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Interdendritic fluid flow Streat, Norman 1974-01-27

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INTERDENDRITIC FLUID FLOW by NORMAN STREAT B.Sc.(Eng.), University of London, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of METALLURGY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1973 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Metallurgy The University of British Columbia Vancouver 8, Canada Date January 14, 1974 i ABSTRACT Fluid flow through liquid interdendritic channels of a partially remelted lead-tin casting has been measured directly, with gravity as the driving force. The results were shown to be consistent with Darcy's Law. The permeability of the dendritic array was found to be a function of the square of the primary dendrite spacing, and was observed to increase with time due to coarsening of the dendritic structure. The formation of casting defects in lead-tin alloys was studied with isothermal and unidirectional solidification experiments. Solute convection was observed when the liquid close to the bottom of the solid-liquid region was less dense than the liquid above, using radioactive tracer techniques. Macrosegregation was shown to be related to the solidification conditions, and channel-type defects, resembling freckles and A segregates, were formed when the rising interdendritic liquid dissolved dendrite branches in its path. A simple mathematical model is proposed, which predicts the composition profiles in vertical, directionally solidified lead-tin castings, as a function of the structure, growth rate, and temperature gradient. The model is shown to agree qualitatively with the experi ments, and can be used to recommend specific changes in casting practice to reduce gravity segregation effects. ii ACKNOWLEDGEMENTS I would like to express sincere thanks to my research supervisor, Dr. F. Weinberg, for his advice, support and encourage ment throughout this work. Thanks are also extended to other faculty members and fellow graduate students for many helpful discussions. In addition the assistance of the technical staff of the Metallurgy Department has been greatly appreciated. Financial aid from the Killam Foundati on in the form of a Predoctoral Fellowship, and from the National Research Council of Canada (Grant Number A-4609) is gratefully acknowledged. Thanks are also due to the programming staff of the U.B.C. Computing Centre, in particular to Mrs. Janet Streat for her invaluable assistance. iii TABLE OF CONTENTS Page CHAPTER 1 Introduction 1 1.1 Interdendritic Fluid Flow in Castings ... 1 1.2 Purpose of the Present Investigation ... 4 1.3 Organization of the Thesis ..... 5 CHAPTER 2 General Experimental Apparatus and Procedures 7 2.1 Apparatus ........ 7 2.2 Preparation of Lead-Tin Alloys .... 9 2.3 Metallography ........ 9 2.4 Measurement of Dendrite Spacing . . . .11 2.4.1 Primary dendrite spacings . . . .11 2.4.2 Secondary dendrite spacings ... 16 2.5 Autoradiography . . . . . . .16 CHAPTER 3 The Measurement of Interdendritic Fluid Flow Rates . . . . . . .17 3.1 Review of Previous Work ...... 17 3.2 General Description of the Technique Used in the Present Work . ...... 22 3.3 Preparation of the Alloy under Test (A) . . 27 3.4 Preparation of Castings B and C .... 30 3.5 Flow Measurement Equipment ..... 31 3.6 Flow Testing Procedure ...... 34 3.7 Precision of the Flow Measurement Technique . 35 CHAPTER 4 Results and Discussion of Flow Measurements 38 4.1 Interpretation Using Darcy's Law .... 38 4.1.1 Laminar flow ...... 40 4.1.2 Interaction effects ..... 42 4.2 Application to the Flow Cell Experiments . . 43 4.2.1 The method for finding the initial permea bility ....... 45 iv TABLE OF CONTENTS (Continued) Page 4.2.2 Results ........ 49 4.3 Dendrite Spacings and Structure .... 52 4.3.1 Autoradiography ...... 54 4.4 Microexamination ....... 59 4.4.1 Negative deviations from Darcy's Law . . 60 4.4.2 Positive deviations from Darcy's Law . . 65 4.5 Permeability and Dendrite Spacing . . . .68 4.5.1 Straight Capillary Model .... 71 4.5.2 Hydraulic Radius Theory: Other Theories . 75 4.6 Dendrite Coarsening ....... 79 4.7 The Scatter of Permeability Results .... 86 CHAPTER 5 The Effect of Density Differences on the Formation of Channels ..... 90 5.1 Introduction and Review of Previous Work ... 90 5.2 Experimental Procedure ...... 98 5.3 Results ......... 102 5.4 Discussion . . . . . . . . .110 CHAPTER 6 Solute Convection and Freckle Formation During Solidification . . . . .115 6.1 Introduction . . . . . . . .115 6.2 Experimental Procedure . . . . . .115 6.2.1 Apparatus . . . . . . .115 6.2.2 Macrosegregation studies .... 117 6.2.3 Determination of composition from activity measurements ....... 120 6.2.4 Solute convection ...... 126 6.3 Results . . . . . . . . .127 6.3.1 Composition profiles . . . . .127 6.3.2 Convection in the liquid .... 136 V TABLE OF CONTENTS (Continued) Page 6.3.3 Freckles 139 6.4 Discussion of Results ...... 139 CHAPTER 7 A Numerical Model for Macrosegregation in Pb-Sn Alloys . 144 7.1 Introduction and Review of Previous Work . . . 144 7.2 Model of the Solidification Process .... 147 7.3 Interdendritic Fluid Flow Model .... 150 7.4 Unidirectional Solidification of a Vertical Casting . 151 7.5 Results of Calculations for Solidification of a Pb-Sn Alloy 155 7.6 Comparison with Experiment . . . . .161 CHAPTER 8 Conclusions ....... 164 8.1 Summary ......... 164 8.2 Conclusions ........ 165 8.3 Suggestions for Future Work ..... 166 REFERENCES 168 APPENDIX I Integration of Darcy's Law for a Falling Head 172 APPENDIX II FORTRAN Program for Processing Fluid Flow Data 175 APPENDIX III The Solidification of Pb-20%Sn -A Table of Solidification Variables . . 179 APPENDIX IV FORTRAN Program for Calculating Macro-segregation in Lead-Tin Castings . . . 181 APPENDIX V Direct Observation of Solidification Using Electron Microscopy . . . . .188 vi TABLE OF CONTENTS (Continued) Page V.l Introduction . • . . . . . . 188 V.2 Experimental Method . . . . . .188 V.3 Results V.3.1 Pure bismuth 189 V.3.2 Other pure metals (tin, aluminum and indium) . . . . . . . 194 V.3.3 Lamellar eutectics ..... 194 vii LIST OF ILLUSTRATIONS Figure Number Page 4 5 6 7 10 11 12 13 14 15 16 (a) Tube furnace and quenching apparatus for producing columnar castings. (b) Tube furnace for ± 0.5°C temperature control ...... Cross section of a group of primary dendrites (schematic) ....... (a) Longitudinal section of a directionally solidified casting. (b) Corresponding cross section Enlarged views of regions A' and B' in Figure 3(b) Three dimensional composite, from which one can estimate that the dendrites in the top corner are tilted approximately 20° ..... Schematic view of the structure in Figure 4(b) Permeability as a function of the square of the volume fraction liquid, using experimental data obtained by Piwonka'^' ..... Schematic diagram showing the principle of a Falling Head Permeameter ..... Sectional views of the flow cell and the lead-tin alloy inserts ....... Three pieces of Pb-Sn alloy used for flow measurement Partially assembled flow cell .... Pb-Sn alloy before and after flow test Flow measurement apparatus ..... Circuit used for recording the position of the probe on the temperature trace Apparatus for testing the precision of the flow measurement technique ...... (a) Flow measurement results; distance of flow up the riser pipe versus time for X = 116 ym. (b) Similar plot for A = 28 pin . . . . . . viii LIST OF ILLUSTRATIONS (Continued) Figure Number page 17 (a) Data from Figure 16(a), replotted according to Darcy's Law, showing a positive deviation. (b) Similar plot for data from Figure 16(b), showing a negative deviation ....... 44 18 Primary dendrite spacing as a function of distance from the chill, for the quenching conditions in Table II 53 19 Autoradiographs from cross sections and longitudinal sections of Pb-Sn samples used for interdendritic fluid flow studies; (a) and (b) show uniform flow, (c) shows flow down a preferential channel .... 20 Cross section autoradiographs at various levels down the casting A, after testing .... 21 An example of an unreliable flow test, showing uneven penetration of tracer .... 22 Microstructures of casting A before and after flow testing X = 175 urn . . 23 Microstructures of casting A before and after flow testing I = 71 ym . 24 Microstructures of casting A before and after flow testing X = 51 ym 25 Microstructures of casting A before and after flow testing X = 28 ym 26 Microstructures of the reservoir (casting B) after testing ......... 27 Relationship between the initial permeability and the secondary dendrite arm spacing for Pb-Sn at 193 C 28 Relationship between the initial permeability and the primary dendrite spacing for Pb-20%Sn at 193°C 3 2 29 Permeability as a function of g^ /(1-g^) using data obtained by Piwonka'^ ...... 30 Growth of a neck during sintering ix LIST OF ILLUSTRATIONS (Continued) Figure Number 'age 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Dendrite coarsening plot for a sample with Kq = 0.152 cm , average primary dendrite spacing 28 urn 3 Calibration curve; density of Pb-Sn alloys (g/cm ) at 25°C as a function of composition The relationship between permeability and temperature Freckle trails in directionally solidified Mar-M200 Freckles in as-cast Inconel 718 The test assembly for isothermal experiments (a) Macrostructure of columnar casting (series I) (b) Corresponding autoradiograph (a) Macrostructure of columnar casting (series II (b) Corresponding autoradiograph Surfaces revealed after transverse sectioning of a sample from series III .... (a) Longitudinal section of same sample as in Figure 39. (b) Corresponding autoradiograph Surfaces revealed after transverse sectioning of a sample from series III .... (a) Longitudinal section of same sample as in Figure 41. (b) Corresponding autoradiograph (a) Macrostructure of columnar casting (series IV) (b) Corresponding autoradiograph Macrostructure from series V ... Split graphite mould for making long cylindrical ingots ....... Spectrum of y emission for Sn 113 204 Spectrum of y emission for Tl ... Calibration curve; activity versus sample weight 85 87 87 91 91 100 103 103 106 106 107 107 109 109 116 119 119 122 X LIST OF ILLUSTRATIONS (Continued) Figure Number 51 62 63 49 Calibration curve; activity versus Sn113 concentration 122 50 Page Calibration curve; specific activity versus alloy composition for constant Sn113 concentration . . . 123 Calibration curve; specific activity versus alloy composition, when Sn113 concentration is proportional to the solute content ....... 123 (b) Cooling conditions . 128 (b) Cooling conditions . 129 (b) Cooling conditions . 130 (b) Cooling conditions . 131 (b) Cooling conditions . 132 52 Composition profile for one ingot using lathe turning treated with nitric acid (open circles), and untreated samples (closed circles) ...... 125 53 (a) Solute distribution, 54 (a) Solute distribution, 55 (a) Solute distribution. 56 (a) Solute distribution. 57 (a) Solute distribution. 58 Autoradiographs showing the extent of tracer movement one hour after tracer was added; (a) directionally solidified, (b) quenched from the liquid . . .137 59 Shrinkage trail, approximately 7 cm long, along the outside of an ingot solidified under conditions given in Table XI. (b) Longitudinal and transverse sections showing a freckle trail on the right hand side . . 140 60 (a) Transverse section of the freckle trail in Figure 59(b). (b) Longitudinal section showing that the trail originates from widening interdendritic channels in the interior of the ingot ...... 141 61 Schematic representation of unidirectional solidification assumed in the model .... 148 Equilibrium diagram for a binary alloy. The non-equilibrium solidus is shown by the dashed line . .148 Directionally solidifying ingot divided into layers. Temperature, composition and density profiles given by the solidification model ...... 153 xi LIST OF ILLUSTRATIONS (Continued) Figure Number Page 64 (a) Assumed flow pattern showing two main flow cells. (b) Resistances Ri_5» and flow rates q^_5 for flow between six layers . . . . .153 65 Solute distribution as a function of the number of layers . . . . . . . . . .157 66 Solute distribution as a function of structure (effective number of channels) ..... 157 67 Solute distribution as a function of ingot height . 159 68 Solute distribution as a function of growth rate . . 159 69 Solute distribution as a function of temperature gradient .......... 160 70 Pb-Sn alloy in the flow cell, after a time t . .172 71 (a-c) Alternate freezing, melting and freezing in pure bismuth, showing evidence of faceted growth. (d) Enlarged view of the solid-liquid interface showing high angle grain boundaries emerging at the interface ......... 191 72 Alternate melting, freezing, melting and freezing in pwre bismuth ........ 193 73 The melting of pure aluminum, photographed from the fluorescent screen using a 35 mm camera . .195 xii LIST OF TABLES Table Page I Dimensions and Composition of Castings Used for Interdendritic Fluid Flow Studies . . . . 26 II Quench Data ........ 29 III Thermal Conditions for Pb-20%Sn Columnar Castings . 29 IV Precision of the Flow Measurement Technique . . 36 V Results of Flow Measurements ..... 50 VI Results of Dendrite Coarsening Calculations . . 82 VII Composition of Superalloys ..... 92 VIII Test Conditions for Isothermal Experiments . . 101 IX Solubility Data for Isothermal Experiments . . 112 X Solidification Variables and Macrosegregation . . 133 XI Cooling Conditions . . . . . .138 XII Solidification Variables Used for Theoretical Plots . 162 1 CHAPTER 1 INTRODUCTION 1.1 Interdendritic Fluid Flow in Castings Nearly all metal products are made by casting and subsequent fabrication of the cast material. Since the vast majority of metal products are alloys of two or more constituents, as the alloy solidifies the compo sition of the solid must be different to the composition of the adjacent liquid from which it grows (except for the special case of congruent solidification). Assuming the solid is of a single phase and solute is conserved, the composition of both solid and liquid must vary during solidification. In the solid state, composition changes can only occur by means of solid diffusion, which is relatively slow even at high temperatures. Accordingly, for all practical purposes, it is impossible to cast homogeneous alloys. In general, alloy castings have composition differences on a microscopic scale, which can affect the mechanical, corrosion and surface properties, depending on the extent and distribution of the segregated con stituents. The effects of composition variations on a microscopic scale may not necessarily be detrimental, particularly if the casting is to be processed further. However, when microscopic variations are concentrated in local regions, they can seriously reduce the strength and ductility of the casting. Large scale composition variations are termed "macrosegregation" and a number of different types are recognized. These include centreline segregation, A and 2 V segregates in large steel ingots, inverse segregation, freckles and solute banding: 1) Centreline segregation is a line of solute rich material along the axis of an ingot which has cooled from the side walls. 2) A segregates have been described as rope-like concentrations of solute which form in the upper regions of the columnar zone of large steel ingots. They are called A segregates because they are inclined a few degrees from the vertical on each side of the axis, giving the appearance of an A or Greek A when seen on a surface sectioned parallel to the axis. 3) Similarly, V segregates are cones of high solute content which form in the lower equiaxed regions of steel castings, and are V-shaped when seen on the sectioned surface. 4) Inverse segregation is a region of high solute content close to the chill face of a casting which can, in some cases, be observed as exudations or beads of solute rich material on the surface. 5) Freckles are patches or vertical lines of solute rich material which occur in a number of different types of castings, in particular, consumable arc melted ingots. They are so named because of their spotty or speckled appearance when seen on the outer surface, or on polished sections of the ingot. 6) Solute banding is the term applied when alternating regions of solute rich and solute depleted material are observed to occur in the columnar regions of an ingot. It is desirable to minimize these defects in castings that are used in critical applications, particularly when the casting is used without sub sequent working. For example, directionally solidified castings of nickel-base 3 superalloys (designed for high temperature service) are used for turbine blades in jet aircraft engines. These castings are susceptible to the formation of freckles, which can seriously affect the strength and creep properties when the blades are in service. The cost of superalloy castings produced in the United States, which are used in many critical applications of this type, and are potentially susceptible to freckling, is presently about $500 million annually. Considering that there is a possible rejection rate of 20%, and the cost of remelting the rejected castings is about 40% of the cost of the product, the potential cost of this one type of casting defect is about $40 million per year. In general, alloys solidify with a dendritic structure, that is, the solid grows in the form of clusters of tree-like spikes with side branches. The dendritic structure in alloy castings can normally be examined by suitably etching a polished surface. The etchant is selected to react with solute rich or solute depleted regions producing preferential attack of the inter dendritic regions, or dendrite centres. The extent and distribution of the segregated solute in the casting is related to the dendritic structure, which in turn is a function of the alloy composition and casting conditions. These would include the crystallo-graphic properties of the alloy constituents, the thermal environment, and in particular, liquid transport, either by forced or natural convection. The driving forces for liquid transport in a casting will be related to the temperature differences which cause natural convection, composition differences which can lead to solute convection, and other factors such as volume shrinkage and gas evolution. 4 The spacing between side branches of dendrites in the centre of large, slowly cooled castings can be of the order of millimetres. For small castings which cool rapidly the spacing can be of the order of ten microns, consequently, liquid transport through the growing dendritic network will be restricted by the narrow, tortuous channels through which the liquid must move. Since macrosegregation is strongly influenced by the extent of fluid flow during solidification, an understanding of the forces acting on the liquid, and the extent of interdendritic fluid flow restrictions is essential to account for, and modify, certain types of macrosegregation in castings. 1.2 Purpose of the Present Investigation The purpose of the present work was to measure interdendritic fluid flow in a metallic system under a known driving force, and determine the relationship between the resistance to flow and the structure of the casting. With suitable measurements, the results would be considered in terms of established empirical relationships for flow through porous media. Since most metals have similar thermal and viscous characteristics, and solidify in a similar manner, as compared to non metals, it is considered that a detailed examination of one metal system can give results applicable to most other systems. In conjunction with the fluid flow considerations, other aspects of macrosegregation would be examined, specifically, the formation of channel-type defects which resemble freckles and A segregates. To combine the results of both the fluid flow and the macrosegregation studies, a simple mathematical model has been derived. The model considers the solidification of an ingot where the driving force for macrosegregation is density differences in the 5 liquid, and the interdendritic fluid flow is a function of the cast structure as established experimentally. The model is compared with the experimental results. 1.3 Organizat ion of the Thesis The thesis is divided into four main sections. The first section (Chapter 2) gives a description of the apparatus and procedures common to all the following sections. The second section (Chapters 3 and 4) deals with the development of the interdendritic flow measurement technique, and the inter pretation of the results in terms of the theory of flow through porous media. The third section consists of the experiments on the effect of density differences in the liquid on a casting held at uniform temperature in the solid-liquid region (Chapter 5) and the study of macrosegregation and defect formation in ingots solidified under known cooling conditions (Chapter 6). The experimental work on macrosegregation is tied together with the results of interdendritic fluid flow measurements in the mathematical model, presented in the fourth section (Chapter 7). A review of previous work relevant to the particular section is presented at the beginning of Chapters 3, 5 and 7. During the course of these experiments an attempt was made to directly observe solidification in a thin film of metal using an electron microscope. The aim of this work was to study solid-liquid interfacial energies, which would have been relevant to the work on interdendritic fluid flow, in relation to the interpretation of changes which take place in a casting held in the solid-liquid region for long periods of time. The results of the elect microscope study were inconclusive. A brief summary of the work is giv< in Appendix V. 7 CHAPTER 2 GENERAL EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 Apparatus Two tube furnaces were constructed and used to produce columnar castings and to heat the alloy samples for fluid flow studies. Columnar castings were produced in the furnace shown in Figure 1(a) which had a central copper tube to withstand thermal shock when water quenches were used inside the furnace. The copper tube was wrapped with asbestos tape, and chromel windings were wrapped over the tape to prevent short circuiting on the metal tube. A control thermocouple (Chromel/Alumel) was placed next to the windings. Temperatures measured in a molten metal charge in this furnace could be held constant to ± 1°C. For better control a second furnace was built of similar design (Figure 1(b)) with a central ceramic tube. In this case the windings were directly in contact with the ceramic tube, which was a better conductor than the asbestos tape, and ± 0.5°C control was possible. This furnace was used for fluid flow studies where it was necessary to heat the samples rapidly to a predetermined temperature without overshooting, and then hold them constant. Unless otherwise stated, all temperature measurements in this work were made using iron-constantan thermocouples calibrated against the melting point of pure tin. Bare thermocouple junctions were used for rapid response. The thermocouple wires were inserted in small diameter ceramic tubing (approx imately 1.6 mm) to ensure that when the thermocouple was immersed in molten metal the reading was representative of conditions at the tip. This was *— Quench Medium (a) (b) (a) (b) Tube furnace and quenching apparatus for producing columnar Tube furnace for ± 0.5°C temperature control. 9 confirmed by breaking apart the ceramic tubes after the test, where it could be seen that molten metal did not rise up the bore. In all tests the thermocouples were connected via a cold junction in ice water to a Honeywell Electronik 194 millivolt recorder. 2.2 Preparation of Lead-Tin Alloys All lead-tin alloys were prepared using high purity Cominco Pb (99.999%) and high purity Vulcan Sn (99.999%). The required compositions were first melted in lots of approximately 1500 g in a stainless steel beaker over a bunsen burner, stirred thoroughly, and cast into graphite moulds to produce starting ingots about 2.2 cm diameter and 5 cm long. Both lead and tin have a low vapour pressure and no composition changes were expected due to evaporation. This was confirmed by plotting the cooling curves of two samples of Pb-20% Sn. The samples were held above the melting point, in air, for approximately 15 minutes and 17% hours respectively. There was no significant difference between the liquidus arrest points seen on the cooling curves. In those cases where radioactive Tl or Sn was added to the lead-tin alloys, the tracer was first dissolved in 100 g of pure Sn in a pyrex test tube with an argon flow to prevent oxidation. This 'master' alloy was then used in the preparation of the required lead-tin alloy. 2.3 Metallography Since Pb-Sn alloys are extremely soft, special techniques for metallographic preparation were used. Samples were cut with a coarse toothed 10 hacksaw to prevent clogging, and care was taken to prevent overheating. Normally the cut samples were mounted in 'Quickmount', a cold setting plastic that did not heat up more than about 30°C during curing. In addition, though 'Quickmount' is harder than the Pb-Sn alloys, it is softer than other mounting materials and can readily be polished. The mounted specimens were machined on a lathe to give a flat surface using a sharp angled tool which cut the surface but did not tear. Properly machined specimens could be taken directly to a 5 micron alumina lapping wheel where sufficient material was polished away to remove machine marks and any flowed layer. Grinding papers were not used since it was found that they would immediately clog and tear at the surface, and particles of abrasive from coarser papers would become embedded in the soft metal surface and be impossible to remove later. Final polishing was done at slow speed with a thick slurry of 1 micron alumina. The specimens were washed in cold water or alcohol, since it was found that hot water etched the structure. The etchant used for high Pb content alloys was 50 parts acetic acid, 15 parts hydrogen peroxide, and 100 parts water. This acted rapidly, both as an etch and as a mild chemical polish to remove remaining polishing marks. High Sn alloys were etched in a ferric chloride base etch. As an alternative, electropolishing was attempted, but since it was frequently necessary to polish relatively large areas, the currents required proved to be impractical. Since some degree of preparation was necessary because sawn surfaces could not be electropolished, the mechanical method above was found to be the most satisfactory. 11 2.4 Measurement of Dendrite Spacing Dendrites in lead (fee) and tin (bet) have orthogonal branches, and primary dendrites are defined as those growing in the general freezing direction, starting near the chill. Secondary branches grow from primary dendrite stalks, and are therefore perpendicular to the freezing direction. Tertiary branches growing perpendicular to secondary branches and parallel to primary stalks can form at large primary spacings (over approximately 200 ym). In lead rich Pb-Sn alloys, the dendrites may contain up to 19% Sn, and the interdendritic regions 62% Sn, the eutectic composition, which forms a fine lamellar structure. 2.4.1 Primary dendrite spacings When sectioned perpendicular to the growth direction, primary dendrites are seen to form the close packed arrangement, shown schematically in Figure 2. The distance between the centres of nearest neighbours is the primary spacing X. One method of measuring this spacing from a section perpendicular to the growth direction is to mark all the dendrite centres on a photograph, and then count the number of centres (n) in a given area (A). The spacing is 2 then given by A A/n. A second method is to measure the distance between the centres of primary stalks on a longitudinal section (parallel to the growth direction). The stalks can be recognized when secondary branches are visible on both sides. The spacing measured in this case will be A' in Figure 2, from which the nearest neighbour spacing can be calculated (A = A'//2). 12 FIGURE 3: (a) Longitudinal section of a directionally solidified casting, dendrites in region A are vertical, and in region B are tilted. (b) Corresponding cross section. Magnification 18x. 13 It was considered important in the present work that the primary dendrite spacings be measured from sections perpendicular to the growth direction* so that the ends of the castings could be polished before the flow tests, and the dendrite spacing of each could then be checked non-destruct-ively. Difficulties were encountered using the method of marking centres because dendrites are not always clearly revealed in cast structures. In particular, difficulties arise with alloy compositions close to the solubility limit of the primary phase, because the small amounts of eutectic present in the cast structure do not always outline the dendrite branches completely, making it difficult to distinguish between stalks and branches. This can be overcome in some cases by using an etchant which reveals composition gradients in the primary phase; however, an etchant of this type was not found that would work reliably on Pb-Sn alloys. Figure 3 shows the cast structure on sections parallel and perpen dicular to the axis of a directionally cast cylinder of Pb-20% Sn. Vertical primary stalks can be seen in the centre of the longitudinal section marked A. The corresponding area on the cross section is marked A'. On either side of the region marked A the primary stalks are tilted away from the vertical, though it is difficult to determine the angle since they are also tilted with respect to the plane of section. An example of such a region is B on the longitudinal section. The corresponding area on the cross section B' shows a periodic pattern of lines (in this case the lines are about 45° left of vertical), and the other areas on the cross section which correspond to tilted dendrites also show a similar pattern of lines. Although it would be most logical to make measurements from the 14 region A', the enlarged view of this area in Figure 4(a) shows the diffi culty in distinguishing between stalks and branches. Although one can see individual dendrites, it is frequently difficult to identify the centres of the neighbouring primary stalks. It is easy, however, to measure the line spacing in region B', shown enlarged in Figure 4(b), for a comparatively large number of lines. A composite of the microstructures was constructed from three orthogonal sections through region B', and when the model was turned the angle of the primary stalks became apparent. A photograph of the composite (Figure 5) is shown oriented such that the primary stalks in the corner of the cube are normal to the plane of the paper. For this example, it was found that the primary stalks were tilted approximately 20 from the vertical, in a plane perpendicular to the direction of the lines. It was found that the pattern of lines is only seen when the dendrites are tilted between approximately 10° and 30° from the vertical. The lines probably appear because the plane of section passes through secondary branches at an appropriate angle. Figure 6 shows a schematic view of the tilted dendrites (secondary branches are slightly elongated in the plane of tilt). The line spacing L can be used to calculate the primary dendrite spacing since X = /2Lcos6, where 0 is the angle of tilt. Since 6 is between 10° and 30°, cos8 is between 0.98 and 0.87, therefore the use of an average value of 0 = 20° introduces an error of less than 8%, which is less than the error involved in measuring L, which is no better than ± 10%. Thus the formula used in this work was X = 1.3L. 15 FIGURE 5: Three dimensional composite, from which one can estimate that the dendrites in the top corner are tilted FIGURE 6: Schematic view of the structure in Figure 4(b) approximately 20 16 Since primary spacings were measured from sections perpendicular to the growth direction, no distinction could be made between primary and tertiary dendrite arms. 2.4.2 Secondary dendrite spacings Secondary spacings were determined by measuring the spacing between a large number of clearly delineated secondary arms on polished sections parallel to the growth direction. This could not be done non-destructively for the columnar castings, and therefore the measurements were made on samples produced under the same cooling conditions as those used in the flow tests. 2.5 Autoradiography Polished specimens (down to 5 micron alumina) which contained radioactive tracer were placed flat against X-ray or orthochromatic film to 204 make autoradiographs. For a concentration of 500 ppm Tl (irradiated to a specific activity of 5 millicuries/gm) satisfactory exposures were obtained in 16 hours on X-ray film, or approximately 14 days on orthochromatic film. Although exposures were long for orthochromatic film, the resolution was appreciably better. All the autoradiographs in this thesis are printed so that dark areas indicate the presence of radioactive material. 17 CHAPTER 3 THE MEASUREMENT OF INTERDENDRITIC FLUID FLOW RATES 3.1 Review of Previous Work The first direct measurements of interdendritic fluid flow rates (1 2) were reported by T.S. Piwonka ' from experiments on Al-4.5%Cu alloys. Samples of the molten alloy were poured into a U-tube and allowed to solidify. They were then reheated to the testing temperature between the solidus and liquidus, where the interdendritic liquid in the alloy sample was expelled by applying pressure to one branch of the U-tube. The fluid flow rates were calculated from the time taken for the displaced liquid to make contact with a probe in the other branch of the U-tube. Two methods of applying pressure were used. In one case the inderdendritic liquid was displaced using an inert gas (nitrogen), and in the other case liquid lead was used in addition to gas pressure. Piwonka acknowledged that surface tension effects at the liquid-gas interface may have caused the gas displacement results to be unreliable, since the pressure required to force gas into the interdendritic regions might have been a significant proportion of the total pressure required to expel the inter dendritic liquid. An approximate estimate of the magnitude of this effect can be made as follows: The surface tension (a) of liquid aluminum is 520 dynes/cm at 750°C (surface tensions for Al-Cu alloys are not readily available). Assuming an interdendritic channel size of 20ym diameter, the pressure required to force gas into such a channel would be equal to the pressure 18 required to blow a hemispherical bubble of radius r = 10 urn. P = |2. = 1.04 x 106 dynes/cm2. According to Piwonka's thesis, the applied pressures in the 4 6 2 nitrogen gas experiments were in the range 3x10 to 1.8 x 10 dynes/cm . If the assumed channel size is reasonable, the surface tension could possibly account for all the resistance to flow that was observed. Indeed, the increasing resistance with decreasing temperature might be due to the increase in the pressure (P) with decrease in the radius (r). Since the aim of the experiments was to determine the resistance to flow caused by fluid drag within the interdendritic channels, the gas displacement results must be considered completely unreliable. The use of liquid lead instead of gas would reduce the surface tension effect and give a better measure of flow rate. However, in a real solidification situation, the flowing liquid will react with the dendritic solid, and this interaction would probably have an important effect on the measurements. This would not be the case when lead is used in the Al-Cu system, since aluminum is insoluble in lead. Piwonka examined some of the alloy samples after testing to ensure uniform penetration of lead into the interdendritic regions; however, no measurements relating to the structure were given. This is an unfortunate omission, since the structure was assumed to be constant in all the calcula tions. He acknowledged that preferential channels were formed by the inter-19 dendritic liquid as it was being displaced in some of the higher temperature tests, which would suggest that uniform flow did not always occur. Piwonka believed that his results were consistent with a model of the solid-liquid region which considered the region to be a bundle of straight capillary tubes. This could be demonstrated by plotting the logarithm of the permeability of the alloy, which is the reciprocal of the resistance to flow (defined in more detail in subsection 4.1 of the present work) against the logarithm of the liquid fraction, calculated from the Al-Cu phase diagram. A straight line of slope 2 should be obtained if the model is applicable (the theoretical relationship is derived in subsection 4.5.1 of (2) the present work). Piwonka's results of liquid lead displacement in Al-Cu are replotted in Figure 7 as permeability versus the square of volume fraction liquid (from the thesis, it appears that Piwonka incorrectly used weight fractions instead of volume fractions). The results agree fairly well with a straight line, for liquid fractions less than about 0.3, and the con clusion can be drawn that the solid-liquid region can be treated in this simplified manner, i.e., as a bundle of straight capillary tubes, within this range, provided the assumption of constant structure is correct. The results of this investigation have led a number of workers to explain certain effects in solidification in a semiquantitative manner, based on the standard equations for flow through porous media. The explanations are semiquantitative in the sense that numerical values for the parameters which describe the structure of the porous medium must be assumed, since they cannot be obtained from Piwonka's experiments. 20 FIGURE 7: Permeability as a function of the square of the volume fraction liquid, using experimental data obtained by Piwonka^^ . 21 Thus Piwonka' ', Campbellw uy and Tien*'''' used these results in theoretical predictions of hydrostatic tensions which could lead to pore formation during solidification. Standish used the results to argue that A segregate formation is not caused by interdendritic fluid flow, in contrast (9) to Mehrabian et al. who have formulated a comprehensive macrosegregation theory which can be used to semiquantitatively explain the formation of A segregates, based on Piwonka's findings. Attempts to measure interdendritic fluid flow in the Pb-Sn system were made by Kaempffer^^'^\ who observed the formation of droplets on the bottom surface of an ingot with a thin layer of eutectic material placed on top, when it was heated above the eutectic temperature. Radioactive tracer was added to the eutectic layer, and the aim of the experiment was to measure the flow rate by measuring the activity of the droplets. Kaempffer found that it was not possible to produce uniform inter dendritic flow with this experiment. At the eutectic temperature no flow was observed, but as the temperature increased drops began appearing on the bottom surface of the ingot. These would coalesce before falling, and, with a constant rate of heating, the first drop would fall at about 230°C, and the remainder would follow rapidly. Examination of polished and etched sections of the ingot showed that one or two wide channels had formed, and autoradio graphs showed that fluid flow was mostly confined to these channels. Kaempffer interpreted his results as follows; as the temperature of the ingot was increased above the eutectic temperature, the lead-rich dendrites became soluble in the superheated eutectic liquid, therefore, as 22 material from the top layer began to flow down, it was able to dissolve dendrite branches in its path, forming the wide channels. This work was therefore not representative of uniform flow through porous media, and it remained to be shown whether one could produce uniform flow of interdendritic liquid which was not superheated. 3.2 General Description of the Technique Used in the Present Work Measurement of the permeability of a packed bed is often done using a Falling Head Permeameter (Figure 8). This consists of two concentric tubes with the porous material packed in the inner tube. A static head of fluid in this tube will cause flow through the porous bed and up the space between the two tubes. The permeability K can be calculated from the time required for the fluid head (h) to fall a given amount, using an integrated form of Darcy's Law t = - f ln(ht/ho) where c and K are constants, h is the head at time t, and h is the head ' t ' o at t = 0. This equation is derived in detail in Appendix I. The same principle is used in the design of the flow cell for measuring interdendritic fluid flow (Figure 9). In this case the two tubes are side by side instead of concentric. The flow cell was made from four pieces of brass and resembles a split mould. This design was chosen so that the various pieces of the alloy under test could be assembled before the test and removed afterwards without damage. The brass was completely covered with a thin coating of graphite (Aquadag) which prevented contact with the molten alloy and also prevented leaks. Steel screws were used to hold the pieces Outflow Porous Bed Inner Tube Outer Tube 23 "TV FIGURE 8: Schematic diagram showing the principle of a Falling Head Permeameter. argon E u Pb-55Sn Pb-20Sn Pb-55Sn ,2-35 cm dio FIGURE 9: Sectional views of the flow cell and the lead-tin alloy inserts (to scale). 24 together. Brass was chosen for its machinability, strength and thermal conductivity (0.27 cal/cm sec°C). High conductivity was desirable to ensure isothermal conditions within the flow cell, and brass was found to be satis factory by making temperature measurements at various locations inside the cell while it was being heated. Copper would have provided higher conduc tivity (0.88 cal/cm sec°C), but it was difficult to machine to the complex shape of the flow cell, and from previous experience it was found that threads tapped in copper did not hold after repeated heating and cooling. To carry out a flow test three pieces of lead-tin alloy were inserted into the cell (Figure 10). The partially assembled flow cell with the lead-tin alloy is shown in Figure 11. A cylindrical casting A of the alloy under test was placed in the appropriate cavity, and two other castings, B and C, of different composition, were placed above and below it. The compositions of these three pieces of lead-tin alloy were chosen such that, at the testing temperature, both B and C would be liquid, and the casting A would be partially liquid. Thus there would be a hydrostatic pressure in the solid-liquid region through A, and the liquid level would tend to fall on the left, and rise in the smaller diameter 'riser' pipe on the right. The level of the liquid metal in the riser could be measured at any time using a copper wire probe, which closed a circuit on contact. Therefore, under known conditions of pressure, temperature, liquid composition and dendrite spacing, graphs of height of liquid in the riser versus time were plotted, from which the permeability could be calculated. At the end of the test the flow cell was chilled and the alloy was removed. Figure 12 shows the alloy before and after testing. The alloy could FIGURE 12: Pb-Sn alloy before and after flow test. TABLE I DIMENSIONS AND COMPOSITION OF CASTINGS USED FOR INTERDENDRITIC FLUID FLOW STUDIES Diameter cm Length cm Composition Comments Casting A 2.46 3.37 Pb-20%Sn Casting B 1.91 0.76 Pb-55%Sn 204 Approx. 500 ppm Tl added in certain tests as tracer Casting C 1.91 0.64 Pb-55%Sn This casting was made using the flow cell as a mould 27 then be examined by sectioning and polishing. Autoradiography of these sections was used to observe the fluid flow directly in some of the castings 204 where radioactive Tl was added as a tracer. The compositions and dimensions of A, B and C are given in Table I. 3.3 Preparation of the Alloy under Test (A) In most of the tests the cylinders of alloy (A) were columnar castings. For the range of primary dendrite spacings from 28 to 83 microns these were produced by remelting the required weight (255 g) of starting ingots in a vertical graphite mould and chilling from the bottom. The furnace, mould and chilling arrangement are shown in Figure 1(a). When the alloy was molten, an iron-constantan thermocouple was placed in the melt and the temperature was adjusted to the required level by adjusting the furnace controller. When the alloy reached a constant temperature (checked by moving the thermocouple around in the liquid) the thermocouple was withdrawn and the alloy solidified unidirectionally from the bottom. Different cooling rates were produced by using either a blast of nitrogen or a constant pressure of water against the chill, and also by changing the size of the nozzle and the thickness of the chill. Careful control of the temperature of the melt and the cooling conditions made it possible to reproduce directional castings with a given dendrite spacing within the precision with which the spacing could be measured (approximately 10%). Four different quenches were used, and the details are given in Table II. 28 Cooling curves at three positions in the casting were plotted by inserting iron-constantan thermocouples in the melt and solidifying. These curves were taken to be representative of the actual castings held at the same temperature and quenched in the same manner. Table III lists the thermal conditions for the four different quenches used. The chill face cooling rate was calculated from the cooling curve of a bare thermocouple in the liquid, placed in contact with the chill. The cooling rate is taken from the slope of the cooling curve at the liquidus temperature of the alloy. The freezing rate is given at two points and is calculated from the estab lished relationship ^^'^"^ x = A/t where x is the distance from the chill, t is the time elapsed from the start of freezing, and A is a constant. The freezing rate is therefore equal to x/2t. The mean primary dendrite spacing is also listed in Table III. Columnar castings with dendrite spacings larger than 83 microns were prepared by cooling the alloy very slowly under a shallow temperature gradient using the apparatus described in section 6.2. Ingots 2.5 cm in diameter and approximately 12 cm long were produced in graphite moulds that were lowered through the two zone furnace. Each of these ingots was machined to produce cylindrical samples with the dimensions of casting A. In'addition, equiaxed castings of different dendrite spacings were produced by pouring molten alloy into a simple graphite mould, 2.5 cm in diameter and approximately 5 cm long, using different mould preheats and alloy superheats. 29 TABLE II  QUENCH DATA Quench Coolant* Pressure of Coolant Thickness of Chill (cm) Nozzle Dia.(cm) Al N2gas 35 lb/in2 1.98 0.31 A2 N2gas 35 lb/in2 0.88 0.57 Wl water 23.5 in. head 1.98 0.31 W2 water 23.5 in. head 2.86** 0.31 * Coolants were at room ** Two copper discs with total temperature. thickness of 2.86 cm were used. TABLE III THERMAL CONDITIONS FOR Pb-20%Sn COLUMNAR CASTINGS Quench Temp. of melt(°C) Chill face Cooling Rate* (°C/sec.) Freezing Rate A B (cm/sec) (cm/sec) Av. primary dendrite spacing (microns) Al 310 0.29 0.015 0.014 71 A2 310 0.18 0.015 0.016 83 Wl 310 6.00 0.042 0.037 28 W2 310 0.68 0.015 0.019 51 A = B = 1.31 cm from chill 2.77 cm from chill * Measured at the liquidus temperature. 30 After cooling to room temperature all castings were machined to the dimensions given in Table I. The ends were polished, etched and examined microscopically to determine the dendrite spacing, and before testing in the flow cell the ends were again polished and cleaned to remove oxide or other extraneous material that might interfere with fluid flow. 3.4 Preparation of Castings B and C The upper cylinder B was machined from starting ingots of the required composition. In the majority of cases approximately 500 ppm of 204 radioactive Tl was dissolved in the alloy. After testing, fluid flow patterns were obtained from autoradiographs of sections taken from samples containing radioactive tracer. The lower casting C was made by using the lower part of the flow cell as a mould which was preheated before pouring alloy of the required composition (radioactive tracer was not added to the lower castings). After cooling, the flow cell was dismantled and the castings were removed and machined to the required length. The .composition of castings B and C used throughout this work was Pb-55%Sn. The choice of composition was based on Kaempffer's exper ience^^ using eutectic material (62%Sn) as the liquid reservoir above the casting. Since the main requirement in this work was that casting B should not become superheated, an off-eutectic alloy was found by experience to be more suitable. The particular composition chosen was found to work best for permeability measurements approximately 10°C above the eutectic temperature. Problems of preferential channelling, similar to those 31 encountered by Kaempffer, arose when attempts were made to use different alloy compositions to measure permeabilities at higher temperatures. 3.5 Flow Measurement Equipment Flow measurements were made as liquid metal rose up the 'riser' pipe of the flow cell, shown in Figure 9. In most experiments the actual distance involved was only 3 cm or less, therefore it was essential to hold the furnace, flow cell and the measuring probe firmly in position, so that accurate measurements could be made. A schematic diagram of the apparatus is shown in Figure 13. The flow cell was held in position inside the furnace from the metal tube which was also connected to the argon supply. The measuring probe consisted of a copper wire in a ceramic tube which was inserted in the flow cell down the riser pipe. When the probe touched the surface of the liquid metal an electric circuit was closed giving the position of the interface. The probe was attached to a long feed screw and crank handle so that it could be accurately positioned at predetermined intervals (usually 0.5 mm) and the time required for the liquid to make contact could be measured. The position of the probe was given on the dial gauge, which was accurate to ± 0.0125 mm. Thus up to 50 data points of distance versus time could be obtained as the flow took place. The temperature of the flow cell was continuously monitored on a chart recorder during the test by means of an iron-constantan thermo couple in a 3 mm diameter glass sheath. A simple circuit (shown in Figure 14) was used to connect the measuring probe to the thermocouple 32 Probe M Flowcell Lamp O 100 K I K —'I 1 |->AMM^\AAN-| .022/zF T/C FIGURE 14: Circuit used for recording the position of the probe on the tempera ture trace. Recorder 33 wires, so that at the instant when contact was made a blip was produced on the temperature trace. Chart speeds between 10 min/in and 10 sec/in were used in this work, so that the time required for liquid to rise between successive positions of the probe could be measured to an accuracy of ± 0.1 sec (if necessary). It should be noted that the circuit merely produced a blip on contact, but did not otherwise alter the thermocouple signal (however a sheathed thermocouple was necessary to electrically insulate the tip, in this case). The equipment described above, which was used for 22 of the total of 30 tests which were done, in fact evolved gradually since the early tests showed that deviations from Darcy's Law occurred in some of the flow tests. It was felt that the nature of these deviations would be better understood if more accurate equipment was used. Early methods of measuring the time when the probe touched the liquid metal involved photo graphing an electronic timer at the same instant that a lamp showed contact had been made. This method was potentially just as accurate and reproducible as the method described previously, however it required the undivided attention of the experimenter over long periods. The position of the probe was measured using a pointer and scale which was less precise than the dial gauge. The results from all the tests were used to determine the relationship between permeability and dendrite spacing, since it was subsequently found that the results from the early tests lay within the observed scatter. However, only the results from the later tests were used in the study of deviations from Darcy's Law. 34 3.6 Flow Testing Procedure Before assembly, the faces of the castings which were to be placed in contact were painted with a thin layer of a soldering-type flux paste, to ensure that when castings B and C became liquid they would completely wet the end surfaces of casting A. The flow cell was assembled with the Pb-Sn inserts, and placed inside the vertical tube furnace. The central tube was connected to an argon supply (1000 ml/min) to provide an inert atmosphere inside the cell which would prevent oxidation that might inhibit fluid flow. The measuring probe was inserted down the riser pipe until contact was made with the surface of the branched portion of casting C. This established the datum level for measuring fluid pressures, and the dial gauge was set to zero for this point. The probe was then moved up the required amount in preparation for flow measurements. The sheathed thermocouple was inserted in the appropriate hole in the flow cell. The power supply to the furnace was adjusted to heat the flow cell rapidly (approximately 5°C/min) to the required temperature, without overshooting, and once this temperature was attained the auto matic controller held the cell at constant temperature while flow measurements were made. It was found that manual control of the power supply was the most effective method of heating the cell in the initial stages, and a reproducible procedure could be developed after two or three 'dummy' trials. The zero point for timing measurements was taken as the instant when the temperature reached 183°C, the eutectic temperature, 35 i.e., the instant melting would be expected to begin. Flow took place relatively slowly, and in the majority of cases the temperature of the cell had stabilized at 193°C by the time the first measurements were made. In those cases where the temperature had not stabilized within ± 3°C of the required temperature, the points were not used until the temperature had stabilized. When the required number of data points had been measured, the flow cell was chilled, either by lowering into a water bath or by using a 50 Psi air blast. In the latter case solidification was complete 45 sees after the air blast was turned on. After cooling to room temperature the flow cell was dismantled and the total height of the Pb-Sn sample was measured. Comparing the height of the column in the riser pipe after cooling, to the measured height when liquid, it was found that thermal and solidification contraction caused a reduction in length of 9% between the testing temperature and room temperature for this composition. This information was therefore used when calculating the height of liquid from room temperature measurements. The majority of the samples were subsequently sectioned both at right angles and parallel to the axis of the cylinder A. Microexamination and autoradiography were used to determine the fluid flow paths and the effect of flow on the micro-structure. 3.7 Precision of the Flow Measurement Technique The method of using a copper wire probe to locate the position of the surface of a rising column of liquid metal was first tested by using the 36 Motor Contact Guide Tube Mercury Probe Glass Tube (7.9 mm ID.) Flexible Tube FIGURE 15: Apparatus for testing the precision of the flow measurement technique. TABLE IV PRECISION OF THE FLOW MEASUREMENT TECHNIQUE No. of observations flow rate (cm/sec) std. error (cm/sec) 95% conf. interval (cm/sec) 95% conf. interval (pet) 20 0.004397 2.06 x 10"5 ± 4.33 x 10"5 ± 1.0% 45 0.004430 0.73 x 10"5 ± 1.47 x 10~5 ± 0.3% 48 0.004417 0.66 x 10~5 ± 1.33 x 10"5 ± 0.3% 50 0.004420 0.68 x 10"5 ±1.37 x 10"5 ± 0.3% 37 the apparatus to measure the flow rate of a rising column of mercury. A constant flow rate was imposed by raising one branch of a flexible U-tube using a low speed synchronous motor (12 rph). The U-tube contained mercury, and the fixed branch was made the same diameter as the riser tube in the flow cell. A schematic diagram of the equipment is shown in Figure 15. Four experimental runs were done ranging from 20-50 observations. Points were taken approximately every 0.75 mm as the mercury rose. The method of least squares was used to fit a straight line to the measured values of distance and time. The flow rate, standard error of the flow rate, and the 95% confidence intervals were calculated and are listed in Table IV. From these tests one may conclude that the error in the flow measurement technique was ± 0.3% over 50 observations for a constant flow rate. 38 CHAPTER 4 RESULTS AND DISCUSSION OF FLOW MEASUREMENTS 4.1 Interpretation Using Darcy's Law The experimental technique supplied data on the distance (&) of fluid flow up the riser pipe versus time (t), and two typical plots are shown in Figure 16. The flow velocity at any point is given by the slopes of the curves, and it is clear that there is approximately an order of magnitude difference between the initial velocities for these two dendrite spacings. The curves show that the flow velocity decreases with time for the larger dendrite spacing, yet it remains fairly constant for the smaller spacing. The results were interpreted by considering the casting A to be a porous medium which obeys Darcy's Law. The classical experiment performed by Darcy in 1856 consisted of measurements of the quantity of water flowing through a sand filter bed. The quantity was found to be directly propor tional to the pressure drop, and inversely proportional to the length of (14) the bed. From dimensional arguments one can deduce the following relationship: v 4.1 where v = bulk velocity of the fluid (measured over the whole area) K = permeability of the porous medium viscosity of the liquid L = length of the porous medium AP = pressure drop 39 o m CNJ E o o — cvi uJ O < -00 5 ° o AVERAGE PRIMARY DENDRITE SPACING 116 microns COLUMNAR 460 600 800 TIME (seconds) 200 1000 FIGURE 16(a): Flow measurement results; distance of flow up the riser pipe versus time for A = 116 ym. Initial slope = 0.0055 cm/sec. o. in (VI So — CVl UJ < Is m 6 AVERAGE PRIMARY DENDRITE SPACING 28 microns COLUMNAR 3000 4000 5000 TIME (seconds) Similar plot for X = 28 ym. Initial slope =0.00039 cm/sec. 1000 2000 6000 7000 FIGURE 16(b): 40 The permeability K is a property of the porous medium and has 2 the dimensions of area (cm ). The minus sign in the expression indicates that flow is in the opposite direction of increasing AP. Darcy's Law has been verified experimentally for flow through many types of porous media, and Carman^"^ has stated that there is good reason to believe that it can always be applied under the following conditions: i) the flow must be laminar ii) the fluid must be inert to the porous medium, i.e., chemical, adsorptive, electrical, electrochemical and capillary effects are absent. 4.1.1 Laminar flow Laminar flow is related to the Reynold's number, a dimensionless group defined as Re = i£i M where V is the (scalar) velocity measured over the whole area of the bed, p is the density of the fluid, u is the viscosity of the fluid, and 6 is a diameter associated with the porous medium, i.e., the average particle or pore diameter, or some length corresponding to the hydraulic radius theory. The representation of the flow by means of the Reynold's number is therefore dependent on the choice of the length 6, which in turn is dependent on the model chosen to describe the porous medium. Many investigations have been directed towards finding the critical Reynold's number where flow through the bed ceases to be laminar. These have 41 been reviewed by Scheidegger, and the range of values reported for the critical :Reynold's number lies between 0.1 and 75. Scheidegger has i commented that the uncertainty of a factor of 750 is probably related to the fact that 6 is not clearly defined, and he points out that the difference between these values and the Reynold's number of ZOOO^which is normally taken as the critical value for turbulent flow in straight tubes, makes the Reynold's number concept somewhat doubtful when applied to porous media. Nevertheless, experiments have shown that the critical range exists, therefore, to check for laminar flow in the present work, the following simple approach was adopted. The maximum observed velocity was calculated from Equation 4.1: K , v = - — pgh —8 2 where K = 8.2x10 cm when X = 175 ym and g - 0.2 Li y = 0.03 poise L = 3.37 cm 3 P = 8.33 g/cm 2 g = 981 cm/sec h(max) = -4.51 cm i.e. v = 0.03 cm/sec As a first approximation, <5 is taken, as equal to the primary dendrite spacing X, then 42 R = ^ e u X = 175 x 10~4 cm u = 0.03 poise i.e. R = 0.14 e This value is approximately equal to the lowest published estimate of the critical Reynold's number. However, it is reasonable to assume that the value of 6 chosen is a conservative estimate, since the effective diameter of flow channels or particles (depending on the model chosen) is likely to be much less than X. Flow is therefore considered to be laminar in all the tests done in the present work, and the first condition is satisfied. 4.1.2 Interaction effects As well as an upper limit to Darcy's Law, there are a number of references in the literature to a lower limit. Carman reviewed the early observations of this behaviour in 1937^^\ and drew the conclusion that the deviations were related to surface forces between the solid and liquid. This has remained the general consensus since then, and the list of possible surface effects reviewed by Scheideggerincludes surface tension, adsorption and molecular diffusion. Electrochemical effects have (18 19) been of interest recently, and work has been published ' explaining the effects in terms of an electrical double layer. Since liquid metals are not ionic, this would not be relevant to the present work. There is no doubt that the interdendritic liquid will interact chemically with the dendrites, consequently deviations from Darcy's Law will 43 be interpreted in terms of the interaction effects. It is assumed that because the dendrites form a rigid network, interaction effects over short time periods will not cause a general collapse of the structure. Therefore it should be possible to use the data from these experiments to draw some conclusions regarding the nature of the effects which cause the deviations. 4.2 Application to the Flow Cell Experiments For the experiments done in the flow cell AP varies with time, therefore Equation 4.1 can be expressed in the following form t - -| ln(ht/ho) 4.2 where h is the head of liquid at time t, h is the initial head and c is t o a constant. The complete derivation of Equation 4.2 is given in Appendix I. The distance versus time data were therefore replotted as In (ht/ho) versus time. When the permeability K is a constant, these plots should be linear, with a slope equal to -c/K. The data from Figure 16 have been replotted in this manner in Figure 17, and in both cases the plots deviate from linearity. The castings were examined metallographically to investigate the reasons for these deviations and the results will be described in detail in a later section (4.5). In both cases the plots show that the mechanisms which caused the deviation are time dependent, therefore the permeability has been estimated from the initial slopes. It is clearly a problem to decide how many points contribute to the initial slope, and at which point the data begin to deviate from linear-FIGURE 17(b); Similar plot for data from Figure 16(b), showing a negative deviation. 45 ity. A statistical argument was therefore developed, and the straight lines in Figure 17 are the best estimates of the initial slope by this method. 4.2.1 The method for finding the initial permeability The tests described in section 3.7 to establish the precision of the flow measurement technique can be used to separate the random experi mental errors inherent in the technique from the systematic effects which cause the deviations from linearity. The method of least squares can be used to estimate the rate of flow of the mercury from the data in section 3.7. Since the flow velocity was constant, the observed scatter was only due to experimental errors in using the copper wire probe to locate the mercury surface. Therefore the data was fitted to a line y = mx where y is the dependent variable (position of the mercury surface) and x is the independent variable (time). The slope of the regression line (m) would therefore be the velocity. The scatter of points about the best fit line is described by the standard error of Y (°y) where °Y = V n-2 (y.-y.) is the difference between the value of the ith point y_. and the 46 value y_^; estimated by the regression line, in other words, the error; and (n-2) is the number of degrees of freedom. Therefore a is a measure of the "goodness of fit" of the data. However, the data from the flow cell experiments are plotted in the form of ln(h /h ) versus time, therefore one must consider a plot of to •. •» . ln(l-y^) versus x_^ instead of the simple y^ versus x^, where hQ, a2 and a^ are constants defined in Appendix I. (From the numerical values of the constants, it follows that y! is normally less than 1.) Similarly, the fitted value is y!^ where The error in the dependent variable, e^, plotted in this manner is: ei - ln(l - yj) - ln(l - v[) It is well known that when data is plotted on a logarithmic graph the points become weighted, in other words, the error e is a function of position along the line. One can take this effect into account by including a weighting factor w^. A suitable weighting factor can be calculated as follows: Let y! = z, and y' - y! = 6z J1 . Ji ^ l 47 then ln(l - y\) = ln{l - (z + 6z)} Since -1<z<1, the logarithm may be expanded 2 3 1 fi / _. * \ \ / • r \ (z + 6z) (z + 6z) lnil - (z + 6z) } = (z + 6z) - ^— - 3— z2 6z2 z3 = z + 6z - — - z6z - —- — Similarly ln(l - yl) = ln{l - z} 2 3 and ln{l-z} = z - ~ y But e± = ln{l - (z + <5z)} - ln{l - z} 2 <5z is small, therefore one may neglect 6z and higher order terms. Also, 2 since -1<z<1, z &z and higher order terms will be small, therefore as a first approximation e. - &z - z6z x i.e. e = (1 - z)6z When z = 0, ln{l - (z + 6z)} - ln{l - z} = ln(l - 6z) = 6z „ j.. . error at z Defxnxng w. = w i error at z = 0 (1 - z)6z i 6z i.e. wi = (1 ~ z) The standard error of Y, which estimates the "goodness of fit" of the logarithmic plot is ZCw^..)2 n-2 48 The value of has been calculated for the constant velocity data from the test in Section 3.7. It represents the "goodness of fit" one would expect if one were to do a least squares fit on this data, and the only errors were random experimental errors. This value may now be compared with the value of Oy for a given number of points from a flow cell experiment. Using a standard test of significance, the F test, one can decide whether the observed scatter for the flow cell data is larger than one would expect if it were due to experimental errors alone. The method used to calculate the best straight line is to start with the first six points from the flow cell data and fit a straight line on the ln(ht/hQ) versus time graph using the method of least squares, and calculate Oy and the variance (equal to a^). Using the F test with a significance level of 0.05, one can say whether the scatter is greater than one would expect from random experimental errors alone. If the scatter is less, seven points would be taken, and a new straight line fitted and a new Oy calculated. This procedure would be repeated with eight points etc. until the scatter is greater than expected for this significance level. In other words, this approach finds the largest number of points which contribute to the straight line portion of the graph within the experimental error associated with the technique. The slope of this line is considered to give the best estimate of the initial permeability. One can also say with certainty that the deviations seen (such as those shown in Figure 17) are due to effects other than random errors, or weighted errors associated with a logarithmic plot. Because of the large amount of data associated with each flow test, the calculations were done on a digital computer which also provided 49 plots showing the best fit line according to the above method. The FORTRAN program for processing the results is given in Appendix II. 4.2.2 Results The results of the permeability calculations are given in Table V. Deviations from Darcy's Law are considered to be positive when the values of ln(ht/hQ) are larger than would be expected, as in Figure 17(a), and negative for the reverse (Figure 17(b)). This means that for a positive deviation the flow rate becomes slower than predicted by Darcy's Law, indicating that the flow is becoming impeded, and for a negative deviation the flow rate is more rapid, indicating that the flow channels are possibly becoming larger. Those experiments where the deviation is listed as zero in Table V are either early experiments which, because they were less accurate, were not used to study deviations, or the resulting plot did not show a clear trend in either direction. From the table, one can see that positive deviations only occur when the dendrite spacing is greater than 71 microns, and negative deviations only when the spacing is less than 51 microns. The height of liquid in the riser pipe when deviations begin is given in the table, and it can be seen that negative deviations begin in a range 4.5 to 12.7 mm, whereas positive deviations begin in a range 9.1 to 22.8 mm. In general, one can conclude that positive deviations start later than negative deviations (also seen in the two examples in Figure 17). The drop in the liquid level in the reservoir when deviations begin has also been calculated and is given in Table V. The calculations TABLE V RESULTS OF FLOW MEASUREMENTS Average Dendrite Structure Initial Tracer Deviations Height of Distance fal Total Spacing (ym) Permea Used? from liquid in len in reser time at Primary Secon bility q Darcy's Lav riser when voir when temp. dary (K x 10 (+ or -) deviations deviations (hrs) 2. cm ) begin (mm) begin (mm) 28 23 COLUMNAR 0.199 (R) 0 1.07 0.239 (R) 0 1.08 " 0.105 - 7.3 2.4 2.52 " " 0.152 - 7.3 2.9 2.13 " 0.156 - 8.3 2.7 1.67 0.136 - 7.6 3.2 2.07 48 21 EQUIAXED 0.0569 (R) - 8.1 1.9 3.81 51 33 COLUMNAR 0.367 _ 12.7 3.4 1.18 II II II 0.346 0 0.72 II II II 0.244 - 4.5 1.9 1.32 II II II 0.407 - 8.5 2.7 1.22 II II II 0.436 (R) 0 0.74 71 49 COLUMNAR 0.499 (R) 0 0.67 " 1.47 * (R) + 17.3 3.9 0.44 0.821 (R) + 12.2 3.4 0.63 II 1.27 * (R) + 17.8 4.0 0.53 " 1.17 * (R) + 15.7 3.8 0.54 1.49 * (R) + 13.2 3.7 0.44 77 - EQUIAXED 0.300 (R) 0 1.86 80 54 EQUIAXED 0.618 (R) + 9.1 2.9 1.19 TABLE CONTINUED TABLE V CONTINUED Average Dendrite Structure Initial Tracer Deviations Height of ;Distance fal Total Spacing (pm) Permea Used? from liquid in len in reser- time at Primary Secon bility Darcy's Law riser when ; voir when temp. dary (K x 10 (+ or -) deviations deviations (hrs) 2 cm ) begin (mm) begin (mm) 83 57 COLUMNAR 0.546 (R) 0 0.62 it II II 1.06 (R) 0 0.52 n II ti 0.780 (R) 0 0.43 ii II it 0.543 0 0.57 ii II II 1.21 * (R) 0 0.41 103 51 EQUIAXED 0.820 (R) + 11.7 3.1 0.60 116 57 COLUMNAR 2.15 (R) + 19.9 4.5 0.29 130 61 EQUIAXED 1.95 (R) + 20.0 4.4 0.32 175 83 COLUMNAR 6.27 (R) + 22.8 4.4 0.15 n II ti 8.20 (R) + 19.1 3.8 0.12 Autoradiography showed evidence of flow channelling, therefore these results are considered less reliable. (R) Radioactive tracer was added to the reservoir (casting B) in these tests. 52 were based on the final height of the Pb-Sn alloy after testing, there fore they do not necessarily correspond to the height of liquid in the riser pipe. This is because despite careful machining to fit the Pb-Sn castings to the flow cell, small spaces inevitably remained, and these were filled by the liquid before any flow was detected in the riser pipe. For this reason the curves in Figure 16 and 17 do not pass through t = 0. Negative deviations begin when the liquid has fallen between 1.9 and 3.4 mm, and positive deviations between 2.9 and 4.5 mm. The initial height of liquid in the reservoir was calculated as 8.4 mm. 4.3 Dendrite Spacings and Structure The relationship between primary dendrite spacing and distance from the chill was determined for columnar castings produced by the four different quenches described in Table II. The results are given in Figure 18, and they show that the dependence on distance is essentially linear, in agreement with previously published work on unidirectionally cast copper alloys j_n the latter work a linear dependence was also found for secondary arm spacings versus distance from the chill. Consequent ly, the average spacings could be determined non-destructively by taking the mean value for the top and bottom surfaces of the casting. Throughout this work, average primary and secondary spacings have been used as the parameters which characterize the structure of the casting with respect to interdendritic fluid flow behaviour. In these experiments, the bulk flow is one dimensional, and in the case of columnar castings, the flow is in the same direction as the primary dendrites. The columnar castings are placed in the flow cell so that the interdendritic channels 53 FIGURE 18: Primary dendrite spacing as a function of distance from the chill, for the quenching conditions in Table II. 54 vary in size only in the direction of flow, parallel to the axis of the casting. Therefore, visualizing the columnar dendritic structure as a stack of resistances in series, the flow rate will be a function of the sum of these resistances, or alternatively, a function of the average resistance. This reasoning would not necessarily hold for the equiaxed castings, nevertheless, the average spacings have been used because the spacing measured in different locations on the top and bottom surfaces was fairly uniform, and the difference between measurements on the two surfaces was relatively small. 4.3.1 Autoradiography Autoradiography made it possible to examine the Pb-Sn alloy samples after testing to determine whether the measurements which had been made were truly representative of uniform flow through the casting A. Previous experiments on interdendritic fluid flow by Kaempffer^^ showed that there is a strong tendency for superheated liquid to form preferential channels by dissolving dendrite branches, and if this occurred to the same extent in the present work it would invalidate the use of Darcy's Law. Those samples which contained radiocative tracer were sectioned at the mid-point of the cylinder A perpendicular to the axis, and the upper half was then sectioned parallel to the axis. Autoradiographs were made from the cross sections and longitudinal sections (examples are shown in Figure 19). In the majority of cases there was either uniform (Figure 19(a)) or no dark ening of the film (Figure 19(b)) for the cross sections^except for five of the tast3 (marked with an asterisk in Table V). The cross section autoradio-FIGURE 19: Autoradiographs from cross sections and longitudinal sections of Pb-Sn samples used for interdendritic fluid flow studies; (a) and (b) show uniform flow, (c) shows flow down a preferential channel. Magnification 2x. 56 graphs of these tests showed a dark spot (Figure 19(c)), and longitudinal sections showed a non-uniform penetration of tracer which indicated that flow had taken place preferentially in this region. Longitudinal sections for the majority of samples showed a relatively uniform penetration of radioactive material, and for large dendrite spacings the structure was revealed. For the smaller spacings the structure was not clearly revealed, probably because it was too fine to be resolved. Figures 20 and 21 show autoradiographs from cross sections taken at various levels for two of the samples. Figure 20 shows uniform penetra tion of tracer, and microexamination showed no evidence of pipes of the type seen by Kaempffer. For the sample shown in Figure 21 the tracer penetration is non-uniform, and it is also evident that radioactive material has pene trated much further down casting A. Microexamination revealed several small pipes close to the top which extended approximately 0.7 cm down. Enlarged views of one of the pipes are also shown in Figure 21 for the levels on which they were seen. The position of the dark patches on the autoradio graphs from lower sections showed that flow had taken place preferentially down the pipes, even though the pipes themselves were no longer visible on the lower sections. From this evidence it was felt that a single cross section and longitudinal section would provide sufficient information to determine whether the flow was uniform or not. Although pipes were seen close to the top of the cylinder A in the tests which showed non-uniform flow, channels which penetrated right through the dendritic region and caused catastrophic flow, as reported by Kaempffer, were never observed. Although it is felt that the permeabilities for the five tests marked with asterisks in Table V are less reliable than the others, 7 FIGURE 20: Cross section autoradiographs at various levels down the casting A, after testing. Tracer has penetrated uniformly as far as the third section, 4.6 mm from the top. ~ = 28 um. Magnification 1.5x. FIGURE 21: An example of an unre liable flow test, showing uneven pene tration of tracer. \ = 83 ym. Magnification of autoradiographs 1.5x. Microstructures from the bottom right hand corner of the top two sections, showing a pipe. Lower sections did not show this defect. Magnification 33x. 59 the results have nevertheless been included because it is of interest that they are larger than the more reliable values by a factor of between only one and two, for the same dendritic spacing. It is not completely clear why pipes should form in a few of the tests when the same procedure was used throughout. One can suggest that the flow cell was not positioned correctly in the furnace for these tests, which resulted in higher temperatures at the top. Since the samples were 2.46 cm in diameter, quenching rates were not as rapid as in some previous experiments where tracer was used to observe (21 22) convective flow patterns ' , therefore one might question whether quenching the flow cell affected the observed flow behaviour. In view of the following reasons, the autoradiographs are believed to be truly repre sentative of the nature of the flow behaviour (uniform or via preferential channels): 1) Both water quenching and air cooling were used, and despite the different cooling rates, there were no obvious differences in flow patterns between similar samples cooled in different ways. 2) The position of the casting A within the flow cell is off-centre (Figure 9), and would cause it to cool more rapidly on one side than the other, yet no patterns were observed which could be attributed to this effect. 4.4 Microexamination Microexamination of the Pb-Sn castings was directed towards finding the reasons for the deviations from Darcy's Law which emerged from the flow calculations. Longitudinal and cross sections from the centre 60 region of casting A before and after testing are shown in Figures 22-25. 4.4.1 Negative deviations from Darcy's Law For the largest dendrite spacing, 175 ym (Figure 22), there is no apparent difference between the structures before and after testing, however, for the smaller spacings, 71, 51 and 28 ym, the differences become more obvious as the spacing decreases. The interdendritic regions appear to coalesce to some extent to form a continuous network, and there is little difference in appearance between the cross sections and the longitudinal sections. The microstructures taken after testing also show an overall background of white dots which resemble spheroidized precipitates in other systems. These white dots exist on a much finer scale in the microstructures taken before testing, and are attributed to the formation of a dendritic (23) substructure, described in the literature The microstructures therefore indicate that a ripening mechanism is taking place, especially for the smallest dendrite spacings (which were held at temperature for the longest times), which increases the effective diameter of the flow channels with time. In Figure 25 the dendritic structure has changed to a type of cellular structure, and some flow channels appear to have grown at the expense of others. The flow paths also appear less tortuous, which would cause the flow velocity to increase. Negative deviations from Darcy's Law, which were observed to occur only when the spacing was less than 51 ym, are therefore attributed to this ripening mechanism. 61 Longitudinal Sections before flow after flow Cross Sections before flow after flow FIGURE 22: Microstructures of casting A before and after flow testing, t = 175 um, time at temperature 0.12 hours. Magnification 60x. 62 Cross Sections before flow after flow FIGURE 23: Microstructures of casting A before and after flow testing. }, = 71 um, time at temperature 0.44 hours. Magnification 60x. 63 Longitudinal Sections before flow after flow Cross Sections before flow after flow FIGURE 24: Microstructures of casting A before and after flow testing. X = 51 ym, time at temperature 1.18 hours, Magnification 60x. 64 Longitudinal Sections before flow after flow Cross Sections before flow after flow FIGURE 25: Microstructures of casting A before and after flow testing. A = 28 um, time at temperature 1.67 hours, Magnification 60x. 65 4,4.2 Positive deviations from Darcy's Law To account for positive deviations from Darcy's Law the micro-structures were examined for evidence that the flow channels for the larger dendrite spacings were becoming more constricted with time. However, no evidence of this type was found. As in Figure 22, these castings were held at temperature for a relatively short time, and showed little change after wards. Microexamination of the top reservoir, which was originally casting B, revealed a structure which probably accounts for the positive deviations. Three examples of this structure are shown in Figure 26. The microstructures are from longitudinal sections through the reservoir, with the casting A at the bottom. It can be seen in Figures 26(a) and (b) that the material in the reservoir has separated into two layers, the lower layer containing spherical precipitates of high lead content, and the layer above in one case has a fine dendritic structure (Figure 26(b), water quenched) and in the other a coarser dendritic structure (Figure 26(a), air cooled). The structure of the upper layer is therefore related to the cooling conditions, but the spherical precipitates are not. The latter material probably consists of dendrites of the primary phase which, owing to their high lead content, fell to the bottom of the reservoir and spheroidized as the alloy was held at temperature. Although the testing temperature was set equal to the liquidus temperature of casting B, it appears that the dissolution of dendrites is time dependent at this temperature. It also appears as though an Ostwald ripening mechanism is causing the average particle size to increase with time. Other workers have recently reported seeing the same type of precipitate in a Pb-Sn alloy heated above its equilibrium liquidus 66 (a) Air cooled, showing coarse dendritic structure in the upper layer. Total time at temperature 1.86 hours. (b) Water quenched, showing fine dendritic structure in the upper layer. Total time at temperature 0.62 hours. (c) Liquid level is equal to the top of the lower layer. Total time at temperature 0.15 hours. This corresponds to a drop in liquid level of 5.4 mm (initial height of reservoir = 8.4 mm). FIGURE 26: Microstructures of the reservoir (casting B), after testing. Magnification 23x. 67 (24) temperature . It does not appear, from the microstructures, as though the layer of spheroidized dendrites would act as an appreciable barrier to fluid flow, since the liquid channels appear to be much larger than in the casting below. However, the presence of the layer means that the liquid level can only fall as far as the top of the layer. It is therefore suggested that the flow rate becomes lower than predicted by Darcy's Law, because the liquid level in the reservoir is constrained by capillary effects to fall at the same velocity as the spherical particles. This capillary effect would not be important when the concentration of particles is low, but would become increasingly important as the concentration increases. Since it was shown earlier that positive deviations begin when the liquid level in the reservoir has fallen between 2.9 and 4.5 mm, this would mean that capillary effects start to influence the flow rate when the reservoir has fallen to approxi mately half its original height. Using Stoke's Law for terminal velocity of a spherical particle, one can make a rough estimate of whether this mechanism is possible: v = 2gr2(p' - P) where v = terminal velocity of particle, radius r g = acceleration due to gravity u = viscosity of liquid p' = density of particle p = density of liquid. An estimated mean radius of spherical particles in Figure 26 is 68 3 0.002 cm, and the density would be approximately 10.0 g/cm . Liquid 3 density and viscosity would be 8.33 g/cm and 0.03 poise. This gives a terminal velocity of 0.048 cm/sec, which is of the same order of magnitude as the flow velocity calculated in section 4.1.1. Ideally, it should be possible to check this mechanism by quench ing a sample when positive deviations begin, and then examining the structure of the reservoir. This is unfortunately impractical, since the point of deviation is not known until the data have been processed by computer. Therefore the sample which came closest to fulfilling these conditions was examined, and is shown in Figure 26(c). The drop in liquid level for this sample was 4.4 mm, and the microstructure represents a drop of 5.4 mm. The liquid level is equal to the top of the precipitate layer, which supports the proposed mechanism. The density of precipitate particles appears to increase downwards, therefore the rate at which the liquid level falls beyond the point where positive deviations begin may be related to a change in packing of the spheres, and dissolution effects, in addition to the terminal velocity for a single particle. 4.5 Permeability and Dendrite Spacing The permeabilities calculated in section 4.2 were plotted against both secondary and primary dendrite spacings (Figures 27 and 28). Vertical bars were used to show the scatter between repeated experiments on columnar castings produced under the same quenching conditions, and the data known to be less reliable (marked with asterisks in Table V) was not included. Since -9 the scatter ranges from 1.93 x 10 for a primary spacing of 175 ym, to _9 0.134 x 10 for a primary spacing of 28 ym, i.e., the scatter is a function 69 «r io'9-r-< LU or bJ Q_ IO-'°-i • columnar A equiaxed 10 T t—r T I I I I I I 100 SECONDARY DENDRITE SPACING (microns ± 10%) FIGURE 27: Relationship between the initial permeability and the secondary dendrite arm spacing for Pb-Sn at 193°C I0"8' ^ 10 I-< LiJ or LvJ Q_ I0-'°H slope = 2 • columnar A equiaxed T T 1—I I I I I I 1 10 100 PRIMARY DENDRITE SPACING (microns ±10%) FIGURE 28: Relationship between the initial permeability and the primary dendrite spacing for Pb-20%Sn at 193°C 71 of the magnitude of K, log-log plots have been chosen as the best method of representing this data. Figure 27 shows that at the higher secondary arm spacings the permeability increases very rapidly for a small increase in spacing. In view of the stated accuracy with which spacings can be measured, it is felt that primary dendrite spacing is therefore the more useful parameter for characterizing the structure in terms of the interdendritic fluid flow behaviour. Plotting permeability as a function of primary dendrite spacing, Figure 28 shows that values of permeability measured for columnar castings were slightly higher than those for equiaxed castings. This difference is probably not significant in view of the size of the error bars for the columnar castings. The permeability is clearly a sensitive function of the cast structure, and theoretical relationships can be calculated, based on models of the structure. The simplest model considers the porous medium (the casting) to be a bundle of straight, parallel, capillary tubes aligned in the direction of flow. 4.5.1 Straight Capillary Model Flow through a single, straight capillary tube of radius r can be described by the well known Hagen-Poiseulle equation 72 where q is the flow rate along a tube of length L. For n capillaries per unit area, the total flow rate per unit area (i.e. velocity) is 4 nirr AP , . v = " r 4-4 Comparing this equation with Darcy's Law: v = - V 4.1 tiL Thus, by analogy K - ^ 4.5 It is common to add a "tortuosity factor" t to this expression to account for the fact that the flow paths are neither straight nor symmetrical, thus: The length TL then represents the "effective length" of the flow channels. The liquid fraction g is given by; , _ liquid volume 'L total volume 2 mrr TLA  8L ~ LA i.e. gL = nrrr x 4.7 2 4 gL From this expression r = 2 2 2 n i T 73 4 Therefore, replacing r in Equation (4.6) K = n7T 2 8T \ 2 2 2 n IT T 2 i.e. K = ^ 4-8 8nir x (1 2) Piwonka ' demonstrated that permeability was proportional to the square of the fraction liquid for Al - 4.5%Cu which was consistent with the capillary model. Therefore, taking the model one step further, one may intuitively set the number of capillaries equal to the number of channels between primary dendrite stalks. The spacing between channels equals the primary dendrite spacing A, therefore 1 sL2*2 and K = 4.9 8TTT Since the permeability measurements in Figure 28 were made at constant temperature, the volume fraction of liquid would be constant. Assuming the tortuosity factor remains constant for the range of dendrite spacings studied, the theoretical relationship would be the straight line of slope 2 which has been drawn. In addition, this line goes through the point K = 0, A = 0; which cannot be represented in Figure 28. This 2 can be checked by showing that the ratio K/A , is constant for all points along the line. ^2 = 1.46 x 10~5 A 74 2 2 A Since x~ = • ^R from Equation (4.9) 3 8L A Therefore, taking gL = 0.19 at 193°C T3 = 99.0 i.e. x = 4.6 A tortuosity factor of 4.6 implies that the "effective length" of the channels is 4.6 times longer than if.they were considered to be straight and parallel. In most other practical applications of flow through filter beds etc., tortuosity factors are usually considered to lie between 1 and 2, the argument being that the average inclination of capillaries around random irregular particles is about 45°, therefore the mean capillary length would be approximately /2L. However, since dendrite branches mesh together in a highly regular manner, a tortuosity value of 4.6 is not unreasonable. Tortuosity has been introduced as a property equal to the average length of the flow path of a fluid particle, and attempts have been made in the literature to measure tortuosity by electrical resistivity measurements based on the concept that current would flow along the same paths as the fluid particles. This work has been reviewed by Scheideggerbut no direct correlation between the electrical and geometrical properties appears to have been shown. Scheidegger points out that the concept of tortuosity is therefore somewhat doubtful, yet it is clear that the capillary model will fit any porous medium if one adjusts the value of x appropriately. 75 4.5.2 Hydraulic Radius Theory: Other Theories (25) A more elaborate model of porous media developed by Kozeny attempts to relate the actual shape of the particles to the flow behaviour in a more systematic manner than by assigning a tortuosity factor. The theory is based on the observation that permeability, in absolute units, has the dimensions of a length squared, therefore, it is argued, there should be a characteristic length which describes the permeability. This length is called the "hydraulic radius" (r ) and is defined as: n pore volume H wetted surface area Alternatively, the characteristic length may be expressed as the specific surface of particles (S ) where P g _ surface area of particle p volume of particle liquid volume since gT = _\ \ z L total volume gL rH S (1 - gT) 4.10 p L. Using this approach, the flow velocity through the bed is described by the Kozeny equation: 3 1 8L 1 AP ... K" s/d - gTT u L P Li Therefore, from Equation 4.1 3 1 gL K = — —= - 5- 4.12 K- s^a - gTr p " 76 K" is generally known as the Kozeny constant, and has the commonly accepted value of 5 for most porous media. This approach is familiar in metallurgy, where it is used to describe the flow of gases and liquid metal in the blast furnace. To apply the approach to interdendritic fluid flow requires an expression for the specific surface of a dendrite, preferably in terms of the dendrite spacing. From Equation (4.12): l K « S 2 P and from Figure 28 2 K = A (approximately) therefore 1 P * There is no accepted method of measuring the specific surface of dendrites that the author is aware of, and although one could postulate a model which would satisfy the above condition, for example, by considering platelike dendrites, this would not really provide more information than the use of a tortuosity factor in the capillary model. In addition, the Kozeny theory is generally recommended only for beds of small particles which are nearly spherical in shape. In particular, (25) deviations occur when the theory is used for beds of fibres , which, if anything, the dendrites most closely resemble, and also very high and very low void fractions should be avoided. For beds of spheres the densest possible packing can give a void fraction in the range 0.2 to 0.3, therefore one would 77 expect poor agreement with the present work where the liquid fraction was held at 0.19. The only experimental evidence which is available to test the applicability of the Kozeny theory is Piwonka's data, discussed earlier in 3 section (3.1). His results have been plotted in the form of K versus gT /(1-j following Equation 4.12, in Figure 29, and it can be seen that this plot does not correspond well to a straight line, compared to the previous plot of 2 K versus gL (Figure 7). Scheidegger^^ has given a comprehensive criticism of the Kozeny theory, drawing attention in particular to its inability to describe aniso tropic permeability, and he reviews other theories which are more applicable to beds pf fibres, which come under the general heading of drag theories of permeability. Unfortunately, a basic assumption in the drag theory is.that the spacing between individual fibres is large compared to the fibre diameters (high void fractions) and that flow disturbance due to adjacent fibres is negligible. This would clearly not be applicable to the present studies on interdendritic fluid flow. Consequently, it was felt that the simple capillary model provided a useful empirical approach for relating permeability to structure despite the limitation that it does not give much information regarding the nature of flow on a microscopic scale. 78 79 4.6 Dendrite Coarsening Negative deviations from Darcy's Law, which occurred when the primary dendrite spacings were less than 51 ym, have been attributed to microstructural changes such as those shown in Figures 24 and 25. Similar effects have been reported by Kattamis et al. on Al-Cu alloys held in the solid-liquid region for various times. The authors observed an apparent increase in the secondary dendrite arm spacing with time, and they proposed two mathematical models for the dissolution of dendrite arms, based on the difference in solubility between two surfaces of different curvature. The driving force for the process was the reduction of surface area, and the rate of dissolution was controlled by diffusion in the liquid. Both mathe matical models gave qualitative agreement with the observations, though it was not possible to choose one model over the others. A similar coarsening process is believed to take place in the Pb-Sn castings used for interdendritic fluid flow studies, although Figures 23-25 appear to show spheroidization of the dendritic structure, rather than the dissolution of secondary branches. This process is related to the work done by Ostwald in 1900, who observed an increase in solubility with a decrease in particle size for various salt solutions in water. Later workers who observed that large particles tended to grow at the expense of smaller particles named the process Ostwald ripening. Diffusion controlled growth of spherical precip-(27 28) itates was analyzed by Greenwood ' , and he derived the following expression for the growth rate with respect to the radius of the particle (r): 80 dr dt 2DS Va 00 kTr 4.13 where diffusion coefficient of solute Sro = solubility of a particle of infinite radius r = mean radius of the system of particles V = molar volume of particle a = interfacial energy k = Boltzmann's constant T = absolute temperature. From this equation Greenwood noted that the maximum growth rate corresponds to the particle with twice the mean radius. Extending his analysis to a consideration of particle size distribution, he arrived at the following expression for the variation of the mean particle radius with time (t) Experimental evidence for a dependence of r on time has been found in studies of the growth of particles in a strain free medium, usually a liquid, but good agreement has also been found for cobalt particles in a copper matrix, copper in iron, manganese in magnesium, and others reviewed ( 2 8} by Greenwood The growth rate derived by Kattamis et al. is essentially the same as Equation (4.13), except that it is based on the growth and dissolution of cylindrical particles rather than spherical ones. The model cannot readily be extended to a consideration of particle size distribution, therefore no relation corresponding to Equation (4.14) has been derived. As a first -3 8DS oVt r 9kT 4.14 81 approximation, therefore, the results of the Greenwood theory have been applied to the present work. The metallographic evidence, especially Figure 25, shows that the interdendritic channels (pores) become more spherical, yet they must of course remain connected or flow would stop. Assuming the mean radius of interdendritic pores is a function of time as derived in Equation (4.14) r3«t From the capillary model of permeability (Equation 4.6) K « r4 it follows that K « t4/3 4'15 A different view of the coarsening of interdendritic channels can be obtained by comparing the process to sintering in ceramics. Considering two spheres of radius P in contact (Figure 30), where the area of contact is (29) a circle of radius x, it can be shown that x 1/5 - 1 t P for diffusion controlled growth. The radius of curvature at the neck r, increases with x: 2 x r = 4p" 2/5 Therefore the radius of curvature grows as a function of t If one pictures the interdendritic channels as sharp crevices FIGURE 30: Growth of a neck during sintering. TABLE VI RESULTS OF DENDRITE COARSENING CALCULATIONS Primary Dendrite Initial Power Constant Spacing (pm) Permeability (K x 109cm2) o (b) (m) 28 0.105 2.6 3.7 X io-17 ti 0.152 2.3 4.8 X io-16 ii 0.156 3.3 4.1 X io"19 II 0.136 2.6 6.0 X io"17 48 0.0569 2.7 4.4 X io"18 51 0.367 1.9 3.9 X io"14 it 0.244 1.8 1.1 X io"13 II 0.407 2.2 8.1 X io"15 From least squares fitting: error in b, approximately 30% error in m, approximately 65% 83 between secondary arms, one can assume that the effective radius of the channels changes in the same manner as the radius of curvature of the necked region. Following the same argument as before: K oc r °c t 4.16 Using either the Ostwald ripening or the sintering approach, the relation between permeability and time may be written as a power function: K K + at1 o 4.17 where Kq is the permeability at t = 0, and 'a' is a constant. Consequently Kq is equal to the value of permeability calculated from the initial slope of a curve such as Figure 17(b), and t = 0 is defined as the time when ln(h /h ) = 0. The differential form of Darcy's Law (Equation 8, to Appendix I) may now be rewritten for a variable K dh. dt = (K + atp)dt o -c (K + at*) t dh. -c 4.18 o (p+1) tP+1= -c In oj i.e. p+1 ,P+1 •c In oJ - K t o 4.19 Plotting t against the right hand side of Equation (4.19) on a log-log scale should give a straight line of slope (p+1), and an example is 84 shown in Figure 31. Alternatively, one may use the method of least squares to fit an expression of the form Y = mX*3 to the data where: X = t m = a/(P +1) and b = P + 1. This has been done for the eight tests in Table V which showed negative deviations, and the results are listed in Table VI. Following the Ostwald ripening argument the power b should equal 2.33, and following the sintering argument b should equal 2.6. Within the accuracy of the experi mental technique, the results show fairly good agreement with either of the proposed mechanisms, though it is not possible to choose one over the other. The constant 'm' which has also been calculated is a function of a large number of parameters in both mechanisms, including the diffusion coeffi cient and interfacial tension, for which accurate data are not available. It would also probably vary with dendrite spacing, but in view of the range of values (6 orders of magnitude of a very small number), it is not possible to attach too much significance to this constant. On the basis of these results, it appears that the change in permeability of the casting when it is heated above the solidus temperature can be compared either to Ostwald ripening, or to the changes which take place during sintering. In either case, diffusion has been chosen as the mechanism by which material is removed from convex solid regions and is deposited on concave regions. This process would take place in all fluid flow experiments, Y K t o FIGURE 31: Dendrite coarsening plot for a sample with KQ = 0.152 cm , average primary dendrite spacing 28 pm. oo 86 but it would have the greatest effect in those tests where the casting was held in the solid-liquid region for the longest times. Conversely, positive deviations, which depend on the velocity with which particles fall in the reservoir, would probably have the greatest effect in those castings which had the highest flow rates. Microexamination of the reservoir from castings with the smallest dendrite spacings revealed that the liquid level had not fallen to the level of the particle layer, therefore positive deviation effects were considered negligible, and ignored in the calculation of the data in Table VI. 4.7 The Scatter of Permeability Results The results have shown that permeability measurements are affected by the dendrite spacing, dendrite coarsening effects, preferential flow due to the formation of pipes, and external effects in the liquid reservoir above the casting. Other factors which could influence the results were variations in composition and temperature. To check whether composition changes had taken place within casting A during the fluid flow experiments, density measurements were made before and after testing on 10 of the samples. The machined cylinder of casting A was used for the first measurement (weighing in air and in water), and a cylindrical sample was machined after the test for the second measurement. Densities of standard specimens were measured and plotted to give the cali bration curve in Figure 32. The densities could be measured to an accuracy of about ± 0.5% by this method, and within this error, no difference in the composition before and after the test could be detected. The effect of temperature variations was tested by making flow WEIGHT FRACTION Sn Ol 0.2 0.3 ATOMIC FRACTION Sn FIGURE 32: Calibration curve; density of Pb-Sn alloys (g/cm3) at 25°C as a function of composition. I 1 • —l 1 1 r Eutectic Temperature 183 °C 51 i , I i I 180 190 200 210 220 TEMPERATURE °C FIGURE 33: The relationship between permeability and temperature. 88 measurements on one sample at different temperatures. After sufficient data points were recorded at one temperature the sample was rapidly heated to a new temperature and another series of points was recorded. This technique was limited by the tendency towards preferential flow channelling (pipe formation) if the interdendritic liquid became superheated, therefore it was only possible to make measurements close to the eutectic temperature. Subsequent metallographic examination of this sample showed that changes in the structure of the casting A were minimal and that the liquid level in the reservoir had not fallen to the top of the particle layer, therefore positive and negative deviations from Darcy's Law were considered to be negligible. The permeability was calculated for each temperature, and the results, plotted in Figure 33, show that the permeability measurements are relatively insensitive to small changes in temperature close to the eutectic. It was for this reason that the tolerance of ± 3°C on the testing temperature was considered acceptable. Using the capillary model described in section 4.5 a theoretical relationship between permeability and temperature may be calculated, since: Using x = 4.6, A = 67 um, and g as a function of temperature from Li the table in Appendix III, the theoretical curve has been plotted in Figure 33 and also shows that the permeability would not be very sensitive to temper ature close to the eutectic. From the theoretical line, a variation of 89 ± 3 C at 193 C would produce an error of about ± 5% in the value of permeability, which is considerably smaller than the error bars in Figure 28. Consequently composition and temperature variations are not thought to be the major factors responsible for the scatter in the results. Since the permeability has been shown to be very sensitive to structure, the major factor influencing the scatter in Figure 28 is probably the uncertainty in the measured value of the dendrite spacing. In addition, there would be errors involved in describing the structure of the equiaxed castings in terms of an average value. However, it is felt that, despite the scatter, the results cover a sufficiently wide range (approximately one order of magnitude in primary dendrite spacing, and two orders of magnitude in permeability) for the empirical relationship to be valid. 90 CHAPTER 5 THE EFFECT OF DENSITY DIFFERENCES ON THE FORMATION OF CHANNELS 5.1 Introduction and Review of Previous Work To apply the results of the interdendritic fluid flow experiments to a practical casting problem, a study was conducted on the formation of channel-type defects, namely, freckles and A segregates. The term freckles has been used for a number of different types of defects which are probably not all caused by the same mechanism. Freckles in iron and nickel base superalloys can appear as distinct trails of equiaxed grains on the surface of directionally solidified castings. These freckle trails are observed to begin at some distance from the chill face, and the total number decreases with distance from the chill^3^. The trails are called freckles because of the speckled appearance of the equiaxed grains. Figure 34 shows examples of freckle lines in ingots of Mar-M200 (compositions of various superalloys are given in Table VII). The composition of the material in the trails has been shown to be rich in those solute elements which segregate normally, and this, coupled with the evidence of feeding shrinkage within the trails (Figure 34(b)), leads to the conclusion that the freckles represent the last liquid in the system to solidify. The photographs in Figure 34 have been taken from the published work of Giamei and Kear^3^. A second type, of freckle defect is illustrated in Figure 35. In this case, material containing higher concentrations of the solute elements has accumulated in patches or spots in the interior of the casting. These 91 FIGURE 34: Freckle trails in directionally solidified Mar-M200 ; (a) 10 cm diameter ingot showing trails of equiaxed grains; (b) 3.8 cm diameter - single crystal showing feeding shrinkage along freckle line. FIGURE 35: Freckles in as-cast Inconel 718, perpendicular to the axis of a directionally solidified ingot. Magnification lOx. TABLE VII COMPOSITION OF SUPERALLOYS^3°^ (wt. pet.) Cr Al Ti W Mo Nb Ta V Mn Si B C Fe Co Ni Mar-M200 (nominal) 9.5 5.0 2.0 12.5 - 1.0 - - - <0.2 0.015 0.15 - 10.0 bal. Inconel 718 19.0 0.6 0.9 - 3.0 (5.0 sum) - - - - 0.10 18.0 - bal. A286 14.7 - 2.1 - 1.25 - - 0.30 1.5 0.70 0.005 - bal. - 25.5 93 patches can be seen on a macroetched surface, but they do not necessarily form trails. Areas of local segregation of this type appear to be a relative ly common feature of some iron and nickel base consumable arc melted ingots. It has been pointed out that defects of this type can be detrimental even (31) in ingots that are to be hot worked (DeVries and Mumau reported that they could not be removed even after 90% hot reduction). In addition to these two types, the term freckles has been used for many casting defects that have a "spotty" appearance, either on the surface or on a polished section, whether or not the spots are accumulations of solute or porosity, and whether or not they arise due to the solidifica tion characteristics of the alloy, or mould wall effects. The present work was therefore restricted to a study of the solidification conditions which could cause the formation of the first (channel) type of freckle. Two main explanations for the origin of channel-type freckles (32) have appeared in the literature. Gould , noting that freckle trails were related to the direction of gravity, suggested that they might be formed by gas bubbles in the liquid. If a bubble were nucleated at the solid-liquid interface, it would rise vertically and its trace would be filled by lower melting point liquid. Gould reported, however, that neither varying the nitrogen and oxygen content, nor eliminating hydrogen in experiments on A-286 or Inconel 718, had any discernable effect on the incidence of freckling. A more comprehensive study of the relationship between blowholes and "spot segregates" (i.e., the second type of freckles) was made by (33) Mukherjee in electroslag ingots. He found that the occurrence of spot 94 segregates could be influenced by varying the oxygen content and by vibration, supporting the concept that freckles may be caused by gas bubbles. He suggested a mechanism similar to that of Gould, to account for freckle trails caused by rising bubbles. (34) Copley, Giamei et al. offered an explanation based on the formation of lower density liquid close to the bottom of the solid-liquid region during solidification. If the lowest melting point interdendritic liquid is less dense than the bulk liquid above, this would cause upward flow of the lighter liquid. As the liquid rises, it would move towards hotter regions of the casting, becoming superheated. This would lead to dissolution of dendrite branches in its path, and the formation of a channel. They examined the solidification of a transparent ammonium chloride-water model in which a "density inversion" of this type occurred, and observed upward flowing pipes through the solid-liquid region, which supported their hypothesis. They also presented a semiquantitative mathematical analysis which considered the maximum driving force for freckling to be the potential energy difference between the unstable liquid configuration, with the most dense liquid layer on top, and a stable configuration, with the most dense layer at the bottom. The model did not consider resistance to flow through the solid-liquid region, but it was possible to make qualitative predictions on the effect of changing the cooling conditions and changing the alloy composition. They explained the location of freckles on the surface of the ingot in terms of the shape of the solid-liquid interface, and by varying this shape in the ammonium chloride-water model they produced either internal or surface pipes. 95 The concept that density differences cause freckles is consistent (35) with the observations of Smeltzer who noted that the defect could be (31) eliminated by changing the alloy composition, and DeVries and Mumau who reported that the accepted industrial practice for reducing freckles was to lower the power input to the consumable furnace. This would have the effect of changing the cooling conditions. However, it is not possible to draw any a priori conclusions regarding the effect on the temperature gradient or the freezing rate. It is interesting to note that they also reported that freckles could be reduced by decreasing the dendrite spacing. Part of the extensive theory of macrosegregation published by (9) Mehrabian et al., which is discussed in more detail in section 7.1, deals with the formation of channel-type defects. Basically, the theory involves the calculation of the interdendritic fluid velocity at every point in the solid-liquid region when the forces acting on the fluid are solidification contractions and gravity. They propose that the critical condition for the formation of channel-type defects is when the direction of the interdendritic fluid flow vector goes from the colder to the hotter regions of the casting. (34) This hypothesis is similar to that of Copley et al., since density, inversions can lead to interdendritic fluid flow in the direction of increasing temperature, but it differs in that the critical condition is not the sign of the density change, but the magnitude and direction of the flow velocity vector. Thus freckles need not merely be upward trails, but they can go in any direction of increasing temperature, depending on the flow pattern within the solid-liquid zone. The model permits calculation of the flow patterns in an idealized 96 ingot which solidifies unidirectionally from one side-wall, and they provide (36) experimental evidence of a channel-type defect in a degassed Al-20%Cu ingot of this type. The authors extend their theory to commercial ingots by suggesting a number of assumed flow patterns which would lead to channel-type defects. (34) The experiments done by Copley et al. were similar to others (37) done by McDonald and Hunt, who examined an ammonium chloride-water model of a conventional casting, and observed that fluid flow occurred through the solid-liquid regions in the form of rising pipes. Using density data for ammonium chloride-water, they suggested that the pipes were formed by lower density liquid in the interdendritic regions, and they considered these pipes to be analogous to A segregates in large steel castings. There are alternative explanations for the origin of A segregates, (3 8) for example the suggestion by Blank and Pickering that a solute enriched layer is formed between the columnar grains at the sides, and the equiaxed grains in the centre of the ingot. This enriched layer is subsequently drawn back into the columnar regions to form A segregates by volume shrinkage during solidification. Explanations of this type are less satisfactory than the mechanism suggested by McDonald and Hunt since they do not account for either the characteristic channel-like shape of the segregates, or their characteristic inclination. The ammonium chloride-water experiments effectively proved that channels closely resembling freckles and A segregates could be formed by the upward flow of less dense liquid in this system. However, the use of trans parent models to simulate solidification in metal castings has been questioned 97 by a number of workersv~"'"1. It has been shown by Stewart} using radioactive tracer techniques, that the convective flow velocity and flow path in water based systems can be markedly different to that which occurs in liquid metals. This is due to the large difference in the Prandtl (21) number (0.013 for liquid tin, 10.0 for water ), a dimensionless parameter which characterizes convective flow. The extent to which thermal convection would influence the mechanism of freckle and A segregate formation observed by Copley, McDonald and others is uncertain when applied to a metal system. Therefore it was considered important to investigate the interdendritic fluid flow behaviour in an alloy, when less dense liquid exists at the bottom of the solid-liquid zone. The following experiments were therefore directed towards establishing whether the proposed mechanisms, based on the obser vation of transparent models, were possible during solidification of an alloy with a similar density configuration in the liquid. Consequently, they would not imply that this is the only mechanism of freckle formation. In spite of the advantages of tracer techniques, the fact that metals are opaque makes it extremely difficult to reproduce the ammonium chloride-water experiments in an all metal system. For example, to establish that pipes flow upwards during solidification, it would be necessary to introduce tracer close to the bottom of the advancing solid-liquid region. Experimentally this would be very difficult, because one would first have to locate the position of the interface between the solid and the solid-liquid region, and then follow the motion of the tracer as solidification progresses. 98 For this reason experiments described in this chapter were designed to establish the following: a) whether a density inversion can cause liquid to flow upward through the mushy zone; b) whether liquid can flow upward through this zone even if there is no density inversion, due to solubility effects; c) whether the rising liquid tends to advance along a smooth front, or breaks down into pipe flow. While these experiments do not simulate the solidification of real castings, the results could establish the principle that density differences cause upward flowing pipes to be formed through the solid-liquid region. This would provide additional support to the mechanisms for freckling and A segregate formation previously proposed. The experimental work in this chapter was published in Metallurg ical Transactions inDecember 1972^*^. Recently, Hebditch and Hunt^4^ have reported experiments where they injected radioactive material into the solid-liquid region of a growing Sn-Zn alloy using a syringe. They observed upward flow of the less dense radioactive liquid, supporting the concept that less dense liquid can rise through the solid-liquid region, but they did not identify actual channels resembling freckles or A segregates associated with this upward flow. 5.2 Experimental Procedure The test assembly used is shown schematically in Figure 36. Casting A is a directionally solidified lead-tin alloy 2.3 cm in diameter 99 and 4.5 cm long with a columnar dendritic structure having a primary dendrite spacing of approximately 50 microns. Material from both ends of the casting was removed by careful machining and replaced by cylindrical inserts B and C of lead-tin alloy having different compositions to the casting A. The casting and two inserts were tightly enclosed in a copper block which had been coated with colloidal graphite, and placed inside the tube furnace illustrated in Figure 1(b). The columnar castings and inserts were made by techniques described in sections 3.3 and 3.4, and the dimensions are shown to scale in Figure 36. The assembly was heated at about 5°C/min to the desired tempera ture T, taking care not to overshoot this value, held at this temperature (± 0.5°C) for 45 minutes, and then quenched. The resultant casting was then sectioned longitudinally, polished and etched. In each experiment 500 ppm of radioactive thallium was uniformly distributed throughout insert B. The flow of liquid from insert B into the casting A could then be determined directly from autoradiographs of polished longitudinal sections. The alloy compositions used for the castings A, and the inserts B and C are listed in Table VIII. The temperature of testing, and the densities of B and G at this temperature, together with the density difference, are included in the table. The holding temperature T was made equal to the liquidus tempera ture at the top insert B. At this temperature, from the phase diagram, the bottom insert C would be entirely liquid and the casting A would have liquid interdendritic channels. THERMO COUPLE COPPER BLOCK TOP INSERT COLUMNAR CASTING BOTTOM INSERT COVER FIGURE 36: The test assembly for isothermal experiments. Magnification 1.5x. 101 TABLE VIII TEST CONDITIONS FOR ISOTHERMAL EXPERIMENTS Test Composition Pb + wt % Sn Temp. T°C Liquid Density^42^ z 3 g/cm Density Difference , 3 g/cm Series I Casting A 20 224 -0.65 Upper insert B 44 8.70 Lower insert C 62 8.05 Series II Casting A 20 239 -0.93 Upper insert B 37 8.97 Lower insert C 62 8.04 Series III Casting A 20 254 -1.21 Upper insert B 30 9.24 Lower insert C 62 8.03 Series IV Casting A 96 206 +0.66 Upper insert B 84 7.41 Lower insert C 62 8.07 Series V Casting A 15 254 Lower insert C 62 8.03 102 The results of these experiments cannot be quantified, and are presented in the form of autoradiographs and microstructures. Consequently, the results from only five sets of experiments (series I-V) are presented. These were part of a more extensive series of observations on different alloy compositions and holding temperatures, all of which gave results compatible with those reported below. 5.3 Results In series I, II and III the top insert had a higher density than the bottom insert (Table VIII), which is a condition of density inversion as (34) defined by Copley et al. . In series IV, the top insert had a lower density, i.e., there was no density inversion. The composition of the bottom insert was fixed at 62% Sn (the eutectic composition) in all the tests. In each of these four cases, the autoradiographs show that liquid from insert B flowed uniformly down through interdendritic channels near the top of the casting A, without the formation of pipes. These results differ from those of Kaempffer^^ who observed downward flow through pipes because, in his experiment, liquid above the mushy zone was heated above the liquidus temperature. One can reasonably conclude that if the tracer showed uniform flow near the top, then the downward flcr-.T of liquid through the whole of casting A was via interdendritic channels, since a uniform temperature was maintained. Consequently, any pipes revealed by polishing and etching are due to liquid flowing upward. A) Series I The results for series I are shown in Figures 37(a) and (b). In (a) FIGURE 38: (a) Macrostructure of columnar casting (series II). (b) Corresponding autoradiograph. Magnification 2x. 104 Figure 37(a) liquid from the sides of the bottom insert has risen in the interior of the casting. The autoradiograph in Figure 37(b) shows that liquid from the top insert B flowed essentially uniformly down through interdendritic channels. Slightly more liquid flowed down the right hand side, but there is no evidence that this is due to pipes. The enhanced flow probably results from slight differences in size or orientation of the interdendritic channels on the right side of the casting. The etched surface of insert B shows two layers which are similar to the separation which occurred in the reservoir of the flow cell, discussed earlier in section 4.4.2. Although the bottom liquid has risen, it does not appear to show a marked breakdown into pipe flow, in the sense that this term is used by previous workers (in the ammonium chloride-water model studied by Copley et al., a typical upward flowing pipe occupied an area of 4 to 9 (34) dendrites ). Apart from the region in the centre of insert C, where the lower insert does not make contact with the casting, the liquid has risen fairly uniformly by dissolving part of the dendritic structure of casting A. B) Series II In series II the temperature T and the density difference were increased, and the results are shown in Figures 38(a) and (b). The uneven interface between the bottom insert C and the casting A shows the beginning of breakdown into pipe flow. The liquid on both sides of insert C has risen in a less uniform manner than series I. The autoradiograph in Figure 38(b) 105 shows uniform downward flow along interdendritic channels. The lack of contact in the centre of insert C in Figures 37(a) and 38(a), and the porosity in some of the upward flowing channels, was probably caused either by volume shrinkage of the liquid on freezing, or by imperfect fitting of the inserts into the casting. C) Series III In series III the temperature and density difference were again increased, and the castings were sectioned transversely and then longitud inally. The results of two samples are shown in Figures 39-42. Figure 39 shows the two surfaces revealed after sectioning one sample normal to the axis. Surface B-B shows a number of channels in section, and some porosity. Two channels can be seen on surface C-C, the 3 smaller of which covers an estimated area of approximately 10 dendrites. The casting was reassembled and sectioned longitudinally along A-A and the structure and corresponding autoradiograph are shown in Figures 40(a) and (b). The autoradiograph shows uniform downward flow, which confirms that upward flowing liquid produced the wide channels in Figure 40(a). Figure 41 shows the two surfaces revealed after transverse sectioning of the second sample. One channel is seen on each surface. The longitudinal section in Figure 42(a) shows a similar structure to Figure 40(a) and the autoradiograph again shows uniform downward flow. Figure 42(a) shows a very uneven surface between the lower liquid and the casting. One FIGURE 39: Surfaces revealed after transverse sectioning of a sample from series III. Magnification 2x. FIGURE 41: Surfaces revealed after sample from series III. transverse sectioning Magnification 2x. of a FIGURE 42: (a) Longitudinal section of same sample as in Figure (b) Corresponding autoradiograph. Magnification 2x. 108 wide channel (which contains some porosity) has risen up one side and then split into smaller channels higher up the casting. The columnar structure near the top of casting A in Figures 40(a) and 42(a) appears to have changed during the test. Such changes were regularly observed in castings held for some time at temperatures where the fraction of liquid was relatively high. This effect is attributed to the beginnings of a general collapse of the dendritic structure, associated with coarsening effects discussed in section 4.5. D) Series IV In series IV the two inserts had the same density difference as series I, with the heavier insert at the bottom of the casting. The result ant structure is shown in Figure 43(a) and the corresponding autoradiograph in Figure 43(b). There is no evidence of pipes in the casting from either the bottom or the top insert. The flat interface between the columnar structure and the bottom insert in Figure 43(a) clearly shows that liquid did not flow upwards in the centre of the casting. Figure 43(b) shows that the tracer has outlined the dendritic structure to a greater extent than the other tests, yet comparing the original autoradiographs, there appears to have been less overall penetration of the casting. The long trails are not due to pipes, but indicate that the tracer has advanced further down grain bound aries. Since there is no evidence of solute convection, this tracer pene tration must be due to liquid diffusion or possibly shrinkage effects during quenching. (a) (b) FIGURE 43: (a) Macrostructure of columnar casting (series IV). (b) Corresponding autoradiograph. Magnification 2x. FIGURE 44: Macrostructure from series V. The material which etches darker is Pb-15%Sn homogenized to remove the dendritic structure. Lower insert is eutectic Pb-62%Sn. Magnification 1.5x. 110 Figure 43(a) shows that some liquid from the bottom insert has risen along the outside of the casting. This probably occurred when the supporting material between the lower insert and the copper block melted and the casting shifted slightly. Possibly some radioactive material from insert B emerged at the sides and dissolved in this outside liquid. This effect is not considered to be related to freckling. E) Series V In Figure 44, a Pb-15%Sn alloy, homogenized for 46 hours at 177°C was used in place of casting A. The lower insert was eutectic material, and the specimen was subjected to the same isothermal treatment as previous samples from series III, which showed long upward flowing pipes. The dimensions of the Pb-15%Sn alloy and the lower insert were the same as in the other samples, however, the top insert was omitted. After cooling the sample was sectioned transversely and longitudinally as before. The transverse section revealed no pipes, and the longitudinal section (Figure 44) showed that dissolution took place at the interface essentially along a smooth, planar front. 5.4 Discussion (34) In the mechanism proposed for freckling (also applicable to (37) A segregate formation ) the liquid at the root of interdendritic channels was considered to be close to eutectic composition. If this liquid was of lower density than the bulk liquid above the mushy zone, solute convection could occur, and the lighter liquid would rise up interdendritic channels. Ill The rising liquid would move from cooler to hotter regions of the casting, becoming superheated, and would eventually merge into discrete pipes by dissolving away dendrite branches. This is a steady state process; as the mushy zone advances during solidification, the pipe also advances, and the composition of liquid entering the bottom of the pipe would be expected to remain the same. The main difference between the arrangement used in the present work and a true solidification process is that these tests were done at uniform temperature, whereas in a real casting there would be a temperature gradient. The rising liquid would be superheated in this case, because of the difference in composition, and would similarly be able to dissolve dendrite branches. However, no steady state process could be established since the mushy zone in these tests did not advance. Consequently, the upward flowing channels shown ^Ln Figures 40(a) and 42(a) are not identical to the pipes formed in the ammonium chloride-water experiments. However, they do demonstrate that a density inversion can cause upward flowing pipes through the mushy zone. In addition, the results of series IV demonstrate that pipes do not form if there is no density inversion. Table IX lists the maximum solubility of the dendrites in eutectic liquid heated to the holding temperature, estimated from the phase diagram. It can be seen that the solubility of dendrites in series IV, where pipes do not form, is about the same as series III. The ability of the bottom liquid to dissolve dendrites is therefore not a sufficient condition for the formation of upward flowing pipes, unless there is also a density inversion. 112 TABLE IX SOLUBILITY DATA FOR ISOTHERMAL EXPERIMENTS Test Dendritic Phase T °C Solubility of Dendrites in Eutectic Liquid at T °C, grams per gram eutectic. Series I a 224 0.66 Series II a 239 0.95 Series III a 254 1.63 Series IV B 206 1.58 Series V a 254 1.25 113 Following the publication of the results shown in Figures 37-43 (series I-IV), Hebditch and Hunt^ suggested that the structures observed were the result of convection within the lower insert C, and had little to do with interdendritic fluid flow through the casting A. This suggestion was tested by substituting a single phase Pb-Sn alloy for the original columnar casting A (series V, Figure 44). In this case there would be no liquid interdendritic channels, yet the alloy would still be soluble in the superheated liquid of the lower insert (Table IX). If Hebditch and Hunt's contention were correct, this experimental arrangement should produce the same type of structures as seen in Figures 37-42. The result in Figure 44 shows that pipes were not formed in the absence of interdendritic channels in the casting A. Convection within the lower insert probably plays a part in the development of the structures seen, but the experimental evidence leads to the conclusion that the flow pattern caused by material dissolving at the interface leads to a smooth front, and would not account for the long channels. Although porosity due to trapped air has been observed in some of these samples, it is not believed that these experiments provide any support ing evidence for the proposed mechanism of freckling by rising gas bubbles. Trapped air would certainly form a bubble above the lower insert C, but only superheated liquid would be able to dissolve dendrite branches and rise. The bubbles thought to be responsible for producing freckle trails are considered to be on top of the advancing mushy zone, rather than on the bottom. Bubbles which become trapped by the advancing solid are thought to be responsible for "spot segregates" and Figure 42(a) appears to show some 114 isolated spots of this type, associated with porosity. However, one cannot rule out the possibility that these "spots" were originally connected with the main channel, but appear isolated because the plane of sectioning has separated them. Alternatively, they may be related to the general collapse of the dendritic structure, mentioned earlier. Therefore these experiments do not provide any conclusive evidence in support of this mechanism. They do, however, establish the principle that upward flowing pipes can be formed by superheated liquid, when density gradients in the liquid cause interdendritic fluid flow. 115 CHAPTER 6 SOLUTE CONVECTION AND FRECKLE FORMATION DURING SOLIDIFICATION 6.1 Introduction The model experiments described in Chaper 5 demonstrated that pipes can form through a solid-liquid region due to solute convection, and that the direction of flow within the pipes was upwards. Accordingly, the logical extension of this work was to study the actual solidification of an alloy system where the liquid at the bottom of the solid-liquid zone was less dense than the liquid above. If the proposed mechanism for the formation of channel-type defects is correct, it should be possible to demonstrate solute convection in such a system, and produce the defects under appropriate cooling conditions. Pb-Sn alloys containing less than 62%Sn were therefore used because they produce the required "density inversion" when solidified from the bottom. The purpose of the following experiments was to experimentally determine the extent of macrosegregation and freckle formation in vertically solidified samples as a function of the solidification variables. 6.2 Experimental Procedure  6.2.1 Apparatus Cylindrical ingots 14 cm long and 1.27 cm in diameter were solidified by lowering the mould through a furnace with two heating zones. The graphite mould is shown dismantled in Figure 45. After assembly, the 116 123456789 JO 1 2345678 III FIGURE 45: Split graphite mould for making long cylindrical ingots. The mould was assembled by sliding the two sleeves over the ends and then wiring the parts together. 117 mould was suspended vertically in the furnace, and lowered at a controlled rate by a low speed synchronous motor. The two-zone furnace was constructed from two tube furnaces, similar to those shown in Figure 1, placed verti cally end to end, and connected to a two-zone temperature controller which maintained a constant gradient. By adjusting the temperature of the furnace zones and the speed of descent, the temperature gradient and growth rate could be varied independently. Solidification rates between 0.0033 and 0.24 cm/sec were used in this work, with temperature gradients between 1.0 and 2.3°C/cm. In some tests, the alloy was quenched during solidification by surrounding the mould with water. For quenching, a thin walled graphite mould was used, having the same internal dimensions as the standard mould. Water was introduced into the quartz furnace tube from the bottom, quenching the ingot in approximately 20 sec. The graphite mould used for quenching was enclosed in a thin walled quartz tube to prevent the mould cracking and water coming in contact with the molten metal. A protective atmosphere was not used. 6.2.2 Macrosegregation studies For each set of solidification conditions, four castings were made. The temperature gradient and freezing rate were established in one casting using three thermocouples positioned along the ingot axis. The cast structure was determined from longitudinal and transverse sections of two of the castings which were polished and etched after sectioning. The fourth casting was used for measurements of macrosegregation. Approximately 500 ppm of radioactive tracer was well mixed into the liquid prior to solid ification. After solidification, the ingot was placed in a lathe, the 118 outside surface was machined to remove the tarnished surface layer, and cuttings were taken in a plane perpendicular to the ingot axis. Starting at one end of the ingot, the cuttings were collected at 0.37 cm intervals, weighed (approximately 4 g), placed into test tubes, and the activity of each tube measured in a Picker Nuclear Twinscaler II automatic scintilla tion counter. 113 The isotopes used were either Sn (half life 112 days, pri marily a low energy y emitter, irradiated to a specific activity of 0.5 204 millicuries/g) or Tl (half life 4.1 years, primarily a 3 emitter, but also some low energy y, irradiated to a specific activity of 5 milli-113 20A curies/g). The Sn was used in the lead rich alloys, and Tl in the tin rich alloys. The y emission spectrum was measured for both these isotopes using the scintillation counter (Figures 46 and 47), and since the activity levels T^ere fairly low, an appropriate window was set to reduce the ratio between sample activity and background. No corrections for decay of the isotopes were made, because the time taken for each set of measure-113 2 OA ments (2-3 hours for Sn , or 12 hours for Tl ) was very short compared to the half lives. Activity profiles, as a function of distance from the bottom of the ingot, for castings quenched directly from the liquid were compared to the directionally solidified castings to confirm that the effects observed were due to the solidification conditions, and not due to poor mixing of the original alloy charge, or other extraneous factors. 22i 20\ 119 'o 181 o o 14 12 10 CL Window d 270-480 Kev 100 200 300 400 ENERGY (Kev) 500 FIGURE 46: Spectnrai of y emission for Sn113. 29 27 25 23 o x in 2! z 3 O " 19 17 15 50 T 1-,--o. Window 50-100 Kev 70 _l_ 90 ENERGY (Kev) 110 130 FT.GURE 47: Spectrum of y emission for TI 204 120 6.2.3 Determination of composition from activity measurements It is usually assumed that the measured activity of the radio active isotope is directly proportional to the solute concentration. The major factors contributing to errors would be counting scatter, which is a function of the number of counts, and geometrical scatter, since the lathe turnings would not always present the same geometry towards the counter. The number of counts per sample was always between 300,000 and 1 million in this work, which would result in an overall scatter of (43) better than ± 0.5% due to counting scatter It was believed that the effect of geometrical scatter would be magnified in the Pb-Sn system due to the high absorption of the radiation -1 (44) by lead (absorption coefficient 1.127 cm for 0.5MeV y rays ). To reduce this effect, and to obtain a more constant geometry, some of the 113 samples containing Sn were treated with a solution of hot 50% HNO^. This formed soluble lead nitrate, and a white precipitate of tin oxide which settled in the bottom of the test tube. Each tube contained 40 ml of solution, and from the published solubility of PbtNO.^ (37.7 g/100 g solution) it was calculated that all the lead would be in solution. The activity measurements would therefore only be affected by absorption in the precipitate layer. i To convert the measured activities to composition, the following series of calibration tests were done: 113 i) activity versus sample weight, for a constant Sn concentration; 113 ii) activity versus Sn concentration, for a constant sample weight; 121 113 iii) activity versus alloy composition, for a constant Sn concen tration; 113 iv) activity versus alloy composition, when the Sn concentration was directly proportional to the alloy composition. The calibration samples were prepared by careful weighing of the constituents and subsequent treatment with nitric acid. The results are shown in Figures 48-51. The measured activities as a function of both 113 sample weight and Sn concentration (Figures 48 and 49) both deviated from linearity when larger amounts of radioactive material were present, which indicated that the scintillation counter began to saturate at these levels. It was for this reason that relatively low concentrations of the radioactive isotopes were used. Figure 50 shows that the specific activity measurements were relatively independent of the alloy composition for a 113 constant Sn concentration. Therefore from Figures 48-50 one can conclude that if the samples do not vary over a wide range of weights or compositions the solute content of the sample can be assumed to be directly proportional tp the specific activity. This was tested and shown to be true over a wide composition range in Figure 51. The method used for calculating solute content was to determine the specific activity of the whole ingot (a. ), which was taken as the sum mg ' of the activities of each sample (a ) from the ingot, divided by the sum of the sample weights (w^) . The specific activity was then set equal to the prepared alloy composition (CQ). Za. ^ _ i _ ing Ew^ ~ o > < 71 1 1 1 1 10 20 30 40 ALLOY COMPOSITION (% Sn) FIGURE 50: Calibration curve; specific activity versus 113 alloy composition for constant Sn concentration. ALLOY COMPOSITION (wt % Sn) FIGURE 51: Calibration curve; specific activity versus 113 alloy composition, when Sn . concentration is proportional to the solute content. 124 The sample composition was then given by: a. C i W. a. 1 mg Although the method of treating samples with nitric acid probably reduced some of the scatter associated with the analysis technique, it introduced other problems. In particular, the disposal of radioactive waste in the form of a precipitate in a strongly acid solution, in a large number of individual test tubes, proved to be very time consuming. It was found that the time required to safely transfer the samples to one, non-corrosive container, and then reduce the bulk of liquid waste by evaporation was much longer than anticipated. Since the main advantage of using the isotope analysis technique was its speed compared to other methods, this reduced its value appreciably. Consequently, composition profiles for one ingot were compared using the results for samples of turnings which had not been treated, and for the same samples which had been treated with nitric acid. Both sets are plotted together in Figure 52. Although there appears to be less scatter associated with the treated samples, neither plot is smooth, because of the effect of microsegregation. Therefore it was felt that the improved accuracy did not merit the time involved in treating all the samples with nitric acid, and the composition profiles for other ingots were calculated using cuttings from the ingots, and the direct proportionality between composition and specific activity was taken to be correct over the range considered. The reproducibility of the activity measurements from untreated FIGURE 52: Composition profile for one ingot using lathe turnings treated with nitric acid (open circles), and untreated samples (closed circles). ho 126 lathe turnings was tested by counting a single sample several times, emptying and refilling the test tube, to vary the geometry of the packing. The error bars used in the following composition profiles (Figures 53-57) are ± 2s limits based on these tests. For castings with a mean composition of Pb-20%Sn, they represent a scatter of ± 0.38%Sn (percentage error + 1.9%) in the analysis technique. This was considered acceptable since the compo sition difference between the ends of the ingots was, in general, signifi cantly larger than the scatter. For Sn-4%Pb castings, the error bars represent a scatter of ± 0.22%Pb, which is a larger percentage error (4.3%), probably due to greater 204 absorption of the low energy emission from Tl 6.2.4 Solute convection To observe convection through the bulk liquid and solid-rliquid 113 regions, 0.1 g pellets of the casting alloy, containing Sn , were placed in the liquid at the top of the mould during solidification. Solidification was continued for one hour; following which the castings were quenched in the furnace, then sectioned and polished. Autoradiographs of transverse and longitudinal sections showed the extent of tracer movement. A similar experiment was performed on a casting of the same composition held completely liquid under the same temperature gradient for one hour. In this case, spreading of the tracer could be attributed to the cumulative effects of disturbances associate^ with adding the pellets and quenching. The fluid flow resulting from the solidification process could be evaluated by comparing the results of these tests. 127 6.3 Results 6.3.1 Composition profiles The composition profiles of the ingots, determined by the radio active tracer analysis, are shown in part (a) of Figures 53-57. The solid lines are theoretical curves calculated from the mathematical model which is developed in Chapter 7. Correspondence between theory and experiment is discussed in section 746. Figure 54(a) is^ in fact, the same composition profile as Figure 52, and the data points used were those for samples treated with nitric acid. Since the error bars were obtained by testing the geo metrical scatter associated with lathe turnings, they probably represent a more conservative estimate of the errors on this particular graph. The results from the cooling curves for each ingot are represented graphically in part (b) of Figures 53-57, according to the method proposed (45) by Flemings and Nereo . Measurements of the time required for the liquidus and solidus isotherms to pass each thermocouple position are plotted on a distance-time graph. If the liquidus and solidus lines are straight and parallel, this indicates that both the growth rate and tempera ture gradient remained constant during solidification. This is shown to be essentially true under the slow freezing conditions imposed in these experi ments . The solidification variables and macrosegregation are summarized in Table X. Macrosegregation is normally defined as AC = C - C 6.1 x o where C is the composition at a particular location (x) and C is the mean 128 o o C\J • C\J AVERAGE TEMPERATURE GRADIENT l-5°C/cm AVERAGE GROWTH RATE 00047 cm/sec "00 2 0 40 60 80 100 DISTANCE FROM BOTTOM OF CASTING (cm) 12 0 14 0 FIGURE 53(a): Solute distribution. 129 o CM CM o ID AVERAGE TEMPERATURE GRADIENT l-5°C/cm AVERAGE GROWTH RATE 0-013 cm/sec "0 0 20 40 60 80 100 DISTANCE FROM BOTTOM OF CASTING (cm) 12 0 14 0 FIGURE 54(a): Solute distribution. FIGURE 54(b): Cooling conditions. 100 TIME (min) 140 130 o o CM CM (_> O rr b1 UJ CM CL O CO' O AVERAGE TEMPERATURE GRADIENT 2-3°C/cm AVERAGE GROWTH RATE 0011 cm/sec "00 —I 1 1 1 1 1 1 1 1 20 40 60 80 100 DISTANCE FR0IV1 BOTTOM OF CASTING (cm) 12 0 14 0 FIGURE 55(a): Solute distribution. FIGURE 55(b): Cooling conditions, 140 TIME (min) 180 131 C\J o c\l • |2N CE O LlJ CM Q. 5 co-o to • AVERAGE TEMPERATURE GRADIENT I 0°C/cm AVERAGE GROWTH RATE 0 24 cm/sec "00 20 4 0 6 0 SO' lOO ' I2"0 • ' |4'0 DISTANCE FROM BOTTOM OF CASTING (cm) FIGURE 56(a): Solute distribution. FIGURE 56(b) : Cooling conditions. 132 ob AVERAGE TEMPERATURE GRADIENT l-9°C/cm AVERAGE GROWTH RATE 0 0033 cm/sec or o-X — |5 o C\J" '00 ™ ' ^ ' iS ' 8^ ^o" ' [£o DISTANCE FROM BOTTOM OF CASTING (cm) FIGURE 57(a): Solute distribution. FIGURE 57(b): Cooling conditions. TIME (min) TABLE X SOLIDIFICATION VARIABLES AND MACROSEGREGATION Figure Number: Alloy Composition (wt-pct) Average Temperature Gradient (°C/cm) Average Growth Rate (cm/sec) Calculated dis tance between liquidus and solidus isotherms (cm) Primary Dendrite Spacing (microns) Structure Macro-segre gation (AC pet) 53(a) Pb-20Sn 1.5 0.0047 62.0 206 colmnar 1.07 54(a) Pb-20Sn 1.5 0.013 62.0 172 half columnar half equiaxed 0.73 55(a) Pb-20Sn 2.3 0.011 40.4 119 equiaxed 0.13 56(a) Pb-20Sn 1.0 0.240 93.0 58 equiaxed 0.27 Pb-20Sn - - Quenched - - equiaxed -0.04 57(a) Sn-4Pb 1.9 0.0033 24.7 - - half columnar half equiaxed -0.35 - - - Sn-4Pb - - Quenched - - - - ' equiaxed 0.11 134 composition. In this case, however, the amount of macrosegregation over the whole casting has been re-defined as the difference in mean composition between the upper and lower halves of the ingot An 111 2 1 1 , „ AC = ~rw. or 6-2 1 i 2 i where £^ and Z^ are the sums from x = H (the total length of the casting) to x = H/2, and x = H/2 to x = 0 respectively, and and are the composition and weight of the ith sample. Macrosegregation is considered positive when the solute concentration increases in the direction of solidification, and negative for the reverse. Comparing the macrosegregation for directionally solidified and quenched castings, listed in Table X, it can be seen that the solute distribution is a function of the solidification conditions. In general, the amount of macrosegregation for Pb-20%Sn alloys decreased as the primary dendrite spacing decreased. Figures 53(a) and 54(a) show that macrosegre gation increases for the slower growth rate at the same temperature gradient, and Figures 54(a) and 55(a) show the same effect for the shallower gradient when the growth rates are almost the same. Figure 56(a), solidified under the lowest temperature gradient and highest growth rate, showed slightly more macrosegregation than Figure 55(a). For the Sn-4%Pb alloy (Figure 57(a)) there was more solute at the bottom of the casting and less at the top, resulting in some negative macrosegregation. An estimate of the significance of the macrosegregation values in Table X can be obtained by using the "Student's t-test". The composition 135 measurements for each sample are subject to microsegregation and geometrical scatter, however, it is reasonable to assume that the mean of the samples over half the ingot will only be subject to geometrical scatter, since microsegregation only extends over relatively short distances. Therefore it is also reasonable to assume that the standard deviation of the mean is equal to the standard deviation of each sample. Thus, knowing the means and standard deviations for the top and bottom halves of the ingot, one may use the t-test to check whether the composition differences are significant. In general x^ -t = s /J .1 + -nl n2 where x^ and x^ are the two means with standard deviations assumed equal to s, and n^ and n^ are the number of samples used to estimate the means. The number of degrees of freedom is n^ + n^ - 2. The number of samples used to calculate the composition of each half of the casting is approximately 15, therefore one may take the number of degrees of freedom as 28. For a significance level of 0.01, the value of t is 2.763, therefore it is unlikely that the mean compositions of the two halves of the ingot come from the same population when: 2.763 < AC since AC = x^ -One can therefore conclude that there is no significant macrosegre gation when AC < 0.19%Sn for Pb-20%Sn ingots; or AC < 0.11%Pb for Sn-4%Pb 136 ingots. Thus, the values for both quenched-ingots show no significant macrosegregation, nor can the value of 0.13%Sn for the ingot in Figure 55(a) be regarded as a significant difference. The value of 0.27%Sn for the ingot in Figure 56(a) is probably only marginally significant since the scatter in the upper half is larger than in the other ingots, making the assumption regarding microsegregation less valid. The remaining ingots, however, show a significant amount of macrosegregation compared to the quenched ingots, and significant differences when compared to one another. 6.3.2 Convection in the liquid The results demonstrating convection in the liquid during solidification are given in Figure 58. The autoradiographs shown are of sections parallel and perpendicular to the freezing direction of a Pb-20%Sn alloy under the conditions listed in Table XI. In Figure 58(a), the regions which are uniformly dark (sections i-iv) indicate that the liquidus isotherm passed through this region after tracer had become mixed through the bulk liquid. In section (v), only the interdendritic regions are dark, indicating tracer penetration into the solid-liquid zone. Figure 58(b) shows autoradiographs for the ingot quenched from the liquid. Tracer has moved less than 3 cm down the ingot as compared to 6.5 cm for Figure 58(a). The difference in penetration is attributed to solute convection associated with the solidification process. There is no indication, however, of the flow pattern which caused mixing. 137 FIGURE 58: Autoradiographs showing the extent of tracer movement one hour after tracer was added; (a) directionally solidified, (b) quenched from the liquid. Magnification 2.2x. TABLE XI COOLING CONDITIONS Figure Number Alloy Composition (wt pet) Average Temperature Gradient (oc/cm) Average Growth Rate (cm/sec) Calculated Distance Between Liquidus and Solidus Isotherms (cm) 58(a) Pb-20Sn 1.9 0.0033 48.9 58(b) Pb-20Sn - - Quenched - - -59(a) Pb-20Sn 1.5 0.0047 62.0 59(b) 60 Pb-20Sn 1.9 0.0033 48.9 139 6.3.3 Freckles Evidence of structures resembling freckles was seen in ingots solidified at the slowest growth rates. In one ingot solidified at 0.0047 cm/sec, the outer surface showed a shrinkage trail approximately 7 cm long near the top (Figure 59(a)). A shrinkage defect of this type indicates that a long channel of eutectic liquid was present just before the ingot became completely solid. This bears a close resemblance to photographs of freckles in nickel-base superalloys shown in Figure 34(b). One casting grown at 0.0033 cm/sec revealed an internal trail which could be classed as a freckle. Figure 59(b) shows transverse and longitudinal sections through the freckle trail which was 4.5 cm long. An enlarged view of a transverse section (Figure 60(a)) shows that the trail has a finer dendritic structure. Figure 60(b) is an enlarged view of the lowest portion of the trail, showing that it originated in the interior of the ingot as an interdendritic channel which widened and moved towards the mould wall, in the same direction as the primary dendrite stalks. 6.4 Discussion of Results The curves in Figures 53-56 showing positive macrosegregation resemble curves for normal segregation with diffusion controlled mixing ahead of a planar solid-liquid interface. There is, however, considerable (2347) experimental evidence in the literature ' to show that only a negligible amount of solute is rejected ahead of dendrite tips when growth is not planar. Normal segregation takes place over a distance of the order of microns in the liquid between dendrite branches, leading to micro-(a) (b) FIGURE 59: (a) Shrinkage trail, approximately 7 em long, along the outside surface of an ingot solidified under conditions given in Table XI. Magnification 1.7x. § (b) Longitudinal and transverse sections showing a freckle trail on the right hand side. Magnification 3x. 141 FIGURE 60(a): Transverse section of the freckle trail in Figure 59(b), showing fine dendritic structure within the trail. Magnification 25x. 142 segregation. However, one would not expect a net movement of solute in the direction of growth unless there was liquid mixing on a macroscopic scale. This mixing could take place either within the solid-liquid zone, or between this zone and the bulk liquid ahead of the dendrite tips. The experiment where tracer was added to the liquid at the top of the casting (Figure 58) confirms that solute convection took place, which is attributed to the formation of lower density liquid in the solid-liquid region. In the case of Pb-20%Sn, the interdendritic liquid becomes enriched in tin, up to the eutectic composition (62% Sn). The density of 3 the bulk liquid at the interface would be 9.7 g/cm , and the eutectic would 3 (42) be 8.2 g/cm , thus there would be a density inversion through the solid-liquid region which causes the less dense liquid to rise. One can therefore conclude that the solute profiles in Figures 53-56 are a function of the growth rate, temperature gradient and dendrite spacing. Since it was only possible to hold two of the three variables constant for any two ingots, one cannot draw any firm conclusions regarding the effect of a variation in any one. For this reason, a simple mathematical model was derived (chapter 7), which included the concept of mass transfer through the solid-liquid region caused by density differences in the liquid. The Sn-4%Pb alloy was chosen as an example of a composition where the interdendritic liquid would be more dense than the bulk liquid above. 3 3 The comparable densities would be 7.1 g/cm in the bulk liquid, and 8.2 g/cm (42) in the eutectic . This density configuration would be stable, and one would not expect any convection. The resulting composition profile (Figure 57(a)) shows a small increase in lead content close to the bottom of the 143 ingot, which is possibly due to the effect of inverse segregation. A small sample (150 g) of the molten alloy was placed in a graphite crucible and observed during solidification under vacuum. No bubbles were seen. This evidence, together with the information in the literature that only oxygen is very slightly soluble in molten Pb-Sn alloys^48\ confirmed that gas evolution could be ignored as a possible freckling mechanism in this system. Consequently, the freckle trails observed in ingots solidified at the slowest freezing rates are attributed to the effect of solute convection. As the less dense liquid towards the bottom of the mushy zone begins to rise, it becomes superheated and can dissolve dendrite branches in its path. If sufficient interdendritic channels widen in this fashion, they can converge and result in the formation of a large vertical pipe. Direct evidence of this mechanism is shown in Figure 60(b). The experimental evidence suggests that freckles do not always form when solute convection takes place, but they appear when the velocity of the rising interdendritic liquid reaches a critical value. 144 CHAPTER 7 A NUMERICAL MODEL FOR MACROSEGREGATION IN Pb-Sn ALLOYS 7.1 Introduction and Review of Previous Work Macrosegregation caused by interdendritic fluid flow has been treated analytically in a number of published papers. Most of these models are for inverse segregation, where an analytical solution can be obtained by considering backflow through a volume element as the solid and liquid contract during solidification. Chill face segregation under these con-(49) ditions was first predicted by Scheil , and his model was later extended by Kirkaldy and Youdelis to predict the solute distribution along the whole «.. (50,51) casting A more general solution for macrosegregation, considering fluid (45) flow in three dimensions, was first published by Nereo and Flemings Their model is based on the use of the Pfann Equation to predict microsegre-gation in binary alloys: C = kC (1 - f )k_1 ' " 7.1 SOS where Cg = solid composition at the solid-liquid interface k = equilibrium distribution coefficient f = weight fraction solid Cq = initial alloy composition. In addition: C = kCT s L and f = (1 - fT) s L 145 where CT and f are the composition and weight fraction of liquid, Li Lt respectively. Considering a constant k, and a constant solidification shrinkage (3), they solve heat and mass balances in a volume element to obtain the following general expression: L Ih. (Lz-A\ I"1 +51yil !S 7 2 9CT \l - k^ L e J 3t where Ps ~ PL ps V = interdendritic flow velocity vector VT = temperature gradient in three dimensions p = solid density s J p^ = liquid density and e is defined as the rate of temperature change (9T/3t). Equation 7.2 is referred to as the "local solute redistribution equation". It is written for the general case of three-dimensional heat and fluid flow, assuming constant solid density during solidification, negligible diffusion and no pore formation. If further assumptions are now made to calculate the flow velocity vector (v), the equation can be used to estimate macrosegregation in castings. Flemings and Nereo used the model to predict solute distributions for inverse segregation in Al-Cu alloys. The model has subsequently been (52 53) refined and extended ' , and has been applied by Mehrabian, Keane and (9) Flemings to predict macrosegregation caused by a combination of solidifi cation contraction and solute convection. They consider the fluid dynamics through a volume element where the forces acting are solid contraction, 146 liquid contraction and gravity. The liquid is of variable density, and the solid-liquid region is treated as a porous medium of variable porosity. Equations are derived relating interdendritic fluid pressure, interdendritic flow velocity, fraction liquid and liquid composition which can, in theory, be solved to give these variables as a function of position. In practice, solutions for the general case are difficult to obtain, since this would involve the solution of simultaneous partial differential equations. They have therefore made the following simplifying assumptions: 1) The fraction liquid varies with position in the mushy zone only as a function of temperature, and is calculated by assuming steady-state, unidirectional heat and fluid flow. 2) Planar isotherms are assumed, so that for a constant liquidus slope, the liquid composition varies linearly with position in one direction. 3) The density of liquid varies linearly with composition. 4) The density of solid is constant. The model has been applied to the special- case of horizontal, unidirectional heat flow and steady-state solidification, where a value for the parameter characterizing the structure and thermal conditions has been assumed. In Chapter 6, macrosegregation experiments were done on vertically solidified Pb-Sn alloys. Since it is difficult to apply the model derived by Mehrabian et al. to this type of ingot and alloy system, a simple mathemat ical model was developed, which would take into account the effect of growth rate, temperature gradient and structure. Some of the assumptions for this model differ from those used by Mehrabian et al., the major differences being: 1) The partition ratio varies as a function of temperature. 147 2) The density of the liquid is a function of temperature and composition, obtained from experimental data. 3) The structure of the mushy zone is characterized by a parameter obtained from the results of the experiments on interdendritic fluid flow (Chapter 4). 4) Since the model is applied to the solidification of Pb-Sn alloys, backflow due to volume shrinkage is neglected. 7.2 Model of the Solidification Process The solid-liquid configuration during the progressive vertical solidification of a small ingot is assumed to be that shown schematically in Figure 61. The actual structure represented by the spikes may be columnar dendritic or equiaxed. The following assumptions are made: 1) The liquid is completely mixed in the horizontal planes. 2) There is no significant diffusion in the solid state. 3) Local equilibrium exists at the interface between the interdendritic liquid and the adjacent solid. Under these conditions, the composition of the solid at the solid-liquid interface is given by the Pfann Equation (Equation 7.1), provided k is a constant. It is also possible to use Equation 7.1 incrementally to describe the solidification of an alloy where k varies with temperature, by assuming that k remains constant over a small temperature interval AT. For the general case of solidification between temperatures T^ and T^, as shown in Figure 62, liquid composition is given by the liquidus line, k is equal to the average distribution coefficient between these two temperature: 148 FIGURE 62: Equilibrium diagram for a binary alloy. The non-equilibrium solidus is shown by the dashed line. 149 C s 7.3 where and refer to compositions at the equilibrium solidus and liquidus. The weight fraction of liquid which solidifies as the alloy cools between T^ and is: J1J -l/(l-k) 7.4 The solid which freezes in this increment is of composition C' S2 and weight fraction (1 - f ). For small increments AT, C' = kC , but a s2 L2 better estimate of C' can be obtained for larger increments by calculating the total weight of solute in both solid and liquid at T^, then applying conservation of solute mass at T^' Tne composition of the entire solid at any temperature will be less than the equilibrium value, because no diffusion in the solid has been assumed. The non-equilibrium solidus (shown dashed in Figure 62) can be calculated by summing the total amount of solute in the solid. The composition of the material which solidifies at T1 is C' , yet 1 sx the composition of the entire solid is C_ . If the weight of liquid at T is W , and the weight and average composition of solid are W and C , solute conservation gives: Sl Sl % (1 " V \ + CL2 fL \ + % W C (W_ + w ) o Ll sx 7.5 150 and the average composition of solid at ls: C W + C (1 - fT) W Sl Sl S2 L Ll s0 W + (1 - f_) WT 7.6 2. Sj L Consequently the liquid composition, average solid composition and the weight fractions of solid and liquid can be defined at any tempera ture between the liquidus and solidus by using data from the phase diagram. In addition, if density data are available as a function of temperature and composition, volume fractions and densities are also defined at any temperature. 7.3 Interdendritic Fluid Flow Model The model of the solidification process describes how solute is redistributed perpendicular to the dendrite stalks. If no interdendritic fluid flow occurs, the ingot would show microsegregation on a scale equivalent to the dendrite spacing, but there would be no net movement of solute over greater distances. The assumption is now made that solute can be moved over much greater distances by interdendritic fluid flow caused by density differences in the liquid. It is further assumed that the network of dendrites in the solid-liquid region produces a resistance to flow, and that this resistance is a function of the volume fraction, and dendrite spacing of the solid. Therefore, using the capillary model described in section (4.5.1), the permeability of the dendritic network is given by: 2 K = 4-8 8niTT This equation was derived for one dimensional flow; however, the 151 redistribution of solute in a casting of the type shown in Figure 61 involves three dimensional flow. In this case, since flow only takes place down one third of the channels in any one direction, Equation 4.8 becomes: 2 K = ^ 7.7 24mTT (1 2) Previous experimental work ' has shown that K is proportional 2 to g when g is less than 0.3. In the absence of a better model, it has J-j Li been assumed that Equation 7.7 holds for all values of g . Values of the Li 3 factor nx have been taken from Figure 28, and used in conjunction with this model. Dendrite coarsening has been neglected. Macrosegregation is therefore determined using Darcy's Law to calculate the flow rate of interdendritic liquid when the permeability varies with temperature, and the driving force (AP) is given by the density differences in the liquid. 7.4 Unidirectional Solidification of a Vertical Casting As an example of the application of the model, consider the solidification of a vertical, cylindrical casting of constant cross section. To simplify the calculations, constant growth rate (R) and temperature grad ient (G) are assumed. In addition, since this model will be applied to the solidification of Pb-Sn alloys where the volume change on freezing is small (of the order of 2%), it has been assumed that backflow due to volume shrinkage is negligible. The casting is divided into a number of horizontal layers of length iy and the temperature within each layer is assumed to be uniform. 152 The temperature difference between adjoining layers is therefore equal to G£. Increments of time (At) are chosen such that the cooling rate is equal to G£/At. After each time increment, the temperature of each layer is then equal to the temperature of the layer below and the growth rate (R) is equal to £/At. The length I can, therefore, be eliminated as a variable, Since it can be defined in terms of R and At. Solidification is considered to begin when the temperature of the bottom layer is equal to the liquidus temperature T . Within each time step At, solidification and fluid flow are treated separately. Conservation of solute within each layer applies during each solidification step, and conser vation of solute through the whole casting applies in each fluid flow step. Fluid flow through a layer stops when the temperature falls below the eutectic temperature T^, and the final composition profile of the casting is obtained when the top layer reaches T . h Figure 63 gives a schematic representation of the casting at an intermediate time, and shows the temperature, composition and density profiles in the liquid obtained using the solidification model for an alloy where the solute rich liquid has a lower density than the initial liquid. The driving force for flow through the solid-liquid region is given by the density difference between T and T and is equal to Ap gh, where Ap is the density difference in the liquid, g is gravity, and h is the Li distance between T and T . Since the liquid is a continuum, one would L hi expect the driving force at every point through the solid-liquid region to be the same. However, since the liquid fraction decreases downwards, the resistance to flow would increase towards the bottom of the solid-liquid zone. 153 Q" - ---->--_TL_ -/-JE_ J I Ki I Jon t \ ^ N—\ jRAt TEMPERATURE LIQUID COMPOSITION LIQUID DENSITY FIGURE 63: Directionally solidifying ingot divided into layers. Temperature, composition and density profiles given by the solidification model. 6 5 « • • R4.Q4 4 R3 .^3 3 3 R2 .<*2 2 R. ,qi | I 1 (a) (b) FIGURE 64: (a) Assumed flow pattern showing two main flow cells. (b) Resistances R1-5» and flow rates for flow between six layers. 154 A flow pattern within the solid-liquid region has, therefore, been assumed where flow can take place vertically from one layer to the next, and horizontally through the layer. For vertical flow, half the cross sectional area contributes to downward flow, and half to upward flow. Figure 64(a) shows the assumed flow pattern with two main flow cells. However, provided downward and upward flow each occupy half the cross sectional area, the actual number of flow cells is unimportant. The resistance of the dendritic network to fluid flow is repre sented schematically in Figure 64(b). The resistance symbols represent porous media of area equal to half the cross sectional area of the casting, and length equal to the length of the layers. It is assumed that there is no resistance to horizontal flow through the layers, since the distances will be short, especially for a large number of flow cells. Porous layers stacked in this manner obey the laws of series resistances; therefore, since the magnitude of the resistance can be calcu lated in terms of the liquid fraction and structure, and the pressure drop is known, the velocity of the interdendritic liquid (v/gL) can be calculated using Darcy's Law. The flow rate of interdendritic liquid (q) is then equal to 2v/Ag where A is the cross sectional area of the ingot. For brief time Li intervals, At, the quantity of liquid which flows between layers will be small, therefore the driving force for flow is assumed to remain constant. In Figure 64(b), six layers are shown, the resistances between the layers are numbered R-^_5» and the flow rates are qj_5« The liquid compositions of each layer, expressed as weight per unit volume, are equal to (PL^T.,^1-6* The volumes of the layers, V, ,, remain constant. 155 Each layer exchanges liquid with the adjoining layers, and following this notation, the volume flow rate across the top" and bottom surfaces of the ith layer are q^ and q^ ^, respectively. A solute mass balance can, therefore, be written for the ith layer: vi it (PLVI = ViWi-i + ^iWi+i- (qi + Vi)(pLVi 7-8 This mass balance can be written for each layer, giving a series of simultaneous ordinary differential equations which can be solved for the composition of each layer, after a time interval At, using standard numerical methods. Thus, the net effect of fluid flow is that the average composition of each layer is no longer equal to Cq, yet on the next solidification step, the liquid composition will be equal to the value given by the liquidus line on the equilibrium diagram. For each layer, this will result in slight differences in the average composition of the primary solid from Equations 7.4 and 7.5 and in the final fraction of eutectic. 7.5 Results of Calculations for Solidification of a Pb-Sn Alloy The model was used to calculate the final solute distribution in a Pb-20%Sn alloy, as an example of a system which shows a density inversion during solidification. Data for the equilibrium distribution coefficient k and the equilibrium liquidus line, as a function of temperature, were obtained from the phase diagram and converted to polynomial expressions using standard curve fitting techniques. Data for the density of liquid Pb-Sn alloys as a function of temperature and composition were available in the 156 form of a table y and intermediate values were obtained by linear (54) interpolations. The value of viscosity (y) was taken as 0.03 poise The final solute profile was obtained by recalculating the values of each solidification parameter for a solidification step followed by a fluid flow step after every time increment At as the casting cooled between T J_i and Tg, using a digital computer. The FORTRAN program is given in Appendix IV. Figure 65 shows the solute distribution for various values of the time interval, i.e., different numbers of layers, when H, G, R, y and NC have the values shown. H is the length of the casting, and NC is the effec-3 tive number of channels (NC = nx A). The curves show that, except for the ends of the ingot, as the number of layers increases (At decreases) the solute distribution converges to a single solution. The compositions at the extreme ends diverge at At decreases, because the assumption that the flow rate (q) is small compared to the amount of liquid in each layer no longer holds when the size of layers becomes very small. The highest value of q would be at the top, therefore one would expect this assumption to break down first in this region of the ingot. The compositions which are calculated at the extreme ends of the ingot are there fore not considered meaningful, but the shape of the curves and the inte grated amount of solute which has moved from the bottom of the casting to the top are a measure of the relative amount of macrosegregation. The structure of the solid-liquid region is expressed in terms of the effective number of channels (NC) , and the solute profiles for different values of NC are shown in Figure 66. It can be seen that the amount of 157 CVI ro O ob CVJ o CVJ o o I o UJ — o CVJ CO, L= 37 length of casting (H) = 14 cm temperature gradient (G) = l-5°C/cm growth rate(R) = 0 005 cm/sec number of channels (NC) =3-3 x I05 viscosity of the liquid = 003 poise 00 20 40 60 80 100 DISTANCE FROM BOTTOM OF CASTING (cm) 120 14 0 FIGURE 65: Solute distribution as a function of the number of layers. FIGURE 66: Solute distribution as a function of structure (effective number of channels). 158 macrosegregation, considered in terms of the amount of solute which moves from the bottom half of the casting to the top, increases as NC decreases. Since NC is related to the dendrite spacing, this means that for larger spacings the resistance to flow through the solid-liquid region decreases, therefore, for the same pressure drop there is more flow. Figure 67 shows the solute distribution as a function of ingot height. As the height increases, so the fluid head will increase, causing more flow through the mushy zone. However, this only applies when the length of the mushy zone is greater than or equal to the ingot height. The theoretical length of the mushy zone is (T - T )/G, which for the conditions used in Figure 67 is 62 cm. Figure 68 shows that the amount of macrosegregation increases as the growth rate decreases. This would be expected, since the amount of time available for flow increases, as R decreases. Figure 69 shows that the amount of macrosegregation increases as the temperature gradient increases, (34 contrary to the semiquantitative theory proposed by Copley, Giamei, et al. This can be visualized when one considers that the composition gradient through the mushy zone will be steeper for the higher temperature gradient. This will lead to a higher density difference and consequently more flow, when all other variables are held constant. The reason why this appears to contradict experience is that high temperature gradients are usually assoc iated with high growth rates, and it is not normally feasible to vary these two parameters independently. 159 FIGURE 67: Solute distribution as a function of ingot height. FIGURE 68: Solute distribution as a function of growth rate. 160 o I m H 1 1 1 i 1 1 1 1 1 1 1 1 1 1 00 20 40 60 80 100 120 14 0 DISTANCE FROM BOTTOM OF CASTING (cm) FIGURE 69: Solute distribution as a function of temperature gradient. 161 7.6 Comparison with Experiment The data used to generate the theoretical curves in Figures 53-56, together with theoretical and experimental values of macrosegregation according to the present definition (Equation 6.2), are given in Table XII. In general, the model predicts profiles of the same shape as the experimental plots, but the compositions at the ends of the ingot do not always agree well with those predicted. This is due to the assumptions used in deriving the model. In addition to those already discussed, Equation 7.1 does not take into account that the liquid composition cannot rise above the eutectic composition. The aim of the experiments in Chapter 6 was to demonstrate that macrosegregation was related to the solidification variables. They were not specifically designed to test the model, consequently only qualitative comparisons have been made. When attempts were made to use the computer program to calculate the solute profiles for hypothetical ingots with very large dendrite spacings, it was found that the composition at the top would rise to a very large value. This was probably due to the breakdown in the assumption that the flow rate between layers is small compared to the amount of liquid in each layer. However, in the case of the data in Table XII, the maximum theoretical composition for each ingot was well below the eutectic, which is an indication that this assumption was reasonably valid for the dendrite spacings involved in these experiments. For Figures 53(a) and 54(a), which were solidified under the same temperature gradient, but with different growth rates and dendrite spacings, both the theoretical and experimental results show more macrosegregation at TABLE XII SOLIDIFICATION VARIABLES USED FOR THEORETICAL PLOTS Figure Number 53(a) 54(a) 55(a) 56(a) Temperature Gradient (G) °C/cm 1.5 1.5 2.3 1.0 Growth Rate (R) cm/sec 0.0047 0.013 0.011 0.24 Length of Casting (H) cm 14 14 14 14 3 Number of Channels (NC) = nr A 2.96 x 105 4.24 x 105 8.86 x 105 3.73 x 106 Viscosity of the Liquid (u) poise 0.03 0.03 0.03 0.03 Number of Layers 28 28 28 29 AC (theoretical) 1.95 0.39 0.31 0.002 AC (experimental) 1.07 0.73 0.13 0.27 163 the lower growth rate. For Figures 54(a) and 55(a), which were solidified at approximately the same growth rate, but with different temperature grad ients and dendrite spacings, the theory predicts slightly more macrosegrega tion at the lower temperature gradient. This is qualitatively in agreement with experiment. It should be noted that although Figure 69 predicts less macrosegregation at lower temperature gradients (when all other variables are held constant), this effect has been outweighed by the difference in (31) dendrite spacing. This also corresponds with the evidence cited earlier that a reduction in dendrite spacing eliminated freckles in consumable arc melted ingots. The theoretical result for Figure 56(a) shows hardly any macroseg regation, which would correspond with the earlier suggestion that it is not possible to claim any significant macrosegregation in this particular experiment, due to the large amount of scatter. Using the model predictions, it is possible to recommend a number of changes in casting practice that would reduce gravity segregation effects in vertical directional castings: 1) Refinement of the dendritic structure will increase the resistance to flow through the mushy zone. 2) Reduction of ingot height for alloys with a wide freezing range will reduce the driving force for flow. 3) Increasing the growth rate will reduce the time available for flow. 4) Decreasing the temperature gradient will reduce the driving force for flow. 164 CHAPTER 8 CONCLUSIONS 8.1 Summary Interdendritic fluid flow rates have been measured in the lead-tin alloy system with gravity as the driving force. The results have been used to calculate the permeability of the dendritic structure, as defined by Darcy's Law - the standard empirical relationship which describes flow through porous media. It was found that the permeability of a dendritic array is a sensitive function of the primary dendrite spacing. The permea bility results were shown to be consistent with a simple model of the porous medium, which considers the interdendritic channels to be equivalent to a bundle of capillary tubes. It was shown that the interdendritic liquid flowed uniformly through the dendritic array, without the formation of preferential channels, by direct examination of the etched structure and with radioactive tracer techniques. Deviations from Darcy's Law, which occurred when the samples were held above the eutectic temperature for long periods of time, were discussed in relation to dendrite coarsening effects, similar to Ostwald ripening, or sintering in ceramics. Lead-tin alloys were used to investigate the formation of channel-type casting defects (freckles and A segregates). Isothermal and unidirect ional solidification experiments were used to study pipe formation and solute convection, caused by density differences in the interdendritic liquid. Macrosegregation was observed in ingots where the liquid close 165 to the bottom of the solid-liquid zone was less dense than the liquid above, and the resulting profiles were shown to be related to the growth rate, temperature gradient, dendrite spacing, and alloy composition. Shrinkage trails and pipes were produced in some of these experiments when the growth rates were very low. These findings support the previously proposed mechanism for the formation of channel-type defects, based on density differences in the liquid causing interdendritic fluid flow. A numerical model is proposed, which predicts the composition profiles in vertical, directionally solidified castings, as a function of the solidification variables. Density differences in the liquid are taken to be the driving force for macrosegregation, and the dendritic structure is considered to be a porous medium of variable porosity. 8.2 Conclusions i) Using the simple capillary model to describe the dendritic array, the permeability is proportional to the square of the primary dendrite spacing. ii) The "tortuosity factor", which allows for the fact that the interden dritic channels are neither straight nor symmetrical is equal to 4.6 iii) Castings held for long periods of time above the eutectic temperature show dendrite coarsening effects which can modify the structure. The permeability of castings held a few degrees above the eutectic temperature was observed to increase due to this effect. iv) Macrosegregation and channel-type defects can be produced in the lead-tin system by solute convection causing upward flow of less dense liquid. The rising liquid becomes superheated and can form a 166 pipe or channel by dissolving dendrite branches in its path, v) Using the relationship between permeability and structure determined from the interdendritic fluid flow measurements, the numerical macro-segregation model is qualitatively in agreement with the directional solidification experiments. The model can therefore be used to recommend changes in casting practice to reduce gravity segregation effects. 8.3 Suggestions for Future Work i) Using the interdendritic flow measurement technique developed in this work, permeabilities could be measured in other alloy systems, in particular, those systems where the dendrites do not have orthogonal branches. ii) With suitable permeability data for high liquid fractions, the funda mental nature of interdendritic fluid flow could be investigated further, leading to a better theory than the simple capillary model used in this work. iii) The effect of dendrite coarsening could be used as a method of modifying the cast structure. iv) Since density differences in the liquid have been shown to produce freckles in ammonium chloride-water models, and in lead-tin alloys, the next step would be to add radioactive tracers to commercial ingots which are prone to this defect. This method would rapidly provide information on solute convection in these castings, since density data is often not available for many liquid alloys used commercially. v) The most important improvement in the mathematical model would be to 167 have a better description of the solid-liquid zone than given by the simple capillary model. Since the model shows that macro-segregation is very sensitive to the structure, this would probably have the most pronounced effect. In addition, the model could be improved by considering backflow due to volume shrinkage, and by using available data on liquid viscosity as a function of temperature and composition, rather than a constant. 168 REFERENCES 1. T.S. Piwonka: "Interdendritic Flow during Solidification of Aluminum Alloys", D.Sc. Thesis, 1963, Mass. Inst, of Technology. 2. T.S. Piwonka and M.C. Flemings: Trans TMS-AIME, 1966^ vol. 236, pp. 1157-65. 3. J. Campbell: Trans TMS-AIME, 1968, vol. 242, p. 264. 4. J. Campbell: Trans TMS-AIME, 1968, vol. 242, p. 268. 5. J. Campbell: Trans TMS-AIME, 1968, vol. 242, p. 1464. 6. J. Campbell: Trans TMS-AIME, 1969, vol. 245, p. 2325. 7. R.H. Tien: J. Appl. Mech., 1972, vol. 3, p. 333. 8. N. Standish: Met. Trans., 1970, vol. 1, p. 2026. 9. R. Mehrabian, M. Keane and M.C. Flemings: Met. Trans., 1970, vol. 1, pp. 1209-20. 10. F.L. Kaempffer: "interdendritic Fluid Flow", M.A.Sc. Thesis, 1970, Univ. of British Columbia. 11. F.L. Kaempffer and F. Weinberg: Met. Trans., 1971, vol. 2, pp. 3051-54. 12. F. Weinberg and R.K. Buhr: I.S.I. Publication 110, 1968, p. 295. 13. M.C. Flemings: R.V. Barone, S.Z. Uram and H.F. Taylor: Trans TMS-AIME, 1961, vol. 69, p. 422. 14. M. Muskat: "The Flow of Homogeneous Fluids through Porous Media", 1937, McGraw-Hill. 15. P.C. Carman: "Flow of Gases through Porous Media", Butterworths Scient ific Publications, London, 1956. 16. A.E. Scheidegger: "The Physics of Flow through Porous Media", Univ. of Toronto Press, 1957. 17. P.C. Carman: Trans. Inst. Chem. Eng. (British), 1937, vol. 15, pp. 150-166. 18. C.L. Rice and R. Whitehead: J. Phys. Chem., 1965, vol. 69, pp. 4017-24. 19. N.F. Bondarenko and V.G. Karmanov: Soviet Physics-Doklady, 1969, vol. 13, No. 8, p. 791. 169 20. F. Weinberg and E. Teghtsooriian: Met. Trans., 1972, vol. 3, pp. 93-111. 21. M.J. Stewart: "Natural Convection in Liquid Metals", Ph.D. Thesis, 1970, University of British Columbia. 22. L.C. MacAulay: "Liquid Metal Flow in Horizontal Rods", Ph.D. Thesis, 1972, University of British Columbia. 23. F. Weinberg: Trans. TMS-AIME, 1961, vol. 221, pp. 844-850. 24. T.E. Strangman and T.Z. Kattamis: Met. Trans., 1973, vol. 4, p. 2219. 25. J.M. Coulson and J.F. Richardson: "Chemical Engineering", 1968, vol. 2, Pergammon Press. 26. T.Z. Kattamis, J.C. Coughlin and M.C. Flemings: Trans. TMS-AIME, 1967, vol. 239, pp. 1504-1511. 27. G.W. Greenwood: Acta Met., 1956, vol. 4, pp. 243-248. 28. G.W. Greenwood: Monograph and Report Series No. 33, 1969, Inst, of Metals, London, p. 103. 29. W.D. Kingery: "introduction to Ceramics", 1960, John Wiley & Sons Inc., p. 375. 30. A.F. Giamei and B.H. Kear: Met. Trans., 1970, vol. 1, pp. 2185-91. 31. R.P. DeVries and G.P. Mumau: J. Metals, Nov. 1968, vol. 20, p. 33. 32. G.C. Gould: Trans. TMS-AIME, 1965, vol. 233, p. 1345. 33. T. Mukherjee: 3rd Int. Symp. on Electroslag and other special melting technology, A.S.M. and Mellon Inst., June 1971, Symposium Proceedings Part II, p. 215. 34. S.M. Copley, A.F. Giamei, S.M. Johnson and F. Hornbecker: Met. Trans., 1970, vol. 1, pp. 2193-2204. 35. C.E. Smeltzer: Iron Age, 1959, vol. 184, No. 11, p. 188. 36. R. Mehrabian, M. Keane and M.C. Flemings: Met. Trans., 1970, vol. 1, pp. 3238-41. 37. R.J. McDonald and J.D. Hunt: Trans. TMS-AIME, 1969, vol. 245, pp. 1993-97. 38. J.R. Blank and F.B. Pickering: "The Solidification of Metals", I.S.I. Publication 110, pp. 370-376. 170 39. H.P. Utech, W.S. Bower and J.G. Early: "Crystal Growth", Proceedings of an International Conference on Crystal Growth, Boston, June 1966, p. 201. 40. N.Streat and F. Weinberg: Met. Trans., 1972, vol. 3, pp. 3181-84. 41. D.J. Hebditch and J.D. Hunt: Met. Trans., 1973, vol. 4, pp. 2008-10. 42. H.R. Thresh, A.F. Crawley and D.W.G. White: Trans. TMS-AIME, 1968, vol. 242, pp. 819-22. 43. J. Kohl, R.D. Zentner and H.R. Lukens: "Radioisotope Applications Engineering", 1961, Van Nostrand and Co. 44. B. Prabhakar: M.A.Sc. Thesis, University of British Columbia, 1973. 45. M.C. Flemings and G.E. Nereo: Trans. TMS-AIME, 1967, vol. 239, pp. 1449-1461. 46. D.J. Hebditch and J.D. Hunt: Met. Trans., 1973, vol.4, p. 2474. 47. T.F. Bower, H.D. Brody and M.C. Flemings: Trans. TMS-AIME, 1966, vol. 236, pp. 624-34. 48. C.J. Smithells: "Metals Reference Book", vol. 2, 4th Edition, Butterworths, London, 1967. 49. E. Scheil: Metallforschung, 1942, vol. 20, p. 69. 50. J.S. Kirkaldy and W.V. Youdelis: Trans. TMS-AIME, 1958, vol. 58, p. 212. 51. W.V. Youdelis: "The Solidification of Metals", I.S.I. Publication 110, December 1967, p. 112. 52. M.C. Flemings, R. Mehrabian and G.E. Nereo: Trans. TMS-AIME, 1968, vol. 242, pp. 41-49. 53. M.C. Flemings and G.E. Nereo: Trans. TMS-AIME, 1968, vol. 242, pp. 50-55. 54. H.R. Thresh and A.F. Crawley: Met. Trans., 1970, pp. 1531-35. 55. M.E. Glicksman and C.L. Void: Acta Met., 1967, vol. 15, pp. 1409-12. 56. M.E. Glicksman and C.L. Void: "The Solidification of Metals", I.S.I. Publication 110, December 1967, pp. 37-42. 57. M.E. Glicksman and C.L. Void: Journal of Crystal Growth, 1972, vol. 13, pp. 73-77. 171 58. M.E. Glicksman and C.L. Void: Acta Met., 1969, vol. 17, pp. 1-11. 59. M.E. Glicksman and C.L. Void: Scripta Met., 1971, vol. 5, pp. 493-498. 60. M. Hansen: "Constitution of Binary Alloys", Second Edition, 1958, McGraw-Hill, p. 302. 172 APPENDIX I INTEGRATION OF DARCY'S LAW FOR A FALLING HEAD area a. FIGURE 70: Pb-Sn alloy in the flow cell, after a time t. Darcy's Law states: v = PL AP A.l where v = bulk velocity K = permeability AP = pressure drop across the porous medium u = viscosity of the liquid . L = length of the porous medium. During a brief time interval dt, the quantity which flows through the porous medium will be dq, therefore: A dt A.2 173 where A = area of the porous medium. -KA Thus dq = ~ pghtdt A. 3 where p = density of the liquid g = gravity h = head at time t. If the volume of liquid which has risen up the riser pipe is a2& (where a2 is the area of the riser, and £ the length risen), an equal volume will have fallen in the reservoir above the bed. The distance fallen will therefore be a2^/a^, where a^ is the area of the reservoir. The distance hfc is therefore given by: h. = h - (a-A/a.) - I t oil i.e. h = h - £(1 + a./a.) A.4 to 2 1 where hQ is the original head at t = 0. Thus for a small change in the head: dhfc = - (1 + a2/ai)d£ A.5 The volume which flows up the riser pipe during time dt is dq, where dq = - a0d£ 2 A. 6 Substituting in Equation A.5 dh = (1 + a9/a.) t / 1 Since the quantity flowing in the riser equals the quantity flowing through the porous medium, Equation A.7 can be combined with Equation A.3: -KApght A2dht dt UL (1 + a.^/a.^) i.e. dt = - (c/K)dh /h A.8 where c = a^L/(1 + a2/a1)Apg Integrating: h t dt = - (c/K) | dht/ht h o t = - (c/K) In (h /h ). t o 175 APPENDIX II FORTRAN PROGRAM FOR PROCESSING INTERDENDRITIC FLUID FLOW DATA THIS PROGRAM CALCULATES THE PERMEABILITY OF A CASTING FROM THE FLUID PLOH MEASUREMENTS. IT READS L AND T DATA (DISTANCE PLOWED UP THE RISER PIPE, AND TIME) WHICH IT CONVERTS TO THE FORM OF EQUATION 4.2 IN THE TEXT. THE INITIAL PERMEABILITY IS THEN FOUND USING THE METHOD OF LEAST SQUARES ITERATIVELY, AS DESCRIBED IN SECTION 4.2.1. THE TIME DEPENDENCE OF THE PERMEABILITY IS SUBSEQUENTLY CALCULATED BY FITTING THE DATA TO AN EQUATION OF THE FORM GIVEN IN EQUATION 4.19 OF THE TEXT. EXTERNAL LINE DIMENSION F (4 5) . WW (60) ,YF(60) ,E1 (2) ,E2 (2) ,P (2) ,T (60) , ALH (60) ,TT (60 • ) ,A(4) ,TU(60) ,W(60) REAL L (60),LN(60),H,K1,LU(60) DATA TS/'SEC. V.TH/'MIN. V.LI/'II'-'/.f/'MH. V»I"F/'/26V C F-TEST TABLE FOR A SIGNIFICANCE LEVEL OF 0.05 DATA F/2.43,2.27,2.16,2.07,2.00,1.94,1.89,1.85,1.82,1.79,1.76, 11.74,1.71,1.69,1.68,1.66,1.64,1.63,1.6 2,1.61,1.60,1.59,1.58,1.57,1 1.56, 1.55, 1.54, 1.53, 1.53, 1.52,1.51,1.51 ,1.50, 1.49,1.49,1.48,1.47, 1. 147, 1.46, 1.46, 1.46, 1.46, 1.45, 1. 45, 1.45/ V=.8740711E-05 C V-WBIGHTED VARIANCE OBTAINED FROM THE CALIBRATION TEST DESCRIBED IN C SECTION 3.7. IT IS USED IN THE F-TEST COMPARISON CA=0.315*0.315 CB=0.75*0.75 C1=CA»1.33/ (0.93*0.93*(1.0+CA/CB) ) *2.54 C C1=LOWER CASE C IN THE TEXT VISC=.03 C VISC=GREEK MU IN THE TEXT C1=C1*VISC/(981.*8.33) WRITE(7,131) 900 HEAD (5,1,END=901)A 1 FORMAT(4A4) READ (5,2)H,HO,TA,TC,LC,IF C H=NUHBER OF DATA POINTS PER TEST C H0=INITIAL HEAD OF LIQUID-LOWER CASE H,SUBSCRIPT 0 IN THE TEXT C TA=TIHE BETWEEN MELTING AND THE ZERO POINT OF FLOW MEASUREMENTS C TC=UHITS OF TIME C LC=0BXTS OF LENGTH C IF=1 OR 0, DEPENDING ON PORMAT OF L AND T DATA 2 F0RMAT(I3,F6.3,F7.2,A4,A3,I1) IP (IF.EQ.O)GO TO 4 C READ L DATA AND T DATA IN DIFFERENT FORMATS READ(5,3) (L(I),T(I),I=1,N) 3 FORHAT(F5.3,P10.3) GO TO 65 4 READ (5,5) (T(I) , 1=1,N) 5 FORMAT (11F7.2) RSAD(5,6) (L (I) ,1=1,N) 6 FORMAT(13F6. 3) 65 IF (LC.EQ.LI) PL=1. IP (LC.EQ.LP)PL*1./26. IP (LC.EQ.LH)PL-1./25.4 176 DO 7 1*1,H T(I)=T(I) *Tk LO(I)*L (I) TO (I)«T(I) 7 L(I)»PL*L(I) 8 IP (TC.BQ.TS) GO TO 9 DO 85 1*1,1 C COBVEBT T TO SECONDS PROH THE INSTANT OF BELTING 85 T(I)=60.»T(I) 9 HO=(1.0+CA/CB)/HO DO 10 I*1,8 HM(I) = 1 • 0-H0*L (I) C LI(I)*LN (HT/HO) IN EQUATION Q.2 OF THE TEXT 10 LI(I)=ALOG(Hli(I)) HBITB (6 , 10 1) A 101 FORMAT(////IX,20A4) WHITE(6,103)LC.TC 103 FOBNAT(/1X,' NO. L(»,*3,') T(«,AU,«) L (IB. ) T(SEC.) LN(1-HO* 1L) •) HBITE(6,102) (I,LU(I) ,T0(I) ,L(I) ,T(I) ,LN(I) ,1=1,«) 102 FOBHAT (1X,I<»#F8.3,2F9.3,F8.0,21,E1U.7) C LEAST SQUARES FITTING ROUTINE (LN(I) VEBSUS T(I)) J=6 JJ=1 111 SH=0. SXYH=0. SXB>0. SYW=0. SWXS=0. DO 12 1=1, J H(I)=HH(I) • WH(I) C » (I) WEIGHTING FACTOR SH=SH + H(I) SXYH=SXYH*T(I)*LN (I)*H (I) SXH=SXN*T(I)*W(I) SYH=SYH*LH(I)*H (I) 12 SBXS=S»XS*B(I)*T (I)*T (I) DBH*SHXS*Sll-SXW*SXi1 IF (DEB.EQ.0.)GO TO 121 NUH=(SXYW*SW-SXH*SYH) R=NUR/DEN GO TO 220 121 H=0. 220 AN*FLOAT(J) SX*0. SI=»0„ DO 13 1=1,J SX=SX*T(I) 13 SI»SY*LN(I) TBAB=SX/AN C BEST FIT VALUES OF LB(I) ARE ALBAR,AND FOB T(I) ARE TBAR AL8AHZSY/AN C=ALBAR-H*TBAR SBES=0„ DO 1»4 1=1,J BES=H(I)*(LN(I)-C-H*T(I))*(LN (I)-C-H*T (I)) 1«» SRES=SRES*RES C CALCULATE VARIANCE VAB=SBES/(AN-2.) IP (VAB.LE.V)GO TO 112 177 C DO P-TEST, AND ITERATE TO FIHD THE MAXIMUM NUMBER OF POINTS WHICH C CAN BE USED FOR THE INITIAL SLOPE PP-VAR/V IP (PP.GE.P(JJ))GO TO 161 IP (J.EQ.SO)GO TO 161 112 J«J*1 JJ-JJM GO TO 111 C CALCULATE THE STANDARD ERROR OP Y PROS THE VARIANCE 161 SEY*SQRT (VAB) SWYS*0. DO 15 I»1,J 15 SWYS<*SWYS«LM (I) *LN (I) *W (I) R«NUH/SQRT (DEN* (SWIS*SW-SYW*SYW) ) WRITE(6,104)H,C,R,SBY 104 PORHAT(//1X,»SLOPE=',E16.7, • INTERCEPT3*',E16.7//IX,'CORRELATION C 10EPPICIBNT='.E16.7//1X,»STD. ERROR OF Y=',E16.7) WRITE (6,105)VAR 105 FORMAT(//IX,'WEIGHTED VARIANCE OF Y=',E16.7) WRITE (6,106) J 106 FORMAT(//IX,'NO. OF DATA POINTS USED TO ESTIMATE SLOPE=',I3) C CALCULATE THE INITIAL PERMEABILITY ( K 1=K IN THE TEXT) K1=-H*C1 TO*-C/H DO 18 I»1,N C FIT DATA TO THE PORM OF EQUATION 4.19 OF THE TEXT TT(I)=T(I)-TO 18 ALH (I)=-C1*LN (I)-K1*TT (I) C ALH»RIGHT HAND SIDE OF EQUATION 4.19 WRITE(6,1061)TO,K1 1061 FORMAT(////1X,'TIME BETWEEN MELTING AND ZERO POINT OF FLOW MEASURE 1 HENTS (TO)=',P7.2,'SEC. '//1X,'PERMEABILITY(K1) AT TIME TO=',E16.7,' • (SQ.CM.) •) WRITE (6,140) 140 FORMAT(/1X,'RESULTS FROM LQF'/) P(1)=0.0 P(2) =0.0 C USE THE LIBRARY LEAST SQUARES FITTING ROUTINE TO CHECK THE LEAST C SQUARES ROUTINE THAT WAS WRITTEN FOR THIS PROGRAM. CALL LQF(T,LN,YF,W,E1,E2,P,1.0,J,2,1,ND,1.E-4,LINE) WRITE(8) A,N,P (1) ,P(2) EM=E2 (1)/P(1) BC=B2(2)/P(2) P(2)=-P(2)/P(1) P(1)=-P(1) *C1 WRITE (8) P (1) ,P (2) WRITE (8) (L(I) ,T(I) ,LN(I) ,I=1,N) WRITE(6,141)El (1),E2 (1) 141 FOR9AT(1X,0 STATISTICAL ERROR IN SLOPE=•,E16.7,3X,•TOTAL ERROR IB S •LOPE=«,E16.7) WRITE (6, 142) E1 (2) , E2 (2) 142 FORMAT (IX,'STATISTICAL ERROR IN INTERCEPT=•,E16.7,3X,'TOTAL ERROR • IN INTERCEPT31' , E16. 7) ETO*SQRT((EM*EM*EC*EC) *P(2)*P(2) ) ETO=2.0*ETO EK1= (C1*E2(1))*2.0 WRITE (6, 143) P (1) ,EK1,P(2) , ETO 143 FORMAT(/1X,'PERMEABILITY (K1) AT TIME TO=•,E16.7,• (SQ.CH.)•,2X,•95* • CONF. INTERVAL3',E16.7//1X,'TIME BETWEEN MELTING AND ZERO POINT 0 •F PLOW MEASUREMENTS(TO)=•,F7.2,'SEC.•,2X,'951 CONF. INTERVAL=•,P7. 178 •3) BR ITE (8) (ALH(I) ,TT(I) ,1=1,N) WRITE (6,107) 107 FORMAT{////IX,'DATA FOR HON LINEAR LEAST SQUARES FITTING'//1X,6X, • •T-TO',10X,•ALH') WRITE(6,108) (I,TT(I) ,ALH (I) ,1=1,N) 108 FORHAT (1I,I3,F7.0,2X,E16.7) WRITE(6,109) 109 FORMAT (//1X,120(**')) WRITE(7,132)A,N,J,R,P(2) ,ETO,P(1) ,EK1 GO TO 900 131 FORMAT(<*5X,'SUMMARY OF TEST RESULTS'/20X,* N',6 X,' J•,7X,• R',81,•TO* •,«X,'ERROR',8X,»K1 ',8X,'ERROR') 132 FORMAT («AU,2X,I3#<*X,I3,UX,F6.ft,ftX,P5.0,2X,P7.3f E13. <*, E12. U) 901 STOP END FUNCTION LINE (P.D.T.LQ) DIMENSION P (2),D (2) D(1)=T D(2)=1.0 LINE=P (1) *T*P (2) RETURN END APPENDIX III 179 THE SOLIDIFICATION OF Pb-20%Sn - A TABLE OF SOLIDIFICATION VARIABLES Values of the partition ratio kQ, and the liquid composition C^, were obtained as a function of temperature from the phase diagram. The solid composition C and the weight fraction liquid f were calculated using the S Li Pfann equation, as described in section 7.2. The volume fraction of (42) liquid was calculated using the density data for Pb-Sn alloys T°C k0 CL Cs fL gL 276.0 0.501 20.005 0.000 1.000 1.000 275.0 0.497 20.488 10.107 0.953 0.956 274.0 0.493 20.974 10.184 0.910 0.915 273.0 0.490 21.461 10.259 0.870 0.877 272.0 0.486 21.951 10.332 0.833 0.841 271.0 0.482 22.442 10.402 0.798 0.808 270.0 0.478 22.935 10.470 0.765 0.776 269.0 0.474 23.430 10.535 0.734 0.747 268.0 0.471 23.926 10.598 0.706 0.720 267.0 0.467 24.424 10.659 0.679 0.694 266.0 0.463 24.923 10.718 0.654 0.670 265.0 0.459 25.423 10.776 0.630 0.647 264.0 0.456 25.925 10.831 0.608 0.626 263.0 0.452 26.427 10.884 0.587 0.606 262.0 0.449 26.931 10.936 0.567 0.587 261.0 0.445 27.436 10.987 0.548 0.569 260.0 0.442 27.941 11.035 0.531 0.552 259.0 0.438 28.447 11.082 0.514 0.535 258.0 0.435 28.954 11.128 0.498 0.520 257.0 0.432 29.462 11.173 0.483 0.506 256.0 0.428 29.970 11.216 0.469 0.492 255.0 0.425 30.478 11.258 0.455 0.488 254.0 0.422 30.986 11.298 0.442 0.475 253.0 0.419 31.495 11.338 0.430 0.464 252.0 0.416 32.004 11.376 0.418 0.452 251.0 0.413 32.513 11.414 0.407 0.436 250.0 0.410 33.022 11.450 0.397 0.426 249.0 0.407 33.531 11.486 0.386 0.416 248.0 0.404 34.039 11.520 0.377 0.407 247.0 0.402 34.548 11.554 0.368 0.398 246.0 0.399 35.055 11.586 0.359 0.389 245.0 0.396 35.563 11.618 0.350 0.381 244.0 0.394 36.070 11.650 0.342 0.373 243.0 0.391 36.576 11.680 0.334 0.365 242.0 0.389 37.081 11.710 0.327 0.358 241.0 0.387 37.586 11.739 0.320 0.351 240.0 0.384 38.089 11.767 0.313 0.344 239.0 0.382 38.592 11.795 0.306 0.338 238.0 0.380 39.093 11.822 0.300 0.332 T°C 237.0 236.0 235.0 234.0 233.0 232.0 231.0 230.0 229.0 228.0 227.0 226.0 225.0 224.0 223.0 222.0 221.0 220.0 219.0 218o0 217.0 216.0 215.0 214.0 213.0 212.0 211.0 210.0 209.0 208.0 207.0 206.0 205.0 204.0 203.0 202.0 201.0 200.0 199.0 198.0 197.0 196.0 195.0 194.0 193.0 192.0 191.0 190.0 189.0 188.0 187.0 186.0 185.0 184.0 "0 0.378 0.376 0.374 0.372 0.370 0. 368 0.367 0. 365 0. 363 0. 362 0.360 0. 359 0. 358 0.356 0.355 0. 354 0.352 0. 351 0.350 0.349 0.348 0.347 0.346 0. 345 0.344 0.343 0.342 0.341 0. 340 0.339 0.338 0.337 0. 336 0. 335 0.334 0.333 0.332 0.331 0. 330 0.329 0.327 0. 326 0.325 0o 324 0.322 0. 321 0. 319 0.318 0.316 0.314 0. 312 0. 310 0.308 0. 306 C. Cs fL 9L 180 'L 39.594 40.092 40.590 41.086 41.581 42.074 42.565 43.054 43.542 44.027 44.511 44.992 45.471 45.947 46.421 46.893 47.362 47.828 48.292 48.752 49.210 49.664 50.1 16 50.564 51.009 51.450 51.888 52.322 52.753 53.180 53.603 54.022 54.437 54.847 55.254 55.656 56.054 56.447 56.836 57.221 57.600 57.975 58.344 58.709 59.068 59.423 59.772 60.1 15 60.453 60.786 61.113 61.434 61.750 62.059 11.849 11.875 11.901 11.926 11.950 11.974 11.998 12.021 12.043 12.066 12.088 12.109 12.130 12. 151 12.171 12.191 12.211 12.230 12.249 12.267 12.286 12.304 12.321 12.339 12.356 12.372 12.389 12.405 12.421 12.436 12.452 12.467 12.481 12.496 12.510 12.523 12.537 12.550 12.563 12.575 12.588 12.599 12.611 12.622 12.633 12.644 12.654 12.664 12.674 12.683 12.692 12.700 12.709 12.716 0.294 0.288 0.282 0.277 0. 272 0.267 0.262 0.257 0. 253 0.248 0.244 0.240 0.236 0.232 0.229 0.225 0.222 0.218 0.215 0.212 0.209 0.206 0.203 0.201 0.198 0. 195 0.193 0. 190 0. 188 0. 186 0. 184 0.181 0.179 0. 177 0.175 0. 173 0.172 0. 170 0.168 0. 166 0. 165 0. 163 0.162 0. 160 0. 159 0. 157 0.156 0. 155 0.153 0. 152 0. 151 0. 150 0.149 0. 148 0.326 0.319 0.313 0.308 0.303 0.298 0.293 0.288 0. 284 0.280 0.275 0.271 0.268 0.264 0.260 0.257 0.253 0.250 0.247 0.251 0.248 0.245 0.235 0.232 0.230 0.227 0.225 0.222 0.220 0.218 0.215 0.213 0.211 0.209 0.207 0. 205 0.203 0.202 0.200 0. 198 0.197 0. 195 0. 193 0. 192 0. 191 0. 189 0.188 0. 191 0.190 0. 189 0.187 0. 186 0.185 0. 184 181 APPENDIX IV FORTRAN PROGRAM FOR CALCULATING MACROSEGREGATION IN LEAD-TIN CASTINGS THIS PROGRAM CALCULATES MACROSEGREGATION ACCORDING TO THE MODEL DESCRIBED IN CHAPTER 7. THE METHOD BASICALLY INVOLVES THE FOLLOWING STEPS: 1) THE CASTING IS DIVIDED INTO A NUMBER OF HORIZONTAL LAYERS 2) THE TEMPERATURE OF THE BOTTOM LAYER IS SET EQUAL TO THE LIQUIDUS TEMPERATURE (SEE SECTION 7.4) 3) KNOWING THE TEMPERATURE GRADIENT, THE TEMPERATURE OF ALL THE OTHER LAYERS ARE CALCULATED. 4) THEREFORE KNOWING THE GROWTH RATE, THE TEMPERATURE OF EACH LAYER AT ANY POINT IN TINE IS DEFINED. THEREFORE, DURING SOLIDIFICATION, ALL OTHER VARIABLES CAN BE DETER MINED AS A FUNCTION OF TEMPERATURE. THUS, LIQUID COMPOSITION AND THE PARTITION RATIO ARE A FUNCTION OF TEMPERATURE FROM THE PHASE DIAGRAM. USING THE PFANN EQUATION (SECTION 7.2) THE FRACTION LIQUID AND COMPOSITION OF SOLID CAN BE CALCULATED FOR EACH LAYER AT EVERY POINT IN TIME. SINCE THE COMPOSITION AND TEMPERATURE OF EACH LAYER IS NOW DEFINED, ITS DENSITY IS GIVEN USING THE SUBROUTINE DENS. DENSITY DIFFERENCES THROUGH THE MUSHY ZONE PROVIDE THE DRIVING FORCE FOR FLUID FLOW, AND THE RESISTANCE OF THE DENDRITIC STRUCTURE IS CALCULATED USING DARCY'S LAW (SECTION 7.3). THE MAIN ROUTINE READS IN THE DATA, AND THEN SETS THE INITIAL TEMPERATURE OP EACH LAYER OF THE CASTING. IT USES THE SOLIDIFICATION MODEL TO CALCULATE SOLID AND LIQUID COMPOSITIONS AND FRACTIONS. THE PHASE DIAGRAM DATA IS CONTAINED IN SUBROUTINE PBSN. EACH SOLIDIFICATION STEP IS FOLLOWED BY A FLUID FLOW STEP: THE LATTER IS CONTAINED IN SUBROUTINE FLOW. TWO DIMENSIONAL ARRAYS ARE USED FOR SOME OF THE VARIABLES. IN THIS CASE THE COLUMNS (1ST DIMENSION) REPRESENT THE POSITION - LAYER NUMBER - IN THE CASTING, AND THE ROWS (2ND DIMENSION) REPRESENT THE POIBT IN TIME. HEAHIBG OF SYMBOLS IN MAIN PROGRAM AK=EQUILIBBIUM DISTRIBUTION COEFFICIENT AKO=AVERAGE DISTRIBUTION COEFF. BETWEEN TWO TEMPERATURES AL,ALAYER=NUMBER OF LAYERS (REAL NUMBER) ALEN=LENGTH OF INGOT AH=NUMBER OF INTERDENDRITIC CHANNELS C=TORTUOSITY FACTOR (EQUALS GREEK TAU CUBED) CL=COHPOSITION OF LIQUID COHP=TOTAL COMPOSITION OF LAYER CS=COHPOSITIOH OF SOLID DIST=DISTANCE FROH BOTTOM OF INGOT DHSTY=DENSITY OF LIQUID DT'TIHE INTERVAL (INCREMENT) FL=1-PS PRLIQ=WT. FRACTION OF LIQUID PS=PBOPORTION OF FRLIQ THAT IS FREEZING 182 G=TEHPERATURE GRADIENT GL=VOLUME FRACTION OF LIQOID L=NUHBER OF LAYERS (INTEGER) P=A POWER PERH=PERHEABILITY OF LAYER R=GROWTH RATE SD=SOLID DENSITY SSOL=WT. OF SOLID SSOLUT=TOTAL WT. OF SOLUTE IN SOLID STATE= STATE OF LAYER (SOLID,MUSHY OR LIQUID) T=TOAL SOLIDIFICATION TIME TB,TB1,TB2=DATA TABLES GENERATED BY THE PROGRAM TE=SOLIDUS TEMPERATURE TIHB=TOTAL TIME AFTER THE START OF FREEZING TEMP=TEHPERATORE OF LAYER TL=LIQUIDUS TEMPERATURE TSOLOT=TOTAL WT. OF SOLUTE IN SOLID AND LIQUIC TWT=TOTAL WT. OF SOLID AND LIQUID VISC=VISCOSITY OF THE LIQUID VL=VOLUHB OF LIQUID VS=VOLUHE OF SOLID WSLIQ=WT. OF SOLUTE IN FRLIQ WSOLID=WT. OF SOLID THAT IS FREEZING WSOLUT=WT. OF SOLUTE IN WSOLID DIMENSION TEMP(100,2),CL(100,2) ,AK (100,2) ,DNSTY (100),FRLIQ (100,2) DIMENSION COMP (100) ,CS (100,2) ,WSLIQ (100) DIMENSION SSOL(100),SSOLUT (100),GL (100) DIMENSION DIST (100) ,PERM (100) INTEGER TB,TB1,TB2 DATA ALIQ/« LIQ•/,*HUSH/«HUSH•/,SOL/, SLD'/ READ (5,203)DT READ(5,201) ALEN,G,R READ (5,202)TL,TE RBAD(5,204) AN,C,VISC READ (5,205)TB1,TB2 201 FORMAT(F6.1,F6.2,F7.tt) 202 FORMAT (2F10.U) 203 FORMAT(F10. 1) 20U FORMAT(E9.3,F6.2,F6.3) 205 FORMAT(215) ALAYER=ALEN/ (R*DT) L=INT (ALAYER) AL=FLOAT (L) T=((TL*AL*G*R*DT)-TE)/(G*R) TEMP(1,1)=TL DO 1 1=2,L TEMP(I,1)=TEHP(1-1,1)•G*H*DT 1 CONTINUE DO 11 1=1,L CALL PBSN(TEMP (1,1) ,AK (I, 1) ,CL (I, 1)) CL(I,1)=CL(1,1) COMP (I)=CL(I, 1) PRLIQ(I,1) = 1. SSOL (I)=0. SSOLUT(I)=0. WSLIQ(I)=CL(1,1) GL (I)=1. CS(I,1)=0. 183 11 CONTINUE TIHE-0. PL-1. WSOLID=0. WSOLUT-0. TSOLUT=CL(1,1) TWT=1. STATE=ALIQ TB=1 WRITE (6,101) TIRE 101 FORMAT (//IX,• TIME=',F7. 1, 'SECONDS -BEFORE FLUID FLOW•/) WRITE(6,102) 102 FORMAT (IX,' L TEMP STATE CL AK PRLIQ DNSTI FL • WSOLID WSLIQ WSOLUT SSOL SSOLUT CS GL TNT • COMP'/) DO 2 1=1,L CALL DENS(TEMP(1,1),CL(1,1),DNSTX (I)) WRITE(6,103)I,TEMP(1,1) ,STATE,CL (1,1),AK(1,1),FRLIQ (1,1),DNSTT (I), • FL,WSOLID.WSLIQ(I),WSOL0T,SSOL(I),SSOLUT(I),CS (I,1),GL (I),TWT,COHP •d) 103 FORMAT (1X,12,F6.1,5X,AO,1X.F8.5,IX,F6.3,12 (2X,F6.3)) 2 CONTINUE TIME=TIHE»DT 7 IF(TIME.GT.T) GO TO 999 IP (TB.GT.TB1.AND.TB.LT.TB2)GO TO 501 WRITE{6,101)TIME WRITE (6,102) 501 CONTINUE DO 3 1=1,L TEMP(1,2)=TEMP(I,1)-G*R*DT AK (I,2)=AK (I,1) CL(I,2)=CL(I, 1) PRLIQ (I,2)=FRLIQ (I,1) CS(I,2)=CS(I,1) STATE=ALIQ CALL DENS(TEMP(I,2),CL(I,2),DNST¥(I)) IP (TEHP(I,2).GE.TL)GO TO 304 302 IF (TEMP (I, 2) . LT. TE) GO TO 303 301 STATE=AHUSH CALL PBSN(TEMP (1,2) ,AK(1,2),CL (1,2)) AKO= (AK (1,2) • AK (I, 1) )*0.5 P=-1./(1.-AKO) FL= (CL (I,2)/CL (I, 1) )*»P FRLIQ(I,2)=FRLIQ(I,1) *FL GO TO 307 303 STATE=SOL AK(I,2)=1. FL=1„ WSOLID=0. WSOLUT=0. GO TO 308 307 WSOLID=(1.-PL)*FBLIQ(1,1) BSLIQ (I)=FRLIQ(1,2)*CL(1,2) WSOLUT=COHP(I)-WSLIQ (I)- (CS (1,1)* (1.-FRLIQ (1,1))) IP (HSOLUT.LE.O.)WSOLUT=0. 308 SSOLUT(I)=SSOLUT(I)*WSOLUT SSOL (I)=SSOL (I)*HSOLID IP(SSOLUT(I).LT.0.00001)GO TO 333 CS (I,2)=SSOLUT (IJ/SSOL (I) GO TO 33H 184 333 CONTINUE CS(1,2)=0. 330 CONTINUE TWT-SSOL(I) *FRLIQ (1,2) TSOLUT=SSOLUT(I)•HSLIQ(I) COMP(I)=TSOLUT/TWT IP (CS (1,2) . LB. 0. )CS (1,2) =0. CALL DENS(TBBP(1,2) ,CS (1,2) ,SD) VS=SSOL(I)/SD VL=FBLIQ (1,2) /DNSTY (I) GL (I)=VL/(VS^VL) IP(TEMP(I,2) .LT.TE)GL(I)=0. GO TO 309 304 CONTINOE STATE=ALIQ FL»1. WSOLID=0. WSOLUT*0. TSOLUT=CL(1,1) TNT* 1. 309 IP(TB.GT.TB1.AND.TB.LT.TB2)GO TO 502 WRITE(6,103) I,TEMP(1,2) ,STATE,CL (I,2),AK(1,2),FRLIQ (1,2),DNSTY (I), •FL,WSOLID,WSLIQ(I),WSOLOT,SSOL(I),SSOLDT(I),CS(1,2),GL(I),TWT,COUP •(I) 502 CONTINUE AI=FLOAT(I) DIST (I)=(AI-0.5)*R*DT PERN (I)=0. 3 CONTINUE CALL FLOW (L , TBBP (1,2) ,DNSTY,GL,R,DT,TL,TE,VISC,AN,C,PERM , CL (1,2) ,C • S (1,2),FRLIQ (1,2),WSLIQ,SSOLDT,SSOL,COHP,TB,TB1,TB2) IF (TB.GT.TB1.AND.TB.LT.TB2)GO TO 503 WRITE(6,1011)TIME 1011 FORMAT (/1X,' TIHE=',F7.1,'SECONDS -AFTER FLUID FLOW'/) WRITE(6,106) 106 FORMAT(1X, * L TEMP PERM CL FRLIQ DIST COM •P') WRITE (6, 107) (I,TEMP (I, 2) ,PERM (I) ,CL (I, 2) ,PRLIQ (1,2) ,DIST (I) ,COMP (I •) #1=1. M 107 FORMAT (1X,I3,F6.1,1X,E14.6,F9.4,F7.3,F7.2,F10.4) 503 CONTINUE DO 6 1=1,L TEMP(1,1)=TEHP(1,2) CL(I,1)=CL(I,2) AK(I,1)=AK(1,2) FRLIQ(I,1)=FRLIQ (1,2) CS(I,1)=CS(I,2) 6 CONTINUE TIB B-TIM E*DT T3=TB*2 GO TO 7 999 WRITE (6,990)ALEN,G,R,DT,AH,C, VISC WRITE(6,992) WRITE(6,991)(DIST(I),COMP(I),1=1,L) 990 PORRAT(/1X,' LENGTH OF CASTING=•,P6.2,'CM. '/1X, • TEMPERATURE GRA • DIENT=',F6.3,'OEG.C/CM.«/1X,« GROWTH RATE=',F7.4,'CM./SEC */IX,« •TIME INTERVAL=',F6.1,'SEC.'/1X,• NUMBER OF CHANNELS= • ,E12.4,/1X,• • TORTUOSITY FACTOR=',F6.2,/1X,• VISCOSITY OP THE LIQUID*',P6.3,• •POISE') 992 FORBAT(/1X,' DIST COMP') 185 991 FORMAT (1X , F6. 2.F9.4) STOP BID; SOBROUTINB DENS (TEMP,C,DNSTY) C THIS SUBROUTINE CALCULATES THE DENSITY OP LIQUID LEAC-TIN ALLOYS C AS A PUNCTION OP COMPOSITION AND TEMPERATURE. THE DATA ARE TAKEN C FROM A TABLE OF VALUES PUBLISHED BY THRESH ET AL, TRANS. TMS-AIME C 1968,PAGE 819. INTERMEDIATE VALUES ARE OBTAINED BY A LINEAR C INTERPOLATION DIMENSION CP(14) ,A(14) ,B(14) DATA CP/0.,10.,20.,30.,32.5,40.,48.75,50.,60.,62.5,70.,83.,85.,100 •./ DATA A/11.06,10.49, 9.956,9.497,9.383,9.079,8.697,8.671,8.321.8.22 • 9,7.995,7.603,7.543,7. 139/ DATA B/12.22, 1 1.582,10.481,10. 109,9.762,9.708,8. 688,8.761,8. 69,8.6 •52,8.443,7.76,7.775,7.125/ DO 100 1=1,14 E=C-CP (I) IP(E.LT.O.)GO TO 101 100 CONTINUE 101 1=1-1 CC= (CP (1+ 1) -C) / (CP (1*1) -CP (I) ) AA=A (IO) • (A (I) - A (1*1) ) *CC BB=B (I)-B (1 + 1) BB=ABS(BB) BB=B (1*1)•BB*CC DNSTY=AA-BB*0.0001*TEHP RETURN END SUBROUTINE PBSN (T,AK,CL) T2=T*T T3=T2*T T4=T2*T2 P1=-13.86709 P2=.2432528 P3=-. 1552426E-2 P4=.435223E-5 P5=-.448334E-8 Q1=-75.10938 02=2.297987 Q3=-. 1115845E-1 Q4 = . 147874E-4 AK=PUP2*T»P3*T2*P4*T3 + P5*T4 CL=Q1*Q2*T+Q3*T2*Q4*T3 RETURN BHD SUBROUTINE PLOW(L,TEMP,DNSTY,GL,R,DT,TL,TE,VISC,AN,C,PERM,CL,CS,PR •LIQ,HSLIQ,SSOLUT,SSOL,COMP,TB,TB1,TB2) C THIS SUBROUTINE CALCULATES THE NEW COMPOSITION PROFILE APTER EVERY C TIME INCREMENT. PERMEABILITY IS CALCULATED FROM EQUATION 7.7 IN THE C TEXT, AND THE SERIES OP SIMULTANEOUS DIFFERENTIAL EQUATIONS (7.8) C ABB SOLVED USING A RONGE-KUTTA TECHNIQUE - LIBRARY ROUTINE DRKC. DIMENSION TEMP (L) ,DNSTY (L) ,GL (L) ,PERM (L) ,CL (L) ,CS (L) ,FRLIQ (L) DIMENSION HSLIQ(L),SSOLUT(L).SSOL (L),COMP (L) 186 DIMENSION BY (99),B(99) REAL*8 X.Z,Y(99) ,P(99) ,H,HflIN,E,G (99) ,S (99) ,T (99) , VLIQ (99) ,QT (99) INTEGER TB,TB1,TB2 EXTERNAL FUNC COHHON /ZVQ/ ?LIQ,QY,LL,K,J J=1 DO 550 1=1,L TH»TL-0.01 IF(TEHP(I).GT.TB)GO TO 560 J«I 550 CONTINUE 560 CONTINUE DO 570 1=1,L K=I IP (TEflP(I).GT.TE)GO TO 580 570 CONTINUE 580 IF(K.EQ.J)GO TO 99 GR=981. PI = 3. 142 YA=.6334 YD=R*DT XA=R*DT*0.5*2.54 DO 1 I=K,J PERM (I)=GL(I) *2.*YA/(24.*AN*PI*C) BY (I)=VISC*YD/(PERN(I)*YA) 1 PBRH (I)=PERH (I) *GL (I) N=J-1 DO 21 I=K,N 21 HY (I) = (HY (I) + HY (I* 1) ) *0.5 DO 100 I=K,N AJK=PLOAT(J-K) 100 B(I)=(DNSTY(J)-DNSTY(K))*GR*AJK*R*DT DO 200 I=R,N 200 QY(I)=B(I)/(2.*RY(I)) X=0.D0 Z=DT H=Z/64.D0 HHIN=H*1.D-3 B=1.D-5 PB=R*DT*9.667*1.27*1.27*PI/«. H=1 DO U I=K,J X(0) =DHSTY(I)*CL (I) VLIQ (I)=PRLIQ(I)*FR/DNSTY(I) U »=H*1 N=J-K*1 LL=N CALL DRKC(N,X,Z,Y, F, H,HHIN,E,PUNC,G,S,T) 11=1 DO 10 I=K,.J CL (I)=Y (H) /DNSTY (I) HSLIQ (I) =FRLIQ (I) *CL (I) TSOLUT=SSOLUT(I)•HSLIQ(I) THT=SSOL(I)•FRLIQ(I) COHP (I)=TSOLUT/TWT !!=!!• 1 10 CONTINUE 99 RETURN END 187 SUBROUTINE FUNC(X,Y,F) C THIS SUBROUTINE SETS OP THE DIFFERENTIAL EQUATIONS (7.8 IN THE C TEXT) POR THE RUNGE-KUTTA TECHNIQUE DRKC. IMPLICIT REAL*8(A-H,0-Z) DIMENSION 1(1) ,F (1) COMMON /ZVQ/ VLIQ (99),QY(99) ,LL,K,J A1-QY (K)/VLIQ (K) F(1)=A1*Y(2)-A1*Y(1) IF (LL.BQ.2) GO TO 2 JJ=LL- 1 KK=K»1 H-2 DO 1 I=KK,JJ A1»QY(I)/VLIQ(I) A2=»QY (I-1)/VLIQ(I) A3= (QY(I) •QY(I-1))/VLIQ(I) F(H)=A1*Y (M*1)*A2*Y (M-1) -A3*Y (M) 1 H=H*1 2 A2=QY (J-1)/VLIQ (J) F(LL) =»A2*Y (LL-1) -A2*Y(LL) RETURN END 188 APPENDIX V DIRECT OBSERVATION OF SOLIDIFICATION USING ELECTRON MICROSCOPY V.l Introduction The aim of this work was to directly observe melting and solidifi cation in thin films of pure metals and alloys, using electron microscopy. The method was essentially the same as that developed by Glicksman and (55-59) Void , who observed melting and solidification in thin films of pure bismutli/"^'"'^ , and a number of dilute alloysand used their observa tions to obtain the absolute value of the solid-liquid interfacial energy .. ,,(58,59) for pure bismuth The present work was first directed towards reproducing Glicksman's experiments on pure bismuth, and then using the technique, to observe solidification in other pure metals, and thereby calculate the solid-liquid interfacial energies. It was hoped that sufficient expertise would be gained to observe the growth of a lamellar eutectic from the liquid. V.2 Experimental Method The thin films, produced by various methods discussed below, were examined using a Hitachi HU-llA electron microscope (the microscope was the same as that used by Glicksman). Both a simple heating stage, and a heating-tilting stage were used, but it was found that temperatures could not be controlled with sufficient precision using the latter stage, therefore the results only apply to work done with the simple heating stage. 189 The specimen was heated slowly using the heating stage, until a small molten zone was produced using the additional heat induced by focusing the lOOkV electron beam. Glicksman estimated that the optimum temperature of the specimen was about 10°C below the melting temperature, however, this could not be accurately determined in the present work using the available equipment. The electron beam simultaneously provided image illumination and local heating to produce the molten zone. It was found that the molten zone could be made to expand and contract by adjusting the current to the second condenser lens. In all experiments, one of the major problems was stability of the molten zone. The available power supply did not provide sufficiently sensitive control to hold the specimen at the required temperature for long periods. For this reason, it was found that the best results were obtained by setting the power supply to heat the specimen very slowly. This usually allowed about ten minutes for observation of solidification and melting while the specimen was in a suitable temperature range. In contrast to Glicksman's findings, it was extremely difficult to hold the solid-liquid interface stable enough for photography during this period. Exposure times of about 1-5 seconds were required, and the image frequently shifted during the course of the exposure. V.3 Results V.3.1 Pure bismuth Thin films were prepared by vacuum evaporation onto carbon support films using standard methods. All metals used in this work were 6-9's purity, 190 and the vacuum system was flushed several times with oxygen-free nitrogen before pumping, and a titanium getter was used before evaporation. The vacuum before evaporation was 2.0 x 10 ^ Torr. The thickness of the carbon films was about and the bismuth thickness was in the range 1000-2000A1 (measured using an interference microscope). Typical results are shown in Figures 71 and 72. Figure 71(a-c) shows freezing, followed by melting, followed by freezing in the same region. Figure 71(d) shows the solid-liquid interface at higher magnification. Faceted growth of the solid is seen in the lower right-hand corner of Figures 71(a) and (c), similar to that seen by Glicksman. The double image of the large grains in Figures 71(b) and (d) is due to the instability of the molten zone. The small grains in the corners show the original structure of the evaporated film. The liquid regions appear uniformly dark because the liquid phase scatters the electron beam randomly. The light patch, which appears to be growing in the centre of the liquid zone, is caused by thinning of the film in this region. Eventually this would lead to de-wetting and the liquid would draw back into globules around a central hole. When this occurred, the liquid became too thick for the electron beam to penetrate, and the inter face could no longer be observed. The advantage of using bismuth for interfacial energy measurements is that it tends to deposit from the vapour phase with the basal plane parallel to the plane of the specimen. Thus the boundary between neighbour ing grain? in the thin film is usually a simple tilt boundary, and the method FIGURE 71: (a-c) Alternate freezing, melting, and freezing in pure bismuth, showing evidence of faceted growth. Magnification 5000x. (d) Enlarged view of the solid-liquid interface, showing high angle grain boundaries emerging at the interface. Magnification lOOOOx. 192 used by Glicksman was to search for low-angle tilt boundaries which emerged at the solid-liquid interface. The angle of tilt could be measured by taking a selected area diffraction pattern across the boundary, or by counting the dislocation spacing along boundaries with very low tilt angles. The interfacial energy was calculated from measurements of the cusp angle, where the grain boundary emerged at the solid-liquid interface^^'"^ . No attempt was made in the present work to repeat these measurements on pure bismuth. All the bismuth samples observed in the present work showed dark speckles over the field of view. These speckles were not seen before heating, but seemed to form when the specimens came close to the melting point. They are seen in Figure 71, and are even more pronounced in Figure 72. They also appear in photographs published by Glicksman, but are not as common as in the present work. Although oxidation was suspected in both the previous and present work, no oxide rings were observed in the diffraction patterns. The nature of the speckles therefore remains unknown. One can speculate that they might be due to some interaction between the bismuth and the carbon substrate,, since liquid bismuth can dissolve minute amounts of carbon (0.0028 atomic percent at 300°C), which it rejects as graphite crystals on solidification^^ . Therefore, the speckles might be crystallites of graphite, which one could not distinguish from the substrate by electron diffraction. The speckles in Figure 72(c) appear to pin the interface, which has a rougher contour than in Figure 71. It is also possible that the speckles may be related to some kind of contamination. The "islands" of FIGURE 72: Alternate melting, freezing, melting and freezing in pure bismuth. Note the high concentration of "speckles" and the interface pinning effect in (b) and (d). The "islands" of solid in (a) and (c) resemble photographs of melting pub lished by Glicksman, and are probably caused by contam ination of the metal film. Magnification 5000x. 194 solid which remain after melting (Figures 72(a) and (c)), are similar to the photographs of Bi-Sn alloy melting published by Glicksman, and they can be considered part of a "mushy" zone. This would indicate that contamination which causes some alloying occurred in the specimen shown in Figure 72. V.3.2 Other pure metals (tin, aluminum and indium) Satisfactory thin films of tin and aluminum were produced by vacuum evaporation onto carbon support films, but it was not possible to produce oxide-free films of indium. Both tin and aluminum behaved in the manner shown in the sequence pf photographs shown in Figure 73. During heating, grain growth would be observed (Figures 73(b) and (c)), followed by melting (Figure 73(d)) and immediate de-wetting (Figures 73(e) and (f)). These photographs were taken using a 35 mm camera to photograph the fluorescent screen, because the usual techniques were too slow to record the rapid events which occurred. The grain size of the screen, plus the use of a fast film account for the poor quality. The high thermal conductivity of aluminum may be responsible for the difficulty in keeping the molten pool localized. Since stable molten zones could not be produced in pure metals other than bismuth, no interfacial energy calculations were attempted. V.3.3 Lamellar eutectics Three techniques for producing alloy films were attempted. These were evaporation of the two constituent pure metals, microtoming, and electro lytic thinning. The work was concentrated on aluminum-copper, but the problem of; de-rwetting persisted, and lamellar growth was not observed. 195 FIGURE 73: The melting of pure aluminum, photographed from the fluorescent screen using a 35 mm camera; (a) original vapour deposited structure; (b) and (c) grain growth (approx. 450°C); (d) and (e) beginning of melting and de-wetting; (f) total de-wetting. Magnification approx. 5000x. 

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