniformly distributed heat source, where Qo,m is the maximum value of the function QQ. 900 -\ 800 -- 3 - 1 Q = 0.0 MWm s o . m - 3 - 1 Q = 27.5 MWm s ^ o , m - 3 - 1 Q. = 60.0 MWm s ^ o , m o ° . 700 -u * 600 -<D o. S 500 -<i> E-400 -300 h 200 I 1 1 1 1 1 1 0 20 40 60 80 100 120 Time, s Figure 4.7: Semi-analytical solution, at the centreline of a 10 mm-dia rod, for Q0,m — 27.5 and 60.0 MWm"3 r - 1 . The curve for Q0 = 0 M W m - 3 s _ 1 is included for comparison. Heat Transfer Model 86 850 825 o o « 800 in 3 Cll P. s a> 775 750 725 20 At = 1 0 ms At = 1 0 s = 600 MWm 3s 1 40 60 Time, s 80 100 120 Figure 4.8: Effect of At on the semi-analytical solution, at the centreline of a 10 mm-dia. rod, for Q 0 , m = 600.0 M W m " 3 s"1. 900 400 Analytical FEM (centre) FEM (surface) 20 40 Time, s 60 80 Figure 4.9: Comparison between the semi-analytical and the numerical (FEM) solution, during cooling of a 10 mm-dia. rod, for Q0,m = 27.5 M W m - 3 s - 1 . Heat Transfer Model 87 Figure 4.10: Comparison between the semi-analytical and the numerical (FEM) solution, during cooling of a 10 mm-dia. rod, for Q0,m = 60.0 M W m - 3 s _ 1 . 500 10 20 30 Time, s 40 50 Figure 4.11: Finite-difference [188] and finite-element solutions for the thermal response at the centreline and surface of a 10 mm-dia. eutectoid carbon steel. The cooling rate at 728 °C was 9.4 °C s _ 1 . Heat Transfer Model 88 0 10 20 30 40 50 Time, s Figure 4.12: Finite-difference [188] and finite-element solutions for fraction transformed to pear lite at the centreline and surface of a 10 mm-dia. eutectoid carbon steel. The cooling rate at 728 °C is 9.4 °C s - 1 . Chapter 5 The Inverse Heat Conduction Problem The accurate determination of the heat transfer boundary condition is a crucial com-ponent of mathematical, models aimed at predicting the evolution of the thermal, mi-crostructural and stress fields during the heat treatment of a metal. This task is difficult to accomplish experimentally due to the high temperatures involved, the uncertainty associated with surface temperatures measured with thermocouples, and, in the case of quenching, the presence of boiling heat transfer. A viable alternative consists of estimat-ing the heat flux and surface temperature from a temperature response measured inside the body; this is known as the inverse heat conduction problem (IHCP). In most heat treatment applications of inverse analysis the determination of surface heat flux constitutes a function estimation problem, i.e., the unknown function is es-timated without any prior knowledge of the actual functional form of the variation of surface heat flux with time. To estimate this function, a large number of surface heat flux components need to be determined. From a mathematical point of view, the IHCP can be classified as an ill-posed problem, i.e., it does not satisfy the requirements of existence, uniqueness and stability of the solution. This translates into inverse solutions being highly sensitive to small fluctuations in the measured temperatures. Current so-lution algorithms for solving the IHCP have been designed to overcome this problem. The method of solution can be either analytical or numerical. It may be applied either sequentially, i.e., by estimating a single component of the surface heat flux function at each time step, or to the whole time domain, in which case all components are estimated 89 The Inverse Heat Conduction Problem 90 simultaneously. Due to the restrictive nature of analytical methods (chiefly their inability to handle nonlinearities), practically all IHCPs need to be solved numerically. In this chapter, the application of the sequential function specification algorithm to the solution of the IHCP is presented, with special emphasis on the modification (to accommodate cylindrical geometry) of an existing computer code. 5.1 Solution Algorithm The one-dimensional IHCP is schematically illustrated in Figure 5.1. A plate of thickness 2L is initially at a uniform temperature, T0. For times t > 0, an unknown, time-dependent heat flux, q(t), is applied at the boundary x = L, while the boundary at x = 0 is that of symmetry with respect to the thermal gradient. In order to estimate q(t), temperature measurements are made at a position inside the body, denoted by x\. It is assumed that no information is available regarding the specific form of q(t) and, therefore, a large number of components of q(t) need to be estimated. In the past, 'brute force' algorithms based on a trial-and-error procedure have been used to solve the IHCP. At each time step, a guessed value of the heat flux is adopted for the calculation of the temperature at the sensor position, and the iterations continue until the difference between the calculated and measured values falls within a predetermined tolerance. Current methods adopted to solve this function estimation class of IHCPs fall into one of two broad categories [191] : 1) function specification and 2) regularization. The algorithm adopted in this investigation was proposed by Beck et al. [192] and can be described as a sequential function specification algorithm. In this algorithm, the surface heat flux is estimated by assuming temporarily that heat flux components up to (r — 1) time steps ahead of the current time are constant (see Figure 5.2). For a single thermocouple, the estimated temperature exactly matches the measured value at each time step, if r is set to 1. However, the solution becomes unstable and very sensitive to The Inverse Heat Conduction Problem 91 measurement errors (specially when small time steps are adopted). The unknown heat flux at the surface is obtained from temperature measurements spanning several future time steps, by minimizing the following least squares expression with respect to the heat-flux component at time t = IM '• S = E E (Yf4*-1 - T^1-1)2 (5.1) »=i i=i where y^M+'-1 J s the measured temperature at the jth sensor at time tM+i-i, T^+l_1 is the corresponding calculated temperature, and r is the number of future time steps adopted for estimating qM. It is assumed that the heat flux is known for t < £ M - I -Through the use of the future data concept, stable solutions, even for small time steps, can be obtained. It should be noted that by minimizing the least square norm, instead of making it zero, the uniqueness of the solution is guaranteed [193]. Differentiating Eq.(5.1) with respect to qM, replacing qM by qM (the estimated heat flux at time £M), and setting the expression equal to zero, one obtains : 2 £ E ( C + i - 1 - 7 f + - ) ( a^if—) .„ = ° (5-2) 8=1 j=i • \ " / ' Assuming that the future temperature at the sensor position j can be calculated from a Taylor series expansion about qM~l : „ , , . , *M+i-l * M+i-1 Tf^~l = T3 +(qM-qM~1)X3 (5.3) where the asterisk implies that the T and X functions are evaluated using the thermal properties and surface heat flux values at time t\i-i- The quantity X^+t_1 is called the sensitivity coefficient and is defined by QrpM + i-l x r - 1 = - f a - <5-4> The Inverse Heat Conduction Problem 92 Note that, by using this definition, the partial differential equation that describes the sensitivity coefficient distribution is obtained as d dx dx k— = pCp— (5.5) dxi dxi dt with the following boundary conditions : 8X B.C . 1 : —— = 0 at X l = 0, t > 0 (5.6) Ox i dX B . C . 2 : -k-z— = 1 at X l = Xu t > 0 (5.7) ox i representing an insulated (or, equivalently, symmetric) boundary and a specified heat flux condition for the inactive and active boundaries, respectively. It is evident that Eq. (5.5) has exactly the same form as the differential equation for the thermal field, the only difference between the two problems, from a mathematical point of view, being the exact form of the boundary conditions. It follows that the same algorithm adopted to solve the direct heat conduction problem can be applied to compute the sensitivity coefficient distribution, resulting in a more efficient code. Introducing Eq. (5.4) into Eq. (5.3) and solving for qM gives 1 r J / *M+i-l\ * M+i-l dM = ^ + T ^ E £ (Y?*-1 ~ T3 ) X3 (5.8) M i = l j = 1 V / where JL-L/* M+i-l\ 2 A M = E E ^ (5-9) ;=ij=i v J Note that estimating T M . . . T M + r - 1 and XM ...XM+r~1 by adopting the thermo-physical properties, k and pCp, which correspond to the previous time step, the problem has been linearized and Eq. (5.8) becomes explicit in qM. The justification for this as-sumption is that, for a small time step, At, the thermal properties change little at a given The Inverse Heat Conduction Problem 93 location from one time to the next even though there may be a large variation in prop-erties from one end of the body to the other; an iterative procedure is then not needed, even for a problem with varying thermal properties. The algorithm is summarized in the flowchart shown in Figure 5.3. Once the response of the heat flux and temperature at the surface have been calcu-lated, they are used to estimate the heat-transfer coefficient. It should be noted that the algorithm calculates both q and Ts, but they are calculated at slightly different times : qM is best associated with ijvf-1/25 while Ts corresponds to tjvf- Therefore, the heat-transfer coefficient at time £M is estimated from : The computer program C O N T A [183] incorporates the sequential function specifi-cation algorithm described above, for the case of a one-dimensional, planar IHCP. This code uses the Crank-Nicholson implementation of the finite-difference method to solve the partial differential equations describing the thermal field and the sensitivity coefficient distribution. It has been successfully applied to estimate the heat-transfer coefficient during pool quenching of stainless steel disks [194] and to characterize interfacial heat transfer during metal solidification [195,196]. 5.2 Modification for Cylindrical Coordinates The code C O N T A was modified [197] to incorporate cylindrical geometry - appropriate for the quenching of bars. Once coded and debugged, the program was tested by compar-ing results obtained using analytical solutions for two cases : 1) a solid cylinder subjected to a constant surface heat flux and 2) a solid cylinder subjected to a medium of constant heat-transfer coefficient and fluid temperature. In both cases the direct problem was (5.10) The Inverse Heat Conduction Problem 94 first solved analytically to obtain the thermal response at a given point in the domain, which was then adopted as input to the numerical solution of the IHCP to estimate both surface heat flux and surface temperature. Case 1 : Solid Cylinder Subjected to a Constant Heat Flux at the Surface. This problem is described mathematically, assuming constant thermophysical properties, by Eq. (4.31), subject to the following boundary and initial conditions : B . C . 1 : B .C . 2 : L C . : dT dr = 0 , dT T(r,t)= T0 at r = 0, t > 0 (5.11) at r = R, t > 0 (5.12) 0 < r < R, t = 0 (5.13) for which an analytical solution, for T0 = 0, is available [198] : r r . 2q0at q0Rf r2 1 ~ / a\mH\ J0{r\m/R) kR k \2R2 4 m=l where A m are the roots of the transcendental equation (5.14) Wn) = 0 (5.15) The results at the centreline (T(0,t)) were used as input for the program and the surface heat flux was recalculated. The input data are given in Table 5.1. Figure 5.4 shows a plot of estimated surface heat flux versus time as well as the input heat flux; good agreement can be observed except at early times. The thermal response at the surface was also computed and compared to the analytical solution; very good agreement was obtained, as shown in Figure 5.5. The Inverse Heat Conduction Problem 95 Case 2 : Solid Cylinder Subjected to a Medium of Constant Heat-Transfer Coefficient. Eq. (4.31) is again applicable in this case, with the following boundary and initial con-ditions : B . C . 1 : B . C . 2 : I.C. : dT dr = 0 dT T(r,0) = T0 The analytical solution is given in [190] as at r = 0, t > 0 (5.16) at r = R, t > 0 (5.17) 0 < r < i?, t = 0 (5.18) T(r,t) 2HT0 00 E Jo{Xmr) R ^ ( A ™ 2 + W)J0(\mR) e x p ( - a A m 2 i ) (5.19) where H — h/k and Amare the roots of the transcendental equation XmJ'0{XmR) + HJ0(\mR) = 0 (5.20) The program was run, using the thermal response at the centreline calculated with the analytical solution as input, and the heat-transfer coefficient was back-calculated. The data used is shown in Table 5.2. Figure 5.6 shows good agreement between the estimated and the input heat-transfer coefficients, except at early times. The estimated thermal response at the surface also agrees well with the analytical solution, as shown in Figure 5.7. The Inverse Heat Conduction Problem 96 5.3 Application to Controlled Air Cooling of Rods An additional test, using experimental information, was conducted to verify the modified code [197]. Measured temperatures at the centreline of rods subjected to forced air cooling were used as input to estimate the corresponding heat-transfer coefficients. The experiments, reported by Campbell et al. [42], involved a range of steel grades, rod diameters and air velocities, and were designed to simulate the Stelmor process. The specimens were long cylinders (L /D > 25 ), thus ensuring one-dimensional heat flow in the radial direction. Assuming a uniform air flow around the specimen, the boundary conditions for the direct problem were symmetry at the centreline (r = 0) and heat transfer by convection and radiation at the surface (r = R). The rods were initially kept at a uniform temperature. To illustrate the calculations, the estimated heat-transfer coefficient plotted against computed steel surface temperature for an 8 mm-dia. rod, air-cooled at 22 m/s, is shown in Figure 5.8, together with results obtained by Campbell et al. [42] using an iterative procedure based on an implicit finite-difference approximation of the direct problem with exact matching of experimental and computed temperatures (see [197] for thermophysical data used in the calculations). Good agreement can be seen, although it is important to note that the calculations based on the sequential function specification algorithm exhibit less scatter. This result is a consequence of the more stable solution achieved through the use of the concept of future time steps. 5.4 Application to Forced Convective Boiling In order to study the behaviour of the inverse analysis code under similar conditions to those expected in the laboratory trials, the thermal response of a thermocouple located near the surface of a 38.1 mm (1.5 in) bar subjected to forced convective quenching was The Inverse Heat Conduction Problem 97 modelled using a finite-element program. Constant thermophysical properties (repre-sentative of steel) were adopted in the calculations; it was assumed that no heat was generated inside the bar. The form of the surface heat flux functional used in the simu-lations was : ' 1 + 0.003 (T, - 900) if Ts > 900 4 - 0.0075 (Ts - 500) if 900 > Ts > 500 q=\ (5-21) 1 + 0.0075 (Ts - 100) if 500 > Ts > 100 0.0133 (T, - 25) if 100 > Ts > 25 where q was in M W m - 2 and Ts in °C. This form of q, as shown in Figure 5.9, was adopted for the sensitivity analysis because of its similarity with preliminary laboratory results. The data used in the calculations is given in Table 5.3. A total of 38 8-node isoparametric elements were used with a time step, A t , of 0.01s; the results were printed every 0.1 s. The calculated thermal response at the centre, surface, and a subsurface position are shown in Figure 5.10. The solution at the subsurface position was adopted as input to the IHCP algorithm to simulate an errorless measured thermal response. For the inverse analysis, the specimens were subdivided into two regions of 1.5 and 17.55 mm, discretized by 5 and 15 nodes, respectively. The time step adopted for the calculation was 0.05 s. A sensitivity analysis was conducted by varying the number of future time steps (parameter r), the thermal conductivity, and the thermocouple position. The results are summarized in Table 5.4. The last column in the table gives the difference, in percentage, between the input and estimated maximum heat flux. The difference between input and estimated heat flux can also be characterized by the root mean square (rms) of the error : where q and q are the actual and estimated surface heat flux values, respectively. The Inverse Heat Conduction Problem 98 The estimated surface heat flux, as a function of surface temperature, is shown in Figure 5.11 for three values of r : 2, 4, and 6. Good agreement can be seen, except at early times and for the peak value of heat flux. The maximum heat flux is underestimated in all cases, with the difference increasing as r increases (from a -4.25 % difference for r = 2 to - 11.75 % for r = 6). The peak heat flux is underestimated because in the vicinity of the surface temperature corresponding to qmax the algorithm anticipates the drastic change in slope sign and starts to decrease earlier. However, the sudden change in the surface heat flux is followed well. It should be noted that only the results obtained with r — 2 showed the early decreasing trend in the heat flux at high temperatures (Ts > 900 °C). For the case of r = 1, i.e., an exact matching of the thermal response at the thermocouple position, an unstable solution was observed. In contrast to the heat flux results, the surface temperature at which the maximum heat flux is observed differs by no more than 3 °C (a difference of only 0.6 %) for the three values of r used; the estimated surface temperature, as a function of time, is shown in Figure 5.12. To study the sensitivity of the analysis with respect to the thermophysical proper-ties, the thermal conductivity adopted for the inverse analysis was allowed to vary from - 20 % to + 20 % of the value used in the finite-element simulation, while using the ther-mal response obtained with the base value as input to the program for all cases. The estimated surface heat fluxes (for r = 4) are shown in Figure 5.13. A higher value of the thermal conductivity results in a higher maximum heat flux and the peak position was shifted to higher surface temperatures, while the opposite was observed when the thermal conductivity was lowered. The effect of using a wrong estimate of the thermocouple position was also studied. To this effect, three values of thermocouple position were used : 16.55, 17.55, and 18.55 mm (the base value used in the finite-element simulation was 17.55 mm); the simulated response using the base value was adopted for input to the inverse analysis in all cases. The estimated surface heat flux, as a function of surface temperature, for the three cases The Inverse Heat Conduction Problem 99 (with r = 4) is shown in Figure 5.14. The effect of adopting a thermocouple position closer to the surface than the actual one, is similar to using a higher value of thermal conductivity; but, for the parameter levels adopted, is less pronounced. Finally, the simulated thermal response at 3 positions in the cylinder (15.55, 16.55, and 17.55 mm from the center) was used to estimate the surface heat flux. The results are shown in Figure 5.15. The results show that a better estimate is obtained as the sensor is positioned closer to the surface. The rms of the error in the estimated heat flux decreases from 0.0552 at 15.55 mm to 0.0402 at 17.55 mm. From the previous results, it is evident that care should be taken in the selection of parameters such as future time steps, thermophysical properties and sensor position. A discussion on filtering noisy data before applying the inverse analysis is left for a later chapter. 5.5 Sensitivity coefficients and experimental design The sensitivity coefficients, Xj, are calculated as part of the solution to the IHCP. They represent a quantitative measure of the sensitivity of the thermal response to changes in the unknown surface heat flux (see Eq. (5.4)), and can, therefore, be used to design experiments. In general, one is interested in large, uncorrelated values of Xj, which means that more non-repetitive information can be extracted from the experimentally determined thermal response. Figure 5.16 shows calculated sensitivity coefficients at sev-eral positions across the radius of a solid cylinder subjected to a constant heat-transfer coefficient and fluid temperature. It can be seen that the largest value of the sensitivity coefficient occurs at the position closest to the surface of the cylinder; for this reason, ev-ery effort should be made to place the thermocouples as close to the surface as practically possible. Lambert and Economopoulos [199] established a linear relationship between the logarithm of the mean error of the determined heat-transfer coefficient and the distance The Inverse Heat Conduction Problem 100 of the point of measurement below the surface for cylindrical samples. They obtained a value of the mean error as high as 10 % for a position 2 mm below the surface, which decreased to less than 1% for a measurement taken at 1 mm below the surface, when long cylinders of 20 mm-dia. were used. This is a direct result of the lagging and damping effects characteristic of transient diffusion problems. In quenching, where steep thermal gradients are present at the early stages, the damping of the transient temperature re-sponse would have an even more adverse effect on the estimation of the surface heat flux from temperature measurements. The Inverse Heat Conduction Problem 101 Table 5.1: Parameters used for Case 1 : constant surface heat flux. Parameter Value R 5 x l O - 2 m a 0.00125 m 2 s- 1 50 W m - 2 k 25 W m " 1 K - 1 Cp 200 J kg" 1 K - 1 P 100 kg m -3 Table 5.2: Parameters used for Case 2 : constant heat-transfer coefficient. Parameter Value h 160 W m~ 2 K - 1 Tf 20 °C T0 850 °C R 5 mm k 25 W m - 1 K _ 1 Cp 625 W m - 1 K " 1 P 7650 kg m~3 The Inverse Heat Conduction Problem 102 Table 5.3: Parameters used for the simulation of a forced convective quenching experi-ment. Parameter Value R 19.05 mm k 25 W m - 1 K - 1 P 7650 kg m -3 Cp 625 J k g - 1 K - 1 T 1000 °C * r 17.55 mm Table 5.4: IHCP algorithm : summary of sensitivity runs. The base values for thermal conductivity and thermocouple position were k0 =25 W m _ 1 K _ 1 and r* = 17.55 mm, respectively. r k T / C position ^max T s,max r^ms % A 9 m a x (W m K - 1 ) (mm) (MW m" 2 ) 2 k0 K 3.83 497.0 0.0203 -4.25 4 k0 K 3.67 500.3 0.0402 -8.25 6 k0 K 3.53 502.5 0.0616 -11.75 1 k0 < - - - -4 k0 - 20 % K 3.44 471.6 0.0979 -14.0 4 k0 + 20 % K 3.82 523.8 0.0527 -4.5 4 k0 K-i 3.63 501.6 0.0491 -9.25 4 k0 K + l 3.64 527.0 0.0471 -9.0 4 k0 r* = 16.55 3.65 498.8 0.0463 -8.75 4 k0 r* = 15.55 3.64 475.8 0.0552 -9.0 The Inverse Heat Conduction Problem 103 B.C. 1 B.C. 2 Y(t) • X 1 ! x = 0 q(t) =' (a) o o O O o o (b) Figure 5.1: (a) Schematic representation of a one-dimensional, single-sensor IHCP in a flat plate of thickness 2L; the sensor is located at position x\. The boundary conditions are : at x — 0, symmetry (dT/dx = 0); and at x = L, unknown time-dependent heat flux, (b) Discrete temperature measurements (Y(ti)) at position x\. The Inverse Heat Conduction Problem 104 LL CO CD I CD O CO CO 0 t. t t M - 1 M M + r - 1 T i m e Figure 5.2: Piecewise aproximation of the surface heat flux as a function of time. The constant heat flux function between tu-\ and tM+r-i is adopted to calculate qM in the sequential function specification algorithm [192]. The Inverse Heat Conduction Problem 105 ( START ) Read input data and initialize variables *M+i-i ^_ Q ) \ < i < r Ti > 1 < i < r * M + t - l Xj , 1 < % < r rpM t > t 7 •( E N D n Af <- Af + 1 Figure 5.3: Flow chart of the sequential function specification algorithm adopted for the solution of the IHCP. The Inverse Heat Conduction Problem 106 70 60 h i 1 1 1 r o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o input o estimated J I L 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time, s Figure 5.4: Comparison of estimated and input surface heat flux for Case 1 : constant surface heat flux. 150 Figure 5.5: Comparison of estimated, using the sequential function specification tech-nique, and analytical thermal response at the surface of the cylinder for Case 1 : constant heat flux. The Inverse Heat Conduction Problem 107 'a 200 160 120 80 40 0* .o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o Input o Estimated 10 15 Time, a 20 25 Figure 5.6: Comparison of estimated and input heat-transfer coefficient for Case 2 constant heat-transfer coefficient. 900 850 800 V 750 700 650 600 Analytical o Estimated 10 15 Time, s 20 25 Figure 5.7: Comparison of estimated, using the sequential function specification tech-nique, and analytical thermal response at the surface of the cylinder for Case 2 : constant heat-transfer coefficient. The Inverse Heat Conduction Problem 108 150 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — L - ' — ' — ' — t — 1 — 1 — 1 — ' — ' — 1 600 650 700 750 800 Figure 5.8: Estimated heat-transfer coefficient for 8 mm-dia. steel rods air cooled at 22 m/s; calculated by using a sequential matching (SM) approach (open circles) and using the sequential function specification (SFS) algorithm (closed circles). The solid curve is a best-fit curve using all the points [197]. 4.5 h o TSurf> C Figure 5.9: The functional q = f(Ts) adopted for the finite-element simulation of the thermal response in a 38.1 mm-dia. cylinder subjected to forced convective quenching. The Inverse Heat Conduction Problem 109 0> 1000, 800 600 400 200 100 Figure 5.10: Calculated thermal response at the centreline, surface, and simulated ther-mocouple position (r* = 17.75 mm), obtained when the heat flux distribution shown in Figure 5.9 was applied at the surface of a 38.1 mm-dia cylinder. surf1 Figure 5.11: Effect of varying the parameter r adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder. The Inverse Heat Conduction Problem 110 1100 FEM -o r = 2 V r = 4 o r = 6 -30 Figure 5.12: Effect of varying the parameter r on the estimated surface temperature response during the simulated forced convective quenching of a 38.1 mm-dia. cylinder. i 1 i 1 r _ i _ r = 4 o input k = k Q -- k = k ° - 20 % k = k ° + 20 % o _1_ 200 400 600 800 1000 Tsurf C Figure 5.13: Effect of varying the value of thermal conductivity adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quench-ing of a 38.1 mm-dia. cylinder. The base value was k0 = 25 W m _ 1 K _ 1 . The Inverse Heat Conduction Problem 111 0 200 400 600 800 1000 T s u r f C Figure 5.14: Effect of varying the value of the thermocouple position adopted in the inverse analysis on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder. The base value was r0 = 17.55 mm. 1000 surf Figure 5.15: Effect of the thermocouple position on the estimated surface heat flux during the simulated forced convective quenching of a 38.1 mm-dia. cylinder. The Inverse Heat Conduction Problem 112 Figure 5.16: Sensitivity coefficients at several radial positions in a solid cylinder subjected to a medium of constant heat-transfer coefficient and fluid temperature. Chapter 6 Stress Model To model the development of the stress field during the quenching operation, a continuum mechanics approach was adopted. The analysis was restricted to small deformations, based on the fact that the deformations caused by thermal and transformation strains are likely to be moderate and no external forces are applied. Given that the bars spend a relatively small time at high temperatures, rate-dependent effects were not considered. In this chapter, the stress-strain relationships for inviscid elastic-plastic materials with work-hardening, including thermal and microstructural effects generated during quenching, are first derived. To obtain the equations, the classic incremental theory of plasticity (a loading-path-dependent formulation) is invoked. Then, the finite-element implementation of these relationships is discussed and two instances of code verification are presented. 6.1 Stress-Strain Relations To illustrate typical material behaviour in the plastic regime, the case of uniaxial loading shall be examined first. Referring to Figure 6.1, the material behaves elastically upon loading until the yield stress (point A) is reached; beyond the yield stress there no longer exists a linear relationship between stress and strain but instead the slope of the stress-strain curve decreases monotonically until, eventually, fracture occurs. If a test is conducted where a specimen is loaded up to point B and then unloaded, the unloading path (BC) will have a slope similar to that of the elastic path but the strain 113 Stress Model 114 at zero stress has a finite value. This residual, or irrecoverable strain, is termed the plastic strain (OC), whereas the recoverable strain (CD) is the elastic strain component of the total strain. Upon reloading (CBE) the material will behave elastically until the previous stress corresponding to point B is reached (this is termed the subsequent yield stress) after which the plastic deformation occurs according to path BE. From the figure it can be observed that the subsequent yield stress of a real material increases as induced plastic strain increases; this effect is known as strain hardening or work-hardening. From the previous discussion it is clear that the mechanical response is a function of both the current state of stress and the loading history, or loading path. Thus, the classic incremental flow theory of plasticity is adopted to describe the stress-strain relationship during plastic loading under a multiaxial state of stress. From the experimental evidence presented above, the fundamental assumption that the individual components of the incremental strain can be separated is expressed as d e ^ d ^ + d ^ + de?- (6.1) where total strain increment elastic strain increment plastic strain increment non-mechanical strain increment For cooling during heat treatments, the non-mechanical strain increment has contribu-tions from thermal- and transformation-related strains. In particular, strains due to volume changes associated with cooling (de* )^ and phase transformations (de*^), as well as those arising from transformation plasticity (de'J), variation of the elastic constants deij = de?--del = de° Stress Model 115 with temperature (defjE), and variation of the flow stress with temperature(degF) need to be considered : de?- = def3 + de% + de% + deg* + teff (6.2) The first four strain increments are present in both the elastic and plastic regimes while the last one is nonzero only during plastic flow. Accordingly, they can be grouped as follows : ds°l3 = de°f + de°? (6.3) where de°f = de j + de* + de* + deg* (6.4) and de°f = de?/ (6.5) In the following, the fundamental characteristics of material behaviour in a uniaxial load test, as described above, are generalized to predict the mechanical response under a multiaxial stress state, taking into account thermal- and transformation-related strains. The discussion is separated into elastic and plastic regimes. 6.1.1 Elastic Stress-Strain Relations To model elastic loading, the material is assumed to be homogeneous and isotropic. Further, a linear relationship between stress and strain is adopted. Such a relationship, for uniaxial loading, is provided by Hooke's law. In the more general case, the so-called generalized Hooke's law is invoked : Oij = Dijki (ek! - e°kf) (6.6) Stress Model 116 or, in incremental form do-;j = D^ki (deki - de°kf) (6.7) where Dijki is the material elastic constant tensor (or stiffness tensor) given by E I 2u SijSki + SikSji + SijSjk (6.8) 2(1+1/) [(1 -2u) where E is the Young's modulus and v is Poisson's ratio. 6.1.2 Plastic Stress-Strain Relations The elastic limit or yield stress under all possible combinations of stresses is defined as a yield function : where ki,k2,... are material constants to be determined experimentally and the state of stress, o~ij, is defined in terms of the invariants of the deviatoric stress tensor. In particular, the von Mises yield criterion is widely adopted : where J2 is the second invariant of the deviatoric stress tensor and k is the yield stress in pure shear. Geometrically, a plot of the yield function in stress space results in a surface, while the stress state is represented by a point. The yield function, Eq. (6.10), is modified to reflect the effects of temperature and 1The Kronecker delta is denned as F(aij,kuk2,...) - 0 (6.9) F(J2) = J2-k2 = 0 (6.10) l 0 if i = j if i j Stress Model 117 transformation changes and includes isotropic work hardening as follows : F = F(aij,K,T,Xk) (6.11) where n(ep) is a hardening parameter and ep is the effective plastic strain. In order to apply the loading function to a real problem, the hardening parameter must be related to experimental uniaxial stress-strain curves. This is done through the definition of the effective stress and the effective plastic strain. A n expression for the effective stress is derived from the results of a uniaxial test : (6.12) The effective plastic strain is given by d£p = \ l D 4 D 4 ( 6 - 1 3 ) The effective stress, <7e, is related to the effective plastic strain by cre = ae(ep) (6-14) Differentiating : dcre = Hp(ae)dep (6.15) where Hp(ep) = dae/dep is the plastic modulus. For a generalized state of stress, the plastic modulus is related to the rate of expansion of the yield surface, while in a uniaxial test it represents the slope of the stress-plastic strain curve at the current value of ae. The effective strain path gives the strain history of the material. Its length at any given point is given by £p = Id£p = Iint)  (6- 16) Stress Model 118 The plastic strain increment under a generalized state of stress can be computed by invoking the flow rule : de?- = d A ^ (6.17) every-where g(o~ij) is the plastic potential function and dA is a positive proportionality constant, termed the loading parameter, which is nonzero only when plastic deformation occurs. The direction of the plastic strain increment tensor, de?-, is defined by the gradient of the plastic potential surface, Bg/Baij, while its magnitude is determined by the loading parameter. A special case of the flow rule is the so-called associated flow rule, in which the yield function and the plastic potential function coincide, i.e., g = F. Thus dF de*- = d A - — (6.18) Oaij Adopting the associated flow rule, the elastoplastic stress-strain relation is derived as follows. Differentiating F in Eq. (6.11) results in BF BF BF BF d F = ^-daa + ^-dK + ^-dT + £ ^rdXk (6.19) daij BK BT Y d X k Given that the hardening parameter, K, can be represented by the amount of plastic work done to the solid during plastic deformation the chain rule can be used to obtain BF BF BK J^iK = d^W,^'1 (6'20) Following the consistency condition2 dF = 0 (6.21) 2The consistency condition ensures that the stress state remains on the yield surface. Stress Model 119 then dF , dF 8K , _ dF :rT1 ^ dF n d ^ - '•' /3K del V d T The stress increment based on the elastic strain increment is : (6.22) der,- Dijki de = D.jW (deij - de ?• - de th Aptr ij U f c i j de'? d e - ) (6.23) Invoking the associated flow rule Eq. (6.18), substituting into the consistency condition and solving for dA results in : dA = S-1 dF daij U dT Dijkl (dei3 - de% - de% - de% - de£ where < i - d F n  d F o — L>ijkl doij da^ dF du dF du de^j da^ (6.24) (6.25) A loading criterion under a multiaxial state of stresses can now be derived. Referring to the consistency condition d F n (A A ° A - L D F A T ± X - d F AY Dijki [deij - de8-j ) + —dT + \^ —-dXk a. SdX = 0 (6.26) For a work-hardening material it can be shown that the scalar function S is always positive. Given that dA is a non-negative quantity (for plastic loading), it then follows that S'dA is always positive. Plastic flow begins when the stress point coincides with the yield surface and continues, if and only if, the stress incremental vector, daij, is directed outwards from the current elastic region. If the stress increment vector is parallel to the yield surface, no additional plastic deformation will occur (i.e., neutral loading); while if it points inward from the current yield surface, unloading will occur. Defining the Stress Model 120 loading criterion function, L , by L = »LDiju ( d £ , _ d 4 / ) + | f dT + E §-AXk (6.27) the following loading criterion is obtained : > 0 loading —> elasto-plastic constitutive eqn. = 0 neutral loading —> elastic constitutive eqn. (6.28) < 0 unloading —>• elastic constitutive eqn. Finally, the elasto-plastic thermo-microstructural constitutive equation is obtained : = de« - A7 ( D 4 + D 4 + D £ ? + KE) -Dljkl (j^dT + £ | ^ d ^ ) (6-29) Note that the last term is different from zero for plastic deformation only. In Eq. (6.29), D%M = Dijkl - Dpjkl (6.30) is the elasto-plastic matrix and Dijkl = DHH S-la'i3Dijkl a'ij (6.31) is the plasticity matrix. The constitutive equation can be written in the following, more compact, form d(T0- = D?itA (de,, - de?-) (6.32) where de?- = de# + de* + deg + de?* + d e , f (6.33) Stress Model 121 deg F = a* (D^Y 1 (^dT + £ ^-dXk^j (6.34) and a = < 0 if elastic 1 if plastic Note that while (6.35) Dijki ^ Diju {o-ij) (6.36) l%i = Oik (^) ( 6- 37) 6.1.3 Finite-Element Formulation Due to the material nonlinearity in Eq. (6.32), an analytical solution for real problems is very difficult to obtain. Instead, a numerical method must be adopted. In this study, a computer program based on the finite-element method was used. The basic components of the finite-element formulation are : 1) finite-element equations and 2) solution algorithm for the nonlinear equations. 6.1.3.1 Finite-Element Equations In the finite-element solution of stress problems, the displacement field at the element level is approximated by the following relationship : {u} « {u} = [N]{ay (6.38) Stress Model 122 where {u} = [u,v]T is the displacement vector for a two-dimensional problem, [N] is the shape function matrix 3 and {a}e is the nodal displacement vector. Note that, in contrast with the thermal analysis, there are now two degrees of freedom (u and v) at each nodal point. Also, the strains and nodal displacements are related by {e} = [L]{u} = [L][N]{ay (6.39) where, for a solid of revolution, [L] d_ dr 8_ 1/7 d_ f dz dr (6.40) Let us define the strain-nodal displacement matrix by [B] = [L][N] (6.41) then {e} = [B}{aY (6.42) By invoking the principle of virtual displacements, a weak form of the equilibrium equations can be obtained, from which, upon substituting Eq. (6.42) and introducing a general constitutive equation, the following equivalent form of the governing equation is obtained4 [ A T W e = {RY (6.43) 3Linear-order (4-node) and quadratic order (8-node) shape functions applicable to two-dimensional isoparametric elements can be found in Ref. [185]. 4 Details of the derivation are given in Appendix C. Stress Model 123 where, for the axisymmetric case, the stiffness matrix, [K]e, and the load vector, {R}e, at the element level are given by 5 [K]e = 2TT / [B)T[D][B]r dr dz JAe » 2irr [ [B]T[D][B] dr dz (6.44) JAe and {R}e = 2TT I [B]T[D}{e°}r dr dz JAe « 27rf / [B]T[D]{e0} dr dz (6.45) •Me For the problem in hand, there are five components of the nonmechanical strain (Eq. (6.2)). Appropriate expressions for each of these terms are given in Appendix D. 6.1.3.2 Solution Algori thm Due to the nonlinear relationship between stress and strain, the governing equation (Eq. (6.43)) is nonlinear on the nodal displacements. Moreover, since the elasto-plastic constitutive equation depends on deformation history, the variation of displacement, strain and stress should be traced along with the loads. The current load vector is computed from 6 m+1{R} = m{R} + {AR} (6.46) The balance between internal forces, {F} , and applied loads, {R}, is expressed in the 5 To facilitate the integration, r is used instead of r. 6In this subsection m + 1 ( ) and m() denote values at the current and previous load increment, respectively. Stress Model 124 equilibrium equation : { F } 77V {R} (6.47) where 771+1 { F } = jv[B]Tm+1{a}dV « 2nf f [B]Tm+1{a}dr dz JA (6.48) In general, Eq. (6.47) is not satisfied, but rather a residual, ^ , is obtained : * ( m + 1 M ) 771 + 1 {F(m+1{u})}-m+1{R} = o (6.49) The elastoplastic constitutive equation derived previously (Eq. (6.32)) provides a rela-tionship between an infinitesimal strain increment and an infinitesimal stress increment, and is adopted to compute the stress tensor at the Gauss points. In a finite-element solution, however, a finite load increment (as opposed to an infinitesimal load increment) is applied at each load step, which results in finite increments of stress and strain. Con-sequently, the incremental constitutive relation given by Eq. (6.32) has to be integrated, usually employing a numerical technique. In virtually all the algorithms used to solve elastoplastic problems, the solution of Eq. (6.47) at each loading step7 is carried out in two stages : Stage 1 The current load increment, m+1{AR}, is computed based on the thermal and microstructural states at the end of the current loading step. The corresponding displacement increment at the global level is obtained by solving Eq. (6.43) at the global level. Stage 2 A trial stress increment, {ACT 6}, is computed assuming elastic behaviour and 7A loading step in the stress problem is equivalent to a time step in the thermo-microstructural solution. Stress Model 125 is used to determine the loading state. In the most general case where the Gauss point has entered a plastic state from an elastic one, the trial stress increment is split into its elastic and elastoplastic components by determining an scaling factor, r, such that f(m{cr} + r{Aae},mk) = 0, see Figure 6.2. Then, the elastoplastic fraction of {A<re} is computed by integrating the elastoplastic constitutive equation (Eq. (6.32)). In the first stage, the load increment is used to predict the stress state (assuming elastic behaviour) while a 'correction' is applied in the second stage to obtain the actual stress increment. For this reason,, the method described above is commonly termed predictor-corrector [200]. In this study, the computer program developed by Nagasaka et al. [47] based on the algorithm proposed by Yamada et al. [201] has been adopted. The algorithm is summarized in the flowchart shown in Figure 6.3. The stress increment at each Gauss point was calculated by first selecting r m i n (the minimum value of the scaling factor r) and multiplying the load, nodal displacement, stress and strain increments by rmin to obtain the deformation state after the previously elastic element with r = r m i n has just yielded. From then on, within the load increment, this element was treated as plastic. The predictor-corrector method was applied in this manner for all elements (at which point J2rmin = 1-0). Then, the deformation state was updated and the next loading step calculated. The scaling factor, r, can be calculated either analytically or numerically [159]. Fol-lowing Yamada et al. [201], r was computed using : r = (6.50) 2({Aa}Y where r = ({Aa}) 2 - 2aeAae - {Aaef (6.51) Stress Model 126 and {A^} = y^{A<7 '}{A<7'} (6.52) In the equations above, Acr e, denotes the increment of effective stress induced by the load increment, {Ai?}. In order to speed up the computations, elements just prior to yield (<re > 0.995a"i?) are considered to be plastic in the next cycle of calculation. 6.2 Verification of Mathematical Model of Stress Generation As mentioned above, the finite-element code developed by Nagasaka et al. [47] to simulate stress generation during spray cooling of steel bars was adopted to model stresses in forced convective quenching. The elastic component of the code was verified by comparing numerical predictions of thermal stresses in an infinite cylinder with constant mechanical properties against an analytical solution. A verification of the elastoplastic component of the computer code has been reported elsewhere [47]. 6.2.1 Infinite Solid Cylinder : Elastic Thermal Stresses In absence of body forces, and considering axial symmetry in the temperature distribu-tion, the thermal stresses in an axisymmetric body are obtained from the equations of equilibrium, which reduce to [202] : ?ZL + °JLZ£L = o (6.53) or r with the following boundary conditions : B .C . 1 u = 0 a t r = 0 (6.54) Stress Model 127 B . C . 2 : -4L= 0 at r = R ( 6 . 5 5 ) The stresses are generated by a temperature gradient which is symmetrical about the z axis and independent of the axial coordinate, z. Assuming constant thermomechanical properties, an analytical solution for the radial (oy), circumferential (erg) and axial (az) stress distributions can be derived by applying the method of strain suppression [ 2 0 3 ] to obtain = r^(£f<r-r->'d'-?r<r<6-56> °> = T ^ ( % 0 T - T ^ R D R - ( T - T ^ ) <6'58> The corresponding radial displacement is u = [ ± ^ a ( ( 1 - 2u)± J\T - T r e f )rdr+1- J\T - T r e f ) r dr) ( 6 . 5 9 ) For a parabolic temperature gradient of the form T(r) = T0- (Arf ( 6 . 6 0 ) and taking T r e f = T0, i.e., zero thermal strain at T — T0, the above equations reduce to aE aE aE ar = i ( f ^ y ( 4 A V - A2R2) ( 6 . 6 1 ) cre = A a E A4A2r2-A2R2-A2r2) ( 6 . 6 2 ) 4(1 - u) u = , ( 4A 2 r 2 - 2vA2R2) (6.63) 4(1-u) " ^ T ^ A ( ( 1 " 2 " ) J R 2 R + R 3 ) ( 6 - 6 4 ) Stress Model 128 To assess the performance of the model, the elastic thermal stresses in an infinitely long cylinder of 0.1 m diameter generated by a parabolic temperature gradient (T(r) = 100 — (50r)2) were calculated. The cylinder was assumed to be initially at a uniform temperature T0 = 100 °C. The thermomechanical properties were assumed constant and are given in Table 6.1. The finite-element mesh consisted of 10 4-node isoparametric (2D4) elements. The results of the comparison between the radial, circumferential and axial stress distributions predicted by the model and by the analytical solution are shown in Figure 6.4. The applied temperature gradient is also shown in the figure. A very good agreement between numerical and analytical predictions was observed. The correspond-ing radial displacements are shown in Figure 6.5. The variation of radial displacement with radial position was also properly predicted by the model. 6.2.2 Infinite Solid Cylinder : Elastoplastic Thermal Stresses There are no analytical solutions for the complete quenching problem or even for elasto-plastic stresses generated by thermal gradients without considering phase transforma-tions. Thus, only comparisons between measured and predicted residual stresses can be used to verify the models. The ability of the stress model to predict elastoplastic thermal stresses was investigated by Nagasaka et al. [47]. In that work 8, the residual stresses predicted by the model were compared with measured (using a boring-out tech-nique [204,205]) residual stresses in a 50 mm-dia. pure iron bar quenched from 850 °C in ice-water. Since pure iron was used, the stresses were generated by thermal strains only. The results are shown in Figure 6.6. The predicted residual stress distributions agreed well with the published measured values. It should be noted that Mitter et al. [205] found that plastic flow had occurred during their measurements, which would explain discrepancies between their values and those reported by Buhler et al. [204]. 8For details of the calculations, the reader is referred to the original reference [47]. Stress Model 129 6.3 S u m m a r y A computer program developed by Nagasaka et al. [47] has been adopted to simulate stress generation during forced convective quenching of steel bars. The code is based on a finite-element formulation of the thermal-microstructural elastoplastic problem, in-cluding non-mechanical strains associated with temperature gradients, microstructural evolution, changes in elastic properties and flow stress with temperature and phase com-position, and transformation plasticity. The thermal-microstructural elastoplastic con-stitutive equation was derived adopting the classical incremental theory of plasticity, considering isotropic hardening and a J2 (von Mises) material behaviour. The code has been verified by modeling 1) elastic thermal stresses in an infinitely long cylinder sub-jected to an axisymmetric temperature gradient (assuming constant thermomechanical properties), and 2) thermal stress generation in a pure-iron bar quenched in ice-water [47]. Stress Model 130 Table 6.1: Input data used for the comparison of finite-element and analytical solutions for elastic stresses generated in an infinite solid cylinder by a temperature gradient. Variable Value R 0.1 m E 1.25 x l O 5 M P a V 0.3 a 2.0 x l 0 ~ 5 ° C _ 1 T ( r ) . 100 - (50r)2 0 C D Figure 6.1: Schematic of loading and unloading paths for a work-hardening material under a uniaxial load. Stress Model 131 Figure 6.2: Schematic representation of the scaling factor, r [159]. Stress Model 132 ( START ) Read input data Initialize variables Read 7 7 1 + 1 T , 7 7 1 + 1 £ Initialize variables for r m i n calc" Calculate m+1{AR} Solve m+1[K] m + 1 { A a } = m+l{AR} Compute m+1{Aae} Calculate r Compute r^r+HAi?}] Compute 7 7 1 + 1 {Aa} n £ r = l ? t <- < + At n * > ' e n d ? ( END Figure 6.3: Flow chart of the stress solution algorithm. Stress Model 133 - 4 0 60 0.00 0.02 0.04 0.06 0.08 Radial Posi t ion, m 0.10 Figure 6.4: Comparison between analytical and numerical (FEM) predictions of stress distributions in a 100 mm-dia. cylinder, produced by the thermal gradient shown as a broken line. 0.00 -o.oi o d V a o CO -0 .02 -0 .03 Analytical O FEM 0.00 0.05 Radia l Pos i t ion , m 0.10 Figure 6.5: Comparison between analytical and numerical (FEM) predictions of radial displacements in a 100 mm-dia. cylinder, produced by the thermal gradient shown as a broken line in Figure 6.4. Stress Model 134 Figure 6.6: Comparison between measurements made by Buhler [204] and Mitter et al. [205] and numerical ( F E M ) predictions of residual stress distributions in a 50 mm-dia. pure-iron bar quenched in ice water from 850 °C [47]. Chapter 7 Laboratory Experiments : Quenching Tests The layout and operation of the industrial equipment allowed little flexibility for instru-mentation to measure the thermal response of the steel rods during the quenching cycle. This information is needed to estimate the heat transfer boundary condition. Thus, a laboratory facility was designed and built to simulate the industrial operation. In this chapter, the apparatus and experimental procedure adopted to study heat transfer in forced convective quenching of steel bars are described. The objectives of the laboratory experiments were twofold : 1) to characterize the surface heat flux as a function of surface temperature, under forced convective boiling conditions, and 2) to produce specimens for metallographic characterization and resid-ual stress measurement (to validate the mathematical models). To accomplish the first objective, an interstitial-free (IF) steel was chosen. The material selection was based on the fact that, during quenching, the IF specimens transformed to ferrite only. Thus, the effect of the heat evolved during the transformation was confined to relatively high temperatures, facilitating the application of the inverse analysis (see Chapter 9). The chemical composition of the IF steel is given in Table 7.1. In order to have as complete a boiling curve as possible, the initial test temperature in these experiments was 1000 °C. Alta Steel furnished three alloyed, near-eutectoid steels, in the form of 100 mm (4 in)-dia. bars. The chemical composition of the three steels is given in Table 7.2. One of these steels (steel A) was quenched in tests aimed at producing data to validate the model. In these runs, the test conditions (water temperature and austenitizing temperature) were 135 Laboratory Experiments : Quenching Tests 136 chosen to simulate the operational conditions. However, the hardenability of this steel, combined with the smaller bar diameter used, produced through-hardened samples and even cracked specimens (at high water velocity/low water temperature). Thus, 1045 car-bon steel specimens, which have a lower hardenability, were also quenched. The chemical composition of the 1045 carbon steel is given in Table 7.3. 7.1 Quenching Appara tus For the laboratory runs it was desirable to have the maximum control possible. Thus, the quenching apparatus was designed to maintain the sample stationary during the heating and quenching cycles. In addition, the initial test temperature had to be uniform across the bar diameter. With these goals in mind, the sample was heated inside a quartz tube by induction; the quartz tube also contained the flowing water during the quench. A 600 mm long, 46 mm I.D. ( 50 mm O.D. ) quartz tube was selected based on the largest inside diameter that would allow for a practical design. The water inlet end of the quartz tube was connected to the rest of the system by an aluminum bronze coupling. A schematic drawing of the test section is shown in Figure 7.1; it consists of a hollow tapered extension (screwed to the front of the sample), a sample, an adaptor (screwed to the back of the sample), and a holder. Once assembled, the adaptor, specimen and ex-tension formed a single piece. The sample geometry is schematically shown in Figures 7.2 (IF steel) and 7.3 (alloyed and 1045 carbon steel). The specimens were instrumented with two type K , Inconel sheated, 0.3 mm (0.010 in) thermocouple wires1. For this purpose, 60 mm long thermocouple holes were drilled parallel to the axis of the test bars. In the case of the IF specimens, one of the thermocouples was positioned at the centreline while the other was located 1.5 mm below the surface. The subsurface thermocouple was used to estimate the surface heat flux through the solution of the IHCP and, therefore, needed 1The recommended range of operation for type K thermocouples is from -200 to 1260 °C [206]. Laboratory Experiments : Quenching Tests 137 to be located as close to the surface as possible. For the alloyed eutectoid steels and 1045 carbon steel samples, the second thermocouple was placed at a position halfway be-tween the surface and the centre (in order to minimize the possibility of crack formation). The thermocouples were spring-loaded to maintain good contact with the specimen; the spring mechanism was attached to an adaptor (Figure 7.4) that was screwed to the sam-ple. To lessen turbulence, and insure a fully developed velocity profile at the point where the fluid comes into contact with the hot sample, a tapered extension made of mild steel (Figure 7.5) was attached to the water inlet end of the sample. In order to eliminate specimen distortion at high temperatures (caused by excessive weight) a hollow extension was used; moreover, this extension was fitted with four supporting points in the quartz tube. The fully assembled test section was rigidly held in position by a 1 m long, 25.4 mm I.D., 31.25 mm O.D. carbon steel tube mounted on a mechanically controlled test bed. The end of the steel tube was sealed with a rubber bung held in place by a brass gland nut. Two slits were made in the rubber bung to allow the thermocouple leads to pass through it. The specimen dimensions were determined based on hydrodynamic considerations, including entry effects. The relevant dimensionless number to insure dynamic similarity is the Reynolds number : Re = ^ (7.1) t1 where v is the velocity of the fluid, L is a characteristic length, p is the fluid density and p is the fluid viscosity. Both, p and p, should be computed at the mean temperature be-tween the rod surface and the fluid. For annular flow the equivalent (hydraulic) diameter is : D, - ^ (7.2) where A± is the cross-sectional area perpendicular to the flow and Pw is the wetted Laboratory Experiments : Quenching Tests 138 perimeter. Then, and the Reynolds number takes the form : where D\ and D2 are the bar and inside tube diameters, respectively. In terms of the volumetric flow rate, Q, one obtains : A(D,-D1)Q with v being the dynamic viscosity (p/p). Typical operational plant data (prototype) are given in Table 7.4. Under those con-ditions : #e = 4 .1x l0 6 i.e., turbulent flow (7.6) To select the sample diameter, D\, the water velocity corresponding to several com-binations of volumetric water flow rate and bar diameter was computed with D2 fixed at 46 mm, and compared with the water velocity used in the industrial operation (5.2 m s _ 1 ) . The results are summarized in Figure 7.6. The nominal volumetric flow rate, rated at 32 m (105 ft), of the Goulds submersible pump (model 70 L G 30) used in this study is also shown in the figure. As can be seen in Figure 7.6, the available pump would not yield high enough velocities for Di— 25.4 mm (1 in) or 31.8 mm (1.25 in); thus a sample diameter of 38.1 mm (1.5 in) was selected. For a water velocity of 5.2 m s _ 1 (the same as in the plant), the corresponding Reynolds number for the laboratory unit was : Re = 3 .2xl0 5 i.e., turbulent flow (7.7) Laboratory Experiments : Quenching Tests 139 A n important design criterion results from considering entry effects. To determine whether the thermocouples were located far enough downstream to allow for fully devel-Calculated values of the hydrodynamic entry length obtained using Eq. (7.8) are shown in Figure 7.7. For the target velocity of 5.2 m s _ 1 and a Reynolds number of 3.2 x l O 5 , a value of Lhyd = 14.8(1)2 — Di) = 117 mm (4.6 in) was obtained. It should be noted that the edge in the tapered extension may cause a separation of the boundary layer [208] and, therefore, must be considered in the design, i.e., the hydrodynamic entry length should be measured from the edge, rather than from the tip of the tapered extension. To avoid thermal entry effects the distance from the thermal entrance to the point of temperature measurement needs to be greater than the thermal entry length, defined as the distance downstream of the thermal entrance necessary for the local Nusselt number to fall to within 5 % of its fully-developed value. The thermal entry length for Reynolds numbers in the range 104 < Re < 106 and Prandtl numbers in the range 0 < Pr < 104 has been computed by Notter and Sleicher [209] by solving, numerically, the energy equation describing the heat transfer to a fluid in a pipe, i.e., the turbulent Graetz problem. Their results, for uniform wall heat flux, are shown in Figure 7.8. Considering that the Prandtl number for water varies from 1 to 10, for the temperature range of interest, the curve for Pr = 3.0 can be taken as representative of the system. Then, the thermal entry length for the Reynolds numbers of interest is confined to Yielding, Lth < 39.5 mm (1.55 in). The assumption of uniform wall heat flux (as op-posed to uniform wall temperature) has been adopted to estimate L t h because thermal LJDe < 5 (7.9) Laboratory Experiments : Quenching Tests 140 entry lengths calculated in this way are slightly longer than the corresponding values for uniform wall temperature [209] and, therefore, represent a worst case scenario. The flow loop of the system is schematically shown in Figure 7.9. In order to handle the large water volumes required during the experiments, the quartz tube discharged into a 95 1 (25 gal) drum mounted horizontally on top of a 170 1 (45 gal) drum. The large drum acted as a water reservoir and housed the submerged pump; it was mounted on a frame to stabilize the entire apparatus. The test bed referred to above was welded to the inside wall of the small drum. During quenching, the water in the small upper drum drained back into the large one through a 150 mm-dia. steel tube, creating a closed circuit. To limit the unsteady-state effects associated with the start-up of the pump, a separate water circuit was established through which the water could be diverted back to the reservoir (valve # 2 open, valve # 3 closed). The water flow rate into the test section was measured with a Fisher & Porter Series B10D1465 C O P A - X industrial magnetic flow meter. Immersion heaters were used to heat the water in the reservoir to the required temperature. To prevent oxidation during heating, provisions were made to allow for evacuating the test chamber and passing H 2 over the sample surface. To accomplish this, a gas tight cap (Figure 7.10) was used to temporarily close the discharge end of the quartz tube. The cap was made of aluminum and fitted with a stainless steel tube that served as gas inlet. O-rings were placed in grooves machined along the surfaces of the aluminum cap in contact with the quartz tube and the sample holder to ensure gas-tightness. Another O-ring was placed at the back face of the seal to prevent direct contact with the quartz tube. When quenching was initiated, the aluminum cap was pushed to the back of the small drum by the flowing water. The data acquisition system consisted of a Metrabyte 16-channel multiplexer (EXP-16/A) and a Metrabyte A / D converter and time/counter board (DAS-800). The data acquisition software was Labtech Notebook (v 7.2.1) running on a P C - A T . The sampling Laboratory Experiments : Quenching Tests 141 rate was 5 Hz 2 . A schematic diagram of the data acquisition configuration is shown in Figure 7.11. The output signal from the magnetic flowmeter was converted from a 4 - 20 mA span to a 0 - 50 mV range by soldering a resistance to the corresponding channel in the multiplexer. The gain in the multiplexer was set to 50. The flowmeter was calibrated by measuring the volume of water collected in a barrel for recorded time intervals and at the same time recording the voltage drop across a 330 0 resistance. Figure 7.12 shows the calculated volumetric flow rate (1 s _ 1 ) and the output signal from the flowmeter (mA) for various valve settings, the latter given as number of turns with respect to the fully closed valve position. The calibration curve for water flow rate as a function of output signal from the flowmeter, in the range of water flow rates equivalent3 to 2 - 8 m s _ 1 , is given by : Q = - 3.9 +0.5867 i (7.10) where Q is given in 1 s _ 1 and i is input in mA. This expression was programmed in the data acquisition software to convert and record the output signal from the flowmeter. Heating was carried out with a 11-turn, 100 mm I.D., 125 mm O.D., 180 mm long, water-cooled copper coil, fabricated with a 12.7 mm O.D., 1.5 mm thick copper tubing. The coil was powered by an Inductotherm induction furnace (model Inducto 15) operating at approximately 10kHz. The relatively low frequency resulted in deep penetration of the magnetic field and, therefore, more uniform heating. To reduce radiative heat losses, a fiberglass tube was placed between the quartz tube and the coil. During preliminary tests it was found that, in order to attain a reasonably uniform temperature across the specimen diameter, heat losses by conduction to the tip and to the adaptor had to be 2The time constant of the thermocouple was estimated following a procedure outlined by Hernandez-Avila [210]. Its value was of the order of x l O - 2 s. 3Water velocity is related to water flow rate by Q = vA, where Q is water flow rate in 1 s - 1 , v is velocity in m s _ 1 , and A is area normal to the flow in m 2 . Laboratory Experiments : Quenching Tests 142 minimized. To accomplish this, freshly heated saffil4 was placed at both ends of the specimen. Typical heating cycles obtained with this arrangement are shown in Figure 7.13 for two initial test temperatures, 1000 and 850 °C. In the first case, the power supplied to the coil was maintained at 4 kW up to ~ 900 °C and then the power was varied (following the thermocouple readings) to reach the initial test temperature smoothly. In the 850 °test, the power was reduced after reaching ~ 500 °C. Very good reproducibility was observed in all cases. The difference between subsurface and centre temperature at the end of the heating cycle, for the IF steel experiments, is plotted in Figure 7.14. As can be seen, the heating cycle produced a uniform temperature field at the end of the heating period, the temperature difference between the centre and subsurface being less than 10 °C. 7.2 Procedure Prior to each test, the thermocouples were calibrated by immersion in boiling water and the test section was assembled5. Once the water in the reservoir was heated to the desired temperature, the test section, quartz tube and induction coil were set in place (making sure they were concentric to each other) and the aluminum cap was attached to the discharge end of the quartz tube. To facilitate the ejection of the cap, silicon grease was applied to the external surface of the quartz tube. The computer was then started and the thermocouples were connected to the data acquisition system and checked for functionality. The pump was started with the valves directing the water flow back to 4Pure alumina fibers. 5Due to the required number of tests and the difficulties associated with machining the specimens, the same specimen was used for several experiments. To restore the original surface condition, previously used specimens were clamped in a lathe and polished with emery cloth. Laboratory Experiments : Quenching Tests 143 the reservoir. The test chamber was then evacuated with a vacuum pump before back-filling it with hydrogen. The induction furnace was turned on and the sample heated to the initial test temperature6 and held for approximately 3 min. When the desired austenitizing condition was attained, the hydrogen flow into the system was stopped, the induction furnace turned off, the thermocouple leads connected to the multiplexer and the data acquisition started. The quench was initiated by diverting the water flow into the test chamber. Data was recorded until the temperature at the centre of the specimen had decreased to the water test temperature. A total of 9 tests were conducted to study the effect of water temperature and velocity on forced convective boiling heat transfer. The test matrix included low, medium, and high water temperatures and velocities, as shown in Table 7.5. In all cases, the initial test temperature was 1000 °C. To check the reproducibility of the experiments, one of the runs (Tf = 50 °C, v — 4.8 m s _ 1 ) was repeated. Three specimens were saved for residual stress measurements and metallographic characterization, one from each of the three types of steel used in the experiments. The quenching conditions are shown in Table 7.6. The effect of surface oxidation was studied by quenching clean as well as oxidized alloyed samples (runs 20 and 32) and IF specimens (runs 10, 27 and 33). To produce a 'light' surface oxide, a clean specimen was heated to 500 °C in air and furnace-cooled; the surface was protected from further oxidation during heating in the quenching apparatus. A 'heavy' surface oxide was produced by reusing a quenched specimen without removing the oxide layer produced by the water quench and heating it up in the quenching appara-tus without a protecting atmosphere. No attempt was made to characterize the surface oxide. The quenching conditions are summarized in Table 7.7. To investigate the effect of initial test temperature, two IF steel bars were quenched with water flowing at 4.8 m s"1 at 32 °C, from 1000 °C (run 19) and 850 °C (run 21). 6During heating, the thermocouples were monitored with two digital recorders. Laboratory Experiments : Quenching Tests 144 7.3 Metallographic Characterization After residual stress measurements were completed on the specimens quenched under the test conditions given in Table 7.6, the quenched bars were sectioned near midlength to produce cylindrical specimens (~ 20 mm tall) for hardness testing and metallography. After polishing to a 1 pm diamond finish, the samples were etched to reveal the final mi-crostructure. Both, the 1045 carbon steel and the alloyed eutectoid steel samples where etched with 2 % nital by rotating a soaked cotton swab on the sample surface for approx-imately 15 seconds; the IF steel sample required a longer etching time of approximately 40 seconds. The alloyed eutectoid steel sample exhibited a fully martensitic structure and was, therefore, used to determine the prior-austenite grain size. To measure the prior-austenite grain size, the alloyed steel cylindrical specimen was further sectioned to obtain a strip parallel to the radius of the sample. This section was cold mounted, pol-ished to 1 pm diamond finish, and etched by immersing the mounted sample in a boiling alkaline picrate solution (2g picric acid, 25 g NaOH and 100 ml water) for approximately 12 minutes, followed by a light 2 % nital etch at room temperature. Several areas were photographed, and the revealed austenite grain boundaries drawn on tracing paper. The prior-austenite grain size was determined by measuring the areas of the grains, according to the Jeffries' method [211]. From the measured mean grain area, a grain diameter was estimated by assuming a perfectly spherical grain sectioned at its equator. The analysis was performed using a Leitz O R T H O L U X 2 microscope, a Leitz O R T H O M A T camera system and a C ' I M A G I N G analysis system running S IMPLE imaging software. Laboratory Experiments : Quenching Tests 145 Table 7.1: Chemical composition of the IF steel used in the forced convective quenching experiments (in weight percent). Element w/o C 0.01 M n 0.14 P 0.007 s 0.013 Si 0.01 Cu 0.02 Ni 0.03 Cr 0.03 Mo 0.001 V 0.004 Nb 0.002 Ti 0.068 Table 7.2: Chemical composition of the alloyed steels used in the forced convective quenching experiments (in weight percent). Element Steel (Heat #) A (E27166) B (D25404) C (E27071) C 0.69 0.68 0.65 Mn 0.88 0.83 0.82 P 0.013 0.009 0.010 s 0.020 0.013 0.020 Si 0.23 0.20 0.20 Cu 0.15 0.20 0.16 Ni 0.08 0.08 0.07 Cr 0.19 0.21 0.19 Mo 0.081 0.138 0.169 V — 0.001 — Nb 0.022 0.020 0.020 Laboratory Experiments : Quenching Tests 146 Table 7.3: Chemical composition of the 1045 steel used in the forced convective quenching experiments (in weight percent). Element w/o C 0.46 Mn 0.78 P 0.007 s 0.021 Si 0.20 Cu 0.02 Ni 0.02 Cr 0.04 Mo 0.001 V 0.002 Nb 0.001 Table 7.4: Typical operational plant data. Parameter Value Dl 100 mm (4 in) D2 200 mm (8 in) Q 0.1262 m 3 s- 1 (2000 USGPM) V 5.2 m s- 1 T0 815 °C 32 °C V 1.29 x l O - 7 m 2 s-1 (at 315 °C) Laboratory Experiments : Quenching Tests 147 Table 7.5: Test matrix used for the boiling heat transfer experiments. In all cases the initial sample test temperature was 1000 °C. Water Temperature, °C Velocity, m s 1 2.8 4.8 6.9 25 RUN08 RUN09 R U N 11 50 R U N 12 RUN10 R U N 13 75 RUN16 RUN14 R U N 15 Table 7.6: Quenching conditions for specimens produced for residual stress measurements and metallographic analysis. Material Water Temperature °C Water Velocity m s _ 1 Initial Temperature ° c IF 25 4.8 1000 1045 50 2.8 1000 Alloyed 75 2.8 860 Laboratory Experiments : Quenching Tests 148 a .2 a "2 ' x o CD U o CD CD Ti fl co o Ti fl o CJ DJO fl ^fl cj fl CD fl C? rjD CD in fl a u CD On " f l °o o CD s-l CD OH CD CD CJ fl CD o O o o o o oo o L O fl CD O fa o o o oq o L O fl .2 Ti " x o rJfl b O CM fa O O o oo o L O fl .2 °-+^  - x O CD co co fa o L O oo oo CM CO fl __CD 6 o CM i d CD >> o o L O oo oo CM CO fl .2 " x O h0 CM CO CD O Laboratory Experiments : Quenching Tests Laboratory Experiments : Quenching Tests CO I. "86 Laboratory Experiments : Quenching Tests 38.1 m m 0) CO o CO T o z £ £ LO CO CO Laboratory Experiments : Quenching Tests 153 Laboratory Experiments : Quenching Tests 154 25 10' 20 15 10 0 0 Prototype O D = 25.4 m m A D ] • D ] 31.8 m m 38.1 m m 46 m m 6 8 10' 0> i o l 10 10 Q, 1 s Figure 7.6: Calculated velocity and corresponding Reynolds number as a function of water flow rate for an annular region with D2 = 46 mm. The output of the submersible pump used in the experiments is also shown. Laboratory Experiments : Quenching Tests 155 Figure 7.8: Calculated thermal entry length as a function of Reynolds number for various Prandtl numbers [209]. Laboratory Experiments : Quenching Tests 156 Sample' Holder Immersion Heater Vaccum Pump Electromagnetic Flowmeter Pressure Gauge Submerged Pump %#2 Figure 7.9: Flow loop of the quenching apparatus. Laboratory Experiments : Quenching Tests 157 m u i 89 UIUI 8^ 7 uimoe o co o —^< +-< o o o C«1 O 2 Pi 13 CD p CD b£) Laboratory Experiments : Quenching Tests 158 110V(a.c.) Power Surge Protection a C O M P U T E R Ribbon cable FLOWMETER 4-20 mA THERMOCOUPLE (Centre) 0-20 mV THERMOCOUPLE (Subsurface) 0-20 mV E X P - 1 6 / A 7K 7K~ 7K-Figure 7.11: Schematic diagram of the data acquisition configuration. Laboratory Experiments : Quenching Tests 159 Figure 7.12: Magnetic flowmeter calibration : water flow rate and current as a function of valve setting, the latter in terms of number of turns with respect to the fully open position. Laboratory Experiments : Quenching Tests 160 Figure 7.13: Temperature response at the centre of the specimen during heating to (a) 1000 °C and (b) 850 °C. Laboratory Experiments : Quenching Tests 161 10 8 h 6 h 4 h 2 h 0 00 o c\2 CO o o Figure 7.14: Difference between centre and subsurface temperature prior to the start of the quench for the IF steel tests. Chapter 8 Laboratory Experiments : Transformation Kinetics To model the microstructural evolution obtained in heat treatments, the transformation kinetics must be characterized experimentally. In this study, the kinetics of the austen-ite decomposition in 3 alloyed, near-eutectoid steels for a range of continuous cooling and isothermal conditions were determined using the G L E E B L E 1500 thermomechanical simulator. The transformation kinetics of continuously cooled 1045 steel samples were also measured. The aim of the isothermal and continuous cooling tests was to describe the transformation in terms of the Avrami equation and to determine the transformation start times, respectively. The prior-austenite grain size resulting from the heating cycle, and the ferrite fraction produced by the transformation were determined metallographi-cally. In this chapter, the details of the material and sample preparation, as well as the procedure followed for the various tests, including the microstructural characterization, are described. 8.1 Material and Sample Preparation Alta Steel provided the alloyed near-eutectoid steel. The material was received as an-nealed 100 mm (4 in)-dia. rods. The 1045 steel samples were machined from commercially available 38.1 mm (1.5 in)-dia. cold-rolled bars. The transformation studies were carried out using thin-wall tubular samples (6 mm I.D., 8 mm O.D.), shown in Figure 8.1. To assure a homogeneous composition, free of any centreline segregation common to 162 Laboratory Experiments : Transformation Kinetics 163 rod material, rough sample blanks were cut from the 100 mm-dia. rods as shown in Figure 8.2. The tubular cylindrical test samples were machined from these blanks and homogenized at 900 °C for 1 hour. To minimize oxidation while being homogenized, the samples were placed in a senpak bag, which was then back-filled with argon. 8.2 Procedure 8.2.1 Continuous Cooling Tests The progress of the austenite decomposition was monitored by measuring the central plane diametral dilation of the cylindrical specimen with a modified Linear Variable Differential Transformer (LVDT) C-strain gauge. The sample temperature was controlled and monitored using a chromel-alumel thermocouple spot-welded on the exterior surface at the same mid-axis location. Typical sampling frequencies ranged from 2.5 to 59.6 Hz for cooling rates varying from 0.2 to 40 °C s _ 1 . Prior to testing, the main test chamber of the Gleeble 1500 was pumped down to 5 x l O - 4 torr and back-filled with prepurified argon (99.998 % pure). A total of 10 tests were conducted for each of the alloyed eutectoid steels. The test conditions are schematically shown in Figure 8.3 and summarized in Tables 8.1 to 8.3. In all cases, the samples were initially heated at 5 °C s _ 1 to a holding temperature of 850 °C, held for 3 minutes and continuously cooled at the test cooling rate. Six tests were done for the 1045 carbon steel samples, following the same procedure previously described. The test conditions are summarized in Table 8.4. 8.2.2 Isothermal Tests The progress of the austenite decomposition under isothermal conditions was also moni-tored by measuring the central plane diametral dilation of the cylindrical test specimen Laboratory Experiments : Transformation Kinetics 164 as described above. Typical sampling frequencies ranged from 7 to 30 Hz for test tem-peratures varying from 665 to 525 °C. The isothermal test conditions are schematically shown in Figure 8.4. In all cases, the samples were heated at 5 °C s _ 1 to a holding temperature of 850 °C, held for 3 minutes, cooled at 8 °C s _ 1 to 750 °C, rapidly cooled (at ~ 140 °C s _ 1 ) to the test temperature and held until completion of the transformation. The two-step cooling path was found to be necessary to minimize errors associated with overshooting the test temperature upon rapid cooling. A series of isothermal tests was also conducted to measure the isothermal transformation kinetics of the austenite-to-bainite transformation in the alloyed eutectoid steels. In this case, the samples were austenitized as described above and rapidly cooled (He quenched) to the desired test temperature. No temperature overshoot was observed. The test conditions for all cases are summarized in Tables 8.5 to 8.8. 8.2.3 Temperature Gradient in a Gleeble Specimen Ideally, the mid-plane (C-strain) diametral dilatometric measurements should be made with little or no temperature gradient across the wall of the cylindrical test specimens. A finite-element simulation of the rapid cooling of a tubular Gleeble specimen was performed to determine if a significant temperature gradient exists across the thickness of the tubular wall. The boundary condition at the inner surface was estimated by solving the inverse heat conduction problem (IHCP). From the measured temperature response obtained during test BCC05-2 ( He quench with a cooling rate of 38 °C s _ 1 ), a set of temperatures was established at constant time intervals (0.1 s), and adopted as input to the inverse heat conduction code described in Chapter 5 (modified for a hollow cylinder). To avoid recalescence due to the austenite-to-bainite transformation, experimental temperatures below 520 °C were not included. Considering that the heat transferred to the helium flowing inside the sample is much higher than the heat loss due to radiation plus natural convection at the external surface, Laboratory Experiments : Transformation Kinetics 165 the external boundary was assumed to be insulated (to simplify the solution of the IHCP). For the calculations, the sample was subdivided into 15 nodal points using a time interval, At, of 0.1 s; the number of future time steps adopted in the inverse analysis was r = 2. The estimated heat-transfer boundary condition at the inner surface was adopted for the finite-element simulation of the direct problem. For the finite-element calculations, the cross-section of the sample wall was discretized using 10 8-node isoparametric ele-ments (the ratio Ar/Az for a given element was kept close to 1.0), and a time interval, A t , of 0.05 s was selected. The calculated results showed a maximum temperature dif-ference of approximately 0.5 % across the specimen thickness, at mid-axis, when the temperature was 700 °C. Because of the approximation of no heat flux at the outside boundary adopted for the IHCP calculations, the actual surface heat flux at the interior boundary is expected to be lower than its estimated value, thus reducing the estimated temperature difference across the specimen wall even further. Also, the helium quench was a rapid cooling condition; lower cooling rates would result in even more uniform temperature profiles in the specimen. Thus, it was concluded that a relatively small temperature gradient within the cylinder wall exists during the transformation kinetic measurements, resulting in an essentially isothermal plane at the specimen mid-axis. 8.3 Microstructural Characterization After each test was completed, the tubular specimen was sectioned near the thermocouple position, cold mounted, polished to a 1 pm diamond finish and etched to reveal the microstructural detail. It was important to determine the prior-austenite grain size produced by the heating cycle during measurements done with the Gleeble 1500 since it affects the transforma-tion kinetics. To reveal the prior-austenite grain boundaries, a fully martensitic specimen Laboratory Experiments : Transformation Kinetics 166 (alloyed eutectoid steel cooled at 58 °C s _ 1 ) was sectioned, cold mounted, polished to 1 pm diamond, and etched by immersing the mounted sample in a boiling alkaline sodium picrate solution (2g picric acid, 25 g NaOH and 100 ml water) for 12 min followed by a light 2 % nital etch at room temperature. The prior-austenite grain size was deter-mined by measuring the mean chord length of the grains. The image analysis system consisted of a Leitz O R T H O L U X 2 microscope, a Leitz O R T H O M A T camera system and a C ' I M A G I N G analysis system running S IMPLE imaging software. The ferrite fraction produced by all the continuous cooling tests done using the 1045 carbon steel was measured metallographically. The ferrite fraction present in alloyed eutectoid steel samples, continuously cooled at the slowest cooling rates, was also deter-mined. To measure the ferrite fraction, the samples were etched with a mixture of 15 ml 2 % nital (2 % nitric acid in alcohol) and 85 ml 5 % picral (5 % picric acid in alcohol) [188]; the etchant was applied by gently swabing the sample surface for approximately 10 to 15 seconds. The ferrite fraction was determined by measuring the area occupied by ferrite in up to 50 separate fields for each specimen, using the image analyzer described above. Laboratory Experiments : Transformation Kinetics 167 Table 8.1: Summary of continuous cooling tests for the alloyed eutectoid steel A . Test Heating Rate °C/s HOLD Cooling Condition Temperature °C Time s ACC01 5 850 180 Air cooling ( C R . = 17.5 °C s- 1) ACC02 5 850 180 1 °C s- 1 ACC03 5 850 180 4 °C s- 1 ACC04 5 850 180 0.25 °C s"1 ACC05 5 850 180 He quench, 1/2 turn ( C R . = 37 °C s- 1) ACC06 5 850 180 He quench, 2 turns ( C R . = 58 °C s- 1) ACC07 5 850 180 2.5 °C s"1 ACC09 5 850 180 8 °C s- 1 ACC10 5 850 180 0.5 °C s- 1 ACC11 5 850 180 He quench, 1 turn ( C R . = 44 °C s- 1) Table 8.2: Summary of continuous cooling tests for the alloyed eutectoid steel B . Test Heating Rate °C/s H O L D Cooling Condition Temperature °C Time s BCC01 5 850 180 Air cooling ( C R . = 16 °C s- 1) BCC02 5 850 180 1 °C s- 1 BCC03 5 850 180 4 °C s- 1 BCC04 5 850 180 0.2 °C s"1 BCC05 5 850 180 He quench, 1/2 turn ( C R . = 38 °C s"1) BCC06 5 850 180 He quench, 0.15 turn ( C R . = 28 °C s"1) BCC07 5 850 180 2.5 °C s"1 BCC08 5 850 180 0.3 °C s- 1 BCC09 5 850 180 8 °C s- 1 BCC10 5 850 180 0.5 °C s- 1 Laboratory Experiments : Transformation Kinetics 168 Table 8.3: Summary of continuous cooling tests for the alloyed eutectoid steel C. H O L D Test Heating Rate Temperature Time Cooling Condition °C/s ° c s CCC01 5 850 180 Air cooling ( C R . = 17 °C s- 1) CCC02 5 850 180 1 °C s- 1 CCC03 5 850 180 4 °C s- 1 CCC04 5 850 180 0.2 °C s- 1 CCC05 5 850 180 He quench, 1/2 turn ( C R . = 40 °C s"1) CCC06 5 850 180 He quench, 1/2 turn ( C R . = 33 °C s- 1 CCC07 5 850 180 2.5 °C s- 1 CCC08 5 850 180 He quench, 1/2 turn ( C R . = 28 °C s"1) CCC09 5 850 180 8 °C s"1 CCC10 5 850 180 0.5 °C s- 1 Table 8.4: Summary of continuous cooling tests for the 1045 carbon steel. Test Heating Rate °C/s HOLD Cooling Condition Temperature °C Time s CC4501 5 850 180 1 °C s"1 CC4502 5 850 180 Air cooling ( C R . = 19 °C s- 1) CC4503 5 850 180 He quench, 1/2 turn ( C R . = 38 °C s"1) CC4504 5 850 180 He quench, 1 turn ( C R . = 47 °C s- 1) CC4505 5 850 180 He quench, 5 turns ( C R . = 104 °C s"1) CC4506 5 850 180 He quench, 2 1/2 turn ( C R . = 250 °C s"1 Laboratory Experiments : Transformation Kinetics 169 Table 8.5: Summary of isothermal tests for the alloyed eutectoid steel A . HOLD Test Heating Rate °C/s Temperature ° c Time s Cooling Condition AIT01 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 625 °C and hold AIT02 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 605 °C and hold AIT03 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 635 °C and hold AIT04 5 850 180 Cool at 8 °C s~l to 750 °C followed by helium quench to 650 °C and hold AIT05 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 665 °C and hold AIT06 5 850 180 Cool at 8 °C s _ i to 750 °C followed by helium quench to 575 °C and hold AIT07 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 525 °C and hold Laboratory Experiments : Transformation Kinetics 170 Table 8.6: Summary of isothermal tests for the alloyed eutectoid steel A (bainite reac-tion). Test Heating Rate °C/s H O L D Cooling Condition Temperature ° c Time s BAI01 5 850 180 Helium quench to 350 °C and hold BAI02 5 850 180 Helium quench to 375 °C and hold BAI03 5 850 180 Helium quench to 400 °C and hold BAI04 5 850 180 Helium quench to 420 °C and hold BAI05 5 850 180 Helium quench to 440 °C arid hold BAI06 5 850 180 Helium quench to 460 °C and hold BAI07 5 850 180 Helium quench to 480 °C and hold Laboratory Experiments : Transformation Kinetics 1 Table 8.7: Summary of isothermal tests for the alloyed eutectoid steel B . Test Heating Rate °C/s H O L D Cooling Condition Temperature ° c Time s BIT01 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 630 °C and hold BIT02 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 600 °C and hold BIT03 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 615 °C and hold BIT04 5 850 180 Cool at 8 °C s- 1 to 750 °C followed by helium quench to 650 °C and hold Laboratory Experiments : Transformation Kinetics 1 Table 8.8: Summary of isothermal tests for the alloyed eutectoid steel C. Test Heating Rate °C/s H O L D Cooling Condition Temperature °C Time s CIT01 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 620 °C and hold CIT02 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 600 °C and hold CIT03 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 635 °C and hold CIT04 5 850 180 Cool at 8 °C s _ 1 to 750 °C followed by helium quench to 650 °C and hold CIT05 5 850 180 Cool at 8 °C s- 1 to 750 °C followed by helium quench to 665 °C and hold CIT06 5 850 180 Cool at 8 °C s- 1 to 750 °C fol-lowed by helium quench 550 °C and hold CIT07 5 850 180 Cool at 8 °C s- 1 to 750 °C followed by helium quench to 500 °C and hold CIT08 5 850 180 Cool at 8 °C s"1 to 750 °C followed by helium quench to 680 °C and hold Laboratory Experiments : Transformation Kinetics 173 - H i h - 3 - H All dimensions in mm Figure 8.1: Geometry of the Gleeble specimen used for characterizing the phase trans-formation kinetics. Rough Sample Blanks 11 mm Note : Not to scale Figure 8.2: Location of the sample blanks taken from the 100 mm-dia. rods. Laboratory Experiments : Transformation Kinetics 3 min _J_ 200 300 Time, s 400 500 Figure 8 . 3 : Schematic representation of continuous cooling tests 200 300 Time, s 500 Figure 8 . 4 : Schematic representation of isothermal tests. Chapter 9 Laboratory Results and Discussion : Quenching Tests The results of the quenching tests are presented in this chapter. The measured temper-ature response in the quenched specimens is first discussed, followed by the estimated surface heat flux (from the inverse analysis). Finally, the results of the microstructural characterization of the quenched samples are presented. 9.1 Measured Temperature Response The data acquired during the forced convective quenching experiments consisted of the temperature response at two positions in the steel bars. Centre and subsurface temper-ature responses were measured during quenching of IF steel specimens, while centre and mid-radius positions were selected for the 1045 carbon and the alloyed eutectoid steels. The quenching conditions were given in Tables 7.5 to 7.7. To assess the reproducibility of the results, two IF steel bars were quenched in separate experiments (runs 10 and 18) with water flowing at 4.8 m s _ 1 at Tj = 50 °C. The temperature response at the sub-surface and centre locations is shown in Figure 9.1. As can be seen, good reproducibility was observed. Visual inspection of the measured cooling curves was used to compare the cooling power of the bath for the various quenching conditions investigated [11]. The mea-sured temperature response at the centre, during forced convective quenching of the 38.1 mm-dia. IF steel bars in water flowing at 2.8, 4.8 and 6.9 m s _ 1 for 3 values of water temperature is shown in Figures 9.2 to 9.4. The comparisons are presented based on 175 Laboratory Results and Discussion : Quenching Tests 176 temperature responses at the centre because the effects of water velocity and water tem-perature are more evident on these curves than on the subsurface temperature responses. When a water temperature of 75 °C was used, the cooling curves showed recalescence near the austenite-to-ferrite equilibrium temperature (910 °C). To confirm this effect, the temperature response during the quench was simulated numerically (see Appendix E). For a given water velocity, the specimen cools faster as the water temperature de-creases (subcooling increases); however, the temperature response for Tj — 75 °C is markedly different than that for 25 and 50 °C, in that a much lower rate of change of temperature with time was observed at the early stages of the quench. This result sug-gests that there is a threshold value that separates two distinctive types of behaviour. A similar observation was made when a stainless steel disk was quenched from 850 °C in still water [194]; in that investigation, curves of heat-transfer coefficient vs surface tem-perature showed different behaviour when the bath temperature was below 60 °C. Note that the difference between the curves corresponding to Tj = 25 and 50 °C decreases as the water velocity increases. On the other hand, when the results were grouped by water temperature (Figures 9.5 to 9.7), the curves for the three water velocities showed a remarkable similarity. As the water velocity increases, the specimen cools faster, due to an improved liquid-solid contact. As can be seen in the figures, the temperature response corresponding to water flowing at 4.8 m s _ 1 approaches that for v = 6.9 m s _ 1 for all levels of water temperature investigated. This observation suggests that there is a limit to the rate at which the specimen can be cooled, dictated by the internal resistance of the bar. Thermal stresses are generated in quenched parts by temperature gradients generated during cooling. It is, therefore, important to investigate the effect of water temperature and velocity on the measured thermal gradients as the quench progresses. The tem-perature difference between the centre and the subsurface temperatures as a function of subsurface temperature during forced convective quenching of IF steel specimens in Laboratory Results and Discussion : Quenching Tests 177 water flowing at 25, 50 and 75 °C for 3 water velocities is plotted in Figures 9.8 to 9.10. Starting with an essentially uniform temperature of ~ 1000 °C, the thermal gradient across the specimen radius increases as the subsurface temperature decreases, reaching a maximum, and then decreasing towards zero as the specimen temperature approaches the water temperature. This behaviour is the result of a combination of two factors : 1) the dependence of the surface heat flux on surface temperature, which shows a maxi-mum at the critical heat flux, and 2) the internal resistance to conduction heat transfer, which causes a damping and lagging effect in the temperature response at the centre. In all cases, the maximum temperature difference occurs at or just above 200 °C; this is an important finding, given that the martensitic transformation in the alloyed eutectoid steel occurs at ~ 220 °C. The results for Tf = 25 °C are essentially identical regardless of water velocity, while smaller thermal gradients were observed as the water velocity decreased for Tf = 50 and 75 °C, i.e., 'softer' quenching conditions. The local cooling rate is an important parameter in determining the microstructural evolution in a quenched part. In this work, the cooling rate was estimated by computing the rate of change of temperature with time, using the commercial package T - S M O O T H [212]. The resulting cooling rate at the centreline as a function of local temperature for tests done with water flowing at 25, 50 and 75 °C, for 3 water velocities is shown in Figures 9.11 to 9.13. As the temperature decreases, the cooling rate increases, until a maximum value is reached, and then decreases to zero. For the highest water temperature investigated (Tf = 75 °C, Figure 9.13), the cooling rate was nearly constant between 1000 and ~ 800 °C before a sudden increase was observed. The maximum cooling rate occurred at local temperatures temperatures between 500 and 700 °C. The maximum cooling rate as a function of water velocity, for the 3 levels of water temperature studied, is shown in Figure 9.14. For a given water temperature, the magnitude of the maximum cooling rate increases as the water velocity increases. The largest cooling rates were obtained for the lowest water temperature (highest sub cooling). Similar calculations were performed Laboratory Results and Discussion : Quenching Tests 178 using the data at the subsurface position. The measured maximum cooling rates as a function of water velocity are shown in Figure 9.15 for 3 values of water temperature. As expected, the maximum cooling rates are higher than the corresponding values at the centreline. Also, the effect of water velocity on the maximum water flow rate was small at the highest water temperature, while a significant influence was observed for Tj = 25 and 50 °C. In the actual plant operation, scale formation occurs during reheating and transporta-tion of the bars before the quench commences. To study the effect of surface condition on heat extraction, IF and alloyed eutectoid steel specimens were intentionally oxidized prior to quenching. The measured temperature response at the centre during forced convec-tive quenching of clean, 'lightly' oxidized and 'heavily' oxidized IF steel bars with water flowing at 4.8 m s _ 1 at 50 °C is shown in Figure 9.16. The presence of an oxide layer increased the rate of heat extraction, but the effect was minimal. It should be pointed out that no attempt was made to characterize the oxide layer and, therefore, references to this layer are only qualitative. The temperature response at the centre, during forced convective quenching of clean and 'heavily' oxidized alloyed steel bars in water flowing at 4.8 m s _ 1 at 32 °C is shown in Figure 9.17. Little difference in cooling behaviour was observed between the two surface conditions. Differences in heat extraction have been reported when clean and oxidized specimens were quenched in pool boiling condi-tions [194,199]. A better adherence of the vapour film when an oxidized layer was present was cited as the reason for the observed difference [199]. Data on the effect of an oxide layer on heat extraction during forced convective quenching is scarce. The results of the present investigation suggest that the enhanced water-solid contact provided by the flowing water overcomes any influence that an oxide layer may have on heat extraction. The effect of initial test temperature was also studied. Two IF steel bars were quenched from 1000 and 850 °C with water flowing at 4.8 m s _ 1 at 32 °C; the tem-perature responses at the centre are shown in Figure 9.18. Except for a short initial Laboratory Results and Discussion : Quenching Tests 179 period, the two curves showed essentially the same behaviour. A similar result was ob-served in boiling curves obtained during forced convective quenching of small-diameter (0.3 or 0.5 mm) platinum wires, for a wide range of initial temperatures [213]. 9.2 Estimated Surface Heat Flux Based on the measured temperature response at the subsurface of IF steel bars, the surface heat flux as a function of surface temperature was estimated by solving the inverse heat conduction problem, as described in Chapter 5. Some of the data recorded during the experiments showed fluctuations that would be amplified by the solution of the IHCP. Therefore, the data were filtered before the inverse analysis was applied. A 11-point Savitzky-Golay filter of order 2 [214] was adopted to remove high frequency fluctuations in the measured temperature responses. The filtering algorithm is based on a polynomial least-square fitting inside a moving window; in general, it provides a better smoothing than a simple moving average [214]. The raw and filtered temperature responses during forced convective quenching of an IF steel bar in water flowing at 2.8 m s _ 1 at 25 °C are shown in Figure 9.19. It should be noted that, for clarity, relatively few raw data points were plotted. The inserts in the figure show magnified views of two areas where sharp bends in the cooling curve were present. As can be seen, the filtering procedure removes fluctuations in the data without losing the main characteristics of the cooling curve. As mentioned before, some cooling curves showed recalescence due to the heat gen-erated during the austenite-to-ferrite transformation in the IF steel. The code adopted to solve the IHCP was not designed to handle heat sources/sinks. Thus, the following strategy was implemented : the temperature response in the range of the transformation was not included in the inverse analysis; the corresponding surface heat flux was then estimated by extrapolating the values obtained after the transformation was complete. Laboratory Results and Discussion : Quenching Tests 180 Given that the transformation occurred at relatively high temperatures (see Appendix E), where film boiling was present, and the recalescence effect was observed only when water at 75 °C was used, this treatment was considered to be adequate. A similar proce-dure has been adopted by Fernandes et al. [127] to estimate the surface heat flux in the austenite-to-pearlite transformation range during the air cooling of 4 mm-dia. carbon eutectoid steel rods. For the calculations, the specimens were subdivided into two regions of 1.5 and 17.55 mm, discretized by 5 and 15 nodes, respectively; this mesh was adopted based on the subsurface thermocouple position (1.5 mm beneath the specimen surface), and the need for a finer mesh near the surface. The actual thermocouple position was measured in few specimens after the tests were completed, by sectioning the sample; it was found that the thermocouple was located within ± 1 mm from its nominal position. The At adopted for estimating the surface heat flux and surface temperature was 0.2 s (the same as the experimental time step). The computer code allows the time step used in the solution to be smaller than the experimental time step; however, no significant differences in the estimated surface heat flux were found when computational time steps of 0.1 s and 0.05 s were adopted. In the latter case, numerical instabilities were introduced in some runs, which then required the use of a larger number of future time steps to stabilize the solution. An attempt was made to simulate data recorded at higher frequency, by interpolating values from a given temperature response; the new data was then input to the inverse analysis code. No significant differences were observed between the results obtained with the original data (measured every 0.2 s) and the simulated data ('measured' every 0.1 s). The thermophysical properties of IF steel were input as explicit functions of temperature by obtaining the best-fit curve of data reported in the literature for pure iron and very low carbon steel [215,216] : k = 81.86 -0.0974 T + 4.3 x 1 0 - 5 T2 (9.1) Laboratory Results and Discussion : Quenching Tests 181 p = 7876 - 0.331 T (9.2) Cp = a(b - T)cexp[d(b - T)], T < 770 °C a = 1756.8, b = 771.9, c= -0.1506, d=-0.00047 (9.3) Cp = a(T - b)c exp[d(T - b)} T > 770 °C a = 2164.6, 6 = 765.4, c = -0.2964, d = 0.00144 (9.4) where T is in °C and k, p and Cp are in SI units. The application of the inverse analysis is illustrated by considering the case of forced convective quenching of an IF steel bar with water flowing at 2.8 m s _ 1 at 25 °C. The experimentally determined temperature response at the centre and subsurface is shown in Figure 9.20. The corresponding estimated surface heat flux as a function of surface temperature is plotted in Figure 9.21. The raw data was filtered using a Savitzky-Golay filter and a value of r = 2 was adopted for the calculations. When r was set to 1 (exact matching of the data), very large oscillations in the estimated heat flux were observed. For comparison, the estimated surface heat flux as a function of surface temperature, obtained with raw and filtered data, is shown in Figure 9.22. By filtering the data, a smoother boiling curve was obtained, while preserving the general characteristics of the curve. The number of future time steps adopted for the calculation of the heat flux in the inverse solution is given by the parameter r. To investigate the effect of future time steps and prefiltering, runs with r = 2, 4, and 6, using raw as well as filtered data were conducted. The results are presented in Figures 9.23 and 9.24 for raw and filtered Laboratory Results and Discussion : Quenching Tests 182 input data, respectively. In the figures, the residuals (difference between measured and estimated subsurface temperature) are plotted as a function of measured temperature, for 3 values of the parameter r. For clarity, the right axis was used to plot the curves corresponding to r = 2. The scales on the left and right axis are identical, except that one of them is shifted with respect to the origin. As can be seen, the magnitude of the residuals increased as the value of r increased. This trend of poorer agreement between the measured and estimated temperature responses for higher values of r, arises from the use of additional future time steps, which tends to 'smooth' the solution of the IHCP. A negative effect of adopting larger values of r, is that sudden changes in the heat flux, such as peaks or valleys, are smoothed. On the other hand, larger values of r may need to be used to stabilize the solution when particularly small time steps are adopted or very noisy data is used [183]. There are no specific rules for choosing r, but general guidelines have been reported [183,217]. For a plate of thickness L, heated at one boundary and insulated at the other, values of r = 2, 3, 4, 5 and 6 were recommended for values of a/S.t j I? = 0.12, 0.07, 0.052, 0.042 and 0.036, respectively, for the case where the thermocouple was located at the insulated boundary. In general, the closer the thermocouple is located with respect to the active heat transfer boundary, the smaller is the required value of r [217]. By computing the thermophysical properties of IF steel at an intermediate temperature of 500 °C, and choosing L = 1.5 mm (the distance between the thermocouple and the specimen surface), a value of aAt/L2 = 0.76 was obtained. Low values of r can then be adopted in the calculations; this is a direct result of having filtered the raw data, which is equivalent to using data with small random errors. At all levels of r, prefiltering of the data reduces the magnitude of the residuals and removes unwanted responses to fluctuations in the experimental data (compare the residuals between 50 and 250 °C for raw and filtered data in Figures 9.23 and 9.24). The running sum of the root mean square (RMS) of the temperatures is a good estimate of the matching between measured and calculated temperatures. The sum of the RMS of the temperatures at the subsurface Laboratory Results and Discussion : Quenching Tests 183 position was 1.7, 4.4 and 8.1 °C for r = 2, 4, and 6, respectively when using raw data; the corresponding values for filtered data were 0.8, 3.8 and 7.6 °C. Clearly, a better agreement between measured and estimated temperatures, especially at low values of r, was obtained by prefiltering the measured data. The critical heat, flux (CHF) is an important parameter in heat transfer processes where boiling is present. Raw and filtered temperature responses from run 8 {y — 2.8 m/s, Tf = 25 °C) were used to assess the effect of filtering the data previous to the inverse analysis on the estimated C H F . The effect of pre-filtering on the estimated magnitude and temperature of occurrence of the C H F is summarized in Table 9.1. When raw data was used, the estimated values of C H F were 5.3, 5.0, and 4.7 M W m - 2 for r = 2, 4, and 6, respectively; the corresponding estimated surface temperatures were 445.4, 410.8 and 373.9 °C. In contrast, the filtered data resulted in C H F values of 5.3, 5.0 and 4.8 M W m~ 2 at 411.1, 412.9, and 414.2 °C for r = 2, 4, and 6, respectively. As can be appreciated, the uncertainty associated with the surface temperature at which C H F occurs is greatly reduced by pre-filtering the data, while the magnitude of the C H F is only slightly increased (for r = 5). It can then be concluded that pre-filtering of the measured data results in a better characterization of the critical heat flux. Boiling curves estimated from subsurface temperature measurements recorded during forced convective quenching of IF steel bars with water flowing at 2.8, 4.8 and 6.9 m s _ 1 for 3 values of water temperature are shown in Figures 9.25 to 9.27. For a given water velocity, the surface heat flux is higher as the subcooling increases (water temperature decreases). At low subcoolings, a film boiling stage was evident in the boiling curves, whereas at subcoolings of 50 and 75 °C, no film boiling stage could be identified; the boiling curves immediately reached the transition boiling stage. The boiling curves ob-tained with subcoolings of 50 and 75 °C were very similar for the intermediate (4.8 m s _ 1 ) and high (6.9 m s _ 1 ) water velocities investigated, i.e.,.the amount of heat extracted ap-proached a limit, defined by the internal thermal resistance of the bar. In some of the Laboratory Results and Discussion : Quenching Tests 184 boiling curves in Figures 9.25 to 9.27, a 'shoulder' can be observed. This observation has also been reported by Ishigai et al. [73] in a study of boiling heat transfer of a water jet impinging on a hot surface. As was the case in the present study, those researchers found that no shoulder appeared in the transition region when the subcooling was low. From visual inspection, they concluded that this sudden decrease of heat flux in the transition boiling region, can be attributed to frequent and instantaneous liquid-solid contacts and that it disapears when the solid surface is wet. The effect of film boiling on the boiling curve can be shown by plotting the surface heat flux as a function of time [40]. The estimated surface heat flux during forced convective quenching of IF steel bars with water flowing at 2.8 m s _ 1 for the 3 water temperatures investigated is plotted in Figure 9.28. The presence of film boiling in the run corresponding to Tj = 75 °C has displaced the start of the transition boiling stage considerably. Very little experimental data on forced convective boiling has been published and, consequently, few comparisons with the results of the present study can be made. Honda et al. [213, 218] have studied the heat transfer during forced convective boiling of thin platinum wires (0.3 and 0.5 mm in diameter), using water and CaC^/water solutions as the quenchant. Their results, in the form of surface heat flux as a function of wall superheat ( A T s a t = Ts — T s a t ) , are shown in Figure 9.29. When quenching in pure water, they identified two local minimum-heat-flux points (shown as Ml and M 2 in the figure), which correspond to the two slope changes they observed in the cooling curve. The first minimum point was associated with the collapse of the vapour film [218]. In contrast, when solutions of CaCl2 in water were used as a quenchant, the Ml point was not observed in the experiments with wire of 0.5 mm-dia. They attributed these results to an enhanced wetting of the wire surface due to deposition of CaCl2. The results presented in Figures 9.25 to 9.27 are replotted in Figures 9.30 to 9.32 for comparison with the boiling curves of Honda et al. [213,218]. A major difference between the results Laboratory Results and Discussion : Quenching Tests 185 of the present study and those shown in Figure 9.29 is the absence of the point Ml of their experiments with pure water. Instead, the boiling curves in Figures 9.30 to 9.32 resemble their results for quenching with CaCl2 solutions. The heat fluxes obtained in this investigation are significantly lower than those reported by Honda et al. [213, 218] due to the much larger diameter used (38.1 mm vs 0.3 - 0.5 mm). As pointed out in Chapter 2, there are very few reported data for heat transfer during forced convective quenching for the conditions of interest. The film boiling heat flux as a function of surface temperature was computed using an integral method [219], based on the solution given by Nakayama and Koyama [220]. Given that the ratio of thickness of the boundary layer to bar radius is small, the assumption was made that the problem could be treated as forced convective boiling (parallel flow) over a flat plate. The results of calculations for an IF steel bar quenched with water flowing at 75 °C at 3 water velocities are shown in Figure 9.33, along with the measured boiling curves. The computed forced convective film boiling curves underestimate the magnitude of the heat flux during the film boiling stage of the measured boiling curves. Both, the theoretical and measured curves showed an increase in heat transfer as the water velocity increased. Similar findings have been reported by Honda et al. [213]; they performed a numerical analysis of film boiling on a horizontal cylinder with upward flow, and found that the computed heat fluxes underestimated the measured values by 30 %. The estimated critical heat flux as a function of water velocity, during forced con-vective quenching of IF steel bars for 3 water temperatures is shown in Figure 9.34 and summarized in Table 9.2. The critical heat flux increases, almost linearly, as the wa-ter velocity increases; it also increases as the water temperature decreases (subcooling increases). The heat extraction obtained under the quench conditions investigated can also be characterized by the average heat flux (q ), which is proportional to the area under the Laboratory Results and Discussion : Quenching Tests 186 boiling curve : avg (9.5) The average heat flux as a function of water velocity for 3 water temperatures is shown in Figure 9.35 and summarized in Table 9.3. The average heat flux increases as the water velocity increases and the water temperature decreases. Once the surface heat flux and surface temperature have been estimated, it is possible to compute the heat-transfer coefficient. Figure 9.36 shows the heat-transfer coefficient as a function of surface temperature, during forced convective quenching of IF steel bars with water flowing at 25 °C for 3 water velocities. It should be pointed out that the heat-transfer coefficients were computed with respect to the boiling point of water (see Eq. (2.1)). In all cases, the heat-transfer coefficient increases monotonically as the surface temperature decreases. The heat-transfer coefficient increases, at all levels of surface temperature, as the water velocity increases. Even though these curves can be easily correlated to the surface temperature, they all tend to infinity as the surface temperature approaches the boiling point of water. Large errors in the calculated heat-transfer coefficient can then be generated at low surface temperatures and, therefore, it is preferable to describe the heat transfer at the active boundary in terms of surface heat fluxes. 9.3 Metallographic Characterization Three specimens, corresponding to each of the materials used during the quenching ex-periments, were saved for metallographic characterization, hardness and residual stress measurements. The quenching conditions were given in Table 7.6. The microstructures at the centre and ~ 3 mm from the surface produced by forced convective quenching of IF and alloyed steel bars, under the conditions given in Table 7.6, Laboratory Results and Discussion : Quenching Tests 187 are shown in Figures 9.37 and 9.38, respectively. In both steels, the microstructure at the centre and near the surface were very similar. The microstructure in the quenched IF steel bar (Figure 9.37) consisted of essentially 100 % ferrite (see Figure 9.37), whereas the alloyed eutectoid steel showed a predominantly martensitic microstructure in Figure 9.38. A low magnification (8X) photograph of the etched macrostructure of the 1045 carbon steel sample (Figure 9.39) clearly showed a ring of martensite (white) at the surface. The microstructure in this steel in the martensitic band, in the transition zone, and at the centre of the sample are shown in Figure 9.40 (a), (b) and (c), respectively. Near the surface, the final microstructure was essentially martensitic (white) with areas of pearlite/bainite (dark) delineating the prior-austenite grain boundaries. In the transition zone (Figure 9.40 (b)), the amount of martensite decreases, the white areas in Figure 9.40 (a) being replaced with pearlite and bainite, the pearlite colonies outlining the original austenite phase boundaries; at the centre (Figure 9.40 (c)), the microstructure consisted of large patches of martensite (white) with narrow white ferrite outlining the original austenite grains, the remaining dark equiaxed phase being pearlite. The hardness profile across the bar diameter of the 3 steels (IF, alloyed eutectoid and 1045) is shown in Figure 9.41. The hardness of the alloyed eutectoid and the 1045 carbon steel specimens was measured in Rockwell C units (left axis) while Vickers units 1 (right axis) were used for the IF sample. To avoid errors near the surface of the sample, hardness measurements extended only to 17.5 mm from the center of the bar. To complete the hardness profile, microhardness measurements were made near the surface. The alloyed eutectoid steel specimen produced the highest level of hardness, followed by the 1045 carbon steel and the IF steel. As expected from the metallographic analysis, the hardness across the alloyed eutectoid and the IF steel specimen was very uniform. In contrast, the 1045 carbon steel exhibited a hardness gradient across the radius of the bar. Near the surface, the hardness was uniform for the first 2.5 mm, and of the order of the value 1 Using a 10 kg f load. Laboratory Results and Discussion : Quenching Tests 188 exhibited by 100% martensite in a 1045 carbon steel [221]; it then decreased sharply (~ 25 HRc units) between 2.5 and 5 mm, and remained relatively constant in the core. This hardness profile agrees well with the microstructures shown in Figures 9.40 (a) to 9.40 (c). For comparison, a typical measured hardness profile in a 100 mm-dia., 6 m long alloyed eutectoid steel rod, after quenching and tempering under industrial conditions, is shown in Figure 9.42. In the figure, the hardness across the radius near both ends and at mid-length, as well as their average, is shown. The fully martensitic alloyed eutectoid quenched bar was used to determine the prior-austenite grain size produced by the heating cycle. To facilitate image analysis, the grain boundaries revealed by the boiling alkaline sodium picrate etch were drawn on tracing paper; a typical example is shown in Figure 9.43. The prior-austenite grain size was determined by measuring the areas of the individual grains according to the Jeffries method [211]. Several frames were analysed and a total of 488 grain areas were measured2. The number of grain areas measured was considered to be statistically significant. The measured grain area distribution is shown in Figure 9.44. The equivalent area grain diameter (EQAD) was calculated assuming the areas in the photomicrographs represent perfectly spherical grains sectioned at their equator : where the Jeffries number, Aj in fim2, is computed as : where Ap is the frame area, in yum2, and NQ is the number of grains measured. The measured equivalent prior-austenite grain diameter was 10.1 pm, which is similar to the grain diameter (10.8 pm) produced in the Gleeble experiments. Thus, the transformation 2According to the Jeffries method, grains located at the border of frames are counted as half. (9.6) Laboratory Results and Discussion : Quenching Tests 189 kinetic data measured in the tubular samples can be applied to compute the microstruc-tural evolution produced in the 38.1 mm-dia. alloyed eutectoid steel bars. Laboratory Results and Discussion : Quenching Tests 190 Table 9.1: Estimated critical heat flux (CHF) using raw and pre-filtered data for a 38.1 mm-dia. IF steel bar quenched in water flowing at 2.8 m s _ I at 25 °C. r Unfiltered Filtered C H F , M W m - 2 1 C H F ' ^ C H F , M W m -2 1 C H F ' ^ 2 5.3 445.4 5.3 411.1 4 5.0 410.8 5.0 412.9 5 4.7 373.9 4.8 414.2 Table 9.2: Estimated critical heat flux (CHF) as a function of water velocity during forced convective quenching of 38.1 mm-dia. IF steel bars in water flowing at 3 water temperatures. v, m s 1 q C H F ' M W m - 2 Tf = 25 °C Tf = 50 °C Tf = 75 °C 2.8 5.0 3.7 2.6 4.8 6.4 5.8 3.8 6.9 8.1 7.1. 4.5 Table 9.3: Average heat flux as a function of water velocity during forced convective quenching of 38.1 mm-dia. IF steel bars in water flowing at 3 water temperatures. v, m s 1 c l a v a , M W m-2 Tf = 25 °C Tf = 50 °C Tf-= 75 °C 2.8 2.98 1.96 1.18 4.8 3.90 3.31 1.71 6.9 4.82 4.27 2.21 Laboratory Results and Discussion : Quenching Tests 191 u «S u CD a. s o H o RUN10 RUNIB IF s t e e l -- 1 V = 4.8 m s o -= 50 C 100 Figure 9.1: Measured temperature response at the centre and subsurface of two 38.1 mm-dia. IF steel bars quenched with water flowing at 4.8 m s - 1 at 50 °C. 0) -p (0 u © p . s a> H 1100 1000 900 800 700 600 500 400 300 200 100 0 0 = 75 C o T, = 50 °c * T f = S5 °c IF steel v = 2.8 m s~ Centre 20 40 60 80 Time , s 100 120 Figure 9.2: Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 2.8 m s _ 1 for 3 values of water temperature. Note the recalescence when water at 75 °C was used. Laboratory Results and Discussion : Quenching Tests 192 6 1100 1000 900 800 700 600 500 400 300 200 100 0 A 0 = 75 °C ° T ( = 50 °C ' T, = 25 ° C I F s t e e l v = 4 . 8 m s C e n t r e 1 O V ' V ^ ° O O Q O O O O O 20 40 60 T ime , s 80 Figure 9.3: Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 for 3 values of water temperature. Note the recalescence when water at 75 °C was used. s <u 1100 1000 900 800 700 600 500 400 300 200 100 0 0 v T f = 75 C ° T f = 50 ° C ' T, = 25 °c I F s t e e l v = 6.9 m s C e n t r e A ^ o o o o o o o o o o o o o 20 40 T ime , s 60 Figure 9.4: Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 6.9 m s _ 1 for 3 values of water temperature. Note the recalescence when water at 75 °C was used. Laboratory Results and Discussion : Quenching Tests 193 0 uoo 1000 900 800 700 600 500 400 h 300 200 100 0 v v = 2.8 m s o v = 4.8 m s » v = 6.9 m s IF steel 0 10 20 30 40 Time , s 50 Figure 9.5: Measured temperature response at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 25 °C for 3 values of water velocity. CD ex, CO 1100 1000 900 800 700 600 500 400 300 200 100 0 v = 3.8 m s v = 4.8 m s v = 6.9 m s IF steel 50 C Centre «*BTttUiHp 20 40 60 Time , s 80 Figure 9.6: Measured temperature response at the centre of a 38.1 mm-dia. bar quenched with water flowing at 50 °C for 3 values of water velocity. Laboratory Results and Discussion : Quenching Tests 194 3 ctf a) s H 1100 1000 900 800 700 600 500 400 300 200 100 0 * O v v = 2.8 rn s v = 4.8 m s v = 6.9 m s 20 IF steel 75 C Centre ***8gi>itii t M i i i l ! ^^ 40 60 Time, s 80 100 120 Figure 9.7: Measured temperature response at the centre of a 38.1 mm-dia. bar quenched with water flowing at 75 °C for 3 values of water velocity. 800 700 600 500 400 300 200 100 0 V V = 2.8 m - 1 s _ i o V 4.8 m s — i V = 6.9 m s r V A O IF steel 25 C I i i i i I 1000 800 600 400 200 T (subsurface), C Figure 9.8: Measured temperature difference between the centre and the subsurface as a function of subsurface temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 25 °C for 3 values of water velocity. Laboratory Results and Discussion : Quenching Tests 195 800 700 V V = 2.8 m - l s o V = 4.8 m s_, °_ 600 A V = 6.9 m s V 500 400 300 200 100 0 V IF steel T ( = 50 C 1000 BOO 600 400 200 T (subsurface) , C Figure 9.9: Measured temperature difference between the centre and the subsurface as a function of subsurface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 50 °C for 3 values of water velocity. 800 700 600 500 400 300 200 100 0 V = 2.8 m - l s o V = 4.8 m s — 1 V = 6.9 m s 3 IF steel T ( = 75 C 1000 BOO 600 400 200 T (subsurface) , C Figure 9.10: Measured temperature difference between the centre and the subsurface as a function of subsurface temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 75 °C for 3 values of water velocity. Laboratory Results and Discussion : Quenching Tests 196 < 50 -50 -100 -150 -200 3 « v v IF steel f Centre 25 C - l v v = 2.8 m s i o V = 4.8 m s - l » V = 6.9 m s 200 400 600 800 o Temperature, C 1000 Figure 9.11: Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 25 °C for 3 values of water velocity. < 50 h -50 -100 -1 V V = 2.8 m s - l o v = 4.8 m s 6.9 - l * V = m s -150 - 2 0 0 IF steel T ( = 50 C Centre 200 400 600 800 o Temperature, C 1000 Figure 9.12: Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 50 °C for 3 values of water velocity. Laboratory Results and Discussion : Quenching Tests 197 < 50 -50 -100 -150 -200 - l V V — 2.8 m s - l o V = 4.8 m s - l 4 V = 6.9 m s v t> v v v o o ° > IF steel 75 C Centre 200 400 600 Temperature, BOO 1000 Figure 9.13: Cooling rate at the centre as a function of local temperature during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 75 °C for 3 values of water velocity. < 200 150 100 50 O o T, = 25 c A = 50 °c • T, = 75 °c 4 6 Water velocity, m s Figure 9.14: Maximum cooling rate at the centre as a function of water velocity dur-ing forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. Laboratory Results and Discussion : Quenching Tests 198 50 h 0 2 4 6 8 Water velocity, m s Figure 9.15: Maximum cooling rate at the subsurface as a function of water velocity during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. SH u 1) 1000 800 600 400 h 200 Clean Light scale Heavy scale IF Steel v = 4.8 m s Tf = 50 °C Centre 80 Figure 9.16: Effect of surface condition on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 4.8 m s _ 1 at 25 °C. Laboratory Results and Discussion : Quenching Tests 199 Clean O Oxidized 20 40 60 80 100 Time, s Figure 9.17: Effect of surface condition on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. alloyed steel bar in water flowing at 4.8 m s- 1 at 32 °C. Figure 9.18: Effect of initial test temperature on the temperature response at the centre during forced convective quenching of a 38.1 mm-dia. IF steel bar in water flowing at 4.8 m s- 1 at 32 °C. Laboratory Results and Discussion : Quenching Tests 200 0 10 ' 20 30 40 50 Time, s Figure 9.19: Raw and filtered temperature response at the subsurface during forced convective quenching of a 38.1 mm-dia. IF in water flowing at 2.8 m s _ 1 at 25 °C. The inserts show the performance of the filtering procedure at two selected regions of the curve. Laboratory Results and Discussion : Quenching Tests 201 1100 0 1 i , , , , i 0 20 40 60 80 T i m e , s Figure 9.20: Experimentally determined temperature response at the subsurface and centreline during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s'1 at 25 °C. 8 h IF Steel - l V = 2.8 m s T f = 25 °C 7 h i Figure 9.21: Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar. The raw data of Figure 9.20 were filtered before being input to the computer program. Figure 9.22: Comparison between estimated surface heat flux as a function of surface temperature, using raw and filtered data, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s - 1 at 25 °C. Figure 9.23: Effect of the parameter r on the residuals obtained during the estimation of surface heat flux during forced convective quenching of a 38.1 mm-dia. IF steel bar when raw data was used. Laboratory Results and Discussion : Quenching Tests 203 u o H I H _ 50 40 30 20 10 0 |- ^ -10 20 30 40 50 r = 2 r = 4 r = 6 200 400 600 800 o ^exp' C _L 80 70 60 50 u o 40 o "5 0 30 H 1 20 ft D 10 H 0 -10 -20 1000 Figure 9.24: Effect of the parameter r on the residuals obtained during the estimation of surface heat flux during forced convective quenching of a 38.1 mm-dia. IF steel bar when filtered data was used. Figure 9.25: Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 for 3 values of water temperature. Laboratory Results and Discussion : Quenching Tests 204 ....... T f 25 0 C -T f = ....... T f = 50 °C -75 o c 7 h 0 200 400 600 800 1000 Figure 9.26: Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 for 3 values of water temperature. Figure 9.27: Estimated surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 6.9 m s - 1 for 3 values of water temperature. Laboratory Results and Discussion : Quenching Tests 205 100 10 5 0.01 v = 2.8 m s T f = 25 °C T ( = 50 °C T f = 75 °C 10 20 30 40 T ime, s 50 60 70 Figure 9.28: Estimated surface heat flux as a function of time, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s" 1 for 3 values of water temperature. 1000 Figure 9.29: Surface heat flux as a function of wall superheat, during forced convective quenching of thin platinum wires [218]. Laboratory Results and Discussion : Quenching Tests 206 10* 10 ' & 10" 10 i o _ 1 . . . . . . . T f _ 25 O c — T f = T f = 50 °c 75 o c v = 2.8 m / s 200 400 AT sat' 600 o c 800 1000 Figure 9.30: Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s - 1 for 3 values of water temperature. 1000 Figure 9.31: Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 for 3 values of water temperature. Laboratory Results and Discussion : Quenching Tests 207 10" 10* 10 10 10"1 . . . . . . . T f = 25 O c T f = 50 °c 75 o c v = 6.9 m / s 200 400 AT 600 o C 800 1000 Figure 9.32: Estimated surface heat flux as a function of wall superheat, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 6.9 m s _ 1 for 3 values of water temperature. Figure 9.33: Estimated and calculated (film boiling) surface heat flux as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 2.8 m s _ 1 for 3 values of water temperature. Laboratory Results and Discussion : Quenching Tests 208 10 - o T f = T f = 25 o C A 50 o c • T f = 75 o c Is 2 h 4 5 6 W a t e r v e l o c i t y , m s Figure 9.34: Estimated critical heat flux (CHF) as a function of water velocity, dur-ing forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. 0 O T f = 25 C 5 A T , = 50 o c o • T, = 75 c f 6 W a t e r v e l o c i t y , m s Figure 9.35: Average heat flux as a function of water velocity, during forced convective quenching of a 38.1 mm-dia. IF steel bar for 3 values of water temperature. Figure 9.36: Estimated heat-transfer coefficient as a function of surface temperature, during forced convective quenching of a 38.1 mm-dia. IF steel bar with water at 25 °C for 3 values of water velocity. A value of r = 2 was adopted for filtered data. Laboratory Results and Discussion : Quenching Tests 210 30 pm 30 pm 00 Figure 9.37: Photomicrographs of a 38.1 mm-dia. IF steel bar quenched with water flow-ing at 4.8 m s _ 1 at 25 °C, at (a) ~ 3 mm from the surface and (b) centre. Magnification : 500 X . Laboratory Results and Discussion : Quenching Tests 211 Figure 9.38: Photomicrographs of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s - 1 at 75 °C, at (a) ~ 3 mm from the surface and (b) centre. Magnification : 400 X . Laboratory Results and Discussion : Quenching Tests 212 2 mm Figure 9.39: Photograph of the macrostructure of the 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. Note the martensitic ring (white) at the surface. Magnification : 8 X . Figure 9.40: Photomicrographs of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. (a) within the martensitic ring, (b) in the transition zone and (c) at the centre. Magnification : 800 X . Laboratory Results and Discussion : Quenching Tests 213 Figure 9.40: Photomicrographs of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. (a) within the martensitic ring, (b) in the transition zone and (c) at the centre. Magnification : 800 X . Laboratory Results and Discussion : Quenching Tests 214 o 05 33 0) Pi 13 u C6 w 70 60 50 40 30 20 10 ' I — 1 — l — 1 — i — ' — i — 1 — l — 1 — i — 1 — i — r A Al loyed Steel O 1045 Steel • IF Steel i i i i i i i 900 -20 - 1 6 - 1 2 - 8 - 4 0 4 8 12 16 20 Rad ia l Pos i t i on , m m Figure 9.41: As-quenched hardness distribution in 38.1 mm-dia. IF, alloyed and 1045 carbon steel bars. Use the right axis to read hardness in the IF steel bar. Figure 9.42: Hardness distribution near the ends and at mid-length in a 100 mm-dia. alloyed eutectoid steel bar quenched and tempered under industrial conditions. Laboratory Results and Discussion : Quenching Tests 215 Figure 9.43: Outline of prior-austenite grain boundaries in a 38.1 mm-dia. alloyed eu-tectoid steel quenched bar, etched in a boiling alkaline sodium picrate solution. Magni-fication : 1000 X . 140 120 100 I 80 21 60 40 20 0 2 Area, /xm Figure 9.44: Measured prior-austenite grain area distribution in a 38.1 mm-dia. alloyed eutectoid steel quenched bar. Chapter 1 0 Laboratory Results and Discussion : Transformation Kinetics The results from the transformation kinetics study are presented in this chapter. First, the transformation kinetics measured under continuous cooling conditions are discussed, followed by the results from the isothermal tests. Calculated phase boundaries, measured prior-austenite grain size and measured ferrite fraction are also presented. 10.1 Phase Boundaries The theoretical phase boundaries for the three alloyed eutectoid steels and the 1045 carbon steel were calculated using a computer program based on the algorithm proposed by Kirkaldy et al. [104]. The start of the austenite-to-martensite transformation, M s , was calculated using the empirical linear chemistry-sensitive formula given by Andrews [116] while the start of the austenite-to-bainite transformation, Bs, was computed using the empirical formula proposed by Kirkaldy et al. [14]. The results are summarized in Table 10.1. 216 Laboratory Results and Discussion : Transformation Kinetics 217 10.2 Continuous Cooling Tests 10.2.1 Alloyed Eutectoid Steels Typical measured cooling curves are shown in Figure 10.1. Note that recalescence was observed for the intermediate cooling rates of 17.5 and 8 °C s _ 1 only. The correspond-ing dilation vs temperature curves are shown in Figure 10.2. The data from each run were filtered by averaging every 5 points, which smoothed the curves and resulted in a reduction of the total number of data points to be analyzed. The analysis of the filtered data was carried out as follows : 1. Compute the change in diameter due to thermal contraction of the parent phase1. This was done by fitting the austenite data obtained above the transformation start temperature by linear regression. 2. Compute the change in diameter due to thermal contraction of the product phase. This was done by fitting the product phase data obtained below the transformation finish temperature by linear regression. 3. Compute the difference between the experimentally measured dilation (thermal contraction plus expansion due to transformation) and the change in diameter due to thermal contraction only, and determine the start and end of the transformation. These calculations are illustrated in Figures 10.3 (steps 1 and 2), 10.4 (step 3 : start of the transformation), and 10.5 (step 3 : end of the transformation) for the 17.5 °C s - 1 of the alloyed eutectoid steel A . dilation data measured with the Gleeble 1500 simulator is given in arbitrary length units. The following relationship was used to transform the data to milimeters : D = 2.88315 D, meas + D0 where D0 is the initial sample diameter (mm) and D] units). meas is the measured dilation (arbitrary length Laboratory Results and Discussion : Transformation Kinetics 218 The experimentally determined C C T diagrams for the alloyed eutectoid steels A , B and C are shown in Figures 10.6 to 10.8. From metallographic observations it was determined that austenite transformed to pearlite for cooling rates up to 17 °C s _ 1 (air cooling); a mixture of bainite plus martensite was obtained at the higher cooling rates. It should be noted that only one test (ACC06-1 : cooling rate = 58 °C s _ 1 ) produced a structure composed of 100 % martensite. The experimentally determined martensite start temperature ( M J was 218 ± 2 °C, 221.5 ± 1.5 °C and 229 ± 2.5 °C for steels A , B and C, respectively, which compares favourably with the corresponding values (216, 221 and 234 °C, respectively) obtained using the empirical formula proposed by Andrews [116]. The transformation start temperature as a function of cooling rate is shown in Fig-ure 10.9 for the three alloyed eutectoid steels. As can be seen, the three steels exhibit very similar behaviour for all the cooling rates examined in the study, with steel A hav-ing the highest start temperature for any given cooling rate. From these results the start temperature was correlated with the cooling rate, for cooling rates up to 44 °C s - 1 , according to the following relations : where T t is the transformation start temperature (in °C) and C R is the cooling rate (in °C s _ 1 ) . Measured and predicted (using Eq. (10.1)) start temperatures are shown in Figure 10.10 (for the alloyed eutectoid steel A); excellent agreement can be observed. To predict microstructural evolution, correlations for the C C T start time as a function of undercooling below the appropriate asymptote are required. From the measured C C T start time (tAVcer), the following correlation was developed : 638.2 - 74.0 C R + 9.9 C R 2 + 7.71 C R 3 , C R < 17 °C s -651.8 + 2868.0 C R - 2081.6 C R 2 + 467.6 C R 3 , C R > 17 °C s - i \ntAVCCT = 27.9 + 0.02 U l - 5.654 l n u x (10.2) Laboratory Results and Discussion : Transformation Kinetics 219 where u\ is the undercooling below TAl- The measured and predicted (using Eq. (10.2)), C C T start times are shown in Figure 10.11; good agreement can be seen. The thermal expansion coefficient of the parent and product phases was computed from the corresponding slopes of the dilation vs temperature curves. The following rela-tion was used : a = 2.88315 J (10.3) where m is the slope of the dilation vs temperature curve (steps 1 and 2 in the data reduction procedure described above) and D0 is the initial sample diameter (mm). The measured average values of the thermal expansion coefficient for austenite, pearlite and bainite obtained using data from the three alloyed eutectoid steels are given in Table 10.2. 10.2.2 1045 Ca rbon Steel A similar analysis was applied to results of the C C T tests conducted for the 1045 carbon steel. The experimentally determined C C T diagram is shown in Figure 10.12. The following correlation between start temperature and cooling rate was obtained from the experimental data : T g t = 700 - 6.35 l n C R + 6.56 l n C R 2 - 8 . 1 0 I n C R 3 (10.4) The transformation start temperature, T t as a function of cooling rate is plotted in Figure 10.13, together with predicted values obtained using Eq. (10.4); good agreement was obtained. Regression equations to predict the C C T start time, tAvccr, as a function of under-cooling were also developed for the different microconstituents. For ferrite, the following equation was obtained : In tAVccT-F = 80.86 + 0.151 u 3 - 20.694 In u3 (10.5) Laboratory Results and Discussion : Transformation Kinetics 220 where u$ is the undercooling below TA3- The results are plotted in Figure 10.14 and, as can be seen, the agreement between measured and predicted values is good. The following regression equation for the C C T pearlite start time as a function of undercooling below TAl, ui, was obtained : In tAVccT-p = 113.17 + 0.26 ux - 30.193 In U l (10.6) The measured and predicted values are shown in Figure 10.15; good agreement was obtained. The thermal expansion coefficient of the parent and product phases was computed from the dilation vs temperature curves, as described above for the alloyed eutectoid steel. The results are given in Table 10.2. For comparison, values of thermal expansion coefficient for a Ck45 (SAE 1045) steel reported in the literature (Ref. [222]) are also included. 10.3 Isothermal Tests The results of the isothermal tests obtained for the 3 alloyed eutectoid steels are presented in this section. Typical experimental isothermal dilation vs time curves for the alloyed eutectoid steel A are shown in Figure 10.16 for the range 525 °C < T < 665 °C . At 605 °C and above, austenite transformed to pearlite, whereas below that temperature the products were pearlite/bainite or bainite only. Note the enhanced transformation rate as the test temperature decreases. Prior to the analysis, data from each run was filtered by averaging it (moving average) every 5 points, which smoothed the curves and resulted in a reduction of the total number of data points to be analyzed. The experimentally determined isothermal diagrams for the three alloyed eutectoid steels are shown in Figures 10.17 to 10.19 along with predicted 1 and 99 % isotransformation curves. The predicted curves were computed from the correlations given by Park and Laboratory Results and Discussion : Transformation Kinetics 221 Fletcher [223]. Their correlations were derived by fitting the results of isothermal tests conducted on 19 eutectoid steels (base composition : 0.75 % C, 0.28 % Si). The prior austenite grain size of all steels studied in their investigation was approximately A S T M number 3, whereas the prior austenite grain size in the present study was 9-10 A S T M (see section 10.4). As can be seen, the experimental results obtained in this work are consistent with the values predicted adopting the formulae given by Park and Fletcher [223]. The Avrami equation (Eq. (2.4)) was adopted to model the kinetics of the isothermal transformation. In order to estimate the Avrami kinetic parameters (n, b, and tAv) for each isothermal test, a weighted, 5-parameter, non-linear regression analysis2 was performed on the experimental data. The 5 parameters adopted in the regression model were : n, 6, tAv, Dmin (measured dilation at the start of the transformation) and Dmax (measured dilation at the end of the transformation). The estimated kinetic parameters, for the pearlitic reaction, obtained with this proce-dure are shown in Figure 10.20. For comparison, estimated parameters for weighted and non-weighted analyses are included. Clearly, the use of weighting functions not only re-sults in better regressions (from visual inspection of measured and estimated D vst plots) but also the estimated parameters show less scatter. Assuming the reaction is additive, the parameter n should be independent of temperature. A value of n = 1.83 ± 0.18 was computed by averaging the estimated n values obtained with the weighted, non-linear regression. Using this average value, the kinetic parameter b was re-estimated by re-applying the weighted, multiple non-linear regression analysis to the data but for a fixed value of n — n. The re-estimated values of b are shown in Figure 10.21, along with a regression line given by : l n o = -5.4 exp(42.8/ui) (10.7) where u\ represents undercooling below TAl. 2See Appendix F for details. Laboratory Results and Discussion : Transformation Kinetics 222 Based on the 1 and 99 % transformation curves, predicted using the formulae pro-posed by Park and Fletcher [223], the Avrami parameters for the austenite-to-pearlite transformation were estimated for the three alloyed steels studied in the present investiga-tion. The average value of n obtained in this fashion was 1.55 (compared with 1.83 ±0.18 obtained in this study). The computed values of the parameter b as a function of under-cooling (derived from the theoretical curves) are shown in Figure 10.22, along with the values estimated in this work. For comparison, values reported by Campbell et al. [43] for the pearlitic transformation in a plain carbon eutectoid steel are also shown. A series of experiments were also conducted to determine the Avrami kinetic param-eters for the austenite-to-bainite transformation in the alloyed eutectoid steel A . The isothermal tests were conducted in the range 350 °C < T < 480 °C . The kinetic pa-rameters were estimated with the 5-parameter, nonlinear equation described above. The resulting values of the kinetic parameter, n, are plotted in Figure 10.23. The average value of n was h = 1.91 ± 0.4. Using this estimate, the parameter b was re-estimated by fixing n = n in the regression analysis. The estimated values are shown in Figure 10.24, along with a regression line given by : In 6= -42.6 exp-540.2/ui (10.8) where u\ represents undercooling below TAl. 10.4 Prior-Austenite Grain Size To measure the prior-austenite grain size produced during austenitizing of the tubular samples, a fully martensitic specimen (alloyed eutectoid steel A cooled at 58 °C s _ 1 ) was prepared metallographically. The microstructure revealed by a boiling alkaline sodium picrate etch is shown in Figure 10.25. To determine the prior-austenite grain size, the mean chord length in a total of 66 grains was measured from several frames. The resulting Laboratory Results and Discussion : Transformation Kinetics 223 mean chord length distribution is shown in Figure 10.26. The computed average chord length is 10.8 /mi, which corresponds to a grain size of 9-10 A S T M . 10.5 Ferrite Fraction Results of the ferrite fraction measured in the continuously cooled 1045 carbon steel specimens as a function of cooling rate at 750 °C are shown in Figure 10.27. Each point in the plot represents the mean of no less than fifty separate fields measured on the image analyzer, as described previously. The standard deviation of each measurement is given as an error bar. For comparison, the values reported by Hawbolt et al. [98] for a 1025 carbon steel are also shown in the figure. The dotted line represents the extrapolation to the equilibrium value. The ferrite fraction decreases monotonically from its equilibrium value as the cooling rate increases. As expected, the ferrite fraction in the 1045 carbon steel is smaller at all cooling rates. The ferrite fraction in two tests on the alloyed eutectoid steel was determined in a similar fashion. The tests selected corresponded to the two slowest cooling rates (0.2 and 0.3 °C s _ 1 ) for steel B . The results showed less than 1 % ferrite for a cooling rate of 0.2 °C s _ 1 ; no significant trace of ferrite was found when the cooling rate was 0.3 °C s _ 1 . Since the primary phase formed was pearlite, the alloyed steels were treated as eutectoid. Laboratory Results and Discussion : Transformation Kinetics 224 Table 10.1: Phase boundaries calculated from the equations by Kirkaldy et al. [104], the empirical formula given by Andrews [116], and the empirical formula given by Kirkaldy et al. [14]. Steel Alloyed Eutectoid A Alloyed Eutectoid B Alloyed Eutectoid C 1045 Plain Carbon Kirkaldy et al. [104] 730 730 731 756 724.4 724 725 725 Andrews' formula [116] Ms 216 221.1 234.3 323.6 Mf 1 6.1 . 19.3 108.6 Kirkaldy et al. [14] Bs 557 558 560 585 Table 10.2: Measured expansion coefficients for the alloyed eutectoid and the 1045 carbon steel. For comparison, values reported in the literature are also given. Phase Thermal Expansion Coefficient, °C 1 Alloyed Eutectoid Steel This Work Ref. [215] Austenite 2.46 x l O - 5 2.05 x l O - 5 Pearlite 1.67 x l 0 ~ 5 Bainite 1.63 x l O - 5 1045 Carbon Steel This Work Ref. [222] Austenite 2.22 x l O " 5 2.1 x l O - 5 Pearlite/Bainite 1.61 x l O - 5 1.4 x l O - 5 Laboratory Results and Discussion : Transformation Kinetics 225 Figure 10.1: Experimentally determined cooling curves for the alloyed eutectoid steel B showing the cooling rate for each test. 100 200 300 400 500 600 700 800 Time.s Figure 10.2: Experimentally determined dilation-temperature curves obtained for con-tinuous cooling of the alloyed eutectoid steel B showing the cooling rate for each test. Laboratory Results and Discussion : Transformation Kinetics 226 ti >> >« a 0.40 0.35 0.30 0.25 0.20 Exper imenta l Due to thermal contract ion 300 400 500 600 700 o Temperature , C 800 900 Figure 10.3: Continuous cooling test showing the thermal contraction of the austenite (high temperature) and the low temperature product phase obtained for the 17.5 °C s _ 1 cooling of the alloyed eutectoid steel B. 800 J3 a a -200 400 500 600 Temperature , 700 800 Figure 10.4: Continuous cooling test showing the procedure to determine the start of transformation for the 17.5 °C s _ 1 cooling of the alloyed eutectoid steel B . The dotted lines represent ± 3 standard deviations. Laboratory Results and Discussion : Transformation Kinetics 227 a 140 120 100 h 80 60 40 £ 20 h a -20 + 3(7 transf ormation finish 3 j i0^f%m 100 200 300 400 o Temperature , C 500 600 Figure 10.5: Continuous cooling test showing the procedure to determine the end of transformation for the 17.5 °C s _ 1 cooling of the alloyed eutectoid steel B. The dotted lines represent ± 3 standard deviations. 10000 Figure 10.6: Continuous cooling diagram for the alloyed eutectoid steel A . Closed cir-cles represent the start of diffusional transformations; open circles represent the end of diffusional transformations. Laboratory Results and Discussion : Transformation Kinetics 228 o ' — — — — ' 1 10 100 1000 10000 Time.s Figure 10.7: Continuous cooling diagram for the alloyed eutectoid steel B . Closed cir-cles represent the start of diffusional transformations; open circles represent the end of diffusional transformations. Figure 10.8: Continuous cooling diagram for the alloyed eutectoid steel C. Closed cir-cles represent the start of diffusional transformations; open circles represent the end of diffusional transformations. Laboratory Results and Discussion : Transformation Kinetics 229 8 0 0 7 0 0 6 0 0 500 p . a C o o l i n g R a t e ° C / m i n 1 0 3 1 0 2 1 0 1 1 1 1 1 1 | M 1 1 1 1 1 1 | 1 1 1 T A l -. A - -_ _ 2 0 0 •DO A Steel A o Steel B • Steel C 1 0 ' 10 1 1 0 " 1 0 " C o o l i n g Rate, C / s Figure 10.9: Transformation start temperature as a function of cooling rate for the alloyed eutectoid steels A , B and C. BOO CD U «-> cS u <D 700 600 \-500 - P (0 400 300 O Data Predicted 200 1 1 10 2 10 1 10° 10" 1 0 . C o o l i n g Rate, C / s Figure 10.10: Transformation start temperature plotted as a function of cooling rate for the alloyed eutectoid Steel A . The line plotted in the figure is based on Eq. (10.1). Laboratory Results and Discussion : Transformation Kinetics 230 800 750 700 650 600 550 500 450 400 1 1 - O Measured Predicted --- o / ° -o -10 u 10 1 10' 1 0 ° 10* Figure 10.11: Measured and predicted C C T transformation start times for the alloyed eutectoid Steel A . The line plotted in the figure is based on Eq. (10.2). 1000 Figure 10.12: Continuous cooling diagram for the 1045 carbon steel. Closed circles repre-sent the start of diffusional transformations; open circles represent the end of diffusional transformations. Laboratory Results and Discussion : Transformation Kinetics 231 500 1000 100 10 1 0.1 o Cooing Rate, C / s Figure 10.13: Transformation start temperature plotted as a function of cooling rate for the 1045 carbon steel. The line plotted in the figure is based on Eq. (10.4). Figure 10.14: Measured and predicted C C T ferrite start time for the 1045 carbon steel. The line plotted in the figure is based on Eq. (10.5). Laboratory Results and Discussion : Transformation Kinetics 232 a. 740 720 h 700 680 660 640 620 600 580 O Measured Predicted 0.1 10 100 Figure 10.15: Measured and predicted C C T pearlite start time for the 1045 carbon steel. The line plotted in the figure is based on Eq. (10.6). I , i , i i 0 100 200 300 T ime , s Figure 10.16: Experimentally determined isothermal dilation-time curves for the alloyed eutectoid steel A . Laboratory Results and Discussion : Transformation Kinetics 233 700 600 500 g 400 300 h This work : A 1 % A 99 % Park and Fletcher : o 1 % • 99 % 200 10 u 10 1 10* 10 J Time, s 10 4 1 0 ° Figure 10.17: IT diagram for the alloyed eutectoid steel A . Closed circles represent the start of diffusional transformations; open circles represent the end of diffusional transfor-mations. • 2 0 0 i i . ' 1 0 ° 10 1 10 2 10 3 10* 10 5 Time, s Figure 10.18: IT diagram for the alloyed eutectoid steel B . Closed circles represent the start of diffusional transformations; open circles represent the end of diffusional transfor-mations. Laboratory Results and Discussion : Transformation Kinetics 234 700 600 of 500 3 cd CD g 400 CD r l u i i—i—i i i 11 h I—AH O h-AH O I—AH I A H / - ^ l i j l - ^ - O • Th is w o r k : -A 1 % -A 99 % P a r k a n d F l e t c h e r : -O 1 % -• 99 % 300 200 10 u 1 0 1 10* 1 0 J T i m e , s 10 4 1 0 ° Figure 10.19: IT diagram for the alloyed eutectoid steel C. Closed circles represent the start of diffusional transformations; open circles represent the end of diffusional transfor-mations. Laboratory Results and Discussion : Transformation Kinetics 235 2.5 2.0 1.5 1.0 0.5 0.0 A e o n = f(T) Fit (no weights) O Steel A • Steel B A Steel C Fit (weighted) • Steel A • Steel B A Steel C 560 580 600 620 640 660 680 700 o Temperature, C a J2 a -2 -6 - 8 -10 -12 Fit (no weights) O Steel A • Steel B A Steel C Fit (weighted) • Steel A • Steel B A Steel C A • a n = f(T) -40 -60 -80 -100 -120 o Undercooling below TA , C -140 (b) Figure 10.20: Kinetic parameters for the pearlitic transformation in the 3 alloyed eutec-toid steels estimated with a 5-parameter, non-linear regression analysis (weighted and non-weighted), (a) n; (b) In b. Laboratory Results and Discussion : Transformation Kinetics 236 ti -10 h -12 -14 o Steel A o Steel B A Steel C Best Fit 60 f = A exp(b/x) A = -5.4 b = 42.8 80 100 120 140 Undercooling below T. Figure 10.21: Kinetic parameter b, as a function of undercooling, for the pearlitic transfor-mation in the 3 alloyed eutectoid steels estimated with a weighted, 5-parameter non-linear regression analysis, for a constant value of n = n = 1.83 ± 0.18. The line plotted in the figure is based on Eq. (10.7). ti 0 -2 -4 -6 -8 -10 -12 -14 -16 This work O Campbell et al. • Park et al. 60 80 100 120 Undercooling below T A , 140 Figure 10.22: Kinetic parameter b, as a function of undercooling, for the pearlitic trans-formation in 3 alloyed eutectoid steels, as estimated in this study (solid line); derived from the regression equations of Park et al. [223] (filled circles); and reported by Campbell et al. [43] (for a eutectoid carbon steel). Laboratory Results and Discussion : Transformation Kinetics 237 3.0 2.5 \-2.0 a 1.5 1.0 0.5 0.0 n = 1.91 i 0.4 Steel A 320 360 400 440 o Temperature , C 480 Figure 10.23: Kinetic parameter n for the bainitic transformation in the alloyed eutectoid steel A , estimated with a 5-parameter, weighted, non-linear regression analysis. - 4 - 8 - 1 0 -12 14 O Steel A Predicted. f = A exp(b/x) A = -42 .6 b = -540.2 240 280 320 360 400 o Undercoo l ing below T A , C Figure 10.24: Kinetic parameter b for the bainitic transformation in the alloyed eutectoid steel A as a function of undercooling, estimated with a weighted, 5-parameter non-linear regression analysis, for a constant value of n = n — 1.91 ± 0.4. The line plotted in the figure is based on Eq. (10.8). Laboratory Results and Discussion : Transformation Kinetics 238 15 jim. Figure 10.25: Microstructure of etched Gleeble specimen (alloyed eutectoid steel A cooled at 58 °C s - 1 ) showing the outline of prior austenite grains. Magnification : 1000 X . 35 Figure 10.26: Measured prior-austenite mean chord length distribution obtained in the alloyed eutectoid steel A Gleeble specimen cooled at 58 °C s - 1 . Laboratory Results and Discussion : Transformation Kinetics 239 T 1 1 I I I I I | 1 1 1—I I I I I | 1 1 1—111 1 I | 1 1 1—I I 1 I I 10"1 10° 101 102 103 0 - 1 C o o l i n g Rate , C s Figure 10.27: Ferrite fraction as a function of cooling rate for the 1045 (open circles) and 1025 (filled circles) [98] carbon steel samples. The uncertainty in the measurements is shown as error bars. Chapter 11 Residual Stress Measurement The residual stress distributions in forced convective quenched IF, 1045 carbon, and alloyed eutectoid steel bars were determined experimentally by means of neutron diffrac-tion. The quenching conditions were given in Table 7.6. Due to the greater depth of penetration of neutrons, complete stress profiles can be measured non-destructively. Diffraction peak profiles were obtained with the E3 neutron diffractometer at the N R U reactor ( A E C L - Chalk River Laboratories) in two separate campaigns. In this chapter, the experimental technique and data reduction adopted are discussed, and the measured strain and stress fields are presented. A discussion on the generation of stresses during quenching is given in Chapter 12. 11.1 Experimental Procedure 11.1.1 Campaign 1 In the first set of experiments, the residual stress distributions in IF and alloyed steel bars were measured. Neutrons of wavelength 2.412 A (1 A = 10 _ 1 nm) were obtained by diffraction from the (113) planes of a squeezed germanium crystal at a take-off angle, 29m, of 90°. This monochromator plane spacing was selected because a wavelength A > 2.34 A and high take-off angle are required to optimize the neutron beam penetration and an-gular resolution of diffraction peaks [224]. The mosaic spread of the monochromator was « 0.2°, which is well-matched to the angular resolution needed in most residual stress 240 Residual Stress Measurement 241 scanning experiments [143]. The typical scattering angle, 26, for the (110) reflection of the material 73°. The intersection of the incident and scattered beams defines the sampling volume. The beams were shaped with slits made of cadmium (which is opaque to thermal neutrons). The resulting sampling volumes were 2 x 2 x 5 m m 3 (for radial and circumferential strain measurements) and 2 x 2 x 10 m m 3 (for axial strain measurements). These sampling volumes were selected to give a good spatial resolution along the radius of the bars while maintaining a volume large enough to achieve an adequate signal intensity. The sample orientation with respect to the incident and diffracted beams for the determination of the axial, radial, and circumferential (hoop) strains is shown in Figure 11.1. The gauge volume was centered on the rotational axis of the spectrometer specimen table by first locating a 2 mm plastic pin on the rotational centre of the table and then translating the incident slit across the neutron beam to find the maximum intensity with the scattering angle set at 90°. The diffracted-beam slit was also optimized in this way. The spatial coordinates of the sample surface were determined by measuring the diffracted intensity as the surface was translated through the gauge volume. The specimens were mounted on a computer-controlled X Y translator. Strain mea-surements were made every 1 mm from the centre along the radius of the specimen, at mid-length, by translating the sample in the appropriate direction. The precision asso-ciated with the radial position of the sampling volume was ± 50 f im . To ensure having the gauge volume inside the specimen, the last radial position was selected to be at 17.5 mm from the centre (the specimen radius is 19.05 mm). Neutrons were detected with a single 3 He detector. A diffraction peak profile with 21 points was obtained by scanning an angular range chosen to get a complete profile. A set of measurements along the bar radius required 7.5 hours (for radial and circumferential scans) and 50 h for (axial scans) to provide diffraction peaks of sufficient counting statis-tics. The corresponding times for the IF steel were 6 and 40 hours, respectively. The Residual Stress Measurement 242 reason for the long counting times is the attenuation of the neutron beam as it travels inside the specimen : the attenuation increases as the sampling volume is translated towards the centre of the specimen. This effect is particularly noticeable in the axial measurements, due to thelonger paths involved (see Figure 11.1 (c)). 11.1.2 Campaign 2 In the second set of measurements, the residual stress distribution in forced convective quenched 1045 carbon steel bars was determined, and measurements on alloyed steel specimens were repeated. Data from the alloyed steel bars (Campaign 1) were difficult to analyze, because of a near-overlap of diffraction peaks from martensite and retained austenite. This difficulty prompted the selection of a diffraction line that did not con-tain any contribution from the austenite retained in the material. Thus, diffraction from (112) planes in the specimens was selected for measurements in Campaign 2. Neutrons of wavelength 1.65 A were obtained by diffraction from the (331) planes of a squeezed germanium crystal at a take-off angle, 29m, of 79°. The mosaic spread of the monochro-mator was « 0.2°. The typical scattering angle, 29, for the (112) reflection of the steel was « 90°. The gauge volume and sample orientation with respect to the neutron beam were the same as described previously for the first series of tests. Strain measurements were made along the radius of the sample, at mid-length, for radii from 0 to 18 mm. As previously described, diffraction peaks in the alloyed steel bar were collected by scanning a single detector through a series of scattering angles and measuring neutron intensity at each angle setting. However, because the diffraction peaks were sufficiently narrow, it was possible to use a multi-wire, position-sensitive detector to acquire data more rapidly. Designed and built at Chalk River Laboratories, the detector has 32 grounded anode wires, suspended vertically, in a single chamber of 3 He gas. Neutrons are captured by the 3 He atom, and a nuclear reaction generates a proton and triton with Residual Stress Measurement 243 high energy. These particles generate a charge cascade that follows a high electric field towards the nearest anode wire. The multiwire detector is set to look through a window in a cadmium mask, which is located as close as possible to the sampling volume, to minimize parallax effects. The array of wires in the detector spans a range of 2.7° in scattering angle, in 32 steps. Thus, the detector collects a complete diffraction peak without any angular motion, so high data-acquisition rates can be achieved. It should be pointed out that this detector was used for the first time in this set of experiments. In order to investigate possible microstructural effects (changes in plane spacing non-related to actual stresses in the material), reference slices approximately 1-2 mm thick (radial direction) x 4 mm 'wide' (circumferential direction) x 20 mm tall were extracted from the steel bars. The slices were cut from a 20 mm tall cylindrical sample as shown in Figure 11.2. The number of slices extracted, and their radial positions, were se-lected based on the hardness distribution across the various specimens (see Figure 9.41). Diffraction peak profiles were obtained for each individual slice in both campaigns. 11.2 Results Typical diffraction peak profiles obtained for the IF, 1045 carbon, and alloyed eutectoid (both campaigns 1 and 2) steel bars are shown in Figures 11.3 to 11.6. As seen in Figure 11.5, diffraction peak profiles for the first set of strain measurements on the alloyed eutectoid steel specimen have the appearance of a double-peak Gaussian curve. This peak profile is the result of a combination of contributions from the (110) B C T martensite and the (111) F C C austenite planes. The retained austenite accounts for the shoulder at the low-angle side of the peak. The measured diffraction peak profiles were fitted with a 5-parameter function con-sisting of a Gaussian peak (defined by its position, width and intensity) superimposed Residual Stress Measurement 244 on a sloping background (defined by its slope1 and y-intercept.). The precision in the peak mean angle and width was ± 0.02 and ± 0.4°, respectively; the precision in the peak intensity, as percentage of the maximum intensity, was typically 5 %. The goodness of fit for the regressions was estimated from the value of x 2 obtained by comparing the measured and predicted intensities [225] : The results of the curve fitting of measured diffraction peak profiles did not show variation of x 2 with radial position, other than that associated with the maximum number of counts; this is an indication that the measurements, and data reduction, were consistent along the radius. Examples of fitted profiles are shown in Figures 11.7 and 11.8 for IF and 1045 car-bon steel bars, respectively. In the figures, the uncertainty in the measured intensities is shown as error bars. The 5-parameter function adopted adequately described the diffraction peaks obtained in both cases. Severe fluctuations in the measured mean an-gle distribution obtained in the first set of measurements in the alloyed eutectoid steel specimen, caused by the presence of retained austenite, prevented any meaningful stress distribution from being deduced from the data. The diffraction peaks obtained for the (112) martensite planes in the alloyed eutectoid steel in Campaign 2 were comparatively broad and weak (see Figure 11.6) but without interference from the retained austenite, resulting in fluctuations in the mean angle along the radial direction. To stabilize the fitting program, an average slope and y-intercept were calculated for each strain com-ponent after a first analysis with the fitting procedure, and the data was subsequently reanalyzed with these fixed values of slope and y-intercept. A typical fit obtained with this procedure is shown in Figure 11.9. The reproducibility of the technique was assessed 1In some cases, the slope of the background was nearly zero. However, the fitting analysis was always done with the 5-parameter function, to retain generality. *2 = £ (Observed - Expected)2 (11.1) Expected Residual Stress Measurement 245 by repeating measurements at a given radial position in the 1045 carbon steel sample. Af-ter fitting the measured diffraction peaks, the same peak parameters (within the quoted uncertainties) were obtained for measurements made at r = 2 and r = 3 mm. The strain component at each radial position was computed by comparing the respec-tive mean scattering angle (20) with a reference angle (260) as follows : sin(0) The average of the mean scattering angles measured for a given stress component (axial, circumferential and radial) was adopted as the reference angle for that component. Figure 11.10 shows the variation of mean scattering angle with radial position for the strain-free reference slices extracted from the 1045 carbon steel specimen. A decrease in the mean scattering angle from —89.903° to RS —89.865° is observed for the two slices extracted near the surface. Notice that this mean scattering angle profile coincides with the microstructure obtained in the quenched bars (Figures 9.39 and 9.40). Given that these slices are essentially strain-free, the change in mean scattering angle can only be attributed to a microstructural effect. This variation in the mean scattering angle gives rise to an isotropic strain of ~ 3.0 ± 0.5 x l 0 ~ 4 mm/mm which must be added to the strains computed with Eq. (11.2) for radial positions r > 17.05 mm (as determined from metallography, see Figure 9.39). In contrast, the mean scattering angle was essentially constant in all the slices extracted from the IF and the alloyed steel specimens. There-fore, no correction of strains calculated using Eq. (11.2) was required. Interestingly, the variation of the estimated F W H M (full width at half of maximum) with radial position in the 1045 carbon steel bar was similar to that observed for the mean scattering angle in the corresponding reference slices. Once the residual strain distributions are know, the residual stresses can be computed as follows. Assuming that the steel bars can be treated as a homogeneous elastic contin-uum, with elastic constants £ ( 1 1 2 ) = 225 G N m - 2 and V(u2) = 0.276, and that the radial, Residual Stress Measurement 246 circumferential, and axial directions correspond to the principal directions of the stress tensor, the principal stress components may be obtained from Hooke's law, which then takes the form : a i i = T ^ £ i i + { l + v)(l-2V) ekkSii'  1 = 3 ( 1 L 3 ) The determination of residual stress distributions via diffraction methods requires a precise characterization of the unstressed state of the material, i.e., either the interplanar spacing or the mean scattering angle. When the determination of these parameters is uncertain, relative to the expected strain values, the overall equilibrium conditions given by elasticity theory can be adopted to characterize the unstressed state of the material [141]. Assuming axisymmetric conditions, the residual stresses normal to any plane must balance : 'azzrdrdz = 0 (11.4) / <roe drdz = 0 (11.5) In addition, at any surface, the stress acting in the normal direction should vanish. Typ-ically, the residual stress is known at only few radial positions and, consequently, a nu-merical procedure needs to be applied to compute the integrals in Eqs. (11.4) and (11.5). In this investigation, there were two factors that made the determination of the mean scattering angle in the unstressed material uncertain : 1) the thickness of the reference slices extracted from the quenched specimens was similar in size to that of the sampling volume, and 2) the neutron beam attenuation observed during internal measurements in the quenched cylindrical specimens did not occur when data was collected from the reference slices. Thus, the following iterative procedure, based on the overall equilibrium conditions, was adopted to calculate the residual stress distribution of each component in the specimens : Residual Stress Measurement 247 1. estimate 0O as the average of measured mean scattering angles; 2. compute the residual stress field using Eqs. (11.2) and (11.3); 3. compute the integrals in Eqs. (11.4) and (11.5); 4. if the integrals approach zero then terminate the iteration, if not, guess a new value of #o and go back to Step 1. The integrals in Eqs. (11.4) and (11.5) were calculated by discretizing the areas normal to the axial and circumferential stresses as shown, schematically, in Figure 11.11. The resid-ual stress acting on each control area was assumed to be constant within that area and a force increment was calculated by multiplying the stress value times the corresponding control area. The resultant force was then computed by adding the force increments. The measured residual strain distributions along the radius of the specimen for the IF, 1045 carbon, and alloyed eutectoid steel bars are shown in Figures 11.12 to 11.14. Typical uncertainty in the measured values of strain was ± 1.3, 2.1, and 2.0 x l O - 4 mm m m - 1 for the radial, circumferential, and axial components, respectively, and is shown in Figures 11.12 to 11.14 as an error bar. The residual axial strain in the IF steel bar (Figure 11.12) increases from a compressive value of - 6 x l O - 4 mm m m - 1 near the surface to a tensile value of ~ + 10 x l O - 4 mm m m - 1 at the centre. The circumferential (hoop) strain was also compressive (-4 x l O - 4 mm m m - 1 ) near the surface and became tensile (+ 2 x l O - 4 mm m m - 1 ) at the centre while the radial strain was tensile near the surface (+ 4 x l O - 4 mm m m - 1 ) , decreased to almost zero and was tensile (+ 3 x l O - 4 mm m m - 1 ) at the centre. The residual axial strain distribution in the 1045 carbon steel bar (Figure 11.13) was qualitatively similar to that observed in Figure 11.12 for the IF steel specimen; however, the magnitudes of the compressive strain near the surface and the tensile strain at the centre were much larger. The circumferential (hoop) strain was compressive (- 12 x l O - 4 mm m m - 1 ) near the surface, increased until reaching a maximum of + 4 x l O - 4 Residual Stress Measurement 248 mm m m - 1 and decreased towards zero at the centre. The radial strain was tensile (+ 20 x l O - 4 mm m m - 1 ) near the surface, increased to a maximum of + 24 x l O - 4 mm m m - 1 and decreased towards zero at the centre. The residual strain distributions in the alloyed eutectoid steel (Figure 11.14) were very different than those obtained for the IF and 1045 carbon steel bars. Both the circumferential (hoop) and axial strain profiles were tensile near the surface, decreasing monotonically to compressive values at the centre. The compressive axial strain at the centre was larger than the circumferential component. The radial strain was slightly tensile near the surface, showed a maximum tensile strain of ~ + 8 x l O - 4 mm m m - 1 at mid-radius and decreased towards zero at the centre. Strains in a given direction can be influenced by stresses acting in other directions through the Poisson's ratio and, therefore, it is difficult to deduce mechanical behaviour from residual strain distributions [143]. The data can be more easily interpreted based on residual stress results. The residual stress distributions, deduced from the strains as described above, along the radius of the IF, 1045 carbon and alloyed eutectoid steel specimens are shown in Figures 11.15 to 11.17. The typical uncertainties in the stress values, deduced by combining the uncertainties in the strain measurements, were ± 40, 50, and 50 M P a for the radial, circumferential, and axial components, respectively. The axial residual stress in the IF specimen (Figure 11.15) was compressive (- 180 MPa) near the surface and tensile (+ 260 MPa) at the centre. The circumferential (hoop) residual stress was also compressive at the surface and tensile at the centre. The radial residual stress was close to zero near the surface and increased towards a tensile value at the centre. As was the case with the measured strains, the axial residual stress profiles obtained with the IF and 1045 carbon steel specimens were similar. The axial residual stress distribution measured in the 1045 carbon steel bar (Figure 11.16) showed a compressive stress (- 430 MPa) near the surface and a tensile stress (+ 500 MPa) at the centre. The circumferential (hoop) residual stress was compressive at the surface (- 350 Residual Stress Measurement 249 MPa), increased to a maximum of + 350 M P a and decreased to a lower tensile value of 200 M P a at the centre. The radial residual stress was tensile near the surface, increased to a maximum of approximately 4- 350 MPa and decreased to a value of + 200 M P a at the centre. In contrast, all the components of the residual stress tensor were tensile near the surface and decreased towards compressive values at the centre in the alloyed eutectoid steel bar (Figure 11.17). Residual Stress Measurement 250 specimen motion incident scattered beam beam (a) (b) Figure 11.1: Specimen orientations with respect to neutron beams to measure (a) radial, (b) circumferential, and (c) axial strain components in force convective quenched steel bars. The gauge volume is represented by the shaded area. For clarity, cross-sectional views are shown. Residual Stress Measurement 251 (c) Figure 11.1: Specimen orientations with respect to neutron beams to measure (a) radial, (b) circumferential, and (c) axial strain components in force convective quenched steel bars. The gauge volume is represented by the shaded area. For clarity, cross-sectional views are shown. Residual Stress Measurement 252 Residual Stress Measurement 253 400 350 h 300 a 3 250 o u a o u CD 200 150 100 50 0 V V 5 D J IF Steel -o r - 17.5 m m • r = 17.0 m m V r = 16.0 m m T r = 15.0 m m D r = 14.0 m m - 7 6 - 7 5 - 7 4 - 7 3 - 7 2 -71 Scat ter ing Angle, degrees -70 -69 Figure 11.3: Measured diffraction peak profiles in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s - 1 at 25 °C. 2000 ri 1000 500 1 > T l 1 i 1 i V V 1045 Steel o r = 16 m m • r = 17 m m v r = 17. 5 m m -T o % v ' ° o • T r = 18 m m v 0 V * ° f 9 • i . i . i . i --92 -91 -90 -89 - 8 8 Scatter ing Angle, degrees Figure 11.4: Measured diffraction peak profiles in a 38.1 mm-dia. 1045 steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. Residual Stress Measurement 254 a 3 © u a o 3 0) Z BOO 700 600 500 400 300 200 100 0 " T " Alloyed Steel o r = 17 m m • r = 15 m m v r = 13 m m - 7 6 - 7 5 - 7 4 - 7 3 - 7 2 -71 - 7 0 Scat ter ing Angle, degrees -69 Figure 11.5: Measured diffraction peak profiles in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C showing the position of the F C C austenite (A(in)) and B C T martensite (M(n 0)) peaks. Data collected during Campaign 1. a 3 o o cl o -p 3 a •z. 900 800 700 600 500 400 300 200 i 1 1 Alloyed Steel o r = 15 m m • r = 16 m m v r = 17 m m »o° o * o» » ° o . , v „ 8s«,v -91 - 9 0 - 8 9 - 8 8 Scatter ing Angle, degrees Figure 11.6: Measured diffraction peak profiles in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2. Residual Stress Measurement 255 o l _ i l i l i i i i i i i l I - 7 5 - 7 4 - 7 3 - 7 2 -71 - 7 0 Scat ter ing Angle, degrees Figure 11.7: Measured and fitted diffraction peak profile in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s - 1 at 25 °C. 2000 o 1 • 1 • 1 • 1 • 1 -92 -91 - 9 0 - 8 9 - 8 8 Scatter ing Angle, degrees Figure 11.8: Measured and fitted diffraction peak profile in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. Residual Stress Measurement 256 900 200 1 ' 1 ' 1 ' 1 ' 1 -92 -91 - 9 0 - 8 9 - 8 8 Scatter ing Angle, degrees Figure 11.9: Measured and fitted diffraction peak profile in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2. -89 .84 i i . | i i i i i i i i i o Set 1 grees - 89 .86 * Set 2 Average 0 o CD T3 co" be a -89.88 / A A < g CD -89 .90 j _ Scatt. Mean - 89 .92 -89 .94 -i i • 1 i i i . 1 . . . . 1 . 0 5 10 15 20 Radia l Pos i t ion , m m Figure 11.10: Mean scattering angle as a function of radial position obtained from the strain-free reference slices extracted from the 1045 carbon steel specimen. Residual Stress Measurement 257 Figure 11.11: Control volumes adopted for force balance calculations (Eqs. (11.4) and (11.5)) : (a) axial component (plan view) and (b) circumferential component. Residual Stress Measurement 258 o Radial -• Hoop -A Axial -3 03 2 4 6 6 10 12 14 16 18 20 Radia l Pos i t ion , m m Figure 11.12: Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C. The uncertainty in the measurements is shown as an error bar. G « 3 T3 a> 03 24 20 16 12 8 4 0 - 4 - 8 -12 -16 -20 -24 o Radial • Hoop A Axial 1 . 1 , 1 , 1 4 6 8 10 Radia l Pos i t ion , m m Figure 11.13: Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. The uncertainty in the measurements is shown as an error bar. Residual Stress Measurement 259 3 S 3 tn CO o Radial • H o o p A Axial Alloyed Steel 2 4 6 8 10 12 14 16 18 20 R a d i a l P o s i t i o n , m m Figure 11.14: Measured radial, circumferential (hoop) and axial residual strains as a function of radial position in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2. The uncertainty in the measurements is shown as an error bar. 5 0 0 h 4 0 0 CD s~ w i-H a 3 "2 cn co « - 3 0 0 ~ i — 1 — i — • — i — • — i — 1 — r IF Steel o Radia l • H o o p A Axial 4 6 8 10 12 14 16 18 20 R a d i a l P o s i t i o n , mm Figure 11.15: Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. IF steel bar quenched in water flowing at 4.8 m s _ 1 at 25 °C. The uncertainty in the measurements is shown as an error bar. Residual Stress Measurement 260 cd CL, CD m "3 !2 tn Hi « 600 400 200 -200 -400 h -600 ~i—1—i—1—i—1—r o Radial • Hoop A Axial _i_ _i_ 0 2 4 6 8 10 12 14 16 18 20 Radial Position, m m Figure 11.16: Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. 1045 carbon steel bar quenched in water flowing at 2.8 m s _ 1 at 50 °C. The uncertainty in the measurements is shown as an error bar. id a. 0> u m "3 tn cp K 600 400 200 -200 -400 -600 0 Radial • Hoop A Axial Alloyed Steel _l i I i I i I i L 0 2 4 6 8 10 12 14 16 18 20 Radial Position, m m Figure 11.17: Measured radial, circumferential (hoop) and axial residual stresses as a function of radial position in a 38.1 mm-dia. alloyed steel bar quenched in water flowing at 2.8 m s _ 1 at 75 °C. Data collected during Campaign 2. The uncertainty in the measurements is shown as an error bar. Chapter 12 Mathematical Analysis of Forced Convective Quenching Results of the numerical (finite-element) simulations of microstructural evolution and stress generation during forced convective quenching of steel bars are presented in this chapter. As a first step in modelling the process, the applicability of the boiling curves (surface heat flux vs surface temperature) estimated using the inverse analysis was as-sessed by comparing predicted and measured temperature responses at the centre and subsurface locations during the quenching of three IF steel bars. Then, the mathematical models were applied to simulate the thermal, microstructural and mechanical responses of selected quenched bars. As mentioned before, the thermal/mi crostructural model was uncoupled from the mechanical module. Thus, the thermal and microstructural responses for the whole time domain were first calculated, and the results adopted as input for the stress model. 12.1 Verification of the Inverse Analysis To verify the results of the inverse analysis (Chapter 5), estimated surface heat fluxes were adopted as the active boundary condition in the finite-element code to calculate the temperature response at the thermocouple locations. The predicted values were then compared against measured temperatures. For this verification, three quenching condi-tions applied to IF steel bars were considered (see Table 12.1). They fall in a diagonal of the test matrix adopted in the forced convective quenching experiments (Chapter 7) and, therefore, are representative of all boiling conditions found during the experimental runs. 261 Mathematical Analysis of Forced Convective Quenching 262 As was the case in the inverse analysis, the bars were assumed to behave as infinitely long cylinders with respect to heat transfer. For the verification runs, the heat evolved during the austenite-to-ferrite transformation was ignored. Thus, the governing equation is given by Eq. (4.30). The computational domain represented one half of a single radial slice (see Figure 12.1). The boundary conditions imposed were : 1) symmetry at the centreline (Eq. (5.11)), and 2) specified heat flux at the surface (Eq. (5.12)). In addition, surfaces perpendicular to the bar axis were considered adiabatic. A schematic repre-sentation of the boundary conditions is shown in Figure 12.2. The initial temperature distribution was assumed to be uniform (Eq. (5.13)). The mesh adopted for the calcu-lations consisted of a row of 21 4-node isoparametric elements as shown, schematically, in Figure 12.3. Note that smaller elements (in the radial direction) were used near the surface, to improve accuracy in that region. The aspect ratio (height/width) of each element varied from 1.33 at the centre to 4 at the surface. These values were within the requirement that the aspect ratio of any given element should not exceed 6 [226]. During each simulation, the surface heat flux was interpolated from a table of es-timated heat flux vs surface temperature values using a cubic spline algorithm [227]. This approach was selected because there is no single equation that can be used to fit all regimes found in the boiling curves obtained in the experiments, and the size of the problem and the efficiency of the interpolation algorithm did not compromise hardware requirements. A disadvantage of this strategy is that parametric studies (see, for example, Marjorek et al. [228]) are difficult to conduct. The thermophysical properties of the IF steel given in Chapter 9 were adopted for the calculations. The calculation time step, A t , was 0.2 s in all cases. A comparison between the numerical (FEM) and measured temperature responses at the centreline and subsurface positions during forced convective quenching of an IF steel bar with water flowing at 4.8 m s _ 1 at 50 °C is shown in Figure 12.4. The boiling curve adopted in the simulation is shown as a broken line in Figure 12.5. During the early stages of Mathematical Analysis of Forced Convective Quenching 263 the quench, the predicted temperature response lagged behind the measured values. It is thought that this behaviour was associated with low values of estimated surface heat flux at the beginning of the quench. To test this hypothesis, the boiling curve was modified in such a way that a minimum value of surface heat flux could be attained immediately (solid line in Figure 12.5). The results of the calculations with the modified boiling curve are shown in Figure 12.6. As can be seen in the figure, the predictive capability of the model improved significantly. It should be pointed out that a similar observation was made by Wiskel [229] when simulating the temperature evolution during the start up phase of aluminum D.C. casting. An alternative solution to this problem would be to acquire data at a much higher rate at the beginning of the quench and then to use smaller time steps in the early stages of the finite-element simulation. Such an approach would require a more flexible code to perform the inverse analysis, so that variable time steps could be selected. Even in that start up value might be needed. The temperature responses during forced convective quenching of IF steel bars with water flowing at 1) 6.9 m s _ 1 at 25 °C, and 2) 2.8 m s _ 1 at 75 °C, were also computed using the approach described above. The boiling curves adopted as boundary condition are shown in Figures 9.27 and 9.25, respectively. For all three conditions simulated, no phase transformation was considered (since only the thermal module was being tested). The predicted and measured temperature responses at the centreline and subsurface positions are shown in Figures 12.7 and 12.8, respectively. Good agreement between predicted and measured temperature responses was also observed for these quenching conditions. For the highest water temperature (75 °C), the boiling curve included the film boiling regime, which resulted in a low rate of heat extraction at temperatures where the austenite-to-ferrite transformation occurs. Since the heat of transformation was ignored in this set of calculations, predicted temperatures in the transformation range were lower than the measured ones. As mentioned above, the three cases considered covered the different boiling regimes found during the laboratory measurements. Thus, Mathematical Analysis of Forced Convective Quenching 264 it can be concluded that the thermal module of the finite-element code, combined with the results of the inverse analysis, correctly predicts the temperature response during forced convective quenching. The sensitivity of the inverse analysis to variables such as thermal conductivity, num-ber of future time steps, and thermocouple position, was presented in Chapter 5. In the following, the results of a sensitivity analysis of the thermal module alone, i.e., when no transformations are included, are presented. When no phase transformations are con-sidered, the evolution of the thermal field depends on the rate of heat extraction at the surface and the ability of the material to transport energy. Accordingly, the variables considered were thermal conductivity and surface heat flux. The case studied was that of forced convective quenching of an IF steel bar with water flowing at 4.8 m s - 1 at 50 °C. The effect of varying the value of the thermal conductivity by ± 10 % on the thermal response at the centre and subsurface of the bar during the quench is shown in Figure 12.9. When a higher value of thermal conductivity was adopted, a faster response to changes in the boundary condition was predicted. The opposite effect was observed when the thermal conductivity was lowered by 10 % with respect to its base value. Given that these results were expected from the physics of the problem, they also serve as an internal consistency check for the model. To study the sensitivity of the thermal response to the magnitude of the surface heat flux, the modified boiling curve shown in Figure 12.5 was allowed to vary by ± 10 % of its base value1. The calculated temperature response at the centre and subsurface positions is shown in Figure 12.10. An increase in surface heat flux resulted in faster cooling, while the opposite effect was observed when a lower value of surface heat flux was adopted. The shape of the temperature vs time curves was not affected, which was expected given that the three boiling curves used in the calculations had the same characteristics. 1 Typical uncertainties quoted for boiling heat transfer correlations are in the order of ± 20 % [89]. Mathematical Analysis of Forced Convective Quenching 265 12.2 Temperature Response and Microstructural Evolution The results of the previous section indicate that the heat-transfer boundary condition at the surface can be applied confidently to predict the temperature response during forced convective quenching. It was then possible to apply this information to study the microstructural evolution in the IF, 1045 carbon and alloyed eutectoid steel specimens prepared for residual stress measurements and metallographic analysis (Chapter 7). To model the temperature response and microstructural evolution during forced convective quenching of the three steel bars, the same mesh and boundary conditions described in the previous section, were used in the finite-element model. The governing equation must include the heat generated during the various transformations (Eq. (4.3)), and boundary conditions of symmetry at the centreline (Eq. (5.11)) and specified heat flux at the surface (Eq. (5.12)). A uniform initial temperature distribution was assumed (Eq. (5.13)). 12.2.1 IF Steel The microstructural evolution during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 at 25 °C was modelled using the finite-element code. The thermophysical properties adopted for the calculations were given in Chapter 9. The incremental time step, At, adopted in the simulation was 0.05 s (the total quench time was 30 s). Due to the limited amount of available data, the kinetics of the austenite-to-ferrite transformation were modelled via an explicit function of measured ferrite fraction transformed as a function of temperature (see Appendix E). The latent heat associated with the austenite-to-pearlite and the austenite-to-ferrite transformations was estimated from the following correlations given by Campbell [188] : A i J 7 _ p = 70651 + 225.23 T - 0.3469 T 2 + 6.755 x l 0 _ 5 T 3 , T > 500 °C (12.1) Mathematical Analysis of Forced Convective Quenching 266 AH 1-F 3.277xl0 6 - 10575.5 T + 11.545 T 2 - 4.244xl0~ 3 T 3 , T > 780 °C -2 .9175xl0 7 + 1.146xl0 5 T - 148.8 T 2 + 0.064 T 3 , 720 < T < 780 °C 2.216xl0 5 - 864.4 T + 1.979 T2 - 1.48xl0- 3 T 3 , T < 720 °C (12.2) where A i 7 7 _ p and AH^-p are the latent heat (J k g - 1 ) of the austenite-to-pearlite and austenite-to-ferrite transformations, respectively. Temperature-independent values of 92 and 82.6 kJ k g - 1 were adopted for the latent heat released by the austenite-to-bainite and austenite-to-martensite transformations [47]. The predicted and measured2 thermal response at the centre and subsurface positions during the quench are shown in Figures 12.11 and 12.12, respectively. Very good agree-ment was observed at the subsurface position, except for temperatures below 150 °C. At the centre, the temperature response was underpredicted for temperatures below ~ 600 °C, which correspond to measured subsurface temperatures below ~ 150 °C. This result suggests that the surface heat flux at the lower temperature end of the boiling curve was overestimated. The corresponding predicted and measured cooling rates at the centre and subsurface are shown in Figures 12.13 and 12.14, respectively. As was the case with the predic-tions of the temperature response, a better agreement was observed at the subsurface. Another feature of interest is the evolution of the temperature gradients inside the bar as the quench progresses, given that thermal strains arise due to non-uniform cooling. Calculated temperature gradients at 5 different times during the quench are shown in Figure 12.15. The surface temperature decreased very rapidly at the start of the quench. After 8 seconds, a large temperature gradient between the surface and the centre had been established. At later times, the temperature gradient decreased steadily to near zero towards the end of the quench. Note that the rate of cooling was much faster at the surface at the beginning of the quench, while the centre cooled more rapidly in the later 2The measured temperature response shown in the plots corresponds to raw values. Mathematical Analysis of Forced Convective Quenching 267 stages. The predicted microstructural evolution at the centre and surface of the IF steel bar is shown in Figures 12.16 and 12.17, respectively. As can be seen in the figures, the austenite transformed completely to ferrite. The results of the metallographic analysis (Chapter 7) also showed 100 % ferrite at all radial positions. Due to the slower cooling rate at the centre, an appreciable difference in the start time for the transformation was predicted between the surface and the centre. The surface started transforming at ~ 1 s after the start of quenching, while the centre started to transform at 8.5 s. 12.2.2 1045 Carbon Steel In the second example, the microstructural evolution during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s _ 1 at 50 °C was simulated. The thermophysical properties of hypoeutectoid carbon steel [188] were adopted for the calculations (Table 12.2). The time step, A t , adopted in the simulation was 0.1 s (the total simulation time was 65 s). To model the transformation kinetics of diffusional transformations, kinetic data reported by Nagasaka et al. [47] for the austenite-to-ferrite, austenite-to-pearlite, and austenite-to-ferrite transformations were adopted. The austenite-to-martensite transformation was modelled using the Koistinen-Marburger equation (Eq. (2.6)). The values of heat of transformation given in Section 12.2.2 were also adopted for this calculation. The predicted and measured thermal response at the centre and mid-radius as the quench progressed are shown in Figures 12.18 and 12.19, respectively. Fair agreement between predicted and measured temperatures was observed, although the model under-predicted the temperature response. Note that, at the centre, the recalescence associated with the diffusional transformations was properly predicted. The corresponding predicted and measured cooling rates at the centre and mid-radius are shown in Figures 12.20 and 12.21, respectively. Fair agreement was observed in both Mathematical Analysis of Forced Convective Quenching 268 cases. The decrease in cooling rate at the centre, just above 600 °C, reflects the heat generated by the diffusional transformations. Calculated temperature gradients at 5 different times during the quench are shown in Figure 12.22. Due to the internal thermal resistance of the material, significant temperature gradients developed at early times. Comparing Figures 12.15 and 12.22, it can be seen that the rate of change of the thermal gradient was lower when a higher water temperature (50 °C) was used for quenching. This is a direct result of the lower rates of heat extraction at the surface obtained under this quenching condition. The metallographic analysis of the quenched bar showed a ring of martensite at the surface (see Figure 9.39). In contrast, the predicted final microstructural distribution showed a mixture of bainite and martensite at the surface. To tune the model to the observed microstructure, the start times for the diffusional transformations were shifted towards larger values, until a ring of martensite comparable in depth to that observed in the quenched sample (Figure 9.39) was obtained. It was found that a shift of the C C T 'nose' from 1.8 to 3.9 s was needed. A different austenite grain size and a slightly different chemical composition could account for this difference. The evolution of the microstructure at the centre and surface of the bar is given in Figures 12.23 and 12.24, respectively. At the surface, martensite started to transform at 9.5 s, and the transfor-mation was essentially complete after 35 s. Lower cooling rates at the centre allowed transformation to ferrite, pearlite and bainite to start at 22, 23 and 25 s, respectively. Martensite started to transform at 30 s. The calculated final microstructural distribution, adopting the modified kinetics, is shown in Figure 12.25. 12.2.3 A l l o y e d Eutec to id Steel The microstructural evolution during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s _ 1 at 75 °C was also simulated using the finite-element code. The thermophysical properties of plain carbon eutectoid Mathematical Analysis of Forced Convective Quenching 269 carbon steel [188] (Table 12.3) were adopted for the calculations. The transformation start times and J M A K (Eq. (2.4)) kinetic parameters derived in Chapter 10 were adopted to model the kinetics of the diffusional transformations. The kinetics of the austenite-to-martensite transformation were modelled using the Koistinen-Marburger equation, Eq. (2.6). The heat of transformation for the reactions was estimated using the same values adopted for the calculations of the IF and 1045 carbon steel specimens (see Section 12.2.1). A time step, At, of 1.0 s (for a total quench time of 135 s) was adopted in the simulation. The predicted and measured thermal responses at the centre and mid-radius during the quench are shown in Figures 12.26 and 12.27, respectively. Good agreement was observed. However, the recalescence associated with the austenite-to-martensite trans-formation was overpredicted by the model. The corresponding predicted and measured cooling rates at the centre and mid-radius of the specimen are shown in Figures 12.28 and 12.29, respectively. Fair agreement was observed in both cases, except in the range of 600 to 700 °C, where the model overpredicted the cooling rates. Given that the bar transformed fully to martensite, this result indicates that the surface heat flux in this temperature range was overestimated. The heat generated by the transformation resulted in a decrease in the cooling rate at ~ 220 °C. Calculated temperature gradients at 5 different times during the quench are shown in Figure 12.30. Since the quench was conducted with water at 75 °C, the rate of change of the temperature profile was the slowest of the three cases considered. The predicted microstructural evolution at the centre and surface of the eutectoid al-loyed steel bar is shown in Figures 12.31 and 12.32, respectively. At the surface, marten-site started to transform after 25 s, and the transformation was essentially complete at 60 s. The lower cooling rates experienced at the centre allowed transformation to a mixture of bainite and martensite. Bainite started to transform at 40 s, followed by martensite at 62 s. The final microstructural distribution is shown in Figure 12.33. The Mathematical Analysis of Forced Convective Quenching 270 model predicted an essentially through-hardened material, which is consistent with the metallographic analysis and hardness measurements. 12.2.4 Sensi t iv i ty Analys i s There are two groups of variables that affect the predictions obtained from the ther-mal/microstructural model. They are related to 1) heat transfer from the specimen, and 2) transformation kinetics. The first group includes such variables as rate of heat extraction, thermophysical properties and specimen geometry. The effect of varying the surface heat flux and thermal conductivity of the bar on the temperature response during forced convective quenching of IF steel bars (when transformation is not considered) has been presented above. To predict transformation kinetics, empirical equations have been used to estimate parameters such as continuous cooling start times and kinetic constants for the J M A K equation, Eq. (2.4). Another variable that plays an important role on the thermal/mi crostructural predictions is the heat released during the transformation. Campbell [188] has studied the effect of transformation kinetics-related parameters on the predicted microstructural evolution during the austenite-to-ferrite and austenite-to-pearlite transformations occurring during air cooling of hypo- and eutectoid carbon steel rods. His results indicated that variations in the transformation start times, within the expected experimental error, have an small effect on the calculated microstructural evo-lution. A n increase in either one of the J M A K kinetic parameters (n and b) resulted in enhanced kinetics, which in turn led to a more rapid release of heat of transforma-tion and, consequently, higher average transformation temperatures. Similar results have been reported by Medina [230] for the cooling of a 1005 carbon steel in a run-out table. To avoid repetition, the sensitivity of the present model to variations of kinetic-related variables for the austenite-to-martensite transformation only, were considered. Mathematical Analysis of Forced Convective Quenching 271 The progress of the austenite-to-martensite transformation is a function of tempera-ture below Ms and is described by the Koistinen-Marburger equation3, Eq. (2.6). Thus, the kinetic variables of interest are the transformation start temperature ( M J and the kinetic constant (A = -0.011). Stress is known to increase the kinetics of the austenite-to martensite reaction by increasing the transformation start temperature. Accordingly, the effect of varying the magnitude of M s was studied by simulating the temperature and mi-crostructural evolution during the quench for values of M s : 1) 20 °C and 2) 40 °C above its measured value under stress-free conditions. The predicted temperature response at the centre and mid-radius are presented in Figure 12.34. The corresponding microstructural evolution is shown in Figure 12.35. By increas-ing M , the reaction started at shorter times (higher temperatures). Referring to Fig-ures 12.28 and 12.29, it can be observed that this resulted in the reaction taking place under higher cooling rates with respect to the predicted conditions when the base value of M g was adopted. Thus, not only the reaction started earlier but also enhanced kinetics at both the surface and the centre were predicted. Note that under conditions of con-stant cooling rate, increasing M s would have resulted in only a shift of the transformation curves towards shorter times but without modifying their shape. The effect of varying the magnitude of the kinetic constant in the Koistinen-Marburger equation on the predicted temperature response and microstructural evolution was in-vestigated by allowing this parameter to take two values that simulated enhanced kinet-ics : 1) -0.015 and 2) -0.020. The calculated temperature response at the centre and mid-radius, and the corresponding microstructural evolution, are shown in Figures 12.36 and 12.37, respectively. Note that the temperature axis in Figure 12.36 has been bro-ken to improve readability. By increasing the kinetic constant, the fraction transformed for a given degree of supercooling, A T , increased. The martensitic transformation is time-independent. However, the temperature at a given location is a function of the 3 E q . (2.6) is : X = 1 - exp[A(M s - T)]. Mathematical Analysis of Forced Convective Quenching 272 local cooling rate and, therefore, there is a direct relationship between supercooling and time. The enhanced kinetics resulted in a larger amount of heat released during the transformation, delaying the start of the transformation at the centre. However, the rate of transformation corresponding to the values of -0.015 and -0.020 was increased to such extent that those reactions were completed at the centre at shorter times than the one predicted using the base value of -0.011. 12.3 Stress Generation The results of the previous section were adopted as input to the mechanical model to predict transient and residual stresses during quenching of the three specimens studied. A row of 4-node elements, like the one used in the previous section, predicts a null axial stress at all radial positions. Thus, a new mesh was created to simulate one-half of the entire bar 4, as shown in Figure 12.38. The mesh consisted of 21 elements in the radial direction (with the same geometry adopted for the thermal/microstructural calculations) and 34 elements in the axial direction. A schematic representation of the computational domain and the thermal boundary conditions applied is shown in Figures 12.39 and 12.40, respectively. Note that the ends of the specimen were assumed to be adiabatic. This is a reasonable assumption, considering the very high rates of heat extraction experi-enced at the surface. The mechanical boundary conditions are schematically shown in Figure 12.41. At the centreline, the symmetry of the specimen dictates that the radial displacement must be set to zero. The front end of the bar was assumed to be free to move in the axial direction. Due to the symmetry of the problem, the only possible rigid-motion was translation in the axial direction. The boundary condition at the back end of the specimen reflects the fact that, as described in Chapter 7, the bar was rigidly held in position by attaching it to a steel tube through an adaptor, restricting both axial 4The symmetry of the problem allows the use of only one half of the specimen. Mathematical Analysis of Forced Convective Quenching 273 and radial displacement at two points in the back end of the specimen. The thermal/microstructural calculations were repeated with the new mesh, and the results adopted as input to the mechanical model. It should be pointed out that the thermal response and microstructural evolution at mid-length were identical to those obtained with the 21 node mesh. In the following, the computed residual stresses are compared to measured values, to assess the predictive capabilities of the model. Then, the mechanisms of stress generation during quenching are illustrated by discussing the evolution of transient stresses. In both cases, only the results at mid-length are presented. Moreover, following a standard practice [231], average stresses were computed at the geometrical centre of the individual elements. 12.3.1 Residual Stresses 12.3.1.1 IF Steel The stress generation during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 at 25 °C was simulated with the mechanical module of the mathematical model. The thermomechanical properties of pure iron given by Mitter et al. [205] were adopted for the calculations, and are given in Table 12.4. The transformation plasticity strain increment was calculated using Eq. (D.4). The At used in the simulation was 0.05 s. The radial, circumferential and axial components of the predicted and measured resid-ual stress distributions are shown in Figures 12.42, 12.43 and 12.44, respectively. Good agreement between predicted and measured residual stress distributions was observed. Near the surface, the model overpredicted (in absolute terms) both the axial and circum-ferential stresses. Two conditions dictated by the equilibrium equations are : 1) the radial residual stress at the surface must be zero, and 2) at the centreline, the circumferential and radial stresses must coincide. This conditions can be used as a consistency check for Mathematical Analysis of Forced Convective Quenching 274 the model. As seen in Figures 12.42 and 12.43, both conditions were correctly predicted by the model. Regarding the predicted residual stress distributions, compressive axial and circumferential stresses were obtained at the surface while tensile stresses were com-puted at the centre. The radial residual stress distribution was zero at the surface and rose steadily to its maximum value at the centre. 12.3.1.2 1045 C a r b o n Steel The model was also applied to simulate the stress generation during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s _ 1 at 50 °C. The thermomechanical properties of 1035 carbon steel given by Nagasaka et al. [47] were adopted for the calculations, and are given in Table 12.5. The poisson ratio, u, was assumed as constant and equal to 0.3 for all phases. The transformation plasticity strain increment was calculated using Eq. (D.4). The At used in the simulation was 0.1 s. The microstructural evolution computed with the modified kinetics was adopted as input for the calculations. The radial, circumferential and axial components of the predicted and measured resid-ual stress distributions are shown in Figures 12.45, 12.46 and 12.47, respectively. The predicted residual stress distributions agree qualitatively with the measured ones. In particular, very good agreement was observed regarding the circumferential component. The radial component was underpredicted near the surface and overpredicted towards the centre. The axial component was underpredicted in the central region. As was the case with the previous simulation, the conditions given by the equilibrium equations were fulfilled by the model (see Figures 12.45 and 12.46). The residual stress distributions were similar to the ones obtained during the simula-tion of quenching of the IF steel, i.e., the axial and circumferential stresses at the surface were compressive while tensile stresses were predicted at the centre. The radial residual stress distribution increased from zero at the surface, reaching its maximum value at the Mathematical Analysis of Forced Convective Quenching 275 centre. 12.3.1.3 Alloyed Eutectoid Steel The residual stress distribution developed after forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s _ 1 at 75 °C was predicted using the mathematical models. There is no data published on the thermomechanical properties of this steel. Instead, the properties of an alloyed steel (3.23 pet Ni-0.83 pet Cr) given by Nagasaka et al. [47] were adopted for the simulation. It should be noted that Henriksen et al. [232] have pointed out that the thermomechanical properties of steels are more sensitive to temperature and the specific phase present than to carbon content. The transformation plasticity strain increment was calculated using Eq. (D.4). A time step, A i , of 0.05 s was used in the calculations. The radial, circumferential and axial components of the predicted and measured residual stress distributions are shown in Figures 12.48, 12.49 and 12.50, respectively. The measured residual stress distributions showed a marked difference when compared to the distributions observed in the quenched IF and 1045 carbon steel specimens. As can be seen in the figures, the model correctly predicted these differences, at least qualitatively. Thus, tensile circumferential and axial stresses were predicted at the surface, while compressive values were obtained at the centre, as a result of the through-hardening of the laboratory test sample. The predicted radial component of the residual stress was zero at the surface, as required by the equations of equilibrium, and became increasingly compressive towards the centre. 12.3.1.4 Sensitivity Analysis Given that the predicted residual stress distributions are based on the thermal and mi-crostructural evolution in the specimen, which are interdependent, and a function of the rate of heat extraction at the surface and the kinetics of the phase transformations, it Mathematical Analysis of Forced Convective Quenching 276 is apparent that a sensitivity analysis based on a factorial design including all variables would result in a very large number of combinations of parameters. Instead, by ana-lyzing the results presented above, it appears that the most significant factor affecting the predicted residual stresses is the microstructural distribution. In particular, the mi-crostructural evolution in the partially hardened 1045 carbon steel quenched bar is the most likely to be miscalculated. Thus, to test the sensitivity of the stress module to microstructural evolution, the residual stress distribution in a 1045 carbon steel bar with the final microstructural distribution predicted based on the kinetic data reported by Nagasaka et al. [47] was computed. The results showed that the model underpredicted the residual stresses. This is a consequence of the smaller amount of martensite predicted by the thermal/microstructural model when the original transformation kinetic data was adopted. The largest volume expansion corresponds to the austenite-to-martensite trans-formation and, consequently, when less martensite was formed, the model predicted lower residual stresses. When the transformation behaviour was modified to account for the microstructure obtained in the quenched sample, much better agreement between the calculated and the measured residual stresses was obtained, as shown in Figures 12.45, 12.46 and 12.47. 12.3.2 Transient Stresses In order to elucidate the role of thermal evolution and phase transformations on stress generation during forced convective quenching, the mechanical response of the bars as the quench progressed was monitored. From the neutron diffraction results, it was evident that the stress field evolved in a similar manner in the unhardened IF and the partially hardened 1045 carbon steel bars, while the behaviour of the through hardened alloyed eutectoid steel was different. Thus, only the stress evolution in the 1045 carbon and the alloyed eutectoid steel were considered. The variation with time of the radial, circumferential and axial components of stress Mathematical Analysis of Forced Convective Quenching 277 at the surface and centre of the alloyed eutectoid steel quenched bar are shown in Fig-ures 12.51 and 12.52, respectively. Note that, as required by the equilibrium equations, the radial and circumferential components at the centre were identical at all times, and the radial component at the surface remained constant and is approximately zero. Also, the stresses did not change appreciably towards the end of the quench, indicating that they represent the residual stress field. At the start of the quench (Figure 12.51), the surface cooled more rapidly than the centre (see Figure 12.30), and the accompany-ing contraction set up tensile stresses which increased steadily from the start of the quench up to point A . At point A (30 s), the surface transformed to martensite (see Figure 12.32). The expansion associated with the transformation overcame the contrac-tion due to cooling, resulting in a reversal in the stress vs time curve between points A and B . The martensitic transformation was essentially completed at point B; from then on, the contraction due to cooling dominated until the end of the quench, resulting in another reversal in the sign of the rate of change of stress. At the start of the quench, the centre (Figure 12.52) did not cool significantly and, therefore, the contraction at the surface was accommodated by the softer core, resulting in compressive stresses until the martensitic transformation started (point A) . At that point, the stresses were reversed and, eventually, became tensile up to the time when martensite started to transform at the centre (point C). The expansion due to the transformation caused another stress reversal, that resulted in compressive stresses. The subsequent cooling did not modify the stress values significantly. Note that the small amount of bainite transformed at the centre (which started to transform at 40 s), had a minimal effect on the stress evolution. The evolution of the radial, circumferential and axial components of stress at the surface and centre of the 1045 carbon steel quenched bar are shown in Figures 12.54 and 12.53, respectively. At early times, the rapid cooling experienced by the surface layers generated tensile stresses at the surface (Figure 12.53), until martensite started to transform (point A) . The volume change produced by the martensitic transformation Mathematical Analysis of Forced Convective Quenching 278 was larger than the contraction caused by cooling and, therefore, the stress increment was negative until the transformation was complete (point B in Figure 12.53). At times greater than thermal contraction, generated a positive stress increment, until the cen-tre started to transform. The accompanying expansion caused another stress unloading at the surface. At the centre, the stresses evolved along a different path. The shrinkage experienced at the surface at the beginning of the quench produced compressive stresses at the softer core. When the surface transformed, the stress at the centre was reversed (point A in Figure 12.54). The positive rate of change of stress at the centre continued as the transformation front travelled through the radius of the bar, until the centre started to transform to ferrite (point C). The diffusional transformations (austenite-to-ferrite,-pearlite, and -bainite) occurred between points C and D. The expansion associated with these transformations resulted in a stress unloading at point C. Once the transformations were completed, the rapid cooling at the centre (relatively to the surface) reversed this trend, until the centre started to transform to martensite (point E), causing another stress unloading. The martensitic transformation was completed at point F. From this time onward, stresses were only generated by the relatively rapid cooling of the core. 12.4 Application to Industrial Conditions The thermal-microstructural module of the mathematical model was also applied to sim-ulate the microstructural evolution in a 100 mm-dia alloyed eutectoid steel bar. The surface heat flux was derived from measurements obtained in the laboratory facility for a 38.1 mm-dia. alloyed eutectoid steel bar quenched from 850 °C with water flowing at 4.8 m s _ 1 at 32 °C. The bar was treated as infinitely long; thus, the computational domain adopted was one half of a radial slice, as in the verification calculations (Figure 12.1). The domain was discretized with a 21 4-node element mesh. The calculational time step, At was 0.5 s. Thermophysical and kinetic data used in the calculations for the 38.1 Mathematical Analysis of Forced Convective Quenching 279 mm-dia. alloyed eutectoid steel bar (Section 12.2.3) were adopted for the simulation. The calculated final microstructural distribution is shown in Figure 12.55. A mixture of pearlite and bainite was predicted in the core while 100 % martensite was predicted at the surface. It should be pointed out that no measurements were conducted under industrial conditions and, therefore, these results should only be considered as qualita-tive. Additionally, there is an uncertainty associated with applying the surface heat flux obtained in the laboratory to the industrial set up. Nonetheless, the microstructural profile shown in Figure 12.55 is consistent with the hardness profile given in Figure 9.42. Mathematical Analysis of Forced Convective Quenching 280 Table 12.1: Quenching conditions simulated in the verification runs of the thermal model. Run Material Water Velocity, m s 1 Water Temperature, °C R U N 11 IF Steel 6.9 25 R U N 10 IF Steel 4.8 50 R U N 16 IF Steel 2.8 75 Mathematical Analysis of Forced Convective Quenching 281 oo 00 03 o S - . c<3 O L O o o co ( -1 03 o ( -1 O H It! O 'cQ >> ^3 O H o C M C M o bO a CD ! - i c$ u 03 a CT1 W a .2 'co CO 03 • S H bO 03 tf ( -1 03 O H O s-. PU Pi 03 CO o O .9 E N o o o o o VI E N VI o E N CM I O i—I X r -o L O CM L O O o o o o VI E N VI o o L O C O bO o .9 E N o o o o o VI E N VI o E N i o 1 — I X L O C O CM O CM + E N co CM C O bO tf 03 +3 o o .9 E N o o o o o VI E N VI o CD I 0 T—I X co c~ CO + CN 1 O 1 — I X CO r— .—i L6 C M CM L O C O I O o o o o VI E N VI o o L O C O bO \> \S \ > \ > V h*H h^H H H h*H H^H .9 .9 .9 .9 .9 E—H E-H E—H E—i E—H O o c-CM L O VI E N VI o O o C M VI E N O o C O VI E N I o X I E N in O i-H X CM O r-T—I L O I E N oo C O o co co i—i C O o L O 7 bO A S tf E N O t~ L O o E N oo co L O + L O o E N C O co CM + C O CM C O o oo VI E N V V V t— r— CT5 CM ( M C O L O E N co •co L O L O r— CO O o r--oo t— A E N X C M L6 + E N oo co co O S L6 + L O C O o o cS 03 PL. 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CN I O i — I X CO oo L O co CO co o L O co E^ co o co <D r 10 0 i — I X C O t ~ co ai + CN 1 o 1—I X CO t— T——I CM L O CM CM L O CO o L O CO t— X + o I—I X CM o l>-I—I L O oo CO o co i E-H 05 o 1—I X I > ->o o + E H oo co L O + L O o • E H CO co + CO CM CO E H co a i , — i OS i — I C O I L O L O t— co E-H a> O i — i X CM L O + E-H oo CO CO °] L O + L O co o o CD a3 CD O H E-H I o 1—I X C O L O + E-H I o X CO L O o CM o L O co o LO u V O H o l-l OH I bO A ! bO A l tf1 I bO I bO AS Mathematical Analysis of Forced Convective Quenching 283 L O o C M 0) CD CO CO _CD HJ ( H CD P H O fH P H o "3 o CD a o S H CD C M CO fl CD O bO fl cc3 tf <D M fl SH CD P H fl .2 fl w fl o CD *-< 1 bO <D tf ? H CD P H O PH O o E-H O o o o o 1—I VI E H V I E H x C O C M T—I 00 C O I E H o 06 X C M O O o CM ce PH co o O o E~H O o o o o 1—I V I V I o o C M E H I o 1—1 X oq C M + E-H C O I—I L O o I LO ^ C M L O C M n3 PH 3> b O o o C M V I E-H E H C O C O o I o O o E H o o o o 1—I V I E-H V I o o co ct3 PH o o c o o C O V E H CN CN E-H E H 1 <N 1 1 0 I 0 1—1 1—1 X X L O L O O O C O C O + + E-H E H 00 co L O 0 a i 1—1 1—1 1 + -tf cn 0 O 1—1 1—1 X X 0 1 1—1 c o C O Mathematical Analysis of Forced Convective Quenching 284 GO Pi o O O o o O o O O rime o .3 o .5 o .5 o .9 o .9 o .9 o .3 o .3 o .3 o .3 Con EN EN EN EN EN EN EN EN EN EN ige o o ige o O O o O o o o O O O ci tf o o o o o o o o o o o o o o O o o o o o o o o o o o o o o o o o o m SH 1—1 o o 1—1  1—1 1—1 o 1—1 o 1—1 1—1 o o 1—1 o 1—1 o 1—1 eratu V I o V I V I V I V I V I V I V I O LO V I V I V I eratu EN V EN EN EN V I LO EN EN EN EN V I o V EN EN EN dm V I o EN V I V I CD V I V I V I EN V I V I V I o o o co o o o o o o CD :arli CD CD LO CD :arli rtensi steni EN EN EN CD PH EN / E N EN H^3 '3 rtensi EN EN d steni t— OJ EN t~ CD" t>- oq EN t— "ci <D EN EN CD rtensi OJ EN t-CT* oj , — i OJ LO OJ oo CM LO PQ LO H ^ co i—i W < i — i co co CO • ^ 1—1 LO CM CO 'in OJ CM 'EH CO oo t — OJ OJ oo SH CM CM LO ]>- SH <V <H-H CO OJ s-l OJ t- i—i a 1 1 T—1 HH 1 1 CM LO CO 1 1—1 1 o i i o 1 I I; O 1 CO o o CO 1 o i ' cn CD i n O o 1 CO (—) i n O i c O I m t—, 1 1 ci i n o I i n O CD i—I T—1 i— I i—I <D CD 1—1 i — 1 U bfj X X CM V X X V 3 o a X LO X CD oo CM cd 7.85: oo co 7.85: ci CD CM cc3 1—1 o t--tf CO CO -OJ 7.85: CD CM r- 7.85: CO CD OJ CO 1—1 . OJ CM OJ T—1 7.85: CM 1>-CM CM 7.85: co o CM i — i i - H >> Properl Properl MPa MPa , MPa MPa MPa , MPa MPa MPa , MPa MPa MPa , MPa b fe b b b Mathematical Analysis of Forced Convective Quenching 285 191 mm Figure 12.1: Computational domain adopted for the finite-element simulation of ther-mal/microstructural evolution during forced convective quenching. Insulated q Insulated Figure 12.2: Schematic representation of the boundary conditions adopted for the ther-mal/microstructural simulations. Mathematical Analysis of Forced Convective Quenching o rjjJ —H UIUI c'o Mathematical Analysis of Forced Convective Quenching 287 T Time , s Figure 12.4: Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. The boiling curve shown as a broken line in Figure 12.5 was adopted as boundary condition. Figure 12.5: Boiling curves adopted as boundary condition to simulate forced convective quenching of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. Mathematical Analysis of Forced Convective Quenching T i m e , s Figure 12.6: Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s - 1 at 50 °C. The boiling curve shown as a solid line in Figure 12.5 was adopted as boundary condition. 0 5 10 15 20 25 30 T i m e , s Figure 12.7: Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 6.9 m s- 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 289 T T i m e , s Figure 12.8: Comparison between predicted and measured temperature responses at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 2.8 m s"1 at 75 °C. T Time , s Figure 12.9: Effect of varying the value of thermal conductivity by ± 10 % adopted in the finite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. Mathematical Analysis of Forced Convective Quenching 290 T T i m e , s Figure 12.10: Effect of varying the surface heat flux by ± 10 % adopted in the fi-nite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. IF steel bar quenched with water-flowing at 4.8 m s _ 1 at 50 °C. 0 10 20 30 40 T i m e , s Figure 12.11: Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s - 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 291 u 3 cd cu 1000 800 h 600 400 200 O Measured FEM IF Steel Subsurface 40 Figure 12.12: Comparison between predicted and measured temperature responses at the subsurface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C. o co" •+J cd « a "3 o o 40 20 0 - 2 0 - 4 0 - 6 0 - 8 0 -100 -120 -140 -160 Measured FEM IF Steel Centre 200 400 600 800 o Temperature , C 1000 Figure 12.13: Comparison between predicted and measured cooling rates at the centre of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 292 20 h - 1 8 0 \-0 200 400 600 800 1000 o T e m p e r a t u r e , C Figure 12.14: Comparison between predicted and measured cooling rates at the sub-surface of a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C. 1000 3 600 200 0 1 t = 1 1 1 1 0 s -t = 8 s -12 s -t = 16 s t = 29 s * 1 , 1 , 1 , 0 5 10 15 Radia l Pos i t ion , m m Figure 12.15: Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. IF steel with water flowing at 4.8 m s _ 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 293 T3 d a o o 6 -1.0 0.8 0.6 0.4 0.2 0.0 Ferr i te Pearl i te Bainite Martensite IF Steel Centre 10 20 T i m e , s 30 Figure 12.16: Calculated microstructural evolution at the centre during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 at 25 °C. CP d u a o '£ o c « S-, 1.0 0.8 0.6 0.4 0.2 0.0 Ferr i te Pearl i te Bainite Martensite IF Steel Surface _ i _ 6 8 10 T ime , s 12 14 Figure 12.17: Calculated microstructural evolution at the surface during forced convective quenching of a 38.1 mm-dia. IF steel bar with water flowing at 4.8 m s _ 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 294 a) u CD ft 1000 800 600 400 200 o Measured FEM Figure 12.18: Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. 0 u 3 CC u CB CM S CO E-i 1000 800 600 400 200 ~i ' r 1045 Steel Mid—Radius o Measured FEM <>-<"> O O P o 50 60 70 Figure 12.19: Comparison between predicted and measured temperature responses at mid-radius of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 - 1 at 50 °C. m s Mathematical Analysis of Forced Convective Quenching 295 40 SO 0 O-' o o - 2 0 - 4 0 h -60 - 8 0 h -100 Measured FEM 1045 Steel Centre 200 400 600 800 o Temperature , C 1000 Figure 12.20: Comparison between predicted and measured cooling rates at the centre of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. 40 20 a a O - 2 0 - 4 0 -60 -80 -100 1045 Steel M i d - R a d i u s 200 400 600 800 o Temperature , C 1000 Figure 12.21: Comparison between predicted and measured cooling rates at mid-radius of a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s - 1 at 50 °C. Mathematical Analysis of Forced Convective Quenching 296 CO P. s CO E-1000 800 600 400 200 1 1 i t = 1 0 s t = 10 s -20 s -30 s -t = 65 s """-•-...^ 1 , 1 , 1 , 5 10 15 Radia l Pos i t ion , m m Figure 12.22: Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s - 1 a t 5 0 ° C . d o o CO 1.0 0.8 h 0.6 0.4 0.2 0.0 1045 Steel Centre 10 Ferr i te Pearl i te Bainite Martensite 20 30 40 T i m e , s 50 60 70 Figure 12.23: Calculated (with modified kinetics) microstructural evolution at the centre during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 2.8 m s"1 at 50 °C. Mathematical Analysis of Forced Convective Quenching 297 1.0 a a u H a o •rH o 0.8 0.6 0.4 h 0.2 h 0.0 Ferri te Pearl i te Bainite Martensite 1045 Steel Surface 10 20 30 40 50 60 70 T i m e , s Figure 12.24: Calculated (with modified kinetics) microstructural evolution at the surface during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar with water flowing at 4.8 m s - 1 at 50 °C. Radial Posi t ion, m m Figure 12.25: Calculated (with modified kinetics) final microstructural distribution in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 4.8 m s _ 1 at 50 °C. Compare with Figure 9.39. Mathematical Analysis of Forced Convective Quenching 298 Figure 12.26: Comparison between predicted and measured temperature responses at the centre of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s- 1 at 75 °C. n CO H 800 600 400 200 O Measured — FEM _1_ 20 40 60 80 100 120 140 T i m e , s Figure 12.27: Comparison between predicted and measured temperature responses at the subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s- 1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 299 cd K bo a "3 o o 40 20 -20 h -40 - 6 0 Measured FEM Alloyed Steel Centre 200 400 600 o Temperature, C 800 Figure 12.28: Comparison between predicted and measured cooling rates at the centre of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. cd « be el 40 20 \--20 -40 - 6 0 Measured FEM Alloyed Steel M i d - R a d i u s 200 400 600 o Temperature, C 800 Figure 12.29: Comparison between predicted and measured cooling rates at the subsur-face of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s"1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 300 800 t = 0 s t = 25 s 600 400 ... t = 35 s 200 t = 65 s t = 135 s 5 10 15 Radial Position, mm Figure 12.30: Calculated temperature gradients at five different times during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s- 1 at 75 °C. 1.0 h 0.8 % 0.6 a 0.4 0.2 0.0 Ferrite Pearlite Bainite Martensite Alloyed Steel Centre _ l _ 20 40 60 80 Time, s 100 120 140 Figure 12.31: Calculated microstructural evolution at the centre during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s _ 1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 301 CD a o a cd u 1.0 r-0.8 h •B 0.6 h cd 1H H 0.4 0.2 0.0 Alloyed Steel Surface — Ferr i te — Pearl ite Bainite — Martensite _i_ 20 40 60 80 T ime , s 100 120 140 Figure 12.32: Calculated microstructural evolution at the surface during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar with water flowing at 2.8 m s _ 1 at 75 °C. 5 10 15 Radial Posi t ion, m m Figure 12.33: Calculated final microstructural distribution in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. Compare with Figure 9.38. Mathematical Analysis of Forced Convective Quenching 302 Figure 12.34: Effect of varying the value of Ms adopted in the finite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s - 1 at 75 °C. Figure 12.35: Effect of varying the value of Ms adopted in the finite-element simulation on the microstructural evolution at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 303 900 BOO 700 300 A = - 0.011 A = - 0.015 A = - 0.020 CU s cu E-i 200 100 Centre Mid—Radius Alloyed Steel 20 40 60 80 Time, s 100 120 140 Figure 12.36: Effect of varying the value of the kinetic constant in the Koisti-nen-Marburger equation adopted in the finite-element simulation on the temperature response at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. XI a> 6 u o <** CD d C8 1.0 0.8 0.6 h .2 0.4 0.2 0.0 Alloyed Steel Surface A = - 0.011 A = - 0.015 A = - 0.020 i 100 120 140 Time, s Figure 12.37: Effect of varying the value of the kinetic constant in the Koisti-nen-Marburger equation adopted in the finite-element simulation on the microstructural evolution at the centre and subsurface of a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 304 19.05 mm Figure 12.38: Finite-element mesh adopted to simulate the thermal/microstructural evo-lution and stress generation during quenching. Mathematical Analysis of Forced Convective Quenching 305 T 38.1 mm JL 191 mm H Figure 12.39: Computational domain adopted for the finite-element simulation of stress generation during forced convective quenching. Figure 12.40: Schematic representation of the boundary conditions applied for the ther-mal/mi crostructural simulations. Mathematical Analysis of Forced Convective Quenching 306 Figure 12.41: Schematic representation of the boundary conditions applied for the stress simulations. Figure 12.42: Comparison between predicted and measured residual stress distribution (radial component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching 307 Figure 12.43: Comparison between predicted and measured residual stress distribution (circumferential (hoop) component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s _ 1 at 25 °C. Figure 12.44: Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. IF steel bar quenched with water flowing at 4.8 m s- 1 at 25 °C. Mathematical Analysis of Forced Convective Quenching cd a. S 600 400 200 f -200 -400 -600 T 1 i 1 i 1 i 1 r Radial Stress o Measured FEM 1045 Steel 308 2 4 6 8 10 12 14 16 18 20 Radia l Pos i t ion , m m Figure 12.45: Comparison between predicted and measured residual stress distribution (radial component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C . Modified kinetics. td PH td 3 xi CO Ct> PH 600 400 200 -200 -400 -600 1045 Steel Hoop Stress • Measured F E M 0 2 4 6 8 10 12 14 16 18 20 Radia l Pos i t ion , m m Figure 12.46: Comparison between predicted and measured residual stress distribution (circumferential (hoop) component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s _ 1 at 50 °C. Modified kinetics. Mathematical Analysis of Forced Convective Quenching 309 CM CO in 600 h 400 200 -200 -400 -600 h ^ A A Axial Stress A M e a s u r e d F E M 1 0 4 5 S t e e l 0 2 4 6 8 10 12 14 16 18 20 R a d i a l P o s i t i o n , m m Figure 12.47: Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. 1045 carbon steel bar quenched with water flowing at 2.8 m s"1 at 50 °C. Modified kinetics. a, P CD « 600 400 h 200 -200 -400 -600 1 — 1 — i — 1 — r Radia l Stress o Measured FEM Al loyed Steel j i i i i i i i i i i_ 0 2 4 6 8 10 12 14 16 18 20 R a d i a l P o s i t i o n , m m Figure 12.48: Comparison between predicted and measured residual stress distribution (radial component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s _ 1 at 75 °C. Mathematical Analysis of Forced Convective Quenching 310 tS CU td •a cn cu « 600 h 400 200 -200 -400 -600 Hoop Stress • Measured FEM Alloyed Steel _L_ 6 8 10 12 14 16 18 20 Radia l Pos i t ion , m m Figure 12.49: Comparison between predicted and measured residual stress distribu-tion (circumferential (hoop) component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s - 1 at 75 °C. tS cu cS CU 600 400 200 -200 -400 -600 h Axial Stress A Measured FEM Alloyed Steel 4 6 8 10 12 14 16 18 20 Radia l Pos i t ion , m m Figure 12.50: Comparison between predicted and measured residual stress distribution (axial component) in a 38.1 mm-dia. alloyed eutectoid steel bar quenched with water flowing at 2.8 m s"1 at 75 °C. I Mathematical Analysis of Forced Convective Quenching 311 CM 1000 800 600 400 200 0 -200 -400 -600 -800 -1000 -1200 -1400 CT r CT z CT. Alloyed Steel Surface _l_ _l_ _ l_ 20 40 60 80 100 120 140 Time , s Figure 12.51: Evolution of radial, circumferential (hoop) and axial stresses at the surface during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar. 140 Figure 12.52: Evolution of radial, circumferential (hoop) and axial stresses at the centre during forced convective quenching of a 38.1 mm-dia. alloyed eutectoid steel bar. Mathematical Analysis of Forced Convective Quenching 312 CM 0) u 400 300 200 100 0 -100 -200 -300 -400 -500 -600 -700 a r CT z 0" = 1045 Steel Surface 10 20 30 40 50 Time , s 60 70 Figure 12.53: Evolution of radial, circumferential (hoop) and axial stresses at the surface during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar. Figure 12.54: Evolution of radial, circumferential (hoop) and axial stresses at the centre during forced convective quenching of a 38.1 mm-dia. 1045 carbon steel bar. Mathematical Analysis of Forced Convective Quenching 313 T ' I 1 I 1 I 1 I ' V J _ 1 I 1 L ^ J I 1 Radial Posi t ion, m m Figure 12.55: Calculated final microstructural distribution in a 100 mm-dia. alloyed eutectoid steel bar quenched from 850 °C with water flowing at 4.8 m s _ 1 at 32 °C. Chapter 13 Summary and Conclusions The research programme was undertaken to study, under laboratory conditions, an in-dustrial operation aimed at producing grinding rods of improved abrasion resistance and toughness. The motivation to manufacture a product of these characteristics, lies on the fact that grinding constitutes the greatest operating cost in mineral processing [2]. The proprietary heat treatment process developed at AltaSteel is based on forced con-vective quenching of steel rods, and produces a material with a martensitic shell and a bainitic/pearlitic core. In order to optimize the process, as well as to assist in future developments, it is desirable to generate a predictive tool that would obviate the need for expensive and time-consuming trial-and-error operations. The work presented involved a comprehensive study of heat transfer and stress gen-eration during forced convective quenching of steel bars and was conducted under the framework of microstructural engineering. Mathematical models of heat transfer, phase transformations, and elasto-plastic stress were applied to predict the thermal response, microstructural evolution and stress generation during the quench. Once debugged and tested, the mathematical models were validated by comparing calculated results with values measured in the laboratory. The effect of internal stresses on transformation ki-netics was not considered. Thus, the models could be effectively uncoupled, i.e., the results of the thermal/microstructural evolution model were adopted as input to the stress generation model. The problem at hand is highly nonlinear and, therefore, a numerical solution needed 3 1 4 Summary and Conclusions 315 to be implemented. In structural analysis, the finite-element method (FEM) has become a de facto standard (due to its ability to handle components of complex geometry) and was adopted for the analysis. Based on geometric considerations, an axisymmetric for-mulation was deemed adequate. A finite-element program developed to simulate thermal stresses in fused-cast monofrax-s refractories [182] was used as a basis for the tran-sient thermal/microstructural model, while a computer program developed for the time-independent, elasto-plastic analysis of stress evolution in water spray-quenching of steel bars [47] was adopted to model internal stress generation. In the thermal/microstructural model, the additivity principle was applied to compute the amount of fraction trans-formed under continuous cooling conditions. Diffusional transformations (austenite-to-ferrite,-pearlite, and -bainite) were described using the J M A K equation. The martensitic transformation was modelled using the Koistinen and Marburger equation. The mechani-cal model includes thermal- and transformation-related strains. In particular, strains due to volume changes associated with cooling and phase transformations, as well as those arising from transformation plasticity, variation of the elastic constants with tempera-ture, and variation of the flow stress with temperature were included. The mechanical properties were obtained from the literature. A critical component of the modelling exercise is the characterization of the heat transfer boundary condition. Due to a lack of data in the literature, for the conditions of interest, the surface heat flux, as a function of surface temperature, was estimated from the measured temperature response of instrumented samples through the solution of the inverse heat conduction problem (IHCP). An existing computer program [183] that implements the sequential function specification technique, was modified for this purpose. The layout and operation of the industrial equipment allowed little flexibility for in-strumentation to measure the thermal response of the steel rods during the quenching cycle. This information is needed to estimate the heat transfer boundary condition. Summary and Conclusions 316 Thus, a laboratory facility was designed and built to simulate the industrial operation. The objectives of the laboratory experiments were twofold : 1) to characterize the surface heat flux as a function of surface temperature, under forced convective boiling conditions, and 2) to produce specimens for metallographic characterization and residual stress mea-surement (to validate the mathematical models). A test matrix consisting of three water velocities (2.8, 4.8 and 6.9 m/s) and three water temperatures (25, 50 and 75 °C) was adopted to obtain the boiling curves using IF steel as test material. Additional tests were conducted using alloyed eutectoid and 1045 carbon steels. The effect of an oxide layer was also studied. The kinetics of the austenite decomposition in 3 alloyed, near-eutectoid steels for a range of continuous cooling and isothermal conditions were determined using the G L E E -B L E 1500 thermomechanical simulator. The transformation kinetics of continuously cooled IF and 1045 carbon steel samples were also measured. The transformation studies were carried out using thin-wall tubular samples (6 mm I.D., 8 mm O.D.). The aim of the isothermal and continuous cooling tests was to describe the transformation in terms of the Avrami equation and to determine the transformation start times, respectively. The results of the heat transfer measurements showed that, for a given water velocity, the surface heat flux is higher as the water temperature decreases. The total amount of heat extracted is, therefore, higher as the water temperature decreases (subcooling increases). The boiling curves obtained for water flowing at 25 and 50 °C showed sim-ilar features. In these cases, no film boiling stage could be identified, i.e., the boiling curves immediately reached the transition stage. The similarity in the boiling curves was particularly noticeable for the intermediate (4.8 m s _ 1 ) and high (6.9 m s - 1 ) water ve-locities investigated. In contrast, the boiling curves corresponding to quenching in water at 75 °C did show a film boiling stage. Thermal responses in clean, 'lightly' and 'heavily' oxidized specimens were not significantly different. Summary and Conclusions 317 To produce specimens for residual stress measurements and metallographic charac-terization, bars of IF, 1045 carbon and alloyed eutectoid steel were quenched in the lab-oratory facility. The residual stress distributions in selected IF, 1045 carbon and alloyed eutectoid steel quenched bars were measured using neutron diffraction. The variation of measured axial and circumferential residual stresses with radial position, in the IF and 1045 carbon steel quenched bars, were similar. In both cases, they were compressive at the surface and tensile at the centre, while the radial component was always tensile. On the other hand, the three components obtained in the alloyed eutectoid steel bar were compressive at the centre. Metallographic analysis of the IF steel bar showed that the austenite had completely transformed to ferrite. The alloyed eutectoid steel exhibited a fully martensitic structure, while the 1045 carbon steel showed an outer ring of martensite and a core consisting of a mixture of diffusional and martensitic products. The measured hardness distributions were consistent with the observed microstructures. Comparisons between measured and model-predicted thermal responses, final mi-crostructure and residual stress distributions have been made. It was found that the position of the 'nose' of the continuous cooling diagram, when a mixture of diffusional and martensitic products was present, has a significant influence on the predicted final microstructure distribution and, therefore, on the predicted residual stress distribution. The difference in the measured residual stress distributions obtained in the alloyed eu-tectoid steel specimen, when compared with those found in the IF and 1045 carbon steel quenched bars, has been explained based on the sequence of transformations that took place during the quench. The models were also applied to obtain a qualitative description of the microstructural evolution under industrial conditions. Given the strong effect of water temperature on boiling heat transfer, it is recom-mended that the water temperature be monitored and controlled tightly in the industrial operation. The difference between inlet and exit water temperatures should be kept to a minimum in order to prevent significant differences in mechanical properties along the Summary and Conclusions 318 length of the bars. Possible modifications to the process can be envisioned, and are now described. An optimum cooling rate that minimizes transient internal stresses, and consequently distortion, while producing the desired final microstructural distribution consists of slow cooling at high temperatures, to minimize thermal stresses, rapid cooling at intermediate temperatures to achieve the targeted microstructural changes, and slow cooling at lower temperatures. At the present time, the inlet water temperature and ve-locity are kept constant throughout the quench; one action that could be taken would be to vary water temperature and/or velocity to modify the cooling path. Another option would involve the use of oil or aqueous polymeric solutions as quench medium; however, issues of recyclability and cost would need to be considered. The quench time controls the maximum temperature attained during self-tempering. Currently, the water flow is stopped completely after a given quench time; alternatively, the water flow rate could be reduced at a controlled rate to modify the self-tempering conditions. Progressive induction hardening, where the workpiece is moved at a constant speed through a coil and cooling ring, has been used to produce parts with a hard surface with compressive stresses and a tough core with tensile stresses. The process is based on the interaction betweeen eddy currents generated by the magnetic field induced by the coil, and the workpiece. The heat area is confined to few millimeters below the surface. This thin layer is subsequently quenched by the cooling ring. For highly alloyed steels, air could be used as the quench medium. Future work should include temperature measurements under operational conditions in order to devise a strategy to scale the laboratory results to the plant operation. 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