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The solidification of ductile cast iron Boeri, Roberto Enrique 1989

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THE SOLIDIFICATION OF DUCTILE CAST IRON By ROBERTO ENRIQUE BOERI Ingeniero Mecanico, Universidad Nacional de Mar del Plata, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDDZS Department of Metals and Materials Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA November 1989 ©Roberto Enrique Boeri  In presenting this thesis in partial fulfilment degree  of the  requirements for an advanced  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 The University of British Columbia Vancouver, Canada  DE-6 (2/88)  - ii -  ABSTRACT  The microsegregation of Mn, Cu, Cr, Mo, Ni and Si has been measured in cast ductile iron and in ductile iron which has been quenched when partially solidified. Effective segregation coefficients have been determined for each of the elements, and used to calculate the segregation on the basis of the Scheil equation. The calculated values agree reasonably well with the values of the solute concentration as a function of the solid fraction measured in quenched samples. The microstructure of the solid phases during the solidification of ductile iron has been observed. Solidification of eutectic ductile iron begins with the independent nucleation of austenite and graphite in the melt. Later the graphite nodules are enveloped by austenite, and further solidification takes place by the thickening of the austenite layers enveloping the graphite. Isolated pockets of interdendritic melt are the last material to solidify. On the basis of the measured segregation of the different alloying elements, the mechanisms by which the segregation affects the microstructure are considered, and an explanation for the effect of segregation on the hardenability of ductile iron is proposed. A mathematical model of the solidification of eutectic ductile iron is formulated which includes heat flow, nucleation and growth of graphite nodules, and the segregation of Si. The model uses equilibrium temperatures given by the ternary Fe-C-Si equilibrium diagram. Using the mathematical model, cooling curves, nodule  - iii -  count and nodular size distribution are determined as a function of position in the casting sample. The results are compared to measured temperatures, nodule count and nodule size in rod castings of 12.5, 20 and 43mm radius. There is good agreement between the calculated and measured values for the 43mm radius rod, and not quite good agreement for the rods of smaller radii. The changes in solidification predicted by the model when some solidification parameters are varied are consistent with experimental observations with the same variation in the parameters.  iv  TABLE OF CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS  iv  LIST OF TABLES  viii  LIST OF FIGURES  ix  LIST OF SYMBOLS  xvii  ACKNOWLEDGMENT  xxi  1  INTRODUCTION  1  2  LITERATURE REVIEW  6  2.1  Cast Iron Microstructure During Solidification 2.1.1  2.2  Summary  Segregation in Cast Iron 2.2.1  2.3  Summary  Mathematical Modelling of Solidification 2.3.1  2.4 2.4.1  Summary  6 8 9 15 15 31  Cooling Curves  32  Summary  38  3  OBJECTIVES OF THE PRESENT RESEARCH  40  4  EXPERIMENTAL METHODS AND APPARATUS  46  4.1  Melting  46  4.2  Sampling  47  4.3  Casting and Temperature Recording  51  4.4  Optical Metallography  55  4.5  Electron Metallography and Microanalysis  55  V  5  SEGREGATION AND MICROSTRUCTURE RESULTS AND DISCUSSION  60  Segregation in Cast Samples 5.1.1 Segregation Pattern in the Vicinity of a Graphite Nodule 5.1.2 Quantitative Values for Segregation in Cast Samples 5.2 Analysis of Quenched Samples 5.2.1 Microstructure of Quenched Samples 5.2.2 Measurements of Solute Concentration as a Function of the Fraction Solid 5.2.3 Estimation of Partition Coefficients of the Alloying Elements 5.3 Analysis of Segregation Results 5.3.1 Comparison of Measured Segregation with Calculations Based on the Scheil Equation 5.3.1.1 Solute Concentration in the Liquid During Solidification Using k,. 5.3.1.2 Solute Distribution in the Solid Using k,. 5.3.1.3 Solute Distribution in the Solid Using 1^ 5.3.1.4 Solute Concentration in the Liquid During Solidification using  60 63 71 73 73 80  5.1  5.3.1.5 Analysis of the Fit Between Calculations and Experiments 5.3.2 Correlation Between the Solidification Structure and the Segregation Pattern Around Nodules 5.3.3 Comparison of the Segregation in Sand-Cast and Quenched Samples 5.4 Effects of the Segregation on the Microstructure of Ductile Iron 5.4.1 Influence of Microsegregation on the Cast Structure 5.4.2 Influence of Solute Segregation on the Hardenability of Cast Irons 6  SOLD3D7ICATION MODEL 6.1  Thermal Model 6.1.1 Assumptions and Boundary Conditions 6.1.2 Heat Conduction Equations 6.1.3 Initial Conditions 6.1.4 Surface Heat Transfer Coefficient at the Metal-Mould Interface  88 93 100 101 107 115 121 126 128 134 134 135 139 141 142 142 142 146 147  vi  6.2  Model For Graphite Nucleation  151  6.3  Growth Model  155  6.3.1  Growth of Graphite in Contact with the Melt  155  6.3.2  Growth of Austenite  159  6.3.3  Growth of Graphite Enveloped by Austenite  160  6.3.4  Calculation of the Fraction Solid and the Release of Latent Heat  163  6.3.5  Calculations of Nodular Size Distribution  165  6.4  Segregation Model  166  6.5  Selection of Material Properties  170  6.5.1  Thermophysical Properties of Ductile Iron  170  6.5.2  Sand Properties  171  6.5.3  Other Properties  173  6.6 Solidification Model 7  M O D E L RESULTS AND APPLICATION 7.1  Sensitivity Analysis  174 178 178  7.1.1  Influence of the Mesh Fineness  178  7.1.2  Influence of the Time Step  179  7; 1.3 7.1.4  Influence of the Initial Temperature of the Melt Selection of Parameters  179 179  7.2  Verification of the Heat Transfer Model  182  7.3  Analysis of the Sensitivity of the Models of Nucleation and Growth  183  7.3.1  Exponential Nucleation  185  7.3.2  Parabolic Nucleation  189  7.4  Model Output  195  7.5  Comparison of the Model Results and Calculations  202  7.5.1  Casting of 86mm Diameter Rods  203  7.5.1.1 Exponential Nucleation  203  7.5.1.2 Parabolic Nucleation  211  7.5.2  Casting of 40mm Diameter Rods  216  7.5.2.1 Exponential Nucleation  216  7.5.2.2 Parabolic Nucleation  220  7.5.3  Casting of 25mm Diameter Rod  223  vii  7.5.3.1 Exponential Nucleation  223  7.5.3.2 Parabolic Nucleation  227  7.5.4  8  Discussion  230  7.6  Application of the Model  232  7.7  Discussion  238  SUMMARY AND CONCLUSIONS  242  REFERENCES  245  Appendix 1  254  Appendix 2  257  Appendix 3  260  Appendix 4  263  Appendix 5  273  Appendix 6  279  Appendix 7  289  Appendix 8  295  viii  LIST OF TABLES TABLE I:  Data concerning cooling curves of cast iron  39  T A B L E II:  Composition of charge materials  47  TABLE HI:  Charge constitution  48  TABLE IV:  Data of fifteen microprobe measurements of elemental standards and test samples  59  TABLE V:  Alloying element content in the ductile irons examined  62  TABLE VI:  Segregation in sand-cast ductile iron  72  TABLE VTJ:  Local Mn concentration in quenched samples  84  TABLE VIII:  Local concentration of Cu in quenched samples  85  TABLE EX:  Local concentration of Cr, Mo, Ni and Si in quenched samples  TABLE X:  Measured and published values of effective segregation coefficients  89  TABLE XI:  Fit factor, F, for calculations based on kj and k^.  127  TABLE XII:  Measurements of austenite shell radius (after[12])  164  TABLE XIII:  Parameters used in the model calculations  204  TABLE XTV:  Variation of cooling curves and nodule counts as a function of the pouring temperature  233  TABLE XV:  Assumed values of mould density and thermal conductivity  236  87  ix  LIST OF FIGURES Figure 2.1:  Manganese concentration versus silicon concentration, for two different section sizes, after [33]  11  Figure 2.2:  Change of equilibrium partition coefficients of some elements with carbon content in Fe-C base alloys, after [34]  12  Figure 2.3:  Partition coefficients of a third element between austenite and liquid iron. Markers indicate experimental values. Lines show calculations. After[35]  14  Figure 2.4:  Cooling curves for varied number of eutectic cells and cooling rate, after Fras [29].  22  Figure 2.5:  Comparison of simulated and measured cooling curves, after [28].  24  Figure 2.6:  Simulated nodular size distribution and Wetterfall's data. After [28].  25  Figure 2.7:  Measured and calculated cooling curve for the center of a 50mm diameter gray iron casting, after [38]  27  Figure 2.8:  Measured and calculated cooling curves for gray iron, after [42].  29  Figure 2.9:  Cooling curves corresponding to different positions within a cylindrical casting, (a) experimental, (b) calculated, after [43].  30  Figure 2.10:  Celling curve illustrating characteristic temperature points.  33  Figure 2.11:  Cooling curves of different cast iron types, after [ 14].  Figure 2.12:  Cooling curves for various types of cast irons poured in a sand cup, after [44].  36  Figure 2.13:  Temperature of eutectic undercooling recorded at the center of cylindrical ductile iron castings, as a function of the section size.  37  Figure 2.14:  Length of the eutectic plateau at the center of cylindrical ductile iron castings, as a function of the section size.  38  Figure 3.1:  Stable and metastable eutectic temperatures of cast iron as a function of the silicon content, after [49].  44  Figure 3.2:  Influence of the silicon content on the eutectic region of the Fe-C-Si equilibrium diagram, after [47].  45  Figure 4.1:  Schematic of the plunger.  49  Figure 4.2:  Sampling and quenching procedure.  50  Figure 4.3:  Schematic of the mould.  51  Figure 4.4:  Position of the thermocouples.  53  Figure 4.5:  Schematic of the long cylindrical mold.  54  34  X  Figure 5.1:  Representation of the solute concentration along a line between graphite nodules in ductile iron, a) k > 1; b) k < 1  61  Figure 5.2:  Representation of equiaxed cellular growth in ductile iron.  62  Figure 5.3:  Schematic of analysis along lines between nodules.  63  Figure 5.4:  Solute concentration along lines 1 to 4. (a) Si, (b) Cu, (c) Mn.  66  Figure 5.5:  Si and Mn segregation along lines between nodules, (a) line 1, (b) line 2, (c) line 3.  68  Figure 5.6:  Solute concentration along line between nodules.  69  Figure 5.7:  Qualitative composition profile along a line between nodules.  70  Figure 5.8:  Concentration of Cu and Mn along circular path around a nodule.  70  Figure 5.9:  Quenched liquid (x 1000)  75  Figure 5.10:  Structure of quenched sample (x 40)  76  Figure 5.11:  Quenched liquid at the bottom of the sample (x 500)  75  Figure 5.12:  Microstructure of quenched sample for solid fraction 18% (x 100)  77  Figure 5.13:  Microstructure of quenched sample for solid fraction 67%; a) (x 100); b) (x400)  78  Figure 5.14:  Microstructure of quenched sample for solid fraction 94% (x 100)  77  Figure 5.15:  Microstructure of quenched sample for solid fraction 100% (x 100)  79  Figure 5.16:  Location of the microstructure at which solute concentration was measured (x500).  Figure 5.18:  Effective segregation coefficient as a function of solid fraction for Mn (a), Cu (b), Cr (c), Mo (d) and Ni (e).  92  Figure 5.19:  (a) Microstructure of a sample quenched during solidification, graphite nodule A is enveloped by austenite, which has transformed into martensite during sample preparation, (b) Detail of nodule A and surrounding solid. Note grooves left by SIMS scans.  95  Figure 5.20:  C dot map for area shown in Figure 5.19 (b). Note that horizontal lines showing low density of points correspond to grooves left by the SIMS line scans.  96  Figure 5.21:  C line scan between points A and C in Figure 5.19 (b).  Figure 5.22:  Microstructure of cast sample. Note austenite patch D.  Figure 5.23:  C map for: (a) Area shown in Figure 5.22. Note correspondence between nodules and high point density zones, (b) Top right comer of Figure 5.23.  80  97 96 98  xi  99  Figure 5.24:  C line scan along lines indicated in Figure 5.22. Vertical axis offset.  Figure 5.25:  Mn concentration in liquid as a function of solid fraction, Mn=1.34%.  102  Figure 5.26:  Mn concentration in liquid as a function of solid fraction, Mn=1.05%.  103  Figure 5.27:  Mn concentration in liquid as a function of solid fraction, Mn=0.73%.  103  Figure 5.28:  Mn concentration in liquid as a function of solid fraction, Mn=0.41%.  104  Figure 5.29:  Cu concentration in liquid as a function of solid fraction, Cu=1.36%.  104  Figure 5.30:  Cu concentration in liquid as a function of solid fraction, Cu=0.50%.  105  Figure 5.31:  Cu concentration in liquid as a function of solid fraction, Cu=0.91%.  105  Figure 5.32:  Cr concentration in liquid as a function of solid fraction, Cr=0.50%.  106  Figure 5.33:  Mo concentration in liquid as a function of solid fraction, Mo=0.83%.  106  Figure 5.34:  Ni concentration in liquid as a function of solid fraction, Ni=0.83%.  107  Figure 5.35:  Mn concentration in solid as a function of solid fraction, Mn=1.34%.  109  Figure 5.36:  Mn concentration in solid as a function of solid fraction, Mn=1.05%.  109  Figure 5.37:  Mn concentration in solid as a function of solid fraction, Mn=0.73%.  110  Figure 5.38:  Mn concentration in solid as a function of solid fraction, Mn=0.41%.  110  Figure 5.39:  Cu concentration in solid as a function of solid fraction, Cu=1.36%.  111  Figure 5.40:  Cu concentration in solid as a function of solid fraction, Cu=0.50%.  111  Figure 5.41:  Cu concentration in solid as a function of solid fraction, Cu=0.91%.  112  Figure 5.42:  Mo concentration in solid as a function of solid fraction, Mo=0.83%.  112  XII  Figure 5.43:  Cr concentration in solid as a function of solid fraction, Cr=0.50%.  113  Figure 5.44:  Si concentration in solid as a function of solid fraction, Si=2.45%.  113  Figure 5.45:  Ni concentration in solid as a function of solid fraction, Ni=0.83%.  114  Figure 5.46:  Schematic showing expected solute concentration in liquid and solid.  114  Figure 5.47:  Mn concentration in solid as a function of solid fraction, Mn=1.34%.  116  Figure 5.48:  Mn concentration in solid as a function of solid fraction, Mn=1.05%.  116  Figure 5.49:  Mn concentration in solid as a function of solid fraction, Mn=0.73%.  117  Figure 5.50:  Mn concentration in solid as a function of solid fraction, Mn=0.41%.  117  Figure 5.51:  Cu concentration in solid as a function of solid fraction, Cu=1.36%.  118  Figure 5.52:  Cu concentration in solid as a function of solid fraction, Cu=0.50%.  118  Figure 5.53:  Cu concentration in solid as a function of solid fraction, Cu=0.91%.  119  Figure 5.54:  Mo concentration in solid as a function of solid fraction, Mo=0.83%.  119  Figure 5.55:  Cr concentration in solid as a function of solid fraction, Cr=0.50%.  120  Figure 5.56:  Ni concentration in solid as a function of solid fraction, Ni=0.83%.  120  Figure 5.57:  Mn concentration in liquid as a function of solid fraction, Mn=1.34%.  121  Figure 5.58:  Mn concentration in liquid as a function of solid fraction, Mn=1.05%.  122  Figure 5.59:  Mn concentration in liquid as a function of solid fraction, Mn=0.73%.  122  Figure 5.60:  Mn concentration in liquid as a function of solid fraction, Mn=0.41%.  123  Figure 5.61:  Cu concentration in liquid as a function of solid fraction, Cu=1.36%.  123  Figure 5.62:  Cu concentration in liquid as a function of solid fraction, Cu=0.50%.  124  xiii  Figure 5.63:  Cu concentration in liquid as a function of solid fraction, Cu=0.91%.  124  Figure 5.64:  Cr concentration in liquid as a function of solid fraction, Cr=0.50%.  125  Figure 5.65:  Mo concentration in liquid as a function of solid fraction, Mo=0.83%.  125  Figure 5.66:  Ni concentration in liquid as a function of solid fraction, Ni=0.83%.  126  Figure 5.67:  Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  130  Figure 5.68:  Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  131  Figure 5.69:  Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  132  Figure 5.70:  Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  133  Figure 6.1:  Casting system, showing the assumed boundary conditions for the thermal model.  Figure 6.2:  Schematic of the volume elements arrangement.  147  Figure 6.3:  (a) Motion of the mold and casting during the soldification of ductile iron in a sand mold, (b) Cooling curves for the same casting in (a), after [67].  148  Figure 6.4:  Calculated heat transfer coefficient (a) with imperfect contact interface; (b) with gap formation, after [43].  150  Figure 6.5:  Nucleation rate in heterogeneous nucleation.  153  Figure 6.6:  Growth rate of the graphite spheroids as a function of time, for interface controlled, (a) and (b), and diffusion controlled growth, (c) and (d).  158  Figure 6.7:  Correction factor applied to the growth rate of graphite enveloped by austenite.  161  Figure 6.8:  Graphite nodule enveloped by austenite.  162  Figure 6.9:  Concentration of C as a function of the solid fraction in: -austenite in equilibrium with graphite, Ca/g; -austenite in equilibrium with liquid, Ca/1; -liquid in equilibrium with austenite, Cl/a; -liquid in equilibrium with graphite, Cl/g.  167  Figure 6.10:  Difference in the austenite C concentration at the austenite/liquid and austenite/graphite interfaces.  169  Figure 6.11:  Specific heat of ductile iron as a function of temperature.  172  143  xiv  Figure 6.12:  Thermal conductivity of silica sand as a function of the temperature, measured by two different methods (after [70]).  173  Figure 6.13:  Flow chart of program SOLI.  175  Figure 6.14:  Flow chart of subroutine FRACSOL.  177  Figure 7.1:  Influence of the number of nodes selected in the casting on the solidification time.  180  Figure 7.2:  Influence of the number of nodes selected in the sand mold on the solidification time.  180  Figure 7.3:  Influence of the time step on the solidification time.  181  Figure 7.4:  Influence of the initial temperature of the melt on the cooling of the center of the casting.  181  Figure 7.5:  Analytical and numerical calculations of the cooling of a solid cylinder.  183  Figure 7.6:  Calculated cooling curve.  184  Figure 7.7:  Model calculations for different values of the constant b. a) Cooling curves; b) Nodule counts.  186  Figure 7.8:  Model calculations for different values of the constant c. a) Cooling curves; b) Nodule counts.  187  Figure 7.9:  Model calculations for different imposed cooling rates, a) Cooling curves; b) Nodule counts.  188  Figure 7.10:  Model calculations for different values of the nucleation constant a. a) Cooling curves; b) Nodule counts.  190  Figure 7.11:  Model calculations for different values of the exponent n. a) Cooling curves; b) Nodule counts.  191  Figure 7.12:  Model calculations for different imposed cooling rates, a) Cooling curves; b) Nodule counts.  192  Figure 7.13:  Model calculations for different values of the critical nucleation supercooling, a) Cooling curves; b) Nodule counts.  193  Figure 7.14:  Model calculations for different values of the segregation coefficient of Si. a) Cooling curves; b) Nodule counts.  194  Figure 7.15:  Calculated cooling curves at points distant 0,10, 21, 33 and 43 mm from the casting centre.  197  Figure 7.16:  Calculated temperature distribution at different times from pouring.  198  Figure 7.17:  Calculated transformation kinetics at points distant 0,10,21, 33 and 43 mm from the casting centre.  199  Figure 7.18:  Calculated number of nodules per unit volume as a function of the distance from the casting axis.  200  XV  Figure 7.19:  Calculated nodular size distribution at the center, mid-radius and edge of a 86mm diameter casting.  201  Figure 7.20:  Calculated and measured cooling curves for casting C12.  206  Figure 7.21:  Calculated and measured cooling curves for casting C13.  207  Figure 7.22:  Variation of the nodule counts as a function of the distance from the casting centre. Markers show measurements.  207  Figure 7.23:  Calculated graphite volume distribution, (a) centre, (b) mid radius, (c) near the edge.  Figure 7.24:  Measured graphite area distribution on casting C12. (a) centre, (b) mid-radius, (c) near the edge.  209  Figure 7.25:  Measured graphite area distribution on casting C13. (a) centre, (b) mid-radius, (c) near the edge.  210  Figure 7.26:  Calculated and measured cooling curves for casting C12.  212  Figure 7.27:  Calculated and measured cooling curves for casting CI3.  213  Figure 7.28:  Variation of the nodule counts as a function of the distance from the casting centre. Markers show measurements.  214  Figure 7.29:  Calculated graphite volume distribution, (a) centre, (b) mid radius, (c) near the edge.  215  Figure 7.30:  Calculated and measured cooling curves for casting C14.  217  Figure 7.31:  Calculated and measured cooling curves for casting C15.  217  Figure 7.32:  Variation of the nodule counts as a function of the distance the casting centre. Markers show measurements.  Figure 7.33:  Calculated graphite volume distribution, average.  218  Figure 7.34:  Measured graphite area distribution on casting C14.  219  Figure 7.35:  Measured graphite area distribution on casting C15.  219  Figure 7.36:  Calculated and measured cooling curves for casting C14.  221  Figure 7.37:  Calculated and measured cooling curves for casting CI5.  221  Figure 7.38:  Variation of the nodule counts as a function of the distance the casting centre. Markers show measurements.  Figure 7.39:  Calculated graphite volume distribution, average.  Figure 7.40:  Calculated and measured cooling curves for casting C14.  224  Figure 7.41:  Calculated and measured cooling curves for casting C15.  224  Figure 7.42:  Variation of the nodule counts as a function of the distance the casting centre. Markers show measurements.  Figure 7.43:  Calculated graphite volume distribution, average.  225  Figure 7.44:  Measured graphite area distribution on casting C14.  226  Figure 7.45:  Measured graphite area distribution on casting C15.  226  208  from  from  218  222 222  from  225  xvi  Figure 7.46:  Calculated and measured cooling curves for casting C14.  228  Figure 7.47:  Calculated and measured cooling curves for casting C15.  228  Figure 7.48:  Variation of the nodule counts as a function of the distance from the casting centre. Markers show measurements.  229  Figure 7.49:  Calculated graphite volume distribution, average.  229  Figure 7.50:  Temperature of eutectic undercooling as a function of the section size, a) exponential nucleation formulation; b) parabolic nucleation formulation; c) experimental.  232  Figure 7.51:  Nodule counts as a function of the casting radius.  234  Figure 7.52:  Nodule counts at the mid-radius as a function of the casting radius, for different values of the nucleation constant.  235  Figure 7.53:  Solidificationtimeas a function of the mould factor.  237  Figure 7.54:  Nodule counts at the center, mid-radius and near the edge of a casting of 86mm diameter, as a function of the mould factor.  Figure A3 -1:  Melt solidified from one end.  261  Figure A4-1:  Axial volume element.  264  Figure A4-2:  Internal volume element of the casting.  265  Figure A4-3:  Volume element at the surface of the casting.  267  Figure A4-4:  Volume element at the internal surface of the mould.  268  Figure A4-5:  Internal volume element of the mould.  269  Figure A4-6:  Volume element at the mold surface in contact with the copper coil.  271  Figure A4-7:  Volume element at the free surface of the mold.  272  Figure A5-1:  Schematic of the eutectic region of the Fe-C-Si equilibrium diagram for a given Si concentration.  274  Figure A5-2:  Curves describing the Fe-C-Si diagram for 2.5% Si.  277  238  LIST O F SYMBOLS  A = atomic weight a = constant b = constant c = constant C = initial solute concentration in the liquid 0  C = concentration of solute in the liquid phase L  Cp = specific heat at constant pressure C = concentration of solute in the solid phase s  C" = carbon concentration in liquid equilibrated with austenite y  C  ,,gr  = carbon concentration in liquid equilibrated with graphite  C^ = carbon concentration in austenite equilibrated with graphite F  C '= carbon concentration in austenite equilibrated with liquid y  C = carbon concentration in graphite gr  D = diffusion coefficient DNU = number of graphite nuclei fs = fraction solid F(%) = fit factor g = solid fraction h = surface heat transfer coefficient H = heat generation J = correction factor ji = solute concentration calculated by the Scheil equation = solute concentration measured by EPMA k = heat transfer coefficient ko = equilibrium segregation coefficient kg = effective segregation coefficient ke,. = fraction solid dependent effective segregation coefficient L = latent heat of solidification  n = constant N = number of measurements for a given alloy sample N = nucleation rate Q = heat flux r = radius r = initial radius 0  REAG = ratio between austenite and graphite radius RHA = rate of heat accumulation RHG = rate of heat generation RFfl = rate of heat input RHO = rate of heat output RNU = size of graphite nuclei s = radius of austenite shell S = diameter from volume from which X-rays are generated t = time tf = local solidification time T = temperature TAL = temperature of austenite liquidus T  A S  = temperature of austenite solidus  T = eutectic temperature E  T  G L  = temperature of graphite liquidus  T = critical supercooling for nucleation N  V = energy of incident electrons (KeV) V = absorption edge of the element analyzed (KeV) k  VGR = total graphite volume VGR'= total graphite volume at room temperature X* = mean value of the sample of n elements Z = atomic number of the specimen (3 = constant y = constant y'= constant 8 = Boltzman's constant  AT = supercooling AG  D  = activation energy for diffusion of attorns accross the interface melt/nucleus  AG* = activation energy for nucleation u. = mean value of the distribution p = density a = standard deviation a* = standarized standard deviation y = Plank's constant  Dedicada a mi Familia (To my Family)  xxi  ACKNOWLEDGMENT  I would like to thank Dr. Fred Weinberg for his advice and encouragement during the course of this work. Help from Peter Musil, Mary Mager and Laurie Fredrick is gratefully acknowledged. This investigation has been part of a joint research project on cast iron technology between the Department of Metals and Materials Engineering of UBC, CANMET, and LEMIT (Argentina). The financial support from IDRC, Canada, is gratefully acknowledged.  -1-  Chapter 1  INTRODUCTION  Cast iron is one of the oldest engineering materials employed by man, its origin going back to the second century BC. Cupola furnaces for producing cast irons have been in use in Europe since the 14 century, and some operational practices applied then are th  still in use today [1]. Gray cast iron is normally used for applications where high strength and ductility are not required. Foundries were often associated with machine shops, providing the cast blocks from which machine components were produced. In the past thirty years there has been major improvements in the properties of the cast iron produced, with increased strength, toughness and reliability. Cast iron has now replaced cast steels and forged steels in some applications, with very significant savings in component costs. Some critical components, such as gears for automotive transmissions, are now made from heat treated ductile iron. Containers for nuclear waste disposal -a critical application- are now made from ferritic ductile iron [2]. The improvement in cast iron properties comes in part  Chapter 1 : INTRODUCTION  -2-  from better control of the quality and reliability of flake iron. The higher strength ductile cast iron follows the discovery and development of a method to produce spheroidal graphite cast iron about forty years ago [3,4]. At the present time, cast iron is primarily produced in electric furnaces. It is probably the cheapest metallic material available for structural and machinery applications. Large amounts of cast iron are produced; more than 37 million tons in 1986, including over 8 million tons in North America [5]. Further increases in the production of high grade cast iron is expected to occur in the next decade, as this material continues to replace steel. Cast iron has unique characteristics, which makes it one of the most complex alloys used in metal casting. For example, depending upon the cooling rate and chemical composition, cast iron can solidify in the manner defined by the stable Fe-C equilibrium diagram, in which an austenite-graphite eutectic is formed. Alternatively, it can solidify under non-equilibrium conditions defined by the metastable Fe-C equilibrium diagram, in which the eutectic formed is of austenite and cementite, forming at a different eutectic temperature and composition than the equilibrium eutectic. Even when the cast iron solidifies under equilibrium conditions, the morphology of the graphite which forms can vary widely, from flakes, to compacted or vermicular, and spheroidal or nodular shapes, depending on the melt chemistry and cooling rate. The mechanical properties of the cast iron are strongly dependent on the size, shape and distribution of the graphite. The process by which the shape and size of the graphite is controlled is called inoculation. In this process specific alloys such as ferrosilicon, magnesium bearing ferrosilicon or other inoculants are added to the melt just prior to casting. The main  Chapter 1: INTRODUCTION  -3-  objective usually is to increase the number of heterogeneous nucleation sites in the melt, increasing the number of graphite particles, and to control the shape of the particles. The effectiveness of the inoculants decrease withtimeafter being added to the melt, which makes analysis and control of the process difficult. Numerous studies have been carried out related to the production and properties of cast iron. In particular, advances have been reported in the following areas: (a) Control of graphite morphology. (b) Inoculation alloys and the inoculation process. (c) Control of the matrix microstructure through alloying. (d) Heat treatment. (e) Moulding design and casting practice. On the basis of the results of these investigations, cast iron foundries can produce, routinely, castings of good quality with the desired structure and mechanical properties. However there are still many aspects of the solidification of cast irons which are not understood and cannot be defined, probably due to the complexity of the solidification process in these materials. For example, although many studies of the nucleation and growth of graphite have been reported in the literature [6-12], and a number of theories formulated, no theory is properly validated, nor is one generally accepted [13]. In addition, although it has been shown that cooling curves contain information that can be used to determine the casting microstructure [14-20], and empirical methods have been developed to use cooling curves for this purpose, there is no theoretical basis on which the cooling curves can be related to specific structural features in the cast iron.  Chapter 1 : INTRODUCTION  -4-  In recent years heat transfer mathematical models have been developed which quantitatively define the solidification process in castings, using numerical methods and computer calculations. The calculations are directed towards the design of molds, risers and chills, and the selection of casting parameters to optimize the casting quality, reduce or eliminate shrinkage porosity, and improve the efficiency of the casting process. The models provide solutions more efficiently, accurately and economically than empirical data based on extensive temperature measurements. However the models require quantitative data on the physical characteristics of all the constituents in the system, some of which are often unknown, and details of the solidification process between solidus and liquidus temperatures which are generally not known. In general mathematical models calculate the isotherms throughout the casting as a function oftime.The calculated temperatures at a given position can then be compared to temperature measurements at the specified point to verify the model, at least at that point. However in cast iron it is equally important to be able to predict the local microstructure, including the graphite size and morphology, and the matrix structure and composition. In addition it is important to know the residual stress after solidification and cooling. Current heat transfer mathematical models do not provide this information; their results are generally confined to predicting local cooling curves. Information concerning the local structure and residual stress in a casting determined by calculations would allow design engineers to optimize casting configurations using the predicted properties. The microstructure in a casting of cast iron is complex and depends on a number of factors. These include the local cooling conditions, melt chemistry, nucleation and morphology of graphite, and in particular solute segregation during solidification. In general the extent of local segregation of the multiple solutes in cast iron as the material  Chapter 1:  INTRODUCTION  -5-  solidifies is not known. Following solidification, solid state diffusion will occur as the cast iron cools, which is rapid for carbon which diffuses interstitially. The solute distribution is the major factor determining the relative amounts of ferrite and pearlite present in the matrix. Solute segregation in cast iron has been examined and reported in the literature. However, the studies have generally been directed toward the characterization of precipitate phases in areas of high solute concentration. Elements have been identified which have positive or negative segregation. However, the basic mechanisms governing the segregation process have not been identified, and the relationship of the segregation pattern with the microstructure of the matrix has not been established. The present investigation was undertaken to experimentally determine the local segregation of the solute elements in cast iron during solidification, and the relationship between the segregation and the microstructure. This information would provide the data necessary to derive equations which could quantitatively describe the segregation during solidification. In the second part of the present investigation, a mathematical model of the solidification of ductile iron of eutectic composition is developed, which combines models of the heat transfer, the nucleation of graphite, the growth of eutectic phases, and the segregation of Si. The model is solved numerically, and the solutions compared with experimental results.  6  Chapter 2  LITERATURE REVIEW  2.1 C A S T I R O N M I C R O S T R U C T U R E D U R I N G  SOLIDIFICATION  In general the evolution of the microstructure of cast iron as it solidifies has been examined by rapidly quenching partially solidified samples and observing the structure of polished and etched sections of the quenched specimens. Wetterfall et al[12] examined the eutectic solidification of Fe-C-Ni alloys inoculated with Mg. Solidification started with the growth of austenite dendrites and the formation of graphite nodules in the melt between the dendrite branches. For samples which were quenched early in the solidification process, nodules were observed which had nucleated and grown in direct contact with the liquid phase. Samples that were quenched later during solidification had nodules which were completely enveloped in an austenite shell. Itofuji et al.[21] studied the growth of graphite in vermicular and spheroidal graphite cast irons. Small specimens were quenched at selected temperatures during the solidification. Obsevation of the sectioned and etched surfaces of the specimens showed that the austenite phase grew dendritically, and that some graphite nodules were  Chapter 2: LITERATURE REVIEW  -7-  entrapped by the dendrites. J. Su et al.[22] examined the appearance of the interfaces between the liquid iron, the austenite phase and the graphite phase in quenched ductile iron grown unidirectionally. They reported that the austenite phase grew dendritically. They observed some graphite nodules, having diameters up to 40 microns, in direct contact with the quenched liquid, without an austenite envelope. D. Stefanescu and C. Kanetkar [23] showed that spheroidal graphite solidifies in a cellular manner, with cells being formed by graphite spheroids enveloped by an austenite shell. J.C. Heindrix et al.[24] and D. Stefanescu [25] examined directionally solidified cast iron, treated with cerium, varying the thermal gradients and freezing rates during solidification. They reported that, for a given Ce content, it is possible to produce spheroidal, compacted, or flake graphite iron, depending on the cooling rate imposed. The solid-liquid interface, in the case of spheroidal graphite iron, is reported to be irregular, and eutectic colonies consisting of single graphite nodules enveloped by austenite are observed growing ahead of the solid interface. A. Rickert and S. Engler [26] examined the solidification morphology of cast irons using quenching, flow-out and tracer techniques to determine the solid/liquid interface. Their results indicated that the structure in ductile iron is dominated by austenite dendrites, with graphite nodules forming initially between the dendrites. R. Hummer [27] found that during the eutectic growth of gray cast iron, graphite flakes and austenite appeared to grow in a coupled manner, forming eutectic cells. For spheroidal graphite cast iron this was not the case, in that the graphite grew independent  Chapter 2: LITERATURE  REVIEW  -8-  of the austenite. The nodules of graphite were observed to nucleate and grow directly from the melt. As they grew, a solid layer of austenite formed arround the nodules. Further growth was then dependent on the diffusion of C through the austenite shell. Comparing the solidification of flake graphite and spheroidal graphite irons it was noted that for flake graphite a solid layer of material, skin formation, develops at the melt surface early in the solidification process. For spheroidal graphite, the dendritic growth of the austenite phase extends the mushy zone such that skin formation only occurs towards the end of solidification. In the heat transfer mathematical models of the solidification of cast irons [28,29,30], it is generally assumed that the solidification of eutectic ductile iron is cellular, each cell consisting of a single spherical graphite nodule enveloped by a solid shell of austenite.  2.1.1  SUMMARY  For flake graphite cast iron there is general agreement that coupled growth of the graphite flakes with the austenite phase occurs forming eutectic colonies. In the case of ductile iron, the growth process is not clear. The majority of the reported observations indicate that the graphite nodules nucleate and grow in the melt initially, followed by the formation of a solid austenite shell surrounding the nodule. The austenite grows dendritically, independent of the graphite nodules, in the first part of the solidification process. The results of other studies suggest that the solidification of eutectic ductile iron is cellular, each cell consisting of a single spherical graphite nodule enveloped by a shell of austenite.  Chapter 2: LITERATURE REVIEW  -9-  2.2 S E G R E G A T I O N I N C A S T I R O N Solute redistribution during solidification results in microsegregation of the alloy components. The microsegregation leads to inhomogeneities in the microstructure, which markedly influence the physical and chemical properties of the cast iron. Alloying elements such as Mn, Cu, Ni, Cr and Mo are frequently added to cast irons. Particular elements are added to improve mechanical and corrosion resistance; to give required levels of hardenability; to improve the graphitization; and to control the microstructure in the cast product. All of the added elements segregate during solidification to some degree. As a result, the composition of the melt as solidification progresses can deviate markedly from the initial melt composition. For the case of Mn, Cr and Mo, concentration of these elements in the residual liquid, as a result of segregation, can lead to carbide formation. The presence of carbides in the microstructure reduces the ductility of the cast iron, and markedly reduces its machinability. In addition, when these cast irons are heat treated to improve their mechanical properties, the local variation in concentration of the segregated elements produces variations in the hardenability of the material, and poorer overall behaviour of the casting. The segregation of alloying elements in cast iron has not been examined extensively. N. Datta and N. Engel [31] studied the distribution of Si, Cu, Mn, Ni, Mo and Cr between the austenite and carbide phases during the isothermal transformation of ductile iron using electron probe microanalysis. Their qualitative observations indicated that graphitizing elements Si, Cu and Ni tend to segregate to the austenite, and the carbide stabilizing elements, Mo, Cr and Mn, tend to segregate to the carbide phase  Chapter 2: LITERATURE REVIEW  -10-  P. Liu and C. Loper [32] used electron probe microanalysis to study the distribution of P, Mo, Mn, Cr, V, Si, Cu, and Ti in as cast ductile iron of several section sizes. The study was devoted to the quantification of the chemical composition of phases precipitated in the intercellular regions. They found that carbide promoting elements tend to concentrate in the residual liquid phase, and graphite promoting elements tend to concentrate in the austenite, in agreement with the results of Datta et al.[31]. It was also found that both Ti and P segregate strongly to the intercellular regions. In a recent publication, K. Hayrynen et al. [33] compared the segregation of Si, Ni, Cu, Mn and Mo in heavy section ductile iron with that in one inch sections. The results indicated that more extensive segregation occurred in the heavier sections. The ratio between the concentration of two different alloying elements at a specific location of the microstructure, appeared to be predictable, and not very sensitive to the section size, as shown in Figure 2.1, for Mn and Si. The observations indicated that Mn and Mo segregate more extensively than Si, Cu and Ni. The segregation of an alloying element is primarily related to the segregation coefficient k, which is the ratio of the solute concentration in the. solid at the solid/liquid interface, to the solute concentration in the liquid at the interface, under equilibrium conditions. For most binary and ternary alloys, equilibrium partition coefficients can be determined from equilibrium phase diagrams. In the solidification of cast irons which contain Fe, C, Si and alloying elements, quaternary and higher phase diagrams are required to determine the equilibrium segregation coefficients of the components in the system. These phase diagrams are not available. In addition, the specific temperatures at which the phase transformations occur in these complex alloys are not known. As a result  Chapter 2: LITERATURE REVIEW  2.0  - 11 -  T • MOOS i-iicx axrtR • MOW  CD W CD C CO  l-INO* ixt  1.0  D) C CO  2  1 0  20  3.0  Silicon (Wt%)  Figure 2.1: Manganese concentration versus silicon concentration, for two different section sizes, after [33].  both the partition coefficients and the phase transformation temperatures can only be determined experimentally for specific alloys, or from basic thermodynamic concepts when this is feasible. Values of equilibrium partition coefficients have been measured by Morita and Tanaka [34] on specimens which were allowed to reach equilibrium at a specific temperature and quenched. Electron probe microanalysis line scans across liquid-solid interfaces were used to determine the relative amounts of solute in each phase. Actual  Chapter 2: LITERATURE REVIEW  LU  0  1  1 005  - 12-  1 0.10  i  0.15  I  Mole fraction of carbon Figure 2.2: Change of equilibrium partition coefficients of some elements with carbon content in Fe-C base alloys, after [34].  compositions were estimated from calibration curves previously established for low concentrations of the ternary elements. In addition to their own results, the authors list partition coefficients reported by other researchers. In Figure 2.2 partition coefficients for Ni, Si, Cu, Co Mn, Cr, Mo and V are plotted as a function of the carbon content in the base alloy. Measurements are indicated by symbols. All the elements investigated have partition coefficients below unity at low C concentrations. As the concentration of C increases, partition coefficients become larger for Ni, Cu and Si, and smaller for Co, Mn,  Chapter 2: LITERATURE  -13-  REVIEW  Cr, Mo and V. Kagawa and Okamoto [35] determined thermodynamically the partition coefficients of third elements in Fe-C base alloys. Calculated values of the partition coefficients of Cr, Mn Si and Ni as a function of temperature and alloying element content are shown by the lines in Figure 2.3. Calculated values of the partition coefficient of C are also shown in the figure. The symbols indicate experimental measurements. The partition coefficient decreases with temperature for Ni and Si, and increases for Mn and Cr. The change in the partition coefficient with the concentration of alloying element is small in all cases. The calculations of Kagawa and Okamoto agree fairly well with the measured values. R. Forrest and I. Hewaidy [36] studied the segregation of alloying elements in the Fe-C metastable eutectic. Electron microprobe linescans across regions containing coarse eutectic structure were used to determine the relative concentration of alloying elements in austenite and cementite. The results indicated that graphitizing elements concentrate in the austenite phase, while carbide promoting elements concentrate in the cementite phase. Gundlach et al. [37] studied the relation between the formation of carbides in gray cast iron and the segregation of the alloying elements during solidification. Partition coefficients for some alloying elements taken from the literature were listed. The values reported are: 1^=1.2, k =1.6, k=0.002, k=0.2, k =0.7, ^=0.85, k =0.6. Si  s  P  Mo  Xi  - 14-  Chapter 2: LITERATURE REVIEW  JW.  Mn  06  (PCJCQM... ....-.••-••"1  06 04 02 * 0  aSi  1  ......^  08 •  0.4 •  ]V.Si  02-  e  o  u 08 0 04 0 21500 Equih bration  1600 Temperature  I'OO  1800  ( K )  Figure 2.3: Partition coefficients of a third element between austenite and liquid iron. Markers indicate experimental values. Lines show calculations. After[35].  Chapter 2: LITERATURE  2.2.1  REVIEW  - 15-  SUMMARY  Graphitizing elements, such as Si, Ni and Cu, tend to segregate to the austenite phase, while carburizing elements, such as Cr, Mn, and Mo, tend to concentrate in the liquid phase during solidification. Measurements and calculations of equilibrium partition coefficients have been reported for ternary elements in Fe-C based alloys. No data has been reported for partition coefficients of quaternary elements in Fe-C-Si based alloys.  2.3  MATHEMATICAL MODELLING OF  SOLIDIFICATION  Few mathematical models of the solidification of cast irons have been reported in the literature [23,28,29,30,38]. The ability of such models to fit the experimental data is varied. Fredriksson and Svensson [30] modeled the solidification of nodular, flake and white cast irons. The model predicts the conditions for the formation of white cast iron during the solidification of gray cast iron. In the model, the temperature is assumed to be uniform throughout the entire casting volume. The heat extraction from the melt is calculated by applying the Chvorinov relation [39], shown in equation 2.1.  dQ _ k/Cp/pf dt  Where: Q = heat flux  {  nt  (T-T ) 0  (2.1)  Chapter 2: LITERATURE  REVIEW  -16-  k = heat conductivity of the mould material f  Cp = heat capacity of the mould f  p = density of the mould f  T = room temperature 0  T, = interface temperature  A heat balance is used to describe the temperature evolution of the volume element, as follows:  A^r = Vp CpJ?dt dt y nn  y  n  (2.2)  Where: A = total mould/casting interface area V = casting volume Cp„ = metal heat capacity p„ = metal density  When the temperature of the casting falls below the liquidus temperature calculated from the binary Fe-C equilibrium diagram, the release of latent heat is accounted for in the heat balance as:  Where: AH = latent heat of fusion  Chapter 2: UTERATURE  - 17 -  REVIEW  f = solid fraction s  Eutectic cells are assumed to be spherical, growing radially. Nucleation of graphite is assumed to be random, and all the nuclei are considered to be formed at the same time. In consequence the growth can be described by the equation of Johnson-Mehl [40] considering early saturation of nucleation sites:  /, = l-exp  ( 4  ^  I 3  )  (2.4)  Where: N = number of nuclei per unit volume S = cell radius Rearranging equation (2.4) and differentiating with respect totimeleads to:  dS  dt  1 \0.33 f j In {367dV  f  \0.66 1 df l-fsdt  s  (2.5)  Equations reported in the literature for different types of cast irons were applied to the calculation of the growth rate: 1) For flake graphite cast iron solidification:  (2.6)  Chapter 2: LITERATURE  REVIEW  - 18-  Where: f = mole fraction of graphite in eutectic g  f = mole fraction of austenite in the eutectic y  D = diffusion coefficient of C in liquid l  c  L = interlamellar spacing L* = critical interlamellar spacing u. = interface reaction constant xH - molar concentration of C in liquid equilibrated with austenite y  Xo' = molar concentration of C in liquid equilibrated with graphite r  The ratio L/L* is given by:  (2.7)  and L* =  1.8(10^) At  (2.8)  2  2) For white cast iron solidification:  — = 30.(10"*) (A7/)  2  (2.9)  Chapter 2: LITERATURE  REVIEW  -19-  3) For ductile iron:  0.2435  Vl  c  (X* -X ) r  igr  (2.10)  Where: V* = molar volume of graphite VI = molar volume of austenite X  gr  =molar fraction of C in the graphite  Based on equations 2.1 to 2.10, Fredriksson and Svensson developed a numerical model, and used it to calculate the influence of eutectic cell number and cooling rate on the microstructure of cast irons after solidification. Although the results of the calculations are qualitatively consistent with the actual characteristics of the solidification of cast irons, the model has not been validated by comparison with the results of experiments performed under conditions similar to those assumed in the model. Fras [29] constructed a model of the solidification of spheroidal graphite iron which accounts for the effect of impinging of the grains during growth. The approach to the nucleation and growth processes is very similar to that of Fredriksson et al. [30]. The temperature change of the liquid metal is assumed to be given by the following relationship:  7 = T exp p  -2bFy[?  (2.11)  Chapter 2: LITERATURE  REVIEW  -20-  Where: T = pouring temperature p  b = coefficient of heat accumulation of the mould F = casting surface area  o = c,v P  C = specific heat of the alloy p  V = casting volume p = density of the alloy The latent heat release, L, is accounted for in the heat balance of the metal volume as follows:  df, dQ dAT L r - ^ - = -4>^= dt dt dt J  (2.12)  L  The substitution of the Johnson-Mehl equation (2.4) and the equation developed by Wetterfall et al. [12] for the growth rate of spheroidal graphite iron, results in a differential equation for the rate of change in the degree of undercooling.  d(AT) _bF(T -AT t  d t  Where: T = eutectic temperature e  T„ = initial temperature of the mould  - TJ -AE^ntjAT)  (t + f,)  3  + fi) (®-A  E^ATt ) 3  Chapter 2: LITERATURE  REVIEW  -21 -  r = starting time for solidification t  t = time elapsed from start of solidification AT = undercooling T - T, A =  2nNVa  3  N = number of eutectic cells a = 0.0561 V = casting volume  (2.14) Once the sample is solidified, its change in temperature is calculated as:  r=r,ex  P  -26F(vr-Vf7) <Wit  (2.15)  Where: T = temperature when f= 0.99 s  t =timefor/, =0.99 s  The results of the model are shown in Figure 2.4. The temperature of the casting decreases until the rate of heat generation by the solidification is equal to the rate of heat extracted by the mould. When equality is reached, recalescence starts. Figure 2.4 also illustrates the predicted influence of the number of cells and the cooling rate on the cooling curves. The model appears to give results qualitatively consistent with the experiments, but, as pointed out by Fras, qualitative differences are expected. The model has not been validated.  Chapter 2: LITERATURE REVIEW  -22-  1280  Figure 2.4: Cooling curves for varied number of eutectic cells and cooling rate, after Fras [29]  Chapter 2: LITERATURE  REVIEW  -23-  Su et al. [28] formulated a mathematical model of the solidification of ductile iron, in which both the temperature evolution and the nodular size distribution are calculated for a two-dimensional geometry. The model consists of three parts, a nucleation model, a growth model and a heat transfer model. The nucleation in the melt was considered to proceed according to the formulation proposed by Oldfield [41], equation (2.16)  N =AAT  2  (2.16)  Where: N = number of graphite nuclei AT = supercooling A = nucleation constant In order to calculate the nucleation rate, Equation (2.16) was differentiated with respect to time and expressed in central finite differences. Nucleation was assumed to stop when recalescence starts. In coincidence with other solidification models examined earlier in this section, Su et al. calculated the growth rate of the eutectic cells on the basis of the equation developed by Wetterfall et al. [12], which assumes that the eutectic ductile iron cells are constituted by graphite spheres enveloped by an austenite shell. In the model, it was assumed that graphite nodules grow enveloped by austenite at all times. The growth rates of both graphite and austenite are then controlled by the diffusion of carbon through the austenite shell. Nucleation and growth models were coupled with a two dimensional transient solution of the heat transfer equation for a cylindrical, sand cast. When the temperature of  Chapter 2: LITERATURE  REVIEW  -24-  a volume element falls below the eutectic temperature, the models of nucleation and growth are used to calculate the number of eutectic cells and its size. On the basis of these values, the fraction solid and the release of latent heat are calculated. The results of this model were compared with temperatures measured in experimental casting. Calculated and measured temperatures are shown in Figure 2.5. It can be seen that the model predicts the solidificationtimefairly well, but the calculated supercooling is larger than the measured values. In addition, calculated and measured nodular size distributions were substantially different, as shown in Figure 2.6. Su and co-workers concluded that the equation of Oldfield is not suitable for the description of graphite nucleation in ductile iron. They also suggested that the nucleation model should allow the nucleation to continue even after recalescence begins.  1300 1270 1240 1210 f  1180  ~Z  '150  a  1090  °  1050  MOLD  VE&SUPSD CALCULATED  1030 1CC0  10  20  30  4 0 50 Time  60  70 80  90 100 110 120 130  ( sec.)  Figure 2.5: Comparison of simulated and measured cooling curves, after [28].  Chapter 2: LITERATURE REVIEW  6  -25-  i ' i  I  • i 'I-.12 1  ' I j' • l  1  I  1  I  1  .  Rag CONSTANT, 2 3109 A = 900 Lt-. 0-2233  i  A-10 AN: A t&Tl &t SO". Fi= I 0 7  fr XX  \ Wettrrlou  i  2  Figure 2.6: Simulated nodular size distribution and We tterf all's data. After [28].  Stefanescu and Kanetkar [23] modelled the solidification of cast irons of eutectic composition. The calculations of nucleation and growth were based on the equation of Johnson-Mehl, assuming that the eutectic cells grow as spheres. Thermal gradients throughout the casting were neglected, and the mould was assumed to be semi-infinite. The heat flow across the metal-mould interface was calculated using the equation of Chvorinov [39]. The nucleation of eutectic cells is assumed to take place when the temperature of the metal reaches a specific supercooling; at which point a specified number of cells form, which remains constant throughout the rest of the solidification. In order to calculate the solid fraction, an equation describing the growth rate of eutectic cells is coupled with the equation of Johnson-Mehl, modified to account for the  Chapter 2: UTERATURE  REVIEW  -26-  impingement of the growing cells. Even though the calculations were not compared with experiments, it is evident that the model does not describe the solidification of ductile iron appropriately, since a cell number approximately five orders of magnitude larger than that usually found in ductile iron had to be used in order to obtain cooling curves qualitatively consistent with the experiments. Kanetkar et al. [38] modelled the solidification of eutectic gray cast iron (flake graphite) in sand molds. Similarly to models described above, the nucleation is assumed to occur instantaneously at a unique temperature, and the equation of Johnson-Mehl is used for the calculation of the solid fraction. In order to characterize the heat extraction imposed on the casting, the temperature of the mould after pouring was measured at several locations. These temperature readings were used to estimate the value of the heat extraction as a function of the time elapsed from pouring. One-dimensional and two-dimensional solutions of the heat transfer model were both tested. Figure 2.7 shows the cooling curve for the center of a cylindrical casting of 50mm in diameter, as measured and as calculated by the one-dimensional model (Eucast), and the two-dimensional model (Bamacast). The two-dimensional model calculations are in good agreement with the measurements. In a more recent article, Stefanescu and Kanetkar [42] reported further work on the mathematical modeling of the solidification of cast irons. As in earlier studies, they assumed that the nucleation proceeds at a unique temperature simultaneously. This assumption was based on experiments performed earlier, which showed that, for the case of gray irons with uniform cell size, all nucleation occurred within a temperature range of 1°C. They also noted that the nucleation temperature and the number of nuclei are functions of the cooling rate. Since Stefanescu and Kanetkar had not found enough  Chapter 2: LITERATURE  -27-  REVIEW  1300' -EXPERIMENTAL •  SIMULATED (BAMACAST)  A SIMULATED (EUCAST)  UJ  s  s  OC 1  Ui  1000. 0  100  200  300  400  TIME, SECONDS  500  600  Figure 2.7: Measured and calculated cooling curve for the center of a 50mm diameter gray iron casting, after [38]  published data to establish the correlation between those variables, they used a simple experiment to roughly estimate the relationship between the cell number and the casting size for gray irons. The growth rate of eutectic cells was calculated applying the equation derived for multidirectional non-isothermic solidification: (2.17) Where: \i =7.25 (10 ) to9.5 (lO") m/sK 8  b  8  2  Chapter 2: LITERATURE REVIEW  -28-  AT = supercooling b  The growth rate, as calculated by Equation 2.17, and the estimated number of eutectic cells, have been used as input in an Avrami type equation to calculate the fraction solid as a function oftime.Measured and simulated cooling curves agreed fairly well, as shown in Figure 2.8. The model was also used to calculate the white/gray transition during solidification. These calculations showed a significant discrepancy with the experiments. The solidification of ductile iron was calculated by the model under assumptions similar to those applied for gray iron. In this case, in order to obtain an acceptable fit between experiments and calculations, Stefanescu and Kanetkar found it necessary to assume cell counts approximately two orders of magnitude larger than the actual counts. Zeng and Pehlke [43] modelled the cooling of gray cast iron during solidification. The latent heat of solidification was released by means of a step-like function, between 1157 and 1143 C. Special care was taken in the selection of both sand mould and metal properties, as well as in the estimation of the surface heat transfer coefficient between the mould and the metal. Figure 2.9 shows temperature-time profiles for eight locations of the casting, as measured and as calculated. There is agreement between the measured and calculated curves, although some differences are evident.  Chapter 2: LITERATURE  REVIEW  -29-  Figure 2.8: Measured and calculated cooling curves for gray iron, after [42].  Chapter 2:  LITERATURE REVIEW  - 30  2200 2100 -  g  7  §1900  £1800 2 H  4  -  1700  -  1600  -  UJ  i i 2  4 3 2 1  •  8  •  1 200  1500  • -  (a) 1  400  600  800  1000  800  1000  TIME (SEC.) 2200  0  200  400  600  TIME (SEC.)  Figure 2.9: Cooling curves corresponding to different positions within a cylindrical casting, (a) experimental, (b) calculated, after [43].  Chapter 2: LITERATURE  2.3.1  REVIEW  -31 -  SUMMARY  In most cases the solidification models for cast iron are composed of two parts, a heat transfer model and a nucleation and growth model. The following heat transfer models have been investigated in the literature: a)  Simplified heat transfer models in which the heat extracted from the casting is calculated on the basis of the Chvorinov equation or similar equations, and the temperature is assumed to be uniform throughout the casting.  b)  Complete heat transfer models, which include a description of the heat transfer at the metal-mould interface, and the calculation of the temperature distribution throughout the casting and mould. Nucleation and growth have been approached in different ways. The following  nucleation models have been assumed: c)  Nucleation proceeds instantaneously at a given undercooling. The number of nuclei is not calculated but estimated a priori.  d)  The nucleation rate is a function of both the supercooling and a nucleation constant. Nucleation stops when recalescence begins. Most studies have considered that the growth rate of ductile iron eutectic cells is  controlled by the diffusion of C through the austenite layer enveloping the graphite nodules. In those cases in which the formation of all nuclei was assumed to proceed simultaneously, Avrami type equations were applied to calculate the fraction solid. In the  Chapter 2: LITERATURE  REVIEW  -32-  cases based on a supercooling dependent nucleation rate, the growth of cells was computed individually, which in turn allowed the nodular size distribution to be calculated. Some of the mathematical models reported in the literature predict cooling curves at the centre of experimental gray iron castings accurately. This is not the case for ductile iron, where calculated cooling curves and nodular size distributions do not entirely agree with the experimental values.  2.4 C O O L I N G C U R V E S The literature includes many articles describing different features of the cooling curves of gray, compacted and ductile irons. Cooling curves are usually characterized by four temperature points, the temperature of the austenite liquidus (TAL), the temperature of eutectic start (TES), the temperature of eutectic undercooling (TEU) and the temperature of eutectic recalescence (TER). These points are schematically shown in Figure 2.10. In the case of cast irons of near eutectic composition T A L does not appear, since no precipitation of primary austenite is expected. Highly hypereutectic cast irons will show the precipitation of primary graphite (TGL), although this is difficult to detect since the graphite is present in small amounts. Many studies have determined these temperatures accurately by using differential thermal analysis (DTA). DTA is particularly useful when an accurate determination of TAL and TES is required. Values of TEU and TER can be readily obtained from conventional cooling curves. L. Backerud et al. [14] studied the cooling curves of different types of cast iron. They reported qualitative and quantitative differences in the solidification of cast irons of  Chapter 2: LITERATURE  REVIEW  -33-  different graphite morphology. In their experiments, melt samples were not poured but extracted from the melt in a cup-like device. The cooling curves of eutectic cast irons of different graphite morphology are shown in Figure 2.11. Eutectic gray iron solidifies with a small supercooling, followed by a recalescence. Ductile iron shows a much larger supercooling than gray iron, followed by a minor recalescence. Compacted graphite iron shows a large supercooling, with the recalescence being more pronounced than that of ductile iron. The differences in nucleation temperature are generally attributed to  Chapter 2: LITERATURE REVIEW  -34-  differences in the chemistry of the irons. In the production of both ductile and compacted irons, the melt is inoculated with alloys containing Mg, Ce, Al and Ti, or combinations of those elements. The addition of these elements is considered to reduce the amounts of oxygen and sulphur in the melt, which in nun diminishes the availability of certain nucleation centres at low supercooling. The activation of new nucleation centers requires a further temperature drop.  Figure 2.11: Cooling curves of different cast iron types, after [14].  The differences in the recalescence are explained by the growth characteristics of the different cast irons. Gray iron eutectic grows with both austenite and graphite in direct contact with the melt. The growth rate of the eutectic is then mainly controlled by  Chapter 2: LITERATURE  REVIEW  -35-  the diffusion of C in the melt, which is fast. Therefore shortly after nucleation the temperature rises to near the eutectic temperature, as a result of the rapid heat generation of the eutectic solidification. On the other hand, ductile iron graphite particles grow mainly enveloped by austenite, therefore the growth rate is controlled by the C transport from the liquid to the graphite through the solid austenite. The diffusion of C is slower in the solid than in the liquid at similar temperature, therefore the release of latent heat is slower for ductile iron than for gray iron. Thus, even though the supercooling is large at the begining of the solidification, the recalescence is small. Eutectic compacted graphite cast iron, on the other hand, is believed to grow with both austenite and graphite phases in direct contact with the melt. Therefore the growth rate is considerably larger than that of ductile iron, and the recalescence is intense. D. Stefanescu et al. [15,44] also reported cooling curves for gray, ductile and compacted irons. In their case, which differs from Backerud et al.[14], samples of the melt were poured into sand cups, which were initially at room temperature. The results are shown in Figure 2.12. In this case the TEU of compacted iron, curve (a), was lower than that of ductile iron, curve (b), and the ductile iron did not show recalescence. Table I summarizes the data in the literature concerning cooling curves of ductile iron. Most of the data was obtained by using eutectometers, in which a small sample of the melt (50 cm ) is poured into a sand cup, and the cooling curve obtained during 3  solidification with an immersed thermocouple. One of the objectives of this investigation is to determine the effect of the cooling rate on the characteristics of the cooling curve. In particular, how do TEU, TER and the length of the eutectic plateau change with the heat extraction. This is not discussed  Chapter 2: LITERATURE  REVIEW  -36-  extensively in the literature. Rao et al.[45] reported cooling curves obtained at the centre of cylinders having different diameters. Values of TEU and the length of the eutectic plateau from the curves as a function of rod diameter are shown in Figures 2.13 and 2.14. The results in Figure 2.13 show that the minimum temperature at the start of solidification, TEU, is low for the smaller rod diameters, and increases for the larger diameters. The length of the eutectic plateau, Figure 2.14, also increases with increasing rod diameter.  270 E . SEC.  Figure 2.12: Cooling curves for various types of cast irons poured in a sand cup, after [44].  Chapter 2: UTERATURE  -n  1  H  20  REVIEW  1  1  40  -37-  1  1  60  1  1  1  80  100  Rod Diameter (mm)  Figure 2.13: Temperature of eutectic undercooling recorded at the center of cylindrical ductile iron castings, as a function of the section size.  Su et al. [28] measured cooling curves at several points of a casting with different cooling rates. The curves are shown in Figure 2.5. None of the cooling curves show recalescence, and the plateau or break in the curve occurs at different temperatures in each case; the lower the plateau temperature, the shorter the plateau length.  Chapter 2: LITERATURE  REVIEW  -38-  Rod Diameter (mm)  Figure 2.14: Lenght of the eutectic plateau at the center of cylindrical ductile iron castings, as a function of the section size.  2.4.1  SUMMARY  There is enough evidence in the literature to conclude that gray iron solidifies with a smaller supercooling than ductile iron, and that ductile iron shows little or no recalescence. For ductile iron, both the undercooling and the temperature of eutectic recalescence depend on the size of the casting and the cooling rate.  Chapter 2: LITERATURE  REVIEW  -39-  Table I: Data concerning cooling curves of cast iron. Author  Cast Iron Type  Mould Type  Section Size(mm)  TEU  TER  Length TEU(s)  Proeutectic  Rao etal  Ductile  Shell  20  1020  1129  nil  1140  [45]  Ductile  Shell  50  1111  1112  85  1150  1135  180  no  Ductile  Shell  80  1134  Ductile  Shell  90  1155  1155  290  no  Ductile  Sand  50  1114  1115  35  1170  Ductile  Sand  80  1139  1139  160  no  Ductile  Sand  90  1144  1144  190  1175  Ductile  Sand  100  1150  1150  280  no  Comments  Cheng and  Ductile Eutectometer  1139  1140  55  1188  eutectic  Stefanescu  Ductile Eutectometer  1140.5  1143  65  1188  hypoeutectic  [17]  Ductile Eutectometer  1154.4  1157  120  1193  hypereutectic  Ductile Eutectometer  1143  1143  140  no  Strong[19]  Ductile Eutectometer  1145  1145  55  no  Monroe and  Ductile  Bates  sand  22  1120  n/a  n/a  n/a  uninoculated  Ductile  sand  22  1133  n/a  n/a  n/a  inoculated  Gray  sand  22  1138  n/a  n/a  n/a  base iron  Stefanescu  Ductile Eutectometer  1140  1143  60  1190  hypoeutectic  etal[15]  Ductile Eutectometer  1151  1151  60  1190  hypoeutectic  Ductile Eutectometer  1140  1132  60  no  eutectic  Stefanescu  Ductile Eutectometer  1143  1145  n/a  n/a  hypoeutectic  et al[25]  Ductile Eutectometer  1146  1148  n/a  n/a  eutectic  Ductile Eutectometer  1143  1145  n/a  n/a  eutectic  Ductile Eutectometer  1143  1143  n/a  n/a  eutectic  1150  1150  240  no  eutectic  [18]  Hummer[27] Ductile  Sand  60  40  Chapter 3  OBJECTIVES OF THE PRESENT RESEARCH  Mathematical modeling of the solidification and microstructure of ductile iron can lead to the production of ductile iron under more controlled conditions and with improved properties. Mathematical modelling, to be effective, requires detailed information of the microsegregation of the alloying elements during solidification. Microsegregation in ductile iron is not clearly understood, and the extent of the microsegregation not well documented. As a result the first part of this investigation will deal with microsegregation, with the following objectives: 1)  To measure the effective partition coefficients of the alloying elements during solidification.  2)  To identify the mechanisms governing the segregation and to develop equations which will enable the segregation to be calculated from first principles.  3)  To correlate the segregation with the microstructure of ductile iron.  Chapter 3: OBJECTIVES  OF .  -41 -  Microsegregation of the solute elements will be measured in ductile iron samples, both after casting and quenched during solidification, using electron probe microanalysis. Examining quenched samples should provide a better understanding of both the microsegregation pattern and the mechanisms leading to the segregation in ductile irons solidified under normal conditions. In addition, the metallographic analysis of quenched samples is essential for an understanding of the microstructural evolution during solidification. Samples will be quenched at progressive stages of solidification with different solid fractions present. A general procedure for doing this is to extract portions of the melt simultaneously in small containers, and place the individual containers in furnaces set at temperatures between the liquidus and solidus temperatures of the melt. When equilibrium between sample and furnace temperatures is reached, the samples are quenched. An alternative procedure is to quench different melts of the same composition at different times from the start of solidification. Both sampling methods described above are laborious and difficult to reproduce. A different sampling technique, described in detail in Chapter 4, has been devised for the present study, in which one sample contained in a quartz tube, when quenched, has the full range of solid fraction. The second part of this research program will model the solidification of ductile iron. The analysis of the literature reviewed in Section 2.3 suggests that aspects of the solidification of ductile iron have been either overlooked or not properly accounted for in the reported models. Some of the main objections to the existing models are listed below: 1)  None of the models take into account the fact that gray irons are, as a first approximation, ternary Fe-C-Si alloys having Si contents between 2 and 3%. As  Chapter 3: OBJECTIVES  OF .  -42-  reported by Heine et al.[46,47,50] and Subramanian et al.[48,49], Si increases the temperature gap between the stable and metastable eutectic temperatures significantly, as shown in Figure 3.1. In addition, Si reduces the solubility of C in austenite, shifting the point of maximum solubility of C in austenite and lowering the carbon concentration at the eutectic point, as shown in Figure 3.2. 2)  Most studies use simple formulations of the heat transfer problem, assuming, for example, that the temperature throughout the casting is uniform. This reduces the applicability of the model since temperature gradients normally exist in castings.  3)  The solidification assumed in the models is cellular growth. Experimental observations show growth of austenite is normally dendritic.  4)  With one exception [28], graphite nucleation is considered to take place instantaneously at a specified supercooling. With this assumption, nucleating sites, and therefore the number of graphite nodules is specified for each calculation. However this does not consider the dependence of the number of graphite nodules on the cooling rate, which is significant. This factor must then be introduced in the model as an input parameter.  5)  The Johnson-Mehl equation is generally used to calculate the solid fraction during solidification. This is incorrect as the equation only applies to the growth of equiaxed spherical cells, which is not normally the case for ductile iron.  6)  Growth rate calculations have been based on equations derived by Wetterfall et al.[12], assuming that graphite nodules are enveloped by austenite immediately after nucleation. Observations have been reported in which graphite nodules have grown in the melt without an austenite envelope [12,22].  Chapter 3: OBJECTIVES  7)  OF .  -43-  None of the models reviewed considers microsegregation of the solute elements. Some authors [28] point out the need to include segregation for more accurate calculations. It is proposed in the second part of this investigation to develop an improved  mathematical model for the solidification of eutectic ductile iron. The model will include the following: 1)  Equilibrium temperatures and compositions will be calculated in the model based on the ternary stable Fe-C-Si equilibrium diagram.  2)  The thermal model will include a complete description of the heat transfer throughout the casting-mold system.  3)  The actual phase morphology during solidification will be included in the model.  4)  Nucleation rate will be a function of the supercooling. Nodule density and nodule size distribution will thus be determined from the model.  5)  Graphite growth in direct contact with the melt will be considered.  6)  The effect of Si segregation will be considered in the model.  Chapter 3: OBJECTIVES OF.  -44-  Figure 3.1: Stable and metastable eutectic temperatures of cast iron as a function of the silicon content, after [49].  Chapter 3:  OBJECTIVES  OF  -45-  2600  2000  .0  2J0  3J0  4.0  PERCENT CARBON OR CARBON EQUIVALENT  Figure 3.2: Influence of the silicon content on the eutectic region of the Fe-C-Si equilibrium diagram, after [47].  5.0  -46-  Chapter 4  EXPERIMENTAL METHODS AND APPARATUS  4.1  MELTING  Cast iron was melted in a 15 KW high frequency (10 KHz) induction furnace using a silica crucible of 90 mm ID and 180 mm height. The charge materials were high quality pig iron, low carbon steel scrap, electrolytic Mn, ferrosilicon, electrolytic copper, electrolytic nickel, chrome, molybdenum and FeSiMgCe. The chemical composition of the charge materials is listed in Table U, and the charge constitution for each of the castings is listed in Table in. In order to produce ductile iron, the melt was inoculated with 2 wt% of FeSiMgCe, and post-inoculated with 0.6 to 1 wt% of FeSi. Charge materials were placed inside the crucible prior to heating. When the temperature reached approximately 1380 C, the FeSiMgCe was added using the plunger system shown in Figure 4.1. After the intense reaction between melt and FeSiMgCe was completed, the slag was removed, and the FeSi added immediately. The melt was then stirred, using a steel rod, to mix the FeSi in the melt.  Chapter 4: EXPERIMENTAL  METHODS  AND .  -47-  Table II: Composition of charge materials.  Charge  Composition (Wt%)  Material  Si  Pig Iron  0.18  Steel Scrap  <0.10  FeSi  76.6  FeSiMgCe  45.1  4.2  C  Mn  Mg  Ce  Al  Ca  P  S  Fe  "— — 0.027 0.015 balance 0.18 0.75 — — — — <0.04 <0.05 balance — — — — 1.16 0.75 — — balance — — 5.97 1.15 0.55 1.08 — — •balance 4.3  0.009  ~  ~  SAMPLING  Samples of partially solidified material were prepared in the following manner. Quartz tubes of 15mm ID. Were inserted in the melt for 20 seconds, Figure 4.2-1, and then filled with liquid metal, Figure 4.2-2. The tube was then slowly pulled from the melt, progressively freezing the liquid from the top, Figure 4.2-3. With the metal in the tube partially solidified, the tube was rapidly removed from the melt and quenched in cold water, Figure 4.2-4. The cooling rate was estimated in one test by placing a Pt/Pt-Rd thermocouple in the melt contained in the tube, and recording the temperature during cooling. The cooling rate of the liquid was observed to be approximately l°C/sec.  Chapter 4: EXPERIMENTAL  METHODS  AND ...  -48 -  Table HI: Charge constitution.  Kg Iron  Steel  FeSi  FeSi  FeSiMgCe  Alloying  (grams)  (grams)  (bath)(g)  (inocXg)  (grams)  (grams)  Cl  4800  700  55  55  110  Mn:60  C2  4200  800  43  50  110  Cu: 72  C3  4500  500  43  50  110  Cr: 23  C4  4500  500  43  50  110  Mo: 40  C5  4500  500  43  50  110  Ni: 52  C6  4500  500  43  50  120  ~  C7  5000  500  48  55  120  Mn: 17  C8  5000  500  48  55  120  Mn: 34  C9  5000  500  48  55  120  Mn: 51  CIO  4800  600  43  55  150  Cu: 25  Cll  4800  600  43  55  150  Cu: 50  C12  4800  600  43  55  150  C13  4500  830  72  53  140  C13  4800  600  43  55  150  C14  4500  830  72  53  140  TEST  — — — —  Chapter 4: EXPERIMENTAL  protective lid  METHODS  AND  -49-  — handle  FeSiMgCe  -steel pipe tapered  stopper  Figure 4.1: Schematic of the plunger.  Chapter 4: EXPERIMENTAL  METHODS  AND  ...  -50  Chapter 4: EXPERIMENTAL  METHODS  AND .  -51 -  4.3 C A S T I N G A N D T E M P E R A T U R E R E C O R D I N G  After samples were removed from the melt, the remaining melt was poured into a resin coated silica sand mould, as shown schematically in Figure 4.3. The sand mould was contained in a steel flask, as shown, to prevent expansion or cracking of the mould due to the large stresses generated in the mould as the graphite nodules precipitate and grow in the ductile iron. The melt was solidified primarily from the vertical side walls of the mould. This was done by placing a refractory brick next to the bottom surface, to reduce the heat flow, and a water cooled copper tube coil next to the vertical walls to increase the heat flow. The copper tube was 10 mm ID and placed 10 mm from the vertical surfaces of the mould.  SAND MOULD  86 mm STEEL FLASK  COPPER COIL  REFRACTORY BRICK  Figure 4.3: Schematic of the mould.  Chapter 4: EXPERIMENTAL  METHODS  AND  -52-  After pouring the top was covered by a lid made of refractory brick. Temperatures during cooling were measured with four thermocouples placed at the positions shown in Figure 4.4. Two bare thermocouples type K were used to measure the temperatures at the mould next to the copper coil, TC4, and at the casting surface, TC3. Two thermocouples type S protected by thin walled quartz tubes, TCI and TC2, were used to measure the temperatures in the central region of the casting. The output of the thermocouples was measured on a multichannel digital voltmeter Hewlett Packard 3480A. An ice/water bath was used for the thermocouple cold junction. The measured voltages were converted to temperatures using the polynomial function specified for the thermocouples [53]. The accuracy of the temperature measurements was estimated to be +/- 3°C for the type S thermocouples, and +/- 7°C for the type K thermocouples.  A long cylindrical mould, shown in Figure 4.5, was also used in which the heat extraction is essentially unidirectional in the central portion of the casting. This mould was made from the same materials used for the water cooled mould. Cooling curves at two different points on a plane perpendicular to the cylinder axis were obtained. The temperature of the casting surface was measured by using a bare type K thermocouple marked TC6 in Figure 4.5. The temperature at the cylinder axis was measured by using a type S thermocouple, TC5, protected by a thin-walled quartz tube. Internal diameters of both 25 and 40mm were used in the tests.  Figure 4.4: Position of the thermocouples.  Figure 4.5: Schematic of the long cylindrical mold.  Chapter 4: EXPERIMENTAL  4.4  OPTICAL  METHODS  AND .  -55-  METALLOGRAPHY  The quenched samples were mounted on a magnetic table and ground to produce a plane section parallel to the sample rod axis, polished and etched for examination and microanalysis. The cast ingot was sectioned at a plane perpendicular to TC2 and containing TC3, polished and etched. The fraction of quenched liquid and solid present at specific locations within the quenched sample were measured using a Leitz Tas Plus Image Analyzer. The phases present in the etched microstructure are identified on the basis of the differences in their gray levels. The surface area occupied by each phase can then be measured. The solid fraction was calculated as the ratio between the surface area occupied by austenite and graphite, and the total surface area examined. Image analysis was also used to measure nodule counts, and to characterize the nodular size distribution at different locations of the cast ingots.  4.5  ELECTRON METALLOGRAPHY  AND  MICROANALYSIS  Quenched and as-cast samples were examined in a Hitachi scanning electron microscope. Quantitative measurements of the composition of the samples at specific locations were made with a MICROSPEC Wavelength Dispersive X-ray Analyzer (WDX) attached to the SEM. Microanalysis was carried out at 20 Kv acceleration potential and 45° take-off angle. Pure elemental standards were used in all cases. The specimen current was carefully controlled during the operation of the probe. Typical currents were 15 nanoampers. Peak and background count times were 10 and 5 seconds  Chapter 4: EXPERIMENTAL  METHODS  AND .  -56-  respectively for the high concentration elements, such as Fe and Si. For low concentration elements (0.1 to 2 wt%), the peak and background count times were extended to 20 and 7 seconds respectively to obtain better statistics. The output of the Wavelength Dispersive X-ray Analyzer is in counts per second at a particular wavelength. This is converted to composition by comparing the output to that of pure elemental standards and using the programme FRAME, supported by the microprobe software. This programme corrects for absorption, fluorescence, atomic number and deadtime.A sample of the microprobe output and analyzed results is given in Appendix 1. The output gives element composition values in weight percent and atomic percent. Weight percents were used in this study. Measurements of carbon concentration were not made since its consideration in the analysis would require frequent changes in the acceleration potential of the microprobe electron beam, which is time consuming, and would make it very difficult to collect the number of readings necessary for this study. Since C diffuses interstitially in Fe at a high rate, extensive diffusion  would be anticipated during cooling of the casting. As a result  the carbon distribution at room temperature could be appreciably different from that at the end of solidification. The reproducibility of the results for specific operative conditions was determined by repeat measurements. Fifteen measurements were made on each sample and the mean value and standard deviation calculated for both the true peak counts and the background counts. Table IV lists the results for the pure elemental standards and samples containing low concentrations of Si and Mn. The dispersion of the values is small for all the standards, the ratio of the standard deviation to the mean value ranging from 0.0063 for  Chapter 4: EXPERIMENTAL  METHODS AND ...  -57-  Si to 0.0031 for Cr. Note that this analysis is testing the performance of the overall probe system, including variations on the acceleration voltage, beam current and detector performance, together with the natural random characteristics of the X-rays. The count readings on samples containing Si and Mn were used to estimate the experimental error of the probe under actual analysis conditions. As shown in Table IV, for samples containing Si and Mn, the ratio between the mean and the standard deviation is approximately one order of magnitude larger than that obtained for the standards. Assuming that the number of readings is large enough to consider the standard deviation calculated as equal to the standard deviation of the random variable "counts per second", the accuracy of the calculation of the median value of counts per second as a function of the sample size can be calculated on the basis of the law of large numbers for a sample mean [54], which states that the probability of the mean value of a sample of n elements will fall outside an interval (u. - e, n + e), for any given e > 0, is given by the following equation:  P(\X*-\L\>E)<,  n  (4.1)  Where: o = standard deviation u, = mean value of the distribution X* = mean value of the sample of n elements The less favorable value of standard deviation in Table IV, which corresponds to sample Mnl, is standardized as follows (4.2)  Chapter 4: EXPERIMENTAL  METHODS  AND  -58-  Assuming a sample of three elements, and an interval X* ± 10%, the probability P is equal to:  PQX*-\i\>0A)<.  = 0.10  (4.3)  These results show that the probability of the median of a sample of three elements is inside the interval u.± 10% is 90 %. It can be then assumed that if the median of a sample of three elements is affected by a 10% error margin, there will be a 90% probability that the median of the distribution is within the range. Following this, all microprobe analysis reported below will be the average of three readings and will carry a 10% error bar, unless otherwise specified. Further contributions to the experimental error can be expected from other factors, such as the dispersion in the standard readings, and, more significantly, the dispersion in the background counts. Nevertheless, the error margin of 10% adopted is considered to be large enough to account for the influence of these factors. For simplicity, the error bars will not be calculated for every point, but will be taken as ten percent of the average composition of the sample.  Chapter 4: EXPERIMENTAL  METHODS  AND .  -59-  TABLEIV: Data of fifteen microprobe measurements of elemental standards and test samples.  Standard Samples (counts/sec)  Test Samples (counts/sec)  Fe  Si  Mn  Cu  Cr  Mo  Mn 1  Mn2  Si  mean value true peak  5690  9817  5154  5134  5187  2083  26.9  37.4  134.3  Std. deviation true peak  22.2  62.0  17.0  2Z7  16.2  9.7  1.5  1.2  5.8  mean/std deviation true peak  0.004  0.0063  0.0033  0.0044  0.0031  0.0038  0.056  0.03240  0.043  mean value background  13.17  2.89  9.97  23.4  9.08  6.1  8.95  8.74  3.55  Std. deviation background  0.817  0.38  0.79  0.90  0.89  0.64  0.632  0.51  1.65  0.062  0.130  0.079  0.038  0.098  0.105  0.071  0.058  0.46  mean/std deviation background  -60-  Chapter 5  SEGREGATION AND MICROSTRUCTURE RESULTS AND DISCUSSION  Segregation was investigated in both as-cast and quenched samples. The compositions of the ductile irons examined are listed in Table V. The irons were all of near eutectic composition, with single alloying elements added, except for casting CO.  5.1  S E G R E G A T I O N IN C A S T S A M P L E S  It has been suggested [51,52] that the concentration of an alloying element with a segregation coefficient, k, smaller than 1, when measured along a line between the centers of two graphite nodules, will show segregation patterns as illustrated in Figure 5.1. If k is smaller than 1, the concentration would increase with distance from each graphite nodule, as shown, until a maximum is reached midway between the nodules. This concentration profile would be produced by the growth of equiaxed spherical cells,  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -61-  as illustrated in Figure 5.2. The cells are formed by graphite spheroids enveloped by austenite. Solidification mechanisms consistent with that shown in Figure 5.2 have been supported by some authors, as described in section 2.2.  Figure 5.1: Representation of the solute concentration along a line between graphite nodules in ductile iron, a) k > 1 ; b) k < 1  Chapter 5: SEGREGATION  AND MICROSTRUCTURE...  - 62 -  Table V: Alloying element content in the ductile irons examined TEST  CO  Alloying  0.78Mn  Concentration  Cl  C2  0.98Cu 1.34Mn 1.36Cu  C3  C4  C5  0.5Cr 0.83Mo 0.83Ni  C6  C7  C8  C9  CIO  cn  2.45Si 0.4 lMn 0.74Mn 1.05Mn 0.50Cu 0.9 lCu  Wt%  Figure 5.2: Representation of equiaxed cellular growth in ductile iron.  In an attempt to verify the solidification mechanisms described above, the segregation pattern in the vicinity of a graphite nodule was measured on samples of cast iron CO, alloyed with both Cu and Mn. These alloying elements were chosen because Cu has k>\ and Mn has  k<\.  -63-  5.1.1  S E G R E G A T I O N P A T T E R N IN T H E V I C I N I T Y O F  A  GRAPHITE NODULE.  Measurements of solute concentration were made using microprobe analysis at points spaced at approximately 8 microns intervals along paths 1-4 between the nodules, as shown schematically in Figure 5.3.  Figure 5.3: Schematic of analysis along lines between nodules.  The results of the analysis are given in Figure 5.4. For Si, Figure 5.4.(a), measurements were started near the nodule/metal interface. The concentration along the four directions, marked 1 to 4, are observed to differ appreciably, exhibiting marked fluctuations in concentration with distance from the nodule. The Si concentration along  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -64-  line 1 is essentially constant, with a small drop at 45 microns. Scans 2, 3, and 4, exhibit a sharp drop initially, then the Si concentration rises to an essentially constant level, the fluctuations differing for each scan. Results for Cu and Mn are shown in Figures 5.4 (b) and (c) respectively. As in the case of Si, composition variations are small along line 1. The concentration of Cu along lines 2, 3 and 4 decreases with the distance from the nodule surface. The Mn concentration increases with the distance from the nodule. A similar series of measurements was done along three lines starting at the surface of a nodule, on a sample containing 0.88% Mn. The concentration variation with distances from the nodule for Si and Mn are shown for each line in Figure 5.5(a-c). As before, the scans show marked fluctuations. The concentration of Mn along line 1, Figure 5.5 (a), shows a marked increase  near the nodule, followed by decrease, reaching a  minimum value of 0.7% Mn near the mid point between the nodules. After the mid point the Mn concentration rises again to 0.95%, then drops to 0.7% as the scan approaches the end graphite nodule. The Si concentration shown in the figure indicates that, with few exceptions, the Si concentration mirrors the Mn concentration, with drops in Si concentration corresponding to increases in Mn concentration. The concentration of Mn along line 2, Figure 5.5 (b), increases with the distance from the nodule, reaching a maximum value of 1.06% at 12 microns, after which the concentration of Mn drops progressively, reaching a value of 0.8% in the vicinity of the second nodule. The concentration of Si along line 2 shows a drop, coincident with the maximum Mn concentration, but remains relatively constant otherwise. The segregation profile along line 3, Figure 5.5 (c), shows relatively small variations in concentration for both Si and Mn.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  ...  2.7 -r 2.6 -  1.7 1.6 1.5  H  1  0  1  1  1  1  20 40 Distance (microns)  1  -  60  Figure 5.4 (a) 1.7 - T 1.6 1.5 -  0.5 0.4 -I 0  1  1  1  1  20 40 Distance (microns)  Figure 5.4 (b)  1  r60  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  Figure 5.4: Solute concentration along lines 1 to 4. (a) Si, (b) Cu, (c) Mn.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  20  0  -67 -  .  40  60  Distance (microns) Figure 5.5 (a)  0  10  20  30  Distance (microns) Figure 5.5 (b)  40  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -68-  5 c g  « c  <D  o c o O  ti  a  u CO  Distance (microns)  Figure 5.5: Si and Mn segregation along lines between nodules, (a) line 1, (b) line 2, (c) line 3. Additional scans were made between adjacent nodules. The results were similar to those illustrated in Figures 5.4 and 5.5 in the majority of the cases. However, in some scans much larger variations in solute concentration were observed. For example, Figure 5.6 shows the variations in the concentration of Cu and Mn along a line between nodules of the same sample from which the curves in Figure 5.4 were obtained. Both Si and Cu exhibit a large drop in concentration near the mid-point between the nodules, and the Mn a larger increase. The levels of segregation of Cu and Mn shown in Figure 5.6 are considerably higher than those shown in Figure 5.4. Similarly, a qualitative analysis of the concentration of Mn and Si along a line between graphite nodules, for the same alloy from which the curves in Figure 5.5 were obtained, is shown in Figure 5.7. Marked  Chapter 5: SEGREGATION AND MICROSTRUCTURE  -69-  .  segregation of Mn is observed approximately at the mid point between nodules, in coincidence with a drop in the concentration of Si. Although this results are qualitative, the segregation shown in Figure 5.7 is clearly greater than that observed in Figure 5.5.  2.6 -• 2.4 2.2 -  I I  2  "  1.8  -  t:  1.6 -  ro c  1-2  1 »•  2  -  1  0.8 -  <  0.6 0.4 0.2 0 -| 0  1 1 20  1  1 1  40  1  60  1 1 80  1  1  100  Distance (microns)  Figure 5.6: Solute concentration along line between nodules. The spherical symmetry of the composition field around a graphite nodule was investigated by measuring the solute concentration along a circular path, concentric with the centre of a nodule. The results are shown in Figure 5.8. Small scale fluctuations in the Cu and Mn concentrations are observed, but no large scale fluctuations are evident for either element.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -71 -  In summary, the analyses of the segregation of Cu, Mn and Si along lines between nodules gives scattered results. In many cases Mn is preferentially concentrated near the centre point between nodules, although the extent of the segregation varies widely from one case to another. In other cases, different composition profiles were observed along lines between nodules, as shown in Figure 5.5 (a), where two Mn peaks are observed along the line, and in Figure 5.5 (c), where almost no variation of the Mn concentration is observed. These observations do not correspond with the model of spherical cells described above. It is believed that the examination of the microstructure of samples quenched during solidification will allow a better interpretation of the results reported in this section.  5.1.2  Q U A N T I T A T I V E V A L U E S F O R S E G R E G A T I O N IN C A S T SAMPLES  The segregation of the alloying elements between adjacent nodules in ductile iron shows appreciable scatter for different pairs of nodules in a given casting, as shown in section 5.1.1. As a result the measured solute distribution along a path between nodules in a casting is not suitable to quantitatively describe the segregation of specific elements. As an alternative the segregation will be characterized by the ratio between the solute concentration in the vicinity of a large graphite nodule, and the solute concentration at locations of the microstructure having the highest segregation. The position of highly segregated areas in the microstructure of carbide-free ductile iron is not evident when the polished surface is examined on the scanning electron microscope. However, after several successful attempts to find areas of high concentration with the help of the WDX,  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -72-  it was noted that either microporosity orfinerpearlite were associated with regions having the highest concentration of carbide promoting elements, and the lowest in graphitizing elements.  Table VI: Segregation in sand cast ductile iron.  TEST  ALLOYING  AVERAGE  ELEMENT  SOLUTE  ADJACENT  LAST TO  RATIO  TO NODULE SOLIDIFY  (l)/(2)  CONCENTRATION  (1) (Wt%)  (2) (Wt%)  CI  Mn  1.34  0.92  2.62  2.85  C2  Cu  1.36  1.87  0.37  0.20  C3  Cr  0.50  0.28  2.58*  9.21*  C4  Mo  0.83  0.23  43.2*  188*  C5  Ni  0.83  0.98  0.54  0.55  C6  Si  2.45  2.63  1.25  0.47  * Contains precipitate phase.  Chapter 5: SEGREGATION AND  MICROSTRUCTURE.  -73-  Solute concentrations adjacent to large nodules and at the highly segregated areas are listed in Table VI. Each value listed is the average of five readings. The concentration of solute in the vicinity of large nodules is lower than the average concentration of the alloy, for Mn, Mo and Cr, and higher than the average for Cu, Ni and Si. At the highly segregated areas the concentration of Mn, Mo and Cr is higher than the average, and the concentration of Cu, Ni and Si is lower. Segregation ratios listed in Table VI are calculated as the ratio of the solute concentration in highly segregated areas over the solute concentration adjacent to a graphite nodule. The segregation ratio is a measure of the maximum segregation present in the sand cast ductile iron. Alloys containing Cr and Mo contain carbides or other phase constituents. Therefore the ratios listed for those elements can be misleading, since they do not correlate solute concentrations at two locations of the same phase.  5.2 A N A L Y S I S O F Q U E N C H E D S A M P L E S 5.2.1 M I C R O S T R U C T U R E O F Q U E N C H E D S A M P L E S The solidification of slowly cooled graphitic cast iron follows the stable Fe-C-Si equilibrium diagram. At the eutectic temperature, austenite, graphite and liquid coexist. If the solidification occurs more rapidly, with larger supercooling of the melt, metastable cementite can nucleate. The growth rate of the metastable eutectic cementite-austenite is much larger than that of the stable eutectic. In consequence, the precipitation of graphite is prevented, and solidification is completed following the metastable Fe-C-Si equilibrium diagram. Because the precipitation of graphite is prevented, it becomes possible to quench the liquid iron without major restrictions in sample size or cooling rate. In particular, for liquid near the stable eutectic temperature, a supercooling of  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -74-  approximately 20°C is enough to result in the formation of metastable eutectic. The appearance of the quenched liquid is illustrated in Figure 5.9. The microstructure consists of a fine mixture of cementite, A, and austenite, B. Microprobe analyses showed that the alloying elements are partitioned between the two phases, with the carbide forming elements like Mn, Mo and Cr, concentrated in the cementite, and Cu, Ni and Si concentrated in the austenite. A typical microstructure of a quenched sample is shown in Figure 5.10. Note that the austenite dendrites, appearing white, are randomly oriented. Microstructures of quenched samples of alloy C6, observed at higher magnification are shown in Figures 5.11 to 5.15. The microstructure of quenched liquid at the bottom of the sample is shown in Figure 5.11. Small spheroidal particles of graphite, (G), are present in the structure. This suggests that the alloy is slightly hypereutectic. Further from the bottom, for a solid fraction of 18%, a higher number of larger graphite spheroids are present, as shown in Figure 5.12. An austenite dendritic structure is also evident, with the branches appearing white, D, and the quenched liquid surrounding the dendrites gray, C. As we move up the sample, Figure 5.13 (a) (fraction solid 67%), the graphite nodules become larger, and the amount of austenite increases. At this point most graphite particles are enveloped by austenite. Nevertheless, as shown in Figure 5.13 (b), some graphite particles remain in contact with the liquid phase. As the fraction of solid increases to 94%, Figure 5.14, the graphite nodules have further increased in size, and the dendritic structure can no longer be resolved. The continuity of the liquid phase is lost, and isolated pockets of liquid are evident. In Figure 5.15, for 100% solid fraction, the solidification was complete before quenching. No second phase precipitates are observed.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  75-  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  Figure 5.10: Structure of quenched sample (x 40)  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  -77  ...  jI Figure 5.12: Microstructure of quenched sample for solid fraction 18% (x 100)  9  • • \  1  "  vv*-\,i vr-^  ;  . v  .••  !-V? • • • •  Figure 5.14: Microstructure of quenched sample for solid fraction 94% (x 100)  Chapter 5: SEGREGATION AND MICROSTRUCTURE  b)(x400)  .  -78-  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  ...  - 79 -  Figure 5.15: Microstructure of quenched sample for solid fraction 100% (x 100)  The appearance of the microstructure of the quenched samples suggests that austenite dendrites and graphite nodules nucleate and grow in the eutectic cast iron independently. As solidification proceeds the austenite dendrite branches meet and envelop the nodules, which progressively distorts the dendritic structure until it can no longer be identified. Further growth occurs as a result of the thickening of the austenite layer surrounding the graphite. At the end of solidification the continuity of the liquid phase is lost, and isolated pockets of liquid are the last to solidify. Although these observations are based on small samples cooled relatively rapidly, they clearly are in agreement with the solidification mechanisms reported by several authors, as described in section 2.1. In view of this agreement, the measurements of segregation in the quenched samples will be assumed valid for cast samples as well. The validity of this assumption will be assessed by comparing the maximum and minimum values of solute concentration measured in quenched samples with those measured in cast samples.  Chapter 5: SEGREGATION  5.2.2  AND MICROSTRUCTURE  .  -80-  M E A S U R E M E N T S O F S O L U T E C O N C E N T R A T I O N AS A  FUNCTION OF T H E FRACTION  SOLID  Quantitative microprobe analyses were made within several areas of each of the quenched rods. For each area analyzed, the composition was measured at three locations of the rrucrostructure, as indicated in Figure 5.16. Position A is in austenite, next to a large graphite nodule, or alternatively, at the centre of the austenite field for the smaller solid fractions. Position B is in austenite, at approximately 1.5 microns from the austenite/quenched liquid interface. Position C is in the quenched liquid.  Figure 5.16: Location of the microstructure at which solute concentration was measured (x500).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -81-  The distance from the quenched interface at which position B is located is selected to avoid possible interference from the neighboring quenched liquid, which has a different solute content. When the electron beam of the microprobe strikes the specimen surface, X rays are produced. The X rays are generated from a nearly spheroidal volume, of diameter, S. This diameter can be calculated using the following equation [55]:  5 =0,033(V ' -V - )4: pZ 1 7  1 7  i  C-) 5  1  Where: S = diameter of region from which X rays are generated (microns) V = energy of the incident electrons [ KeV ] V = absorption edge of the element analyzed [ KeV ] k  A = atomic weight of the specimen p = density of the specimen Z = atomic number of the specimen For pure iron, equation 5.1 gives a spot diameter of 1.2 microns. This calculation does not account for the dimension of the incident electron beam. The true spot size can be found by adding the electron beam diameter to the calculated spot size. Under the present operational conditions the electron beam of the scanning electron microscope has a diameter of approximately 0.2 microns, which gives an effective spot diameter of 1.4 microns. Thus, measurements of the composition of the austenite can be made with the electron beam positioned at 1 micron or morefromthe interface between austenite and quenched liquid. This can be related to Figure 5.17, in which Mn counts are plotted as a function of distance along a line between the vicinity of a nodule and quenched liquid. A  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  -82-  .  gradual increase in Mn counts is observed between 0 and 3 microns. When the beam is positioned within a micron of the interface, count readings are higher than expected, as a result of the radiation from the quenched liquid. On the basis of the above considerations, routine measurements of the composition of austenite in positions A and B, were made at points located within 1 and 1.5 micronsfromthe quenched interface. 80  30  H 0  1  1  1  2  H 4  Distance (microns)  Figure 5.17: Mn counts intensity along a line crossing an austenite/quenched-liquid interface. As described above, quenching the melt results in a structure consisting of a fine mixture of austenite and cementite. The presence of two different phases complicates the analysis of the quenched liquid, since the area analyzed is not homogeneous. This difficulty can be overcome by analyzing, whenever possible, large areas of the fine structure of approximately 50 square microns.  Chapter 5: SEGREGATION  AND  MICROSTRUCTURE  -83-  Measurements of Mn concentration were made at three locations in the quenched irons, A, B and C, for areas having different solid fractions before quenching. Four different Mn levels were investigated. The results are listed in Table VU. The results, position A, show that as the solid fraction increases, there is little change in the Mn concentration in the austenite adjacent to the nodules. Near the quenched interface, position B, a gradual increase in the Mn concentration is observed for the smaller solid fractions, with a more pronounced increase for solid fractions greater than 80%. The concentration of Mn increases by a factor of 2 from the start to the end of solidification. In the quenched liquid, position C, the concentration of solute is appreciably higher than in positions A or B, and increases to higher levels as the solid fraction increases. In Table VU the Mn concentration at position A is listed for 0 solid fraction. This composition is equivalent to the composition at the centre of an austenite dendrite near the start of solidification. The average alloy composition, listed in Table VII, is measured by electron probe microanalysis in which an area at the bottom of the sample, consisting entirely of quenched liquid, was scanned at 20,000 magnification to give the average composition of the area. The concentration of Cu at points A, B and C in quenched samples having three different copper levels are listed in Table VIII. The concentration of Cu near the graphite nodules, column A, is essentially independent of the solid fraction. There is a decrease in the Cu concentration for the higher solid fractions in alloys C2 and C10. Near the quenched interface, column B, a gradual decrease in the solute concentration is observed as the solid fraction increases, the concentration of Cu decreasing by a factor of 2.2 to 2.5. In the quenched liquid, C, the concentration of Cu is appreciably lower than that in A or B, and decreases with increasing solid fraction.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE ...  Table VII: Local Mn concentration in quenched samples. TEST (Alloy)  CI  (1.34%Mn)  C7 (0.41%Mn)  C8 (0.73 %Mn)  C9  (1.05%Mn)  Solid fraction  Position A  Position B  Position C  Ratio B/C  0  0.95  -  --  0.70  30  0.95  0.92  1.40  0.66  31  1.07  0.94  1.51  0.62  34  0.93  0.99  1.50  0.66  48  1.02  1.21  1.60  0.75  73  0.95  1.29  2.25  0.57  86  0.91  1.23  2.60  0.47  94  0.95  2.01  3.47  0.58  0  0.27  --  -  0.66  36  0.22  0.26  0.44  0.59  51  0.27  0.32  0.52  0.61  64  0.28  0.32  0.54  0.59  83  0.28  0.49  0.81  0.60  96  0.28  0.52  0.99  0.52  0  .54  --  --  0.73  33  0.50  0.56  0.91  0.62  54  0.55  0.53  1.00  0.53  83  0.58  0.71  1.29  0.55  88  0.57  0.84  1.59  0.53  95  0.57  0.93  1.98  0.47  0  0.75  ~  0.72  44  0.78  0.86  1.44  50  0.89  0.90  1.61  0.55  80  0.77  1.00  1.77  0.54  86  0.84  1.23  2.27  0.56  92  0.87  1.25  2.40  0.55  97  0.73  1.70  3.71  0.46  0.60  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  -85-  Table VTU: Local concentration of Cu in quenched samples.  TEST (alloy)  C2 (1.36%Cu)  CIO (0.50%Cu)  Cll (0.91%Cu)  solid fraction  Position A  Position B  Position C  Ratio B/C  0  1.87  --  --  1.37  32  1.93  1.91  1.37  1.39  38  1.78  1.77  1.18  1.50  67.  1.84  1.59  1.03  1.54  84  1.84  1.30  0.64  2.03  92  1.88  0.92  0.47  1.95  96  1.57  0.79  0.40  1.98  97  1.48  0.89  0.38  2.34  0  0.66  --  1.32  27  0.60  0.67  0.45  44  0.66  0.57  0.40  1.43  69  0.66  0.54  0.37  1.46  76  0.62  0.46  0.24  1.92  89  0.58  0.34  0.22  1.53  94  0.55  0.29  0.21  1.40  0  1.12  --  -  1.25  53  1.14  1.03  0.63  1.63  78  1.16  0.87  0.53  1.64  84  1.12  0.69  0.37  1.86  92  1.00  0.41  0.27  1.52  96  1.17  0.40  0.25  1.60  1.48  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  -86-  Local solute concentrations for alloys containing Cr, Mo, Ni and Si are listed in Table DC Similar to Mn, the concentration of Cr remains almost unchanged next to the nodules, and increases at positions B and C as the fraction solid increases. The quenched liquid, C, is always richer in solute than A or B. In samples of alloy C3, massive ledeburite was present for solid fractions greater than 75%. With ledeburite present, measurements of solute concentration in the quenched liquid were not significant. Carbide precipitation was evident in quenched samples of alloy C4, which contains Mo, for solid fractions larger than 90%. The concentration of Mo at position A is approximately constant with solid fraction, with some scatter. Positions B and C become richer in solute as the solid fraction increases. The austenite next to the quenched interface, B, for the largest solid fraction measured, is approximately 2.5 times richer in solute than the austenite next to the graphite nodules, A.  *-  Sample C5, alloyed with Ni, shows segregation characteristics similar to the samples containing Cu. The concentration of Ni in both B and C decreases with increasing solid fraction. The concentration of Ni at B, for the highest solid fraction measured, is approximately half of the concentration at A. Although Si is present in every sample, its segregation behavior was only accounted for in detail in a sample of unalloyed iron C6, to avoid interference from other solutes. Table DC lists solute concentrations for Si at position B only. The concentration of Si tends to decrease as the solid fraction increases, but the amount of segregation is small.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  ...  Table DC: Local concentration of Cr, Mo, Ni and Si in quenched samples. TEST (alloy)  C3 (0.5%Cr)  C4 (0.83%Mo)  C5 (0.83%Ni)  C6 (2.45%Si)  Solid Fraction  Position A  Position B  Position C  Ratio B/C  0  0.30  --  --  0.60  19  0.31  0.31  0.66  0.47  21  0.31  0.29  0.70  0.41  48  0.31  0.35  0.72  0.49  66  0.31  0.36  0.81  0.44  71  0.28  0.44  0.98  0.45  0  0.21  --  -  0.26  9  0.21  0.20  1.04  0.19  24  0.17  0.16  0.85  0.19  37  0.19  0.24  1.06  0.23  38  0.18  0.25  1.26  0.20  57  0.21  0.30  2.00  0.15  67  0.22  0.40  2.13  0.19  72  0.25  0.43  4.15  0.10  89  0.19  0.49  3.26  0.15  0  1.02  --  1.23  40  1.03  1.01  0.74  43  1.08  0.95  0.69  1.38  50  1.08  1.06  0.63  1.68  71  1.10  0.98  0.51  1.92  89  1.05  0.88  0.45  1.95  94  1.02  0.69  0.31  2.22  98  1.02  0.63  0.34  1.85  0  2.67  --  --  1.09.  22  --  2.45  -  --  22  -  2.72  --  -  29  --  2.48  ~  --  52  --  2.31  --  --  82  ~  2.18  --  --  91  -  1.96  -  --  1.36  -88-  5.2.3 E S T I M A T I O N O F P A R T I T I O N C O E F F I C I E N T S O F T H E ALLOYING  ELEMENTS  Equilibrium segregation coefficients, k , are defined as the ratio of the 0  concentration of solute in the homogeneous solid phase over the concentration of solute in the liquid in equilibrium with the solid. Values of k can be determined from phase 0  diagrams. As shown in sections 5.2.1 and 5.2.2, during the solidification of ductile iron solute is segregated within the austenite. Therefore solidification does not take place under equilibrium conditions, where k is applicable. Under non-equilibrium conditions 0  an effective segregation coefficient is used, k , defined as the ratio of the solute e  concentrations in the solid and the liquid at the solid/liquid interface at a given stage during solidification. The value of k can only be determined experimentally, and depends e  on the solidification conditions and the amount solidified. An approximate method to determine the effective segregation coefficient has been adopted for this study. The value of k is calculated as the ratio of the concentration of e  solute at the centerline of the dendrites which first appear in a quenched sample, over the average solute concentration in the melt [66]. Values of k , calculated from the solute a  concentrations listed on Tables VI, VII and VIII, are listed on Table X. In this table, the equilibrium segregation coefficients for Fe based binary alloys and three sets of segregation coefficients reported in the literature for ternary Fe-C based alloys are listed, followed by the values obtained in this investigation. The values calculated in this study agree fairly well with the reported coefficients listed for Mn and Cr. For Si, Ni and Cu the present results are closer to unity than the reported values. For Mo the present result was farther from unity than the reported values. The equilibrium segregation coefficient  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  -89-  .  for Fe based binary alloys taken from the corresponding phase diagrams, are listed at the top of Table X. It is apparent that the effective segregation coefficients differ markedly from the equilibrium coefficients.  Table X: Measured and published values of effective segregation coefficients.  COMMENTS  Effective Segregation Coefficient For:  AUTHOR Mn  Cu  Cr  Mo  Ni  Si Fe based  Equilibrium segregation  0.64  0.62  0.99  0.53  0.75  0.87  binary alloys  coefficient Morita and  0.75  1.5  0.60  0.40  1.30  1.17  Ternary alloys  0.70  n/a  0.50  n/a  1.40  1.50  Ternary alloys  0.85  n/a  n/a  0.7  1.20  1.60  Ternary alloys  0.70  1.37  0.60  0.26  1.23  1.09  Multicomponent  Tanaka[34] Kagawa and Okamoto[35] Gundlach et al [37] Present  alloys  Study  An alternative way of defining k is the ratio of the solute concentrations at e  positions B and C in quenched samples for a given solid fraction. This ratio, B/C, is listed on Tables VI, VII and VUI for the samples investigated. For the purposes of this study, segregation coefficients determined in this manner will be referred to as k . The variation tt  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -90-  of the effective segregation coefficient, k„, with the solid fraction is shown in Figure 5.18(a-c), for Mn, Cu, Cr, Mo and Ni respectively. For simplicity, and given the scatter in the results, a linear regression was used to determine the best fit curve. The equation corresponding to the plotted lines for each of the alloy additions is included in the corresponding figure. The segregation coefficient decreases with solid fraction for alloying elements having k„<l (Mn, Cr and Mo), and increases for Cu and Ni, having k >\. The values for the effective segregation coefficients shown in Figure 5.18 for Mn ee  and Cu, were obtained from a number of alloys of different solute content. However, since sets of values for different alloy contents followed similar patterns, and in view of the scatter in the values, linear regressions for these elements were based on all values listed, without regard to the alloy content.  Figure 5.18 (a)  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  ...  2.4 2.3 2.2 2.1 -  5  c o c <D  8 o O o O  + Cu:1.36% A Cu:0.91% o Cu:0.50%  2  1.9 1.8 1.7 -  1.286+ 0.0054*fs  1.6 1.5 1.4 1.3 1.2  20  40  60  Solid Fraction Figure 5.18(b)  £= O  c 8 c o O 2  o  Solid Fraction Figure 5.18(c)  so  100  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  -92-  o o c o O E c <D 3  s  o  2  Solid Fraction  Figure 5.18(d)  15 C  c o O <s 8  3  Solid Fraction  Figure 5.18: Effective segregation coefficient as a function of solid fraction for Mn (a), Cu (b), Cr (c), Mo (d) and Ni (e).  Chapter 5: SEGREGATION  5.3  AND MICROSTRUCTURE  .  -93-  ANALYSIS O F SEGREGATION RESULTS  The segregation of solute in directionally solidified binary alloys has been extensively studied, and analytical expressions have been derived for the segregation as a function of the extent of diffusion in the solid and mixing in the liquid [56-58]. The diffusion of solute in the solid can be neglected if ak « 1, where a is a function of both e  the diffusion coefficient of the solute in the solid, the solidification rate and the coarseness of the structure. This is shown in Appendix 2. Based on the calculations done in Appendix 2, in the present investigation the liquid will be assumed fully mixed at all times, and the diffusion of solute in the solid will be neglected. Carbon will be assumed to be distributed within the microstructure as dictated by the equilibrium diagram at all times, since it diffuses interstitially in the solid at a high rate. To verify the uniform distribution of C in the iron, line scans and dot maps for C were obtained on both cast and quenched samples. The detection of C in a high density matrix is complicated, and EPMA does not have adequate sensitivity to detect the C distribution. Secondary Ion Mass Spectroscopy (SIMS), is sensitive enough to detect very small concentrations of C, but the data produced is qualitative, not quantitative. Both SIMS dot mapping and line scanning were used to analyze the distribution of C. The results are shown in Figure 5.19 (a,b) and 5.20. The area examined consists of graphite nodules enveloped in austenite shells and surrounded by quenched liquid, Figure 5.19(a). The appearance of one nodule at higher magnification is shown in Figure 5.19(b). The bright lines, A,B,C, are grooves left by the SIMS scans. A carbon dot map of the area in Figure 5.19(b), using SIMS is shown in Figure 5.20. High concentrations of  Chapter 5: SEGREGATION  AND  MICROSTRUCTURE.  -94-  carbon produce a brighter image in the dot map. The round bright area A, is the graphite nodule, the austenite ring around the nodule, B, is darker, and the quenched liquid, C, lighter. The brightness of the austenite ring is uniform, indicating there is no significant C segregation in the austenite as it solidified. Some inhomogeneities in the concentration of C are observed within the quenched liquid. These correspond to the austenitic regions of the quenched liquid, which contain less carbon then the cementite phase in the quenched liquid. A SIMS carbon scan was made between points A and C in Figure 5.19(b) giving the results shown in Figure 5.21. The concentration of carbon does not change significantly within the austenite, point A to point B, but varies appreciably in the quenched liquid, points B to C, reaching higher concentration levels. The large fluctuations in carbon level in the quenched liquid are due to the presence of cementite (higher signal) and austenite (lower signal) in the microstructure of the quenched liquid. A similar analysis was carried out on a sample of sand cast ductile iron produced in the following manner. After solidification was complete, the cast ingot was extracted from the sand mold and quenched in water when the temperature was approximately 900° C. As a result, the microstructure consists of martensite and retained austenite. No carbides were present. Mn was found to be segregated in the regions of retained austenite. Since Mn has an effective partition coefficient of 0.70 and C is also less than 1, both should segregate in a similar manner. To determine if the C did segregate with Mn, SIMS dot maps and line scans for C were made in the areas indicated in Figure 5.22. The dot maps, Figure 5.23(a,b), show high C levels in the nodules, as expected, but no variation in level in the matrix. Three line scans across the matrix, shown in Figure 5.24, exhibit small scale fluctuations in C level, but no longer range segregation pattern.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -95-  Figure 5.19: (a) Microstructure of a sample quenched during solidification, graphite nodule A is enveloped by austenite, which has transformed into martensite during sample preparation, (b) Detail of nodule A and surrounding solid. Note grooves left by SIMS scans.  Chapter 5:  SEGREGATION  AND MICROSTRUCTURE  .  Figure 5.22: Microstructure of cast sample. Note austenite patch D.  -96-  Chapter 5: SEGREGATION  AND  -97-  MICROSTRUCTURE.  CO  e  g  V  1  15 microns  QUENCHED  NODULE ! PRIOR AUSTENrTE  LIQUID  B  Distance  Figure 5.21: C line scan between points A and C in Figure 5.19 (b).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -98-  Figure 5.23: C map for: (a) Area shown in Figure 5.22. Note correspondence between nodules and high point density zones, (b) Top right corner of Figure 5.23.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -99-  110 microns Distance Figure 5.24: C line scan along lines indicated in Figure 5.22. Vertical axis offset.  In summary, the SIMS results show that carbon is distributed uniformly through the solidified austenite prior to the eutectoid transformation. The carbon level in the quenched liquid is higher than that in the austenite. This suggests that the last austenite to solidify should contain higher carbon levels. This was not observed due, it is believed, to the high rate of solid diffusion for carbon in solid iron.  Chapter 5: SEGREGATION  5.3.1  AND MICROSTRUCTURE  .  -100-  COMPARISON OF MEASURED SEGREGATION WITH CALCULATIONS BASED ON T H E SCHEIL EQUATION  The solute distribution in a directionally solidified binary alloy, assuming no diffusion in the solid and complete mixing in the liquid, is given by the Scheil equation:  C, = * C ( 1 - £ ) 0  (5.1)  V 1  0  Where: C, = concentration of solute in the solid k = equilibrium segregation coefficient 0  C = initial solute concentration in the liquid 0  g = fraction solid Equation 5.1 can be rearranged to calculate the concentration of solute in the residual liquid, C , as follows: L  ^ = C = C (l-g) *°  (5.2)  1_  t  0  Under non-equilibrium conditions it is possible that the equation may describe the solute segregation if k is replaced by an effective segregation coefficient, k . This was 0  e  shown to be the case by Burton, Primm and Slichter [59] for vertical crystal growth with the solid rotating during solidification. To determine whether the Scheil equation can be applied to the present observations of segregation, the experimental results were compared to the equation, using the effective segregation coefficients determined experimentally (Table X).  Chapter 5: SEGREGATION  AND  MICROSTRUCTURE  - 101 -  The fit between measurements of the solute concentration as a function of the solid fraction, and the calculations based on the Scheil equation, has been quantified by a parameter F, calculated according to the following expression:  F(%)=^I|^ | i  (5.3)  Where: F(%) = parameter characterizing the fit, perfect fit for F=0 j) = solute concentration calculated by the Scheil equation /,' = solute concentration measured by EPMA N = number of measurements done for a given alloy sample  5.3.1.1  Solute concentration in the liquid during solidification using k  e  The calculated solute concentration of the liquid as solidification progresses is plotted in Figures 5.25 to 5.34. In each figure the solid line is the calculated curve using equation 5.2 and k in Table X. The experimental EPMA measurements from Tables VI, e  VII and VIII are indicated by the points, including error bars as determined in section 4.5. The horizontal axis corresponds to the fraction solid measured by Image Analysis. These measurements should also include an experimental error bar, but the determination of the magnitude of this error is difficult under the present experimental conditions, since the fraction solid is changing continuously through the sample, and in consequence the areas measured are small, and only one field can be analyzed for each experimental point. The melt composition at the start of solidification is indicated by the horizontal line. Comparing the experimental points with the calculated curves, in general there is a good  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 102-  correlation between the two in all the figures. The measured concentrations increase or decrease in conformity with the calculated curves, and nearly all the points are close or on the calculated curves.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -103 -  +1  3.5 3 2.5 -  § •a  V  2 -  ca  a  a  r-  o CA  I  00  S  2  20  40  60  80  100  Solid Fraction (%)  Figure 5.26: Mn concentration in liquid as a function of solid fraction, Mn=1.05%. 2.4 2.2 2 1.8 1.6  § •a a  I  U  so  1.4 1.2 1 0.8 0.6 0.4  n  1 1 1 1 1 1 20  40  60  Solid Fraction (%)  r  100  80  Figure 5.27: Mn concentration in liquid as a function of solid fraction, Mn=0.73%.  Chapter 5: SEGREGATION  0 -f 0  AND MICROSTRUCTURE  .  -104-  1 1 1 1 1 1 1 1 1 20  40  60  80  1  100  Solid Fraction (%)  Figure 5.28: Mn concentration in liquid as a function of solid fraction, Mn=0.41%.  2  -.  :  1  1  1  1.8 1.6  o  H  H  0  1  1  20  1  1  1  40  1  60  1  1  80  100  Solid Fraction (%)  Figure 5.29: Cu concentration in liquid as a function of solid fraction, Cu=1.36%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 105-  100 Solid Fraction (%)  Figure 5.30: Cu concentration in liquid as a function of solid fraction, Cu=0.50%.  §  •3  s § C  3o U  100 ,  Solid Fraction (%)  Figure 5.31: Cu concentration in liquid as a function of solid fraction, Cu=0.91%.  Chapter 5: SEGREGATION  0  AND MICROSTRUCTURE  20  40  -106-  .  60  80  100  Solid Fraction (%)  Figure 5.32: Cr concentration in liquid as a function of solid fraction, Cr=0.50%.  Solid Fraction (%)  Figure 5.33: Mo concentration in liquid as a function of solid fraction, Mo=0.83%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  - 107-  .  1.2 -r 1.1 1 0.9 -  0.1 0  4 0  1  1  20  1  1  1  1  1  40 60 Solid Fraction (%)  1  80  1  100  Figure 5.34: Ni concentration in liquid as a function of solid fraction, Ni=0.83%.  5.3.1.2 Solute distribution in the solid using k  e  The concentration of solute in the solid as a function of fraction solid has been calculated on the basis of equation 5.1, using the effective segregation coefficient k . The e  results for the alloying elements considered are plotted in Figures 5.35 to 5.45. The results for Mn at four melt composition levels are shown in Figures 5.35 to 5.38. The calculated curve and the experimental points show some agreement, with best fit obtained for the 1.36%Mn alloy. In general the calculated values are higher than the experimental values above 50% solid fraction. The results for three alloys containing copper, in which k >\, are shown in Figures 5.39 to 5.41. Some agreement is obtained between the t  calculated and measured values. In this case the calculated values tend to be below the experimental values.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -108-  A reasonably goodfitbetween analysis and experiment is observed for the Mo containing alloy, Figure 5.42, but not as good fit was evident for alloys containing Cr, Figure 5.43, and Si, Figure 5.44. The calculations for the alloy containing Ni, Figure 5.45, follow the experimental results, but these are appreciably larger than the calculations. Consideration of the results in Figures 5.35 to 5.45 shows that the measured solute concentration tends be higher than the calculated concentrations for elements having effective segregation coefficients greater than one, with the exception of the Si. The measured concentrations of elements having segregation coefficients less than one are generally lower than the calculated values. This difference between measurement and calculation can be attributed in part to a systematic error. The composition of the austenite next to the interface is actually measured at 1 to 1.5 microns from the interface position, for reasons discussed in section 5.2.2. The expected segregation in a specimen quenched during solidification is shown schematically in Figure 5.46. Curve 1 represents the solute distribution for an alloying element of k>l. The concentration of the alloying element measured near the interface, A', is higher than the actual concentration at the interface, A. For elements of k<l, curve 2, the concentration of the alloying element near the interface, B', is lower than the concentration at the interface, B. Therefore, the experimental procedure introduces an error that can account for some of the misfit between measurements and calculations. The anomalous behavior observed in Si containing alloys can not be accounted for.  Chapter 5: SEGREGATION  0  20  AND MICROSTRUCTURE  40  60  ...  -109 -  80  100  Solid Fraction (%)  Figure 5.35: Mn concentration in solid as a function of solid fraction, Mn=1.34%.  0.7 0.6  H  0  1  1 1 1  20  40  1  1  60  1 1 1 80  100  Solid Fraction (%)  Figure 5.36: Mn concentration in solid as a function of solid fraction, Mn=1.05%.  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  - 110-  Solid Fraction (%)  Figure 5.37: Mn concentration in solid as a function of solid fraction, Mn=0.73%.  0.2  H  0  1  1 1 20  1  1 1 1 1 1  40  60  80  1 100  Solid Fraction (%)  Figure 5.38: Mn concentration in solid as a function of solid fraction, Mn=0.41%.  Chapter 5: SEGREGATION  20  AND MICROSTRUCTURE  40  60  .  - Ill -  100  Solid Fraction (%)  Figure 5.39: Cu concentration in solid as a function of solid fraction, Cu=1.36%.  100 Solid Fraction (%)  Figure 5.40: Cu concentration in solid as a function of solid fraction, Cu=0.50%.  Chapter 5: SEGREGATION  0.2 -i 0  1  AND  1  20  MICROSTRUCTURE.  1  1  40  1  1  Solid Fraction (%)  60  1  1  80  1  100  Figure 5.41: Cu concentration in solid as a function of solid fraction, Cu=0.91%.  0  20  40  60  80  100  Solid Fraction (%)  Figure 5.42: Mo concentration in solid as a function of solid fraction, Mo=0.83%  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  §  •a  I g U o  100 Solid Fraction (%)  Figure 5.43: Cr concentration in solid as a function of solid fraction, Cr=0.50%.  e o •a  8 c o  U c  8  100 Solid Fraction (%)  Figure 5.44: Si concentration in solid as a function of solid fraction, Si=2.45%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -114-  Solid Fraction Figure 5.46: Schematic showing expected solute concentration in liquid and solid.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  5.3.1.3 Solute distribution in the solid using k  .  - 115 -  ee  The segregation coefficient k„, depends on the fraction solid, different from k  e  which is constant through solidification. The variation of the effective segregation coefficient,  with the fraction solid was considered in section 5.2.3. Equations  representing the change in the effective segregation coefficient k with the fraction solid te  are shown in Figure 5.18(a-e). The following equation, describing the segregation when the segregation coefficient changes linearly with the solid fraction, has been derived in Appendix 3:  C = (Y+ P*) C exp{(|3 + y) ln(l - g) + pg} s  0  (5.4)  The calculated results, shown as lines, are plotted, together with the experimental points, in Figures 5.47 to 5.56. The results for Mn containing alloys are shown in Figures 5.47 to 5.50. The calculated curve is observed to fit the experimental points fairly well. For solid fractions above 80% the curve tends to be higher than the points. For Cu containing alloys, Figures 5.51 to 5.53, the calculated values are generally lower than the measured values, but the shape of the analytical curve tends to follow the path of the measurements. For Mo, Figure 5.54, the calculated solute distribution is considerably higher than the measurements. In the case of Cr, Figure 5.55, the fit is fairly good for solid fractions up to 50%, but becomes poor for larger solid fractions. Similarly, for Ni, Figure 5.56, the fit is good only for solid fractions below 50%.  Chapter 5: SEGREGATION  0  AND MICROSTRUCTURE  20  40  .  60  - 116-  80  100  S o l i d Fraction (%)  Figure 5.47: Mn concentration in solid as a function of solid fraction, Mn=1.34%.  0  20  40  60  80  100  Solid Fraction (%)  Figure 5.48: Mn concentration in solid as a function of solid fraction, Mn=1.05%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 117 -  0.4 0.3  H  0  1  1  1  20  1  40  1  Solid Fraction (%)  i 60  1 1 1 80  100  Figure 5.49: Mn concentration in solid as a function of solid fraction, Mn=0.73%.  0 H 0  1—:  1  20  1  1  40  1 1 1 1 1  Solid Fraction (%)  60  80  100  Figure 5.50: Mn concentration in solid as a function of solid fraction, Mn=0.41%.  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  I § u  T  20  40  60  100  Solid Fraction (%)  Figure 5.51: Cu concentration in solid as a function of solid fraction, Cu=1.36%.  g •a  s I  u 100 Solid Fraction (%)  Figure 5.52: Cu concentration in solid as a function of solid fraction, Cu=0.91%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  -119-  .  0.8 0.7 -  0  20  40  60  80  100  Solid Fraction (%)  Figure 5.53: Cu concentration in solid as a function of solid fraction, Cu=0.50%.  Chapter 5: SEGREGATION  0.2  -I 0  AND MICROSTRUCTURE  1 1 20  1  1  1  .  1 1  40 60 Solid Fraction (%)  - 120-  1  80  1  100  Figure 5.55: Cr concentration in solid as a function of solid fraction, Cr=0.50%. 1.5  -i  1  1.41.3 1.2 1.1 -  Solid Fraction (%)  Figure 5.56: Ni concentration in solid as a function of solid fraction, Ni=0.83%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 121 -  5.3.1.4 Solute concentration in the liquid during solidification using k  ee  The solute concentration in the liquid during solidification was calculated using the following equation, derived in Appendix 3: C = C exp{(f3 - r ) ln(l - g) + L  fe}  0  (5.5)  The results are plotted as lines in Figures 5.57 to 5.66. For Mn, Figures 5.57 to 5.60, the calculated values show a fairly good agreement with the experimental points plotted. For Cu, Figures 5.61 to 5.63, the calculated values are generally lower than the experimental points. For Cr, Figure 5.62, there is a fairly good fit. In the case of Mo, Figure 5.63, the fit is generally good, with the exception of the point corresponding to a solid fraction of 72%, which is probably erroneous. For Ni, Figure 5.64, the fit is only good for solid fractions below 50%.  1  H  0  1  1  20  1 1 1 40 Solid Fraction (%)  1  60  1  1  80  1 100  Figure 5.57: Mn concentration in liquid as a function of solid fraction, Mn=1.34%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 122-  4 3.5  c o  S  c 8 c o O c rt o c> <D CO CD  3 2.5 2 1.5 1  I  ~"~  0.5 0 20  40  60  80  100  Solid Fraction (%)  Figure 5.58: Mn concentration in liquid as a function of solid fraction, Mn=1.05%.  c o  1h3 _  c 8 c o O <D  %  c  ? as  2  100 Solid Fraction (%)  Figure 5.59: Mn concentration in liquid as a function of solid fraction, Mn=0.73%.  Chapter 5: SEGREGATION  0.1  -I  0  H  0  1  1  20  AND MICROSTRUCTURE  1  .  -123-  1 1 1 1 1  40 60 Solid Fraction (%)  80  1  100  Figure 5.60: Mn concentration in liquid as a function of solid fraction, Mn=0.41%. 1  2  1.8 -| 1.6  H  0  20  40  60  80  100  Solid Fraction (%)  Figure 5.61: Cu concentration in liquid as a function of solid fraction, Cu=1.36%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 124-  1  0  20  40  60  80  100  Solid Fraction (%)  Figure 5.62: Cu concentration in liquid as a function of solid fraction, Cu=0.50%. 0.6 n  0  1  20  40  60  80  100  Solid Fraction (%)  Figure 5.63: Cu concentration in liquid as a function of solid fraction, Cu=0.91%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 125-  § •a  3  §• I u E o  100 Solid Fraction (%)  Figure 5.64: Cr concentration in liquid as a function of solid fraction, Cr=0.50%.  c  u  I  s 100 Solid Fraction (%)  Figure 5.65: Mo concentration in liquid as a function of solid fraction, Mo=0.83%.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 126 -  1.2 1.1 -  0.9 0.8 €  0.7 -  ~  0.6-  |  0  4  "  <-> 0 . 3 -  0.1 0 + 0  20  40  80  60  100  Solid Fraction (%)  Figure 5.66: Ni concentration in liquid as a function of solid fraction, Ni=0.83%.  5.3.1.5  Analysis of the fit between calculations and experiments  Values of the fit factor, F, equation 5.3, comparing the calculated and measured values of the liquid and solid phases during solidification, based on the two types of segregation coefficients considered, k and k„, are listed in Table XI. F=0 for perfect fit. e  For the samples containing Mn, F values are small in all cases, with a maximum of 15.4, and a minimum of 4.5. The fit is in general slightly better for the values calculated using k . For the alloys containing copper, the value of F is between 15 and 31 for the te  calculations of solute concentration in the solid based on k , but increases to higher values t  for calculations using k„. F factors are low for the solute concentration in the liquid using k , with a maximum error of 11.6. On the other hand, the error is considerably larger e  Chapter 5: SEGREGATION  AND  MICROSTRUCTURE  - 127-  when k is used. For Cr, F values are between 15 and 10 in all cases. For Mo, F values ee  for the liquid are small for both segregation factors. For the solid, the values using k  t  show smaller error. For Ni, the fit is much better for the values calculated using k . e  The experimental error of the EPMA concentration measurements has been estimated to be 10% (section 4.5). As a result values of F equal or smaller than 10 indicate very good fit. It is likely that a better fit would be obtained if the error in the measurement of the solid fraction would have been considered in the value of F determined. Table XI: Fit factor, F, for calculations based on k and k, e  F for SOLID  F for LIQUID  (alloy)  K  K  Cl(Mn)  11.7  7.9  4.5  4.7  C7(Mn)  12.4  10.6  7.5  6.8  C8(Mn)  13.9  10.7  10.3  9.7  C9(Mn)  8.6  8.6  15.4  12.7  C2(Cu)  31.0  69.7  6.9  52.2  ClO(Cu)  19.5  33.8  6.9  34.3  Cll(Cu)  15.0  29.4  11.6  35.2  C3(Cr)  13.1  10.3  14!4  15.2  C4(Mo)  19.9  29.6  12.7  11.8  C5(Ni)  26.7  >100  8.4  93  C6(Si)  6.04  —  —  —  SAMPLE  Chapter 5: SEGREGATION  5.3.2  AND MICROSTRUCTURE  .  -  128-  C O R R E L A T I O N B E T W E E N T H E SOLIDIFICATION STRUCTURE AND T H ESEGREGATION PATTERN AROUND NODULES  The manner in which ductile iron solidifies, described as the equiaxed growth of spherical eutectic cells, as shown in section 5.2.1, is not consistent with the observations of the solidification morphology of quenched samples, illustrated in section 5.2. In order to understand the segregation patterns obtained during the analysis of cast samples, described in section 5.1.1, the microsegregation must be correlated with the microstructure during solidification. It has been shown in section 5.3.1, that the concentration of solute in the residual liquid can be calculated with good accuracy by using the Scheil equation and effective segregation coefficients. Considering the segregation of an alloying element of k,<\, in which case solute concentration in the residual liquid increases with solid fraction, the following types of segregation can occur along lines between nodules: i)  Two nodules are enveloped by adjacent austenite dendrite secondary branches, as shown schematically in Figure 5.67 (a), during solidification. A plot of solute concentration along the A-B after solidification is complete, will show an increase in the concentration of solute at the mid-distance between nodules, as illustrated in Figure 5.67 (b), since the liquid is richer in solute.  Chapter 5: SEGREGATION  ii)  AND MICROSTRUCTURE  .  - 129-  Two nodules are separated by a dendrite, as illustrated in Figure 5.68 (a), during solidification. A solute concentration plot along line C-D after solidification is complete, will show a decrease in the solute concentration at the mid-distance between nodules, as shown in Figure 5.68 (b).  iii)  Two nodules are enveloped by austenite secondary dendrite arms, which are in turn separated by another dendrite arm, as illustrated in Figure 5.69 (a). A plot of solute concentration along line E-F after solidification is complete, will show two peaks of solute concentration, as illustrated in Figure 5.69 (b).  iv)  Two nodules are separated by an area corresponding to the last liquid to solidify, as illustrated in Figure 5.70 (a). The concentration along line G-H will be as shown in Figure 5.70 (b). The intensity of the segregation is in this case greater than that shown in Figure 5.67. When line scans of solute concentration between two nodules are made by EPMA,  the specific solidification process and structure between the nodules is in general not known. Considering the four possibilities outlined above, and the markedly different solute segregation obtained in the four cases, the variation in the results shown in section 5.1 could be expected. From the results, it is concluded that the analysis of the extent of solute segregation in ductile iron after solidification should not be based on composition scans between randomly chosen pairs of nodules, but should rely on the identification and analysis of highly segregated areas, as done in section 5.1.2. Note that if precipitate phases are present in a highly segregated area, this area should not be used to calculate a segregation ratio, since this would compare solute concentrations of different phases. The stoichiometry of the precipitate phase may be independent of the segregation.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -  130-  Figure 5.67: Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 131 -  Figure 5.68: Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -132-  Figure 5.69: Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -133-  Figure 5.70: Schematic representation of microstructure during solidification (a), and corresponding segregation profile (b).  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 134-  5.3.3 C O M P A R I S O N O F T H E S E G R E G A T I O N I N S A N D - C A S T AND QUENCHED SAMPLES  It has been reported that large section ductile iron castings contain a higher density of precipitate compounds and carbides than small section castings of the same alloy composition [80]. This is attributed to higher segregation levels in the large casting. The results of the present investigation suggest that this is not necessarily the case. The concentration of solute next to the graphite nodules and in the last material to solidify in sand mould castings, as listed in Table VI, can be compared to the concentration of solute at points A and C of the quenched samples, listed in Tables VIII, DC, and X. Comparing the two it is evident that the segregation in the quenched samples is of the same magnitude as that in the sand-cast samples. This result supports the assumption that similar segregation mechanisms operate in cast and quenched samples, and that the effective partition coefficients determined in quenched samples, can be used for the calculation of segregation in ductile iron solidified at a slower rate than in a quenched sample  5.4 E F F E C T S O F T H E S E G R E G A T I O N O N T H E M I C R O S T R U C T U R E O F DUCTILE IRON  Recently, increasing attention is been paid to the understanding of the process leading to the formation of the microstructure in cast irons. This requires the microsegregation of alloy components to be known quantitatively. Nevertheless, no analytical formulations suitable to predict segregation in cast iron have been published, to the best of our knowledge, except for the present author [60].  Chapter 5: SEGREGATION AND MICROSTRUCTURE .  The influence of the microsegregation on  - 135-  both the microstructure formation in  ductile iron and its heat treatment, will be considered based on the segregation characteristics of different alloy components determined in this investigation.  5.4.1  INFLUENCE OF MICROSEGREGATION ON THE  CAST  STRUCTURE  The microstructure of ductile iron as it evolves during solidification under normal conditions has been described in section 5.2.1. This section will deal with the phase transformations in the solid state as the cast iron cools. It can be assumed that immediately below the eutectic temperature the microstructure consists of austenite saturated in carbon and graphite spheroids. The silicon, always present in the composition of cast irons, has been shown to have a weak tendency to segregate towards the first solid to form during solidification. This inhomogeneity in the concentration of Si will be neglected at this point, and it shall be initially assumed that the austenite is chemically homogeneous. As the casting cools from the eutectic to the eutectoid temperature, the solubility of C in austenite diminishes. If the cooling rate is sufficiently slow, and the cast iron unalloyed, the diffusivity of C in austenite is high enough to allow the carbon to deposit on the graphite nodules, which act as carbon sinks. Therefore, no large C gradients are expected to exist within the austenite. When the eutectoid temperature is reached, the transformation proceeds according to the stable Fe-C-Si equilibrium diagram, which indicates that almost all the C dissolved in the matrix must precipitate in the form of graphite. At this point, for temperatures below the eutectoid, the matrix is fully ferritic.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -  136-  Let us now restrain one of the initial assumptions, and consider a faster cooling rate. In this case, some inhomogeneity can be present in the austenite immediately above the eutectoid temperature, with C being concentrated in the austenite located far from graphite nodules. The eutectoid ferrite will preferentially nucleate near the graphite, where the C concentration is smaller. As the eutectoid transformation takes place, the imposed high cooling rate can avoid the completion of the transformation predicted by the equilibrium diagram, and metastable pearlite will form in C rich areas far from the graphite. As the cooling rate is further increased, less carbon is transferred from the austenite to the nodules. As a result, the amount of pearlite formed increases. When very little time for the diffusion of C is allowed at the eutectoid temperature, only minor amounts of ferrite form surrounding the graphite, producing the structure called "bull's eye". The presence of an additional alloying element will affect the microstructure mainly due to the following effects: i)  Alloying elements will change both the solubility and the diffusivity of C in austenite.  ii)  Alloying elements segregate throughout the matrix. Inhomogeneities in the C distribution can be therefore expected, since areas of different element concentration will have different equilibrium C concentrations. Alloying elements such as Mn, which increases the solubility of C in austenite and  segregates in the last liquid to solidify, will affect the microstructure formation as follows:  Chapter 5: SEGREGATION  iii)  AND MICROSTRUCTURE  .  - 137-  During cooling from the eutectic temperature, Mn segregated areas formed during solidification will bericherin C than the bulk of the matrix. Since such areas are generally located far from graphite particles, the transport of C to the graphite will be more difficult than for unalloyed iron. In consequence, even for slow cooling, metastable Fe-Mn carbides can precipitate in the segregated areas when Mn is present in the alloy.  iv)  At the eutectoid temperature, the alloyed austenite has a higher C content than the unalloyed austenite. In consequence, more C has to be transported to the graphite particles in order to produce a fully ferritic matrix. Under normal cooling conditions, and above a certain Mn level, it becomes essentially impossible to obtain fully ferritic matrices. Mn alloyed irons have less ferrite than unalloyed irons solidified under similar  conditions. The ferrite will preferentially form around the graphite nodules, where the Mn concentration is at a minimum and C can be readily transported to the graphite. Alloying elements are often used to produce as-cast pearlitic microstructures. It has been shown [61,62] that strong pearlitizing elements such as Mn and Mo, cannot produce a fully pearlitic matrix free from carbides. This can be explained on the basis of the segregation characteristics of those elements. Both Mn and Mo are strongly concentrated in the last material to solidify, and depleted in the areas surrounding the graphite nodules. Therefore, if it is desired to obtain pearlite next to the graphite nodules, it becomes necessary to use an alloy concentration large enough to bring the concentration of solute in the vicinity of a nodule to a level necessary to avoid the formation of ferrite, ie. to avoid completely the stable eutectoid transformation.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 138 -  Nevertheless, when such a high concentration of the alloying element is used, the solute concentration in the last liquid to solidify is enough to result in the precipitation of carbides. Therefore, successful pearlitizing alloying usually consists of the combination of two or more alloying elements having opposite segregation characteristics. For example, considering combined Cu and Mn additions, Cu tends to concentrate in the first solid to solidify, and thus is concentrated in the vicinity of the graphite nodules. Mn is concentrated in the last material to solidify. The Cu enriched regions produce pearlite in the vicinity of the nodule, and the Mn enriched regions in the areas remote from the nodules. This can lead to a ductile iron which is fully pearlitized. Thus, multiple alloy additions are generally used when pearlitization of the matrix is required. If the production of fully ferritic structures is desired, the alloying elements must be kept as low as possible. In practice, ferritic irons are generally unalloyed, and special care is given to the selection of the charge materials. The utilization of steel scrap is limited to low manganese grades. Silicon is always present in gray cast iron, and therefore it is not usually referred to as an alloying element. The Si presents unique characteristics; it is a graphitizing element and diminishes the solubility of C in austenite. The larger the amount of Si present in the cast iron, the larger the amount of ferrite in the structure at room temperature. In practice the Si content is kept low in ferritic cast iron because of its tendency to embrittle the ferrite. Nickel, a graphitizing element in cast iron, is also a pearlitizing alloy, since it increases the solubility of C in austenite. Similar to the effect of Mn, the austenite in Ni alloyed cast iron has higher C content than unalloyed cast iron at the same temperature.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  -139-  This effect becomes critical when the eutectoid temperature is reached during cooling, since Ni alloyed austenite must reject a large amount of C in order to transform into the stable eutectoid. It is possible to obtain pearlitic structures in ductile iron by using only Ni, but this is highly inefficient when compared to the use of combined amounts of Mn, Mo, Cu and Ni.  5.4.2  INFLUENCE  OF SOLUTE SEGREGATION  HARDENABILITY OF CAST  ONT H E  IRONS  All alloying elements commonly used in cast iron increase the hardenability of the material [62,63]. Mo and Mn additions increase the hardenability the most. Cu and Ni are less effective, and Si has little effect. Lee and Voigt [63] studied the influence of combined amounts of alloying elements on the hardenability of ductile iron. They found that in some cases the combination of two or more elements has greater effect on the hardenability than that expected from the individual elements. In particular, combinations Mo-Ni, Mo-Cu and Mo-Cu-Ni, were reported to markedly enhance the hardenability of ductile iron. This interaction effect was referred by Lee and Voigt as a synergistic effect, with no further explanation given for the rational behind such interactions. Bearing in mind the segregation characteristics of the alloying elements described earlier in this chapter, it is evident that the interactions reported by Lee and Voigt always occur between elements which have effective segregation coefficients greater and less than unity. This suggests that the interactions may be related to the distribution of the alloying elements within the matrix. To contemplate this possibility consider first the criterion used to quantify hardenability. It is customary to define hardenability as the distance from the end of a Jominy test bar at which pearlite ,(or lamellar constituents) is first observed.  Chapter 5: SEGREGATION  AND MICROSTRUCTURE  .  - 140-  The first pearlite is generally formed next to graphite nodules. The alloying elements Mn and Mo, which strongly influence the hardenability of ductile iron, are concentrated in the last material to solidify, leaving the regions surrounding the larger graphite nodules depleted in solute. The depleted areas have lower local hardenability, and will be the site for the first pearlite to precipitate. When the alloying element has a segregation coefficient greater than one, as in the case of Cu, the first pearlite in the Jominy test does not necessarily appear next to the graphite nodules [63], where the concentration of Cu is generally higher than the average concentration in the matrix. Therefore, it can be concluded that the increased hardenability obtained in samples containing two alloying elements having segregation coefficients greater and less than unity, results from the segregation of each element in the iron. Elements having k>l contribute to the hardenability of the first material to solidify; elements having k<l increase the hardenability of the last material to solidify in the interdendritic regions.  141  Chapter 6  SOLIDIFICATION MODEL  The mathematical model of the solidification of ductile iron developed in this investigation consists of four parts:  1)  A thermal model that calculates temperatures throughout the casting and mold.  2)  A nucleation model, which calculates the nucleation rate as a function of the supercooling.  3)  A growth model, which calculates the growth of both austenite and graphite.  4)  A segregation model, which calculates the composition of the residual liquid during solidification.  These models, when coupled together into an overall solidification model should be able to predict the cooling curve for a casting of a given geometry, and to predict the number of nodules and their size distribution.  Chapter 6:  SOLIDIFICATION  - 142-  MODEL  6.1 T H E R M A L MODEL A heat transfer mathematical model is applied to the casting system described in section 4.3, in which the iron is cast in a water cooled cylindrical sand mould. The case of non-cooled sand moulds is considered in Appendix 4.  6.1.1 ASSUMPTIONS AND BOUNDARY CONDITIONS The following assumptions are made: 1)  No heat is lost from top and bottom surfaces of the metal.  2)  The heat transfer between the casting and the mould is characterized by a surface heat transfer coefficient, h.  3)  The water cooled copper coil acts as a constant temperature boundary surface. The accuracy of this assumption was determined experimentally by measuring the sand temperature between two turns in the copper coil ( TC4 in Figure 4.4) during casting. The temperature did not exceed 25°C, which validates the assumption of constant temperature boundary surface. The boundary conditions are shown in Figure 6.1.  6.1.2 HEAT CONDUCTION EQUATIONS The general equation for heat conduction in cartesian coordinates x, y, z is [64]: BT  V(fcV7) + / / = pc ^p  (6.1)  Chapter 6: SOLIDIFICATION MODEL  -143-  In the case of a cylindrical system, it is more appropriate to express equation (6.1) in cylindrical coordinates by expressing x and y as: x = r cos 0  (6.2)  y=r sinG  (6.3)  Replacing equations (6.2) and (6.3) in (6.1): (6.4)  t  Q =0  Figure 6.1: Casting system, showing the assumed boundary conditions for the thermal model.  Chapter 6: SOLIDIFICATION  MODEL  - 144-  In accordance with the boundary conditions shown in Figure 6.1, the temperature field is axisymmetric, and the temperature is uniform along z axis, therefore: dT 30  =0  (6.5)  (6.6) Substituting equations (6.5) and (6.6) in (6.4) gives:  rdr[  dr  +H = p  dT Cl  dt  (6.7)  The first term at the left side of equation (6.7) can be expanded to give: lf.drT}  if  dkdT^  r  drdr  (6.8)  The derivative of k with respect to r can be written as: dk_dk_dT  (6.9)  dr~dTdr  Substituting equation (6.9) in (6.8) results in: f  ^\ 1 +-  r  d±dT^  r y*Tdr\  dr'  (6.10)  For most materials, the variation of k with temperature is small, and can be neglected in (6.10). Therefore equation (6.7) can be written as: 1, d*T „ dT -k — + H = pc — r dr dt 2  p  (6.11)  Chapter 6: SOLIDIFICATION  - 145-  MODEL  The fact that k, c and p are temperature dependent greatly complicates the p  analytical solution of the differential equation (6.11). It becomes necessary to solve the heat conduction problem numerically. It has been shown in the literature [65] that the method of implicit finite differences is suitable for the solution of equation (6.11). The numerical solution of equation (6.11) requires the system to be divided into volume elements. The arrangement of volume elements adopted is shown in Figure 6.2. The volume elements are hollow cylinders of wall thickness dr. Six types of volume elements can be found in the system. Heat transfer equations for each volume element type have been derived in Appendix 4:  i) Axial volume element  2  Akdt  (6.12)  Ti -—ji =-7^Ti+4rRHG(i) +l  dr  +1  dr  1  4kdt  2  1  4k  ii) Internal volume element of the casting 1  —T + J+ 1  dr  dr  2  2  (i-l.5)dr  2  PCp 77 +kdt  +1  1  _  1  {dr  (i-l.5)dr  2  j+i _ P P j C  T  2  T  kdt  1  1  k  RHG(i)  (6.13) iii) Volume element at the surface of the casting LTV+I  r i - i ^. P P) y kdr L/-1.5J mkj C  r  +  dr  2 +  K  + i  h kdr  f  i - l ^  li-1.5j  (6.14)  Chapter 6: SOLIDIFICATION  - 146-  MODEL  iv) Volume element at the internal surface of the mould.  kdr  kdr  +  t  a  drl  _1_  1  J_  ry + l  +  2r dr e  2kdt  t  1  dr?  2r dr  +  c  l  M+2  -  k  a  2dt  Vu+1  (6.15) v) Internal volume element of the mould ( —T  J  +  1  dr.  +  2 1 -r + — — dr; r dr m  PC,  7Y +1  kdt  a  drl  r dr m  kdt  i + 1 aj  (6.16) vi) Volume element at the external surface of the mould  —„Tdr,  pc  1 ^drl  r dr m  +  a  kdt  kdt '  drl  y  ^ 1  1  1  p  +  r dr n  aJ  (6.17) Equations (6.12) to (6.17) form a tridiagonal system.  6.1.3 INITIAL CONDITIONS The solution of the system of equations requires the specification of the initial conditions. The temperature of the melt at time t=0 is assumed to be uniform and equal to the pouring temperature minus a temperature drop to account for the heat loss during pouring. The initial temperature of the mold is taken as equal to the temperature of the water in the copper coil.  Chapter 6:  SOLIDIFICATION  MODEL  -147-  ]  Mold  Figure 6.2: Schematic of the volume elements arrangement.  6.1.4 SURFACE HEAT TRANSFER COEFFICIENT AT T H E METAL-MOULD INTERFACE The formation of air gaps between ductile iron castings and sand molds has been studied by Winter et al. [67]. Their investigation was carried out in a sand mold similar in shape and dimensions to that used in the present investigation, but without steel flasks. They measured the displacements of the mold and the casting with transducers, giving the results shown in Figure 6.3(a). Shortly after pouring, the casting and the mold expand. Since the expansion of the mold is larger than that of the ductile iron casting, a gap between the mold and the casting is formed. Approximately five minutes after pouring  Chapter 6:  SOLIDIFICATION  - 148-  MODEL  the dimension of the gap reaches a stationary value of about 0.06 mm. Cooling curves at several points within the casting are shown in Figure 6.3(b). Comparing Figures 6.3(a) and (b) show that the expansion of the ductile iron starts simultaneously with the beginning of solidification. The volume of the casting remains almost unchanged until the end of solidification, following which the ductile iron contracts as it cools. In the present investigation the use of steel flasked molds, and copper coils in one case, prevents the mold dilation, which in turn suppresses the formation of a gap between the mold and the casting during solidification. However, a gap could form after the completion of solidification, as a result of the contraction of the casting following solidification. Based on Winter's measurements, Zeng and Pehlke [43] calculated the heat transfer coefficient at the mold-casting interface for imperfect contact interfaces, Figure 6.4(a), and for air gap interfaces, Figure 6.4(b). The calculated coefficient for imperfect contact interfaces varies from 4.5 to 2.8 W/cm K, depending on thetimefrom pouring. When a gap forms, 2  the calculated coefficients are two orders of magnitude smaller than those calculated for imperfect contact In the present investigation mold and casting are assumed to remain in contact during solidification. Therefore, the heat transfer coefficient at the mold/casting interface in the calculations will vary between 4.5 and 2.8 W/cm K. Since thetimeat 2  which a gap forms at the casting/mold interface after solidification is complete is not known, and can not be estimated with reasonable accuracyfromthe results of Winter et al, the possible formation of a gap will be neglected. This may lead to some misfit between calculated and measured temperatures after the end of solidification.  Chapter 6: SOLIDIFICATION  MODEL  -149-  2500  Figure 6.3: (a) Motion of the mold and casting during the solidification of ductile iron in a sand mold, (b) Cooling curves for the same casting in (a), after [67].  Chapter 6: SOLIDIFICATION  -150-  MODEL  TIME (SEC.)  Figure 6.4: Calculated heat transfer coefficient (a) with imperfect contact interface; (b) with gap formation, after [43].  151  6.2  MODEL FOR GRAPHITE NUCLEATION  Su et al.[28] investigated the applicability of the nucleation model developed by Oldfield [40] to the solidification of ductile iron. They concluded that Oldfield's nucleation formula is not appropriate for ductile iron. No other relationship between graphite nucleation rate and supercooling, for ductile iron, has been reported in the literature, to the best of our knowledge. R. Heine et al.[47] developed the following equation describing the variation of the eutectic temperature as a function of the Si content, for the stable Fe-C-Si system: 7/ = 1155 + 6.5%Si  [°C]  £  (6.18)  For 2.5% Si, the eutectic temperature is 1171°C. It was shown in section 2.4, Table I, that, for ductile iron, both eutectic arrest and eutectic plateau temperatures occur well below 1171°C. This suggests that the nucleation of the eutectic requires significant supercooling. In addition, it has been shown by Wetterfall et al [12] that graphite nucleation continues in the eutectic liquid well after the start of eutectic solidification. In summary, eutectic nucleation starts at a given supercooling, and continues during solidification. Under normal solidification conditions, for most multicomponent alloys, the nucleation of solid phases from the liquid is heterogeneous. This is the case for the eutectic graphite. In heterogeneous nucleation, the rate of formation of nuclei per unit volume, N, as a function of the absolute temperature T, isgiven by equation (6.19) [77].  .  oW nXT  N =— = dt  V  exp  AG +AG' D  XT  (6.19)  Chapter 6: SOLIDIFICATION  MODEL  -  152-  Where: AG = activation energy for diffusion of atoms across the interface melt/nucleus D  AG * = activation energy for nucleation n = number of atoms per unit volume of the liquid phase X = Boltzman's constant \|/ = Plank's constant In liquids AG « AG*; therefore the nucleation rate depends on AG*. D  In the present investigation the nucleation rate has been calculated using equation 6.20. dN . .„ N = — = b AT dt  (  c ^  K  AT.  exp  (6.20)  Where b and c are constant values, for a given substrate and melt composition. The temperature at which the nucleation rate becomes important depends on the value of the interfacial energy between the nucleating phase and the substrate. Different nucleating substrates and changes in the melt composition will influence the nucleation rate. The variation in the nucleation rate with the supercooling given by equation (6.20) is represented in Figure 6.5. The nucleation rate remains very small, until a supercooling, generally referred to as critical supercooling, is reached. For greater supercooling the nucleation rate becomes significant. In the mathematical model, the nucleation rate will be calculated on the basis of equation (6.20). This can be discretized as:  Chapter 6: SOLIDIFICATION  MODEL  - 153-  •5 c p  "•S  I Supercooling Figure 6.5: Nucleation rate in heterogeneous nucleation. AN — — =ftAT  Ar  exp  (6.21)  AT  The number of nodules nucleated in a node /, during a time step j, DNU(i,j), is calculated by the following expression: DNU(i,j)  = b AT exp - —  (1  -g(i))At  (6.22)  The term (l-g(i)), where g(i) is the solid fraction in node i, is included in order to account for the progressive decrease in the amount of liquid phase in the volume element. Equation (6.22) will be referred to as "exponential nucleation equation".  Chapter 6: SOLIDIFICATION  - 154-  MODEL  Alternatively, it is proposed to test another formulation of the nucleation rate. A variation of the nucleation rate with the supercooling similar to that predicted by equation (6.20) is given by Equation (6.23). oW  -^  (6.23)  = a(T -T -T)  H  E  N  Where: a = nucleation constant T = eutectic temperature, from equation (6.18) E  T = critical supercooling for nucleation N  T = temperature of the melt n =exponent As solidification advances the amount of residual melt diminishes, therefore, equation (6.23) must be modified to: dN  = a(T -T -T) E  N  H  (1 -g)  (6.23.2)  In the model, the number of graphite nodules nucleated in atimestep are calculated as follows: DNU(i,j)  = a(T (i)-T -T(i)T(l-g(i))At E  N  (6.23.3)  Equation (6.23) will be referred to as "parabolic nucleation equation". A nucleation model similar to that given by equation (6.23) was proposed by Su et al[28], but they did not use a critical nucleation supercooling, and based the calculations on the temperatures given by the binary Fe-C equilibrium diagram.  Chapter 6: SOLIDIFICATION  -155-  MODEL  It is important to note at this point that T depends on the Si content in the residual E  liquid. The residual liquid is progressively depleted in Si, therefore the value of T  E  decreases as solidification progresses, resulting in a smaller effective supercooling. In consequence, the segregation of Si will lead to slower nucleation rates than those expected when segregation is not considered.  6.3  GROWTH  MODEL  The growth model calculates the growth of austenite and graphite. The growth of graphite can take place with the graphite surface in direct contact with the liquid phase, or with the graphite enveloped by solid austenite. The following assumptions are made in the formulation of the growth model: i)  Carbon is distributed throughout the cast structure according to the Fe-C-Si equilibrium diagram.  ii)  The proportion of carbon between austenite and graphite present in the casting during solidification is given by the Fe-C-Si equilibrium diagram at the eutectic temperature.  iii)  The austenite/liquid and austenite/graphite interfaces are at equilibrium.  iv) Silicon segregates, affecting the equilibrium at the austenite/melt interface.  6.3.1  G R O W T H O F G R A P H I T E IN C O N T A C T W I T H T H E M E L T  The growth of graphite in contact with the melt can be controlled by either of the following processes:  Chapter 6: SOLIDIFICATION  MODEL  - 156-  i)  Rate of carbon transport at the graphite/liquid interface (interface controlled).  ii)  Rate of carbon transport from the bulk of the liquid to the interface graphite/liquid (diffusion controlled). Subramanian et al. [48,49] proposed the following equation for the interface  controlled growth of graphite nodules in ductile iron: ^ = 5(10"*) (AT) [mm/sec] 2  dt  (6.24)  The integration of equation (6.24), assuming AT constant, leads to: r = r +5(10"*) (AT) r 2  0  (6.25)  On the other hand, Wetterfall et al [12] applied Zener's [69] equation for growth controlled by diffusion, in order to estimate the growth rate of graphite spheroids in direct contact with the melt. For our case, Zener's equation can be expressed as follows:  Where: C"* = carbon concentration in the austenite in equilibrium with liquid 1  C  l,gr  C  = carbon concentration in the liquid in equilibrium with graphite = carbon concentration in the graphite  p, and p = density of liquid and graphite respectively gr  D = diffusion coefficient of the carbon in the liquid c  o Chapter 6: SOLIDIFICATION  MODEL  -157-  ValuesforC" andC"« are derived in Appendix 5. Equation (6.26) can be r  r  rearranged as: (6.27)  (6.28)  The growth rate of the graphite spheroids was calculated on the basis of equations (6.24) and (6.26), for interface controlled and diffusion controlled growth respectively. The results are plotted in Figure 6.6, for different values of supercooling. For a supercooling of 10°C, and very small time, the interface controlled growth rate, curve a, is smaller than the diffusion controlled growth rate, curve c. After a very short period, the diffusion controlled growth rate becomes smaller, and consequently the growth can be considered to be controlled by the diffusion of C in the liquid during most of the period in which the graphite is in direct contact with the liquid. For greater supercoolings, both growth rates increase, but the diffusion controlled growth continues to be the smaller after a very short initial period. In practice, as shown in Table I, the solidification of ductile iron takes place at temperatures below 1150°C, which in turn involves supercoolings greater than 20°C. As a result the growth of graphite in contact with liquid is calculated using equation (6.26). In order to calculate the size increment of a graphite particle in a given time interval, on the basis of equation (6.26), it is necessary to derive an equation for Ar as a function of r and At. The expected radius of a graphite particle after time t and t is given t  by equations (6.29) and (6.30) respectively.  2  Chapter 6: SOLIDIFICATION  -158 -  MODEL  Time (sec) Figure 6.6: Growth rate of the graphite spheroids as a function oftime,for interface controlled, (a) and (b), and diffusion controlled growth, (c) and (d). rl = F{T)t,  (6.29)  r =F{T)t  (6.30)  2  2  2  The size increment can be calculated as: Ar =  r -r 2  1  (6.31)  r = r + Ar 2  l  Subtracting equation (6.29) from (6.30), and substituting in (6.31) gives: (r + A T ) - r = F ( r ) ^ - f ) 2  1  2  I  1  (6.32)  Chapter 6: SOLIDIFICATION  MODEL  - 159-  Rearranging equation (6.32) 2rjAr + Ar = F(T)At  (6.33)  2  Where: &t =  h-t  x  Rearranging (6.33): Ar + 2r, Ar -F(T)At 2  =  0  (6.34)  Equation (6.34) can be solved as: Ar = -r + ^rf + F(T)At t  (6.35)  Based on the observations of Wetterfall et al. [12], it will be assumed that graphite nodules are enveloped by austenite when their radii reach 6 microns.  6.3.2  GROWTH OF AUSTENITE  In the model it is assumed that the ratio of the amount of austenite and the amount of graphite is constant during solidification. Therefore, the amount of austenite is calculated on the basis of the amount of graphite present. The ratio between austenite and graphite volumes, REAG, is calculated in Appendix 5, and gives the following:  R  E  A  G  ~ 2.16-0.1015/  ( 6 3 6 )  Chapter 6: SOLIDIFICATION  MODEL  -160-  It is therefore possible to calculate the volume of austenite, V as a function of the Y  volume of graphite, V , as: gr  V = y  6.3.3  (6.37)  REAG.V  i  G R O W T H OF GRAPHITE ENVELOPED BY AUSTENITE  The cast structures described in section 5.2.1 indicate that as the solid fraction increases, graphite nodules become enveloped by austenite. Further growth of graphite requires the diffusion of C from the liquid to the graphite surface. The simplest case to be considered is that of a single graphite nodule enveloped by a spherical austenite shell. This case has been studied [12], and the following equation describing the growth rate of the graphite is reported:  dr  [C^-C \ nr  9PI  (6.38)  Equation (6.38) assumed that the external surface of the spherical austenite shell is totally in contact with the melt. In reality, as shown in sections 2.1 and 5.1, the austenite envelope results from the interaction of growing austenite dendrites and graphite nodules In the example illustrated in Figure 6.8, only the thinner portions of the austenite envelope, along path A, will significantly contribute to the graphite growth. In order to account for this effect, a correction factor, J, is introduced in equation (6.38), based on the formula used bu Su et al.[28], multiplied by 0.9. The correction factor is intended to represent the reduction in surface-area of the austenite liquid interface actually  Chapter 6: SOLIDIFICATION  MODEL  -161 -  contributing to the graphite growth. The correction factor used in the model is: 2/3  (6.39)  / =0.9(l-s)  Equation (6.39) is represented in Figure 6.8. The correction factor can be introduced in equation (6.38), resulting in equation (6.40):  * dt  a 9  l c^'-c^p^a-gf {C -C* gr  gr  3  (6.40)  o o co UL c o 'o <u \— o o  0.4 0.6 Solid Fraction  Figure 6.7: Correction factor applied to the growth rate of graphite enveloped by austenite.  Chapter 6: SOLIDIFICATION  - 162-  MODEL  Figure 6.8: Graphite nodule enveloped by austenite.  The solution of equation (6.40) requires S, the radius of the austenite envelope to be known. Under the assumption of a spherical austenite shell and steady state conditions, the ratio S/r is calculated from the relationship REAG derived above, to be 2.61. However, our case is more complex, since not all the austenite is enveloping nodules; some is forming dendrites. Wetterfall et al. [12] measured maximum and minimum radii of austenite shells enveloping graphite nodules on quenched specimens. Their results are shown in Table XII. On the basis of these results, the average of the ratio between the austenite shell radius, S, and the graphite radius, r, is calculated.  -=1.89 r  (6.41)  Chapter 6: SOLIDIFICATION  MODEL  -163 -  Equation (6.41) can be substituted into (6.40), leading to: r  .2/3  C-C*")  (6.42)  pr gr  Equation (6.42) can be expressed in finite differences as:  r-  +1  = r- + 1.911  (cT'-C^PyDctt-gf*  V  ^  (6.43)  Note that the growth rate is a function of the nodule size and fraction solid. In addition, since the carbon concentration depends on the temperature and melt composition, both temperature and composition influence the growth rate. It is therefore necessary to calculate the growth rate for each nodule size and at each temperature during solidification.  6.3.4  C A L C U L A T I O N OF T H E FRACTION SOLID AND RELEASE OF LATENT  THE  HEAT  The release of latent heat per unit volume andtime,RHG(I), is calculated from the increment in the fraction solid during atimestep, equation (6.44) RHG(i) =  A pL gi  Agi = increment in fraction solid during atimestep p = density of the cast iron L = latent heat of solidification  (6.44)  Chapter 6: SOLIDIFICATION  -164-  MODEL  Table XII: Measurements of austenite shell radius (after [12]). Nodule Radius  c  c  min  "-"avge  S e/r avg  7.  15  16  15.5  2.21  10  20  26  23  2.30  14  26  33  29.5  2.11  19  36  42  39  2.05  21  36  46  41  1.95  24  32  52  42  1.75  25  48  56  52  2.08  17  26  28  27  1.59  14  22  22  22  1.57  17  26  34  30  1.76  21  37  40  38.5  1.83  21  43  43  43  2.04  25  48  56  52  2.08  32  62  65  63.5  1.98  13  23  24  23.5  1.80  18  30  35  32.5  1.80  19  31  36  33.5  1.76  22  32  42  37  1.68  25  41  42  41.5  1.66  Mean value of S /r = 1.89 avge  Chapter 6: SOLIDIFICATION  MODEL  - 165 -  The information about number and size of graphite nodules in each volume element is contained in two two-dimensional arrays, DNU(i,k) and RNU(i,k). DNU stores the number of nodules of a given size existing in a volume element i. RNU stores the size of the graphite nodules. Therefore, at time j, the total volume of graphite in node i, VGR, can be calculated as:  VGRJ  = I ^nRNU(i,k) DNU(i,k) t = i3 3  (6.45)  Note that VGR is dimensionless; in other words, it is measures volume of solid per unit volume, which in turn is the solid fraction. Applying the relation between austenite and graphite volumes shown in equation (6.38), the solid fraction of a volume element i attimej is calculated from: (6.46)  gj = VGRj(l +REAG)  The change in solid fraction can be then calculated as: Ag=*/ -*/  (6.47)  +1  6.3.5  C A L C U L A T I O N S O F N O D U L A R SIZE  DISTRIBUTION  Information concerning nodular size distribution is contained in arrays DNU(i,k) and RNU(i,k). In the present formulation, the growth of graphite is assumed to stop when solidification is complete. In reality, graphite particles keep growing after the solidification is complete, as a result of the decrease in the solubility of C in austenite. At the eutectic temperature, austenite containing 2.5% Si dissolves 1.3 Wt% carbon. The. eutectic alloy contains 3.49% carbon, therefore, only 63% of the C in the alloy is in the  Chapter 6: SOLIDIFICATION  MODEL  - 166-  form of graphite at the eutectic temperature. Depending on the cooling rate and chemical composition, varying amounts of C will precipitate in the form of graphite during cooling below the eutectic temperature. For ferritic irons, only a negligible amount of carbon remains dissolved in the matrix; therefore, the volume of graphite increases approximately 59% as room temperature is reached. The volume of graphite at room temperature, VGR', can be calculated by the following equation: = 1.59VGR = 1.59 £ ^nRNU(i,kfDNU(i,k)  VGR'  (6.48)  *=i3 Equation (6.49) can be rearranged as follows: VGR = J  £^rc[1.167/?M/(i,fc)]DWl7(;,£) 3  k=l  (6.49)  3  In consequence, the precipitation of C in the solid state can be accounted for by increasing the diameter of the nodules by a factor of 1.167. Smaller correction factors should be applied for irons showing pearlitic or mixed matrices.  6.4  SEGREGATION  MODEL  As shown in Chapter 5, the concentration of solute in the residual liquid can be calculated with good accuracy by applying the Scheil equation and using the effective segregation coefficients listed on Table X. C^k.d-g)^  1  (6.50)  Detailed information about the influence of the alloying element on the equilibrium temperatures and compositions was found only for Si. As a result, the calculations are  Chapter 6: SOLIDIFICATION  MODEL  -167 -  only for segregation of this element. As Si segregates, it influences the equilibrium carbon concentration at the interfaces in the solidifying alloy. Figure 6.9 shows the changes in the equilibrium concentrations of C in austenite equilibrated with graphite, C * , austenite equilibrated with liquid, C \ and liquid equilibrated with both graphite, y  r  C , ilgr  y  and austenite, C" , as a function of the solid fraction, when the Si segregation is 1  considered. As the solid fraction increases the concentration of Si in the residual liquid decreases; therefore the equilibrium concentration of C increases with the solid fraction for all interfaces, as illustrated in Figure 6.9.  0  20  40 60 Solid Fraction  80  100  Figure 6.9: Concentration of C as a function of the solid fraction in: -austenite in equilibrium with graphite, Ca/g; -austenite in equilibrium with liquid, Ca/1; -liquid in equilibrium with austenite, Cl/a; -liquid in equilibrium with graphite, Cl/g.  Chapter 6: SOLIDIFICATION  MODEL  -168-  It has been shown in Chapter 5 that the diffusion of Si in austenite is small and can be neglected for this analysis. In consequence, the Si concentration in the austenite adjacent to a graphite nodule can be considered to remain constant through the cooling of the casting to room temperature. It is shown in Appendix 5 that the carbon concentration in austenite equilibrated with graphite is given by the following equation:  C  igr  =  T-1154.6-6.55/ (1.5 - 0.2165/) + 2.1 - 0.2165/ . 354.6 + 6.55/  (6.51)  Assuming that the austenite surrounding most of the graphite nodules was formed early during solidification, the Si content in the austenite next to graphite nodules can be determined by: 5/^ = 5/ .k  (6.52)  Si  The carbon concentration at the austenite next to the graphite nodules, C , can be ,gr  determined on the basis of the Si concentration, by substituting equation (6.52) in (6.51). For an average Si concentration of 2.5%, the concentration of C in the austenite next to the graphite is 1.48%. Since the concentration of Si in the residual liquid is a function of the solid fraction, the C concentration gradient within the austenite, C ' - C * , which is in turn the driving y  y  r  force for the diffusion of C from the melt to the nodules, depends on the solid fraction. Figure 6.10 shows the variation of C ' - C y  ygr  as a function of the solid fraction, for  negligible Si segregation, curve B, and Si segregation calculated by the Scheil equation, curve A. For curve A the concentration of carbon in the austenite next to the graphite has  Chapter 6: SOLIDIFICATION  -169 -  MODEL  been considered equal to 1.48%. Comparing curves A and B, it is evident that the C gradient within the austenite becomes larger when the segregation of Si is considered, resulting in an increased driving force for C diffusion. In consequence, a faster growth rate of the graphite would be expected when Si segregation is included in the calculations.  Solid Fraction  Figure 6.10: Difference in the austenite C concentration at the austenite/liquid and austenite/graphite interfaces.  170  6.5 6.5.1  SELECTION OF MATERIALS PROPERTIES THERMOPHYSICAL PROPERTIES OF DUCTILE IRON  The thermophysical properties of ductile iron have not been studied extensively. A database of properties of materials commonly used in castings has been published by R.D. Pehlke et al.[70]. Since no values of the heat conductivity of ductile iron have been reported for the temperature range of interest in our calculations, the thermal conductivity of solid ductile iron will be assumed to be 29 W/m°C, which is the average thermal conductivity of carbon steels between 800 and 1200 °C [81]. The variation of specific heat with temperature is shown in Figure 6.11. For temperatures between 777 and 1130°C, which include the range of interest in our calculations, the specific heat of the ductile iron is given by the following function:  C  p  =  0.61 + 1.214(10^)7  [kJ/kg°C]  (6.55)  For higher temperatures the ductile iron is in a liquid state. The specific heat of the liquid does not change much with the temperature, with a value of 0.915 kJ/kg C. Values of thermal conductivity of liquid cast iron have not been reported in the literature to the best of our knowledge. It has been shown [71] that the effective thermal conductivity of the liquid phase in the presence of a thermal gradient is increased by the fluid flow resulting from natural convection. For our calculations it is assumed that the thermal conductivity of fully liquid regions is equal to 100 W/m°C.  Chapter 6: SOLIDIFICATION  -171 -  MODEL  The liquid in partially solidified nodes is considered stagnant. Since no values are reported for the conductivity of liquid cast iron, the thermal conductivity of partially solidified portions of material will be considered equal to 20 W/m°C, which is smaller than the thermal conductivity of the ductile iron at the solidification temperature. The densities of liquid and solid ductile iron are very similar, since the precipitation of the less dense graphite compensates for the liquid/solid contraction. The density will be taken as 7 g/cm. 3  Careful measurements of the latent heat of solidification of ductile iron have been carried out by Upadhya et al.[72]. They reported a latent heat value of 258 kJ/kg.  6.5.2  SAND PROPERTIES  The thermal properties of the sand used in the calculations were taken from the work of Kubo and Pehlke [73]. The specific heat of silica sand is given by the following equations: ForT<846°/i: C = 0.782 + 5.71(10~*)r - \.%%{\0 )T [Ulkg°K\ A  2  P  (6.56)  FOTT>%46°K  C  P  = 1.00 + 1.35(lCT )T[kJ/kg°K] A  (6.57)  The mould density has been measured to be 1.5 g/cm. 3  The thermal conductivity of moulding sands depends on a number of factors, which include sand density, binder type, moisture content, temperature, etc. Measurements of  Chapter 6: SOLIDIFICATION  - 172-  MODEL  the variation in the thermal conductivity of silica sands have been reported in the literature. The results of different experimental methods are different, as shown in Figure 6.12. In view of the discrepancy in the values, it has been chosen to use the temperature independent value of the heat conductivity of the silica sand reported by Kubo and Pehlke [70]. The thermal conductivity of silica sands having a density of 1.5 g/cm, is equal to 3  0.85W/m°C.  0.9  -  0.8  -  0.7  -  0.6  -  O)  < Ul X  y  LL  o  UJ  0.5 0  —i  1 0.2  1  1—-i 0.4  1 0.6  1  1  1  0.8  1 1  1  1  1—  1.2  (Thousands) TEMPERATURE (C) Figure 6.11: Specific heat of ductile iron as a function of temperature.  1.4  Chapter 6: SOLIDIFICATION  MODEL  -173-  2 JO-  1.5  E  05  0 250  500  1000  750  T  1250  1500 1750  ( K )  Figure 6.12: Thermal conductivity of silica sand as a function of the temperature, measured by two different methods (after [70]).  6.53 OTHER PROPERTIES Values of other properties required for the calculations have been obtainedfromthe literature [75]: Density of graphite = Pgr = 1-92 g/cm  3  Density of austenite = p = 7 g/cm  3  T  Diffusion coefficient of C in the melt = D = 0.50 (10' ) cm /s S  2  c  Diffusion coefficient of C in austenite = D = 0.90 (10") cm /s 6  c  2  Chapter 6: SOLIDIFICATION  6.6  SOLIDIFICATION  MODEL  - 174-  MODEL  The model of the solidification of ductile iron calculates cooling curves, nodule counts and nodular size distributions at points along the radius of an infinitely long cylindrical ductile iron casting. The temperature distribution is calculated using equations (6.12) to (6.17). The resulting tridiagonal system of equations is solved by using the subroutine TRISOLV, available in the UBC main frame computer. The release of latent heat for a given volume element, RHG(i), which is necessary for the calculation of the temperature distribution, is calculated by solving equation (6.44). Calculations of graphite nucleation, and growth of austenite and graphite are performed by the subroutine FRACSO, which also calculates the fraction solid, rate of latent heat release, and composition of the residual liquid. Temperature calculations are made with the program SOLI. Flow charts of the program Soli and the subroutine FRACSO are shown in Figures 6.14 and 6.15 respectively. The computer programs, written in Fortran IV, are listed in Appendix VI. The variables are defined at the beginning of the listing of program SOLI.  Chapter 6: SOLIDIFICATION  MODEL  START DEFINE VARIABLES  I  INPUT  Initial Temperatures: TP, TO System Dimensions: RAO, RADF Material Properties: KL, RHO, CPL, L, KA, RA Limit time and Time steps: DTI, DTF, NDTI, NMAX Nucleation and Growth parameters: DTCR, AA, EXP, DCL, DCA, RNUO Segregation Parameters: SIO, KSI Initial Heat Transfer Coefficient: H Mesh Data: M, MA Printing Data: PN1,PN2,PN3,PN4,PN5,PN6,PN7  INITIALIZE  I  Nodal Temperatures: TV(I) Solid Fraction: FS(I) Counters: JO(l), Sl(l), RNU(I,J), DNU(I.J)  I  CALCULATE AUXILIAR VARIABLES t  DR, DRA, TL, REAG*  I  START COUNTED LOOP  EVALUATE HEAT TRANSFER COEFRCIENT AND TIME STEP BASED ON TIME  Figure 6.14: Flow chart of program SOLI.  Chapter 6: SOLIDIFICATION  MODEL  - 176 -  ASSIGN LIQUID PROPERTIES  ASSIGN SOLID PROPERTIES  CALL FRACSO Evaluate properties, solid fraction and rate of heat generated  EVALUATE MATRIX COEFFICIENTS A, B, C, R  I  SOLVE TRIDIAGONAL SYSTEM OF EQUATIONS  YES  INCREASE COUNTER  I  CALCULATE AND PRINT NODULAR SIZE DISTRIBUTION FOR SELECTED VOLUME ELEMENTS  T END  Figure 6.14: Continued.  Print selected temperatures and solid fractions  Chapter 6: SOLIDIFICATION  -177-  MODEL  START DEFINE VARIABLES  HZ  EVALUATE AUXILIAR PARAMETERS CLA, CLG, CAL, CAG BASED ON SI(IO AND TV(I)  EVALUATE NUMBER OF MEW NUCLEI  r  EVALUATE GROWTH OF GRAPHITE NODULES, ON THE BASIS OF THEIR SIZE  I  CALCULATE FRACTION SOLID AND RATE OF LATENT HEAT GENERATED  *  EVALUATE THE CONCENTRATION OF SI IN THE RESIDUAL LIQUID BASED ON THE FRACTION SOLID RETURN  Figure 6.14: Flow chart of subroutine FRACSOL.  178  Chapter 7  MODEL RESULTS AND APPLICATION  7.1  SENSITIVITY ANALYSIS  The sensitivity of the model to changes in some of the input parameters was evaluated on the basis of the change in the solidificationtime,for the solidification of ductile iron in an 86mm diameter sand mold. The results, for the parabolic nucleation law follow.  7.1.1  INFLUENCE O F T H E M E S H FINENESS  The sensitivity of the model to the number of volume elements in the casting and in the sand was evaluated. Increasing the number of nodes in the casting produces an increase in the solidification time, as shown in Figure 7.1. When more than 40 nodes are defined within the casting, the changes in the solidificationtimeas the number of nodes increases is small. Changing the number of nodes in the sand has little influence on the solidification time, as shown in Figure 7.2.  179  7.1.2  INFLUENCE OF T H ETIME  STEP  The implicit formulation of the heat transfer equations in finite differences is stable for any given time step. However, the accuracy of the calculations is influenced by the time step. Solidification times calculated for different time steps are shown in Figure 7.3. The solidification time decreases for decreasing time steps. Refinement of the time step below 1 sec has little influence on the solidification time.  7.1.3  I N F L U E N C E O F T H E INITIAL T E M P E R A T U R E O F T H E  MELT  The influence of the initial temperature of the melt on the cooling curve of the < center of the casting is shown in Figure 7.4. As expected, larger initial temperatures result in increased solidification times. The length of the eutectic plateau and the supercooling do not change markedly.  7.1.4  SELECTION OF  PARAMETERS  Following the results shown in sections 7.1.1 and 7.1.2, the calculation of the solidification of an 86 mm diameter casting are done for 60 nodes in the casting and 10 in the sand; with a time step of 1 second. Calculations of the solidification of ductile iron in the smaller diameter cylindrical molds are done for smaller volume elements in the casting, and smaller time steps. The number of nodes in the sand will be taken as one per millimeter of radius.  Chapter 7: MODEL  RESULTS  AND APPLICATION  - 180  o CD  -52CD  E o  1 [g O CO  Number of Nodes in the Casting  Figure 7.1: Influence of the number of nodes selected in the casting on the solidification time. 580 570  J  560 CD  tn_ CD  E  550 -  i -  c o  1 TJ  540 530 -  "o  C/5  520 510 500 -  -r 7  -  ~\  1— —i r -1— T 15 11 13 9 Number of Nodes in the Sand Mold  r~  -1  17  r  —i— 19  Figure 7.2: Influence of the number of nodes selected in the sand mold on the solidification time.  Chapter 7: MODEL  565  RESULTS  H  1  AND  1  0  APPUCATIONS  1  1  2  1  4  -181 -  1  1  1  6  1  8  10  Time Step (sec)  Figure 7.3: Influence of the time step on the solidification time.  0.8 -\ 0  1  1  200  1  1  400  1  1  600  1  800  Time (sec)  Figure 7.4: Influence of the initial temperature of the melt on the cooling of the center of the casting.  182  7.2  COMPARISON OF T H ERESULTS OF T H EHEAT TRANSFER M O D E L WITH ANALYTICAL SOLUTIONS  The results of the thermal model are compared to an analytical solution, in order to verify the accuracy of the model calculations. This is done for simplified boundary conditions. Analytical solutions of the heat conduction equation for an infinitely long solid cylinder of outside radius r have been reported in the literature [74] for the 0  following boundary conditions: - The cylinder axis is an insulated boundary. - The surface heat extraction is given by: Q=Ah(T -T ) s  0  - The initial temperature of the cylinder is uniform. The solutions are given in the form of dimensionless charts. The cooling curves of the center, surface, and mid-radius of the cylinder were calculated on the basis of the charts, for the following values of the relevant parameters: Cylinder radius = 50 mm Heat transfer coefficient = 0.04 W/cm C 2o  Heat conductivity = 0.3 W/cirTC Density = 7 g/cm  3  Specific heat = 3 J/g°C The analytical results are shown by the points in Figure 7.5. The model calculations for the same test conditions are shown by the solid lines. The maximum difference  Chapter 7: MODEL  RESULTS AND  APPLICATIONS  -183-  between the calculations and the analytical solution is less than 1.8%. In particular, the analytical and numerical solutions of the cylinder surface temperature are in very good agreement.  0  200  400  600  800  Time (sec) Figure 7.5: Analytical results (points) and numerical calculations (lines) of the cooling of a solid cylinder.  7.3  ANALYSIS O F T H E SENSITIVITY O F T H E M O D E L S O F  NUCLEATION AND GROWTH  Tests of the formulations for the calculation of nucleation, growth and segregation were initially carried out under simplified heat transfer conditions. The melt, consisting  Chapter 7: MODEL  RESULTS  AND  APPLICATION  - 184-  of a single volume element, was assumed to have uniform temperature at alltimes.The rate of heat extraction .from the melt was assumed to be constant. A listing of the computer program GROWTH is included in Appendix 7. A typical cooling curve is shown in Figure 7.6. The calculated temperature of the volume element falls below the critical nucleation temperature, until the rate of heat generated by the phase change compensates and surpasses the rate of heat extraction, giving place to the temperature recalescence. At the end of solidification, the rate of heat generation becomes smaller, and the cooling rate increases to its initial level. Tests of the sensitivity of the models were carried out for both the exponential and parabolic nucleation laws.  1.05 104  H 0  r  1 20  1  1 40  1  1  1  60  TIME (s) Figure 7.6: Calculated cooling curve.  1 80  1 1 00  Chapter 7: MODEL  7.3.1  RESULTS  EXPONENTIAL  AND  APPLICATIONS  - 185-  NUCLEATION  Calculations of cooling curves and nodule counts based on the exponential nucleation, equation 6.20, were done for different values of the constants b and c, and different cooling rates. The following changes were observed.  1-  An increase in the nucleation constant, b, causes an increase in both TEU and TER, as shown in Figure 7.7(a), and an increase in the number of nodules per unit volume, as shown in Figure 7.7(b).  2-  Increasing the value of the constant c, decreases TEU and TER, as shown in Figure 7.8(a), and decreases the number of nodules per unit area, as shown in Figure 7.8(b). The increase in c from 450 to 550 produces a decrease in the nodule counts of approximately 20%.  3-  Increasing the heat extraction decreases both TEU and TER, as shown in Figure 7.9(a). The nodule counts increase almost linearly with the heat extraction within the range examined, as shown in Figure 7.9(b). Changing the cooling rate from 100 to 300 J/sec, increases the nodule counts approximately three times.  Chapter 7: MODEL  RESULTS  AND  APPLICATION  -186-  Time (sec) 12 - • 11  -  10  -  1 H 0  1  1  1  2  1 4  1  1 6  (Thousands)  Nucleation Constant Figure 7.7: Model calculations for different values of the constant b. a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  0  AND  - 187-  APPLICATIONS  20  40  60  80  100  120  Time (sec) 17  300  340  380  420  460  5  O  0  S  4  0  S  8  0  Nucleation Constant Figure 7.8: Model calculations for different values of the constant c. a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  0  AND  20  APPLICATION  40  -188-  60  80  1 00  Time (sec)  Heat Extraction from the Mould (J/sec) Figure 7.9: Model calculations for different imposed cooling rates, a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -189-  7.3.2 P A R A B O L I C N U C L E A T I O N  Calculations of cooling curves and nodule counts based on parabolic nucleation, equation 6.23, were done for different values of a and n, and for different nucleation temperatures, T . The influences of cooling rate and the segregation coefficient of Si N  were also evaluated. The results are listed below. 1)  Increasing the nucleation constant, a, increases both TEU and TER, Figure 7.10(a), and increases the number of nodules per unit volume, Figure 7.10(b). Increasing the nucleation constant ten times, from 100 to 1000, causes a 70% increase in the nodule count.  2)  Increasing the nucleation exponent, n, increases both TEU and TER, Figure 7.11(a), and the number of graphite nodules, Figure 7.11(b). Doubling the nucleation exponent, from 1 to 2, increases the nodule count by 120%.  3)  Increasing the cooling rate decreases both TEU and TER, and reduces the solidification time, Figure 7.12(a). The nodule count increases for increasing cooling rate, Figure 7.12(b). Changing the cooling rate from 100 to 300 J/sec increases the nodule count by 140%.  4)  Increasing the critical nucleation supercooling, DTCR=TL-TN, decreases TEU and TER, as expected, and increases the recalescence, (TEU-TER), Figure 7.13(a). The nodule counts decreases with increasing DTCR, Figure 7.13(b)  5)  Increasing the partition coefficient of Si, results in a small decrease in TER, Figure 7.14(a), and a small increase in the nodule count, Figure 7.14(b). This test was done only to examine the sensitivity of the model to changes in the segregation coefficient.  Chapter 7: MODEL  RESULTS  AND  APPLICATION  TIME (s)  CO  E c f o o  _l  ¥ = 2  Q o  NUCLEATION CONSTANT A  Figure 7.10: Model calculations for different values of the nucleation constant Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS AND  APPUCATIONS  - 191 -  120  TIME (s)  I 0.4  i  i 0.8  i  I 1.2  I i 1.8  I  i 2  I ;  i 2.4  i  i 2.8  NUCLEATION EXPONENT [EXP]  Figure 7.11: Model calculations for different values of the exponent n. a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  0  AND  - 192-  APPUCATION  200  400  600  COOLING RATE (J/sec)  Figure 7.12: Model calculations for different imposed cooling rates, a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  (a) TIME (s) 30  CO  E o _  to  lion  ICO UN  -  26  -  24  -  22  c  1—  28  20 18 16  LU _l  14  Z  12  Q O  — _ _  -  10  0  (b)  6 0  10  20  30  40  CRITICAL NUCLEATION SUPERCOOLING [DTCR] (Celsius)  Figure 7.13: Model calculations for different values of the critical nucleation supercooling, a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS  AND  APPLICATION  -194-  o LU DC Z> I— < DC LU 0LU  TIME (s)  SEGREGATION COEFFICIENT [KSI]  Figure 7.14: Model calculations for different values of the segregation coefficient of Si. a) Cooling curves; b) Nodule counts.  Chapter 7: MODEL  RESULTS AND  APPUCATIONS  -195-  7.4 M O D E L O U T P U T  A sample of the model output is shown in Appendix 8. The output lists: 1-  Temperatures at six positions within the casting, PN1 to PN6, and two points within the mould, PN7 and MF. One hundred temperature values, corresponding to different times, are listed for each volume element.  2-  Values of the fraction solid for PN1 to PN6.  3-  Number of nodules per unit volume in each volume element.  4-  Nodular size distribution at five volume elements, PN1 to PN5.  5-  Temperature and fraction solid distribution within the castings at seven time values, PR1 to PR7. Typical cooling curves at six points within the 86mm diameter casting, calculated  . by the model, are plotted in Figure 7.15. The markers indicate the beginning and end of solidification. The cooling curves show that volume elements closer to the cylinder axis have a longer local solidificationtime(given by the time difference between beginning and end of solidification), and solidify over a larger temperature range (given by the temperature difference between the start and end of solidification) than the volume elements close to the periphery of the casting. Temperature distributions throughout the casting at different times from pouring are shown in Figure 7.16. The temperature gradients after pouring are large, progressively decreasing for the first 200 s, when the first layer of fully solid material is formed in the periphery of the casting. After 200s, with further solidification larger gradients develop in the solid shell. When solidification is complete, after 810s, the temperature gradients  Chapter 7: MODEL  RESULTS  AND  APPLICATION  - 196-  become small again. The calculated transformation kinetics at different distances from the centre of the casting are shown in Figure 7.17. The initial transformation rate, for solid fractions less than 40%, is higher for the volume elements closer to the casting surface. For larger solid fractions, the transformation rate of the axial volume element, radius=0, is greater than the transformation rate of the volume elements close to the mid-radius. The calculated nodule count per unit volume as a function of the distance from the casting axis is shown in Figure 7.18. A large peak in the nodule count is observed at the casting surface, resulting from the fast cooling rate at this point in the casting. Moving towards the centre of the casting, the nodule count reaches a minimum value, and then progressively increases. The nodular size distribution at the centre, mid-radius and edge of the casting is shown in Figure 7.19(a-b). The largest number of small nodules is present in the centre of the casting. There are more large nodules in the centre and mid-radius positions in the casting, than at the edge. Comparing Figures 7.15 and 7.17, shows that for most volume elements, particularly those near the centre of the casting, most of the solidification occurs below the plateau temperature during subsequent cooling.  Chapter 7: MODEL RESULTS AND APPUCATIONS  -197 -  1.3  0.9 H 0  1  1  200  1  1  400 Time(sec)  1  1  600  1  ,  1 800  Figure 7.15: Calculated cooling curves at points distant 0, 10, 21, 33 and 43 mm from the casting centre.  Chapter 7: MODEL  RESULTS  AND  APPUCATION  - 198-  Figure 7.16: Calculated temperature distribution at different times from pouring.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -199-  800 Time(sec)  Figure 7.17: Calculated transformation kinetics at points distant 0, 10, 21, 33 and 43 mm from the casting centre.  Chapter 7: MODEL  RESULTS  AND  APPLICATION  -200-  Figure 7.18: Calculated number of nodules per unit volume as a function of the distance from the casting axis.  -201 -  Chapter 7: MODEL RESULTS AND APPUCATIONS  3. a -  o  u5 •3  "8 6  10  14  18  22  26  30  34  38  42  46  38  42  46  38  42  46  N o d u l e Radius (microns)  6  10  14  18  22  26  30  34  N o d u l e Radius (microns) 4  -i  3.5 32.5 2-  as i •8 z  1.5 10.5 0 -  6  10  14  18  22  26  30  34  Nodule Radius (microns)  Figure 7.19: Calculated nodular size distribution at the centre, mid-radius and edge of a 86mm diameter casting.  Chapter 7: MODEL  RESULTS  AND  -202-  APPUCATION  7.5 COMPARISON OF MODEL RESULTS AND CALCULATIONS The mathematical model was used to calculate the solidification of ductile iron of eutectic composition cast into cylindrical sand moulds of 86,40 and 25 mm diameter. The calculated cooling curves, nodule counts and graphite volume distributions were compared with measured cooling curves from the four unalloyed castings, C12 to C15. Comparisons of cooling curves for 86mm diameter castings C12 and C13 were made, as well as 40 and 25 mm diameter castings C14 and C15. A direct comparison of calculated and measured nodule counts and nodular volume distributions cannot be made because the calculated values are volumetric, and the measured values are obtained on a planar section. Noguchi et al.[78] proposed that the number of nodules per unit volume, N , as a function of the number of nodules per unit v  area, N , could be calculated from equation 7.1. A  iV =10.6 v  (N )  135  A  (7.1)  Noguchi et al found that the graphite area and graphite volume distributions are similar, and that the volume curve is slightly shifted to the larger nodule sizes. Secondary peaks observed in the distributions for the larger nodule sizes are more evident in the volume distribution than in the area distribution. Consequently, in the present investigation the calculated volume distributions of graphite is compared to the measured area distribution of graphite.  Chapter 7: MODEL RESULTS AND APPUCATIONS  -203-  The model calculates the size of the nodules at near eutectic temperature. It was shown in section 6.3.5 that the radius of each nodule can be assumed to increase by 16.7 % in order to account for the secondary precipitation of graphite. In the calculations of graphite volume distribution, the nodule radius given by the model has been increased by 15%. This is smaller than 16.7% because small amounts of pearlite are present in the samples. In comparing the measured and calculated results, the parameters for each calculation will be identified by a RUN number, as listed in Table XITJ, in each figure. Both the exponential and the parabolic nucleation laws were used in the calculations.  7.5.1  C A S T I N G O F 86mm D I A M E T E R R O D S  7.5.1.1 E x p o n e n t i a l N u c l e a t i o n  a) Cooling Curves The calculated cooling curves at three positions within the casting are shown by the solid lines in Figures 7.20 and 7.21. The temperatures measured during the solidification of casting C12, at the same positions in the casting are shown by the points in Figure 7.20. The agreement between calculated and measured cooling curves is very good at the edge of the casting. The fit between measured and calculated values at points within the casting is also good, although the calculated solidification time is slightly longer than the measured solidification time. The cooling curves for the center and mid-radius of C12 show a short temperature plateau at 70 seconds after pouring. This can be attributed to the precipitation of proeutectic austenite, which indicates that there is a small deviation from  Chapter 7: MODEL RESULTS AND APPUCATION  - 204 -  Table XIII: Parameters used in the model calculations.  RUN  Pouring  Number  Temperature  Nucleation Parameters a  n  CC)  T -T E  N  b  c  CQ  (xlO ) 5  1  1245  —  —  ~  400  340  2  1245  50  3  20  ~  ~  3  1200  —  ~  —  400  340  4  1200  50  3  20  —  —  5  1180  —  —  —  400  340  6  1180  50  3  20  ~  —  7  1300  —  ~  400  340  8  1300  400  340  9  1300  — — —  400  340  —  — —  the eutectic composition in the casting. The temperatures measured during the solidification of casting C13 at three positions are shown by the points in Figure 7.21. Measured and calculated values do not agree very well during the first 150 s after pouring. After 150s the fit is reasonably good. The calculated solidificationtimeis a little larger than the measured solidification time. The cooling curves of C13 do not indicate the precipitation of proeutectic phase components.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -205 -  b) Nodule Count The calculated nodule count as a function of the distance from the casting centre is shown in Figure 7.22. The values of the nodule counts measured at the central, intermediate and external zones of sections of castings C12 and C13, calculated using equation (7.1), are shown by the points. The measured nodule counts are in general higher than the calculated values for casting C12 and lower for casting C13.  c) Graphite Volume Distribution The calculated graphite volume distribution at the centre (a), mid-radius (b), and near the edge (c) of casting C12, are shown in Figure 7.23. The volume distribution at the center shows a prominent peak for nodules of 11.5 micron radius, and a secondary peak for 32.2 micron radius. At the mid-radius position, two peaks are also present for 12 and 28 micron radii nodules. At the center of the casting, most of the graphite volume is made up of small nodules, and a smaller contribution to the volume from nodules of the larger radius than that observed at the casting centre is evident. Near the edge of the casting, only one peak is observed for 20.7 microns. The graphite area distributions, measured on samples C12 and C13 are shown in Figures 7.24 and 7.25 respectively. The measured area distributions at the center, (a), and mid-radius, (b), of casting C12 show a main peak at the nodule radius 18 microns. The distribution is not symmetric with respect to this peak; the larger nodules contributing more to the graphite area. The area distribution near the edge of the casting, (c), shows a peak for a nodule radius of 22 microns. The distribution is approximately symmetric about the peak.  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -206-  The graphite area distribution for casting C13, shown in Figure 7.25, differs to some extent from the results for casting C12. The smaller nodules contribute more to the graphite area in C13, as compared to C12. The peak in the distribution at the centre of both C13 and C12 occurs at a smaller nodule radius than at the edge of the casting.  Figure 7.20: Calculated and measured cooling curves for casting C12, shown by the lines and symbols respectively, RUN1.  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -207 -  E c  II  o5 —1 ©  3 73  O  z  Distance from Rod Axis (mm) Figure 7.22: Calculated variation of the nodule counts as a function of the distance from the casting centre, shown by line, RUN1. Measured nodule counts shown by points for castings C12 and C13.  Chapter 7: MODEL  RESULTS  AND APPLICATION  . 208  >  s o  •3  6.9 11.5 16.1 20.7 25 3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  6.9 11.5 16.1 20.7 25.3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  6.9 11.5 16.1 20.7 25.3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  Figure 7.23: Calculated graphite volume distribution, RUN1. (a) centre, (b) mid radius, (c) near the edge.  Chapter 7:  MODEL  RESULTS  AND APPLICATIONS  .  ! s o  O  o •a o 2  6  10 14 18 22 26 30 34 38 42 46  Nodule Radius (microns) JS  a.  fl  o O  fc! < "A  O  c o  •3 10  14 18 22 26 30 34 38 42 46  Nodule Radius (microns)  fl  I "fl  22  26 30 34 38 42  Nodule Radius (microns)  Figure 7.24: Measured graphite area distribution on casting C12. (a) centre, (b) mid-radius, (c) near the edge.  209 -  Chapter 7: MODEL RESULTS AND APPUCATION  - 210 -  I s o o  « H  e  o  \3 U  2  6  10 14 18 22 26 30 34 38 42 46 Nodule Radius (microns)  Q.  2  a O  A3  B < O  eo  •3 u n  tt.  2  6  10 14 18 22 26 30 34 38 42 46 Nodule Radius (microns)  2  6  10 14 18 22 26 30 34 38 42 46 Nodule Radius (microns)  Figure 7.25: Measured graphite area distribution on casting C13. (a) centre, (b) mid-radius, (c) near the edge.  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -211 -  7.5.1.2 P a r a b o l i c N u c l e a t i o n  a) Cooling Curves The calculated cooling curves for three positions within the casting are compared with the temperatures at the same positions measured during the solidification of C12 and C13, Figures 7.26 and 7.27. In both cases the calculated cooling curves at the centre and mid-radius of the casting agree reasonably well with the measured values, particularly during the first 600 seconds. The calculated temperatures at the edge of the casting are lower than the measured temperatures, particularly after 200 seconds from the beginning of the calculation.  b) Nodule Counts The calculated nodule counts as a function of the distance from the casting axis is shown by the curve in Figure 7.28. The measured nodule counts are indicated by the points. The measured nodule counts are observed to be higher than the calculated values for casting C12 and lower for casting C13.  c) Graphite Volume Distribution The calculated graphite volume distributions at the centre (a), mid-radius (b), and near the edge (c) of the casting are shown in Figure 7.29. The volume distributions at positions (a) and (b) show two peaks. The more prominent peaks are observed for nodules of 11.5 microns of radius, a large influence of the nodules of 9.2 microns on the volume distribution is evident. Secondary peaks are observed at 29.9 microns, (a), and 26  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -212-  microns (b). The volume distribution near the edge, c, shows only one peak for 11.5 microns. A greater contribution to the volume from the larger nodules is observed in (a) and (b) than in (c).  Chapter 7: MODEL RESULTS AND  APPUCATIONS  -213-  Centre  •radius  800  Time (sec) Figure 7.27: Calculated and measured cooling curves for casting C13, RUN2. 20 19 18 17 18  CO  15  E  14 13  c to  •+-» o C  :=  §1 o 03 O  12  "  1 0  9  7 6 5 4 3 2 1  H  o  —r40  Distance from Rod Axis (mm) Figure 7.28: Calculated variation of the nodule counts as a function of the distance from the casting centre shown by line. Measured nodule counts shown by points for casting C12 and C13.  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -215-  o  > o.  e  C3  o  u  § •a  6.9 11.5 16.1 20.7 25 3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  o > Q.  s o f-  <*» o c o •3  6.9 11.5 16.1 20.7 25.3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  > •g.  2 O  2.3 6.9 11.5 16.1 20.7 25.3 29.9 34.5 39.1 43.7 Nodule Radius (microns)  Figure 7.29: Calculated graphite volume distribution, RUN2. (a) centre, (b) mid radius, (c) near the edge.  Chapter 7: MODEL  RESULTS AND  APPUCATION  -216-  7.5.2 C A S T I N G O F 4 0 m m D I A M E T E R R O D S  7.5.2.1 E x p o n e n t i a l N u c l e a t i o n  a) Cooling Curves The calculated cooling curves for the centre and the edge of the 40mm diameter casting are shown by the solid lines in Figures 7.30 and 7.31, and the measured temperatures for castings C 14,-40 and C15-40 are shown by the points. The calculated and measured temperatures are similar, but the measured values are generally lower than the calculated values.  b) Nodule Count The calculated nodule count as a function of the distance from the casting centre is shown in Figure 7.32. The nodule count initially decrease with the distance from the rod axis, and then increase to higher values next to the casting surface. The average nodule count measured on samples C14-40 and C15-40, as shown in Figure 7.32, are slightly smaller than the calculated values.  c) Graphite Volume Distribution The average calculated graphite volume distribution is shown in Figure 7.33. The distribution is essentially uniform between nodule radii of 9.2 and 23 microns. The graphite area distributions measured on samples C14-40 and C15-40, shown in Figures 7.34 and 7.35 respectively, show a peak at 19 microns.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -217-  Time (sec) Figure 7.31: Calculated and measured cooling curves for casting C15-40, RUN3.  Chapter 7: MODEL  RESULTS  0  1 0  1  1 2  AND  1  1 4  APPLICATION  1  1 6  1  1 8  r~1 10  -218-  1  1  1  12  1 14  1  1 16  1  1 18  1— 20  Distance from Rod Axis (mm) Figure 7.32: Calculated variation of the nodule counts as a function of the distance from the casting centre, RUN3. Points show measured values.  Nodule Radius (microns) Figure 7.33: Calculated graphite volume distribution, average, RUN3.  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  +-*  lc  k_ Q. ca  O o  OJ  o>  <  15 o c g 'TJ ca  Nodule Radius (microns) Figure 7.34: Measured graphite area distribution on casting C14-40.  -»-»  Jz  Q.  ro CD 03  © k  < "ro  •  o c o  rj 03  Nodule Radius (microns) Figure 7.35: Measured graphite area distribution on casting C15-40.  -219-  Chapter 7: MODEL RESULTS AND  APPLICATION  -220-  7.5.2.2 P a r a b o l i c N u c l e a t i o n  a) Cooling Curves The calculated cooling curves and the measured temperatures at the center and the edge of the 40mm diameter castings C14-40 and C15-40 are shown in figures 7.36 and 7.37 respectively. The calculated and measured temperatures are similar, but the measured values are generally lower than the calculated values during the alloy solidification.  b) Nodule Count The calculated nodule count as a function of the distance from the casting axis is shown in Figure 7.38. The nodule count measured on castings C14-40 and C15-40 are shown by the points. There is good agreement between the calculated and measured nodule count.  c) Graphite Volume Distribution The average calculated graphite volume distribution is shown in Figure 7.39. The distribution is effectively uniform between 9.2 and 23 microns.  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -221 -  Time (sec) Figure 7.37: Calculated and measured cooling curves for casting C15-40, RUN4.  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -222-  Distance from Rod Axis (mm) Figure 7.38: Calculated variation of the nodule counts as a function of the distance from the casting centre, RUN4. Points show measured values for castings C14-40 and C15-40.  Nodule Radius (microns) Figure 7.39: Average calculated graphite volume distribution, RUN4.  Chapter 7: MODEL  RESULTS AND  APPUCATIONS  -223-  7.5.3 C A S T I N G O F 2 5 m m D I A M E T E R R O D  7.5.3.1 E x p o n e n t i a l N u c l e a t i o n  a) Cooling Curves The calculated cooling curves and the measured temperatures at the centre and edge of the 25mm diameter castings C14-25 and C15-25 are shown in Figures 7.40 and 7.41 respectively. The calculated and measured temperatures are similar, but the measured values are generally lower than the calculated temperatures during the first 60 seconds after pouring.  b) Nodule Count The calculated nodule count as a function of the distance from the casting centre is shown in Figure 7.42. The average nodule counts measured on samples C14-25 and C15-25, shown in Figure 7.32, are slightly smaller than the average of the calculated counts.  c) Graphite Volume Distribution The average graphite volume distribution calculated by using the model is shown in Figure 7.43. Only nodules having radii between 4.6 and 13.8 microns contribute to the volume distribution. The graphite area distributions measured on samples C14 and C15, shown in Figures 7.44 and 7.45 respectively, show a peak at a nodule radius of 16 microns.  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -224-  Time (sec) Figure 7.41: Calculated and measured cooling curves for casting C15-25, RUN5.  Chapter 7: MODEL  RESULTS  AND  -225 -  APPUCATIONS  co E c  1 I o I ©  3 "D O  Distance from Rod Axis (mm) Figure 7.42: Calculated variation of the nodule counts as a function of the distance from the casting centre, RUN5. Points show measured values on castings C15-25 and C14-25.  CD  E  O >  CO CL  ca O "ca  o  1 i 2  1  1 11 1 1 1  1  Nodule Radius (microns)  11 5 13.8 16.1 18.4 20.7  23  1  i  1  1  25.3 27.6 29.9 32.2 34.5 36.8 39.1 41.4 43.7  r  46  Figure 7.43: Calculated graphite volume distribution, average, RUN5.  Chapter 7: MODEL  RESULTS  AND APPUCATION  - 226  «C —D »  lc Qo  CO CD  <  CO  o c o  'XS CO  i  r  i  i  i  i  i  I  i  r  10 12 14 1« 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46  Nodule Radius (microns) Figure 7.44: Measured graphite area distribution on casting C14-25.  CD  ••—* !c Q. CO  O  "6 CO CD < "CO  o to c o  Tj co  ~l—I—I—I—I—I—I—I—I—  8 1 0 12 14 18 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46  Nodule Radius (microns) Figure 7.45: Measured graphite area distribution on casting C15-25.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -227-  7.5.3.2 P a r a b o l i c N u c l e a t i o n  a) Cooling Curves The calculated cooling curves and the measured temperatures at the centre and the edge of the 25mm diameter castings C14-25 and C15-25 are shown in Figures 7.46 and 7.47. The calculated and measured temperatures are similar, but the measured values are generally lower than the calculated values during the alloy solidification.  b) Nodule Count The calculated nodule count as a function of the distance from the casting axis is shown in Figure 7.48. The nodule counts measured on castings C14-25 and C15-25 are shown by the points. There is good agreement between the calculated and measured nodule counts.  c) Graphite Volume Distribution The average calculated graphite volume distribution is shown in Figure 7.49. Only nodules having radii between 6.9 and 23 microns contribute to the graphite volume.  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -228 -  Time (sec) Figure 7.47: Calculated and measured cooling curves for casting C15-25, RUN6.  Chapter 7: MODEL  RESULTS  AND  APPUCATIONS  -229-  30 26 26  CO  cm  24 22 20  c  3  o O  .  18  on  <= 3^ ,  16  v>  —  14  10  3 T3  8  O  _  -  12  CO  z  -  -  6 4 2  -  0  Distance fron Rod Axis (mm) Figure 7.48: Calculated variation of the nodule counts as a function of the distance from the casting centre. RUN6. Points show measured values.  2.3  4.6  6.9  9.2  11.5 13.8 16.1 18.4 20.7  23  25.3 27.6 29 9 32.2 34.5 36.8 39.1 41.4 43.7  46  Nodule Radius (microns) Figure 7.49: Average calculated graphite volume distribution, RUN6.  Chapter 7: MODEL RESULTS AND  APPUCATION  -230-  7.5.4 D i s c u s s i o n  The cooling curves calculated with the mathematical model are in good agreement with the temperatures measured in the cylindrical castings of 86mm diameter. In general, the exponential formulation of the nucleation process fitted the experimental results slightly better than the parabolic equation. Nevertheless, none of the nucleation models used was able to predict with some degree of accuracy the extent of the supercooling during the solidification of the small diameter castings. Measured and calculated values of TEU are shown in Figure 7.50. Note that two different melts of ductile iron were used for the 86mm diameter casting and the smaller diameter castings, because of the limited size of the melting facility available. Small amounts of ductile iron of similar composition and graphite morphologies are difficult to produce. As a result, different melts may produce materials having different characteristics, even when the production methods used and the charge materials are the same. Because of the difficulty in producing identical small heats of ductile iron, no attempt was made to carefully measure the supercooling TEU and the recalescence TER as a function of casting diameter. Accurate values would be of value in improving the accuracy of the calculated cooling curves. The present results, which show that the measured nodule count at the centre of the 86mm diameter casting is greater than at the mid-radius is in agreement with results of other investigation [79]. The calculated nodule count, as a function of the distance from the centre, are greater than the measured values, but show the same variation with distance from the centre as the measured values. The difference between the measured and the calculated values is attributed in part to the high solidification rate at the centre of  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -231 -  the casting when one-dimensional heat extraction is assumed, as described in section 7.4. The calculate large nodule count near the casting surface is in agreement with measured nodule counts at the surface reported by Piaskowsky [79]. In considering the graphite volume distribution in a casting, it was pointed out that the calculated volume distribution cannot be directly compared to the measured distribution based on area of graphite on a sectioned surface. The measured and calculated distributions were not similar. In the case of rods of 86mm diameter, the maximum and minimum nodule size measured and calculated agreed well, but the calculated distribution showed in general two peaks, while the measured showed only one peak, and. As the rod diameter decreases, the differences between measured and calculated distributions become more marked. For example, the calculated distribution for the 25mm diameter rod using the exponential nucleation, presents nodules between 4.6 and 13.8 microns radii, while the measured distributions on C14-25 and C15-25 show peaks for 16 microns radii. The models of graphite nucleation and growth are very much dependent on the supercooling during solidification, which is not predicted accurately by the model. Therefore, it is possible that the differences between measured and calcualted distributions are due to the inaccuracy in the predictions of the solidification temperature.  Chapter 7: MODEL  RESULTS  AND APPUCATION  - 232 -  1.15  1.1 H 12  1  1  16  1  1  20  1  1  1  24  1  28  1  1  32  1  1  1  36  1  1  40  44  Cast Rod Radius (mm) Figure 7.50: Temperature of eutectic undercooling as a function of the section size, a) exponential nucleation formulation; b) parabolic nucleation formulation; c) experimental.  7.6 A P P L I C A T I O N O F T H E M O D E L In this section the quantitative effect of varying a number of casting parameters on the calculated values of nodule count, supercooling, TEU, recalescence, TER, and solidificationtime,TISOL, will be examined. The casting parameters considered are pouring temperature, casting section size, inoculation and mould material. a) Pouring Temperature Calculations of temperatures and nodule count were made for a 43mm radius rod of eutectic ductile iron for pouring temperatures between 1250 and 1400°C, giving the results listed in Table XIV. No significant variation in the nodule count, TEU or TER are  Chapter 7: MODEL  RESULTS  AND  -233 -  APPLICATIONS  evident when the pouring temperature is increased. The solidification time, TISOL, increases with increasing pouring temperature. The results are in agreement with published data [83] which showed no major change in the nodule count with changes in pouring temperature between 1310 and 1450°C. Table XIV: Variation of cooling curves and nodule counts as a function of the pouring temperature.  Pouring  Nodule Counts (xl0 unit/cm )  TEU  TER  TISOL  6  3  Temperature  Centre  Mid-radius  Near edge  °C  °C  (s)  1250  11.25  8.64  7.07  1145  1146.9  589  1300  11.25  8.65  7.04  1145  1146.9  647  1350  11.26  8.64  7.08  1145  1146.9  705  1400  11.25  8.64  7.12  1145  1146.8  763  b) Section Size Calculations of temperature and nodule count were made for casting of 12.5 to 40 mm radii. The variations in nodule count at the centre, the mid-radius, and the edge of the casting with the rod radius are shown in Figure 7.51. The nodule count per unit volume increases by a factor of between 3 and 4.5 as the rod radius increases from 12.5 to 40mm. This is in good agreement with the results reported by Piaskowsky [79], who found an increase of 2.5 to 3.5 times in the nodule count per unit area. This is equivalent to an increase of 3.4 to 4.4 times in the nodule count per unit volume, according to formula (7.1). The model predictions are also in good agreement with the measurements made in  Chapter 7: MODEL RESULTS AND APPUCATION  -234-  the present investigation, in which the nodule count per unit volume in the castings of 12.5 mm of radius were found to be 2.2 to 3.9timeslarger than the nodule count measured in the 43mm diameter castings. Note that these measurements were made on samples obtained from different melts; and thus the comparison can only be considered as semiquantitative.  CO  E  o O 3  TJ O  Cast Rod Radius (mm) Figure 7.51: Nodule counts as a function of the cast rod radius, RUN7. c) Nucleation Rate The nucleation rate depends on the inoculation practice. The influence of the nucleation rate on the nodule counts at the mid-radius of cylindrical castings has been calculated for samples of 12.5 to 40 mm radii, giving the results shown in Figure 7.52. For rods of 12.5 mm radius the nodule count increases by approximately 25% as the  Chapter 7: MODEL RESULTS AND  APPUCATIONS  -235-  nucleation constant, b in equation 6.20, increases from 200 to 600. The increase in the nodule counts is smaller for the larger samples. For rods of 40mm radius, the nodule count does not change with an increase in the nucleation constant from 200 to 600.  12  16  20  24  28  32  36  40  Cast Rod Radius (mm)  Figure 7.52: Nodule counts at the mid-radius as a function of the casting radius, for values of the nucleation constant b of 200, 400 and 600, RUN8.  d) Thermophysical properties of the mould The heat flow through a semi infinite mould is proportional to a parameter F [82], which value is given by: F=(kC p) p  (7.2)  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -236-  The solidification time and nodular size distribution in a rod of 43mm diameter has been calculated for the values of thermal conductivity, k, and density of the mould, p, given in TABLE XV.  TABLE XV: Assumed values of mould density and thermal conductivity.  Thermal Conductivity  Density  Mould Factor  (W/cm°C)  (g/cm)  (k p )  0.0085  1.3  0.105  0.01  1.5  0.122  0.012  1.7  0.143  3  1/2  The variation of the solidification time with the mould factor, (k p ),is shown in a  Figure 7.53. The variation in the solidificationtimeas a result of small changes in the mould properties is significant. In addition, the nodule counts increase with the mould factor, as shown in Figure 7.54. This results show that accurate values of the thermophysical properties of the mould material are necessary if good values of the solidification time are to be determined from the model. Accurate determination of the properties of the moulding sands is difficult, particularly when moisture or reactive binders are mixed with the sand. The heat evolved in the vapourization of the water and the combustion of the binder must be accounted for in the calculations.  Chapter 7: MODEL  f*»u -i 0.105  RESULTS  1  AND  1 0.115  APPLICATIONS  1  1 0.125  Mould Factor  -237 -  1  1 0.135  1  1  0.145  Figure 7.53: Solidification time as a function of the mould factor, RUN9.  Chapter 7: MODEL  RESULTS  0.105  AND  APPUCATION  0.115  0.125  -238 -  0.135  0.145  Mould Factor  Figure 7.54: Nodule counts at the center , mid-radius and near the edge of a casting of 86mm diameter, as a function of the mould factor.  7.7 DISCUSSION The ability of the mathematical model to predict the temperature variation with time at several points within a casting, the nodule count and the nodular size distribution for eutectic ductile iron, has been evaluated by comparing the results of the calculations with observations made on cast rods of three different radii. Such an evaluation has not been made for any models in the literature for ductile iron, to the best of our knowledge. The evaluation is a demanding test of a model. The comparison of the calculated and experimental values shows good agreement in some cases, and not as good agreement in others. The best agreement was obtained with  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -239-  exponential nucleation, for the cooling curves and nodule count of rods of 43mm radius. The fit was not as good for the cooling curves of the rods of smaller diameter, but the nodule count remained in good fit. In most cases the changes in temperature and nodule count predicted by the model when some solidification parameter is changed, are consistent with the corresponding changes observed experimentally. Note that the present calculations for rods of different radii used the same values of the parameters governing the heat extraction and nucleation and growth of the solid phases. The only parameter changed was the initial melt temperature. The effects of the segregation of Si on the equilibrium temperature and equilibrium C concentration have been accounted for in the solidification model. The effect of the Si segregation on the cooling curves and nodule count is negligible under the modelled conditions. In the case of other alloying additions which segregate more extensively than Si, this will not be the case. When the results of the present model are compared with the results of other published models of the solidification of ductile iron, the following differences are noted: 1)  The prediction of the cooling curve at a single point within a casting, which is the objective of some published models, has not been done in the present investigation, since its utility is questionable.  2)  In those cases in which cooling curves have been calculated for several points within a casting [28,43], the calculations were done for one sample size, although the models were two-dimensional. The calculated cooling curves for a casting of 43mm radius rod in the present investigation, Figure 7.20, are in better agreement with the experimental temperature measurements, than the calculations of Su et  Chapter 7: MODEL  RESULTS  AND  APPUCATION  -240-  al[28], Figure 2.5. The calculated nodular size distributions in the present investigation, fit the experimental results poorly, but are still in better agreement with the experimental values than the calculations of Su et al.[28]. The model developed by Zeng and Pehlke [43] releases the latent heat of solidification linearly between two preset temperatures, and as a result did not calculate the nodule count or the nodular size distribution. The agreement between their calculations and measurements, shown in Figure 2.9, is not as good as that observed in the results of the present investigation. 3)  In those cases in which the solidification kinetics and the release of latent heat have been calculated by applying Avrami type equations, nodule counts two orders of magnitude greater than the counts observed experimentally were required to obtain cooling curves similar to those measured during the solidification of ductile iron. In the present study, the solidification kinetics was calculated on the basis of formulations of the nucleation and growth of solid phases which were dependent on the supercooling of the melt. As a result, nodule counts of the correct order of magnitude were obtained for three different rod radii. On the basis of the above comparisons, the present mathematical model constitutes  a step forward in the objective of formulating an accurate model of the solidification af ductile iron of general applicability. However, the model is not complete since more factors must be taken into account before the effect of the many variables involved in the production of ductile iron casting can be entirely accounted for. More detailed models are required to account for the solidification of non-eutectic alloys; to consider complex  Chapter 7: MODEL  RESULTS  AND  APPLICATIONS  -241 -  casting geometries; to calculate heat transfer during mould filling; to predict the precipitation of carbides; to calculate the solid state transformations during cooling; to account for the segregation of the alloying elements other than Si; and others. It is believed that the exponential formulation of the nucleation of graphite proposed in the present investigation can predict the solidification of ductile iron with good accuracy in a wide range of casting sizes. This will require the identification of the best values of the constants, b and c. As pointed out above, this is the objective of a future investigation. For a given casting size and mould type, the supercooling and the nodule counts in ductile iron depend on the chemical composition of the melt, the type and amount of materials used for the inoculation and post-inoculation of the melt, the time after inoculation, and other factors. The probability of finding a theoretical explanation for the effects of these factors is still remote. Alternatively, experiments can be carried out in which careful measurements of the solidification parameters are made under specific cooling conditions. Such experiments should be designed to obtain enough information to establish the nucleation constants b and c. These, in turn, could be applied to the calculation of the solidification of castings of other size and shape.  -242-  Chapter 8  Summary and Conclusions  SUMMARY  1)  The microsegregation of Mn, Cu, Cr, Mo, Ni and Si has been measured in cast ductile iron and in ductile iron which has been quenched when partially solidified.  2)  From the microsegregation measurements effective partition coefficients have been determined for each of the elements.  3)  Using the measured effective segregation coefficients, the solute distribution between nodules in cast samples has been calculated, using the Scheil equation, and compared to measured solute distribution.  4)  Observations of the morphology of the solid phases during solidification were made on samples quenched during growth.  5)  The measured solute distribution between adjacent graphite nodules was related to morphology of the solid phases present in the material as solidification progressed.  6)  On the basis of the measured segregation of the different alloying elements, the mechanisms by which the segregation affects the microstructure of ductile iron have been discussed, and an explanation for the influence of the segregation on the hardenability of ductile iron has been proposed.  Chapter 8: SUMMARY  7)  AND  CONCLUSIONS  -243-  A mathematical model of the solidification of eutectic ductile iron has been formulated which includes heat flow, nucleation and growth of graphite nodules, and the segregation of Si. The model uses equilibrium temperatures given by the ternary Fe-C-Si equilibrium diagram.  8)  Using the mathematical model, cooling curves, nodule count and nodular size are determined as a function of position in the cast sample. The results are compared to measured temperatures, nodule count and nodule size, in rod castings of 12.5, 20 and 43mm radius.  9)  Using the mathematical model, the effect of pouring temperature, inoculation, casting size and mould thermal properties on die cooling curves and the nodule count has been determined.  CONCLUSIONS  1)  Mn, Cr and Mo segregate to the residual liquid. Cu, Ni and Si concentrate in the first material to solidify. This is in agreement with other investigations.  2)  Segregation measured on as-cast samples along lines between nodules, show poor reproducibility and extensive scatter.  3)  The amount of segregation in as-cast ductile iron can be characterized by the ratio of the solute concentration in the first material to solidify over the solute concentration in the last material to solidify.  4)  The microstructure of samples quenched during solidification indicates that the solidification of eutectic ductile iron begins with the independent nucleation of  Chapter  8: SUMMARY  AND  CONCLUSIONS  -244-  austenite and graphite phases in the melt. Later, the graphite nodules are enveloped by the austenite, and further sohdification takes place by the thickening of the austenite layers enveloping the graphite. Isolated pockets of interdendritic melt are the last material to solidify. 5)  Effective segregation coefficients were determined for Mn, Cu, Mo, Cr, Ni and Si.  6)  Measured values of the concentration of the alloying elements as a function of the solid fraction agree reasonably well with solute concentrations as a function of the solid fraction calculated using the Scheil equation and the measured effective segregation coefficients.  7)  A mathematical model of the solidification of ductile iron which includes the nucleation and growth of graphite and accounts for the segregation of Si during solidification has been formulated, and solved numerically.  8)  The calculated temperatures for a cast 43mm radius rod are in good agreement with experimental temperature measurements and nodule count for castings of the same size. 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Smithless, "Metals Reference Book", Third Ed, Vol2, pp 583-609, Butherworths, London, (1962).  References  -253  -  [76] CF. Gerald and P.O. Wheatley, "Applied Numerical Analysis". Addison-Wesley Publishing Co. Third Ed. (1984), pp 465-476. [77] I. Minkoff, "The Physical Metallurgy of Cast Iron". John Wiley and Sons Publisher, (1983), pp35. [78] T. Noguchi and K. Nagaoka, "Three dimensional distribution of graphite nodules in spheroidal graphite cast iron". Transactions of the AFS, 85-13, (1985), pp 115-122. [79] J. Piaskowsky and A. Jankowsky, Pr. Inst. Odlew, Vol 21(2), pp422, (1971). [80] G. Jolley and G. Gilbert, "Segregation in nodular iron and its influence on mechanical properties", British Foundryman, pp 79-92, (march 1967). [81] B. Thomas, PhD Thesis, The University of British Columbia, (1985). [82] J.P. Holman, "Heat Transfer", 4th edition, Mc Graw-Hill Book Co., (1976), pp 103. [83] J.F. Wallace, P. Du, H-G Su, R. Warrick and L.R. Jenkins, "The influence of foundry variables on nodule count in ductile iron", Transactions of the AFS, 85-134, (1985), pp 813-834.  Appendix 1  -254-  Appendix 1  S a m p l e O u t p u t of the W a v e l e n g t h D i s p e r s i v e X - r a y A n a l y z e r  -255 -  Appendices  Procedure 11 Aug  Lin  Sym  Z  Si Fe Cu  14 Ka 26 Ka 29 Ka  sicu-f e 1989 —  Std  Lambda X t a l (Ang)  Si Fe Cu  7.1261 PET 1.9374 L i F 1.5418 L i F  Standard PkSk PkCt BkCt Time Time Time (Sec)(Sec)(Sec) 40 40 40  Acquired 11 Aug SEM Beam V o l t a g e Stage T i l t E l e v a t i o n Angle  Lin  sicu-f e 1989 —  10 10 10  Unknown PkSk PkCt BkCt Time Time Time (Sec)(Sec)(Sec)  5 5 5  10 10 20  5 5 7  Data 15:23  = 20 k.V = 4 5 Deg 0 Deg —  Beam Cur (nA)  Peak Counts  Sym  Z  Si Fe Cu  14 Ka 26 Ka 29 Ka  15.0 15.0 15.0  104425 53840 49980  Sym  Z  Beam Cur (nA)  Peak Counts  Si Fe Cu  14 Ka 26 Ka 29 Ka  15.0 15.0 15.  1090 50225 1 173  Lin  15:23  PkCt Ti me (Sec) 10 10 10 PkCt T i me (Sec) 10 10 20  Standard Bkgd Counts 16 58 11 1 Un known Bkgd Counts 6 59 113  BkCt T i me (Sec)  Counts Per Sec  5. 5 5  10439.3 5372.4 4975.8  BkCt T i me (Sec)  Counts Per Sec  5 5  —> i  107.8 5010.7 42. 5  Appendices  -256-  Comprehensive R e s u l t s 11 Aug SEM Beam V o l t a g e = Stag e T i l t E l e v a t i o n Angle = Z  Si Fe Cu  14 Ka 26 Ka 29 Ka  Sym  Z  Si Fe Cu  14 Ka 26 Ka 29 Ka  Sym  Z  Si Fe Cu  14 Ka 26 Ka 29 Ka  Unknown  Standard  K Ratio  107. 8 5010.6 42. 5  10439.3 5372.4 4975.7  0.0103 0.9327 0.0085  Lin  Lin  15:23  20 kV 45 Deg 0 Deg  Lin  Sym  sicufe 1989 —  Weight Percent  Z Corr  A Corr  1.113277 0.997821 0.967802  0.524417 0.999683 0.936338  1.001362 1.000931 1. (1)00000  Atomi c Percent  N o r m a l i zed At P e r c e n t  Normalized Wt P e r c e n t  F  Corr  1.77 93.60 0.94  1.83 97.19 0.98  3.45 92. 04 0.81  3. 59 95.57 0. 85  96. 31  100.00  96.31  100.00  Appendices  -257 -  Appendix 2  Solute Diffusion During Solidification  Brody and Flemings [58] studied the segregation in dendritic solidification when the diffusion in the solid phase is not negligible. If the solid/liquid interface advances with a constant rate, the concentration of solute in the solid is given by equation (A2-1). When the advance of the interface is proportional to 1/VT, where t is time, the concentration of solute in the solid can be calculated by using equation (A2-2).  c =kch-—$—^  (A2-1)  s  C, = kC (l - (1 - 2 a J c ) y - ™ g  0  Where:  a  =  —  L  D = diffusion coefficient of solute in solid s  t =local solidification time f  L = one half dendrite spacing  t  (A2-2)  Appendices  -258 -  If the condition given by equation (A2-3) is satisfied, and the partition coefficient k has a small value, equations (A2-1) and (A2-2), reduce to the Scheil equation, in which the segregation in the solid in neglected. In order to determine the adequacy of the use of the Scheil to describe the segregation values of a shall be calculated for the solidification conditions of the melt samples taken in quartz tubes. 2ak<\  (A2-3)  At solidification temperature, the diffusion coefficient for alloying elements, which diffuse substitutionally is approximately 10' m/sec [75]. The distance between 12  2  dendrites in quenched samples is approximately 150 microns, and the local solidification time 30 seconds. Therefore: a = 0.005  (A2-5)  Assuming k = 2 results: 2a* =0.02 which can be considered much smaller than 1. The solidification of ductile iron in sand molds will proceed at a slower rate than that observed in the quenched specimens before quenching. Nevertheless, the influence of the increased local solidification time will be compensated, at least in part, by the coarsening of the dendritic structure. Therefore, equation (A2-3) may still be satisfied.  Appendices  -259-  Carbon diffuses interstitially in the solid austenite, with a diffusion coefficient of 10" m/sec, two orders of magnitude larger than that for substitutional diffusion [75]. 10  2  Therefore, the value of a is significant, and extensive diffusion of C in the solid is expected. The diffusion coefficient for the alloying elements in the liquid phase is approximately 10" m/sec [75], this is three orders of magnitude larger than the 9  2  coefficient for diffusion in the solid. This supports the assumption of a fully mixed liquid.  Appendices  - 260 - •  Appendix 3  Segregation i n P l a n a r - F r o n t Solidification w i t h L i n e a r V a r i a t i o n of the P a r t i t i o n Coefficient  An equation will be derived describing the solute segregation observed after a melt is solidified with a plane front from one end, as shown in Figure A3.1. The following assumptions are made: 1) The ratio of the composition of the liquid and solid at the solid/liquid interface is defined by a segregation coefficent k . tt  (A3-1)  2) Diffusion in the solid is negligible. 3) The liquid phase is homogeneous at all times. Under the stated conditions, the mass balance when the solid fraction increases by a small amount, dg, is:  Appendices  dg  ^ i i i l l l M l i i i i l l ^ i i i i i i l  SOUD  LIQUID  J3-  ±9-  Figure A3-1: Melt solidified from one end.  dg{C -C ) = (\-g)dC L  s  L  Substituting equation (A3-1) in (A3-2): dgC {\-k ) = {\-g)dC L  tt  L  Assuming that k„ is given by:  Where y and P are constants. Equation (A3-3) can be rearranged i  Calling: Y = i-y  equation (A3-5) can be rearranged:  Appendices  -262-  By -f*-dg  Y -f—dg  1-g  1 -g  dC  =7^ L  C  (A3-6)  L  Which can be integrated:  0  0  6  *  *~  which results in: l n ^ = (B-r)ln(l-g) + Bg  (A3-8)  Q = C exp{(B-Y)ln(l-g) + B£}  (A3-9)  Which reduces to: 0  Equation (A3-9) describes the concentration of solute in the liquid as a function of the solid fraction. An equation for the solute distribution in the solid, (A3-10), is derived by substituting equations (A3-1) and (A3-4) in (A3-9).  C = (Y+ Bs )C exp{(B - Y) ln(l - g) + Bg} s  0  (A3-10)  Appendices  -263 -  Appendix 4  H e a t T r a n s f e r E q u a t i o n s i n F i n i t e Differences  Six types of nodes with particular heat transfer equations can be identified within the system, as illustrated in Figure 6.2. We shall formulate the equations for each node based on the heat balance, which gives accurate results when small volume elements and time steps are used [76].  1) Axial Volume Element The volume element containing the cylinder axis is schematized in Figure A4-1. If a unitary dimension is assumed along the cylinder axis, the components of the heat balance result: (A4-1)  Rate of heat input = RHI = 0 Rate of heat output = RHO = nk (T[  - T  +1  J +1 2  1  (A4-2)  ) (7V 2 1 1  Rate of heat accumulation =RHA =-TipC dr' p  1 Rate of heat generated =RHG  The heat balance is:  9  =-ndr RHG(i) 2  + 1  T'\  ~ 1 1  (A4-3)  (A4-4)  Appendices  -264-  Figure A4-1: Axial volume element.  RHI - R H O + R H G = RHA  (A4-5)  Substituting equations A4-1 to A4-4 into A4-5 results:  -k (j{  +1  - Ti *)+^ +  dr RHG (i) = 2  <Ti - T{)  (A4-6)  RHG(i)  (A4-7)  +1  Equation (A4-6) can be rearranged as:  f , P^V»__Lr/*i.P^ri, 1  ^dr* 4kdtf  l  dr  2  2  4*dr  1  1  4*  2) Internal volume element of the casting An internal volume element of the casting is shown in Figure A4-2. The components of the heat balance are:  Appendices  -265  RHI = 2n^-kr(j{*l-7/ ) +1  (A4-8)  dr  RHO  = 2n^-k(r dr  + dr) (TJ +l  TJ?})  (A4-9)  RHA=2npC j rdr[Tr -T.^  (A4-10)  RHG=2nrdrRHG(i)  (A4-11)  l  p  t  r Figure A4-2: Internal volume element of the casting.  The heat balance gives: _r'  +  1  -  VTJ +  dr  2  rdr  kdt  dr  2 +  1  _  _  pC  • /?//G(Q  rdr  (A 4-12)  Appendices  -266-  The radius r can be expressed as a function of i and dr r=  (A4-13)  (i-L5)dr  Equation (A4-13) is substituted into (A4-12), giving:  pC„V,., ( 1 dr  2  -  1  +  [dr  (i-\.5)dr  2  2  kdt)  1  {dr  2  (i-1.5)dr (A4-14)  3) Volume element at the surface of the casting The surface volume element of the casting is shown in Figure (A4-3). The components of the heat balance are: RHI = In-^kriT^-Ti; ) 1  dr  (A4- 15)  RHO = 2nh\r +— (TM'-TLW)  (A4-16)  RHA=TxpC j rdr(Ti -Ti )  (A4-17)  RHG = nrdrRHG  (A 4 - 18)  +1  p t  t  l  (Af)  The heat balance results:  Appendices  -267-  J*  \  MOULD  A  dr/2 Figure A4-3: Volume element at the surface of the casting.  L ; +i dr  dr  2  * dr K  +  2r j  2kdt  ^  M  +  l  P ^  T  ,  ~2kdt  RHGjM)  M  2k  +  (A4-19)  The radius r can be calculated by using equation (A4-14), which substituted in (A4-19) gives:: L7V+1 ,  dr  2 +  kdr{i-1.5  +2kdt  ( i - l 1  m  P  kdr {i-1.5) ~2kdt lM+1  lM+  C P  T  /  RHG(M)  M  2k  (A 4-20)  4) Volume element at the internal surface of the mold  -268 -  Appendices  The volume element at the internal surface of the mold is represented in Figure (A4-4). The radial size of the volume elements of the mold, dr , is different from that of a  the volume elements of the casting. The components of the heat balance are: RHI =  (A4-21)  2ithr (Tl; -Ti; \) l  c  Ink  RHO =  y  +  dr  (A 4 - 2 2 )  2,  a  RHA =JcpC ,^r dr fTiV -ri )  (A 4 - 2 3 )  RHG=0  (A 4 - 2 4 )  J  e  -  1  +1  OULD  dra/2  Figure A4-4: Volume element at the internal surface of the mold.  The heat balance results:  Appendices  kdr  T  -269-  +  i + X  J_  h  kdr drt +  a  a  1  2r dr  +  c  a  pC  p  2kdt  +  177 + 1 1  M +l  1  -J  \dr  2 a  ^7-77 ++11_ _J  — 2rtedr » ' aj c  a  i M + 2  p_~,y  ~2^r  i M + 1  (A 4-25)  5) Internal volume element of the mould The internal volume element of the casting is represented in Figure A4-5. In this case the radius, r , can be calculated as: m  r =r + m  c  (i-M-l.5)dr  a  (A 4-26)  The terms of the heat balance are similar to those for the internal volume element of the casting, although in this case:  -270-  Appendices  RHG(i)  dr=dr  (A4-27)  =0  (A 4-28)  a  The governing equation for this type of volume element is obtained by substituting equations (A4-26), (A4-27) and (A4-28) in (A4-13) •y + l _ r  P  T  j  kdt  (A 4-29)  6) Volume element at the external surface of the mold Two different conditions are modeled: 6. (a) Constant surface temperature This condition applies to the water cooled mold. The volume elements next to the copper coil are shown schematically in Figure (A4-6). The heat transfer equation for node (F-l) can be obtained by replacing T  M  by T in equation (A4-29) F  F  (A 4-30)  6.(b) Free surface exposed to air  Appendices  -271 -  This condition applies to the non-water-cooled cylindrical moulds. The surface node of the casting is represented in Figure  = 2nk  RHI  (A4-7).  F~  r  ^1 2  RHO =  RHA  *F  J\  ~ F-\ l  dr  (A 4 - 3 2 )  +1  F  aP  1  p  ^ (T  F  J +1 F  (A4-31)  J  >  2nh*r (T -T )  = nr dr C F  The terms of the heat balance are:  A  - T' ) F  (A4-33)  Where: surface heat transfer coefficient between the mould external surface and the surrounding air = 20 W/m °C 2  Appendices  room temperature = 20°C The heat balance results:  dra/2 Figure A4-7: Volume element at the free surface of the mold.  Appendices  -273 -  Appendix 5  Equations Describing the Fe-C-Si Equilibrium Diagram Near Eutectic Temperature  The models for graphite nucleation and growth assume that the carbon concentrations at the different interfaces present within the microstructure during solidification are defined by the equilibrium diagram Fe-C-Si. The model requires equations defining the concentration of C as a function of the Si content and the temperature. In particular, equations are required for: (a) the carbon concentration in the austenite in equilibrium with liquid, C '; (b) the carbon concentration in the austenite in r  equilibrium with graphite, C ; (c) the carbon concentration in the liquid in equilibrium ygr  with austenite, C \ (d) the carbon concentration in the liquid in equilibrium with l  graphite, C" . In addition, it is necessary to have an equation describing the variation in gr  the eutectic temperature, T , with the Si content. The eutectic region of the equilibrium E  diagram Fe-C-Si, for a given Si concentration, is represented schematically in Figure A5-1. The curves of interest are marked 1 to 5. Equations representing curves 1, 2 and 3 of Figure A5-1, in the form of temperature as a function of C and Si concentration, have been reported by Heine [47]. The temperatures of the austenite liquidus, T^, the austenite solidus, T , and the graphite AS  liquidus, T , representing curves 1 to 3 respectively, are given by the following GL  equations:  Appendices  -274-  Carbon  Concentration  Figure A5-1: Schematic of the eutectic region of the Fe-C-Si equilibrium diagram for a given Si concentration.  1 ^ 7/^ = 1569-97.3 C+-Si f  \  T  GL  4  (A5-1)  J  = 1528.4- 177.9(C +0.185i)  (A5-2)  = 389.11 C +-Si I - 503.2  (A5-3)  Where:  -275 -  Appendices  C = carbon concentration in weight percent Si= silicon concentration in weight percent T = temperature in degrees Celsius  Equations (A5-1) to (A5-3) can be rearranged, considering the temperature as the independent variable. Expressions for the C concentration are then derived. From equation (A5-1): C'^ = ^  (1569-7/-24.325/)  C" =^  (1528.4 - T - 325/)  (A 5-4)  From equation (A5-2):  1  (A 5 - 5)  From equation (A5-3): C"  gr  = —\— (T - 129.75/ + 503.2)  (A 5 - 6)  An equation describing the C content of the austenite in equilibrium with graphite, C , has not been found. Minkoff [77] reproduced a cut of the ternary Fe-C-Si diagram, il  for 2.4% Si, due to Piwowarsky. The eutectoid point is situated at 800 C and 0.6% C. In the present calculation, it will be assumed that the line indicating the maximum solubility of C in austenite is straight, starting at the eutectic temperature, and ending at the eutectoid point. The solubility limit of C in austenite at eutectic temperature, C^, is obtained by intersecting the curves defining T  AS  and T . Heine [47] defines the eutectic E  Appendices  -276-  temperature as: T = 1154.6 + 6.5S/  (A 5-7)  E  Equating (A5-2) and (A5-7) and rearranging gives: C  TE  = 2.1 -0.2165/  (A5-8)  The carbon concentration in the austenite in equilibrium with graphite results: c  n  , JT-1154*-«^)(lJ-CUl«n (354.6 + 6.55i)  Equations for C , C ', C' , C ,/T  y  /fr  r?r  + 2 t  _  Q n 6 5  .  and T are plotted in Figure A5-2. E  The ratio between the amounts of austenite and graphite forming the equilibrium eutectic, REAG(%), is given by the ratio:  Appendices  1 -f  1  -277-  1  1  1.4  1  1  i  1  1.8  r—*—i  2.2  1  2.6  r  1  3  1  1  3.4  r  3.8  Carbon Content (Wt%) Figure A5-2: Curves describing the Fe-C-Si diagram for 2.5% Si.  REAG(%) =  100-C, Cp-C  (A5-10)  TE  The carbon content of the eutectic, C is given by: E  C = 4.26 -0.3175/ E  (A5-11)  Replacing Equations (A5-11) and (A5-8) into (A5-10) gives: (A5-11)  Appendices  -278 -  REAG(%) is a weight ratio. It can be converted into volume ratio, REAG, by ltiplying by the density of graphite, p , and dividing by the density of the austenite, r  REAG=REAG{%)^-  Pr  (A5-12)  Taking: p -1.92 r  p =7 7  REAG results: (A5-13)  Appendices  Appendix 6  L i s t i n g of P r o g r a m S O L I a n d S u b r o u t i n e  FRACSO  Appendices  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  C C C C C C C C C  -280-  LISTING T H I S PROGRAM S I M U L A T E S OF A C A S T I R O N INGOT  DEFINITION  OF  PROGRAM S O L I  THE U N I D I R E C T I O N A L  OF V A R I A B L E S  AND  SOLIDIFICATION  ARRAYS  REAL'8 DT,DR.TL,KL,KM,KS,RHO,CPL,CPM,CPS,H,L,RAD,KP REAL * 8 R A D F , K A , R A , C P A , D R A , T O , T P , R R , D T I , D T F , S I O , A A , D C L , R N U O REAL"8 REAG,NUCLEI,DCA,EXP,DTCR,KSI,TNUC,TCH,TISOL R E A L * 4 R A I , RAM INTEGER N , M , Q , M F , N M A X , P , M M , M A , M L , M M M , M F M , T O T , P A R , J O ( 8 1),NDTI INTEGER MFF,PN1,PN2,PN3,PN4,PN5,PN6,PN7,RADIO,RADI01,RADIOM DIMENSION T N ( 1 0 1 ) , T V ( 1 0 1 ) , A ( 1 0 1 ) , B ( 1 0 1 ) , C ( 1 0 1 ) . R ( 1 0 1 ) D I M E N S I O N F S ( 8 1 ) , R H G ( 8 1 ) , D N U ( 8 1 . 7 0 0 ) , R N U ( 8 1 , 7 0 0 ) , I N D I ( 8 1 . 51 ) D I M E N S I O N K( 101 ) , C P M 0 1 ) , ROC 1 0 1 ) . AL ( 1 0 1 ) . SI ( 1 0 1 ) DOUBLE P R E C I S I O N TN.TV,A,B,C,R,K,CP,RO,AL,FS,DNU,RNU,RHG,INDI DOUBLE P R E C I S I O N SI C C C  DESCRIPTION  OF  VARIABLES  C  C C C C C C C - C C C C C C C C C C C C C C C C C C C C C C C C C C C  T0= I N I T I A L T E M P E R A T U R E OF C O O L I N G WATER AND MOULD TP= P O U R I N G T E M P E R A T U R E OF THE MELT TNUC= T E M P E R A T U R E AT WHICH N U C L E A T I O N R A T E I S CONSIDERABLE TN AND TV = A R R A Y S C O N T A I N I N G THE T E M P E R A T U R E OF THE VOLUME E L E M E N T S TL= E U T E C T I C TEMPERATURE DTCR= C R I T I C A L N U C L E A T I O N S U P E R C O O L I N G DT= T I M E S T E P DTI= I N I T I A L TIME STEP DTF= F I N A L TIME S T E P N D T I = NUMBER OF C A L C U L A T I O N S U S I N G D T I TISOL= SOLIDIFCATION TIME TCH= T I M E L I M I T DR= S I Z E OF C A S I N G VOLUME E L E M E N T S DRA= S I Z E OF MOULD VOLUME E L E M E N T S RAD= C A S T I N G R A D I U S RAOF= MOULD R A D I U S RNUO= I N I T I A L R A D I U S OF N O D U L E S R A I AND RAM= A X U L I A R V A R I A B L E S FOR C A L C U L A T I O N OF NODULAR DISTRIBUTION K L . K M , K S = THERMAL C O N D U C T I V I T Y OF L I Q U I D , MUSHY AND S O L I D C P L , C P M , C P S = S P E C I F I C HEAT OF L I Q U I D , MUSHY AND S O L I D RHO= C A S T I N G D E N S I T Y H= S U R F A C E HEAT T R A N S F E R C O E F F I C I E N T AT THE C A S T I N G / M O U L D I N T E R F A C E L= L A T E N T H E A T OF SOLIDIFICATION K A = THERMAL C O N D U C T I V I T Y OF THE MOULD RA= D E N S I T Y OF THE MOULD C P A = S P E C I F I C HEAT OF THE MOULD D C L = D I F F U S I O N C O E F F I C I E N T OF C I N THE L I Q U I D D C A = D I F F U S I O N C O E F F I C I E N T OF C I N THE A U S T E N I T E SIO= AVERAGE S I L I C O N CONCENTRATION SI (I )= A R R A Y FOR THE S I C O N C E N T R A T I O N I N THE R E S I D U A L M E L T AA AND E X P = N U C L E A T I O N C O N S T A N T S K S I = P A R T I T I O N C O E F F I C I E N T OF S I N U C L E I = COUNTER OF THE NUMBER OF N U C L E I  -281 -  Appendices  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76  77  78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 111 112 113 1 14 1 15 1 16  C C C C C C C C C C C C C C  c c c c c c c c  10 11 12 14  c c c c c c  N= NUMBER OP T I M E S T E P S I N THE RUN P A R = A U X I L I A R C O N T A I N I N G THE R E A L T I M E M= NUMBER OF NODES I N THE C A S T I N G MA= NUMBER OF NODES I N THE MOULD MF= T O T A L NUMBER OF NOOES M M , M L , M M M , M F M , M F F = A U X I L I A R V A L U E S I D E N T I F Y I N G SOME NODES J 0 ( I ) = INTEGER ARRAY PN1 TO P N 7 = NODEL FOR WHICH V A L U E S A R E S T O R E D AND P R I N T E D RADIO, R A D I O !:, R A D I O M = USED FOR THE C A L C U L A T I O N OF NODULAR S I Z E DISTRIBUTION A , B . C , R= COMPONENTS OF THE M A T R I X F S ( I ) = A R R A Y C O N A T I N S THE S O L I D F R A C T I O N FOR E A C H N 0 0 E R H G ( I ) = R A T E OF HEAT G E N E R A T E D I N A NODE P E R U N I T T I M E AND VOLUME D N U ( I . J ) = NUMBER OF NODES C R E A T E D I N NODE I AT T I M E J R N U ( I , J ) = S I Z E OF NODES C R E A T E D I N NODE I AT T I M E J I N D I = A U X I L I A R COUNTER  INPUT  PARAMETERS  FROM  DATAFILE  READ(5,10) READ(5,10) READ(5,10) READ(5,10) READ(5,10)  DTF DTI RAD DTCR KL  READ(5.10) READ(5,10) READ(5,10) READ(5,10) READ(5,10) READ(5,10) READ(5,10) R E A D ( 5 , 10) R E A D ( 5 , 10) READ(5,10) READ(5,10) READ(5,10) READ!5,10) READf 5 , 1 1 ) R E A D ( 5 , 11) READf 5 , 1 0 ) FORMAT(F10. FORMAT(E12. READ!5,12) FORMAT(4G5) READ(5.14) FORMAT(7G5)  RHO CPL H L KA RA RADF TP TO AA SIO KSI EXP DCL DCA RNUO 4) 3) M,NMAX,MA,NDTI  PRINT  PN1,PN2,PN3,PN4,PN5,PN6,PN7  PARAMETERS  WRITE(6,700)  DTI  -282-  Appendices  117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  700 701 702 703 704 705 706 707 708 709 710 711 712 C C C C C C  W R I T E ( 6 , 7 0 1 ) DTF W R I T E ( 6 , 7 0 2 ) NDTI W R I T E ( 6 , 7 0 3 ) TP W R I T E ( 6 , 7 1 2 ) DTCR WRITE(6,704) SIO W R I T E ( 6 . 7 0 5 ) AA W R I T E ( 6 , 7 0 6 ) DCL W R I T E ( 6 , 7 0 7 ) DCA W R I T E ( 6 . 7 0 8 ) RNUO WRITE(6,709) M W R I T E I 6 . 7 1 0 ) MA W R I T E ( 6 , 7 1 1 ) EXP FORMAT('DTI=',F6.3) FORMAT('DTF=',F6.3) FORMATt'NDTI=',G4) F O R M A T ( ' T P = ' ,F7 . 1 ) FORMAT('SI=',F4.2) F O R M A T ( ' A A = ' , F 7 . 1) FORMAT('DCL= ,E12.3) F O R M A T ( ' D C A = ' ,E 1 2 . 3 ) FORMAT('RNUO=',F7.5) FORMAT('M=',G4) FORMAT('MA=',G4) FORMAT('EXP=',F4.2) FORMAT('DTCR=',F6.3) 1  CALCULATE :  AUXILIAR  VARIABLES  MF=M+MA- 1 MFM=MF- 1 C DR=RAD/(M-1) D R A = ( R A D F - R A D ) / ( M A - 1) C C C C C C  20 C  21 22 C C  INITIALIZE  NODAL  TIS0L=20.0 DO 2 0 1 = 1 ,M TV(I)=TP JO(I)=0 FS(I)=0 SI(I)=SI0 CONTINUE DO 2 2 J = 1 , M DO 21 1 = 1 , 7 0 0 RNU(J,I)=0 DNU(J,I)=0 CONTINUE CONTINUE  TEMPERATURES,  SOLIO  F R A C T I O N AND COUNTER  -283 -  Appendices  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  C C C C  c c c c c c c  30  CALCULATE  TL  AND  REAG  TL=1154.6+6.5*SI0 REAG=(26.26+0.078"SI0)/(2.16-0.101*SI0)  SET  TEMPERATURE  OF MOULD  NODES  MM=M+1 DO 3 0 I=MM,MF TV(I)=T0 CONTINUE TV(MM)=TO  C C  c c c c 170  WRITE  HEADINGS  OF  PRINTOUT  W R I T E ( 6 , 1 7 0 ) P N 1 , P N 1 , P N 2 , P N 2 , P N 3 , P N 3 , P N 4 , P N 4 , P N 5 , P N 5 , PN6 , PN7 ,MFM FORMAT('TIME=',4X,'T(',G3,')',3X,'FS(',G3,')',3X,'T(',G3,')', 13X, ' F S ( ' ,G3, ' ) ' ,3X, ' T ( ' , G 3 , ) ' ,3X, ' F S ( ' , G 3 , ' ) ' ,3X, ' T ( ' , G 3 , ' ) ' , 2 3X, ' F S ( ' , G 3 , ' )' ,3X, ' T ( ' , G 3 , ' )' ,3X, ' F S ( ' ,G3, ' )' , 3 X , ' T ( ' , G 3 , 3' )' ,3X, ' T ( ' ,G3, ' ) ' ,3X. ' T ( ' , G 3 , ' ) ' ) 1  C  c c c  START  SOLUTION  c  c c c c  DO  500  ADJUST  N=1,NMAX  HEAT  TRANSFER  COEFFICIENT  c  c  909 919  C C C C c  c  81  IFtN.LE.NDTI) GOTO 9 0 9 pAR=NDTI*DTI+(N-NDTI)*DTF GOTO 9 1 9 PAR=DTI"N I F ( P A R . L T . 4 0 0 ) G 0 T 0 81 H=3 . 0 I F ( P A R . L T . 5 0 0 ) GOTO 81 H = 3. 0  ADJUST  TIME  STEP  IF(N.GT.NDTI) DT=DTI GOTO 9 5  GOTO  90  Appendices  233 234 235 236 237 238 239 240 24 1 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  90 C C C C C •C 95 C C C C  100 810 820  841 830 840  102  108  112 114 110 C C c  0T=0TF CALCULATE PROPERTIES FOR EACH NODE DO 108 1=1,M SORT NODES ACCORDING TO CONDITION "SOLID, LIQUID OR MUSHY" TL=1154.6 + 6.5'SKI) TNUC=TL-OTCR IF((TV(I).GT.TNUC).AND.(FS(I) .LT.0.000001)) GOTO 100 IF(FS(I).GT.0.999) GOTO 102 K(I)=0.35*FS(I)+0.20'(1 -FS(I)) R0(I)=RH0 CP(I)=FS(I)'(0.61+1.214E-4"TV(I))•<1-FSd))'CPL AL(I)=RO(I)'CP(I)/K(I) CALL FRACSO(TV,TL,FS,DNU,RNU,DT,RHG.L.I,AA,SI,RNUO.DCL,JO.REAG 4,DCA,EXP,KSI.SIO.DTCR) GOTO 108 IF(N.LE.NOTI) GOTO 810 PAR=NDTI*DTI+(N-NDTI)"DTF GOTO 820 PAR=DTI*N TCH=5 IF(PAR.LT.TCH) GOTO 830 IF(PAR.GT. 10) GOTO 841 K(I)=KLM5-4*(PAR-TCH)/5) GOTO 840 K(I)=KL GOTO 840 K(I)=KL * 5 CP(I)=CPL R0(I)=RHO AL(I)=RO(I)'CP(I)/K(I) RHG(I)=0 GOTO 108 K ( I)=0.29 R0(I)=RH0 CP(I)=0.61+1.214E-4*TV<I) AL(I)=RO(I)*CP(I)/K(I) RHG(I)=0 CONTINUE DO 110 I=MM,MF IF(TV(I).GT.573) GOTO 112 CP(I)=0.782 + 5.71E-4*(TV(I) + 273)- 1.88E4*(TV(I)+273)*"(- 2) GOTO 114 CP(I) = 1 + 1.35E-4MTV(I)+273) K(I)=KA R0(I)=RA AL(I)=RO(I)'CP(I)/KA CONTINUE  -284-  -285 -  Appendices  291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348  C C C C C  CALCULATE  MATRIX C O E F F I C I E N T S  1=1 B(I)= 1/DR"2+AL(I)/(4'DT) C ( I ) = •1/DR* * 2 R ( I ) = A L ( I ) / ( 4 * D T ) * T V ( I ) + R H G ( I ) / ( 4 * K( I ) ) C  120 C  ML=M-1 DO 1 2 0 1=2,ML A(I)=-1/DR''2 B(I)=2/DR*"2+1/((I - 1.5)*DR*•2)+AL(I)/DT C( I > = - 1/( ( I - 1 . 5 ) * D R " 2 ) - 1 / D R " 2 R ( I ) = A L ( I ) / D T * T V ( I ) + RHG(I)/K ( I) CONTINUE I=M A(I) = -1/DR*'2 B( I ) = 1 / D R " 2 + H ' ( I - 1 )/(K( I ) * D R ' ( I - 1 .5) )*AL( I ) / ( 2 ' D T ) C(I)=-H*(I-1)/(K(I)*DR'(1-1.5)) R ( I ) = ( ( A L ( I ) / D T ) ' T V ( I ) + R H G ( I ) / K ( I ) )/2  C I=M+1 A(I)=(-H/(K(I)'DRA))'K(I) B( I ) = ( H / ( K ( I ) ' D R A ) + 1/DRA"* 2 + 1 / ( 2 * R A D * D R A ) + A L ( I ) / ( 2 * D T ) ) * K ( I ) C( I ) = ( - 1 / D R A " 2 - 1 / ( 2 ' R A D ' D R A ) ) *K ( I ) R(I)=(AL(I)/(2*DT)*TV(I))*K(I) C  130 C  MMM=M+ 2 MFM=MF- 1 DO 1 3 0 I=MMM,MFM RR=RAD+(I-M-1.5)'DRA A(I) = -1/DRA"2 B(I)=2/DRA * * 2+1/(RR'DRA)+AL(I)/DT C(I) = -1/DRA"2-1/(DRA*RR) R(I)=AL(I)/DT'TV(I) CONTINUE I=MF RR=RAD+(I-M-1.5)*DRA A ( I ) = - 1/DRA* * 2 B( I ) = 2 / D R A ' ' 2 + 1 / ( R R ' D R A ) + A L ( I ) / D T R(I)=AL(I)'TV(I)/DT+(1/DRA " 2*1/(RR'DRA))'TO  C C C C C  S O L U T I O N OF T R I D I A G O N A L M A T R I X  CALL C C C C C 150  TRISLV  ( I , A , B . C , R , 0 , & 150) :  A S S I G N V A L U E S G I V E N BY T R I S L V  DO 1 4 0 1=1,MF TN(I)=R(I)  TO TN  Appendices  349 350 351 352 353 354 355 356 357 358 359 360 361 362 36 3 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406  140 C C C C C  c  c 220 160 C C  c  400 450 C C C C  c c  497 500 C C C  c c c  510 501 C  -286-  CONTINUE  WRITE  RESULTS  IF(N.LE.NDTI) GOTO 4 0 0 PAR=(N-NDTI)"DTF+NDTI"DTI DO 1 6 0 J = 1 , 1 0 0 NP=(NMAX-NDTI)*J/100-(N-NDTI) I F ( N P . N E . O ) GOTO 1 6 0 WRITE(6,220)PAR,TN(PN1) ,FS(PN 1 ) ,TN(PN2) , F S ( P N 2 ) , T N ( P N 3 ) , F S ( P N 3 ) 2,TN(PN4),FS(PN4),TN(PN5),FS(PN5),TN(PN6),TN(PN7),TN(MFM) FORMAT(G4,3X,F8.2,F7.3.F8.2,F7.3,F8.2,F7.3,F8.2.F7.3,F8.2, 3F7.3.3F8.2) CONTINUE ASSIGN  VALUES  521  TN TO TV  DO 4 5 0 1 = 1 , M F TV(I)=TN(I ) CONTINUE  CHECK  FOR  SOLIDIFICATION  COMPLETION  AND S T O R E  IF((FS(D-.GT.0.998).AND.(TISOL.LT.30)) GOTO 5 0 0 TISOL=PAR CONTINUE  PRINT  NODULAR  SIZE  GOTO  TIME  TISOL  497  DISTRIBUTION  DO 5 0 1 1=1,M DO 5 1 0 J=1 , 5 0 INDI(I,J)=0 CONTINUE CONTINUE DO 5 2 0  545  OF  1=1.M  IF((I.EQ.PN1).OR.(I.EQ.PN2).OR.(I.EQ.PN3).OR.(I.EQ.PN4). 7 0 R . ( I . E Q . P N 5 ) ) GOTO 5 4 5 GOTO 5 2 0 NUCLEI=0 PAR=JO(I) DO 5 3 0 J=1 , P A R RADIO=0 RADI0=RADI0+2 RAI=RNU(I.J)*10000 RADI0I=RADI0+1  -287 -  Appendices  407 408 409 410 4 1 1 412 413 414 415 416 417 418 419 420 421 422 433 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464  550 530 520 C C C C C C  610 555 56 7 522 570  590 C C 580 C  c c  c c c c c c c c c c  c c  RADI0M=RADI0-1 IF((RAI.GT.RADIOI).OR.(RAI.LT.RADIOM)) I N O I ( I , RAD 1 0 ) = IND I ( I , RADIO)-"-DNU ( I , J ) I F ( R A D I 0 . G E . 5 0 ) GOTO 5 3 0 GOTO 5 2 1 CONTINUE CONTINUE  CALCULATE  AND P R I N T  T O T A L NODULE  GOTO 5 5 0  COUNTS  DO 5 6 7 1 = 1 , M NUCLEI=0 PAR=J0(I) DO 6 1 0 J = 1 . P A R NUCLEI=NUCLEI*DNU(I,J) CONTINUE WRITE<6,555) I,NUCLEI F O R M A T ( ' N U C L E I IN NODE' , 1 X , G 3 . ' = ' , F 1 2 . 0 ) CONTINUE WRITE(6,522> TISOL FORMAT('SOLIDIFICATION TIME='.F10.2) W R I T E ( 6 , 5 7 0 ) PN 1 . P N 2 . P N 3 , P N 4 , P N 5 FORMAT('SIZE',4X,G3,4X,G3,4X,G3,4X,G3,4X,G3) DO * 8 0 J = 2 . 5 0 , 2 Wf< . 6, 5 9 0 ) J , I N D K P N 1 , J ) , I N D I ( P N 2 , J ) , I N D I ( P N 3 . J ) , I N D K P N 4 , J ) , 2 I N L . . r>N5 , J ) FORMAT(G4,3X,F11.1,2X,F11.1,2X,F11.1.2X,F11.1,2X,F11.1)  CONTINUE  STOP END  LIST  OF S U B R O U T I N E F R A C S O THIS CALCULATES NUCLEATION. AND R A T E OF HEAT G E N E R A T E D  GROWTH,  SEGREGATION  SUBROUTINE FRACSO(TV,TL,FS,DNU,RNU,DT,RHG,L,I,AA,SI,RNUO,DCL, 8J0.REAG.DCA.EXP.KSI,SIO,DTCR) INTEGER P E , I , J 0 ( 8 1 ) REAL * 8 V G R , F S V , D F S . C L A , C L G , C A L , T L , D T , L , A A , S I O . R N U O , D C L , K K , R E A G REAL * 8 K G , C A G , O C A , E X P . K S I , S U P C , D T C R D I M E N S I O N T V ( 101 ) , F S ( 101 ) , D N U ( 8 1 . 7 0 0 ) , R N U ( 8 1 , 7 0 0 ) , R H G ( 8 1 ) DIMENSION S I ( 101) D O U B L E P R E C I S I O N TV , F S , D N U , R N U , R H G , S I  Appendices  465 466 467 468 469 470 471 472 47 3 4 74 4 75 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522  C C C C C C C  -288 -  CALCULATE AUXILIAR  VARIABLES  I F ( T V ( I ) . G T . T L ) GOTO 3 3 0 CLA = ( - T V ( I 1 + 1 5 6 9 - 2 4 . 3 2 * S I ( I ) )/97 . 3 CLG=(TV(I)+503.2- 1 2 9 . 7 ' S K I ) )/389. 1 C A L = ( - T V ( I ) + 1 5 2 8 . 4 - 3 2 " S I ( I ) ) / 177 . 9 CAG=(TV(I)- 1154.6-6 5 ' S I ( I ) ) • ( 1 . 5 - 0 . 2 1 6 ' S I ( I ) ) / ( 3 5 4 . 6 8 + 2. 1 - 0 . 2 1 6 ' S K I ) C C C C C C  COMPARE  T E M P E R A T U R E WITH  SUPC=TL-DTCR-TV(I) I F ( S U P C . L T . 0 . 0 1 ) GOTO C C C C C  NUCLEATION  CRITICAL  + 6.5•SI(I))  SUPERCOOLING  300  PROCEEDS  J0(I)=J0(I)+1 D N U ( I , J 0 ( I ) ) = A A * S U P C ' * E X P * ( 1 - F S ( I ) ) * DT RNU(I,JO(I))=RNUO C C C C C C 300  GROWTH OF  GRAPHITE  PE=JO(I) DO 3 0 5 J = 1 , P E  C  306 305 C C C C C 310  315  I F ( R N U ( I , J ) . G E . 0 . 0 0 0 6 ) GOTO 3 0 6 KK=(CLA-CLG)/(100-CLG)'3.646*DCL R N U ( I , J ) = R N U ( I . J ) + (- 2 * R N U ( I , J ) + ( ( 2 " R N U ( I , J ) ) • * 2 + 84*KK'DT)*'0.5)/2 GOTO 3 0 5 KG=(CAL-CAG)'DCA*3.646/(100-CAG) RNU(I,J)=RNU(I,J) + 1.911*KG'(1 - F S ( I ) ) * ' 0 . 6 6 *DT/RNU(I,J) CONTINUE  C A L C U L A T E NEW F R A C T I O N  SOLID  VGR=0 PE=J0(I) DO 3 1 5 J = 1 , P E VGR=VGR+4.19'DNU(I.J)*RNU(I CONTINUE FSV=FS(I) FS(I)=VGR*(1+REAG)  .J ) * * 3  Appendices 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 54 3 544 545 546 547  C C C C C C 406 C C C C C C 408 330 C C C 340 C  DFS=FS(I)-FSV I F ( F S ( I ) . LE . 1 .0)GOTO 406 FS(I)=1.0  CALCULATE SILICON CONCENTRATION AT RESIOUAL LIQUID SI(I)=SI0'(1-FS(I))"(KSI-1)  CALCULATION OF RATE OF HEAT GENERATED AT NODE RHG(I)=DFS*L*7/DT GOTO 340 RHG(I)=0.  RETURN END  -289-  Appendix 1  Appendix 7  Sample of the Output from Program SOLI  -291 -  Appendices  O T I » 0.500 OTFa 2.000  NDTI« 50  TP« 1230.0 DTCR= 0 . 0 0 0 SI-2.50 kkm 400.0 OCL» 0.500E-05 OCAa 0.900E-06 RNUO=0.00010 Ms 80 UA> 10 EXP> 340.0 COOLING CURVES ANO SOLIDIFICATION KINETICS TIME 33 41 49 57 65 73 81 89 97 10S 113 121 129 137 146 153 181 169 177 185 193 201 209 217 225 233 241 249 257 285 273 281 289 297 305 313 321 329 337 345 353 361 369 377 385 393 401 409 417  TC 1239 1235 1231 1228  ) 71 99. 56 54 1221 12 1215 54 1209 99 1204 83 1199 55 1194 78 ttao 34 1186 20 1182 32 1178 68 1175 22 1171 95 1168 87 1166 00 1163 36 1160 93 1158 72 1156 89 1154 83 1153 13 1151 57 1150 17 1148 92 1147 83 1148 91 1148 17 1145 61 1145 23 1145 02 1144 97 1145 05 1145 22 1145 46 1145 73 1148 02 1148 29 1148 52 1148 70 1146 82 1146 67 1146 85 1148 76 1146 80 1146 38 1148 10  FS( 1) 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 001 0 001 0 002 0 003 0 004 0 008 0 008 0 Oil 0 014 0 017 0 020 0 024 0 028 0 032 0 038 0 041 0 046 0 051 0 057  T( 151 1238 57 1232 18 1227 14 1221 88 1216 02 1210 40 1204 98 1199 85 1195 06 1190 59 1186 43 1162 55 1178 90 1175 44 1172 15 1189 05 1186 18 1163 49 1161 05 1158 82 1156 79 1154 92 1153 21 1151 65 1150 23 1148 96 1147 85 1148 91 1146 13 1145 53 1145 10 1144 83 1144 70 1144 70 1144 80 1 14497 1145 17 1145 40 1145 81 1145 79 1145 93 1146 00 1148 00 1145 94 1145 60 1145 59 1145 31 1144 97 1144 57  FS(15) 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 001 0 001 0 002 0 003 0 005 0 007 0 009 0 012 0 015 0 019 0 023 0 026 0 032 0 037 0 043 0 049 0 055 0 082 0 069 0 077  T(35l  1214 53 1206 98 1199 93 1193 71 1188 41 1183 91 1180 02 1178 58 1173 40 1170 45 1187 84 1184 87 1162 08 1159 43 1157 07 1155 02 1153 25 1151 89 1150 29 1148 99 1147 77 1146 85 1145 64 1144 77 1144 05 1143 49 1143 07 1142 77 1142 55 1142 40 1142 27 1142 16 1142 03 1141 87 1141 67 1141 40 1 14105 1140 61 1140 07 1139 41 1138 63 1137 7 1 1136 66 1135 44 1134 07 1132 48 1130 68 1128 65 1128 35  T(42) FS(35) 0 000 1 198 43 0 000 1 190 11 0 000 1 183 22 0 000 1 177 89 0 000 1 173 82 0 000 1 170 59 0 000 1 167 85 0 000 1 165 39 0 000 1 163 06 0 000 1 160 80 0 000 1 158 56 0 000 1 156 10 0 000 1 153 60 0 000 1 151 44 0 000 1 149 72 0 000 1 148 37 0 000 1 147 26 0 000 1 148 26 0 000 1 145 36 0 000 1 144 45 0 000 1 143 57 0 000 1 142 77 0 000 1 142 07 0 001 1 141 50 0 001 1 141 07 0 003 1 140 74 0 005 1 140 48 0 008 140 20 0 012 1 139 90 0 017 1 139 56 0 024 1 139 14 0 033 1 138 61 137 97 0 043 0 056 1 137 22 0 070 138 31 0 087 135 24 134 01 0 106 0 128 132 59 130 99 0 (53 0 182 129 18 0 215 127 18 124 93 0 254 0 299 122 54 352 120 0 01 0 415 1 117 38 0 489 1 114 70 0 571 111 97 0 684 109 21 0 763 106 40  FS(42) 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 001 0 002 0 004 0 006 0 011 0 017 0 026 0 038 0 054 0 073 0 097 0 126 0 160 0 200 0 247 0 302 0 385 0 437 0 519 0 805 0 697 0 789 0 874 0 941 0 983 0 999 1 000 1 000 1 000 1 000  T(54) 1157 03 1150 93 1148 11 1147 27 1147 07 1146 79 1146 28 1145 44 1144 35 1143 06 1141 30 1138 59 1137 07 1136 45 1136 33 1136 22 1 13594 1135 39 1134 51 1133 39 1132 06 1130 63 1129 18 1127 65 1128 10 1124 52 1122 87 1121 15 1119 39 1117 55 1115 67 1113 72  1111 73  1109 70 1107 62 1105 51 1103 37 1 10119 1098 97 1096 71 1094 40 1092 06 1089 88 1087 26 1084 79 1082 29 1079 76 1077 20 1074 57  FSi54) 0 000 0 000 0 000 0 000 0 000 0 000 0 001 0 001 0 002 0 003 0 005 0 010 0 022 0 048 0 084 0 135 0 198 0 274 0 361 0 459 0 561 0 684 0 764 0 852 0 921 0 988 0 992 0 989 0 999 0 999 0 999 0 999 0 999 0 999 0 999 0 999  0 999 0 999 0 0 0 0 0 0 0 0 0 0 0  999 999 999 989 999  999 999 999  999  999 999  T<60) 1130 93 1128 57 1130 00 1132 24 1133 99 1134 88 1135 04 1134 62 1133 74 1132 53 1129 36 1126 11 1124 53 1123 78 1123 37 1122 90 1122 26 1121 33 1120 05 1118 52 1116 86 1115 14 1113 46 1111 80 1110 19 1108 80 1106 97 1105 30 1103 57 1101 78 1089 94 1098 04 1096 09 1094 10 1092 07 1090 01 1087 91 1085 76 1083 59 1081 37 1079 12 1078 82 1074 49 1072 12 1069 70 1087 24 1064 82 1062 28 1059 70  T(61) 1127 74 1125 58 1127 11 1129 42 1131 22 1132 15 1132 34 1131 94 1131 08 1129 89 1126 77 1123 51 1121 93 1121 16 1120 76 1120 29 1119 65 1118 73 1117 45 1115 93 1114 27 1112 57 1110 89 1109 24 1107 63 1106 03 1104 41 1102 75 1101 02 1099 24 1097 40 1095 51 1093 57 1091 58 1089 56 1087 50 1085 40 1083 27 1081 10 1078 69 1076 64 1074 35 1072 02 1069 65 106 7 24 1064 79 1061 59 1059 04 1058 47  T( 88)  191 208 219 227 231 235 237 238 239 240 240 240 240 240 239 239 239 239 239 238 238 238 238 237 237 237 236 238 236 235 235 235 234 234 234 233 233 232 232 232 231 231 230 2 30 229 229 228 228 227  19 75 85 05 68 14 33 79 73 29 58 61 41  13  65 62 43 27  10  91 69 44 15 85 53 20 B8 55 22 88 54 18 82 44 06 67 28 85  43  01 57 13 88 22  75  28 80 30  77  Appendices  425 433 441 449 457 465 473 481 488 497 505 513 521 529 537 545 553 551 589 577 565 593 001 609 617 625 633 641 649 857 665 673 681 689 697 705 713 721 729 737 745 753 761 769 777 785 793 801 809 817 825  1145 78 1145 41 1 14502 1144 59 1144 15 1143 89 1143 22 1142 73 1142 21 1141 85 1141 05 1140 36 1139 56 1138 88 1137 60 1136 27 1134 80 1132 23 1129 19 1124 50 1115 80 1091 48 1071 32 1056 88 1048 73 1039 78 1031 61 1023 95 1016 65 1009 62 1002 76 996 10 989 53 983 06 976 88 970 32 984 05 957 82 951 64 945 49 9 39 39 933 33 927 30 921 31 915 36 909 43 903 55 897 89 891 88 886 09 880 34 NOOULE  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  063 069 078 083 091 100 109 120 132 145 160 177 197 221 250 268 331 391 470 588 779 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000  -292-  1144 1143 1143 1142 1141 1141 1140 1139 1138 1138 1136 1135 1133 1131 1129 1125 1121 1115 1107 1099 1089 1076 1064 1053 1043 1035 1027 1020 1013 1006  999  992 986 979 973 967 960 954 948 942 936 930 924 918 912 906 900 894 888 883 877  12 63 11 54  93 29  59 82  97  00 86 51 85 81 21 86 42 50 95 21 22 90 03 27 91 48 60 16 02 11 37 75 24 81 45 15 90 70 54 43 35 31 30 33 39 49 83 79 99 23 49  COUNTS  NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI  IN IN IN IN IN IN  NOOE NODE NOOE NOOE NOOE NOOE  NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI  IN IN IN IN IN IN IN IN IN IN IN IN IN IN  NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NOOE  1= 2= 3= 4s 5» 8« 7= 8» 9« 10« 1 ID 12* 1314a 15» 16s 17 = 18* 19= 20°  14778807 14686524 14531554 14363988 14228945 13873778 13601985 13229953 12888770 12591899 12326239 12055747 11769205 11525600 11274167 11058400 10825873 10827886 10434807 10222505  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  086 095 108 117 130 145 161 181 204 231 265 307 361 428 515 629 773 924 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000  1123 1120 1117 1114 1111 1108 1104 1101 1097 1093 1090 1088 1082 1078 1074 1070 1085 1061 1058 1051 1048 1040 1034 1027 1020 1013 1007 1000 994 987 961 975 966 962 958 950 944 936 932 926 920 914 908 902 896 890 885 679 873 868 882  77 91 84 63 36 01 61 13 58  95 23 42  51  49 36 10 69 12 38 37 08 38 06 39 63 91 28 73 25 83 47 16 90 68 51 37 28 20 16 17 20 27 38 52 89 89 13 40 71 04 41  0 0 0 1 1 1  860 939 988 000 000 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 000 1 .000 1 000 1 000 1 .000 1 .000 1 .000 1 000 1 000 1 000 1 .000 1 000 1 000 1 .000 1 000 1 .000 1 000 1 .000 1 000 1 000  1103 1100 1097 1094 1091 1088 1085 1081 1078 1075 1071 1068 1064 1060 1057 1053 1049 1044 1040 1036 1031 1026 1020 1015 1009 1002 996 990 984 977 971 965 959 953 947 941 935 929 923 917 911 905 899 893 888 882 876 870 885 859 854  53 62 65 63 56 42 22 96 64 24 77 22 58 88 03 09 04 98 52 00 27 25 84 04 01 85 64 41 18 98 75 58 42 30 21 15 12 12 15 22 32 45 81 81 04 31 80 93 29 88 10  1 .000 1071 1 .0001069 1.000 1066 1.000 1063 1.000 1080 1 .000 1057 1 .000 1054 1 .0001051 1 .000 1048 1.000 1045 1.000 1042 1.000 1039 1.000 1035 1.000 1032 1 .000 1028 1.000 1025 1.000 1021 1 .000 1017 1.000 1013 1.000 1009 1.000 1005 1.000 1000 1.000 996 1.000 991 1.000 966 1.000 980 1.000 974 1.000 988 1.000 963 1.000 957 1 .000 951 1 .000 945 1.000 939 1 .000 933 1.000 927 921 1.000 1 .000 915 1 .000 909 1.000 903 1 .000 898 1 .000 892 1 .000 886 1 .000 880 875 1.000 1 .000 869 863 1.000 1 .000 858 1 .000 852 1 .000 847 1 .000 841 1 .000 836  90 18 41 59 72 80 82 79 70 54 33 04 88 24 72 10 39 58 64 57 34 92 25 28 01 49 80 98 09 15 19 22 25 30 36 45 56 70 87 07 30 56 85 17  53 92 34 79 27 78 32  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  999 1057 08 1053 85 999 1054 41 1051 19  999 999 999 999 999 999 999 999 999 999 999 999  999 999  999 999 999 999 999 999 999 999 999  999 999 999 999 999  999 999  999 999 999 999  999 999  999 999 999 999 999 999 999 999  999  999 .999 999 999  1051 1048 1048 1043 1040 1037 1034 1031 1026 1024 1021 1018 1014 1011 1007 1003 1000 998 991 987 983 976 973 967 962 956 950 944 939 933 927 921 915 909 904 898 892 686 881 875 889 864 858 853 847 842 838 831 825  89 93 12 25 34 37 34 25 10 88 59 23 78 25 62 89 OS 08 96 68 13 32 21 85 30 80 82  99 12 24 37 50 65 61 00 22 47 74 04 38 74 14  57  03 52 04 60 18 79  1048 1045 1042 1040 1037 1034 1031 1028 1024 1021 1018 1015 1011 1008 1004 1000 997 993 988 984 980 975 970 964  959  953 947 942 938 930 924 918 912 907 901 695 889 884 878 872 887 881 856 850 845 839 834 828 823  48 73  93  07 17 21 19 11 97 76 49  13  70 16 57 85 02 07 98 88 17 38 29 95 42 75 99 17 33 47 61 76 93 11 32 56 82 11 43 78 16 58 03 51 01  55 12 72 34  227 226 226 225 225 224 223 223 222 222 221 220 220 219 219 218 217 216 218 215 214 213 213 212 211 210 209 208 207 208 205 204 202 201 200 199 198 196 195 194 193 192 191  24 70 16 82 07 51  95 37  79  20  59  97 35 70 05 38 69  99 27 53 77  99  17 33 45 51 53 51 44 33 19 04 88 68 49 29 10 90 71 52 34 16 00 189 84 188 70 187 55 186 40 185 26 184 13 183 00 181 87  Appendices  NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI NUCLEI  IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN  NOOE NODE NOOE NOOE NOOE NOOE NOOE NODE NOOE NOOE NOOE NOOE NODE NOOE NOOE NOOE NOOE NOOE NOOE NODE NOOE NOOE NOOE NODE NOOE NOOE NOOE NOOE NOOE NOOE NOOE NODE NOOE NOOE NOOE NODE NOOE NOOE NOOE NOOE  21a 22s 23= 24* 25* 28s 27s 28s 29s 30=  31 =  32= 33= 34= 35= 38= 37 = 38= 39= 40a 41 = 42= 43= 44= 45= 46 = 47= 48= 49= 50= 51 = 52= 53= 54= 55= 58= 57 = 58= 59= 80=  SOLIDIFICATION TIME-  10042735 9892259 9720866 9553520 9390269 9253512 9094705 8955847 8799745 8684432 8526804 8388397 8250299 81 16473 7980909 7844658 7734062 7808123 7505564 7 390605 7301787 7225366 7157975 7101581 7055845 7034914 7011106 7013849 7046300 7093162 7144472 7161429 7138027 7083790 7133401 7607367 7643899 8545600 6028130 9968371 593.00  NOOULAR SIZE SIZE 2 4 8 8 10 12 14 16 16 20 22 24 26 28 30 32 34 36 38 40 42 44 48 48 50  1 0 .0 0.,0 397700 .5 10401773 .7 2079769 .0 710498 .9 480115 7 268094 3 149119 2 7 3083 5 38321 .2 24770 .0 42402 .4 78430 . 1 34425 0 303 1 0 .0 0 0 0 0 0 .0 0 0 0 0 0. 0 0 .0 0 .0  15 0 0 0 0 53889 8 6513769 9 2265385 5 933684 . 1 689391 3 306302 0 178587 3 83171. 5 43099 a 44346 7 73639. 8 78026. 6 10672 .7 0. 3 0. 0 0. 0 0 0 0 0 0 0 0. 0 0 0 0 .0 0 .0  DISTRIBUTION 35 0 .0 3092 8 335317. 8 3187134 1 1707976. 6 809775. 1 70882 V 5 371086. 9 273307 3 227934 4 206008 8 129812. 3 20583. 6 57 5 0. 0 0 0 0 0 0 0 0 0 0..0 0 0 0 0 0.,0 0..0 0 0  42  0 .0  2485 . 3 486782 9 2278240 .8 1405317 6 788363 .9 7 35924 8 535628 .5 432930 1 350641..8 195712 .5 34599 9 538 8 0. 0 0 .0 0 .0 0 .0  0 .0  0 0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0  54 0 .0 1977 5 375227. 3 1537744..8 1 185848. 6 1015855 8 1061568. 8 1149220..7 652068. 5 68387. 9 18269 2 23..7 0 0 0. 0 0 .0 0..0 0 .0 0 .0 0 .0 0 .0 0 .0  0 .0 0 .0 0 .0 0 .0  Appendices  TEMPERATURE DISTRIBUTION NOOE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 34 35 38 37 38 39 40 41 42 43 44 45 48 47 48 49 50 51 52 53 54 55 58 57 58 59 60  TPR1 1242 1242 1242 1242 1241 1241 1241 1241 1241 1241 1240 1240 1240 1239 1239 1238 1238 1237 1237 1236 1236 1235 1234 1233 1232 1232 1231 1229 1228 1227 1228 1224 1223 1221 1220 1218 1216 1214 1212 1210 1207 1205 1202 1200 1197 1193 1190 1187 1183 1180 1176 1172 1187 1163 1159 1154 1149 1145 1140 1135  0 0 0 0 9 8 8 5 3 0 8 5 1 8 4 9 4 9 3 7 1 4 6 8 9 0 0 9 8 6 3 9 5 9 2 5 6 6 5 3 9 4 7 0 0 9 7 3 7 0 1 1 9  8 2 8 9 1 2 3  TPR2  TPR3  1214 1 184 1214 1 184 1214 1 184 1214 0 184 1213 8 184 1213 6 183 1213 3 183 1213 0 183 1212 6 183 1212 1 182 1211 6 1182 1211 1 1182 1210 4 181 1209 8 181 1209 0 180 1208 2 180 1207 4 1179 1208 5 179 1205 5 178 1204 5 177 1203 4 178 1202 2 176 1201 0 1175 1199 a 174 1198 5 173 1197 1 172 1195 7 171 1194 3 170 1192 8 169 1191 2 168 1189 7 167 1188 0 166 1186 3 165 1184 6 184 1182 9 163 1181 1 162 1179 3 161 1177 5 159 1175 6 158 1173 7 157 1171 8 156 1169 9 154 1167 9 153 1188 0 152 1 164 0 150 1182 1 149 1160 1 148 1158 2 148 1156 2 145 1154 3 143 1152 3 142 1150 4 140 1148 6 139 1148 7 137 1144 8 1 136 1143 0 1 134 1141 2 132 1139 4 130 1137 3 127 1135 0 125  2 2 2 1 0 8 7 4 2 a 5 1 7 2 7 2 6 0 3 8 9 1 3 5 7 a 9 9 9 9 9 a 7 6 5 3 1 9 7 4 1 8 5 2 8 4 0 6 1 7 2 7 2 7 1 4 5 3 8 2  TPR4 151 151 151 151 151 151 151 151 151 151 150 150 150 150 150 150 149 149 149 149 148 148 148 147 147 147 147 148 148 145 145 145 144 144 144 143 143 142 142 142 141 141 140 140 139 138 137 137 135 134 132 130 128 126 123 120 118 115 112 110  TPR5  TPR8  8 1145 6 1145 8 8 1145 6 1145 8 8 1145 8 1 145 8 5 1 145 8 1 145 7 5 1145 6 1145 7 4 1145 8 1145 6 4 1145 6 1145 5  3 2 0 9 8 6 4 2 0 8 8 3 1 8 5 2 9 8 3 0 8 3 9 a 2 8 4 1 7 3 a 4 0 5 1 6 0 4 7 9 0 9 5 9 9 6 1 4 7 0 4 8 2  1 145 5 1 145 5 1145 5 1145 5 1145 4 1145 4 1145 3 1145 3 1145 2 1145 2 1145 1 1145 0 1144 9 1144 a 1144 7 1 144 8 1 144 4 1144 3 1 144 1 1143 9 1143 6 1143 4 1143 1 1142 7 1142 4 1141 9 1141 4 1140 8 1140 2 1139 4 1138 8 1137 5 1136 3 1135 0 1133 3 1131 4 1129 3 1126 8 1124 2 1121 4 1118 6 1115 8 1113 0 1110 3 1107 6 1104 9 1102 3 1099 7 1097 1 1094 5 1091 9 1089 4 1086 8  1 145 4 1145 3 1 145 2 1 145 0 1 144 8 1 144 8 1 144 4 1144 1 1143 8 1143 5 1143 2 1142 a 1142 4 1141 9 1141 4 1140 8 1140 1 1139 4 1138 6 1137 7 1138 8 1135 5 1 134 1 1 132 5 1 130 7 1128 7 1126 4 1123 8 1120 9 1118 0 1115 0 1112 1 1 109 2 1106 3 1103 5 1100 8 1098 0 1095 3 1092 6 1089 9 1087 3 1084 7 1082 1 1079 5 1077 0 1074 4 1071 9 1069 4 1086 9 1084 4 1062 0 1059 5 1057 1  TPR7 1039 1039 1039 1039 1039 1039 1039 1038 1038 1038 1037 1037 1036 1038 1035 1034 1034 1033 1032 1031 1030 1029 1028 1027 1026 1025 1024 1023 1022 1020 1019 1018 1018 1015 1013 1012 1010 1009 1007 1006 1004 1002 1001  8  a 7  6  5 3 1 a 5 1 7 2 7 1 5 8 1 3 5 6 a 8 a 8 7 8 5 3 1 8 5 1 8 4 9 4 9 4 8 2 5 9  1  999 4 997 6  995 994 992 990 988 986 984 982 980 978 978 974 972 970 967  9 0 2 3 4 5 5 5  5 5 4 3 2 0 9  Appendices  NODE 1 2 3 4 5  S  7 a 9 10 11 12 13 14 15 18 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 34 35 38 37 38 39 40 41 42 43 44 45 48 47 48 49 50 51 52 53 54 55 58 57 58 59 80  TPR1 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 0000 0 .0000 0 0000 0 0000 0..0000 0 .0000 0..0000 0 0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0..0000 0. 0000 0. 0000 0. 0000 0. 0000 0. 0000 0. 0000 0 0000 0. 0000 0. 0000 0. 0002 0. 0020  TPR2  TPR3  0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0..0000 0 0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 0000 0 0000 0..0000 0..0000 0 0000 0. 0001 0. 0004 0. 0023 0 0114 0..0467 0 1591 0. 4090 0. 7976  0. 0000 0. 0000 0. 0000 0. 0000 0 .0000 0 0000 0..0000 0 .0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0 0000 0 .0000 0..0000 0 .0000 0 0000 0 0000 0 .0000 0 0000 0 0000 0 .0000 0 .0001 0. 0003 0..0009 0 .0022 0 0058 0. 0149 0 0394 0. 1061 0. 2685 0. 5833 0. 9217 0. 9998  TPR4  TPR5  TPR8  TPR7  1. 1 .. 1 1 . 1.  0. 0000 0. 0097 0. 0627 0000 0. 0000 0 0097 0. 0628 10000 . 0. 0000 0. 0097 0. 06 30 0000 0 0000 0. 0098 0. 0833 0000 0. 0000 0 0099 0 0639 0000 0. 0000 0 0100 0. 0847 0000 0 0000 0 0102 0. 0657 10000 . 0 .0000 0 .0105 0. 0670 10000 . 0 0000 0 0107 0. 0686 10000 . 0 .0000 0 0110 0. 0704 0. 9993 0 .0000 0 .0114 0. 0728 0000 0 0000 0 0119 0. 0752 10000 . 0 .0000 0 0124 0. 0783 0. 9990 0 .0000 0 0130 0. 0818 1. 0000 0 .0000 0 .0137 0. 0858 0000 0 .0000 0 .0145 0 0905 1. 0000 0 .0000 0 0154 0 0960 0000 0 0000 0 .0184 0 1023 1. 0000 0 .0000 0 .0178 0 1096 0000 0 .0000 0 .0189 0 1182 0000 0 .0000 0 0205 0 1283 1 0000 0 .0000 0 .0223 0 1402 1 .0000 0 .0000 0 .0244 0 . 1544 1.0000 0 .0000 0 .0289 0 .1713 0 .9999 0 .0001 0 .0297 0 . 1918 0 .9998 0 .0001 0 .0330 0 .2187 1.0000 0 .0001 0 .0370 0 2472 1.0000 0 .0001 0 .04 17 0 .2851 0 .9991 0 .0002 0 .0472 0 3322 0 9998 0 0003 0 .0539 0 .3911 .0000 0 .0004 0 .0620 0 4634 0 .9999 0 .0005 0 .0717 0 .5475 0 9995 0 .0007 0 .0837 0 6482 0 .9991 0 .0010 0 .0983 0 . 7548 1.0000 0 .0015 0 . 1185 0 .8800 1.0000 0 .0021 0 . 1392 0 .9432 0 .9991 0 .0029 0 . 1678 0 . 9888 0 .9994 0 .0041 0 .2040 0 .9997 0 .9997 0 .0056 0 .2501 0000 1.0000 0 .0083 0 .3089 0 .9996 0 .9996 0 .0119 0 . 3837 0 9991 0 .9991 0 .0170 0 .4773 0 9999 0 .9999 0 .0245 0 .5858 0 9998 0 .9998 0 .0353 0 7064 0 9998 0 .9998 0 .0511 0 .8272 0 9999 0 .9999 0 .0741 0 .9280 0 9993 0 .9993 0 . 1079 0 .9838 0 9997 0 .9997 0 . 1572 0 .9993 0 .9993 0 .9993 0 .2282 0 .9991 0 .9991 0 .9991 0 .3283 0 .9990 0 9990 0 .9990 0 .4802 0 .9992 0 .9992 0 .9992 0 .6189 0 .9995 0 .9995 0 .9995 0 7836 0 0 .9991 0 .9991 .999 0 9992 0 .9992 0 .9215 0 .9992 0 9909 0 .9997 0 .9997 0 .9997 0 9997 0 .9997 0 9997 0 .9997 0 .9998 0 .9998 0 .9996 0 .9998 0 .9998 0 .9998 0 .9998 0 .9996 0 .9992 0 .9992 0 .9992 •0 .9992 0..9998 0 .9998 0 .9998 0 .9998  1. 1 1. 1 1.  1  1  1  -296-  Appendices  Appendix 8  L i s t i n g of P r o g r a m m e  GROWTH  Appendices  1  2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33  -297-  c  ,  C C C C C C C C C C C C C  LIST OF PROGRAM GROWTH  THIS PROGRAM CALCULATES THE COOLING CURVE AND NODULAR SIZE DISTRIBUTION OF A SINGLE VOLUME ELEMENT UNDER IMPOSED COOLING CONDITIONS BASED ON NUCLEATION AND GROWTH DEFINE  VARIABLES  REAL * 8 DT,TL,RHO.CP,L.VOL,TP,TI,DFS,A,B,RNUO.KSI,TMAX REAL*8 SIO,SI,AA,DCL,DCA,DTCR,EXP,T,FS,FSV,CLA,CLG,CAG,CAL,REAG REAL'8 SUPC,KK.KG,TNUC,TOTNOD INTEGER JO DIMENSION RNU(500).DNU(500) DOUBLE PRECISION RNU.DNU  C C C C C C  INPUT VALUE READ(5,10) READ( 5.10) READ(5,10) READ!5,10) READ(5,10) READ(5,10) READ(5,10)  DT RHO L VOL TP SIO AA  34  R E A D ( 5 , 1 0 ) EXP  35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  READ(5,14) READ15,14) READ(5.10) READ( 5,10) READ(5,10) READ(5,10) READ!5,10) READ(5,10)  C 10 14 C C C C C C  DCL DCA KSI DTCR TMAX A B RNUO  F0RMATIF10.4) FORMAT(E12.3)  .  PRINT PARAMETERS WRITE(6,200)DT WRITE(6,201)VOL WRITE(6,202)TP WRITE(6,203)SI0 WRITE(6,204)AA WRITE(6,205)EXP WRITE(6,206)DTCR  ,  Appendices  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 1 1 1 1 12 113 1 14 115 1 16  WRITE(6,207)A WRITE(6,208)8 WRITE(6,209)KSI C 200 201 202 203 204 205 206 207 208 209 C C C C C C  C  c c c c c  c c c c c c  15  c c c c c c  c  30  FORMAT('DT=' ,F6 . 3) FORMAT('VOL=',F6.3) FORMAT('TP='.F10.3) FORMAT('SI=',F6.3) FORMAT('AA=' ,F8 . 1 ) FORMAT('EXP=',F6.3) FORMAT('DTCR=',F6.3) FORMAT('A=',F8.3) FORMAT('_=',F6.3) FORMAT('KSI=' ,F5. 3)  CALCULATE AUXILIAR VALUES TL=1154.6+6.5'SI0 REAG=0.274«(100 - 4.26 + 0.317 *SIO)/((4 . 260.317*SIO)-(-TL 1+1528.4-32'SIO)/177.9)  INITIALIZE COUNTERS TI=0 FS=0 TOTNOD=0 J0=0 T = TP SI=SIO START CALCULATION  TI=TI+DT IF(TI GT.TMAX) GOTO 300 TL=1154.6+6.5*SI  SORT NODES ACCORDING TO CONDITION TNUC=TL-DTCR IF((T.GT .TNUC).AND.(FS.LT.0.001)) GOTO 30 IF(FS.GT.0.999) GOTO 40 GOTO 50 DFS=0 CP=0.915 GOTO 100  Appendices  1 17 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  C 40 C 50  C C C C C C  90 C  C 70 C 80 C C C C C C  60 C  400  . 299  DFS=0 CP=0.61+1.214E-4*(T+273) GOTO 100 IF(T.GT.TL) GOTO 100 CLA=(-T+1590.5-24. 32*SI)/97. 3 CLG=(T + 503.2- 129.7*SI )/389 . 1 CAL=(-T+1528.4-32*SI)/177.9 CAG=(T-1154.6-6.5'SI)*(1.5-0.216*SI)/(354.6+6.5"SI)+2.1-0.216*SI  CALCULATE NUCLEATION AND GROWTH SUPC=TL-DTCR-T IF(SUPC.LT.O.OI) GOTO 90 J0=J0+1 DNU(JO)=AA* 100000*SUPC*2.718* *(-EXP/SUPC)•(1-FS)*0T RNU(J0)=RNU0 DO 80 J=1 ,JO IF(RNU(J).GE.0.0006) GOTO 70 KK=(CLA-CLG)/( 100-CLG)"3.646*DCL RNU(J)=RNU(J) + (- 2 * RNU(J) + ((2*RNU(J))**2 + 4*KK*DT)**0.5)/2 GOTO 80 KG=(CAL-CAG)/(100 - CAG ) *DCA* 3.646 RNU(J)=RNU(J) + 1 .911*KG'(1-FS)* *0.66*DT/RNU(J) CONTINUE  CALCULATE FRACTION SOLID AND RHG VGR=0 DO 60 J=1.JO VGR=VGR + 4.19*DNU(J)*RNU(J)* * 3 CONTINUE FSV=FS FS=VGR*(1+REAG) DFS=FS-FSV IF(FS.LE.1.0)GOTO 400 FS=1.0 CP=(1-FS)*0.915+FS*(0.61+1 SI=SIO'( 1-FS)**(KSI-1)  214E-4*(T+273))  C C C C CALCULATE TEMPERATURE C C 100 T=T + DFS*L/CP-(A -B * TI)*DT/(RHO* CP* VOL) C  Appendices  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 '  C C C C C 199 C 300 C 130 120 131  PRINT RESULTS WRITEI6.199) TI.T.FS FORMAT(3F10.4) GOTO 15 DO 120 1 = 1 ,JO WRITE(6.130) DNU(I),RNU(I) TOTNOD=TOTNOD+DNU(I) FORMAT(F10.1,5X,F10.8) CONTINUE WRITE(6,131) TOTNOD FORMAT('TOTAL NUMBER OF NODES=' ,F 15 . 1 ,'[NOD/CM* * 2 ] ' ) STOP END  -300-  

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