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The columnar to equiaxed transition in Pb-Sn and Sn-Pb alloys Lowe, Geoffrey T. 1990

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T H E C O L U M N A R T O E Q U I A X E D TRANS IT ION IN Pb-Sn A N D Sn-Pb A L L O Y S by Geoffrey T. Lowe B.A.Sc.(Metallurgical Engineering), University of British Columbia, 1978 A THESIS IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metals and Materials Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1990 ©Geoffrey T. Lowe In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) ii ABSTRACT The columnar to equiaxed transition has been studied experimentally using accurate temperature measurements and with a heat transfer mathematical model in Pb-1.5%Sn and Sn-10%Pb alloys. The behaviour of the two alloys was markedly different In the Pb-1.5%Sn alloy, the transition did not occur until a large portion of the liquid thermally supercooled. At this point the temperature gradient in the liquid is very low, approximately O'CAnm. For Sn-10%Pb, the transition occurred at a positive temperature gradient of approximately 0.12°C/mm which confirms earlier work by Mahapatra and Weinberg. Supercooling was also found to occur in this alloy but it is not certain whether the supercooling is constitutional or thermal. The difference in the behaviour of the two alloys is related to the density of nuclei prior to directional solidification. Altering the method to release latent heat by using the Scheil equation or the Lever rule instead of the traditional linear assumption in the mathematical model did not have a significant effect on the predicted temperatures or temperature gradients in the Pb-1.5%Sn alloy. The effect was seen when the Sn-10%Pb alloy was considered. The difference is attributed to the total amount of latent heat to be released by a particular alloy. iii TABLE OF CONTENTS ABSTRACT ii Table of Contents iii List of Figures vi List of Tables xi ACKNOWLEDGEMENTS xii 1.1 INTRODUCTION 1. 1.2 LITERATURE REVIEW 2. 1.2.1 Constitutional Supercooling 3. 1.2.2 Big Bang Theory 4. 1.2.3 Showering of Dendrites 4. 1.2.4 Dendrite Remelting 5. 1.2.5 The Influence of Fluid Flow on the CET 5. 1.2.6 Crystal Separation 6. 1.2.7 Competitive Growth Equiaxed vs. Columnar 6. 1.2.8 Critical Temperature Gradient 11. 1.2.9 Theories of Dendritic Growth Into Supercooled Alloy Melts 12. 1.2.10 Discussion 18. 1.2.11 Objectives of the Present Investigation 19. 2.EXPERIMENTAL PROCEDURES 40. 2.1 Columnar to Equiaxed Transition 40. 2.2 Temperature Measurement 43. 2.3 Directional Solidification 44. 2.4 Mathematical Modelling 47. 2.4.1 Objectives 48. 2.4.2 Solution Method 48. 2.4.3 Assumptions 49. 2.4.4 Boundary Conditions 50. 2.4.5 Initial Conditions 51. 2.4.6 Latent Heat Release 52. 2.4.7 Post Iterative Heat Balance 54. 2.4.8 Representation of the Phase Diagram 56. 2.4.9 Definition of G L and Velocity Calculations 58. iv 2.4.10 Validation of the Model 60. 3. EXPERIMENTAL RESULTS 76. 3.1 Columnar to Equiaxed Transition 76. 3.2 Temperature Measurements 80. 3.2.1 High Purity Lead (99.999%) 80. 3.2.2 Commercially Pure Lead (99.99%) 83. 3.2.3 Commercially Pure Tin (99.89%) 84. 3.3 Directional Solidification of Pb-1.5%Sn 85. 3.3.1 Furnace Off And No Insulation On Top 85. 3.3.2 Furnace Off; Top of Melt Thermally Insulated 91. 3.3.3 Furnace Power On; Melt Top Insulated 92. 3.4 Directional Solidification of Pure Lead (99.999%), Pb-3%Sn, Pb-6%Sn 98. 3.4.1 Directional Solidification of Pure Lead (99.999%) 98. 3.4.2 Directional Solidification of Pb-3%Sn and Pb-6%Sn 99. 3.5 Directional Solidification of Sn-10%Pb 101. 3.6 Mathematical Heat Transfer Model 102. 3.6.1 Directional Solidification of Pb-1.5%Sn 102. 3.6.2 Latent Heat Release 108. 3.6.3 Pb-3%Sn and Pb-6%Sn 108. 3.7 Sn-10%Pb 109. 3.8 Heat Flux at the Bottom of the Mould 111. 3.9 Sensitivity Analysis 113. 4. DISCUSSION 163. 4.1 Temperature Measurements 163. 4.2 Sn-10%Pb 167. 4.3 Pb-1.5%Sn 169. 4.4 Mechanism for the CET in Pb- 1.5%Sn and Sn- 10%Pb 172. 4.5 Mathematical Modelling 174. 5. CONCLUSIONS 180. REFERENCES 183. Appendix 1 186. Appendix 2 190. Appendix 3 194. C O M P U T E R P R O G R A M M E S vi LIST OF FIGURES 1.1 An Example of the CET in a Pb-2%Sb Alloy 21. 1.2 Probable Sequence of Events for the Constitutional Supercooling Theory 22. 1.3 The Relationship Between Columnar Zone Length, Equiaxed Grain Size and Pouring Temperature 23. 1.4 The Relationship Between Alloy Content and Columnar Zone Length 23. 1.5 Secondary Dendrite Branches Detaching in an Carbon Tetrabromide-Salol Solution 24. 1.6 Solidification Phenomena of Pure Metals and Alloys 25. 1.7 Dendrite Tip Temperature Measurement Technique 26. 1.8 The Dendrite Tip Temperature as a Function of Growth Rate and Temperature Gradient 27. 1.9 The Relationship Between Undercooling, (Supercooling), and Alloy Content 28. 1.10 The Hunt Model Predicted Structure Regions For An Al-3%Cu Alloy as a Function of Growth Rate, (V), and Temperature Gradient 29. 1.11 The Gradual Transition Between Columnar Grains and Equiaxed Grains as Visualized by Hunt 30. 1.12 The Relationship Between Columnar Zone Length and The Bottom Heat Transfer Coefficient for Sn- 10%Pb 31. 1.13 The Position of the CET in a Sn-10%Pb Alloy at Three Superheats 32. 1.14 The Predicted Temperature Gradient in the Liquid for Sn-10%Pb as a Function of Distance from the Chill 33. 1.15 The CET in Three Directionally Solidified Al-3%Cu Ingots 34. 1.16 The Cast Structure Appearance at Three Different Concentrations ofTiB2forAl-3%Cu 35. vii 1.17 The Three Components of Supercooling Depicted on a Generic Phase Diagram 36. 1.18 Concentration and Temperature Fields Ahead of a Dendrite Tip 37. 1.19 Predicted Versus Experimental Dendrite Growth Rate as a Function of Alloy Content and Supercooling 38. 1.20 Predicted Versus Experimental Dendrite Tip Radius as a Function of Alloy Content and Supercooling 39. 2.1 The Experimental Apparatus for the CET Observations in Pb-1.5%Sn 62. 2.2 Experimental Apparatus for the Thermocouple Calibrations 63. 2.3 Experimental Apparatus for Directional Solidification of Pb-1.5%Sn 64. 2.4 Various Chill Assemblies Used for Directional Solidification of Pb-1.5%Sn 65. 2.5 Modified Experimental Apparatus for Directional Solidification of Pb-1.5%Sn 66. 2.6 Dimensions of Graphite Cap 67. 2.7 Thermocouple Arrangement for Temperature Measurements During Directional Solidification 68. 2.8 Node Configuration for the Heat Transfer Mathematical Model 69. 2.9 Thermocouple Arrangement for the Determination of the Top Boundary Condition 70. 2.10 Vertical and Radial Temperature Gradients at the Top Part of the Ingot . 71. 2.11 Assumed Top Boundary Condition for the Heat Transfer Model 72. 2.12 The Initial Temperature Distribution Used in the Model 73. 2.13 The Fraction Solidified in a Node for Pb-1.5%Sn as a Function of Temperature 74. 2.14 Nodal Arrangement at the Leading Edge of the Solid/Liquid Interface 75. 3.1 Cast Structure of Test 1 115. viii 3.2 Cast Structure of Test 2 115. 3.3 Cast Structure of Test 4 116. 3.4 The Position of The CET as a Function of the Number of Stainless Steel Sheets 117. 3.5 Cooling Curves for Test 1 117. 3.6 The Duration of the Temperature Plateau as a Function of Distance in Test 1 118. 3.7 Cooling Curves for Test 2 118. 3.8 Cooling Curves for Test 3 119. 3.9 The Duration of the Temperature Plateau as a Function of Distance in Test 3 120. 3.10 Cooling Curves for Test 4 121. 3.11 The Plateau Temperature in Tests 1, 2, and 3 as a Function of Distance 122. 3.12 A Typical Cooling Curve for TCI in 99.999%Pb 123. 3.13 Cooling Curves for TCI, TC2, TC3 in 99.99%Pb 124. 3.14 Cooling Curves for TCI, TC2, TC3 in 99.89%Sn 125. 3.15(a),(b) Cast Structures for Tests 34, and 38 126. 3.16 Cooling Curves for Test 34 127. 3.17 Cooling Curves for Test 34 Expanded About the Liquidus Temperature 128. 3.18(a),(b) Cast Structures for Tests 40, and 42 129. 3.19 Cooling Curve for Test 43 Expanded About the Liquidus Temperature 130. 3.20(a), (b), (c), Cast Structures for Tests 46,47, and 48 131. 3.21 Cooling Curves for Test 47 Expanded About the Liquidus Temperature 132. ix 3.22 Cooling Curves for Test 48 Expanded About the Liquidus Temperature 133. 3.23 Cast Structure of Test 54 134. 3.24 Cooling Curves for Test 54 Expanded About the Liquidus Temperature 135. 3.25 Cast Structure of Test 55 136. 3.26 Cooling Curves for Test 55 137. 3.27 Cooling Curves for Test 57 Expanded About the Melting Point 138. 3.28 Cooling Curves for Test 59 Expanded About the Liquidus Temperature 139. 3.29 Cast Structure of Test 60 140. 3.30 Cooling Curves for Test 60 141. 3.31 Predicted Versus Experimental Cooling Curves for Test 45 with Linear Assumption 142. 3.32 The Thermal Field As Predicted By the Linear Model at Two Different Distances 143. 3.33 G L as a Function of Distance 144. 3.34 Predicted Versus Experimental Cooling Curves Using Incorrect Boundary Conditions 145. 3.35 Predicted Versus Experimental Cooling Curves Using Correct Boundary Conditions 146. 3.36 G L as a Function of Distance for Tests 45 and 48 147. 3.37 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 33 mm from the Chill, Test 45 148. 3.38 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 61 mm from the Chill, Test 45 149. 3.39 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 91 mm from the Chill, Test 45 150. 3.40 G L as a Function of Distance Using the Three Methods to Release Latent Heat 151. X 3.41 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 36 mm from the Chill, Test 59 152. 3.42 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 66 mm from the Chill, Test 59 153. 3.43 Predicted Cooling Curves Using the Three Different Methods to Release Latent Heat at 93 mm from the Chill, Test 59 154. 3.44 Predicted and Experimental Cooling Curves Using the Linear Assumption for Test 60 at 34 mm from the Chill 155. 3.45 Predicted and Experimental Cooling Curves Using the Linear Assumption for Test 60 at 64 mm from the Chill 156. 3.46 Predicted and Experimental Cooling Curves Using the Linear Assumption for Test 60 at 93 mm from the Chill 157. 3.47 Predicted and Experimental Cooling Curves Using All Three Methods to Release Latent Heat for Test 60 at 93 mm from the Chill 158. 3.48 G L as a Function of Distance Using all Three Methods to Release Latent Heat for Test 60 159. 3.49 Measured Temperature Differences in the Three Chill Assemblies as a Function of Time 160. 3.50 Predicted Versus Experimental Cooling Curve for Test 45 at 33 mm from the Chill Using the Calculated Heat Flux Instead of a Heat Transfer Coefficient 161. 3.51 G L as a Function of Distance Using Three Different Values for the Thermal Conductivity 162. xi LIST OF TABLES 2.1 Values of Parameters Used in the Mathematical Models 61. 3.1 Directionally Solidified Pb-1.5%Sn Results 77. 3.2 Cooling Curve Data For TCI with 99.999%Pb 81. 3.3 Cooling Curve Data For TC2 with 99.999%Pb 82. 3.4 Cooling Curve Data For TC3 with 99.999%Pb 83. 3.5 Cooling Curve Data For TCI, TC2, TC3 with 99.99%Pb 84. 3.6 Cooling Curve Data For TCI, TC2, TC3 with 99.89%Sn 85. 3.7 Pb-1.5%Sn Data, Directionally Solidified with Furnace Off and No Insulation 87. 3.8 Pb-1.5%Sn Data, Directionally Solidified with Furnace Off and No Insulation 90. 3.9 Pb-1.5%Sn Data, Directionally Solidified with Furnace Off and Insulation 92. 3.10 Pb-1.5%Sn Data, Directionally Solidified with Furnace On and Insulation 94. 3.11 Radial Temperatures Measurements during Directional Solidification with Furnace On 97. 3.12 Radial Temperatures Measurements during Directional Solidification with Furnace Off 97. 3.13 Directional Solidification Results of Pure Lead 99. 3.14 Directional Solidification Results of Pb-3%Sn and Pb-6%Sn 100. 3.15 Directional Solidification Results of Sn-10%Pb 101. 3.16 Predicted Values ofG L and Vat the CET for Pb-1.5%Sn 107. 3.17 Predicted Values of G L Using the Three Methods to Release Latent Heat for Sn-10%Pb 111. xii ACKNOWLEDGEMENTS I would like to thank the Natural Science and Research Council for the support in putting this thesis together. In addition, I would like to express my heartfelt thanks to Dr. Fred Weinberg for his patience and understanding while this document was being prepared. Finally, this thesis could not have been done without the support of my fellow students whom I owe a great deal of thanks as well as to my family and friends. This thesis is dedicated to the memory of my father, Robert A. Lowe. Chapter 1 1.1 INTRODUCTION The columnar to equiaxed transition, (CET), is a classical problem in the study of the solidification behaviour of cast alloys. The problem lies in the inability to fully understand why the structure changes abruptly from columnar to equiaxed along a plane, parallel to the mould wall, at some distance from the mould wall as shown by the example in Figure 1.1. The CET is sensitive to many of the parameters associated with the casting process, including the alloy composition, cooling rate, mould material, trace elements present in the melt, casting size and others. The transition from columnar to equiaxed may be sharp or gradual, the position of the transition can vary widely with respect to the mould wall, and the size of the columnar grains can vary appreciably. Since the mechanical properties of the casting are directly related to the cast structure all of these variables are important. At present there is no quantitative way of predicting the 1 2 cast structure from first principles, and particularly the CET. Even on a qualitative basis, the mechanism accounting for the CET is not clearly established. Optimum mechanical properties are usually associated with an equiaxed fine grained structure. This is generally the case for ferrous or non ferrous permanent mould castings, as well as the continuous casting of steel and the DC casting of Al slabs. In particular for the continuous casting of steel, cracking of the billet or slab often occurs at the grain boundaries between columnar grains during casting33. As a result, casting conditions are adjusted to minimize the size and length of the columnar structure starting at the mould wall by electromagnetic stirring of the melt during casting. A fine grain structure in the DC casting of Al is promoted, to obtain better mechanical properties, by adding a nucleating agent to the melt such as TiB2. An attempt is being made to establish the mechanism which accounts for the CET and to define a parameter or equation which can be used to predict the position of the CET by examining the CET in a simple binary alloy and using a heat transfer model to define the complex thermal field which occurs during the solidification of a cast alloy. 1.2 L I T E R A T U R E R E V I E W The structure of castings, which includes the CET, has been examined and reported on for many years. Most of the reports are described and reviewed in standard textbooks on casting1,2 and will not be considered here. A number of mechanisms have been proposed for the CET which are described and considered below. All of the models assume that the critical parameter in the CET is the presence and growth of nuclei ahead of the advancing columnar grains. The models then consider the source of the nuclei from 3 which the equiaxed grains grow. There are four mechanisms to account for the presence of the nuclei in the equiaxed zone. These are, constitutional supercooling, the big bang theory, dendrite remelting and dendrite showering. 1.2.1 CONSTITUTIONAL SUPERCOOLING A columnar to equiaxed transition is not observed in pure metals under growth conditions where the structure is initially columnar. As a result, the CET must be associated with the alloying element. This led Winegard and Chalmers4 to propose that the structure changes from columnar to equiaxed as a result of the entire central region becoming constitutionally supercooled. This would enable the existing heterogeneous nuclei to grow in a supercooled melt. The sequence of events as postulated by the authors is shown in Figure 1.2. Further consideration by Chalmers5 led him to discard the concept of the entire central region becoming constitutionally supercooled. Solute is concentrated in a relatively small region ahead of the advancing columnar grains, as a result of segregation of the solute during solidification. The thickness of the region is in the order of microns, and remains essentially constant with a constant growth rate and constant fluid flow. As a result the entire central region cannot be constitutionally supercooled. In addition there is no clear evidence that heterogeneous nuclei grow more readily in a constitutionally supercooled liquid. 4 1.2.2 BIG BANG THEORY Experimental observations showed that both the length of columnar grains and equiaxed grain size increased with increasing pouring temperature, Figure 1.3. In addition, the columnar length decreased with increasing solute concentration, Figure 1.4. These observations suggested that nuclei were present in the melt and led to the Big Bang theory. In this theory, the effective nuclei in the melt are generated at the mould wall as the chill mould is filled. Small solid grains are generated at the chill and are swept into the melt by the vigorous fluid flow in the melt at the start of solidification. At high pouring temperatures most of the grains do not survive and therefore there is a small equiaxed zone. The CET occurs when the melt supercools and allows the surviving grains to grow. The equiaxed grains ahead of the columnar grains block the columnar growth and the CET occurs. 1.2.3 SHOWERING OF DENDRITES Southin6 suggested that the presence of nuclei is the important step in the formation of the equiaxed zone. The nuclei are produced by the showering of dislodged surface dendrites which generally have a higher density than the melt. The surface dendrites result from a loss of heat to the ambient temperature by the air/melt surface. Preventing the loss of heat by hot topping or preventing the dendrites from showering into the central region of the casting by using a basket produced a predominantly columnar structure This mechanism for the CET is unique in that Southin proposes that supercooling is not a 5 necessary criteria for equiaxed grain growth. The equiaxed grains can grow after they touch the advancing columnar dendritic front. The latent heat of solidification is carried away by conduction through the solid columnar dendrites 1.2.4 DENDRITE REMELTING To test the big bang theory and the constitutional supercooling theory Walker7 conducted tests on Ni-Cu alloys and showed that these alloys could be supercooled 200° C and still produce an equiaxed zone during chill casting. This was contrary to what was expected for both theories. Jackson et Al 8 proposed a new theory based on observations made with transparent "metal like" materials. They showed that the secondary branches of the advancing columnar dendrites can become detached during growth, Figure 1.5. The detachment of the secondary branches is attributed to the melting of the solute enriched base of the secondary branch by currents of slightly hotter melt. The secondary branches move into the central region of the melt as a result of fluid flow or buoyancy forces. The secondary branches form the nuclei necessary for equiaxed growth. This mechanism produces a transition which is gradual rather than abrupt. The growth of nuclei into equiaxed grains was considered to occur in constitutionally supercooled liquid as postulated by Winegard and Chalmers4. 1.2.5 THE INFLUENCE OF FLUID FLOW ON THE CET Cole and Boiling910 examined the effect of fluid flow on the formation of the CET in a number of alloys. They found that by suppressing the fluid flow due to natural 6 convection in the liquid by placing physical barriers to flow in the melt that columnar growth dominated. Increasing fluid flow in the melt by using rotating magnetic and electric fields reduced columnar growth producing a structure that was primarily equiaxed grains. Fluid flow was believed to change the structure and the CET by lowering the temperature gradient in the liquid, G L , to a point where the nuclei that are present can survive and grow into equiaxed grains. A secondary effect of fluid flow is the enhancement of dendrite remelting following Jackson et Al 8 . 1.2.6 CRYSTAL SEPARATION AT THE MOULD WALL Ohno11 proposed that nuclei for the equiaxed grains originate on the mould wall prior to solidification which is similar to the "Big Bang theory". However, Ohno explains that growth occurs along the mould wall in the fashion shown in Figure 1.6. Due to solute rejection a neck forms at the junction between the equiaxed grain and the mould wall. The equiaxed grain breaks at the neck and drifts into the central zone which is constitutionally supercooled. Support for this mechanism came from direct observation of the solidification of Sn-Bi in which small grains were observed to leave the mould surface on top of the melt. 1.2.7 COMPETITIVE GROWTH: EQUIAXED VERSUS COLUMNAR GROWTH A completely different approach to the problem of the CET has been taken by Hunt and his co-workers12'17. Hunt recognized that for equiaxed grains to grow they must 7 release the latent heat of solidification into liquid that is cooler than the equiaxed grain. In other words, the liquid has to be supercooled. Since the CET occurs only in alloys, supercooling is constitutional. While most of the other studies consider constitutional supercooling, they have not considered how the constitutional supercooling develops. Hunt has examined in detail how constitutional supercooling is produced in the melt and the resultant structure on solidification based on nucleation and growth models. The CET model developed by Hunt is based on the results of Burden and Hunt1617 who measured the columnar dendrite tip temperature in an Al-Cu alloy under different imposed temperature gradients, G, and growth velocities, V. The columnar dendrite tip temperature was determined by directionally solidifying the alloy in a small rod containing a thermocouple. When the leading edge of the solidifying front passed the thermocouple, the rod was rapidly quenched. The rod was then sectioned and the distance from the quenched solid interface and the thermocouple measured. The temperature of the solid/liquid interface was determined at the time when the interface was coincident with the thermocouple by back calculating using the growth velocity, the distance between the interface and the thermocouple at the time of the quench, and the cooling curve obtained from the thermocouple. This is shown in Figure 1.7. They used calibrated Pt-Pt/Rh thermocouples and measured temperatures to an accuracy of ±0.1°C. The results showed that the columnar dendrite tip is supercooled and varies depending on the imposed temperature gradient, G, and growth velocity, V, as shown in Figure 1.8, and the alloy composition as shown in Figure 1.9. An equation was developed to calculate the amount of supercooling at the dendrite tip using the above variables and is given below. 8 V where: AT = Columnar dendrite tip supercooling G = Temperature gradient in the liquid D = Liquid diffusion coefficient V = Velocity of the columnar dendrite tip CQ = Alloy composition A' - Constant The Hunt model only considers the case where: AT=^-+A'(C0Vf (1.1) ^-«A\C0Vf (1.2) Which gives the supercooling at the dendrite tip as. AT = A' (1.3) Nucleation and growth of the nuclei into equiaxed grains are then assumed to occur in the supercooled zone ahead of the dendrite tip. Hunt assumes that the nucleation rate is given by the expression. / = ( A ^ - A % e x p j - ^ j (1.4) where: I = Nucleation rate per unit volume. 9 N 0 = Total number of heterogeneous nuclei available per unit volume. N = Number that have already produced grains. AT = Supercooling at the dendrite tip. u = A constant for a given nuclei. Io = A constant = 1020 estimated using the liquid diffusion coefficient of the melt. The value of u is calculated on the basis of the heterogeneous nucleation temperature, ATN, and is given by: u=-(A7)2ln(iVoxl020) (1.5) The competitive process Hunt considers in the model involves the competition between equiaxed grain growth and columnar grain growth in a constitutionally supercooled zone just ahead of the columnar dendrite tip. For the model to predict the growth of equiaxed grains in the supercooled zone, the concept of an extended volume fraction is used to account for the impingement of equiaxed grains. The extended volume fraction is then used to determine whether or not at a specific columnar dendrite tip supercooling the structure is equiaxed or columnar by introducing an arbitrary criteria which states that a structure is fully equiaxed when: (b£ > 0.66 (1.6) where: <J)£ = The extended volume fraction. The extended volume fraction, <j)£, is calculated using the following equation. 10 <t>£ = 4 J C T X 3 (1.7) where: r = Radius of the equiaxed grain. The radius of the equiaxed grain is determined by relating the growth velocity of the equiaxed dendrite tips, V e, at supercoolings greater than the heterogeneous nucleation supercooling, ATN, so that: Equation 1.8 is rearranged so that the velocity of the dendrite tip, V, and the alloy composition, COJ are removed by substituting(A 'AT,.)2 for VC„ from equation 1.3. In addition the radius of the equiaxed grain, r, is removed by using equation 1.7 at <}>£ greater than 0.66 to give the criteria for fully equiaxed growth as a function of the dendrite tip supercooling, the heterogeneous nucleation supercooling, the nuclei density, and the temperature gradient giving the following expression. For fully equiaxed growth: Columnar growth is determined in a similar fashion except that it is taken to dominate when <|>E < 0.0066 giving the following equation for fully columnar growth. r = A'[(ATf-(ATNf] 3VGC0 (1.8) (1.10) 11 The structure is predicted by assuming NQ is 1/mm3 and ATN is 0.75 K, which gives the columnar and equiaxed regions in a directionally solidified Al-3%Cu alloy shown in Figure 1.10 as a function of G and V. The CET using Hunt's model is assumed to be gradual as shown schematically in Figure 1.11. 1.2.8 CRITICAL TEMPERATURE GRADIENT Mahapatra and Weinberg18 investigated the CET in Sn-Pb alloys. They observed that the position of the CET was dependent on the rate of heat extraction on the bottom of a directionally solidified ingot, Figure 1.12. Experimental observations indicated that the position of the CET was independent of the melt superheat, as shown in Figure 1.13 contrary to previous reports, (Figure 1.3). As shown in Figure 1.13 the CET is abrupt and the transition plane flat contrary to the model proposed by Hunt. The temperature field during solidification was determined for a directionally solidified Sn-Pb melt using a one dimensional heat transfer model. The model assumes the linear release of latent heat, negligible radial heat loss, and negligible melt/air heat loss. The model determined a specific temperature gradient in the liquid, G L, that occurred at the position of the CET. For example, for a Sn-10%Pb alloy, the CET occurred at a G L of 0.1 l'C/mm as shown in Figure 1.14. Thus, it was termed to be a critical temperature gradient for the formation of the CET in Sn-Pb alloys. There was very little composition dependence on the value of the critical temperature gradient as G L for a Sn-5%Pb alloy was 0.10°C/mm and for a Sn-15%Pb alloy G L was 0.13°C/mm. The authors proposed that the CET occurred when the constitutionally supercooled region ahead of the advancing dendrite tip was sufficiently large enough to allow a 12 heterogeneous nucleus in the melt to grow. This region increased as the temperature gradient ahead of the melt decreased. Once equiaxed grains formed ahead of the advancing columnar dendrite tip, columnar dendritic growth would stop, producing a CET. In a similar study, Ziv and Weinberg19 showed that the CET occurs when G L = 0.06°C/mm for an Al-3%Cu alloy. The CET is also very sharp for this alloy as shown in Figure 1.15 when directionally solidified. The experimentally determined value of 0.06oC/mm agreed generally with the theoretical prediction of 0.073°C/mm using the Hunt model assuming a nuclei density of 4.6/cm3 and a calculated columnar dendrite tip supercooling of 0.71°C. Increasing the number of nuclei in the melt by adding up to 100 ppm TiB 2 did not change the position of the CET as seen in Figure 1.16. Adding 171 ppm of TiB 2 produced a fine grained structure at a very high G L of 0.88°C/mm. From these observations the authors proposed that there is a minimum large number of nuclei required to have an effect on the position of the CET. Below that number, the nucleant has very little effect on the CET and its position is determined by the temperature gradient in the liquid. 1.2.9 THEORIES OF DENDRITIC GROWTH INTO SUPERCOOLED A L L O Y MELTS Equiaxed and columnar grains are both manifestations of dendritic growth. As such it is felt to be worthwhile to discuss some of the theories of dendritic growth into supercooled alloy melts even though at the present time they do not explain the CET. They do however have potential to explain the CET if the critical parameters controlling 13 the CET are fully defined. A model by Lipton, Glicksman, and Kurz , assimilates the current theories very well to describe the behaviour of free dendritic growth in a supercooled liquid. The model splits the total supercooling at a dendrite tip into three components, so that: AT = ATt + ATc + ATR (1.11) where: AT = The total supercooling AT t = Thermal supercooling ATC = Constitutional supercooling AT R = Supercooling associated with the curvature of the dendrite tip. The three components of the total supercooling are illustrated in Figure 1.17 relative to a schematic phase diagram, and in Figure 1.18 relative to the temperature of the dendrite tip and the liquidus temperature of the alloy for a given solute composition profile ahead of the dendrite tip. The three components of the total supercooling in Figure 1.18 are contrasted with the case of a pure metal where C c = 0 and the total supercooling consists of only two components, the thermal component and the component associated with the curvature of the dendrite tip. Lipton, Glicksman, and Kurz developed expressions which give the growth rate, V, and the dendrite tip radius, R, of a free growing dendrite for a given alloy having a specified supercooling. 14 One expression is the Ivantsov equation which is the general solution for the diffusion field around a shape preserving paraboloid of revolution. This shape approximates the shape of a dendrite tip. For a steady state dendrite tip radius, the Ivantsov equation is: where: Q. = Dimensionless supersaturation Et(P) = Exponential integral function Iv(P) = Ivantsov function P = Peclet number For the thermal field, the Peclet number is: where: V = Interface growth rate R = Dendrite tip radius a = Thermal diffusivity For the solutal field ahead of a dendrite tip: Q = P EXP(/> )£,(/>) = IV(P) (1.12) ID (1.14) 15 where: D = The interdiffusion coefficient The dimensionless supersaturation, Q, is given a thermal and solutal component. The thermal component, Q, is given by: a>=ijf ( U 5 ) where: AH/cp = The designated unit thermal supercooling Combining equations 12 to 15, gives the thermal supercooling: AH AT,=—Iv(Pt) (1.16) CP where: AH = Latent heat cp = Specific heat It can also be shown that the constitutional supercooling is given by: •^-"4-(i^ kffj} <in) where: m = Liquidus slope C 0 = Initial alloy composition k„ = The equilibrium partition coefficient 16 The depression of the liquidus temperature due to the curvature of the tip is given by: IT ATR= — (1.18) where: T = The Gibbs-Thomson parameter Combining equations 16 to 18 gives: AT--AfuP,) + m c \ l - ^ j ^ + f (1.19) This equation relates the product of the growth velocity, V, and the dendrite tip radius, R, to the supercooling, AT. To obtain unique values for V, and R, a stability criterion is introduced. la (mGc-G)\ where: \ = Shortest wavelength which can form at the tip a* = Stability constant G c = Concentration gradient G = Temperature gradient Mean gradients were assumed in the model and defined as: 17 (1.21) — PcC\\-kQ) (1.22) where: C* = the solute concentration at the dendrite tip Thus, the stability criterion is written in the form. R = (1.23) PtAH PcmC0{\-k0) cp l-(l-ko)UPc) The equations for the tip radius and the total supercooling were used to determine how the growth rate, V, and the tip radius, R, varied with total supercooling and solute concentration for a succinonitrile-acetone solution. The model results agreed well with the experimental measurements as shown in Figure 1.19 and Figure 1.20. In the CET, a necessary condition for the equiaxed grains to grow ahead of the advancing columnar dendrites is that the liquid be supercooled. The model and analysis of the growth of dendrites described above is the most recent attempt to quantify the dendritic growth process. The model develops expressions for the dependence of supercooling at the dendrite tip on growth rate, tip curvature, and solute segregation. Since the dependence is a basic component of the CET process, the analysis could be of value in developing a model for the CET from basic principles. 18 1.2.10 DISCUSSION It appears that the temperature gradient in the liquid, G L is a critical factor in the development of the CET in cast alloys considering the work by Hunt12"17, Weinberg18,19, and the earlier work by Cole and Boiling9,10. To predict the position of the CET using Hunt's model requires specific knowledge on the number of nuclei present in the supercooled zone, the heterogeneous nucleation temperature of the nuclei, and the amount of supercooling at the columnar dendrite tip. These values are not readily available which limits the applicability of the Hunt model to predict the CET under normal casting conditions. In addition Hunt's model is for a gradual transition from columnar to equiaxed which has been shown not to be the case in work by Weinberg. The concept of a critical temperature gradient, G L, in the melt in front of the columnar dendrite tip requires detailed knowledge of the temperature field in a casting during solidification and the thermophysical properties of the metal. Heat transfer mathematical models are being developed extensively for castings and can be validated with temperature measurements. This can provide the detailed thermal field during the solidification which can be used to establish where in the casting the critical G L will occur. The values of G L for Sn-Pb are different from the Al-Cu alloys, which indicates G L is different for different alloy systems. However, there is no mention of thermocouple calibration reproducibility by Mahapatra and Weinberg18 in the Sn-Pb work while the G L for Al-Cu alloys could not be determined by a mathematical model due to the poor fit between experimental temperatures and predicted temperatures. The values for G L were 19 taken from the cooling curves which would have to be produced by thermocouples with a known accuracy relative to each other and an absolute accuracy relative to the melting point of the pure Al. 1.2.11 O B J E C T I V E S O F T H E P R E S E N T INVEST IGAT ION The nucleation theories to account for the CET are not clearly established, and cannot be used to predict when the CET will occur in a casting. The Hunt model, as it stands, requires information concerning the density and effective nucleating temperature of nuclei in the melt as well as the supercooling at the dendrite tip, all of which are generally not known in most alloys. In addition, the theory assumes a gradual transition at the CET which is generally not the case. The model which associates the CET with a critical temperature gradient ahead of the dendrite tip is critically dependent on accurate temperature measurements and a precise heat transfer mathematical model since the gradients are continuously changing and are very small at the CET. The result for Sn-Pb in which the experimental measurements were not fitted well by the heat transfer model indicate that better measurements are required as well as a better heat transfer model, particularly in establishing boundary conditions used in the model. Similar problems were present in the Al-Cu observations. The objectives of the present investigation are: 1. Experimentally determine the temperature distribution in the liquid ahead of the columnar dendritic front for Pb-Sn alloys using accurate temperature measurements. 20 2. Determine if a critical temperature gradient, G L , in the melt exists for Pb-Sn alloys at the CET and establish the value of the gradient. 3. Investigate various procedures used to account for the release of latent heat in a one dimensional heat transfer model. These procedures for releasing the latent heat include the following, 1) Linear release of latent heat using the specific heat method, 2) Release proportional to fraction solid determined by the Scheil equation, 3) Release proportional to fraction solid determined by the Lever rule. 4. With the mathematical model, establish the temperature gradient in the melt as a function of distance from the chill. Determine the critical temperature gradient, G L , at the CET. Examining the Pb-Sn system will establish G L for a third alloy system. In addition it will consider the possibility of equiaxed grain movement during solidification. When Sn-Pb solidifies, the initial grains are Sn rich and as a result are lighter than the melt. In the Pb-Sn system, the initial grains are heavier than the melt and they will not rise. Accordingly if upward movement of grains is significant in the CET, the results for Pb-Sn should be markedly different than Sn-Pb. 21 Figure 1.1 The columnar to equiaxed transition in a cast Pb- %Sb alloy at planes A-A mdA'-A' Figure 1.2 The sequence of events for the CET according to the constitutional supercooling theory. 23. Alloy content Figure 1.4 Variation in grain size with alloy content 24 Figure 1.5 Secondary dendrite branches showing neck formations and detachment during growth. Carbon tetrabromide with added salt Mag. X75 Figure 1.6 Mechanism for the formation of the equiaxed grains initiating at the mould wall following Ohno. 26 OI$T*MCI • <*Om T H f U H O C O U m TO TIM K K S U O I O O T >C»UT THIH««OC0U»Vt Sp«cifTitn grown at constant vtlocity v And q u t n c h t d T [ « P t « « r u i i ( F i g . I. A tempertture/lime trace of thermocouple in specimen. Figure 1.7 Measurement of temperature of dendrite tips at constant velocity, V, and quenching the sample during growth. 27 S ; 0 . i t 'CM • •• , u. G R O W T H R A T E C M / S E C Figure 1.8 Dendrite tip temperature vs. growth rate at three temperature gradients. 28. Figure 1.9 Supercooling at dendrite tip vs. weight percent Cu. 29. Figure 1.10 Predicted columnar and equiaxed regions in a directionally solidfied Al-3%Cu alloy as a function of the growth velocity V and temperature gradient G. The dotted lines use a more accurate analysis than the solid lines. Figure 1.11 Gradual transition from columnar to equiaxed structure following the model by Hunt 31 Figure 1.12 Columnar zone length vs. heat transfer coefficient for Sn-10%Pb (a) 19°C (b) 31°C (c) 36 Figure 1.13 The independence of the CET position on superheat for Sn-10%Pb. Figure 1.14 G u vs. distance from the chill for Sn-10%Pb. The four vertical lines are CET positions for tests 7,12,10, and 14. f 34 Figure 1.15 The CET in an Al-3%Cu alloy directionally solidified for various bottom heat transfer coefficients. The transition is very sharp in each case. 76 pj>*i Figure 1.16 The influence of TiB2 concentration on the CET in an Al-3%Cu alloy. 36 Figure 1.17 Phase diagram showing the elements of supercooling. Figure 1.18 Concentration and temperature fields ahead of the dendrite tip. 38 Figure 1.19 Growth rate V for two supercoolings as a function of acetone concentration C A C B (=Cn) in succinonitrile-acetone alloys, symbols are measured values and the line is calculated. 39 Figure 1.20 Dendrite tip R for two supercoolings as a function of acetone concentration C A C E (=Q) i succinonitrile-acetone alloys, symbols are measured values and the line is calculated. Chapter 2 EXPERIMENTAL PROCEDURES 2.1 COLUMNAR TO EQUIAXED TRANSITION Four directional solidification tests were carried out to establish that a columnar to equiaxed transition,(CET),occurred in a Pb-1.5%Sn alloy. The experiments were performed using the apparatus shown in Figure 2.1. The apparatus consists of a copper mould filled with a Pb-1.5%Sn alloy and contained in a quartz tube. The quartz tube is surrounded by a resistance heating furnace. The bottom of the mould is cooled by a water cooled copper chill block. The rate of cooling is controlled by layers of 1 mm thick stainless steel sheets between the mould and the copper chill block. 40 41 The mould is made from copper pipe with a copper plate brazed on one end. The outside and inside diameter of the mould are 42 and 37 mm respectively and the height of the mould is 110 mm for tests 1 to 3 and 200 mm for test 4. The inside of the mould is coated with a dispersion of graphite in alcohol,(Acheson DAG 154). The inside and outside vertical walls are lined with 3 mm thick Fibrefrax sheet to minimize radial heat flow during solidification. Weighted amounts of 99.99%Pb and 99.89%Sn are melted in the mould to produce a Pb-1.5%Sn alloy. The alloy is homogenized by vigorously stirring the melt for six minutes using a stainless steel plunger. Temperature measurements are made in the alloy during solidification with thermocouples at equidistant positions along the vertical axis of the mould. The thermocouples are held in position by a rubber bung and brass cap as shown in Figure 2.1. Four thermocouples are used in test 1 at approximately 20, 40, 60, and 80 mm from the mould bottom. Three thermocouples are used in tests 2, 3, and 4 located at approximately 25, 50 and 75 mm from the bottom. The thermocouples are made from chromel and alumel wires of 0.00254 mm (0.0001") diameter, separated by ceramic sleeves and sheathed in a 3 mm O.D. Pyrex tube. The thermocouples are calibrated against the temperature plateau at the freezing point of 99.99%Pb using an ice water cold junction. 42 The voltage signal from the thermocouples is measured with two Kipp and Zonen chart recorders for test 1 and an HP 3480A digital voltmeter for tests 2, 3, and 4. With the digital voltmeter, measurements are made at 1 second intervals for the three thermocouples in a single scan. The time interval between scans was dependent on the temperature being measured. For a test, the number of stainless steel sheets is selected and placed between the chill block and the copper mould. The mould is then filled with weighted amounts of Pb and Sn to produce a Pb-1.5%Sn alloy and the furnace turned on. After the metal has melted, it is stirred and the system equilibrated to the required temperature. The furnace is then turned off and water passed through the chill block to initiate solidification. For test 3, the furnace power is reduced to 50% instead of being turned completely off. The test is completed when all the thermocouples show a temperature below the liquidus temperature of the alloy. The solid metal is removed from the mould, the position of the thermocouples measured, and the height of the ingot determined. The height varied between 75 to 110 mm. The ingot is sectioned on a vertical plane containing the central axis. The cut surface is filed flat with an aluminum type A file, and etched to remove the worked material on the surface. The etch used is 4-3-16 part solution of nitric acid, acetic acid, and distilled water at 40°C. Etching is continued until the grain structure is clearly delineated. The surface is then ground on 80 to 400 SiC paper, lubricated with paraffin saturated kerosene, and given a final polish on a 5|i alumina polishing wheel. The surface is then lightly etched using the same etching solution used to remove the cold worked material. 43 2.2 TEMPERATURE MEASUREMENT It became evident in the initial tests that the accuracy of the temperature measurements is a critical factor in the observations since attempts are being made to measure very small temperature differences in a transient temperature field over relatively long time periods. As a result, a series of tests are conducted to calibrate the thermocouples more accurately and to determine the reproducibility of the temperature measurements. Accurate temperature measurements are made using the apparatus shown in Figure 2.2. The apparatus consists of a copper mould similar to that described in Figure 2.1 with graphite coating and fibrefrax insulation on the inside and outside walls. The bottom of the mould is thermally insulated with two, 3 mm thick sheets of fibrefrax. The mould height is 200 mm. The thermocouples are calibrated against the temperature plateau at the freezing point of 99.999%Pb (327.502°C42). The thermocouple to be calibrated is positioned in the centre of the melt and its voltage signal referenced to 0 °C using an ice-water cold junction. The voltage signal from the thermocouple is measured and recorded with a 16 bit analog to digital,(A/D), data acquisition board in an IBM PC. Temperatures are recorded intermittently at time intervals dependent on the temperature being measured. Each temperature measurement recorded is the average of 32 individual measurements made at 10 ms intervals. 44 For a normal test, the entire system is equilibrated at a specified melt temperature, and the furnace turned off. The melt slowly cools and solidifies from the outer vertical walls to the center of the mould. The cooling curve exhibits a clearly defined and flat temperature plateau at the freezing point of pure Pb which is related to the thermocouple output. The Pb is remelted and the procedure repeated. This is done a number of times to establish the accuracy and reproducibility of the measurements. Three calibration measurements are also made using 99.99%Pb and 99.89%Sn to establish the change in the shape and temperature of the temperature plateau using these metals. 2.3 DIRECTIONAL SOLIDIFICATION Directional solidification tests are carried out to carefully measure the temperature gradient in the liquid ahead of the solid/liquid interface ,G L , in the vicinity of the CET during solidification. The metals examined are pure Pb(99.999%), alloys of Pb-1.5%Sn, Pb-3%Sn, Pb-6%Sn, and Sn-10%Pb. The apparatus for these tests is shown in Figure 2.3 which is a modification of the apparatus shown in Figure 2.2. In Figure 2.3, a brass support ring is brazed to the bottom of the mould, the ring resting on an alumina sleeve which supports the mould in the furnace. In this way, the mould is separated from the chill block by an air space while the alloy is being melted and the furnace equilibrated. Cooling water flows through the chill block at all times. The chill system consists of a water cooled copper chill block and three copper discs which are positioned on top of the copper chill block, (Figure 2.4) This configuration is referred to as CA1 in this thesis. Each disc is 45 mm in diameter and 6.4 45 mm thick. The top and bottom discs contain a keyhole slot to accommodate a thermocouple. The center disc is reduced in cross sectional area by 39 holes of 3 mm diameter evenly distributed in the disc which reduces the rate of heat transfer from the mould. This basic configuration is altered by inserting a 6.4 mm thick stainless steel disc on top of the bottom copper disc, (Figure 2.4), and is referred to as CA2. The final alteration of the chill system consists of an additional 5,1 mm stainless steel sheets inserted on top of the bottom copper disc, (Figure 2.4), and is referred to as CA3. The chill system is altered in some tests to further reduce the rate of heat transfer from the mould bottom. The thermocouples in the top and bottom discs are made from chromel and alumel wires of 0.00254 mm (0.0001") diameter and are sheathed in ceramic tubes, positioned to measure the surface temperature in the center of the disc. The temperature difference, AT, between the top and bottom copper disc is used to estimate the heat flux from the bottom of the mould. Tests were carried out following three procedures. 1. Weighted amounts of Pb and Sn are added to the mould and melted. After vigorous stirring, three thermocouples are positioned in the melt at approximately 35, 65 and 95 mm from the bottom of the mould. No insulation is used at the top of the melt. When the melt has reached equilibrium at the desired melt temperature, the chill is raised with a laboratory jack until it comes into firm contact with the bottom of the mould. The current to the furnace is then reduced to a predetermined level and left at that position until solidification is complete. In this procedure, each temperature recorded on the (A/D) board from each thermocouple, including the 3 in the melt and the 2 in the chill system, is 46 the average of 8 measurements taken 50 ms apart. 2. Comparing the temperature measurements to predicted temperatures from a heat transfer model indicated that heat is being lost from the vertical side walls of the mould and from the top of the melt. To determine the extent of radial heat loss and heat lost from the top of the melt, the apparatus shown in Figure 2.3, is modified to that shown in Figure 2.5. The melt is prepared using procedure 1 except that loosely packed insulation is added to the top of the melt, and a graphite cap placed on top of the copper mould. The graphite cap has 6, 3 mm diameter holes drilled through it as shown in Figure 2.6. This allows thermocouples to be located at the positions in the melt shown in Figure 2.7, to measure both the vertical and the radial temperature gradients during directional solidification. Temperature measurements are made using those in procedure 1. 3. To further improve the accuracy and reproducibility of the temperature measurements, the calibration and measuring procedure is modified in the following way. A melt was prepared in a graphite crucible in the furnace and thoroughly stirred. The melt is then poured into a copper mould and allowed to solidify. Fibrefrax insulation is then loosely packed on top of the ingot and the graphite cap, containing the thermocouples, is placed on top of the mould. A quartz tube is positioned around the mould and the entire assembly lowered into the furnace to remelt the alloy. The three thermocouples are then lowered into the melt to a position approximately 65 mm from the mould bottom and the system allowed to reach equilibrium at the desired melt temperature. The furnace is then turned off and cooling curves are obtained for each thermocouple using the A/D data acquisition board and IBM PC. In this case, each temperature recorded from each thermocouple in the melt is the average of 8 47 measurements taken at 30 ms intervals. The alloy is then remelted and the three thermocouples positioned at approximately 35, 65, and 95 mm from the bottom of the mould The alloy is then resolidified directionally using procedure 1 described above. After solidification, the ingots are removed from the mould, sectioned vertically along the ingot axis, etched, ground, polished and reetched in the manner described previously. A Sn-10%Pb alloy is directionally solidified following procedure 3 used for the Pb-Sn alloys. In this case, after directional solidification, the ingot is removed, sectioned vertically, the surface is ground, polished and etched in a solution of 80-10-10 glycerol, nitric acid, and acetic acid. The sample is immersed in the solution for 10 minutes at 40°C. 2.4 MATHEMATICAL MODELLING The experimental temperature measurements during directional solidification are confined to three positions in the metal at which the three thermocouples are located. What is required is the full temperature field as a function of time and position as the metal solidifies. In particular, local temperatures and temperature gradients are required at the moving solid/liquid interface. This information can be obtained by developing a heat transfer model for the directionally solidified alloy system. The mathematical model predictions can then be compared to the experimental temperature measurements to verify the model. Additionally, the temperature measurements can be used to establish the boundary conditions required for the model. 48 2.4.1 OBJECTIVES The objective of the mathematical model is to predict the temperature in the metal as a function of time and distance during the directional solidification. From the temperature field, the temperature gradient at the solid/liquid interface, G L, can be determined as well as the growth velocities. 2.4.2 SOLUTION METHOD A one dimensional mathematical model in cylindrical coordinates is used to determine the temperature distribution in the directionally solidifying ingot. The governing heat conduction equation is: U 1 = PC'37 <21) where: k = the thermal conductivity of the alloy T = the temperature of the alloy z = the distance from the bottom of the mould p = the density of the alloy Cp = the specific heat of the alloy t = time 49 Equation 1 is solved using the implicit finite difference method for the node configuration scheme shown in Figure 2.8. The development of the finite difference equations based on a heat balance is oudined in Appendix 1. 2.4.3 ASSUMPTIONS The following simplifying assumptions are made to solve equation 2.1. 1. Heat flow is axisymmetric. 2 = 0 (2.2) where: 9 = the angular direction 2. The radial heat flow is zero. f = 0 (2,, where: r = the radial direction 3. There is no convection in the liquid ahead of the solid/liquid interface. 4. There is no melt supercooling prior to solidification. 5. The thermophysical parameters, k, p, and Cp, are linear functions of temperature between the liquidus and solidus temperatures. 6. The thermophysical parameters, k, p, and Cp, for the alloy can be determined from the parameters of Pb and Sn on the basis of the weighted averages of the Pb and Sn in the alloy. 50 2.4.4 BOUNDARY CONDITIONS Two boundary conditions need to be defined to solve the governing heat conduction equation, one at the top of the ingot where z = 1 and the other at the bottom of the ingot where z = N. At both boundaries, a constant heat transfer coefficient is used to calculate the heat flux from the corresponding surface. 1. At the bottom of the ingot where z = N: -k^H^-Tun) (2.4) 2. At the top of the ingot where z = 1: — = HTop(T\-TPurn) (2.5) where: H B = the heat transfer coefficient at the bottom of the ingot H X o p = the heat transfer coefficient at the top of the ingot 7"N = the temperature of the bottom node 7"! = the temperature of the top node T A M B = the ambient temperature taken to be 25°C T F u m = the furnace temperature immediately above the top node The furnace temperature, T F u m , is determined experimentally using the thermocouple configuration shown in Figure 2.9. Thermocouple 1 is in the loose fibrefrax insulation 10 mm above the melt surface; thermocouple 2 is in the melt 95 mm from the bottom of the mould; and thermocouple 3 is at the same height as thermocouple 2 51 adjacent to the furnace wall. The temperature difference between thermocouples 1 and 2, and thermocouples 3 and 2, as a function of time during solidification with the furnace power off, is shown in Figure 2.10. The results show that there are both vertical and radial temperature gradients between the melt and the furnace which change with time in a non-linear fashion. For the first 1500 seconds, there is a positive temperature gradient in both directions indicating heat is flowing into the melt from furnace. After 1500 seconds, the vertical temperature gradient is zero and continues to drop becoming negative at which point heat is flowing out of the melt. The radial temperature gradient decreases more rapidly than the vertical temperature gradient and becomes zero at 960 seconds after which it is negative again indicating that heat flows out into the furnace. This thermal behavior is incorporated into the model by assuming the temperature difference between the top of the melt and the insulation above the melt changes with time as shown in Figure 2.11. This is a simplified form of the experimental results. The radial heat losses are not incorporated in the model as it is assumed that the radial isotherms in the alloy during directional solidification are flat. 2.4.5 INITIAL CONDITIONS The initial temperature distribution in the melt at the start of cooling in the model is taken from the temperatures recorded by the three thermocouples in the melt immediately after the chill block contacts the bottom of the mould. A typical temperature distribution used in the model is shown in Figure 2.12. The temperature at the bottom and top of the melt is taken as the extrapolation of lines 1 and 2 respectively. Each line can be represented by the linear equation: 52 T{ = Slope (Disti) + Axis (2.6) where; Tj = the temperature of any node i Slope = the slope of the line Distj = distance between the node i and the bottom of the mould Axis = the temperature of the bottom node or the intercept of line 2 2.4.6 LATENT HEAT RELEASE Any mathematical model involving solidification must account for the latent heat of solidification. While there are a number of well established techniques to incorporate the latent heat into mathematical models41, these techniques still assume that the latent heat is released linearly between the liquidus and the solidus lines. Like any assumption, releasing latent heat linearly may take the model away from reality. The assumption's validity or usefulness is only tested by comparing the model's predicted temperatures to experimental temperatures. As such, the assumption on how the latent heat is released can be changed and tested in the same manner. In this investigation the assumption of how the latent heat is released is changed by assuming that the alloy solidifies according to the Scheil equation, and then changed again to assume that the alloy solidifies according to the Lever rule. Thus, there are three basic methods on how the latent heat is released. Each method incorporates the latent heat of solidification into the model in different ways. For example, to release the latent heat linearly the specific heat of the alloy is artificially raised following equation 2.7» 53 Cp{eff) = Cp(mushy) + L (2.7) where: Cp(eff) = the effective specific heat of the alloy in the mushy zone Cp(mushy) = the specific heat of the alloy in the mushy zone L = the latent heat of solidification of the alloy TU q = the liquidus temperature of the alloy T w l = the solidus temperature of the alloy The Scheil equation and the Lever rule do not direcdy release the latent heat but are used to calculate the fraction solidified within a single node. The latent heat is released in proportion to the chtnge in the fraction solidified as the node cools through the mushy zone. The Scheil equation, shown in equation 2.8, assumes that during solidification, there is complete solute mixing in the liquid and no mixing in the solid. Cs=kC0(l-gf-1 (2.8) where: Q = solute concentration in the solid Co = the bulk alloy composition c k = = the equilibrium partition coefficient g = the fraction solid The Lever rule, equation 2.9, assumes that solidification is at equilibrium in which: where the solid and liquid are homogeneous during solidification with solute concentrations of C s and C L respectively. The fraction solid as a function of temperature during solidification for the linear method, the Scheil Equation, and the Lever Rule are shown in Figure 2.13 for a Pb-1.5%Sn alloy between the liquidus and solidus temperatures only. In Figure 2.13, the fraction solid given by the Scheil equation at the solidus temperature is only 0.75. The node does not completely solidfy until the temperature reaches 183°C. This is not shown in Figure 2.13 for the sake of clarity. There is a significant difference in the shape and position of the curves for the three alternative procedures for releasing latent heat in the model. 2.4.7 POST ITERATIVE HEAT BALANCE To incorporate the Scheil equation and the Lever rule into the implicit finite difference mathematical model to release latent heat requires including three more unknown variables directly into the each nodal equation, namely, C s , CL, and g. Inserting these unknown variables directly into the nodal equations would make the simultaneous solution of the equations impossible since there are now too many unknowns per nodal equation. To overcome this problem, a correction technique is employed whereby the nodal equations are first solved simultaneously without latent heat being introduced and then the temperatures corrected afterwards to account for the latent heat. The correction 55 technique revolves around two heat balance equations to account for a node going from liquid to mushy or for a node going from one mushy state to another. For a node going from liquid to mushy, the heat balance is given by: After the initial solidification step, further solidification through the mushy zone is corrected using the following heat balance. where: T3 = the new corrected temperature after the latent heat has been brought in. T 2 = the new temperature after the time has been advanced a single time step, but no latent heat has been accounted for. TU q = the liquidus temperature of the alloy. Q = the specific heat of the node in the mushy state. C, = the specific heat of the node in the liquid state. Ag = the change in the fraction solidified of the node L = the latent heat of solidification The development of the post iterative heat balance equations is shown in Appendix The Secant convergence technique is used to arrive at the new corrected temperature for the Lever rule and the Scheil equation method. (2.10) (2.11) 1. 56 2.4.8 REPRESENTATION OF THE PHASE DIAGRAM Equations for the liquidus and solidus lines from the Pb-Sn phase diagram are required in the model to calculate C s , and C L for a node which is solidifying. The liquidus and the solidus lines are represented as a series of linear equations connected by two boundary temperatures. For the liquidus line in the Pb rich end of the phase diagram between the eutectic temperature, 183°C and the melting point of Pb, 327.5"C, the following linear equations apply. Between 183°C and 207° C CL = 61.9 -0.29(7,- -183°C) (2.12) Between 207°C and 300°C CL = 55.0- 0.48(7; -207°C) (2.13) Between 300°C and 327.5°C CL = 0.37(7,. - 300°C) (2.14) For the solidus lines the following linear equations apply. Between 183°C and 269°C Q = 19.2-0 .11(7 , -183°C) (2.15) Between 269°C and 327.5°C Cs = 10.0 - 0.17(7,.-269°C) (16) 57 For the Sn rich end of the phase diagram, with the melting point of 232 °C, the following linear equations apply for the liquidus line. Between 223°C and 232°C C t = 5.0-0.56(7; -223°C) (2.17) Between 215°C and 223°C CL = 10.0-0.63(7,.-215°C) (2.18) Between 209°C and 215°C CL = 15.5 - 0.92(7; - 209°C) (2.19) Between 183°C and 209°C Q = 38.1 - 0.87(7, - TEUT) (2.20) , The following linear equations apply for the solidus line. Between 217°C and 232'C Cs = 1.5- 0.1(7,. -217°C) (2.21) Between 200°C and 217°C Cs = 1.9 - 0.024(7,. - 200°C) (2.22) Between 183°C and 200 °C Cs = 2.5 - 0.035(7; - 183°C) (2.23) where: 58 Tj = the temperature of the node i. C L = the solute concentration in the liquid phase within a node. C s = the solute concentration in the solid phase within a node. 2.4.9 DEFINITION OF G L AND VELOCITY CALCULATIONS In order to calculate G L , the temperature gradient in the liquid immediately ahead of the advancing solid/liquid interface, the position of the interface must be located. With the assumption that there is no supercooling at the interface, the front of the interface can be associated with the liquidus temperature, T U Q . However, as a node begins to solidify, the predicted temperature may not be at the liquidus temperature exacdy, but, because of the length of the time step, it may be slightly lower. Therefore, to ensure that the position of the solid/liquid interface is determined to within a single node, the following criteria is used to define its position, measured from the bottom of the mould and given the arrangement shown in Figure 2.14. The solid/liquid interface is at node i when: UQ T';<T, UQ i + 1 <T, UQ i-1 >T, UQ where: 59 Tj = the temperature of node i at the current time step 7", = the temperature of node i at the future time step T'i + l = the temperature of node i+1 at the future time step i - l = the temperature of node i-l at the future time step The temperature gradient in the liquid ahead of the solid/liquid interface is then given by: where: AZ = the nodal distance of 0.5 mm. The velocity of the solid/liquid interface, V, is calculated by first assuming that its position at any time, t, is given by: where: x = the position of the solid/liquid interface measured from the bottom of the mould A and B are constants t = time The velocity is therefore given by: AZ (2.24) x =A + B^T (2.25) dx V = — dt (2.26) 60 fAB{{ <2-27> To obtain B in equation 2.27, the interface position, as determined by the model, is plotted first as a function of ytime. A best fit line is used to determine the slope of the line which is B. 2.4.10 VALIDATION OF THE MODEL The mathematical model is validated by considering that the heat transfer coefficients are constant at the top and bottom of the mould during solidification The two constants are adjusted until a good fit is obtained between the experimental temperatures and the calculated temperatures for each of the three thermocouple positions for the entire cooling and solidification period. Good fit is taken as ±1"C in the region near the liquidus temperature and ±3*C in the remaining regions. The validation was made first for the model using the specific heat method and then, with the same heat transfer coefficients and thermophysical parameters, with the Scheil equation, and finally with the Lever rule. The values of the parameters used in the model calculations are listed in Table 2.1. 61 T A B L E 2.1 V A L U E S O F P A R A M E T E R S U S E D IN M O D E L C A L C U L A T I O N S INPUTS A L L O Y Pb-1.5%Sn Pb-3%Sn Pb-6%Sn Sn-10%Pb LIQUID SOLID LIQUID SOLID LIQUID SOLID LIQUID SOLID T H E R M A L CONDUCTIVITY J/sec-cm °C 0.164 0305 0.168 0313 0.177 0328 0324 0.585 SPECIFIC HEAT J/°Cgm 0.155 0.141 0.158 0.144 0.163 0.150 0.251 0.251 DENSITY gm/cm3 10.46 11.27 1036 11.16 10.18 10.96 7.18 7.47 L A T E N T HEAT J/gm 24.50 24.82 2532 56.02 LIQUIDUS TEMPERATURE °C 322.9 318.9 310.8 215.0 SOLIDUS TEMPERATURE °C 318.3 309.6 292.2 183.0 NODE SIZE cm 0.05 0.05 0.05 0.05 TIME STEP sec 0.5 0.5 0.5 0.5 TOP HEAT TRANSFER COEFFICIENT J/cm 2-°C-sec 0.02 0.02 0.02 0.02 B O T T O M HEAT TRANSFER COEFFICIENT J/cm 2-°C-sec 0.01 0.01 0.01 0.01 RUBBER BUNG BRASS CAP THERMOCOUPLES QUARTZTUBE FTBREFRAX INSULATION FURNACE ELEMENTS COPPER M O U L D (FTBREFRAX INSULATED) Pb-Sn ALLOY STAINLESS STEEL SHEETS CHILL BLOCK PLATFORM WATER OUT "L——J" WATER IN Figure 2.1 Experimental apparatus for column ar/equiaxed transition observations 63 RUBBER BUNG GRAPHITE RINO FIBREFRAX INSULATION FURNACE ELEMENTS THERMOCOUPLE - COPPER MOULD (FIBREFRAX INSULATED) 99.999* Pb FIBREFRAX INSULATION CERAMIC BRICKS PLATFORM Figure 2.2 Experimental apparatus for thermocouple calibration and reproducibilty experiments 64 WATER OUT WATER IN RUBBER BUNG BRASS CAP THERMOCOUPLES QUARTZ TUBE FTBREFRAX INSULATION FURNACE ELEMENTS COPPER MOULD (FTBRFRAX INSULATED) Pb-Sn ALLOY BRASS SUPPORT RING AIR SPACE CHILL ASSEMBLY ALUMINA TUBE - PLATFORM CHILL BLOCK LAB JACK Figure 2.3 Experimental apparatus for directional solidification CAl CA2 CA3 SOLID COPPER DISK H I COPPER DISK (WITH HOLES) STAINLESS STEEL DISK STAINLESS STEEL SHEET Figure 2.4 Various chill assemblies used for directional solidification Figure 2.5 Modified experimental apparatus for directional solidification 67 30.5 mm 50.1 mm 7.5 mm © H 15.0 mm h Figure 2.6 Dimensions of graphite cap and positions of holes to accommodate thermocouples 68 0 0 c £ in <5 S E 9\ </"> N O R M A L T H E R M O C O U P L E A R R A N G E M E N T THERMOCOUPLE ARRANGEMENT FOR RADIAL HEAT FLOW DETERMINATION Figure 2.7 Thermocouple arrangement for radial heat flow determination and for normal temperature measurements TOP N BOTTOM AZ = 0.05 cm Figure 2.8 configuration scheme used for one dimensional heat transfer model 70 QUARTZTUBE THERMOCOUPLE 1 FTfiREFRX INSULATION COPPER MOULD FTBREFRAX INSULATION FTBREFRAX INSULATION THERMOCOUPLE 2 THERMCOUPLE3 Pb-Sn ALLOY Figure 2.9 Thermocouple positions for measurement of vertical and radial temperature gradients at the top of the Pb-Sn alloy during solidification 71 Figure 2.10 Vertical and radial temperature gradients in the top part of a Pb-Sn alloy as a function of time during directional solidification 72 F i g u r e 2.11 A s s u m e d v e r t i c a l temperature gradient at the top o f the melt for the heat transfer m o d e l 375 370 365 E-< g 360 S AXIS 2 cS"'" 355 AXIS 1 350 LINE 2 LINE 1 I . I . I L _ J — i — L _ i _ TOP .A SLOPE 2 I . I . I . 0 10 20 30 40 50 60 70 80 90 100 110 120 DISTANCE FROM CHILL FACE (mm) Figure 2.12 Temperature distribution in the melt at the start of cooling 74 TEMPERATURE C Figure 2.13 Fraction solid as a function of temperature for a Pb-1.5%Sn alloy calculated using the Scheil equation, the Lever rule, and assuming a linear function 75 z o H U ai oC 3 o a: LIQUID LEADING FRONT OF N SOLID/LIQUID INTERFACE SOLID/LIQUID ZONE AZ = 0.05 cm Figure 2.14 Nodal arrangement about the solid/liquid interface Chapter 3 EXPERIMENTAL RESULTS 3.1 COLUMNAR TO EQUIAXED TRANSITION To establish that a columnar to equiaxed transition, (CET), occurred in the Pb-1.5%Sn alloy, four directional solidification tests were carried out as listed in Table 3.1. Stainless steel inserts were used between the chill and the mould to change the cooling rate from one test to the next. At the slowest cooling rate, (test 1), with 7 stainless steel sheets, no CET is observed. A completely equiaxed structure is produced as shown in Figure 3.1. Increasing the cooling rate by decreasing the number of stainless steel sheets to 3, (test 2), results in the cast structure shown in Figure 3.2. A CET is observed to occur sharply on a plane that is approximately perpendicular to the vertical plane containing the central 76 77 31 mm from the chill face. Increasing the cooling rate further by not using any stainless steel sheets, (test 4), results in the cast structure shown in Figure 3.3. A CET is seen to occur on the left hand side approximately 85 mm from the chill face. For these four tests, the distance between the CET and the chill face decreases as the number of stainless steel sheets between the mould and the chill block increases in the manner shown in Figure 3.4. TABLE 3.1 DIRECTIONALLY SOLIDIFIED Pb-1.5%Sn A L L O Y RESULTS TEST N° AT H •c INGOT HEIGHT mm N ° O F STAINLESS STEEL SHEETS %FURNACE POWER ON DURING COOLING CET N ° O F EQUIAXED GRAINS/cm2 DISTANCE FROM CHILL mm 1 40 92 7 0 NO - 1 2 25 75 3 0 YES 34 6.7 3 25 80 3 50 YES 31 9.1 4 45 111 0 0 YES 85 12.5 The equiaxed grains in the region beyond the CET change in grain size with different cooling rates. The average number of grains per cm2, (N), measured on the polished samples for the four tests are also listed in Table 3.1. The value of N is observed to increase appreciably from test 1 to test 4 accompanying an increase in cooling rate for tests 1, 2, and 4. Test 3 should have a slower cooling rate than test 2 since the furnace power is kept on but it is not clear why N is larger in test 3. 78 Cooling curves are obtained in each test using the melt thermocouples (TC). At the slowest cooling rate, test 1, with the furnace power off during cooling, the cooling curves from the four TC's unexpectedly show four distinct temperature plateaus as seen in Figure 3.5. This is unexpected as all the plateaus should coincide since they are from TC's measuring temperature in the same alloy. If there are thermal second order effects this might result in a progressive change in the plateau with distance from the chill face but this is not the case. The duration of the plateau as a function of distance is shown in Figure 3.6. The duration of the plateau is seen to increase, then remain constant before unexpectedly decreasing as the distance is increased from 23, to 43,.to 62, to 81 mm from the chill face. The cooling curves for test 2 are shown in Figure 3.7. The increased cooling rate in this test elirninates the plateau at 27 mm from the chill face. A plateau is obtained for the TC at 51 mm from the chill face. The temperature of the plateau, (Tp), at this position is 324°C which is above the liquidus temperature, (TU Q), for the alloy and lasts for 240 seconds, appreciably shorter than the plateaus in test 1. At 77 mm from the chill face a plateau is obtained at 321°C, well below T L I Q with a short duration of 60 seconds. The results for test 3 with 50% furnace power on but the same number of stainless steel inserts as test 2 are shown in Figure 3.8. In this case, plateaus are observed at 27, 54, and 71 mm from the chill with T p being nearly coincident with T U Q at 27 and 54 mm but appreciably higher than T U Q at 54 mm from the chill face. The duration of each plateau increases with distance from the chill in this case as shown in Figure 3.9, as might be anticipated. 79 The cooling curves for test 4, having the highest cooling rate are shown in Figure 3.10. The curves do not exhibit a temperature plateau for any of the thermocouples. The cooling curves do exhibit a positive temperature gradient in the melt as the temperature increases with distance from the chill throughout cooling and solidification. Considering the cooling curves from tests 1-3 which have plateaus, the T p recorded are not consistent with each other or what might be expected from the phase diagram. The T p as a function of distance from the chill are plotted in Figure 3.11. For equilibrium conditions, one would expect the start of the plateau should occur at the liquidus and the curve should slope downward terminating at the solidus. This is clearly not the case since the plateaus are horizontal and T p for tests 1 and 3 are above or at T U Q . Also in test 1, T p varies in a cyclic manner between 326°C and 323.5°C with distance from the chill in the same Pb-1.5%Sn alloy. These results may be attributed to either errors in the temperature measurements of at least several degrees, or non-equilibrium solidification. In the case of non-equilibrium solidification, this would require the melt composition to be reduced to Pb-l.l%Sn for a liquidus temperature of 326°C. It is difficult to account for a reduction in the melt composition to l.l%Sn at 43 mm from the chill, and to assume no fluid flow during solidification. Accordingly the results indicate appreciable absolute and relative deviations of the measured temperatures during solidification. Following this conclusion a series of calibration and reproducibility tests were undertaken to delineate the errors in the temperature measurements for the specific thermocouples used in the measurements. 80 3.2 TEMPERATURE MEASUREMENTS 3.2.1 HIGH PURITY LEAD (99.999%) In the previous section in which the CET was examined, it was concluded that the measured temperatures were not sufficiently accurate or reproducible to determine the low temperature gradients associated with the CET. As a result, a series of careful measurements were undertaken to calibrate the thermocouples used and establish their accuracy and reproducibility. This was done by obtaining cooling curves for pure lead (99.999%) and comparing the freezing point, (TMP), determined from the plateau in the cooling curve to that of pure lead, defined as 327.502°C. The cooling curves were obtained from a thermocouple positioned in the center of a cylindrical graphite mould containing the molten lead, which was slowly cooled as previously described in section 2.2. A typical cooling curve for TCI is shown in Figure 3.12. The liquid initially supercools to 328.1°C, then the temperature rises to 328.5°C and remains constant for approximately 660 seconds. Compared to the melting point of pure Pb, the TC in this case, registers a temperature which is high by 328.5-327.5=1.0°C. This experiment, repeated 11 times for TCI, gives the results listed in Table 3.2. In all the tests, the melt supercooled for about 210 seconds before solidification with an average supercooling of 0.6°C, the supercooling varying between 0.2°C and l .TC. Horizontal temperature plateaus are observed in all cases with the scatter in the plateau within ± 0.1°C in each test. The average plateau temperature, T P , for all 11 tests is 328.5°C, 1.0°C above the Tm for pure lead. The difference between T P and T M P varied between 0.6°C and 1.3°C. 81 TABLE 3.2 COOLING CURVE DATA FOR TCI WITH 99.999% Pb. TEST N° SUPERCOOLING PLATEAU Tp-TM P ATS DURATION T P DURATION °C °C SECS. °C SECS. 5 0.6 240 328.7 960 1.2 6 0.6 180 328.6 1080 1.1 7 0.9 180 328.7 1080 1.2 8 1.1 210 328.2 780 0.7 9 1.0 240 328.3 990 0.8 10 0.6 180 328.1 720 0.6 11 0.7 240 328.1 660 0.6 12 0.5 180 328.4 750 0.9 13 0.7 240 328.5 810 1.0 14 0.4 240 328.6 690 1.1 15 0.2 210 329.2 780 1.7 16 03 210 328.8 480 13 AVG. 0.6 210 328.5 810 1.0 The results shown in Table 3.3 pertain to TC2 using the same procedure to obtain the results for TCI except the experiment was repeated only seven times. As with TCI, the melt supercools before solidification an average of 0.6°C. Horizontal temperature plateaus are observed in all cases which last for an average of 690 seconds. Unlike the case for TCI, the average T P is 328.3°C, 0.2°C lower than the average T P for TCI. The variation in the difference between T P and T M P for TC2 is also different as T p - T ^ for TC2 varies from -0.1°C in test 19 to 1.50C in test 21. The average difference is 0.8°C. 82 TABLE 3.3 COOLING CURVE DATA FOR TC2 WITH 99.999% Pb TEST N° SUPERCOOLING PLATEAU Tp-TM P °C AT. °C DURATION SECS. T P °C DURATION SECS. 17 0.6 240 328.6 750 1.1 18 0.7 240 328.6 900 1.1 19 0.7 180 327.4 720 -0.1 20 0.8 180 327.5 780 0.0 21 0J 180 329.0 780 1.5 22 0.4 240 328.8 510 13 23 0.4 180 328.3 480 0.8 AVG. 0.6 210 328.3 690 0.8 In a similar manner, nine repeat experiments were performed to obtain the results shown in Table 3.4 for TC3. This thermocouple registers an identical average supercooling to TCI and TC2 of 0.6°C but indicates a higher average T P than the previous two thermocouples at 328.8°C which is higher than the T ^ for pure Pb. The difference between T p and T ^ for TC3 is always positive and varies between 0.9°C and 1.9°C with an average difference of 1.3°C which is the largest difference of the three thermocouples tested. In summary, the three thermocouples indicate absolute temperatures which are too high when compared to T ^ for pure lead. Using the average values of T P for TCI, TC2, TC3 as listed in Tables 3.2, 3.3 and 3.4 respectively, the corrections would be -1.0°C, -0.8°C, and -1.3°C. Even with these corrections, an individual test would not produce highly accurate results as the scatter of individual measurements is close to 0.7°C from 83 the average values. These results indicate that each thermocouple should be calibrated prior to a test since the absolute correction is in the order of 1°C, and varies between thermocouples. The correction should be subtracted from any experimentally measured values. T A B L E 3.4 C O O L I N G C U R V E D A T A F O R TC3 W I T H 99.999% Pb TEST N° SUPERCOOLING PLATEAU Tp-Tivn. °C AT, C DURATION SECS. T P °C DURATION SECS. 24 0.9 180 328.7 840 1.2 25 0.9 180 328.5 840 1.0 26 0.6 180 328.4 660 0.9 27 0.8 180 329.2 1170 1.7 28 0.5 180 328.9 690 1.4 29 0.5 210 329.0 660 1.5 30 0.2 240 329.4 780 1.9 31 0.3 180 329.1 540 1.6 32 0.4 300 328.4 480 0.9 AVG. 0.6 210 328.8 750 1.3 3.2.2 COMMERCIALLY PURE LEAD (99.99%) Cooling curves were obtained for commercially pure lead, (99.99%), under the same test procedures as the high purity lead. One cooling was obtained for each of the three thermocouples giving the results shown in Figure 3.13. The liquid is again seen to 84 supercool but this time to 326.9°C, and then gradually recalesce until the temperature plateau is reached at 327.5°C, which is at the for lead. The plateau lasts for approximately 450 seconds until the temperature decreases. The test results from all three thermocouples are listed in Table 3.5. TABLE 3.5 COOLING CURVE DATA WITH 99.99% Pb FOR TCI, TC2, TC3 TC N° SUPERCOOLING PLATEAU T -T AT, C DURATION SECS. TP °C DURATION SECS. °c 1 0.6 390 327.5 450 0 2 0.4 300 327.5 360 0 3 N.M. N.M. 327.7 450 0.2 N.M.= Not Measured 3.2.3 C O M M E R C I A L L Y P U R E TIN (99.89%) Cooling curves were measured for commercially pure tin, (99.89%), for the three thermocouples. The cooling curves are shown in Figure 3.14 and the data are given in Table 3.6. For TCI, a supercooling of 1.1°C is measured before the temperature rises to the freezing plateau of 230.3°C which is well below the T M P for high purity tin at 231.97°C 4 2. For TC2, and TC3, the measured supercooling is smaller than TCI while the temperature rises to a freezing plateau slightly above for tin. In these measurements 85 TCI is reading abnormally low for reasons unclear at this time. TABLE 3.6 COOLING CURVE DATA WITH 99.89% Sn FOR TCI, TC2, TC3 T C N° SUPERCOOLING PLATEAU Tp-T M P AT, °C DURATION SECS. TP °C DURATION SECS. °C 1 1.1 570 230.3 660 -1.7 2 0.8 540 232.1 720 0.1 3 0.6 540 232.3 840 0.3 3.3 DIRECTIONAL SOLIDIFICATION OF Pb-1.5%Sn 3.3.1 FURNACE OFF AND NO INSULATION ON TOP Twelve tests were carried out following procedure 1 as outlined in section 2.3 using the apparatus shown in Figure 2.3. No insulation was at the top of the melt and the furnace was turned off during cooling. For seven of the tests, (33-39), chill assemblies CA1 and CA2 and a wide range of melt superheats, ATH, varying between 7°C and 42°C were used. The test conditions and results for these tests are listed in Table 3.7. A CET was observed in all the tests at an average distance of 67 mm from the chill face, varying between 62 and 72 mm. The position of the CET is the same for both CA1 and CA2 chill assemblies and is independent of the melt superheat within the range examined. The cast structure of the ingot section from test 34 using the CA1 chill assembly is shown in 86 Figure 3.15(a) and from test 38 using CA2 in Figure 3.15(b). In both cases the transition from columnar to equiaxed is observed to be irregular, as compared to the CET in Figure 3.2. Some columnar grains continue to grow into the equiaxed region, which makes the measurement of the distance of the CET from the chill face a little subjective. The equiaxed grain size is observed to be large with occasional isolated clusters of small grains among the large grains. Cooling curves were obtained from the output of the three thermocouples. An example of the resulting three cooling curves are shown for test 34 in Figure 3.16. At 34 mm from the chill face, the liquid cooling rate for the first 300 seconds is rapid. After 300 seconds, the cooling rate decreases until sohdification starts. A horizontal temperature plateau is not observed from a TC at this distance from the chill face, but only a sharp inflection point at the liquidus temperature. Below the liquidus, the cooling rate is rapid and constant. The cooling curves for TC2 and TC3 positioned at 63mm and 94 mm from the chill have slower melt cooling rates than TCI and both exhibit a horizontal temperature plateau very near the liquidus temperature. The temperature at 94 mm from the chill in the melt is higher than at 63 mm, the temperature difference decreasing as the liquidus is approached. At the liquidus, the temperatures at 63 and 94 mm appear to merge for a short time before the temperature at 63 mm decreases rapidly. The temperature at 94 mm remains constant for a longer period of time. This thermal behavior is more clearly illustrated in Figure 3.17 where the temperature scale has been expanded. The liquid at both TC positions supercools 0.5°C with the liquid at 63 mm from the chill face supercooling first followed quickly by the liquid at 94 mm. At approximately 87 TABLE 3.7 DATA FOR Pb-1.5%Sn FURNACE OFF DURING COOLING NO INSULATION ON TOP TEST N° CHILL ASSEMBLY ATH TC SUPERCOOLING PLATEAU CET INGOT HEIGHT °C mm "C/mra AT8 "C DURATION SECONDS TP °C DURATION SECONDS mm mm 3 0.25 - - - -33 CAl 29 61 0 0.2 120 322.8 120 62 132 94 - 0.6 210 323.4 240 34 0.06 - - - -34 CAl 42 63 0.0 0.5 120 323.1 150 67 130 94 - 0.5 60 323.2 420 38 0.09 - - - -35 CA2 38 67 0.0 - - 3223 120 71 141 95 - 0.7 150 322.9 60 35 0.12 - - - -36 CA2 9 72 0.02 - 322.6 90 62 134 104 - 03 60 323.4 300 37 0.10 • - - -37 CA2 32 64 -0.03 03 120 322.7 30 67 136 92 - 0.9 240 322.6 180 30 0.15 - - - -38 CA2 13 60 0.03 - - 322.7 30 72 132 92 - 0.2 90 322.9 210 41 0.17 - - - -39 CA2 7 65 0.0 - - 322.5 120 70 135 98 - 03 90 322.7 630 AVERAGE 0.45 120.0 322.8 180.0 67 134 1200 seconds, the liquid at both TC positions is supercooled. There is a gradual rise in temperature from supercooling to the temperature plateau at both positions. The plateau 88 at 63 mm from the chill occurs first at 323.1°C, 0.2°C above the liquidus, followed approximately 60 seconds later by the plateau at 94 mm at 323.2°C, 0.3°C above the liquidus. For the next 60 seconds, the temperature at 63 mm and at 94 mm from the chill remain at these two constant values. After this time, the temperature 63 mm decreases rapidly, while the temperature at 94 mm remains at 323.2°C. The temperature at 94 mm from the chill remains constant until approximately 1620 seconds where the temperature starts to decrease rapidly. Considering the overall results for tests 33-39, listed in Table 3.7, no supercooling is observed in any of the tests for the TC closest to the chill, which is similar to test 34. The amount of supercooling at the other two thermocouples varied between 0.2°C and 0.9°C. The plateau temperatures varied between 322.5°C which is 0.5° below the liquidus temperature and 323.4 which is 0.4°C above the liquidus temperature. Estimates of the temperature gradient in the liquid ahead of the solid/liquid interface, G L X , were made assuming a linear gradient between TCI and TC2, and between TC2 and TC3 and assuming the interface is coincident with the liquidus temperature. The results are also listed in Table 3.7, and are observed to be very small. With the interface at TCI, varies between 0.06°C/mm and 0.17°C/mm as measured between TCI andTC2 for tests 34 to 39. At TC2, G L X varies between -0.03°C/mm and +0.03C/mm, a range which is below the sensitivity of the temperature measurements. At 3 mm from the chill face in test 33, G^x is 0.25°C/mm. Tests 40-44 were carried out in the same manner as tests 33-39 except that chill assembly CA3 was used and the superheat was kept low. The chill assembly CA3 puts a higher thermal resistance between the chill and the mould compared to CA1 and CA2. 89 The test conditions and results are listed in Table 3.8. From Table 3.8, the distance from the chill that the CET occurs varies between 39 and 51 mm with an average of 45 mm for tests 41-44. This is appreciably shorter than the corresponding distances for tests 33- 39 with an average distance of 67 mm. The CET position appears to be independent of the melt superheat. The cast structure for tests 40 and 42 are shown in Figure 3.18(a) and (b). A large grained highly irregular structure is observed for test 40, and no clear CET is delineated. The structure for test 42, also does not have a sharply delineated CET, but columnar grains are evident at the bottom and the CET is estimated to be the solid horizontal line drawn in the figure. One columnar grain continues well into the equiaxed region to 100 mm from the chill face. The cooling curves for tests 41-42 are generally similar to those shown in Figure 3.15 for test 34. as shown in Figure 3.19 for test 43. The curve for TCI was the same as TCI in Figure 3.15 with no supercooling and no temperature plateau and therefore is not shown. For TC2 and TC3 at 61 and 94 mm from the chill face, there is 0.2°C supercooling at 61 mm and 0.4°C supercooling at 94 mm. In the other tests in this series listed in Table 3.8 supercooling is only observed at 94 mm distance ranging between 0.2"C and 0.4°C. This is appreciably less than the supercooling observed in tests 33-39. A temperature plateau is observed for both TC2 and TC3, with the plateau temperature a little below the liquidus for TC2, (0.3°C), and appreciably below the liquidus for TC3 (1-2°C). It is significant that the plateau temperature at 94 mm is below that at 61 mm and is preceded by supercooling. This was generally the case for tests 40 to 44, but not for tests 33-39 where the plateau temperature near 94 mm was higher than that near 61 mm. 90 TABLE 3.8 DATA FOR Pb-1.5%Sn FURNACE OFF DURING COOLING NO INSULATION ON TOP T E S T N ° C H I L L A S S E M B L Y A T H T C S U P E R C O O L I N G P L A T E A U C E T I N G O T H E I G H T •c m m °C/ m m A T , •c D U R A T I O N S E C O N D S T P •c D U R A T I O N S E C O N D S m m m m 35 0.04 - - 322.7 60 40 CA3 6 65 -0.03 - - 323.3 180 0 136 98 - 0.2 330 322.4 180 40 0.07 - - - -41 CA3 14 75 -0.04 - - 322.9 210 51 136 101 - 0.2 120 322.1 300 31 0.07 - - - -42 CA3 6 59 -0.04 - - 322.7 240 44 136 94 - 0.4 300 321.9 240 32 0.06 - - - -43 CA3 14 61 -0.04 0.2 30 322.7 210 47 141 94 - 0.2 180 321.8 150 36 0.1 - - - -44 CA3 11 65 -0.04 - - 322.1 150 39 135 95 - 0.4 120 321.5 240 AVERAGE 0.3 180 322.4 210 45 137 Values of G^ for tests 40 to 44 were determined in the same way as tests 33 to 39, giving the results listed in Table 3.8. In general, the values of G^ shown are very low. The region where the CET occurs near 44 mm corresponds to the gradient determined between TCI and TC2 with values of G ^ ranging between 0.04°C/mm and O.rC/mm. In the upper part of the casting, between TC2 and TC3 the values of G L X are all negative and range between -0.03°C/mm and -0.04°C/mm. 91 The low temperatures at the top of the melt suggest that significant heat is being lost from the top of the melt. To overcome this effect a series of tests were carried out with insulation added to the furnace above the melt to reduce or eliminate heat loss from the top of the melt. 3.3.2 FURNACE OFF TOP OF M E L T THERMALLY INSULATED Four tests, (45-48) were carried out in a manner similar to tests 33 and 34 using chill assembly C A l but with insulation added to the mould above the melt. In addition, the thermocouples were calibrated "in-situ", following procedure 3 in section 2.3 to improve the accuracy of the temperature measurements. The test conditions are listed in Table 3.9. A CET was observed in each of the tests at distances from the chill ranging between 60 to 91 mm. The cast structure for tests 46,47 and 48 are shown in Figure 3.20 with the position of the CET indicated by the horizontal dark line. As with the previous tests on Pb-1.5%Sn the transition to equiaxed grains is not flat and well defined. The equiaxed grains have large diameters except for test 48 where clusters of small grains are present among the large grains near the transition. The cooling curves for tests 45 to 48 were generally similar to the curves for tests 33 to 44. Typical curves for TC2 and TC3 of test 47 are shown in Figure 3.21. For TC2, supercooling and a temperature plateau are not observed. Similar results were obtained for tests 45, and 46. In all cases, no supercooling or plateaus are observed for TCI. The cooling curve for TC3 shows supercooling of 0.4°C and a freezing plateau at 323.2°C, 92 TABLE 3 9 COOLING CURVE DATA FOR Pb-1.5%Sn FURNACE OFF DURING COOLING INSULATION ON TOP USING CHILL ASSEMBLY CAl TEST N° A T H °C T C mm °C/mm SUPERCOOLING PLATEAU CET mm INGOT HEIGHT mm AT S °C DURATION SECONDS T P °C DURATION SECONDS 45 38 33 0.35 - - - -76 118 61 0.07 - - - -91 - 0.3 60 323.1 150 46 20 - - - - - -91 122 66 0.09 - - - -95 - 0.4 150 323.0 150 47 37 35 .34 - - - -76 126 66 .11 - - - -95 - .4 150 323.2 150 48 41 36 .14 - - - -60 118 61 .01 0.6 120 322.7 90 93 - 0.5 120 322.9 360 0.2°C above the liquidus. Similar supercooling between 0.3°C and 0.5°C was observed for tests 45 and 46 and plateaus at a temperature of 323.1°C +0.TC which is very close to the liquidus temperature of 323.0"C. The results for test 48 are a little different than for tests 45-47. The cooling curves for TC2 and TC3 for this test are shown in Figure 3.22. Supercooling is observed at both thermocouples, 0.6°C and 0.5°C respectively. A short freezing plateau is observed for 93 TC2,0.3°C below the liquidus, and a much longer plateau for TC3 at 0.1 °C below the liquidus. It is not clear why the cooling curves for test 48 differed from the other three tests in the series. Estimates of the temperature gradient in the melt, G ^ , were made using the procedure described previously, giving the results listed in Table 3.9. At 33 mm from the chill, (TCI), two values are near 0.35°C/mm and one at 0.14"C/mm. No value is listed for test 46 due to a defective thermocouple. At about 65 mm from the chill, GLX, is much smaller varying between O.OrC/mm and 0.1 VC/mm. These values are obtained using TC2 and TC3 values covering the region where the CET occurs. The presence of supercooling in the upper region of the melt in these experiments, even with insulation above the melt suggest that significant radial heat is being lost during cooling. As a result tests were carried out in which the furnace power was reduced instead of being turned off during cooling in an attempt to reduce the radial heat loss. 3.3.3 FURNACE POWER ON DURING COOLING M E L T TOP INSULATED Tests 49 to 55 were carried out in which the furnace was left on during cooling at various power settings. The Pb-1.5%Sn alloy was solidified following procedure 3 in section 2.3 which includes "in-situ" calibration of the thermocouples, and chill assembly CAl. The top of the melt was thermally insulated. In test 50, thermocouples were positioned at the centre and near the mould wall at the same distance from the chill to measure the radial temperature gradient as described in section 2.3, procedure 2. The test conditions and results are listed in Table 3.10. 94 TABLE 3.10 COOLING CURVE DATA FOR Pb-1.5%Sn FURNACE ON DURING COOLING INSULATION ON TOP USING CHILL ASSEMBLY CA1 TEST N° FURNACE CURRENT LEVEL AMPS. AT„ •c TC mm "C/mm SUPERCOOLING PLATEAU CET mm INGOT HEIGHT mm AT S •C DURATION SECONDS •c DURATION SECONDS 4 0.32 - - -49 3.0 43 64 0.0 - - 322.9 120 85 123 95 - 0.4 210 323.4 240 60 0.11 - - - -50 4.4 46 89* - - - - - 119 93" - - - -31 0.26 - - - -51 5.0 38 60 0.14 - - - - - 126 89 - - - - -34 0.28 - - - -52 4.4 40 64 0.12 - - - - - 119 93 - - - - -- - - - - -53 3.5 39 67 0.20 - - - - - 137 95 - - - - -- - - - -54 3.0 41 56 .02 0.3 90 323.3 270 88 129 98 - 0.5 180 323.9 270 29 0.29 - - - -55 5.0 43 62 0.10 - - - - - 123 92 - - - - -•THERMOCOUPLE IN THE MIDDLE OF THE MELT * "THERMOCOUPLE ON THE EDGE OF THE MELT 95 A CET is observed in only two of the seven tests, 49 and 54. In these tests, the furnace current was set at 3.0 amps, the lowest value used in this series. The structure of the test sample from test 54 is shown in Figure 3.23. The structure consists of very large grains with the CET taken at the marked horizontal line 88 mm from the chill face. The structure for test 49 was similar with the CET at 85 mm from the chill. The cooling curves for TC2 and TC3 from test 54 are shown in Figure 3.24. Both curves exhibit supercooling (0.3'C and 0.5°C respectively) and temperature plateaus above the liquidus temperature (0.3°C and 0.9°C respectively). Values for TCI are not included as the thermocouple was defective during cooling. The curves for test 49 were similar to 54. The value of G L X for test-54 at TC2 was small, 0.02°C/mm. For test 49, G^ is large close to the chill, 0.32°C/mm, 4 mm from the chill, and 0°C/mm at 64 mm from the chill. The remaining tests in this series 51 to 53 and 55 all had higher power inputs to the furnace during cooling compared to tests 49 and 54 as listed in Table 3.10. The structure of the test samples consisted of large columnar grains. An example is shown in Figure 3.25 for test 55. No CET is evident in the structure. A typical set of coohng curves for TCI, TC2, and TC3 from test 55 is shown in Figure 3.26. Supercooling and a horizontal temperature plateau are not evident, which is similar to that for tests 50 to 53. As a result, no data is included for these values in Table 3.10. The temperature of TC3 is always higher than TCI and TC2 during cooling and solidification, and TC2 is always higher than TCI. A positive temperature gradient is thus maintained throughout the sample during cooling. The temperature gradient 96 listed in Table 3.10 is relatively high nearest the chill for tests 51, 52, and 55 at 0.26°C/mm, 0.28°C/mm and 0.28°C/mm. At about 65 mm from the chill G L X decreases to between 0. l°C/mm and 0.2°C/mm. Measurements of the radial temperature gradient were made in test 50 with one thermocouple at the centre of the melt, and the second adjacent to the mould wall separated by 15 mm, 90 mm from the chill face. The results are listed in Table 3.11. The maximum radial temperature gradient in the liquid occurs at the start of solidification and is approximately 0.07°C/mm. As the liquid cools and the temperatures approach the liquidus temperature the radial temperature gradient decreases to 0.02°C/mm which is realistically beyond the sensitivity of the present measurements. After the alloy solidified at this vertical position the radial temperature gradient increased to approximately 0.09°C/mm. The test was repeated with the power to the furnace off giving the results listed in Table 3.12. The results are very similar except that the time taken to cool to the liquidus temperature is shorter and now a period of supercooling and a temperature plateau are observed. The decline in the radial temperature gradient follows the pattern in Table 3.11. 97 T A B L E 3.11 T E M P E R A T U R E S A N D T E M P E R A T U R E D I F F E R E N C E B E T W E E N C E N T R E O F M E L T A N D A D J A C E N T T O M O U L D W A L L F U R N A C E P O W E R O N A T 4.4 A M P S TIME (SECONDS) STATE TC2 CENTRE °C TC3 MOULD W A L L °C AT °C 277 LIQUID 361.3 362.3 1.0 516 LIQUID 351.9 352.7 0.8 1597 LIQUID 331.7 332.0 03 2677 LIQUID 325.3 325.6 03 4073 SOLID 317.4 318.8 1.4 4305 SOLID 314.0 315.2 1.2 T A B L E 3.12 T E M P E R A T U R E S A N D T E M P E R A T U R E D I F F E R E N C E B E T W E E N C E N T R E O F M E L T A N D A D J A C E N T T O M O U L D W A L L F U R N A C E O F F D U R I N G C O O L I N G TIME (SECONDS) STATE TC2 CENTRE °C TC3 MOULD W A L L °C AT °C 227 LIQUID 360.2 361.2 1.0 467 LIQUID 352.7 353.7 1.0 1067 LIQUID 332.3 333.1 0.8 1523 SUPERCOOLED LIQUID 321.7 322.1 0.4 1847 FREEZING PLATEAU 323.1 323.4 0.3 2391 SOLID 310.2 311.8 1.6 98 3.4 DIRECTIONAL SOLIDIFICATION OF PURE LEAD (99.999%) AND Pb-Sn ALLOYS In this series of experiments the cooling curves and cast structure are discussed for directionally solidified pure Pb and Pb containing 3% and 6% Sn. 3.4.1 DIRECTIONAL SOLIDIFICATION OF PURE LEAD Pure lead, (99.999%) was solidified using procedure 1 of section 2.3 with no thermal insulation above the melt and with two melt superheats. The test conditions and results are listed in Table 3.13, and the cooling curves for test 57 are shown in Figure 3.27. These results in general are similar to the Pb-1.5%Sn alloy. Since a CET was not expected to occur in pure lead, the sample was not sectioned and etched. The melt supercools 0.5°C at 64 mm from the chill face in both tests and approximately 0.5°C at 93 mm. A short temperature plateau is observed at 64 mm from the chill, and a much longer plateau at 93 mm from the chill. The plateau temperatures are 0.1° C and 0.4" C above the melting point of pure lead for TC2 and TC3 respectively which is within the experimental error of the measurements. The melt temperature gradient, G L X is relatively steep close to the chill, 0.29°C/mm, between 5 and 64 mm from the chill, and effectively zero between 64 and 93 mm from the chill. 99 TABLE 3.13 COOLING CURVE DATA FOR PURE LEAD(99.999%) FURNACE OFF DURING COOLING USING CHILL ASSEMBLY CA1 TEST N° AT„ TC SUPERCOOLING PLATEAU °C mm °C/mm ATS °C DURATION SECONDS TP °C DURATION SECONDS 5 0.30 - - - -56 79 64 0.01 0.5 90 327.6 180 93 - 0.5 120 327.9 510+ 5 0.28 - - - -57 20 64 0.0 0.5 210 327.6 90 93 - 0.4 150 327.9 600 3.4.2 DIRECTIONAL SOLIDIFICATION OF Pb-3%Sn AND Pb-6%Sn ALLOYS Two tests, (58 and 59), were carried out, one on a Pb-3%Sn alloy and the second on a Pb-6%Sn alloy. Procedure 3 of section 2.3 was used in which the thermocouples were calibrated "in-situ", the furnace turned off during coohng, chill assembly CA1 was used and thermal insulation was present above the melt. The test conditions and results are listed in Table 3.14 and the cooling curves for test 59 in Figure 3.28. The test samples were sectioned and etched to determine the cast structure. Unfortunately, a suitable etch to bring out the cast structure was not found for either alloy so that the presence or position of the CET could not be established. The etching solutions tried, in addition to that described in section 2.3, and are listed in Appendix 3. 100 T A B L E 3.14 C O O L I N G C U R V E D A T A F O R Pb-3%Sn, A N D Pb-6%Sn F U R N A C E O F F D U R I N G C O O L I N G I N S U L A T I O N O N T O P USING C H I L L A S S E M B L Y C A l TEST N° SOLUTE CONTENT AT„ TC SUPERCOOLING PLATEAU %Sn °C mm °C/mm AT, C DURATION SECONDS T , °C DURATION SECONDS 33 0.19 - - - -58 3.0 20 62 0.0 - - 318.7 60 94 - 0.5 150 319.3 210 36 0.14 - - - -59 6.0 12 66 0.0 0.2 30 310.5 30 92 - 0.4 90 311.0 180 In both tests the melt supercools between 0.2*C and 0.5*C before solidification. The supercooling is followed by a short plateau at 66 mm from the chill at 0.3*C below the liquidus, and a much longer plateau at 92 mm from the chill 0.2*C above the liquidus. The cooling curves for test 58 for the Pb-3%Sn alloy were similar to Figure 3.28.With a liquidus temperature for Pb-3%Sn of 318.9*C, the plateau at 62 mm from the chill was below the liquidus by 0.2*C, and above the liquidus at 94 mm from the chill by 0.4*C. The measured temperature gradient, G±x is 0.19*C/mm and 0.14°C/mm for the two alloys at 33 mm from the chill and O'C/rnm in both cases at 62 mm from the chill. 101 3.5 DIRECTIONAL SOLIDIFICATION OF Sn-10%Pb ALLOY In a previous investigation18, the CET in Sn-10%Pb alloys was examined extensively. It was considered desirable to do at least one test with this alloy using the better temperature measuring apparatus developed in this investigation. One sample of Sn-10%Pb alloy was prepared and directionally solidified with the furnace off during cooling using chill assembly CA1, and with thermal insulation above the melt. The thermocouples were calibrated "in-situ" against the T U Q of the alloy. The test conditions and results are listed in Table 3.15, while the cast structure is shown in Figure 3.29, and the cooling curves in Figure 3.30, TABLE 3.15 COOLING CURVE DATA FOR Sn-10%Pb FURNACE OFF DURING COOLING INSULATION ON TOP USING CHILL ASSEMBLY CA1 T C mm G L X C/mm S U P E R C O O L I N G P L A T E A U C E T mm I N G O T I H E I G H T mm I A T S D U R A T I O N TP D U R A T I O N C S E C O N D S C S E C O N D S 34 0.14 0.3 60 213.6 60 64 0.10 1.7 240 216.1 210 45 127 93 - 2.8 300 218.8 300 A clearly defined CET is observed at 45 mm from the chill, as shown in Figure 3.29, the transition occurring sharply on a horizontal plane, similar to that reported previously18. The cooling curves in Figure 3.30 show a small supercooling, (0.3*C), 102 nearest the chill and large supercooling occurring simultaneously, (1.7°C and 2.8°C), at 64 and 93 mm from the chill respectively. Following supercooling the temperature rises to a very short plateau from the TC nearest the chill, and progressively longer plateaus from the TC's away from the chill. Plateau temperatures vary appreciably with respect to the liquidus temperature. At 33 mm the difference is -1.4°C, at 64 mm, +1.TC and at 93 mm, +3.8°C. Values of GLX , based on the temperature difference between the thermocouples is always positive. The values are 0.14°C/mm at 34 mm and 0.10°C/mm at 64 mm. Since the transition occurs at 45 mm the gradient at that point should be between these two values. This is consistent, to a first approximation, to the value of the critical gradient at the CET of 0.1 rC/mm determined previously for Sn-10%Pb. 3.6 MATHEMATICAL HEAT TRANSFER MODEL 3.6.1 DIRECTIONAL SOLIDIFICATION OF Pb-1.5%Sn It is necessary to know values for G L , the temperature gradient in the liquid ahead of the solid/liquid interface, for all points in the melt as the solidification front progress from the chill face to the top of the melt and how it changes with time in order to determine the G L at the CET. This information cannot be derived solely from cooling curves using thermocouples at fixed positions in the melt. The cooling curves data from the three thermocouples can only provide two measured values assuming a linear gradient 103 in the liquid between each thermocouple. In addition, the position of the CET is not known apriori, and therefore the CET may not coincide with the position of a fixed thermocouple. A solution to this problem is to use a heat transfer mathematical model which predicts temperatures in the ingot as a function of time and distance from the chill face. These temperatures in turn are used to determine how G L changes throughout the course of the experiment using equation 2.24. In order to gain confidence in the model, the predicted temperatures must first agree closely with accurate temperature measurements. The boundary conditions needed to solve the transient heat conduction equation, (2.1), are generally unknown ahead of time but are assumed to have the form given by equations 2.4 and 2.5, see section 2.4. The unknowns in equations 2.4 and 2.5, are the heat transfer coefficients at the top and bottom of the alloy, H B and H X o p , and the furnace temperature above the melt, T p ^ , . The heat transfer coefficient at the bottom of the alloy was determined first by iteration using H X o p equal to zero. This produced calculated temperatures which were close to the experimental temperatures at the first TC position but did not fit well with experimental temperatures at the top two thermocouple positions during cooling and solidification. In order to obtain a good fit between calculated and predicted temperatures at all three thermocouple positions, H T o p was arbitrarily selected, and T F u m calculated using the simplifying assumption of how T ^ R N changes with time as shown in Figure 2.10 and discussed in section 2.4. The heat transfer coefficient at the bottom of the alloy was then adjusted until the calculated temperatures closely coincided with the measured temperatures at all three thermocouple positions. 104 The results of this process are shown in Figure 3.31, for test 45, in which the experimental results are shown by symbols for the three indicated distances from the chill, and the calculated temperatures by the solid and dashed lines. In these calculations it was assumed that the latent heat was released linearly between the liquidus and the solidus temperatures. Best fit between the measured and the calculated values is at 33 mm. At distances farther away from the chill the calculated values above the liquidus temperature tend to be a little lower than the measured values. As the calculated temperatures approach the liquidus, the fit improves while below the liquidus, the calculated values are higher than the measured values. On average, the fit at all points for the three curves is within ±3°C while the fit at the liquidus temperature is within ±1°C. In addition, the predicted time the liquidus is reached is within 60 seconds of the experimental time. The temperature gradient in the melt immediately ahead of the liquidus was determined from the predicted temperatures. In general, a linear gradient is calculated for distance up to 10 mm ahead of the interface. This is shown in Figure 3.32 for test 45 at two different positions, one, when the liquidus has reached a distance of 61 mm from the chill face, where TC2 is located, and two, at a distance of 76 mm from the chill where the CET occurred for test 45. At these two distances the gradient in the liquid is different and has decreases as the position of the liquidus temperature increasingly progresses away from the chill. The gradient, G L , assumed to affect the CET, is obtained over one node length,(0.5 mm), ahead of the interface and for test 45 is shown as a function of distance in Figure 3.33. Included in Figure 3.33, are the two measured values, G^, and the 105 position of the CET. As expected, G L is high at distances close to the chill face and decreases with increasing distance from the chill until finally at 76 mm reaching a value of 0°C/mm. The predicted values of GLat 33 and 61 mm from the chill are close to the measured values at those two distances. At 33 mm from the chill, the measured value of G^ is below the calculated value by 0.05°C/mm while at 61 mm, the two values coincide. At the CET, the calculated G L is 0°C/mm. The same values for H B , H X o p , and T F u r n , were used in the model and compared to the experimental results from tests 46 and 47 in the same way as described for test 45 with similar results. The same is not true for test 48. As can be seen in Figure 3.34 which shows the calculated and experimental temperatures at TCI which is closest to the chill at 36 mm and for TC3 which is furthest from the chill at 93 mm, the results are very different. The experimental temperatures for TC3 are appreciably lower than the calculated temperatures by up to 20°C. At 36 mm from the chill, the discrepancy between the experimental temperatures and the calculated temperatures is not as large and shows a reversal at 300 seconds. The experimental temperatures are higher than the calculated temperatures at the start but as the temperatures decrease the experimental temperatures become lower than the calculated temperatures. The calculated and experimental temperatures for TC2 at 61 mm from the chill are not shown for sake of clarity. In order to bring the predicted temperatures in line with the experimental temperatures, it was necessary to increase the heat loss from the top of the melt by changing T p ^ in equation 2.5 until a reasonable fit was again obtained between the calculated and the experimental temperatures. The fit between the experimental and the 106 calculated temperatures shown in Figure 3.35 was obtained using the same constant heat transfer coefficient at the bottom used for test 45, 46 and 47. These results show that the cooling curves are very sensitive to the heat loss from the top of the melt. The calculated temperature gradient in the melt ahead of the liquidus, G L , from test 48 also started high at distances close to the chill face but decreases much more rapidly than G L for test 45 as shown in Figure 3.36. The measured values, G L X , for test 48, also shown in Figure 3.36, are also much lower than those for test 45, but, they still agree with the predicted values. The position of the CET in test 48 was observed to occur at 60 mm from the chill where G L is calculated to be less than 0.0rC/mm. Mathematical modelling was done for every experiment with the furnace off during cooling and solidification. The top and bottom boundary conditions were adjusted as discussed above in order to obtain a good fit between predicted and experimental temperatures at all three thermocouple positions. The calculated temperature gradient, G L , and growth velocity, V, at the CET are listed in Table 3.16 along with the top and bottom boundary conditions and the position of the CET. The measured temperature gradients, G ^ , 3 1 6 n o t listed in Table 3.16 because in general, the measured values are taken at positions not coincident with the CET. In most cases, the calculated value of G L at the CET is very small, 0.02*C/mm. The exceptions are tests 42 to 44 where the calculated G L was higher and varied from 0.04 to O.lO'C/mm. In test 40, the structure was completely equiaxed, therefore no CET occurred and no G L calculated. The growth velocities, V, at the CET are calculated using equation 2.28 in section 2.4.9. The position of the interface, x , was plotted against the ^ Itime to give a slope B, 107 TABLE 3.16 PREDICTED VALUES OF G L AND V AT THE CET FOR Pb-1.5%Sn [ T E S T N° CHILL ASSEMBLY HTC J/cmVC-sec AT Tp»r. -Txo* °C CET mm °C/mm V mm/sec START FINISH 33 CA1 0.0105 10 -7.5 60 0.02 0.05 34 CA1 0.0090 10 •7.5 67 <0.01 0.07 CA2 0.0105 0 -5.0 71 <0.01 0.07 36 CA2 0.0105 0 -73 62 0.01 0.06 37 CA2 0.0105 0 -73 67 <0.01 0.07 38 CA2 0.0105 0 -73 72 <0.01 0.07 39 CA2 0.0105 0 -73 70 <0.01 0.06 40 CA3 0.0080 0 -73 - - -41 CA3 0.0080 0 -73 51 0.02 0.08 42 CA3 0.0095 0 -73 44 0.04 0.08 43 CA3 0.0095 0 -73 47 0.09 0.08 44 CA3 0.0095 0 -73 39 0.10 0.09 45 CA1 0.0100 20 -5.0 76 <0.01 0.06 46 CA1 0.0100 15 -73 91 0 0.06 47 CA1 0.0105 20 -5.0 76 <0.01 0.06 48 CA1 0.0105 0 -5.0 60 <0.01 0.08 H ^ O l E J / c r n ^ C - s e c H T C LISTED A T C H I L L F A C E F U R N A C E O F F DURING C O O L I N G with B ranging in values from a low of 2.7 mm/sec to a high of 5.1 mm/sec . The average value of B is 4.2 mm/sec1/2. The growth velocity of the interface at the CET is calculated based on the slope B, and the time it took for the interface to reach the CET. As seen in Table 3.16^  the velocities are relatively constant varying between 0.05 and 0.08 mm/sec with an average of 0.07 mm/sec. 108 3.6.2 LATENT HEAT As described in section 2.4.6, the latent heat release in the calculations can be based on a linear release between the liquidus and solidus temperatures, or it can be released in a non-linear fashion using either the Scheil equation, or the Lever rule to calculate the fraction solidified as a function of temperature. All three methods were used to produce the cooling curves at 33, 61, and 93 mm from the chill for test 45 as shown in Figure 3.37, 3.38 and 3.39 respectively. The cooling curves in Figure 3.37 are observed to be superimposed in the liquid as expected. Below the liquidus, the Lever rule gives slightly higher temperatures than the Scheil and linear curves which are still superimposed. At 61 mm from the chill, Figure 3.38, similar results to those at 33 mm are observed. At 93 mm, Figure 3.39, the cooling curves for all three methods are virtually identical. The effect of changing the method to release latent heat on the temperature gradient, G L , was examined using the calculated temperatures shown in Figures 3.37 to 3.39. The calculated values of G L are shown in Figure 3.40 as a function of distance from the chill. The difference between the procedures is observed to be very small, the Lever rule giving slightly lower values than the other two. 3.6.3 Pb-3%Sn and Pb-6%Sn The effect of changing the method to release latent heat on Pb-6%Sn, (test 57), at three distances from the chill is shown in Figures 3.41 to 3.43 inclusive. Similar to the Pb-1.5%Sn alloy, the calculated difference between the three methods is small with the largest difference at 36 mm from the chill face. At this position, the Lever rule 109 temperatures are higher than the linear values which are higher than the Scheil values. All of the calculated temperatures are higher than the observed values, with the Scheil values being the closest. At 66 mm from the chill, Figure 3.42, the temperatures using the linear method are below the other two, with all the calculated values being higher than the observed values at decreasing temperatures below the liquidus. At 93 mm, the three cooling curves are effectively superimposed and coincident with the observed values along the liquidus line. The results for Pb-3%Sn are similar to those for the Pb-6%Sn alloy. 3.7 Sn-10%Pb Cooling curves are calculated for the Sn-10%Pb alloy, (test 60), using the phase diagram temperatures given in section 2.4.8. The same boundary condition values are applied at the bottom for the Pb-1.5%Sn alloy. At the top surface of the melt, q was taken as zero. The results for the linear release of latent heat at 34, 64 and 93 mm from the chill face are shown by the solid lines in Figures 3.44 to 3.46 respectively. The experimental results are indicated by the square symbol in the figures. In general, there is poor fit between the calculated and experimental values. Above the liquidus, there is good fit at 34 mm but poor fit at 64 and 93 mm. The calculated values at the top two TC positions are lower than the experimental temperatures. The experimental cooling curves at all three positions deviate substantially from the calculated cooling curves after the start of solidification. The experimental points show supercooling followed by a horizontal temperature plateau at all three thermocouple positions. The calculated curves show no supercooling or horizontal temperature plateau but show continuous cooling which 110 results in a large difference between the calculated and measured temperatures. A comparison is made in Figure 3.47 between the calculated cooling curves using all three methods to release latent heat for a distance of 93 mm from the chill face. At temperatures well above the liquidus temperature, the three calculated cooling curves are very similar and are all below the experimental results. As the liquidus temperature is approached, the cooling curve from the linear method to release latent heat diverges from the curves produced using the Scheil equation and the Lever rule. The linear method produces temperature below those produced by the other two methods. At the liquidus temperature, the curves from the Scheil and the Lever rule exhibit a brief horizontal temperature plateau but the calculated results are still lower than the experimental results. In contrast to the Pb-Sn results the manner of latent heat release influences the temperature gradient, G L for the Sn-10%Pb alloy. This is shown in Figure 3.48 where G L is shown as a function of distance from the chill for the three methods. Releasing the latent heat using the Scheil equation and Lever rule give the same calculated values of G L , which are significantly lower than the calculated values using the linear procedure for distances up to 100 mm from the chill face. The measured values of G L X taken from the cooling curves for test 60 do not show the same relationship with distance from the chill as the calculated values. For example, the experimental point at 34 mm from the chill is between the two calculated curves at 0.14*C/mxn, while the experimental point at 64 mm from the chill is well above all three calculated curves at 0.10°C/mm. The CET in test 60 occurred at 45 mm from the chill face, see Figure 3.48. At this position, the calculated values of G L for the three procedures to release latent heat are I l l listed in Table 3.17. The linear model gives a value for G L of 0.12*C/mm, which agrees with the interpolated value of 0.125'C/mm assuming a linear temperature gradient between the thermocouples. The calculated value of G L for the Scheil and the Lever rule procedures was 0.06'C/mm, significantly lower than the linear procedure and the estimated measured value. TABLE 3.17 G, AS CALCULATED AT 45 mm FROM CHILL FACE FOR TEST 60 USING THE THREE PROCEDURES TO RELEASE LATENT HEAT CET POSITION mm LINEAR MODEL G L 'C/mm SCHEIL MODEL G L "C/mm LEVER MODEL C/mm LINEAR APPROXIMATION 'C/mm 45 0.12 0.06 0.06 0.125 3.8 HEAT FLUX AT BOTTOM OF MOULD The mathematical heat transfer model uses as a bottom boundary condition a constant heat transfer coefficient, H B , equal to 0.0105 J/cm2-*C-sec and an ambient temperature of 25"C to solve the transient heat conduction equation, (2.1). An alternative method was attempted which utilized the temperature measurements from the two TC's in the chill assembly described in section 2.3 and illustrated in Figure 2.4. The temperature measurements were recorded along with those temperatures in the melt with the A/D board. Since the distance between thermocouples is known as well as the thermal conductivity of copper, and stainless steel, the temperature difference, A T C A , between the 112 TC in the top disk, and the TC in the bottom disk could theoretically be used to determine the heat flux, tf, from the bottom of the mould and through the chill assembly if the following assumptions are valid: 1. There is intimate contact at all surfaces. 2. Heat flow is unidirectional. 3. Heat flow through the copper disk with holes is evenly distributed. 4. The thermocouples are positioned exacdy in the middle of the disk. 5. Steady state exists. 6. The furnace does not affect the heat flux in the chill assembly. The heat flux is calculated through the chill assembly, CA1, using the following equation: ATCA where: tf = Heat flux through the chill assembly. AT C A = The temperature difference between the top and bottom solid copper disks. Ax = The thickness of the copper disks. kc = The thermal conductivity of copper. AQ , = Cross sectional area of the solid copper disks. A E F F = Effective cross sectional area of copper disk with holes. Measurements of the temperature difference with time across the three chill assemblies were made under normal test conditions. The results are shown in Figure 3.49. The temperature difference AT is observed to increase sharply when the chill is put into 113 contact with the bottom of the mould, reach a maximum, and then slowly decrease with time for the three assemblies. As expected, the temperature difference at a given time increases with an increased thickness of stainless steel from CAl to CA3. A comparison of the calculated melt temperatures using equation 2.29 as the bottom boundary condition and the values of AT as a function of time from test 45 and the experimental melt temperatures is made in Figure 3.50. The cooling curve at 33 mm from the chill face determined in this way is shown as a solid line while the corresponding experimental values are indicated by the points. The calculated results shown by the solid line are lower than the experimental results right from the start. After 1020 seconds of cooling, the calculated results are 50°C below the experimental values. Similar results between calculated and experimental values at 61 and 93 mm were obtained but are not shown. The results clearly indicate that the measured AT values cannot be used to establish the heat flux from the bottom of the mould. This is most likely due to the inapplicability of assumption 1 regarding intimate contact, and assumption 3 regarding heat flow through the copper disk with holes. Heat flow through this disk may be more complex than assumed making the true effective area for heat flow unknown. 3.9 SENSITIVITY ANALYSIS The results shown in section 3.6.1 for the directional solidification of Pb-1.5%Sn were obtained using a node size of 0.5 mm and a time step of 0.5 seconds. Increasing the node size to 1.0 mm and the time step to 1.0 seconds, did not significantly change the calculated temperatures or temperature gradients. 114 Thermophysical parameters such as k, p, and Cp had to be assumed for the Pb-Sn alloys and the Sn-10%Pb alloy as this data is not readily available in the hterature. It was assumed that the values for k,p, Cp, for the alloy could be determined from the values for Pb and Sn on the basis of the weighted averages of the Pb and Sn in the alloy. Therefore, it is worthwhile to perform a sensitivity analysis on these parameters to see their influence on the results of the model. There is thermal conductivity data for Pb-10%Sn at 54*C which can be used to provide some basis for the error encountered in the model using the weighted averages assumption. At 54°C, the thermal conductivity of Pb-10%Sn is reported as 0.360 W/cm-K43. Using the weighted averages method, and values for Pb as 0.352 W/cm-K, and for Sn as 0.742 W/cm-K, the thermal conductivity of Pb-10%Sn at 54*C is 0.415 W/cm-K, 15% higher than the reported value. Thus, the sensitivity analysis is performed with this error limit. The values of G L as a function of distance from the chill were calculated using three values for the thermal conductivity, the original k, 0.85k and 1.15k giving the results shown in Figure 3.51. The largest effect of k on G L is in the region between 10 and 40 mm from the chill. Decreasing k by 15% increases G L in this region by up to O.rC/mm, while increasing k by 15% decreases G L by up to 0.07°C/mm. The values of G L for the three values of k all converge to 0°C/mm at approximately 76 mm from the chill which is in the vicinity of the CET for this particular test. Therefore, for Pb-1.5%Sn alloys, a 15% error does not significandy affect the results at the CET. Figure 3.1 Cast structure of test 1 Figure 3.2 Cast structure of test 2 Figure 3.3 Cast structure of test 4 1 2 3 4 5 6 7 8 9 10 NUMBER OF STAINLESS STEEL SHEETS Figure 3.4 The position of the CET as a function of the number of stainless steel sheets. 330 329 328 327 a sc 326 12 325 < at 324 U J IS 323 322 321 320 1200 1500 1800 2100 2400 TIME (SECONDS) 2700 3000 Figure 3.5 Cooling curves from test 1 at the indicated distances from the chill 1200 1 TC2 T C 3 300 i ' i i—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1 0 10 20 30 40 50 60 70 80 90 100 DISTANCE FROM CHILL FACE (mm) Figure 3.6 The duration of the plateau as a function of distance from the chill in test 1 Figure 3.7 Cooling curves from test 2 at the indicated distance from the chill 1800 TIME (SECONDS) Figure 3.8 Cooling curves from test 3 at the indicated distances from the chill 120 Figure 3.9 The duration of the plateau as a function of distance from the chill in test 3 121 300 600 900 1200 1500 1800 TIME (SECONDS) Figure 3.10 Cooling curves from test 4 at the indicated distances from the chill 122 Figure 3.11 The plateau temperature from tests 1, 2, and 3 as a function of distance from the chill 123 Figure 3.12 A typical cooling curve for TCI using 99.999%Pb 124 Figure 3.13 Cooling curves from TCI, TC2, and TC3 using 99.99%Pb 125 - e - TCI TIME (SECONDS) Figure 3.14 Cooling curves from TCI, TC2, and TC3 using 99.89%Sn (a) Figure 3.15 (a) Cast structure of test 34 using chill assembly CA1 (b) Cast structure of test 38 using chill assembly CA2 127 Figure 3.16 Cooling curves for test 34 at the indicated distances from the chill 128 Figure 3.17 The plateau region for test 34 at the top two TC positions at the distance (a) Figure 3.18 (a) Cast structure of test 40 using chill assembly CA3 (b) Cast structure of test 42 using chill assembly CA3 130 1500 TIME (SECONDS) Figure 3.19 Cooling curves for test 43 at the plateau region at the indicated distances from the chill (a) (b) (c) Figure 3.20 (a) Cast structure of test 46 using chill assembly CAl and top of melt insulated (b) Cast structure of test 47 using chill assembly CAl and top of melt insulated (c) Cast structure of test 48 using chill assembly CAl and top of melt insulated 132 TIME (SECONDS) Figure 3.21 Cooling curves for test 47 at the plateau region at the indicated distances from the chill 133 Figure 3.22 Celling curves for test 48 at the plateau region at the indicated distances from the chill f 1 Figure 3.23 Cast structure of test 54 using chill assembly C A l and the furnace power on during cooling, 1=3.0 Amps. 135 Figure 3.24 Cooling curves for test 54 at the plateau region at the indicated distances from the chill; cooling occurred with 1=3.0 amps to the furnace I Figure 3.25 Cast structure of test 55 using chill assembly CA1 and the furnace power on during cooling; 1=5.0 Amps. 137 | , . . . i • i , . . . I i i i i i i i i • i * i ' 0 300 600 900 1200 1500 1800 2100 2400 TIME (SECONDS) Figure 3.26 Cooling curves for test 55 at the indicated distances from the chill 138 2400 TIME (SECONDS) Figure 3.27 Cooling curves for test 57 (99.999%Pb) at the plateau region at the indicated distances from the chill 139 Figure 3.28 Cooling curves for test 59 at the plateau region at the indicated distances from the chill I 140 Figure 3.29 Cast structure of test 60 (Sn-10%Pb) using chill assembly CAl and the top of the melt insulated 141 1500 TIME (SECONDS) Figure 3.30 Cooling curves for test 60 at the indicated distances from the chill 142 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.31 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 45. H T C g ^ = 0.0105J/cm2-*C-sec H T C ^ = 0.02J/cm2-*C-sec 143 Figure 3.32 The thermal field ±10 mm from the liquidus temperature at two positions (1) when the liquidus is at TC2 61 mm for test 45 and (2) when the liquidus is at 76 mm from the chill where the CET occurs. CALCULATED O COOLING CURVE DATA CET I . I . I . — I — i _ L I i I i 0 10 20 30 40 50 60 70 80 90 100 110 120 DISTANCE mm Figure 3.33 G L as a function of distance from the chill for test 45. 145 Figure 3.34 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 48. HTC B o a o m = 0.0105J/cm2-*C-sec HTC^ = 0.02J/cm2--C-sec 146 Figure 3.35 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 48 using the same heat transfer coefficients but with different values. H T C B ^ = 0.0105J/cm2-*C-sec H T C T ^ = 0.02J/cm2-*C-sec 147 Figure 3.36 G L as a function of distance from the chill for test 45 and for test 48. 148 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.37 Predicted cooling curves for test 45 using the Linear model, the Scheil equation model and the Lever rule model. Distance from chill = 33 mm 149 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.38 Predicted cooling curves for test 45 using the Linear model, the Scheil equation model and the Lever rule model. Distance from chill = 61 mm 150 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.39 Predicted cooling curves for test 45 using the Linear model, the Scheil equation model and the Lever rule model. Distance from chill = 91 mm 151 DISTANCE FROM CHILL (mm) Figure 3.40 G L as a function of distance from the chill for test 45 using the three procedures to release latent heat. 152 Figure 3.41 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 59 (Pb-6%Sn), using the three procedures to release latent heat for the predicted curves. H T C B o a o m = 0.0105J/cm2-*C-sec HrCx^ = 0.02J/cm2-*C-sec 153 290 '—1—1—1— 0 300 600 900 1200 TIME (SECONDS) Figure 3.42 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 59 (Pb-6%Sn), using the three procedures to release latent heat for the predicted curves H T C B o 0 o i n = 0.0105J/cm2-*C-sec HTCrop = 0.02J/cm2-*C-sec 154 330 300 290 LIQUTDUS • TC3 93 mm LEVER SCHEIL LINEAR 0 300 600 900 TIME (SECONDS) 1200 Figure 3.43 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 59 (Pb-6%Sn), using the three procedures to release latent heat for the predicted curves. H T C B o o o m = 0.0105 J/cm2-*C-sec HTCrop = 0.02 J/cm2-'C-sec 155 Figure 3.44 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 60 (Sn-10%Pb), using the linear method to release latent heat for the predicted curves. H T C B ^ = 0.0105 J/cm2-'C-sec HTCrop = 0 J/cm2-*C-sec 156 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3:45 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 60 (Sn-10%Pb), using the linear method to release latent heat for the predicted curves. HTC B o t t o m = 0.0105 J/cm2-'C-sec HTC^ = 0 J/cm2-*C-sec 157 240 < UJ 230 220 210 200 • TC3 93 mm A D \ D \ n v t i 4 = I = Q = t ° n u n LIQUIUDS ' ' L_1 1 1 1— PREDICTED (LINEAR) i i i i l — i — 1 _ • 1 1 1 1 1 1 1 1 1 L _ l _ 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.46 Predicted cooling curves (lines) and experimental cooling curves (symbols) for test 60 (Sn-10%Pb), using the linear method to release latent heat for the predicted curves. HTC B o a o m = 0.0105 J/cm2-*C-sec HTC^ = 0 J/cm2-*C-sec 158 2oo 1 ' ' ' 1 1 1 ' 1 1 1 1 1 1 1 1 1 i i i I i i i i I i—i—i—i—i—i—i i i 0 300 600 900 1200 1500 1800 2100 TIME (SECONDS) Figure 3.47 Predicted cooling curves (lines) and exr^ rirnental cooling curves (symbols) for test 60 (Sn-10%Pb), using the three procedures to release latent heat for the predicted curves. H T C B O ^ = 0.0105 J/cm2-*C-sec KTCr^ = 0 J/cm2-*C-sec 159 Figure 3.48 G L as a function of distance for the chill for test 60 (Sn-10%Pb) using the three procedures to release latent heat 160 3.49 Measured Temperature Differences in the Three Chill Assemblies Function of Time 161. 3.50 Predicted Versus Experimental Cooling Curve for Test 45 at 33 mm from the Chill Using the Calculated Heat Flux Instead of a Heat Transfer Coefficient 162 3.51 G L as a Function of Distance Using Three Different Values for the Thermal Conductivity Chapter 4 DISCUSSION 4.1 TEMPERATURE MEASUREMENTS Thermocouples are inherently inaccurate and their use for precise, and accurate temperature measurements requires careful calibration. It is clearly evident in tests 1-4 where the thermocouples were calibrated once, prior to test 1 using 99.99% Pb, that the thermocouples were not calibrated carefully enough. The correction factors applied to the thermocouple output, in °C, produced erroneous results as shown in Figure 3.11. The different plateau temperatures, T P , should all be the same given that the alloy composition was held constant at Pb-1.5%Sn. In addition, it is expected that all the horizontal temperature plateaus occur at the liquidus temperature for the alloy, approximately 323°C. 163 164 The errors which occurred in tests 1-4 can be explained by reviewing the results from tests 5 to 32, ie, the temperature measurements made with high purity lead, 99.999%. In these tests the melting point of the lead, TMP, is constant at 327.5°C and so the horizontal temperature plateau observed in the cooling curves should be at this precise value. Any deviation from this value , (TP-TMP), may be attributable to errors in the chemical composition in the thermocouple wires, errors associated with the cold junction, and the degree of cold worked material after the thermocouple is made21. It is quite clear that these errors are not constant as the difference between T P and T^p changes from test to test for a given thermocouple. Therefore, it is reasonable to expect that this occurred in tests 1-4 as well so that the correction factor established independently, no longer applied. The end result is that of Figure 3.11 with the measured T P for Pb-1.5%Sn ranging from 321°C to 326°C, an error of approximately ±2.5°C. Improving the accuracy further using a single calibration experiment is not possible. In order to use the results of tests 5 to 32 and improve the accuracy of the temperature measurements beyond the value mentioned above, it was decided to use a mean correction factor based on the values of Tp-T^ in tests 5 to 16 inclusive for TCI, in tests 16 to 23 inclusive for TC2, and in tests 24 to 32 for TC3. In using this procedure, it is assumed that the correction factor in subsequent directional solidification tests do not deviate outside the range established in Tables 3.2, 3.3, and 3.4. In this way, the absolute error using the three thermocouples is expected to be in the order of ±0.7°C while the relative error between two thermocouples is expected to be ±1.4°C. This approach was met with limited success for subsequent directional soldification tests using Pb-1.5%Sn, in that the T P in Table 3.7 for tests 33 to 39 does not deviate from the T U Q of the alloy by more than 0.5°C. However, in later tests shown in Table 3.8, the 165 T P deviates from the T U Q up to 1.4°C indicating that the correction factors established in tests 5 to 32 may no longer apply. When more than one T P was measured from the three thermocouples, see Figure 3.19, the relative error did not exceed 1.4°C. This means that the difference in the two temperatures can be taken as negligible. Further improvements in the accuracy of the temperature measurements using the three thermocouples are possible by eliminating the correction factor which is established independently of the directional solidification test. In this way, the three thermocouples are calibrated simultaneouly to the liquidus temperature of the alloy by performing immediately prior to directional solidification, an "in-situ" calibration test as described in section 2.3. The correction factor for each thermocouple becomes specific for this test and is not applied to any other tests. The deviation between T P and T U Q in Pb-1.5%Sn tests is ±0.2°C as shown in Table 3.9. Applying this technique to the Pb-3%Sn and the Pb-6%Sn produced a deviation between T P and T L I Q between 0.2°C and 0.4°C which is still better than the results from Table 3.7 for Pb-1.5%Sn using the mean correction factor. In establishing a mean correction factor for TCI, TC2, and TC3 using the 99.999% Pb melting point as a basis of comparison, an average supercooling of 0.6°C is measured. This is a small amount and in order to establish its credibility, the precision of the temperature measurements must be briefly discussed. In Tables 3.2 to 3.4 inclusive, the temperature measurements are reported using a precison of ±0.1°C. This is based on the temperature range recorded while the temperature was on the plateau. On average, the temperature range was 0.2°C, with an equal balance of temperatures above and below the 166 average T P . Thus, each T P can be determined within +O.TC. The same external and internal electrical noise is expected to influence temperatures not on the plateau with the same degree of precision. Since each recorded temperature is an average of 32 measurements taken 10 ms apart, a statistical analysis can be made using the mean, the standard deviation, and the variance of each measurement to confirm the precision of the temperature measurements observed on the temperature plateau. The analysis concerns the significance of the difference of the means of two sets of independent data, the analysis, shown in Appendix 2, determines that the smallest significant difference is 0.07°C. This is very close to ±0.1°C and well below 0.6°C. Therefore, the supercooling is indeed present and at the value reported. In the directional solidfication tests of Pb-1.5%Sn, the measured supercooling decreased to a range from 0.2°C to 0.4°C. While this is still above the ±0.1°C limit set for the pure Pb tests discussed above, the precision of the temperature measurements decreases because the number of measurements per temperature decreases, see Appendix 2. The difference becomes 0.16"C which is now close to 0.2°C, the bottom range of the supercooling. This may raise some questions as to the realilty of the measurements. However, even with the very small supercooling there is a consistent rise in temperature from the maximum supercooling value to the plateau and not just a single spike which indicates that supercooling does exist at that point. 167 4 . 2 S n - 1 0 % P b In a previous study18, there is a degree of uncertainty regarding the value of G L of 0.1 rC/mm at the CET for Sn-10%Pb. The value was determined primarily on the predicted temperatures from a one dimensional heat transfer mathematical model validated by fitting the predicted temperatures to within +3°C of the experimental temperatures taken over a 600 second time period. On this basis, the relative error between two thermocouples positions could be as high as 6°C. The error in the determination of the temperature gradient over 20 mm, the distance between two thermocouples, could therefore be as high as 0.3°C/mm which is three times the value of the critical temperature gradient. There was no mention of an experimentally determined temperature gradient to assist in validating the model. An attempt is made in this study to address this uncertainty as the Sn-10%Pb alloy is reexamined, test 60, using carifully calibrated thermocouples with a high degree of accuracy to produce experimental values of G L in front of two fixed positioned thermocouples straddling the CET. It was found that the CET occurred between experimental values, denoted in this study as GLX, of 0.14°C/mm and 0.10°C/mm. Linearly interpolating between these two values results in an experimentally determined value at the CET of 0.12°C/mm which is very close to the original value found by Mahapatra and Weinberg18. Therefore, to a first approximation, this verifies the earlier finding. It is also interesting to note that the one dimensional mathematical heat transfer model using the linear assumption of latent heat release predicted a value of G L of 0.12°C/mm even though the fit between the predicted cooling curves and the experimental cooling curves is not good. The redeeming feature of the fit could be in that 168 all the predicted values are lower than the experimental values rather than being on either side of the experimental cooling curve. In this way, the absolute temperatures are incorrect but the relative temperatures are close but only for times before the CET takes place. After that time, the model predictions are not valid. The model predicts that the G L continues to decrease until reaching 0°C/mm at 100 mm from the chill face whereas there is some indication from Figure 3.30 that the temperature gradient remains positive. The three cooling curves show that the temperature in the ingot increase as the distance from the chill increases until solidification is complete at 1320 seconds. An important feature of the CET in the Sn-10%Pb test performed in this study is that it resembles the CET in the previous study. That is, the CET is abrupt and flat and is very easy to identify. In addition, the entire cast structure is reminiscent of the cast structure found by Mahapatra and Weinberg18. The equiaxed grain size is small and decreases as the distance from the CET increases. However, two very prominent features in the cooling curves found in this study were not reported by Mahapatra and Weinberg. They are, supercooling followed by a horizontal temperature plateau. It was found that both the amount of supercooling and the temperature of the plateau increased as the distance from the chill increases. The reason this was not previously reported could lie in the method and duration of the temperature measurements made by Mahapatra and Weinberg. In their study, four thermcocouples were used to measure temperatures during cooling and solidification and were connected to a two pen Kipp and Zonen chart recorder via a switching device. Therefore, only two thermocouples could be monitored simultaneously leaving open the possibilty of important information such as the small amount of supercooling and short plateaus at 169 distances close to the chill being missed. In addition, terminating temperature measurements after 600 seconds would not allow the large supercoolings and significant temperature plateaus observed in this work at 780 seconds to be observed. Unfortunately, no repeat work was done using Al-3%Cu alloy and the carefully calibrated thermocouples to verify the findings of Ziv and Weinberg19 that the G L is 0.06°C/mm at the CET. 4.3 Pb-1.5%Sn The results of the Pb-1.5%Sn work show that the G L does not influence the position of the CET as it does for Sn-10%Pb. There are many instances in the experimental data where the G L goes to 0°C/mm before the CET occurs. Test 35 is an example where at 67 mm from the chill, the experimental value, GLX, is 0°C/mm while the CET occurs at 71 mm from the chill. However, there are cases where this trend is not followed. This is most clearly seen in the results of test 40 which features a completely equiaxed structure yet the GLX does not go to 0°C/mm until somewhere between 35 and 65 mm from the chill face. With inconsistent results such as these, it is difficult to detect a critical temperature gradient for Pb-1.5%Sn. It would be much easier to determine a temperature gradient at which the CET occurs if the structures in test 33 to 48 resembled those in tests 1 to 4 where the CET is abrupt and flat. Unfortunately, the poor temperature measurements prevent this from happening. The major differences are in the grain size and and the shape of the CET. Therefore, the difficulty in detecting a critical temperature gradient for Pb-1.5%Sn is -attributed to the irregular shape of the CET and the very large equiaxed grain size. The 170 large equiaxed grain size infers that there were very few effective nuclei in the melt ahead of the advancing solid/liquid interface prior to directional solidification. The small number of nuclei are also responsible for the irregular shape of the CET in tests 33 to 48. The appearance of the CET in the Pb-1.5%Sn tests is described as being irregular as some columnar grains continue to grow into a predominantly equiaxed zone as shown in Figures 3.15(a) and (b). The columnar grains which continue to grow do so because there are no equiaxed grains growing immediately ahead them. Adjacent to these columnar grains, which are generally situated in the middle of the ingot, are equaixed grains which have stopped the columnar growth at a distance closer to the chill. In order to have an equiaxed grain growing ahead of the columnar grain two conditions must be met. The liquid has to be supercooled and must contain nuclei with a nucleation temperature, AT N , less than the amount of supercooling. The first condition is always met for alloys, as a small region of constitutionally supercooled liquid is present ahead of the dendrite tip due to the rejection of solute, since the partition coefficient is less than 1. Intuitively, it is felt that equiaxed grains should nucleate in this region if condition two is met. With Pb-1.5%Sn, this zone is especially small since the amount of solute used is low and given the results it is concluded that it is not met for every columnar dendrite. Therefore there are an insufficient number of nuclei in the melt to produce a flat CET. Another reason for the irregular shape of the CET is possible. This could occur if there was a very steep radial temperature gradient such that temperatures in the middle of the ingot are much higher than on the outside edge. This possibility is discarded given the results shown in Table 3.12. There is a small temperature gradient observed between the outside edge of the ingot and the centre starting at 0.06°C/mm, (1.0°C/15 mm), which decreases gradually with time to 0.02° C/mm when the temperature plateau is recorded. 171 The gradient though is in the opposite direction with temperatures being hotter on the outside and cooler in the centre. Since there is not a steep radial temperature gradient, the isotherms across the face of the ingot are considered to be flat which leaves the lack of nuclei responsible for the irregular shape of the CET in Pb-1.5%Sn alloys. Supercooling is observed consistendy in the cooling curves for Pb-1.5%Sn at distances of approximately 95 mm from the chill. Since directional solidification of an alloy is occurring, it is reasonable to consider at first that the supercooling is constitutional. This conflicts with the above argument. However, there are results which indicate that the supercooling is thermal. First, there is the fact that when directionally solidified, pure Pb, (99.999%), supercools by approximately the same amount and in the same general location, ie, towards the top half of the ingot, see Table 3.13. Second, if enough power to the furnace is supplied such that heat is added to the melt to maintain a positive temperature gradient, the supercooling disappears as shown in Table 3.10. This would not occur if the supercooling were constitutional. It is surprising to see a ciirectionally solidifed alloy thermally supercool. This is only possible if there were another avenue for the heat to escape other than from top of the ingot to the chill block. The two other sources of heat loss are radially through the fibrefrax and into the furnace, and vertically out from the top surface of the melt and into the furnace assuming that a layer of fibrefrax has not been put on the top surface. There is some experimental evidence to show that heat is being lost in these directions, see Figure 2.10. It is difficult though to determine the magnitude of each but a good fit can be obtained in the model assuming heat is lost only from the top surface of the melt. This suggests that the radial heat loss is much lower than the heat lost vertically out the top of the melt. 172 4.4 MECHANISM FOR THE CET IN Pb-1.5%Sn AND Sn-10%Pb ALLOYS The irregularity of the CET and the existence of thermal supercooling in tests 33 to 48 are explained by the lack of nuclei in the melt prior to solidfication, yet it is difficult to account for the presence of sufficient nuclei in tests 1 to 4 without considering the showering of dendrites theory proposed by Southin to produce nuclei for all the Pb-1.5%Sn tests. An integral part of the showering of dendrites theory is heat loss from the top surface of the melt. This clearly happened in test 2 as shown in Figure 3.7. The thermocouple situated at the top of the melt, 77 mm from the chill face, cools faster than the other two thermocouples situated closer to the chill face. It is felt that this occurs in test 1 as well because the duration of the horizontal temperature plateau 81 mm from the chill face, see Figure 3.5, is shorter than the other three. It is felt that heat loss from the top surface occurred in tests 33 to 48 since thermal supercooling is observed in these particular tests yet in these tests there were a lack of nuclei. The difference to account for the small grain size and abrupt CET in tests 1 to 4 is in the amount of heat being lost from the top surface. In tests 1 to 4 a greater amount of heat is being loss from the top surface than in tests 33 to 48. In only one test, (test 35), is the duration of the horizontal temperature plateau at the top thermocouple shorter than than the duration of the plateau measured at a distance closer to the plateau. In addition, the cooling rate of the thermocouples located 95 mm from the chill face is approximately equal to the cooling rate of the thermocouple located 65 mm from the chill face. Thus in accordance to the showering of dendrites theory, increasing the heat loss from the top surface increases the likelihood of surface dendrite formation. This in turn should increase the number of 173 nuclei that detach and increase the number of nuclei in the melt ahead of the advancing columnar grains. It is also felt that the smaller size ingots used in tests 1 to 3 are more conducive to the survival of detached surface dendrites as the distance they travel to the growing columnar front is much shorter. This would make them less likely to melt. Not all aspects of the showering of dendrites theory is supported in this work. Growing equiaxed grains still have to release their latent heat into a supercooled environment. In the case of Pb-1.5%Sn the liquid thermally supercools which allows this to occur. Therefore, it is proposed that thermal supercooling is required for the showering of dendrites mechanism to take effect which conficts with the original theory proposed by Southin. It is felt that the melt in tests 1 to 4 also thermally supercooled but that it could not be measured due to improper techniques and equipment. As such, the showering of dendrites mechanism was active for these tests. It must be emphasized that thermal supercooling of an alloy is phenomenon that probably only occurs in a laboratory experiment as described in this study and may not occur in a commercial casting application such as the continuous casting of steel billets. The difficulty is in visualizing how the central portion of a steel billet loses heat other than to the surrounding solid shell in order to thermally supercool. Therefore, the mechanism to account for the CET in Pb-1.5%Sn alloys can not be extended to commercial applications. It is also felt that this mechanism is not applicable even to the case of Sn-10%Pb performed in this study. In this test the large number of small equiaxed grains in the equiaxed zone indicate that there were a sufficient number of nuclei present in the melt prior to directional solidification such that the nuclei were activated in the constitutionally supercooled zone which is ahead of the dendrite tips in the columnar grains. In addition, the cooling curve data indicate that there was no loss of heat from the 174 top surface. First of all, the duration of the plateau increases with distance from the chill as discussed previously. Second, the slope of the cooling curves obtained from T C 2 and T C 3 are almost identical which means that the cooling rate is almost identical at these two positions. Without a loss of heat from the top surface, the showering of dendrites mechanism can not take place. Finally, the relative density of the initial equaixed grains to the solute enriched liquid causes them to rise and not fall. The rise of the inital equiaxed grains is seen in the Sn-10%Pb test in the variation of the temperature of the horizontal plateau, TP, and perhaps in the decrease in grain size in the equiaxed zone with distance from the CET. Equiaxed grains which rise cause a depletion of solute towards the top of the ingot, and an enrichmnet of solute toward the CET. This is easily calculated based on the TP, the composition at 93 mm from the chill is calculated to be Sn-7.6%Pb while at 64 mm, it is Sn-9.3%Pb. It is expected that if the initial equiaxed grains do move upward after nucleation then one would see more and smaller equiaxed grains toward the top of the ingot which is exactly what is shown in Figure 3.29. In contrast to this if the initial equiaxed grains do not move then there should be no relationship between distance from the CET and equiaxed grain size. It is unfortunate that there are so few grains in the Pb-1.5%Sn ingots that this relationship can not be explored. 4.5 M A T H E M A T I C A L M O D E L L I N G In order to fulfill the mandate to fit the predicted temperatures to the experimental temperatures as closely as possible, the supercooling has to be comnpletely ignored while the temperature at the top of the melt, T p ^ , has to be changed. Fortunately, for 175 Pb-1.5%Sn the supercooling is small, <0.5°C, and the changing of Tpu^ only illustrates the lack of precise knowledge on how heat is lost or gained using a vertical mould in a resistance heating furnace. General knowledge on the fact that heat is first gained by the melt and then lost by the melt to the furnace is all that is needed to use the model to interpolate temperatures between the fixed position thermocouples. Even though the model does have its shortcomings it is still able to show that in most cases, at the CET the G L is essentially 0°C/mm, see Table 3.13. There are however a few cases where the G L is substantially higher ranging from 0.04"C/mm to O.lO'C/mm. With this kind of inconsistency it is still difficult to determine a critical G L at the CET for Pb-1.5%Sn. It appears that given the very few effective nuclei in the melt, it is a hit and miss situation to form the CET. The experimentally determined values, G ^ , assumed that the temperature gradient between two thermocouples is linear. The results shown in Figure 3.32 show that this is a good assumption as the temperature gradient in front of TC2 is linear for up to 10 mm. It is not known if the gradient remains linear beyond this distance. The very close match between G ^ and G L in Figures 3.33 and 3.36 suggest that it does remain linear. The effect of releasing latent heat in a different fashion for Pb-1.5%Sn is observed to have very little effect. Using the Scheil equation or the Lever rule to calculate the amount of latent heat released does not result in significatntly different predicted temperatures from the linear assumption as shown in Figures 3.37 to 3.39 inclusive. In both the Scheil equation and the Lever rule model, a greater proportion of the latent heat is released closer to the liquidus temperature. In turn, one would expect that the predicted temperatures around the liquidus be higher for models incorporating the Scheil equation 176 or the Lever rule. It is obvious that this is not so. If the temperatures are higher at the liquidus the one would expect that the G L would be lower than using the linear assumption. As shown in Figure 3.40, this does happen but again the difference is negligble. There are two principal reasons that proportioning the latent heat release differently does not result in different predicted temperatures. The first reason is due the low amount of latent heat released overall. The calculated value listed in Table 2.1 is only 24.5 J/gm. The second is due to the very short freezing range of the alloy which is only 4.6°C. Increasing the solute content of the alloy does increase the freezing range over from 4.6°C to 9.3"C for Pb-3%Sn and 18.6°C for Pb-6%Sn. This does have some effect on the predicted temperatures as shown in Figure 3.41. The predicted temperatures from the three models are now separated by a greater amount. The change in the way latent heat is released does not change the overall shape of the cooling curves. For example, a horizontal temperature plateau does not appear using the Scheil equation or the Lever rule when one does not appear using the linear assumption. It is also interesting to note that at very low temperatures in Figure 3.41 the Scheil equation model predicts the lowest temperatures. This is due to the fact that using the Scheil equation means that the temperature range over which the alloy solidifies is extended to the eutectic temperature and at some temperature within the liquidus to solidus temperatures, the fraction solid versus temperature plot levels off. As a result, the linear method releases more latent heat at the lower temperatures than the Scheil equation which produces higher predicted temperatures. Changing the alloy system to Sn-10%Pb greatly affects the ability to fit the predicted temperatures to experimental temperatures using all three methods to release latent heat as shown in Figures 3.44 to 3.46 inclusive. The predicted temperatures are 177 always lower than the experimental temperatures but more importantly, the shape of the predicted cooling curve does not resemble the experimental cooling curve. The difference in the shape of the curve is magnified as the distance from the chill face increases. The predicted cooling curves have a concave up shape while the experimental cooling curves show a change in the concavity from concave up while the alloy is liquid to concave down while the alloy is solidifying. This is felt to be important as it signifies that the model is not solidifying in the same way as the alloy solidifies. The situation is improved only slightly when the method to release latent heat is changed to using the Scheil equation or the Lever ruleThis is shown for the case at 93 mm from the chill face, where the predicted cooling curve from the Scheil equation and the Lever rule do show some change in the concavity of the curve but this does not significandy improve the fit between predicted and experimental temperatures. The predicted temperatures are still lower than the experimental temperatures at the liquidus temperature. The problem in obtaining a good fit for Sn-10%Pb is not one of latent heat release but one concerning the supercooling and the deviation of the horizontal temperature plateau from the liquidus temperature observed in the experimental cooling curve. Assuming that no supercooling is required for solidification to occur is extremely erroneous in this case as the supercooling in the Sn-10%Pb alloy is measured up to 2.8°C which is now a very significant amount when compare to <0.5*C for Pb-1.5%Sn. The experimental conditions state that supercooling must be considered and that it varies with distance from the chill. In order to incorporate this condition into the model, the nature of the model must be changed from one involving heat transfer alone to one involving nucleation and growth processes. An example of such a model is the Hunt model which provides a nucleation condition, equation 1.4, and a growth condition, equation 1.3. In 178 addition, for the model to be successful, it must also account for the increase in the horizontal temperature plateau caused by the movement of light equiaxed grains and heavy solute enriched liquid. Accomodating these observations into a model is difficult as not much is known about why the supercooling varies with distance, and how fast the nuclei rise. Despite the inaccuracies of the linear, the Scheil and the Lever rule models, comments should be made on the influence of the release of latent heat on the predicted G L for the Sn-10%Pb alloy. In Figure 3.48 releasing more of the latent heat closer to the liquidus temperature, as the Scheil equation and Lever rule methods do, decreases the predicted G L significantly. This does change the value of G L predicted to occur at the CET. However, increasing the amount of latent heat released closer to the liquidus temperature does not prevent the G L from decreasing to 0°C/mm. In fact, the Scheil equation and the Lever rule models show that the G L reaches 0°C/mm faster than the linear assumption. The major difference between the Sn-10%Pb system and the Pb-1.5%Sn system, to produce the differences observed in the predictions from the three methods to release latent heat, is in the total amount of latent heat released. The Sn-10%Pb alloy is calculated to release 56J/gm which is twice as much as the Pb-1.5%Sn alloy. Thus, any change in the proportioning of the latent heat is expected to have a greater influence on the amount that is released at various temperatures between the liquidus and the solidus temperatures. In addition, the Sn-10%Pb alloy has a much wider temperature range than the Pb-Sn alloys. For a given amount of latent heat using the hnear assumption, increasing the temperature range decreases the amount of latent heat released per °C. The Sn-10%Pb alloy also shows the Lever rule and the Scheil equation model to predict 179 identical temperatures and temperature gradients whereas the Pb-1.5%Sn shows a small difference. This is due to the fact that with Sn-10%Pb the solidus temperature and the eutectic temperature are the same and therefore the range over which the latent heat is released is the same. This is not true for Pb-1.5%Sn which has a solidus temperature of 318.3°C and an eutectic temperature of 183°C. It is expected given the difference in the predicted temperatures and temperature gradients of the Pb-Sn and Sn-Pb alloys using the three methods to release latent heat that the differences would be even more with an alloy that has a very large amount of latent heat released over a very wide solidification range. For example, Al-3%Cu has a calculated latent heat of 382J/gm and solidifies over a 55°C range. Both of these values are greater than Sn-10%Pb alloy's values so that the predicted results using a linear assumption of latent heat should be vasdy different than the predicted results using either the Scheil equation or the Lever rule. Chapter 5 CONCLUSIONS 1) . Temperature measurements as accurate as ±0.2*C are possible using chromel-alumel thermocouples if an "in-situ" calibration is performed immediately prior to the experiment. 2) . The "in-situ" calibration technique is complimented by taking multiple data points and using the mean of the data points as the recorded temperature when using an A/D data acquisition board. 3) . Calibrating thermocouples once prior to testwork performed over a prolonged period results in inaccurate temperature measurements because of the errors associated with the thermocouples change with time in a non predictable fashion. 180 181 4) . The CET in a Sn-10%Pb alloy has been found to occur at approximately O.rC/mm as determined by accurate temperature measurements. This confirms to a first approximation the results of Mahapatra and Weinberg18. 5) . Supercooling is observed in directionally solidified Sn-10%Pb and varies with distance from the chill ranging from 0.3'C to 2.8'C. 6) . A horizontal temperature plateau is also observed in Sn-10%Pb and also varies with distance from the chill due to the initial equiaxed grains rising and the Pb enriched solute sinking during equiaxed growth. 7) . The CET in a Pb-1.5%Sn alloy must wait for the liquid ahead of the solid/liquid interface to thermally supercool because of the lack of nuclei in the melt prior to directional solidification. The G L at the CET is usually close to 0°C/mm as determined experimentally and verified from a heat transfer model. 8) . The maximum amount of thermal superooling is approximately 0.4*C in directionally solidified a Pb-l.%Sn. 9) . Horizontal temperature plateaus are also seen with the Pb-1.5%Sn alloy but unlike the Sn-10%Pb alloy, the temperature of the plateau does not vary with distance from the chill. The temperature of the plateau occurs at the liquidus temperature of the alloy. This is due to the fact that the initial equiaxed grains do not rise in the liquid during their growth but probably remain motionless. 10) . The CET in Pb-1.5%Sn occurs following the showering of dendrites theory as proposed by Southin6. However, thermal supercooling is intrinsically involved in the CET for this alloy which does not follow the original theory. 182 11) . The showering of dendrites mechanism for Pb-1.5%Sn is not transferable to Sn-10%Pb because of the difference in the number of nuclei in the melt and their behaviour after the initial grains form. 12) A heat transfer model is successful in predicting temperatures for Pb-1.5%Sn because of the small amount of supercooling and no movement of equiaxed grains during growth. 13) . A heat transfer model is not successful in predicting temperatures for Sn-10%Pb because of the large amount of supercooling and the movement of light equiaxed grains after they initially form. A better approach to take to model this alloy when directionally solidifying this alloy is to use a nucleation and growth model in conjunction with the heat transfer model. 14) . The effect of changing the method to release latent heat on the predicted temperatures and temperature gradients is dependent on the amount of latent heat to be released and the freezing range of the alloy. Increasing both has a tendancy to increase the effect as higher temperatures and lower gradients are predicted using the Scheil equation or the Lever rule compared to the results using a linear assumption. 183 REFERENCES 1. B.Chalmers, Principles of Solidification, John Wiley and Sons, 1964 2. G.J.Davies, Solidification and Casting, Applied Science Publishers Ltd., 1973 3. B.Chalmers, Journal of Metals, Transactions AIME, May 1954 4. B.Chalmers, W.C.Winegard, Transaction of The ASM, Vol. 46, 1954, pl214 5. B.Chalmers, D.Walton, Transactions of The Metallurgical Society of Aime, Vol. 215, June 1959 6. RT.Southin, Transactions of The Metallurgical Society of Aime, Vol.239, Feb. 1967, p220 7. J.L. Walker, Phys. Chem. of Proc. Met. LT, G.R. St. Pierre ed, p 845, 1961 8. K.A.Jackson, J.D.Hunt, D.R.Uhlman, T.P.Seward III, Transactions of The Metallurgical Society of Aime, Vol. 236, Feb. 1966, pl49 9. G.S.Cole, G.F.Bolling, Transactions of The Metallurgical Society of AIME, Vol.239, Nov. 1967, pi824 10. G.S.Cole, G.F.Bolling, Transactions of The Metallurgical Society of AIME, Vol.233, Aug. 1965, pi568 11. A.Ohno, T.Motegi, H.Soda, Transactions ISIJ, Vol.11,1971, p 18 12. J.D.Hunt, Material Sciences and Engineering, Vol. 65, 1984, p75 13. J.D.Hunt, S.C.Flood, Modelling of Welding and Casting II, 1983, p207 14. J.D.Hunt, S.C.Flood, Modelling of Welding and Casting Processes, 1986, p607 15. M.H.Burden, J.D.Hunt, Metallurgical Transactions A, Vol. 6A, Jan. 1975, p240 16. M.H.Burden, J.D.Hunt, Journal of Crystal Growth, Vol. 22,1974, p99 17. M.H.Burden, J.D.Hunt, Journal of Crystal Growth, Vol. 22, 1974, pl09 184 18. R.B.Mahapatra, F.Weinberg, Metallurgical Transactions B, Vol. 3, N°2, June 1987, p425 19.1. Ziv, F. Weinberg, Metallurgical Transactions B, Vol 20B, 1989, p731 20. J.Lipton, M.E.Glicksman, W.Kurz, Material Science and Engineering, 65, (1984), 57-63 21. P.A.Kinzie, Thermocouple Temperature Measurement, John Wiley and Sons, 1973 22. RJ.Schaefer, M.E.Glicksman, Transactions of The Metallurgical Society of AIME, Vol. 239, Feb. 1967, p257 23. G.S.Cole, K.W.Casey, G.F.Bolling, Metallurgical Transactions, Vol. 1, May 1970, pl413 24. G.S.Cole, G.F.Bolling, Transactions of The Metallurgical Society of AIME, Vol. 242, Jan. 1968,pl53 25. J.Lipton, W.Henemann, W.Kurz, Arch. Eisenhuttnwes, 55, 1984,pl95 26. L.A.Tarshis, J.L.Walker, J.W.Rutter, Metallurgical Transactions, Vol. 2, Sept. 1971, p2589 27. J.Lipton, W.Kurz, W.Henenmann, Concast News Technology, Vol. 22, N°2, 1983, p4 28. H.Fredriksson, M.Hillert, Metallurgical Transactions, Vol. 3, Feb. 1972, p565 29. HFredriksson, M.Hillert, Material Sciences and Technology, Vol. 2, May 1986, p508 30. S.Witzke, J.P.Riquet, Acta Metallurgica, Vol. 30, 1982, pl717 31. S.Witzke, J.P.Riquet, F.Durand, Acta Metallurgica, Vol. 29, 1981, p365 32. L.Beitelman, Transactions of The ISS, Aug. 1988, p43 33. J.K.Brimacombe, K.Sorimachi, Metallurgical Transactions B, Vol. 8B, Sept. 1977, p489 34. G.A.Chadwick, Metals and Materials, March 1969, p77 185 35. T.W.Clyne, Materials Science and Engineering, 65, 1984, p i l l 36. C. Schesvov, M.A.Sc. Thesis, 1983, U.B.C. 37. S.E.Kisakurek, Journal of Material Science, Vol. 19, 1984, p2289 38. D.R.Croft, D.G.Lilley, Heat Transfer Calculations Using Finite Difference Equations, Applied Science Publishers Ltd., 1977 39. L.Backerud, B.Chalmers, Transactions of The Metallurgical Society of AIME, Vol. 245, Feb. 1969, p309 40. G.S.Cole, W.CWinegard, Journal of The Institute of Metals, Vol.93, 1964-65, pl53 41. D. Poirier, M. Salcudean, ASME Conference, December, 1986 42. Handbook Of Chemistry and Physics, 57th Edition 43. Thermophysical Properties of Matter, Vol. 1, Thermal Conductivity, LFI/Plenum, New-York, Washington, 1970 186 APPENDIX 1 The heat balance on the interior node, ie: i=2, N, shown below is given as: Heat In - Heat Out + Heat Generated - Heat Consumed = Heat Accumulated Heat In: (T'd-D-T'd)) (k&yiy ^ Heat Out: ( M y l ) ( r ( 0 - r ( 0 ) For a node in the liquid state cooling then, Heat Generated - Heat consumed = 0 Heat Accumulated: After all terms are combined, allowing Ax = Ay, and dividing by Ax2, the heat balance equation for the interior nodes become: - ^ r ( i + l ) -Ax \ Ax2 At j Ax2 At The heat balance on the bottom node where i=N is given by: Heat In: ,(T'(/-l)-r(0) (*Ayl)-Ac 187 Heat Out: //BAy 1(7X0-7^) Heat Accumulated: pcUAxAyl V ( 0 - r ( i » A/ Again, if all terms are combined and Ax = Ay, and divide by Ax2, the bottom heat balance equation becomes: 2k (2k 2HB p C . V -pC p 2HB z?r« - » - [ A ? + " S T + * f ( 0 * R ( I ) " A T T -Similarly, the heat balance at the top node where i=l is given by: Heat In: *Ari£ ' a + I ) - r W Ax Heat Out: HTopAyl(T'(i)-TFltrH) Heat Accumulated: The final heat balance at the top node where i=l after all terms are combined and Ax = Ay and the entire equation is divided by Ax2 is: 188 2 * ™ - (2k 2HTop p C , V f / . x -pC_ / # x 2H T O P / T , -ro + i)-[^ +^ +^ jr-o)--^ ro)--sfrili. As stipulated, the above equations do not consider the situation where a node releases a portion of the latent heat of solidification upon cooling through the mushy zone. To incorporate the latent heat of solidification term, consider heat a node loses as it cools: H = pCVAT If a node is undergoing solidification, the heat lost is given as: H = pCVAT +LpVAFS To bring in the latent heat into the implicit finite difference equations, the temperature of the node is first allowed to cool from the liquid state to the mushy state without the latent heat being brought in as depicted below. The heat lost is then given by: H = pVC(Tl-TJ Reality states that T 2 is not reached at this time because latent heat is released after the temperature falls below T u < r Therefore, the heat lost by the node undergoing solidification should be given by: H = pVCfT , - T^) + pVC2(7^ - r,)+LpVAFS The amount of heat lost by the node is identical, therefore the heat balance is given by: pVC.CT, - TJ = pVC.fT, - 7^) + pVC 2(T^ - T3) +LpVAFS 189 The equation reduces to: A problem appears in that to calculate AFS using the Scheil equation one has to know what the C s, and C, at the new corrected temperature T 3 are, to solve the heat balance. This is accomplished by using the Secant convergence technique which uses two simultaneous guesses at T3 to calculate C„ and Q, which produce two guesses at AFS in order to solve the heat balance. The procedure is repeated until convergence is reached. When convergence is reached, T3 becomes the new temperature for the node. The same situation is encountered when the node goes from one mushy state to another as depicted below. The total heat lost by the node in this situation is given by equation XX while reality states that the heat lost by the node is given by: H = p VC2(TX - T 3)+L pV&FS The heat balance is: pVC^T, - TJ = pVC2(r, - TJ+LpV&FS This equation reduces to: The Secant convergence technique is also used to solve the heat balance. APPENDIX 2 This analysis is concerned with two hypotheses at a 1% level of significance regarding the ratio of variances from two sets of samples. The hypotheses one is testing is whether or not the ratio of: or —=\* I where: o^, oi are the variances of the two sets of samples in question. The hypotheses are tested by calculating a critical region of values which are given by: S? — <F C2 ' ^ - l , 1 4 - 1 , 1 - 0 / 2 or S 2 — > F 2 where: ^ - l . / i j - i . o r t i s m e Snedecor F ratio with (n, - 1, n2 - 1) degrees of freedom at the a level of significance, and a = 0.01. If S]2 and S22, satisfy either of the above inequalities then the variances of the two sets of samples are significantly different. The data from two selected temperatures on the cooling curve shown in Figure 17 corresponding to a temperature in the supercooled portion of the curve and to a temperature on the plateau are chosen for the analysis, and are shown below. Set 1. (from the supercooled portion of the cooling curve) n, = 32 x"1 = 13.3821mV = (328.15°C) sx = 0.0051 mV Set 2. (from the temperature plateau) n2 = 32 x2 = 13.3987m V = (328.39°C) S j = 0.0037 where: n,, n2 = the number of samples taken for data points 1 and 2 xux2, = the mean of the 32 samples for data points 1 and 2 su S2 = the standard deviation of the 32 samples for data points 1 and 2 '? The above data produces a ratio of — = 1.9 si The value of F31 i31;.OTs~/r3o,30;.oos = 2.6278 is from the F m l j n 2 Distribution Table in Guttman, Wilks and Hunter, while the value of F 3 l 3 1. ^ = •=—- 7—^— = .38. 

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