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Macrosegregation in solidification of A356 Khadivinassab, Hatef 2018

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Macrosegregation in Solidification of A356byHatef KhadivinassabM.Eng., University of Birmingham, 2012a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Materials Engineering)The University Of British Columbia(Vancouver)April 2018c© Hatef Khadivinassab, 2018AbstractA combined experimental / numerical approach has been applied to investigate thebulk transfer of solute due to liquid metal feeding during shape casting of aluminumalloy A356 (Al-7Si-0.3Mg). A series of dumbbell-shaped experimental casting geome-tries have been developed, which promote solute redistribution due to liquid metalfeeding. Three of the castings were produced in small moulds with natural cooling,forced cooling and insulated conditions and one casting was made in a large mouldwith natural cooling. The redistribution of solute in the castings has been evaluatedusing a novel image processing technique based on the area fraction of silicon. Theresults show that the casting with the forced cooling configuration exhibited a largerdegree of macrosegregation.In the numerical model, silicon segregation during solidification is calculated as-suming the Scheil approximation, and is coupled with a macro-scale transport modelthat considers resistance in the mushy zone and feeding flow. The model has beenimplemented within the commercial CFD software, FLUENT, which simultaneouslysolves the thermal, fluid flow fields and species segregation on the macro-scale. Theresults from the simulation agree with the experimental results, except for the caseswhere significant liquid encapsulation occurs. The model predicts high levels of enrich-ment when liquid encapsulation is present in the joint section of the dumbbell-shapedcastings.Finally, a constitutive behaviour relationship was developed based on the Ludwik-Hollomon equation to predict the flow stress of Al-Si-Mg alloys with varying siliconcomposition and Dendrite Arm Spacing (das) in the as-cast (ac) or T6 conditionwith high accuracy. This model was then used with the results of the segregationmodel to predict yield strength distribution in the aforementioned dumbbell-shapedcasting. The results show that silicon segregation has a more significant effect on theiiyield strength than das.iiiLay SummaryOne of the on-going challenges in the automotive industry is to control defects soas to improve product quality and reduce production cost. Macrosegregation relateddefects can lead to casting rejection in the automotive industry because they aredetrimental to mechanical performance, due to the variation of mechanical proper-ties throughout the casting. The particular emphasis of this research is to developa methodology to quantitatively describe macrosegregation during solidification ofshape castings of A356 aluminum alloy and to characterize the effects of macroseg-regation on mechanical properties. Ultimately, through this research, the ability topredict macrosegragation and its effect on mechanical properties results in an im-proved ability to predict final performance of a casting.ivPrefaceThe following journal submissions have been extracted from the body of work pre-sented in this dissertation. My supervisor, Prof. Daan Maijer provided experimentalinsight, results interpretation and editorial support covering all aspects of my re-search. Aside from my supervisor, and key secondary contributors, I am the primarycontributor to these works:1) Khadivinassab H., Maijer D. M., Cockcroft S. L., “Constitutive Behviour ofMacrosegregated A356”, (2017) – under revision2) Khadivinassab H., Maijer D. M., Cockcroft S. L.,“Characterization of Macroseg-regation in Eutectic Alloys”, Materials Charcterization, (2017)3) Khadivinassab H., Fan P., Reilly C., Yao L., Maijer D. M., Cockcroft S. L.,Phillion A. B.,“Study of the macro-scale solute redistribution due to liquid metalfeeding during the solidification of A356”, Light Metals Production, Processingand Applications Symposium, The 53rd Annual Conference of Metallurgists,(2014)Prof. Steve Cockcroft provided experimental insight, editorial support and ininterpretation of results for items 1, 2 and 3. Dr. Phillion provided editorial supportand in interpretation of the results for item 3.Chapter 2 contains material from item 1. Chapter 3 is based on the materialdrawn from item 2. Chapter 4 contains material presented in items 2 and 3. Chapter5 contains material from item 3. These chapters contain footnotes mirroring theabove information.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Macrosegregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Experimental investigations . . . . . . . . . . . . . . . . . . . 51.1.2 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Image processing for defect quantification . . . . . . . . . . . . . . . . 151.3 Solidification of A356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Mechanical properties of A356 . . . . . . . . . . . . . . . . . . . . . . 201.5 Scope and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Constitutive Behaviour of hypoeutectic Al-Si-Mg Alloys . . . . . . 252.1 Experimental methodology . . . . . . . . . . . . . . . . . . . . . . . . 262.1.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 262.1.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Constitutive equation development . . . . . . . . . . . . . . . . . . . 342.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42vi3 Characterization of Macrosegregation in Eutectic Alloys . . . . . . 433.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.1 Analysis overview . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.2 Image segmentation . . . . . . . . . . . . . . . . . . . . . . . . 453.1.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.4 Area fraction to mass fraction conversion . . . . . . . . . . . . 473.1.5 amd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Methodology verification . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.1 Image segmentation verification . . . . . . . . . . . . . . . . . 513.2.2 Meshing validation . . . . . . . . . . . . . . . . . . . . . . . . 563.2.3 amd validation . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Macrosegregation in Shape Castings . . . . . . . . . . . . . . . . . . 664.1 Experimental methodology . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.1 Casting with natural cooling . . . . . . . . . . . . . . . . . . . 744.2.2 Casting with insulated joint . . . . . . . . . . . . . . . . . . . 794.2.3 Casting with forced cooling on the joint . . . . . . . . . . . . 834.2.4 Large casting . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935 Modeling of Macrosegregation in Shape Castings . . . . . . . . . . 955.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 955.1.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 1025.1.4 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 1075.1.5 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2.1 Casting with natural cooling . . . . . . . . . . . . . . . . . . . 1095.2.2 Casting with insulated joint . . . . . . . . . . . . . . . . . . . 1135.2.3 Casting with forced cooling on the joint . . . . . . . . . . . . 1155.2.4 Large casting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 Discrepancy in the enriched region . . . . . . . . . . . . . . . . . . . 1235.4 Yield strength prediction . . . . . . . . . . . . . . . . . . . . . . . . . 1285.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1366.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140viiList of TablesTable 1.1 Composition of unmodified A356 . . . . . . . . . . . . . . . . . . . 17Table 2.1 Composition of fabricated alloys used in constitutive behaviour anal-ysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Table 2.2 Average yield strength and standard deviation determined for eachalloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 2.3 Values of fitted coefficients in equation 2.1 for n, K and 0. . . . . 36Table 2.4 Values of fitted coefficients in equation 2.1 for K. . . . . . . . . . . 37Table 3.1 Composition of the alloys used in Khajeh’s research . . . . . . . . 53Table 3.2 Comparison of the three image segmentation methods based on ac-curacy and execution time. . . . . . . . . . . . . . . . . . . . . . . 56Table 3.3 Accuracy of meshes presented in Figure 3.8. . . . . . . . . . . . . . 59Table 3.4 Execution time for meshes presented in Figure 3.8. . . . . . . . . . 59Table 3.5 Accuracy of meshes presented in Figure 3.9. . . . . . . . . . . . . . 61Table 3.6 Execution time for meshes presented in Figure 3.9. . . . . . . . . . 61Table 4.1 Macrosegregation value for each casting. . . . . . . . . . . . . . . . 91Table 5.1 Thermo-physical properties used in the mathematical model. . . . 107Table 5.2 Comparison of predicted and measured macrosegregation values foreach casting condition. . . . . . . . . . . . . . . . . . . . . . . . . 131Table 6.1 Relative significance of terms I to V on parameter n in variousconditions. Note that the values are in percentages. . . . . . . . . 148Table 6.2 Relative significance of terms I to V on parameter K in variousconditions. Note that the values are in percentages. . . . . . . . . 149Table 6.3 Relative significance of terms I to V on parameter 0 in variousconditions. Note that the values are in percentages. . . . . . . . . 149viiiList of FiguresFigure 1.1 Various defects in a sample casting . . . . . . . . . . . . . . . . . 3Figure 1.2 Mechanisms of macrosegregation formation . . . . . . . . . . . . . 5Figure 1.3 Macrosegregation in Sn-Pb from Ridder et al. . . . . . . . . . . . 10Figure 1.4 Geometry and segregation profile from Voller et al. . . . . . . . . 11Figure 1.5 Composition results from Vreeman et al. . . . . . . . . . . . . . . 13Figure 1.6 Segregation results from Zhang et al. . . . . . . . . . . . . . . . . 14Figure 1.7 Composition results from Vreeman et al. . . . . . . . . . . . . . . 16Figure 1.8 A356 microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 1.9 Al-Si phase diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 1.10 Yield strength variation for ac and T6 heat treated A356 alloyswith fitted expressions. . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.1 Illustration of the plate casting mould. . . . . . . . . . . . . . . . 27Figure 2.2 Illustration of the automatic pouring device. . . . . . . . . . . . . 28Figure 2.3 Drawings of the location and naming of each tensile sample cutfrom the plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.4 Microstructure of fabricated alloys TAL01-07 . . . . . . . . . . . . 30Figure 2.5 Example of das measurement technique. . . . . . . . . . . . . . . 31Figure 2.6 Stress-strain curves for five selected samples. . . . . . . . . . . . . 33Figure 2.7 das values at each location in the plates . . . . . . . . . . . . . . 34Figure 2.8 Average porosity over all the plates. . . . . . . . . . . . . . . . . . 35Figure 2.9 Stress-strain curves with fitted models. . . . . . . . . . . . . . . . 36Figure 2.10 Representative experimental stress-strain curves with a fitted model. 37Figure 2.11 Predicted vs. measured flow stress for cross-validation data-set. . 38Figure 2.12 Sensitivity analysis of parameter n. . . . . . . . . . . . . . . . . . 39Figure 2.13 Sensitivity analysis of parameter K. . . . . . . . . . . . . . . . . . 40Figure 2.14 Sensitivity analysis of parameter 0. . . . . . . . . . . . . . . . . . 40Figure 2.15 Sensitivity analysis of yield point. . . . . . . . . . . . . . . . . . . 41Figure 3.1 Sample image and its negative mask. . . . . . . . . . . . . . . . . 46Figure 3.2 Eutectic phase diagram. . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.3 An element with its neighbouring elements. . . . . . . . . . . . . 51Figure 3.4 X-ray Micro-Tomography (xmt) section images of fabricated alloysE01-05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.5 Sample xmt image and the respective segmented images analysis. 54Figure 3.6 Measured vs. actual copper mass fraction values. . . . . . . . . . 55Figure 3.7 Geometries for mesh comparison analysis. . . . . . . . . . . . . . 57ixFigure 3.8 Mesh comparison for Figure 3.7a. . . . . . . . . . . . . . . . . . . 58Figure 3.9 Mesh comparison for Figure 3.7b. . . . . . . . . . . . . . . . . . . 60Figure 3.10 Artificial microstructure images generated to assess Average Max-imum Difference (amd) analysis. . . . . . . . . . . . . . . . . . . 62Figure 3.11 amd curves, respective contour plots and gradient comparison curvesfor images shown in Figure 3.10. . . . . . . . . . . . . . . . . . . . 64Figure 4.1 Geometry of dumbbell-shaped casting with natural cooling condition. 68Figure 4.2 Geometry of dumbbell-shaped mould with insulated central joint. 68Figure 4.3 Geometry of dumbbell-shaped mould with forced cooling on thecentral joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.4 Distribution of local heat transfer coefficient around a circularcylinder for flow of air. . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.5 Geometry of large dumbbell-shaped mould with natural coolingcondition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.6 Sectioning of dumbbell-shaped castings for polishing and analysis. 73Figure 4.7 Temperature data recorded for the dumbbell-shaped casting withnatural cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 4.8 High resolution montage for the dumbbell-shaped casting with nat-ural cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.9 Image segmentation for microstructure images from natural coolingsetup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 4.10 amd curve for the dumbbell-shaped casting with natural cooling. 78Figure 4.11 a) Contour image of Si composition on cross-section and b) plot ofSi composition along the centreline of the dumbbell-shaped castingwith natural cooling. . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.12 Temperature data recorded for the dumbbell-shaped casting withinsulated joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.13 High resolution montage for the dumbbell-shaped casting with in-sulated joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 4.14 Image segmentation for microstructure images from the castingwith insulated joint. . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.15 amd curve for the dumbbell-shaped casting with insulated joint. . 82Figure 4.16 a) Contour image of Si composition on the cross-section and b)plot of Si composition along the centreline of the dumbbell-shapedcasting with insulated joint. . . . . . . . . . . . . . . . . . . . . . 82Figure 4.17 Temperatures data recorded from the dumbbell-shaped casting withforced cooling on the joint. . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.18 High resolution montage for the dumbbell-shaped casting with forcedcooling on the joint. . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 4.19 Image segmentation for microstructure images from the castingwith forced cooling on the joint. . . . . . . . . . . . . . . . . . . . 85Figure 4.20 amd curve for the dumbbell-shaped casting with forced cooling onthe joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86xFigure 4.21 a) Contour image of Si composition on cross-section and b) plot ofSi composition along the centreline of the dumbbell-shaped castingwith forced cooling on the joint. . . . . . . . . . . . . . . . . . . . 86Figure 4.22 Thermocouple data from the large dumbbell-shaped casting. . . . 87Figure 4.23 High resolution montage for the large dumbbell-shaped casting. . 88Figure 4.24 Image segmentation for microstructure images from the large casting. 89Figure 4.25 amd curve for the large dumbbell-shaped casting. . . . . . . . . . 90Figure 4.26 a) Contour image of Si composition on cross-section and b) plot ofSi composition along the centreline of the large dumbbell-shapedcasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 5.1 Geometry, mesh and boundary conditions of model. . . . . . . . . 101Figure 5.2 Depiction of a part of the casting mould and respective resistancemodel for modelling Heat Transfer Coefficient (htc) . . . . . . . 103Figure 5.3 Effective heat transfer coefficient due to natural and forced convec-tion in a copper mould. . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 5.4 Temperature dependant material properties. . . . . . . . . . . . . 108Figure 5.5 Thermocouple data from experiment and simulation for the dumbbell-shaped casting with natural cooling conditions. . . . . . . . . . . 110Figure 5.6 Temperature contours from the dumbbell-shaped casting with nat-ural cooling conditions. . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 5.7 Predicted liquid encapsulation for the dumbbell-shaped castingwith natural cooling conditions. . . . . . . . . . . . . . . . . . . . 111Figure 5.8 Simulated silicon segregation in the dumbbell-shaped casting withnatural cooling conditions. . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.9 Silicon mass fraction along the centerline from the experiment andthe simulation for the dumbbell-shaped casting with natural cool-ing conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 5.10 Measured and predicted temperatures at the thermocouple loca-tions for the insulated dumbbell-shaped casting. . . . . . . . . . . 114Figure 5.11 Temperature contours from the insulated dumbbell-shaped casting. 114Figure 5.12 Predicted liquid encapsulation results for the insulated dumbbell-shaped casting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.13 Predicted silicon segregation in the insulated dumbbell-shaped cast-ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 5.14 Measured and predicted silicon mass fraction along the centerlineof the insulated dumbbell-shaped casting. . . . . . . . . . . . . . 116Figure 5.15 Measured and predicted temperatures at thermocouple locationsfor the dumbbell-shaped casting with forced cooling. . . . . . . . 117Figure 5.16 Temperature contours from the dumbbell-shaped casting with forcedcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 5.17 Predicted liquid encapsulation for the dumbbell-shaped castingwith forced cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 5.18 Predicted silicon composition in the dumbbell-shaped casting withforced cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119xiFigure 5.19 Measured and predicted silicon mass fraction along the centerlinefor the dumbbell-shaped casting with forced cooling. . . . . . . . 119Figure 5.20 Measured and predicted temperature at the thermocouple locationsin the large dumbbell-shaped casting. . . . . . . . . . . . . . . . . 120Figure 5.21 Temperature contours from the large dumbbell-shaped casting. . . 121Figure 5.22 Liquid encapsulation results from simulation for the large dumbbell-shaped casting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.23 Simulated silicon segregation in the large dumbbell-shaped casting. 122Figure 5.24 Measured and predicted silicon mass fraction along the centerlineof the large dumbbell-shaped casting. . . . . . . . . . . . . . . . . 123Figure 5.25 Development of over-prediction of enrichment in the bottom of thejoint region for the casting with forced on the joint. . . . . . . . . 125Figure 5.26 Centerline composition for instances presented in figure 5.25. . . . 126Figure 5.27 Centerline composition for under-relaxed model in several instances.127Figure 5.28 Predicted silicon composition in the dumbbell-shaped casting withforced cooling on the joint with an under-relaxation factor. . . . . 127Figure 5.29 Measured and predicted silicon mass fraction along the centerlineof the casting with forced cooling on the joint with an under-relaxation factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Figure 5.30 Predicted silicon composition in the dumbbell-shaped casting withnatural cooling with an under-relaxation factor. . . . . . . . . . . 129Figure 5.31 Measured and predicted silicon mass fraction along the centerlineof the casting with natural cooling with an under-relaxation factor. 129Figure 5.32 das vs. cooling rate for A356. . . . . . . . . . . . . . . . . . . . . 130Figure 5.33 Predicted yield strength for the simulated castings in ac condition. 132Figure 5.34 Predicted yield strength for the simulated castings T6 condition. . 133Figure 6.1 Sample synthetic microstructure generated using Blender . . . . . 151Figure 6.2 Sample cross-sectional images acquired from synthetic microstruc-tures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Figure 6.3 Error distribution plot for area fraction to volume fraction conver-sion analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153xiiNomenclatureRoman Symbols (page introduced) Units∆H Incremental enthalpy per unit mass (page 92) J/kgg Gravitational acceleration (page 92) m/s2K Permeability tensor (page 93) m2u Superficial velocity (page 91) m/sC Concentration of solute (page 94) %wtC0 Initial composition (page 46) %wt or %atCe Eutectic composition (page 46) %wt or %atCp Specific heat (page 101) J/kg ·KD Diffusion coefficient (page 94) m2/sH Phase enthalpy per unit mass (page 92) J/kgh Sensible Enthalpy per unit mass (page 91) J/kgk Thermal conductivity (page 91) W/m ·KLf Latent heat fusion (page 101) J/kgMi Atomic mass (page 46) uP Pressure (page 92) PaPr Prandtl number (page 98)Ra Rayleigh number (page 98)Re Reynolds number (page 99)T Temperature (page 91) Kt6 Binary T6 condition (page 34)Greek Symbols (page introduced) Unitsxiiiβ Liquid fraction (page 93) True strain (page 21)κ Partition coefficient (page 94)µ Viscosity (page 92) Pa · sρ Density (page 91) kg/m3σ Stefan-Boltzmann constant (page 99) W/(m2 ·K4)σH Ludwik-Hollomon predicted flow stress (page 21) MPaσy Yield strength (page 19) MPaε emissivity (page 99) −ϕ Area fraction (page 47)ϕe Eutectic area fraction (page 46)xivGlossaryaas Atomic Absorption Spectroscopyac as-castamd Average Maximum Differencecafe Cellular Automata-Finite Elementcfd Computational Fluid Dynamicsclahe Contrast Limited Adaptive Histogram Equalizationdas Dendrite Arm Spacingdaq Data Acquisition systemedm Electro-Discharge Machiningedx Energy-Dispersive X-ray Spectroscopygmm Gaussian Mixture Modelshtc Heat Transfer Coefficientihtc Interfacial Heat Transfer Coefficientsem Scanning Electron Microscopyxmt X-ray Micro-Tomographyxrf X-ray FluorescencexvAcknowledgmentsFirst, I would like to thank my supervisor Prof. Daan Maijer for his support andguidance throughout the course of this work. I would like to thank Jacob Kabelfor his valuable help on imaging and sample preparation. I would also like to thankProf. Steve Cockcroft for his valuable comments and fruitful discussions. A big thankyou to Dr. Carl Reilly for his assistance on experimental work and data acquisition.Especial thanks to all colleagues and officemates for providing a friendly environmentthat I was always pleased to work in. Finally, a big thank you to my friends, familyand partner, who have put up with me throughout this.xvi“Should I kill myself, or have a cup of coffee?"—Albert Camus.Dedicated to the dearest reader.xviiChapter 1IntroductionApplications for light metal usage in the automotive and transportation sectors con-tinue to expand to enable improved fuel efficiency and performance. Cast aluminiumcomponents, such as cylinder heads, engine blocks and wheels, have shown good mar-ket penetration by replacing iron based products due to weight saving and improvedaesthetics. Despite the advantages of using light metal alloys such as A356 in the au-tomotive industry, continuous improvement is needed in the manufacturing processesto meet increasing product quality standards and to reduce manufacturing rejectionrates [1]. The main cause for rejecting these types of components is casting defects.Throughout the years, numerous remedial actions have been developed to minimizethe formation of these defects utilizing trial and error methods [2]. However, oftenthese actions lack the necessary understanding of how and why defects form and onlytarget the elimination of such flaws.There are six types of casting related defects observed in light metal castings:shrinkage defects, gas-related defects, filling-related defects, undesired phases, distor-tion and metal/die interaction defects [3]. Controlling and fully eliminating castingdefects is not an easy task as each of the aforementioned categories might have over-laps. For instance, in industry, macroporosity, which is mainly a feeding relateddefect, is typically controlled by adjusting the heat transfer to ensure directional so-lidification. However, this may result in macrosegregation defects, which has recentlybeen reported in automotive wheel castings [4]. This suggests that there is a linkbetween the parameters producing such defects and understanding how each defect1CHAPTER 1. INTRODUCTIONis formed is a necessity to be able to control/reduce the defects overall.Shrinkage induced defects can be divided into three main types; macrosegregation,macroporosity and surface depressions. These defects are related to feeding, wherefeeding refers to the flow of liquid metal within the developing structure of a casting.The extent of solidification, thermo-physical properties of the metal, and pressure andtemperature history in the casting are factors that influence the formation of thesedefects. Similar to other defects, these defects have adverse effects on the mechanicalperformance of the cast parts [5]. The characteristics of these shrinkage-related defectsare as follows:• Macroporosity (Figure 1.1a) defects form when there is inadequate feeding ofthe liquid metal to compensate for the volumetric shrinkage that occurs duringsolidification [6]. The pores form when gas (dissolved or metal vapour) is evolvedat locations within the casting in regions where liquid metal becomes isolatedor encapsulated. To be considered a macro-pore, the voids that form shouldbe greater than 0.3mm in diameter [4]. Macroporosity can negatively impactcasting quality in various ways such as reduced fatigue life and poor aestheticsif the pore is exposed through machining.• Surface depressions (Figure 1.1b) are caused by the deformation of the solidshell of a casting due to the internal pressure drop caused by solidificationshrinkage [7]. The extent to which depressions develop is dependent on theformation of liquid encapsulation and the thickness of the solid shell.• Macrosegregation (Figure 1.1c) occurs when the solute elements in the alloyare transported over length scales larger than the grain size [8, 9]. This causesdepletion or enrichment in solute levels within the casting [10]. These changesin composition lead to variation in microstructure and mechanical propertieswithin the casting [11].2CHAPTER 1. INTRODUCTIONFigure 1.1: A sample casting depicting various shrinkage induceddefects; (a) macroporosity in the top section of the casting, (b) asurface depression due to feeding phenomenon and (c)macrosegregation along the centerline of the casting.In the next section, macrosegregation will be discussed in detail followed by areview of previous experimental and modelling studies. This will be followed by abrief description of the microstructure and solidification of A356. This information isessential to understand how macrosegregation occurs. Finally, the mechanical proper-ties of A356 will be presented to appreciate the potential effects of macrosegregationon the alloy.1.1 MacrosegregationMacrosegregation refers to the variation of composition within a casting over lengthscales ranging from a few millimetres to several meters in large ingots [12]. Macroseg-3CHAPTER 1. INTRODUCTIONregation is considered to be a defect for several reasons. First, in the case of eu-tectic alloys, the increased amount of eutectic caused by variation of compositionand increased number of hydrogen pores associated with higher amount of eutecticcan severely reduce the fatigue life and performance of the cast components [11, 13].Second, the compositional variations generated due to this phenomenon cannot becorrected practically using heat treatment, due to the low solid state diffusion rates ofthe solute and the large length scales. Third, macrosegregation results in the variationof mechanical properties within the casting due to the spatial variation of composi-tion. Moreover, it can affect the final cost of the material in the case of expensivealloys. For instance, in the case of gold alloys, there should be a minimum of 58wt% gold everywhere in a 14 carat gold alloy. However, macrosegregation can causevariations in the composition which result in an increase or decrease of the value indifferent sections [11].To understand macrosegregation, it is first important to consider microsegregationand how interdendritic liquid is enriched. Referring to the growing dendrite shown inFigure 1.2, several isothermal control-volumes have been drawn from the tip to thebase of the dendrite. The temperature in each control volume decreases with distancefrom the tip, while the fraction solid increases. Referring to the Scheil approximation,solute will be rejected into the liquid as solidification proceeds for hypoeutectic alloysresulting in enrichment of the solute element in the liquid as solidification progresses.In the case of near-eutectic alloys, enrichment continues until the liquid reaches theeutectic composition. The local enrichment of solute is a form of microsegregation.If the locally enriched liquid is moved over longer distances within the casting, seg-regation occurs on the macro-scale. There are three main causes of macrosegregationassociated with movement of enriched liquid [11]:• Macro-scale transport of fluid due to solidification shrinkage (Figure 1.2b);• Fluid motion due to convection (Figure 1.2c); and4CHAPTER 1. INTRODUCTION• Fluid motion due to deformation of the semi-solid (Figure 1.2d).Figure 1.2: (a) Depiction of a dendrite. Moving down from the tip tothe base of the dendrite, the local liquid is enriched in solute.Mechanisms of movement of the locally enriched liquid via (b)compensatory flows, (c) convection and (d) deformation ofsolid [11].1.1.1 Experimental investigationsThere have been many experiments reported that were aimed at understanding dif-ferent types and aspects of macrosegregation. One of the first experiments was per-formed by Mehrabian [14]. In this research, the sole driving force for macrosegregationwas assumed to be the flow due to solidification shrinkage and experiments were de-signed in a way that this flow was the only driving force operating. The casting usedin these experiments had a rectangular geometry. Macrosegregation occurred in thelength of the casting spanning to 25cm. This study concluded that a variety of ap-parently different types of macrosegregation in binary alloys were due to mass flow ofinterdendritic liquid to feed solidification and thermal contraction. In further studies,this group focused on macrosegregation of ternary alloys and macrosegregation due5CHAPTER 1. INTRODUCTIONto convective flow [15, 16]. It was concluded from these studies that gravity inducedflows, generated due to changes in density in the liquid metal, play an importantrole in the resulting macrosegregation of large ingots where directional solidificationis present [15].Streat et al. studied macrosegregation in a directionally solidified lead-tin alloyusing a radioactive tracer technique in 14cm long cylindrical moulds with 1.27cmdiameter. It was shown that macrosegregation in this case resulted from the upwardflow of less dense tin-rich interdentritic liquid during solidification [17]. Boettinger etal. investigated the effects of the solute gradient on macrosegregation during castingof a Pb-Sn alloy [18]. The macrosegregation in this study was characterized along thelength of the solidified samples using two methods: X-ray Fluorescence (xrf) anda titration method. It was found that the convection caused by the solute gradientcaused extensive macrosegregation.In a study by Wang et al., two sets of small cylindrical samples (5cm x 19mm)were prepared from a directionally solidified cylinder of length 5cm and diameter9cm with compositions of Pb-15 wt% Sn and Pb-85 wt% Sn. Using detailed chemicalcomposition measurements, segregation in the Pb-rich samples was observed with Sncontent being higher at the top section. Macrosegregation in these alloys was foundto increase with increasing thermal gradients and slower cooling rates. No significantsegregation was observed in the Sn-rich samples since neither the temperature northe composition gradients were favourable for convection [19].Prescott carried out a series of experiments to investigate the effects of magneticfields on macrosegregation using a lead-rich Pb-Sn alloy. In this research, a magneticfield was used to augment or oppose the thermal and solutal buoyancy forces. Asimple hollowed-out cylindrical mould with length of 15cm, internal radius of 16mm and outer radius of 63.5mm was used to conduct the experiments. When largeenough (∼ 5mT) upward magnetic forces were induced the circulation of the melt6CHAPTER 1. INTRODUCTIONwas reversed and the solidification began at the top of the sample, which was a resultof the magnetic field. The magnetic forces also enhanced solutal buoyancy, whichwas a significant source of macrosegregation in the alloy considered in this study. Inanother variant of the experiment downward magnetic forces were also applied, whichaided thermal buoyancy and caused the initial solidification to occur at the bottomof the casting. In this case, the magnetic forces opposed solutal buoyancy and causedreduced macrosegregation [20].Krane investigated the formation of macrosegregation in binary (Pb-Sn) andternary (Pb-Sn-Sb) alloys in a rectangular mould of dimensions 254mm x 127mmx 89mm. In this study, the dominant cause of the macrosegregation was found tobe buoyancy induced convective flows. He also found that the strength of shrinkageinduced flow increased with solidification rate, but even for the highest solidificationrates, macrosegregation effects were confined to small regions near the chilled wall.Krane employed rectangular-shaped castings and vertically sectioned them. He thenremoved specimens, dissolved them using nitric and hydrochloric acid and analyzedthe composition using Atomic Absorption Spectroscopy (aas) [21].In an interesting study, Leon-Torres and co-workers investigated the effects ofincreasing gravity on solidification and macrosegregation of Al-Cu alloys. The exper-iments under increased gravity were conducted during parabolic flights where gravitywas increased from 2 to 10g. A cylindrical mould was used in the experiments. In allcases positive segregation was observed next to the chilled wall, and similar to earlierstudies, this was attributed to shrinkage induced flow [22].Macrosegregation in higher melting point, ternary Al-Cu-Si alloys was studied byFerreira et al. In this research samples with a cylindrical cross-section with lengthof 90mm and diameter of 20mm were directionally solidified. The results showedinverse segregation of copper and no segregation of silicon except for a short lengthof negative segregation at the bottom of the casting. These results were attributed7CHAPTER 1. INTRODUCTIONto the short mushy zone length [23].In another study, Ojha and Tewari investigated the effects of mushy zone char-acteristics on macrosegregation using a quartz tube with length 30cm and diameter6.5mm as a mould. They directionally solidified a Pb-Sn alloy with growth ratesvarying from 1 to 10µm/s in a positive temperature gradient. Large amounts ofmacrosegregation were observed in the castings. It was shown that macrosegrega-tion increased with decreasing growth rates. They concluded that segregation in themushy zone is a result of the combined effects of diffusion and convection. How-ever, for low permeability conditions at higher distances from liquid solid interface,convection can be neglected [24].Recently, the number of the detailed experimental investigations of macrosegre-gation being published per year has slowed. As modelling of this phenomenon hasattracted more attention, experimental findings were generally concentrated on vali-dating the results from these numerical simulations. Furthermore, to date, there hasnot been a systematic study reported on shrinkage-induced segregation as it appliesto shape castings of commercial alloys. As discussed in previous paragraphs, majorityof the conducted experiments were based on simple rectangular or cylindrical moulds.1.1.2 Numerical studiesFundamentally, the main task to predict the occurrence of macrosegregation is tomodel the composition distribution at the macro-scale in the presence of heat transferand fluid flow phenomena. For a casting process, the heat transfer analysis involvespredicting the temperature distribution and phase change throughout the casting ateach time-step. As fluid flow and diffusion are the major causes of macrosegregation,these phenomena have to be modelled in parallel to predict solute distribution on themacro-scale. Depending on the focus, models of casting processes solve the relevantmass, momentum, heat and species transport equations.8CHAPTER 1. INTRODUCTIONFlemings et al. were one of the earliest contributors to the numerical analysisof macrosegregation [25]. They suggested that the main cause of macrosegregationin their experiments was due to interdendritic fluid flow caused by compensatoryflow rather than by convection resulting from buoyancy. Thus, buoyancy effects wereneglected in their study. They developed a model of macrosegregation for a 2Drectangular geometry considering only the flow of interdendritic fluid through a fixeddendritic network. This model was then validated with a number of carefully designedexperiments. Mehrabian et al. extended Flemings’s model to account for convectiveflow in the mushy zone [15]. The numerical results showed good agreement with theexperimental results.Ridder et al. coupled phenomena observed in the mushy zone to the bulk liquid ina rectangular 2D axisymmetric domain with 120mm in length and 80mm in diameter[26]. In this study, the bulk melt was coupled with the mushy zone using a multi-domain approach in which separate equations were solved for the bulk liquid andmushy zone regions. This method involved matching the pressure and velocity ofboth the mushy zone and bulk liquid regions at their interface. Although there wasgood agreement between the experiments and numerical simulation presented in thisstudy (refer to Figure 1.3), this model was unable to predict the phenomena occurringin the casting such as remelting and double-diffusive convection, where two differentdensity gradients (thermal and solutal) with different diffusion rates drive convectiveflow.In a novel study, Bennon and co-workers proposed a method to implicitly couplethe mushy zone (coexisting solid and liquid) regions for binary alloys by a set of mo-mentum, energy and species equations [27,28]. In this so called continuum model, themushy zone was treated as a solid - liquid mixture and individual phase conservationequations were integrated into a set of mixture conservation equations. This methodpartially solved issues occurring in multi-domain models, such as the need to calculate9CHAPTER 1. INTRODUCTIONFigure 1.3: Comparison of experimental and theoretical segergationprofiles in Sn-21wt% Pb ingot from Ridder et al. [26].the liquid/solid interface shape and double-diffusive convection in the liquid resultingfrom thermal and solutal buoyancies. Nevertheless, comparing the numerical resultsto experimental data, the agreement was only fair. Voller et al. reported a similarmethod for binary alloys, however, their model incorporated a better approximationof solute redistribution at the micro scale [29]. Voller et al. used a simple square ge-ometry to apply this model on solidification of Ammonium Chloride (refer to Figure1.4). It was concluded that the model developed by Flemings and co-workers can beused to analyze solidification under equiaxed solidification conditions, but a contin-uum model is necessary for columnar dendritic regions. A hybrid of the two modellingapproaches has been proposed as a best practice for robust macrosegregation analy-sis. As the previous models mainly concentrated on 2D problems, Chakraborty andDutta extended the hybrid model to consider macrosegregation in 3D [9]. The mainfocus of this study was to model the three-dimensional double-diffusive convectionin a cubic enclosure. It was shown that three-dimensional convective flows cause asubstantial solute redistribution in the transverse section of the casting, which couldnot be described with a two-dimensional model.10CHAPTER 1. INTRODUCTION(a) (b)Figure 1.4: Geometry (a) and the macrosegregation profile(b) from Voller et al. [29].Ni and Beckermann presented a different method for modelling solidification. Inthis model, the solidifying melt is considered to have two distinct phases, in which eachphase is treated separately and interactions between the two phases are consideredexplicitly [30]. The two-phase model allows for relaxed assumptions regarding thermalequilibrium and species diffusion in the solid phase. However, the drawback of thismethod compared to continuum models is that twice as many partial differentialequations must be solved, which can be time-consuming especially in the case oflarge problems.Ni and Incropera proposed additional modifications to the equations of the contin-uum and two-phase models, to consider more sophisticated phenomena such as solutalundercooling, nucleation and solid movement in the form of floating or settling crys-tals [31, 32]. By coupling the continuum model with the two-phase model, severalassumptions in the original continuum model were relaxed to consider the aforemen-tioned phenomena. The main focus of this work was to describe solute transportthrough the motion of floating crystals. In similar work, Vreeman et al. studiedthe evolution of macrosegregation caused by the redistribution of alloying elementsthrough the movement of free floating dendrites in direct chill casting of aluminumalloys [33,34]. They also modified and coupled the transport equations from the con-11CHAPTER 1. INTRODUCTIONtinuum model and the two-phase model. However, in this research, the two-phaseregion was characterized as a slurry of free-floating dendrites and a rigid permeabledendritic matrix saturated with interdendritic liquid. For each of these phases, dis-tinct and separate momentum equations were solved. Vreeman et al. applied thismethod on an axisymmetric casting with height of 350mm and diameter of 400mmon Al-4.5wt%Cu. The results are shown in Figure 1.5Krane and Incropera investigated the solidification of ternary alloys [35, 36]. Inthis study, they modified the binary continuum mixture equations for the transportof mass, momentum, heat and species to account for a third component. Numericalsimulations were performed to observe the convective flows and macrosegregationpatterns in the Pb-Sn-Sb system. The results from numerical analysis was comparedthe experimental results and fair agreement was observed.Rappaz and Gandin presented a new method to model grain structure formationduring solidification [37]. The foundation of this model was based upon the use ofa cellular automata technique to model nucleation and grain growth. They thencoupled this method to an enthalpy based finite element heat flow calculation. Theso-called Cellular Automata-Finite Element (cafe) analysis interpolates the temper-ature at a cell location from the macro-scale finite element predictions to calculatethe nucleation and growth of grains [38]. This method was further developed byLee and co-workers to model both grain growth and pore formation during solidi-fication [39]. They reported good correlation with experimental results. Guillemotet al. further developed the original cafe model to account for transport and sed-imentation of equiaxed grains [40]. They also verified the results, however, it wasconcluded that refined experimental data were required to further validate segrega-tion profiles. Zhang and colleagues coupled cellular automata with the finite volumemethod instead of finite element method to investigate the macrosegregation occur-ring in a 2D domain [41]. A 10mm x 10mm square geometry was used to model12CHAPTER 1. INTRODUCTION(a)(b)Figure 1.5: Copper segregation field (a) and coppercomposition variation on z=150mm from Vreeman etal. [34].13CHAPTER 1. INTRODUCTIONthe solidification of Al-7wt%Si, results of which are shown in Figure 1.6. The modelwas verified by using previous experimental solidification studies. The incorporationof a cellular automata model requires finer mesh (e.g. 50 micron elements), as it isemployed to predict phenomena at the micro scale. Therefore, it is significantly timeand comutationally intensive [38].Figure 1.6: Concentration contour for solidification of Al-7wt%Si fromZhang et al. [41].To sum up, there has been extensive research done on modelling solidification andmacrosegregation. The hybrid model, among the modeling approaches proposed, hasproven to be time efficient and accurate due to the number of differential equationssolved. Other more accurate methods such as the two-phase model and especially thecafe model require more time to simulate the solidification and macrosegregation.Therefore, many commercial software packages, such as CFX and FLUENT, utilizethe hybrid model to simulate solidification.14CHAPTER 1. INTRODUCTION1.2 Image processing for defect quantificationThe quantification of defects observed in castings is a challenging task. Most of thereported quantification techniques for pore detection and classification, and macroseg-regation quantification (e.g. X-ray Micro-Tomography (xmt), Energy-Dispersive X-ray Spectroscopy (edx) and xrf) are confined to small sections, which may not berepresentative of the overall casting. Some studies report results from larger sections,however, these results lack accuracy since discrete points were used to generate defectmaps of sections. [42].Image analysis is an effective method to classify defects. One of the first studiesto quantitatively characterize defects in castings was reported by Tewari et al. wherethey utilized a digital image analysis-based experimental technique to characterize thespatial arrangement of microporosity [43]. In this research, several overlapping opticalmicrographs were taken from a 3mm x 4mm area and stitched into a high-resolutionmontage for use in analyzing the pores. In a similar study, Prakash and colleaguesused computational microstructure analysis to characterize and quantify porosity ina high pressure die-cast magnesium alloy over a 2mm x 19mm sample [44, 45]. Thesize and distance distribution and clustering tendency was quantified by this imageprocessing technique. The method was reported to be quick and efficient to detectand distinguish between gas and shrinkage porosity.Quantification of macrosegregation in steel was investigated by Straffelini et al.using image analysis. Continuously cast steel was vertically sectioned and then severalimages were taken from the transverse section. Two parameters were defined, based onthe dimension of segregated areas and distance between two nearest segregated areas.The microstructure was observed to be more homogeneous as these two parametersdecreased [46]. In a different context, Roy designed an apparatus to take discretemicro images from predefined positions of a large aluminum sample, shown in Figure1.7a [47]. He then determined the eutectic area fraction in every micrograph and15CHAPTER 1. INTRODUCTIONlinearly interpolated between two consecutive points to construct an overall eutecticmap for the sample, shown in Figure 1.7b.(a)(b)Figure 1.7: Profiling apparatus consisting of a CNC stageand digital-SLR camera installed (a) and resultingeutectic area fraction profile on a wheel sample (b),from the work of Matthew Roy [47].1.3 Solidification of A356Aluminum alloy A356 is a hypoeutectic Al-Si-Mg alloy widely used in the automotiveand transportation sectors. The characteristic composition of this material is givenin Table 1.1.16CHAPTER 1. INTRODUCTIONTable 1.1: Composition (wt%) of unmodified A356, balanceAlSi Mg Fe Ti Na Ni Cu Zn Ca Zr6.50-7.50 0.20-0.45 <0.13 <0.12 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01The as-cast (ac) microstructure consists of primary aluminum dendrites (α-Al),surrounded by an Al-Si eutectic (Figure 1.8). Other tertiary phases, such as α-intermetallics or β-intermetallics (β-Al5FeSi and pi-Al8FeMg3Si6), may be present dueto melt impurities. The solidification sequence of this alloy starts with the nucleationand growth of primary dendritic α-Al. This is followed by the β-Al-Si eutectic andβ-intermetallics. The remaining liquid, enriched in Si, Mg and Fe, forms Mg2Siprecipitates and engages in complicated ternary and quaternary reactions, producingpi-intermetallics [10,11].Figure 1.8: Example of A356 microstructure in the ac condition,displaying (a) α-Al, (b) Al-Si eutectic and (c) intermetallic and(d) a secondary Mg-Si rich region [47].The solidification of A356 is often modelled using the binary Al-Si system (Figure17CHAPTER 1. INTRODUCTION1.9) [1, 48]. Starting from the fully liquid phase, as the temperature falls belowthe liquidus temperature, α-Al begins to form with low silicon composition. As thetemperature decreases, the volume fraction of α-Al increases, while the surroundingliquid becomes enriched in silicon until it reaches the eutectic temperature. At thistemperature, α-Al has the highest silicon content (1.65 wt%Si) and the liquid is at theeutectic composition (12.6 wt%Si). Further decrease in temperature will cause theenriched liquid to undergo eutectic transformation, where it solidifies into lamellae ofaluminum and silicon [10,11].Figure 1.9: Al-Si phase diagram.A356 is rarely employed in the ac condition owing to a lack of homogeneity andthe detrimental effects on the mechanical properties of coarse plates of Si present inthe eutectic. Several heat treatment schedules are commercially employed, with themost prominent being T6. Most of these schedules consist of solutionizing, waterquenching and then a combination of natural and artificial aging. Both the durationand temperature at which these treatments are carried out decide the final mechanical18CHAPTER 1. INTRODUCTIONproperties. The T6 schedule used in this research is as follows:• Solution treat at 540◦C for three hours.• Quench in water at 60◦C• Artificially age at 170◦C for 6 hours with no natural aging.The solution treatment is applied to induce three phenomena to occur: dissolutionof Mg2Si particles, chemical homogenization and eutectic-Si structure modification.The Mg2Si precipitate that forms during the last stages of solidification is readilysoluble in α-Al at the typical solutionizing temperatures and will dissolve given enoughtime. In the ac state, solute elements are typically highly segregated due to dendriteformation. Solution treatment serves to chemically homogenize the casting, therebyimproving solid solution strengthening [47].The changes to the eutectic-Si structure imparted by solution treatment also playan important role in determining the final mechanical properties. While modifiedAl-Si-Mg alloys contain fairly refined fibrous eutectic-Si, this is further refined dur-ing solution treatment by the processes of fragmentation and spheroidization. Theac fibres break into particles at elevated temperature and gradually spheroidize inorder to minimize surface energy of the Al-Si interface. With longer treatment times,coarsening occurs. Larger Si particles develop facets and coalesce with other nearbyparticles to minimize surface area in regions of high Si concentration [47].For the quench operation, the water temperature is selected to maximize coolingrate while concurrently limiting thermal stress development. A high cooling rate isnecessary to suppress precipitation when cooling from the solution treatment tem-perature to room temperature. This produces a high degree of solute supersaturationas well as retaining a larger number of matrix vacancies. If the cooling rate is tooslow, non-uniform precipitation will occur, localized at grain boundaries or sites ofhigh dislocation density [47].Artificial aging is a precipitation heat treatment process. It consists of taking19CHAPTER 1. INTRODUCTIONpreviously solution treated components and holding at a static temperature for aperiod of time. The process is necessary to precipitate small particles coherent withthe surrounding matrix which are finely dispersed particles to resist dislocation glide.In Al-Si-Mg alloys, the supersaturated solid solution resulting from the solutionizingprocess transforms to a stable phase plus a metastable precipitate phase, β. The rateof precipitation, as well as the precipitate morphology, is dependent on temperature,time degree of supersaturation and diffusivity. At high temperatures, diffusion occursrapidly even when supersaturation is low. The inverse is true for low temperatures[47].1.4 Mechanical properties of A356In previous sections, the microstructure and solidification path of A356 were dis-cussed. This section will discuss the impact of microstructure on mechanical proper-ties of A356. This will be done by looking at the results from several studies wherethe effects of Dendrite Arm Spacing (das) and composition were investigated onmechanical properties.Microstructure refinement, which results in a corresponding strength increase,can be achieved by decreasing the solidification time during casting. By decreasingsolidification time, the cooling rate during solidification is increased which results indecreased primary and secondary das. A number of studies have investigated theeffects of das variation on the yield strength in aluminum alloys [49–53]. Figure 1.10shows the yield strength versus das from five separate studies. These studies considerA356 alloys in the ac and T6 heat treated state with a range of das. As shown inFigure 1.10, the yield strength decreases as the microstructure becomes coarser. Thiseffect is often expressed in terms of the Hall-Petch equation [54]:σy = a+ b · d−1/2 (1.1)20CHAPTER 1. INTRODUCTIONwhere a and b are constants and d is spacing. The definition of spacing, however,changes from grain size for pure aluminum to dendrite cell size (the width of individualdendrite cells) in aluminum alloys with 1.6 - 2.5wt%Si to nearest-neighbour distancebetween Si particles in alloys with 5.3 - 25wt%Si [55]. The majority of the publishedstudies on this topic use the inverse square root of the das to construct a relationshipbetween microstructure and yield strength [56–58]. An expression in the form ofEquation 1.2 has been fit to the data presented for A356.σy = a+ b ·DAS−1/2 (1.2)As depicted in Figure 1.10, the expression exhibits a reasonable correlation (R2equal to 0.62 and 0.81) to both the ac and T6 heat treated data.Figure 1.10: σy versus das for ac and heat treated A356alloys with fitted expressions.Elzanaty [59] and Kalhapure et al. [60] studied the effect of variation in siliconcontent on yield strength in aluminum alloys in the as-cast condition. In both studies,yield strength and silicon content were shown to exhibit a linear correlation over thehypoeutectic region. A 50-100% increase in the yield strength was observed for silicon21CHAPTER 1. INTRODUCTIONcontents ranging from 4 to 11wt%. It should be noted that the hypoeutectic region hasbeen studied extensively since most of the commercial Al-Si-Mg casting alloys, suchas A356, are from this family. Barresi et al. [61] and Moller et al. [62] investigated theeffect of magnesium content on yield strength. These studies showed that a ∼0.2%increase in the yield strength occurs over the range of 0.2 to 0.4wt%Mg. Since theeffects of silicon content and das on yield strength are more dramatic, the effect ofmagnesium content can be neglected for compositions between 0.2 and 0.4wt%.Roy et al. conducted a comprehensive study to characterize the constitutive be-haviour of ac A356 over a range of temperatures and strain rates [63]. In this studyseveral phenomenological and physically-based constitutive expressions were fit tothe experimental data. It was found that the modified Hollomon expression (Equa-tion 1.3) was the most versatile expression for fitting the results over the range ofconditions tested [63,64].σH = K(0 + )n (1.3)where K and n are constants corresponding to strength and strain-hardening, re-spectively. 0 is a constant indicating the yield strain, which is essential for correctlypredicting the flow stress [64].In a more recent study, Haghdadi et al. used artificial neural networks to predictthe hot deformation behaviour of an A356 alloy. They used a series of compressiontests in various temperature ranges and strain rates to train the artificial neuralnetwork. The predicted results were then compared with a strain-compensated typeconstitutive equation. They concluded that the artificial neural network model isstatistically accurate and is a robust tool to predict high temperature flow behaviourof A356 aluminum alloy [65].22CHAPTER 1. INTRODUCTION1.5 Scope and objectivesAs discussed in the previous sections, there has been little attention given to macroseg-regation resulting from the movement of enriched liquid caused by compensatory flow,especially for the case of industrially-relevant shape castings such as wheels and en-gine blocks. The objective of this research is to study the macrosegregation in shapecastings of aluminum alloy A356 and assess its effect on the localized constitutivebehvaiour. To accomplish this objective, four main tasks have been identified:• To develop an experimental casting set-up that will produce macrosegregation;• To develop and verify a method to quantify solute redistribution that has oc-curred on the scale of the casting;• To develop and validate a mathematical model capable of predicting the forma-tion of macrosegregation; and• To develop and apply correlations between silicon mass fraction, microstructureand constitutive behaviour.The first step in this research is to design an experimental apparatus to isolateand exaggerate macrosegregation. A dumbbell shaped casting with adjustable coolingrates is used to this end. Designing the casting apparatus is critical to the successof the research. The concept for this casting is based on two cylindrical volumeslinked by a joint pipe. The mould is fabricated from thin walled copper tube stockwhich has a low thermal mass relative to the casting. This limits the initial heatremoval / solidification and also allows rapid heat extraction when augmented withadditional cooling where required. Each casting is instrumented with thermocouplesat various locations to support validation of the thermal-fluid flow model frameworkdescribed below. The rationale behind this design is that by controlling the coolingrates on the different sections of the casting one can partially solidify the mid-section,23CHAPTER 1. INTRODUCTIONso that there is enriched liquid available for transport. As solidification occurs in thebottom of the casting the enriched liquid in the middle section will compensate theshrinkage. This will cause a variation in composition as the middle part will becomesolute deplete and the bottom section will be solute enriched.The second task in this project is to develop a method to provide a detailed mapof segregation throughout the casting. Previous methods of segregation mapping,such as edx, are not suitable for this task because the size of the samples used toconduct this type of analysis are fairly small, and usually in order to map a largesection, several small samples are cut from the section and analyzed. Therefore, animage processing technique is adopted to quantify solute redistribution. The intendedmethod uses pixel-based analysis to calculate the silicon area fraction throughout thecasting. To evaluate the accuracy of this analysis technique, the initial analysis is per-formed using idealized artificial microstructures. Following this initial development,this technique is used to assess real microstructure.A numerical simulation is developed to further understand the formation of macroseg-regation caused by movement of enriched liquid. An incremental approach to modeldevelopment is taken where physical phenomena and boundary conditions are addedto the base model step by step, validating the model on each stage with simple casesfrom literature. The numerical analysis is formulated using ANSYS FLUENT, a com-mercial CFD software capable of solving the relevant governing equations to predictthe fluid flow/heat transfer occurring during solidification.Finally, a correlation between silicon area fraction, microstructure and constitutivebehaviour is developed by performing careful tensile tests on samples from a seriesof castings with different silicon content. Specialized plate castings are particularlyuseful in this experiment as the microstructure varies steadily with distance fromthe walls. These correlations can be very useful to predict the tensile strength as afunction of location in industrial components.24Chapter 2Constitutive Behaviour ofhypoeutectic Al-Si-Mg Alloys1As the overall objective of this thesis is to study macrosegregation and its effect onthe local constitutive behaviour in A356 alloy shape castings, this chapter examinesthe effects of silicon content and microstructural variation on the mechanical proper-ties of hypoeutectic Al-Si-Mg alloys (off-spec A356 alloys) in the ac and heat-treatedconditions. A series of plate castings were produced from Al-Si-Mg alloys where theSi content was varied and the constitutive behaviour was characterized. The resultswere then used to establish an empirical expression correlating das, silicon contentand the heat-treated state with flow stress based on a modified Hollomon equation.This expression, combined with a numerical model predicting macrosegregation andsolidified microstructure, provides an essential tool to predict the mechanical proper-ties throughout a geometrically complex component.1Portions of this chapter have been published in:• Khadivinassab H., Maijer D. M., Cockcroft S. L., “Constitutive Behviour of MacrosegregatedA356”, Material Science and Engineering A, (2017) – under revision25CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYS2.1 Experimental methodology2.1.1 MaterialStarting from unmodified A356 alloy, six alloys were made by either adding purealuminum2 or Al-Si master alloy with 50wt% silicon. The resulting alloys, summarizedin Table 2.1, have a range of silicon contents.. It should be noted that the additions ofAl and Al-Si master alloy to A356 alloy meant that the nominal Mg content varied inthe range of 0.2wt% - 0.35wt% for each alloy which is within the ASTM specification(i.e. 0.20 to 0.45wt% Mg)3. It should also be noted that the presented compositionsin Table 2.1 are the predicted compositions. A standard deviation of 5% has beenassessed as feasible in these alloys because the Si composition in the base alloy wasreported with a standard deviation of 5% according to Table 1.1.Table 2.1: Composition of fabricated alloys used inconstitutive behaviour experiments.Alloy name wt% Al wt% Si wt% MgTAL01 95.80 4.00±0.20 0.20±0.01TAL02 94.75 5.00±0.25 0.25±0.01TAL03 93.70 6.00±0.30 0.30±0.01TAL04 (A356) 92.65 7.00±0.35 0.35±0.02TAL05 91.66 8.00±0.40 0.34±0.02TAL06 90.67 9.00±0.45 0.33±0.02TAL07 89.67 10.00±0.5 0.33±0.022.1.2 Experimental setupCastings in the form of plates were produced using the casting setup shown in Figure2.1. The casting setup consists of an insulated, steel pour basin and runner systemconnected to a steel mould (via the bottom) to generate the plate castings. Thecharge for each casting was produced by cutting small pieces of melt stock, weighed2Aluminum 10603ASTM B26/B26M - 0926CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSon a small scale, depending on the recipe which were combined in a Silicon Carbidecrucible. The crucible was then placed in a resistance furnace, set to 750◦C, to meltthe charge. Prior to each casting, the melt was degassed to reduce the hydrogencontent by bubbling Ar gas through the melt for 20 minutes. Meanwhile the mouldwas preheated to 300◦C. The crucible was removed from the furnace and allowed tocool while monitoring the temperature with a handheld thermocouple. The metalwas poured into the pour basin once it reached a temperature of 700◦C using anautomated pouring device, shown in Figure 2.2, to ensure reproducibility during thepouring stage. The main design criterion for the plate casting was to produce differentcooling rates at the mid-plane, horizontally across the plate. The variation in coolingrates results in a variation in das within the plate. It is expected that each plateproduces samples with finer das closer to the sides and coarser das in the middle.Figure 2.1: Illustration of the plate casting mould.Overall, ten plates were cast using this method. The composition of the castalloys are summarized in Table 2.1. Seven plates, fabricated using alloys TAL01 - 07,27CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.2: Illustration of the automatic pouring device.were used in the ac condition. The three other castings, cast from TAL01, A356 andTAL07, were heat-treated to a T6 condition. The T6 schedule used in this researchis as follows:• Solution treat at 540◦C for three hours.• Quench in water at 60◦C• Artificially age at 170◦C for 6 hours with no natural aging.2.1.3 CharacterizationTensile test samples were extracted from each plate using a water jet cutter. Figure2.3 shows the number, location and size of the samples cut from each plate. A totalof eleven samples were obtained from each plate. Tensile tests were performed on anInstron 8872 machine equipped with a 100 kN load cell. The tests were conductedat a fixed displacement rate of 2 mm/min until failure. A 1-inch extensometer wasattached to each sample in order to measure the gauge length displacement during the28CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYStests. The conversion of the raw force-displacement data, outputted by the device,to true stress-true strain data was based on Dieter [66] and the yield strength wascalculated based on 0.2% offset method [66]. It should be noted that these methodsare in accordance with the ASTM standard 4.Figure 2.3: Drawings of the location and naming of eachtensile sample cut from the plates and dimensions ofthe samples. Note that all the samples have athickness of 12.5mm.Metallographic samples were extracted from each tensile sample location in theplate using different combinations of handsaw, Electro-Discharge Machining (edm)and bandsaw. Specimen size permitting, samples were polished using an automaticpolishing machine5 with 240, 320, 400, 600 and 1800 grit Si-C paper. This was fol-lowed by two secondary polishing steps with 6 and 1 µm diamond paste. A NikonEclipse MA200 inverted microscope and Nikon DS Fi1 digital camera, with NIS-Elements software were used to take images of the microstructure for follow-on anal-ysis. Examples of the microstructural images from each of the alloys cast for thisstudy are shown in Figure 2.4. The eutectic area fraction increases from TAL01 toTAL07 consistent with the increase in Si content.4ASTM E646 - 165Buehler Phoenix BETA Grinder Polisher with Vector Power Head29CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYS(a) (b)(c) (d)(e) (f)Figure 2.4: Sample microstructure of created alloys, TAL01(a), TAL02 (b), TAL03 (c), TAL05 (d), TAL06 (e)and TAL07 (f) taken from position 2.30CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSdas measurements were performed manually using ImageJ [67]. The procedureapplied to measure the das is shown schematically in Figure 2.5. After loading amicrograph into the software, a line is manually drawn across secondary dendritearms, intersecting at least 5 dendrite arms. The length of the line is then divided bythe number of inter-dendrite arm spaces and scaled accordingly. This procedure wascarried out three times for each image and the values were then averaged. It shouldbe noted that this method is in accordance with GM standards 6Figure 2.5: Example of das measurement technique,showing three different measurements.Porosity and silicon area fraction measurements were all conducted by image anal-ysis using a Python code developed for this purpose. First, the images were segmentedby grey-scale into three parts; pores, primary and secondary. In order to determinethe porosity area fraction, the area of the image segment representing pores was di-vided by the overall area of the image. To determine the silicon area fraction, the6GMW1643631CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSarea of the image segment representing silicon was divided by the total non-porousarea.2.2 Experimental resultsFive curves from samples with a range of silicon compositions and different heat-treated states have been selected to represent the 110 stress-strain curves generatedduring the testing and are shown in Figure 2.6. These representative curves willbe used in this section to discuss the fitting procedure and to assess the goodnessof fit of the proposed phenomenological expression. The results shown in Figure2.6 are for samples cut from the same location in each plate (i.e. position 2 inFigure 2.3). Qualitatively, Figure 2.6 indicates that the tensile strength increaseswith increasing silicon content. Furthermore, as expected, the T6 heat-treatmentresults in a significant increase in tensile strength of the material. It can also be seenthat the slope of the curves in the elastic region exhibits more or less the same value.This value was calculated to be 72.4GPa with 1.3% error across all the samples, whichis consistent with the elastic modulus of the material [68]. Since the elastic modulus isan intrinsic property of the material and the measured value here exhibits a low erroracross all the samples, it was considered a constant and treated as being independentof Si content in follow-on analysis.The yield strength for each tensile sample was determined using the 0.2% offsetmethod. The average yield strength for each plate and the standard deviation aresummarized in Table 2.2. Looking at the quantitative data, the average yield strengthincreases with increasing silicon content.Figure 2.7 shows the average das calculated from all the samples measured ateach location in the plates. As discussed previously, the casting process used in thisstudy generates plates with varying das; coarser in the middle and finer on the sides.Because of the symmetry in the shape of the plate about the centerline of the inlet,32CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.6: Stress-strain curves for five selected samples.Table 2.2: Average yield strength and standard deviation determinedfor each alloy.Alloy name TAL01 TAL02 TAL03 A356 TAL05 TAL06 TAL07σy 85.01 86.24 90.27 96.68 94.72 97.71 99.66SD [MPa] 5.81 8.19 8.92 2.70 6.91 4.06 2.01Alloy name TAL01-T6 A356-T6 TAL07-T6σy 152.36 197.24 237.08SD [MPa] 11.37 7.07 8.05symmetric das data was expected from the castings. The dashed line drawn on Figure2.7 is the average das calculated at each position from the centerline and can be usedto assess this symmetry. Overall, the das is observed to be higher than the averageon one side and lower on the other. This may be due to asymmetry in the coolingprocess due to the structure of the mould. A small decrease in the das was observedin the mid-plate location. The dip in das was present in every plate and may be theresult of refinement caused by solidified particles being transported to the middle ofthe casting by convection.33CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.7: das values at each location in the plates, wherethe red circles represent the average das, blue barsindicate the variability and dashed line shows thesymmetric average values of das in each location.Figure 2.8 illustrates the porosity volume fraction averaged across all samples ateach location in the plates. Due to degassing, the porosity values observed were verysmall and were not expected to affect the strength of the material [69]. Thus, theeffects of porosity were not considered in this study.2.3 Constitutive equation developmentIn order to develop a constitutive equation relating the flow stress to the siliconcontent and das, the stress-strain data was randomly divided into two groups. Atraining set consisting of 80% of the data and a cross validation set consisting of therest. In order to develop an expression for the flow stress, first Equation 1.3 was fitto each of the training data sets. Then an equation was developed for n, K, and 0as a function of Si wt%, das and T6 state. Afterwards, the overall fit was assessedusing the cross-validation data set.After calculating the yield stress for each sample using 0.2% offset method, a34CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.8: Average porosity over all the plates.linearized form of Equation 1.3 was fit to each data set separately. Values for thework hardening, n, and strength, K, constants were found from the slope and y-intercept of ln σ-ln  curves. The value of 0 was extracted directly from the yieldstress data. Figure 2.9 shows the representative experimental data and the calculatedflow stress based on the fits to Equation 1.3. The average R2 obtained for the fit toeach of the training data sets individually was 99.2%In the second stage of fitting the data, each of the parameters, n, K and 0,were fit to Equation 2.1. This equation is a combined form of the modified Hall-Petch equation [54] and a linearly varying silicon content correlation. A binary t6parameter has been incorporated and multiplied through the equation, to considerwhether the t6 heat treatment has been applied. t6 is one if a T6 heat-treatmenthas been applied or 0 if the metal is in the ac condition.f = a+ b× t6+ (t6+ 1)(c× CSi + d× das−1/2 + e× CSi × das−1/2) (2.1)35CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.9: Stress-strain curves with fitted models.Table 2.3: Values of fitted coefficients in equation 2.1 for n,K and 0.Parameter a b c d e R2n 0.185 -0.226 0.013 0.456 -0.069 92.7%K -240.454 -746.425 87.187 3318.584 -414.728 56.8%0 1.973e-3 -6.437e-5 1.43e-4 4.46e-3 -3.968e-4 93.5%where CSi is the silicon weight percent, and the parameters a, b, c, d and e areconstants. It should be noted that constants c, d and e control the influence ofsilicon content, das and combined effect of these parameters, respectively.Table 2.3 shows the calculated constants and the R2 of the fit for each parameter.The fit for both n and 0 exhibit high R2 values, but the fit for K is poor. To resolvethis issue, the modified Hollomon equation was fit to each curve again. However, inthis step, only the K value was adjusted and the values of n were based on Equation2.1. The revised coefficients for K are shown in table 2.3 along with the new R236CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSTable 2.4: Values of fitted coefficients in equation 2.1 for K.Parameter a b c d e R2K -89.096 -598.938 56.984 2440.074 -243.955 83.8%for the fit to this parameter. The R2 value for K increased significantly by carryingout the second step. The average R2 for the multi-step fit to the training dataset is96.5%.Figure 2.10 compares the experimental data from the representative tests with thecalculated flow stress curves based on the das, silicon content and heat treatmentstate. Figure 2.11 illustrates the measured versus predicted values for all the datapoints in the cross-validation data set. The R2 for the final model evaluated with allavailable data is 95.1%. It is noticeable that the R2 for the training dataset is higherthan that of cross-validation dataset. The reason for this difference is that the fit wastailored for the training dataset, therefore, it is expected that this fit would exhibita higher R2.Figure 2.10: Representative experimental stress-straincurves with a fitted model.37CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSFigure 2.11: Predicted vs. measured flow stress forcross-validation data-set.2.4 Sensitivity analysisA sensitivity analysis has been conducted to assess the significance of each term inequation 2.1. The details of this analysis is presented in Appendix 6.2. The sensitivityresults show that the parameters are more sensitive to changes in Si content ratherthan das. This suggests a similar behaviour for the flow stress. Furthermore, applyingthe T6 heat treatment to the alloy seems to have a small effect on the contributionsof the terms dependent on Si content. This is expected since applying a T6 heattreatment does not change the silicon content and its contribution to the flow stress.Overall, the contribution of all the parameters are significant in the equation.The sensitivity of parameters n, K and 0 based on changes in das and Si contentare shown in Figures 2.12-2.14. The sensitivity is defined as percent change from thebase value. For this study, the base values for das and CSi are 35µm and 7wt%,respectively. The data shows that there is a high variability in parameter 0. Pa-rameters n and K, on the other hand, show lower sensitivity to changes in das and38CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYSSi content. it should also be noted that for low das values, parameters n and Kare less sensitive to changes in Si content. Conversely, these parameters show highsensitivity for changes in Si content when das is high. 0, on the other hand, showshigh sensitivity to Si content over the whole analyzed das range. Considering the rel-ative importance of these parameters, since n is an exponentiation factor even smallchanges in this parameter are magnified in the overall flow stress results. On theother hand, small changes in 0 would not have significant effect on the final stressresult as it is a summation factor.(a) (b)Figure 2.12: Sensitivity analysis of parameter nfor (a) ac and (b) T6 condition.Figure 2.15 shows the sensitivity of the calculated yield strength based on theaforementioned das and Si content conditions. As can be seen, in both ac andT6 conditions with low das, yield strength exhibits low sensitivity to changes in Sicontent and vice versa.2.5 DiscussionParameter n can be viewed as a indicator of the work-hardening and formabilityof a material, where increasing n increases the formability [70]. The results fromFigure 2.12 suggest that although applying a T6 heat treatment increases strength,39CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYS(a) (b)Figure 2.13: Sensitivity analysis of parameter Kfor (a) ac and (b) T6 condition.(a) (b)Figure 2.14: Sensitivity analysis of parameter 0for (a) ac and (b) T6 condition.it reduces the formability of the material. Additionally, the effects of das and CSiare intertwined. High das-high CSi and low das-low CSi conditions yield betterformability than high das-low CSi and low das-high CSi conditions in both the as-cast and T6 conditions.Parameter 0 can be considered to be the amount of strain-hardening that thematerial received prior to the tensile test [66]. The results from Figure 2.14 suggestthat initial strain-hardening is highly dependent on the variation of Si content inboth T6 and as-cast conditions; the higher the silicon content, the higher the initial40CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYS(a) (b)Figure 2.15: Sensitivity analysis of yield pointfor (a) ac and (b) T6 condition.strain-hardening. This shows that initial strain-hardening is an intrinsic property ofthe alloy, mostly depending on the composition of the alloy rather than the coolingcondition.Experimental results as well as the developed model suggest that yield strengthvariation is higher for the T6 case compared to the ac case. Looking at table 2.3 themaximum yield strength variation for the ac case is 14.65 MPa where, this value isis 84.72 MPa for the T6 case. This means that by applying T6 heat-treatment to apart where there is macrosegregation would result in a much higher variation in yieldstrength.The developed and verified constitutive equation can be utilized in solidificationsimulations to predict the mechanical behaviour of an Al-Si-Mg alloy. This couldbe especially useful where casting manufacturers are interested in incorporating thistechnique in their simulations to show that their parts can satisfy the designer’sstrength requirements throughout the geometry. Furthermore, the proposed modelcould fit into a through-process modeling methodology where the complete manufac-turing process is simulated to provide a detailed prediction of the state of the partwhich can then be used as an input to a model of in-service performance.41CHAPTER 2. CONSTITUTIVE BEHAVIOUR OF HYPOEUTECTIC AL-SI-MGALLOYS2.6 SummaryThe constitutive behaviour of hypoeutectic Al-Si-Mg alloys in both the as-cast and T6heat treated conditions has been experimentally characterized through an extensiveset of tensile tests. The data was used to fit a modified Ludwig-Holloman expressionwith parameters n, K and 0 as a function of das, Si content and T6 state. The finalequation exhibits a fit with R2 of 95.1% over the test dataset. Analyzing the resultsthe following can be concluded:• Apart from das, silicon variation in A356 castings, especially castings wheremacrosegregation is present, plays a crucial role in the strength of the material.• The maximum yield strength occurs when das is low and Si content is high andthe minimum yield strength occurs when das is high and Si content is low.• High das-high Si content and low das-low Si content results in better forma-bility in both ac and T6 conditions.• Application of a T6 heat treatment results in reduced formability of the mate-rial.• The Initial strain-hardening (0) is mostly dependent on Si content.• T6 heat treatment results in a higher yield strength variation in a cast partwhere macrosegregation has occurred.With the development of this expression, the local flow stress behaviour in an A356casting where macrosegregation is present can now be characterized. This equationcan also be used in conjunction with solidification simulation as a predictive tool toestimate the flow stress distribution after solidification.42Chapter 3Characterization of Macrosegregationin Eutectic Alloys1As explained in the literature review, results from other methods, such as edx, arenot practical for assessing the variation of composition over a large cross-sectionalarea. This chapter presents an image processing method that has been developed tocharacterize the macrosegregation occurring on cross-sections of the castings producedfor this study. The method utilizes a combination of image segmentation, pixel-to-pixel analysis and tessellation techniques to construct a quantitative map of the solutedistribution on large samples. Compared with methods reported previously in theliterature, the current method is robust and can be applied to samples with largeirregular cross-sections. The accuracy and validity of the method has been assessedthrough a series of artificially designed micrographs.3.1 Methodology3.1.1 Analysis overviewThe initial step to analyze the spatial variation of segregation in a sample is toconstruct an image montage of the microstructure over the entire area that is to beanalyzed. This can be done through different techniques, the most common one being1Portions of this chapter have been published in:• Khadivinassab H., Maijer D. M., Cockcroft S. L.,“Characterization of Macrosegregation inEutectic Alloys”, Materials Charcterization, (2017)43CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSoptical microscopy. In order to make a montage, the cross-section is first prepared(i.e. a suitable section identified, extracted and polished). Then, a sequence ofimages of the microstructure covering the entire surface of the sample cross-section isacquired. These images are then segmented into the desired phases. Several methodscan be utilized to segment the images, such as the Gaussian Mixture Model and K-means [71]. Although these methods are quite accurate, when it comes to a large setof images, they are not time-efficient [71]. The recommended method in this case isto use Otsu thresholding, then implement appropriate morphologies to eliminate the"salt and pepper" noise in images [71].In order to visualize the variation of segregation, a tessellation map is overlaidon the montage and the area fraction of the alloying element is then calculated ineach mesh element. The area fraction is then converted to mass fraction based on amethod that have been developed. The spatial variation of segregation in the contourmap is strongly dependent upon the number of elements used in the mesh. For smallnumbers of elements, the generated contour map represents the area average of themass fraction and is therefore quite coarse. An increase in the number of elementsreduces the size of each element, potentially to the point where each element maycontain only one microstructural phase. In this case, the measured area fractionbecomes a binary representation because it is either completely filled with one phaseor not. The optimal mesh size for the tessellations in this study were determined bycalculating a quantity referred to as the Average Maximum Difference (amd) using asimple algorithm.After the optimization stage, the image is divided into small triangular sectionsbased on the determined mesh size. Subsequently, the mass fraction of the desiredphase is determined in each triangular section. The data then gets written into aninput file for the Tecplot360 visualization software 2, which can be used to visualize2Tecplot360 website: http://www.tecplot.com/products/tecplot-360/44CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSand further analyze the segregation map.The following sections will describe the analysis procedures in a greater detail.3.1.2 Image segmentationImage segmentation is one of the more challenging steps in conducting macrosegre-gation characterization analysis. Previous studies on this particular problem utilizedimage histograms to segment the microstructure [45,47]. This method, however, hasits shortcomings as different phases might have large overlaps, which then results inan inaccurate segmentation map.To overcome this issue, a method based on Otsu thresholding has been used inthis study. Otsu thresholding is a histogram based thresholding method. Assuming abi-modal histogram, this algorithm searches for a threshold that minimizes intra-classvariance [72].In the approach applied in this work, the dark pixels representing pores were firstidentified manually and extracted from the data, leaving only the lighter pixels be-longing to microstructural phases. Otsu thresholding was then applied to cluster thepixels representing primary and secondary phases. Nonetheless, the histogram for theremainder of the data points might not be of bi-modal form. In these cases, Otsu willidentify only one cluster instead of two. In order to resolve this issue, the histogramwas equalized using the Contrast Limited Adaptive Histogram Equalization (clahe)method before the application of the Otsu method. After successful thresholding,a combination of morphological operations were applied to the segmented image tofine-tune the clusters.3.1.3 MeshingAfter segmenting each of the individual micrographs and stitching them together, thehigh resolution montage was divided into smaller sections. This can be done throughdifferent algorithms depending on the accuracy and efficiency needed. A truss-based45CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSmeshing algorithm has been used in this work. Truss-based meshing, developed basedon Persson’s work [73], is a method that utilizes an iterative technique to refine amesh defined using a physical analogy of a truss structure, where points in the meshare nodes of the truss structure. Considering a force-displacement function for themembers that makeup the truss, the code solves for equilibrium at each step. At everyiteration, the nodes are moved by the calculated force and Delaunay triangulation isused to adjust the edge topology [73]. After calculating the displacements, the nodesoutside the bounds of the shape are pushed back to the boundary. This process iscarried out until the overall force in the truss system is within an acceptance criteria,i.e. a pseudo-equilibrium state.One of the challenges in this technique is to properly define the geometry. Perssonused a signed distance function to define simple geometries and combined them torepresent more complex geometries [73]. This, however, lacks the ability to representreal-life geometries in detail. Therefore, a new geometry representation technique wasdefined based on the negative mask of the image. Figure 3.1 shows a sample imageand its automatically generated negative mask which is suitable for meshing.(a) (b)Figure 3.1: Sample image (a) and its negativemask (b).46CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYS3.1.4 Area fraction to mass fraction conversionThe first step to convert the phase area faction, calculated from the segregated phaseassessment, to mass fraction is to know its relation to volume fraction. From astereological point of view, there are only a few bulk microstructural parameters thatcan be assessed by analyzing a 2D surface, where volume fraction is one of them [74].According to Kaplan [74], the volume fraction is equal to the area fraction if thecross-sectional plane is randomly positioned.The next step in determining the mass fraction is to convert volume fraction.This operation is highly dependent on the magnification and resolution of the mi-crograph. For low magnification images, constituents of the eutectic phase can bedifficult to distinguish. Thus, the approach proposed here is to determine the eutec-tic area fraction first and calculate the mass fraction of the alloying element. For highmagnification images where the constituents of eutectic phase can be distinguished,the area fraction of the alloying element can be calculated directly and converted tomass fraction. It should be noted that, this method assumes the alloying elementis present as a pure phase. For the case of A356 and similar Al-Si-Mg alloys, sincethe initial composition of Magnesium is small relative to the composition of Silicon,percentage of contribution of Mg2Si precipitates can be neglected and Silicon can beregarded as a pure phase.The conversion of eutectic area fraction to mass fraction of the alloy element in thecase of low resolution images can be accomplished by using the Scheil equation [11].After manipulating the Scheil equation, the initial composition can be extracted as(Equation 3.1):C0 =Ceϕ(Cs/Ce)−1e(3.1)where Cs is the maximum solubility of the alloying element at the eutectic temperature47CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSand ϕe denoted eutectic area fraction. It should be noted that C0, Ce and Cs are allin atomic percentages. Equation 3.2 is then used to convert C0 to a commonly usedweight percentage.wt%B = 1/(1 +100− at%Bat%BMAMB) (3.2)where MA and MB are the atomic masses of components A and B in a binary alloy,respectively.To convert the area fraction of an alloying element to mass fraction in the caseof high resolution images, the lever rule may be applied based on a eutectic phasediagram (shown in Figure 3.2). The volume fraction of secondary phase (i.e. β) canbe calculated at room temperature by the lever rule shown in Equation 3.3. Notethat compositions are in atomic percentages rather than weight percentages. Thelocal composition can then be determined by modifying Equation 3.3 to Equation3.4.Figure 3.2: Eutectic phase diagram.48CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSVf = (C0 − Cα)/(Cβ − Cα) (3.3)C0 = ϕ× (Cβ − Cα) + Cα (3.4)where Cα and Cβ are maximum solubility of component B in phase α and componentB in phase β at room temperature, respectively.Lastly, the calculated atomic percentage needs to be converted into weight per-centage. This conversion is carried out using Equation 3.2.3.1.5 amdOne of the issues with dividing the segregation map into smaller areas (i.e. meshing)is that the final segregation map is sensitive to the mesh size. To combat this in thecurrent study, the optimal mesh size has been determined by calculating a quantityreferred to as the amd using a simple algorithm. In this algorithm, the differencebetween the area fraction of the phase of interest in an element with its nearbyelements is first calculated using a specified kernel. The kernel in this case is a squarematrix centering around an element. The area fraction in each element is calculatedby counting the number of pixels of the desired phase and then dividing by the totalnumber of non-black pixels (Eq 3.5).f ei =neinetot − nebk(3.5)where f ei is the area fraction of the desired phase i in an element e, ne is the numberof pixels in an element e, where subscripts i, bk and tot indicate the pixels of thedesired phase, black pixels and the total number of pixels, respectively.The maximum difference of each element with its neighbours is then determinedby calculating the difference of the area fraction of each element with its adjacent49CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSelements (Eq 3.6) and selecting the maximum value (Eq 3.7).Dei = |f ei − Arkerefi | (3.6)MDei = max(Dei ) (3.7)where Dei is an array of the difference values for an element e, Arkerefiis an arraycontaining the area fractions for all the elements in the kernel centering around anelement e, and MDei is the maximum difference of an element e.Figure 3.3 shows an element and its adjacent elements with three different kernelssuperimposed on them. A kernel with a width smaller than the mesh size will onlycapture the element itself. On the other hand a kernel with a width larger or equalto the mesh size will capture three or more elements. In this research, a kernel witha width of 1.2 times the mesh size has been chosen. This is the smallest kernel thatcaptures all the adjacent elements, while ensuring that only the adjacent elementsare picked. Applying this process to the entire image, the maximum differences ofeach element and its neighbours are calculated. These maximum differences representhow sharply the area fraction is changing with respect to the neighbouring cells. Bycalculating the average of these values over the whole image, a single value, calledthe AMD, representing the overall change in the gradient can be calculated (Eq 3.8).This process can be repeated for different tessellations. The optimal mesh size toevaluate the spatial gradient of a sample is achieved when the calculated amd is aminimum (Eq 3.9).AMDhi =Nh∑e=1MDei/Nh (3.8)hopt = x 3 (AMDxi = min(AMDhi )|hfh=hl) (3.9)where Nh is the number of elements in the mesh with spacing h, and AMDhi is the50CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSFigure 3.3: An element with its neighbouring elements. Redboxes indicate the kernels overlaid on the mesh withthree different sizes, where h indicates the mesh size.Average Maximum Difference of mesh with spacing h. The subscripts opt, l and ffor h, indicate the optimal mesh size, the lower bound for mesh size and the higherbound mesh size for AMD analysis.3.2 Methodology verificationBefore applying the segregation characterization methodology to microstructural sam-ples, a series of verification operations were performed to test and verify the individualsteps of the methodology.3.2.1 Image segmentation verificationPrior to selecting the Otsu thresholding method (explained in section 3.1.2) for seg-mentation, it was compared to two other techniques in an effort to determine theapplicability and efficacy of these techniques. The factors considered in the com-parison were accuracy and execution time. The two other segmentation methodsconsidered in this evaluation were:51CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYS• K-means: a simple unsupervised learning algorithm that can be applied tosolve the clustering problem. Given a number of clusters k known a priori, thealgorithm defines k centroids, one for each cluster. In the next step, each pointin the data set is associated to the nearest centroid. At this point, the centroidsare recalculated as the centers of the clusters resulting from the previous step.These steps are then iterated until the sum of distances of each point from theircentroids is minimized [75].In the case of segmentation of the microstructure, three clusters were defined;pores, primary phase and secondary phase. K-means was then applied to clusterdifferent pixel grey-scale values into these segments.• Gaussian Mixture Models (gmm): data is clustered by assigning the datapoints to a number of normal distributions. This method assumes that the dataconsists of several normally distributed components and strives to identify thesegroupings [76].Similar to previous case, three clusters were defined in order to segment theimages into pores, primary and secondary phases.Khajeh in his work on permeability, fabricated a series of Al-Cu alloys with varyingCu contents [77]. In order to characterize the microstructure, Khajeh performedhigh resolution xmt scans of his samples. In the current research, this data wasreconstructed and utilized to calibrate the image segmentation stage. This data waschosen first, due to the similarity of Al-Cu and Al-Si microstructure and second, dueto the fact that the composition of the analyzed alloys were known.The compositions of the alloys are summarized in Table 3.1 and sample xmtsection images are shown in Figure 3.4.Figure 3.5 shows a sample xmt microstructure segmented using the three differ-ent clustering methods. As can be seen, the visual quality of the segmentation for52CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSTable 3.1: The details of the alloys used in Khajeh’sresearch [77]Alloy name wt% Al wt% CuE01 93.7 6.3E02 92.8 7.2E03 91.5 8.5E04 86.5 13.5(a) (b)(c) (d)Figure 3.4: Sample xmt section images of alloys fabricatedby Khajeh, E01 (a), E02 (b), E03 (c) and E04 (d) [77].53CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSall three cases is comparable to the original image. However, there are subtle differ-ences between each segmentation. For this microstructure, Otsu thresholding slightlyoverestimates the size of the eutectic. The K-means result identifies pores that areslightly larger and generates some black noise (visible on the segmented image). Froma visual point of view, gmm method seems to slightly underestimate the amount ofthe eutectic phase.(a) (b)(c) (d)Figure 3.5: A sample xmt image (a) and the respectivesegmented images using the (b) K-means, (c) gmmand (d) Otsu techniques.These methods were applied to more than 8000 images available from the xmt54CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSscans. The area fraction data, which is in the form of eutectic area fraction due tothe image magnification, were then converted to Cu mass fraction using the tech-nique described in section 3.1.4. Figure 3.6 shows the summarized results for thesecases. As can be seen, overall, the measured values are reasonably close to the actualcomposition values.Figure 3.6: Measured vs. actual copper mass fraction valuesfrom K-means, gmm and Otsu thresholding for alloysE01-04.The average absolute error for each method and the execution time per sampleare summarized in Table 3.2. The data gives a quantitative comparison of the threemethods. The average error for the gmm segmentation is below 10%, where K-means results are on average 18% different from the actual composition values. Theexecution time for K-means is the highest, followed by gmm. This is due to the factthat these methods utilize high-cost optimization algorithms to segment the images.55CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSTable 3.2: Comparison of the three image segmentationmethods based on accuracy and execution time (basedon an i7 CPU with 16Gb of RAM).Alloy name Average error [%] Execution time per sample [s]K-means 18.56 4.12gmm 9.63 2.88Otsu thresholding 13.38 0.10Otsu thresholding, on the other hand, uses a less costly optimization algorithm andresults in an order of magnitude difference in execution time. This is especially usefulin the case where a large dataset of several thousand images is being segmented. Dueto reasonable accuracy and low execution time, Otsu thresholding was chosen for usein the remainder of this work to segment the images for macrosegregation analysis.3.2.2 Meshing validationTwo meshing techniques were compared to truss based meshing (explained in section3.1.3). The techniques were evaluated based on mesh accuracy. The following is ashort description of the meshing techniques:• Simple triangulation: was performed by first populating the shape with meshpoints in a grid and then utilize Delaunay triangulation to generate the mesh.This method is easily applied for convex shapes, since Delaunay triangulationgenerates only convex geometries [73]. For the case of concave geometries, thetriangles outside the geometry need to be identified and discarded. The removalof triangles outside the geometry results in topology errors.• Rectangular mesh: is similar to simple triangulation, where the geometry ispopulated with mesh points. However, instead of utilizing Delaunay triangula-tion, the mesh is generated using rectangular patches. It should be noted that,convex and concave geometries are treated similar to the previous method.56CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSA comparison analysis was carried out on two geometries, shown in Figure 3.7, using10, 20 and 50 pixel (px) mesh sizes.(a) (b)Figure 3.7: Geometries for mesh comparison analysis.Figure 3.8 shows the results of applying the three meshing techniques based onthree mesh sizes for the image shown in Figure 3.7a. Table 3.3 shows the accuracy ofthe generated meshes. Accuracy in this case is defined as the error in calculated areashown in Eq. 3.10.accuracy = (1− areaoriginal − areameshareaoriginal)× 100 (3.10)Table 3.4 summarizes the execution time for each mesh shown in Figure 3.8. De-spite having low execution times, the rectangular mesh approach fails to capture theboundary of the geometry even at small mesh sizes. This would result in inaccuratepredictions in future steps. Triangular meshing, on the other hand, captures theboundaries more accurately. However, the elements adjacent to the boundary exhibitlarge aspect ratios. The Truss based method captures the boundary well, especially57CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSfor small mesh sizes. This method, however, due to its iterative nature takes muchlonger time to process.(a) Rectangular-50px (b) Triangular-50px (c) Truss based-50px(d) Rectangular-20px (e) Triangular-20px (f) Truss based-20px(g) Rectangular-10px (h) Triangular-10px (i) Truss based-10pxFigure 3.8: Mesh comparison for Figure 3.7a, with meshing technique andmesh size indicated under each image.Figure 3.9 shows the results from the three meshing techniques based on three58CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSTable 3.3: Accuracy of meshes presented in Figure 3.8.Mesh size Rectangular Triangular Truss based50px 58.39% 68.28% 94.69%20px 80.98% 89.54% 99.03%10px 90.13% 94.61% 99.73%Table 3.4: Execution time (based on an i7 CPU with 16Gbof RAM) for meshes presented in Figure 3.8.Mesh size Rectangular Triangular Truss based50px 0.114 s 0.113 s 0.433 s20px 0.433 s 0.122 s 4.442 s10px 0.998 s 0.130 s 12.610 smesh sizes applied to a more complex shape, shown in Figure 3.7b. Table 3.5 showsthe accuracy of the generated meshes. Table 3.6 summarizes the execution time forgenerating respective meshes. Analyzing the results of these meshing techniques ap-plied to a more complex shape further highlights the inefficiency of the rectangularmesh technique. The issues with this technique are readily apparent for larger meshsizes, as the mesh is broken into two shapes. Although the triangular meshing tech-nique was able to adequately capture the boundaries of the more complex region, theelements adjacent to the boundary are misshaped. The truss based method, despiteits high execution time, captures the boundaries very well and exhibits a nearly uni-form element structure. Due to the considerable differences in accuracy of the threemeshing methods and the need to prioritize accuracy over execution time, the Trussbased method has been applied in the follow-on analysis in this thesis to mesh thecross-sectional geometry for macrosegregation analysis.3.2.3 amd validationIn order to assess the amd technique, a series of carefully designed, artificial mi-crostructure images were generated and analyzed using the amd method. To con-59CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYS(a) Rectangular-50px (b) Triangular-50px (c) Truss based-50px(d) Rectangular-20px (e) Triangular-20px (f) Truss based-20px(g) Rectangular-10px (h) Triangular-10px (i) Truss based-10pxFigure 3.9: Mesh comparison for Figure 3.7b, with meshing technique andmesh size indicated under each image.60CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSTable 3.5: Accuracy of meshes presented in Figure 3.9.Mesh size Rectangular Triangular Truss based50px 57.06% 84.45% 98.72%20px 81.64% 93.67% 99.83%10px 90.93% 97.73% 99.89%Table 3.6: Execution time (based on an i7 CPU with 16Gbof RAM) for meshes presented in Figure 3.9.Mesh size Rectangular Triangular Truss based50px 0.230 s 0.163 s 1.958 s20px 0.847 s 0.164 s 5.441 s10px 2.227 s 0.174 s 57.908 sstruct images with the area fraction changing in 1D, a python code was used togenerate a number of thin rectangular images with constant area fraction. This wasdone by first initializing the matrix of values representing the image section with ze-ros and then populating the matrix with ones at random positions until the ratio ofthe number of ones to the total number of pixels met the criteria of the desired areafraction. The rectangular regions were then joined together to create a square regionwith a known gradient in area fraction. The same method has been used to createimages with 2D area fraction variations. However, instead of stacking rectangularmatrices, a series of co-centric squares with different sizes, each having a constantarea fraction were stacked on top of each other.The generated images, shown in Figure 3.10, have a range of phase area fractionsbetween 0.05 and 0.15. Figure 3.10a has a negative to positive 1D phase area fractiongradient, where the area fraction first decreases and then increases with distance fromthe top edge. Figure 3.10b has a constant positive 2D area fraction gradient, wherethe area fraction increases moving toward the center of the image. Figure 3.10c has apositive to negative 2D area fraction gradient, where the area fraction first increases61CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSand then decreases moving towards the center. Figures 3.10d-3.10f show schematicimages of the variation in area fraction in each image, where black represent the higharea fraction and white represent the low area fraction.(a) (b) (c)(d) (e) (f)Figure 3.10: Artificial microstructure images generated foramd analysis with (a) 1D decreasing-increasing areafraction, (b) 2D increasing area fraction and (c) 2Dincreasing-decreasing area fractions.(d)-(f) schematicrepresentation of the intended gradient for (a)-(c),respectively. Where black represents high areafraction (0.15) and white represent low area fraction(0.05).Figure 3.11 shows the variation of the amd values calculated for the test imagesas a function of the mesh size, the contour images of the area fraction of the artificialphase, and the calculated phase area fraction as a function of the position along avertical line bisecting the images. The mesh dependency of the amd value exhibitsan initial decrease with increasing mesh size before transitioning to increase with62CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSincreasing mesh size. However, a minimum is visible in each plot, which indicatesan optimal mesh size for each microstructure. This is consistent with the discussionpresented in the methodology, where it was suggested that the segregation gradientwould be less accurately calculated when a non-optimal mesh size is used. The optimalmesh size was found to be 41, 41 and 32 pixels for the test images shown in Figure3.10a to 3.10c, respectively.In order to evaluate whether the mesh size reasonably represents the gradient,the phase area fraction values, extracted along a vertical line bisecting the contourimage, have been compared with the actual area fraction values in the generatedmicrostructures (refer to Figures 3.11g-3.11i). The calculated gradient using the op-timal mesh size closely matches the intended gradient. It should be noted that thecalculated gradient was observed to deviate from the actual value when non-optimalmesh sizes were used. Moreover, from the contour plots shown in Figures 3.11d-3.11f,the intended variations of 1D decreasing-increasing, 2D increasing and 2D increasing-decreasing are detectable. This assessment using artificial microstructures suggeststhat the proposed method and the AMD value are useful for determining the optimalmesh size for visualizing segregation.3.3 SummaryOverall a method was developed to visualize segregation in eutectic alloys using animage montage. This method utilizes a tessellation technique to mesh the image, thencalculates the mass fraction of the desired phase in each element. The current methoduses a continuous map of micrographs to calculate segregation map which results inmore accurate results, where previous methods used discrete number of samples inorder to do so. The following conclusions can be drawn from this study:• Three image segmentation methods were compared based on accuracy and ex-ecution time to segment eutectic micrographs. The Otsu method was found to63CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYS(a) (b) (c)(d) (e) (f)(g) (h) (i)Figure 3.11: amd curves (a) to (c), respective contour plots(d) to (f) and gradient comparison curves (g) to (i)for test images shown in Figure 3.10. Where dashedblue lines in figures (g) to (i) show the intended areafraction and red circles show the calculated areafraction.64CHAPTER 3. CHARACTERIZATION OF MACROSEGREGATION IN EUTECTICALLOYSbe more efficient in segmenting these type of images.• A method was developed to convert area fraction of the secondary/eutecticphase to mass fraction of the alloying element in eutectic alloys.• Three meshing techniques were developed and compared based on accuracyand execution time. A Truss based method was found to be more suitable forthis research as it exhibits much higher accuracy in comparison to the othertechniques.• A method called amd has been developed to determine an optimal mesh sizefor calculating the segregation map. Results presented in this study illustratethe validity of this method for a set of artificial designed microstructures.65Chapter 4Macrosegregation in Shape Castings1A series of dumbbell shaped castings with different sizes and cooling conditionswere produced with the intent of either exaggerating or limiting shrinkage-inducedmacrosegregation. This shape was selected to enable variable cooling rates on sectionsof the casting to cause bulk motion of enriched liquid via compensatory flow. Thefollowing sections are dedicated to explain in detail the experimental procedure andthe respective results for macrosegregation in shape castings.4.1 Experimental methodologyA dumbbell-shaped casting geometry was selected for this work because it was hy-pothesized that, by controlling the cooling condition on the neck of the casting, onecan produce or eliminate macrosegregation caused by shrinkage induced flows. Forexample, if it were possible to pause solidification / cooling when the neck sectionis solidified halfway through, the remaining liquid in the neck would be enriched insolute. This is due to microsegregation which was explained in detail in section 1.1. Ifthis liquid were then pulled into the bottom volume due to compensatory flow, afterfull solidification, the bottom volume of the casting would be enriched and the top1Portions of this chapter have been published in:• Khadivinassab H., Maijer D. M., Cockcroft S. L.,“Characterization of Macrosegregation inEutectic Alloys”, Materials Charcterization, (2017)• Khadivinassab H., Fan P., Reilly C., Yao L., Maijer D. M., Cockcroft S. L., Phillion A.B.,“Study of the macro-scale solute redistribution due to liquid metal feeding during thesolidification of A356”, Light Metals Production, Processing and Applications Symposium,The 53rd Annual Conference of Metallurgists, (2014)66CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSvolume would be deplete of solute.Four variants of the experimental setup were produced for this experiment; adumbbell-shaped mould with natural cooling condition, a dumbbell-shaped mouldwith an insulated neck part, a dumbbell-shaped mould with forced cooling on theneck and a large dumbbell-shaped mould with natural cooling condition. The mouldgeometry and cooling configuration for each of the castings are as follows:• Dumbbell-shaped mould with natural cooling: The mould configurationfor the standard sized dumbbell-shaped casting is shown in Figure 4.1. Themould for this casting was manufactured from standard copper fittings (1.5mmwall thickness). An end cap with a 7mm wall thickness was fabricated for thebottom of the casting to promote directional solidification from the bottom tothe top. The overall height of the mould is 163mm with outer diameters of54, 32 and 54mm for the top, middle and bottom sections, respectively. Fivetype-K thermocouples were embedded in the casting at different axial locationsto monitor the temperature.• Dumbbell-shaped mould with insulated joint: The mould and insulationconfiguration are illustrated in Figure 4.2 for this casting. The mould used forthe dumbbell-shaped mould with natural cooling was reused with insulation onthe central joint. A piece of 15mm thick fibreglass blanket was wrapped aroundthe central joint of the mould and held in place with steel sheet (1mm thick).Similar to the previous case, 5 type-K thermocouples were embedded in thecasting at different axial locations to monitor the temperature.• Dumbbell-shaped mould with forced cooling on the neck: The mouldand cooling configuration for this casting are shown in Figure 4.3. The basemould for the dumbbell-shaped casting was reused with changes to the centraljoint cooling configuration. Forced air cooling was applied to the central joint67CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.1: Geometry of dumbbell-shaped casting withnatural cooling condition.Figure 4.2: Geometry of dumbbell-shaped mould withinsulated central joint.68CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSusing compressed air. To ensure that cooling was applied only to the centraljoint, a steel plate assembly was designed and manufactured to shield the topand bottom sections of the mould. Similar to the previous cases 5 type-K ther-mocouples were embedded in the casting at different axial locations to monitorthe temperature. It should be noted that although the nozzle is directed to-wards one side of the mould, the cooling is still expected to be symmetrical.This is due to the high thermal conductivity of the thin copper mould allowingthe heat to be extracted uniformly.Figure 4.3: Geometry of dumbbell-shaped mould withforced cooling on the central joint.Since the air flow is applied from one specific direction, the variation of the heattransfer coefficient around the periphery of the joint section and the resultingcross-sectional cooling conditions were assess to determine if axisymmetric con-ditions occurred. Figure 4.4 shows the distribution of local Nusselt numberaround a cylinder for forced air cooling conditions [85].Considering an air flow rate of 30 m/s and a kinematic viscosity for air equalto 50× 10−6 m2/s, the Reynolds number of the air flow is 30000. According to69CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.4: Distribution of local heat transfer coefficientaround a circular cylinder for flow of air [85].figure 4.4, this Reynolds number results in a maximum Nusselt number of 120.Equation 4.1 shows the relationship between the Nusselt number and the heattransfer coefficient.htc =Nu× kL(4.1)where Nu is Nusselt number, k is thermal conductivity and L is the character-istic length. Using a characteristic length of 0.05 m (the diameter of the jointsection) and thermal conductivity of air at high temperatures (0.0515 W/mK)in equation 4.1, the maximum heat transfer coefficient is calculated to be 121.65W/m2K.Biot number (shown in 4.2) gives a simple index of the ratio of the heat transfer70CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSresistances inside of and at the surface of a body. This ratio determines whetheror not the temperatures inside a body will vary significantly in space, while thebody heats or cools over time, from a thermal gradient applied to its surface.For Biot numbers below 0.1 the heat conduction inside the body is much fasterthan the heat convection away from its surface, and temperature gradients arenegligible inside of it.Bi =L× htck(4.2)Using the diameter of the joint section as characteristic length, 121.65 W/m2Kas htc and the thermal conductivity of liquid aluminum at liquidus temperature(75 W/mK), the Biot number is calculated to be 0.08. As this number isbelow 0.1, heat transfer variations around the periphery of the cylinder can beneglected.• Large dumbbell-shaped mould with natural cooling: The mould for thelarge dumbbell-shaped casting is shown in Figure 4.5. This mould was manu-factured from large copper fittings with thicker gauge thickness (2.5 mm wallthickness). An end cap with increased wall thickness (13mm) was fabricated forthe bottom of the casting to promote directional solidification from the bottomto the top. The overall height of the mould is 260mm with outer diameters of110, 74 and 110mm for the top, middle and bottom sections, respectively. Seventype-K thermocouples were embedded in the casting at different axial locationsto monitor the temperature.Prior to instrumentation and casting, the moulds were cut in half and reattachedusing several hose clamps in order to facilitate extraction of the castings after cooling.The castings were hand-poured using unmodified A356 (introduced in section 1.3)melted in a resistance furnace. Prior to pouring each casting, the melt was degassed to71CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.5: Geometry of large dumbbell-shaped mould withnatural cooling condition.reduce the hydrogen content by bubbling Ar gas through the melt for 20min. Duringthe degas procedure and prior to pouring, the melt was allowed to cool in the crucibleuntil the temperature reached 680◦C. The moulds were at room temperature (24◦C)when the castings were poured. After pouring, the castings were allowed to cool untilfully solidified. The temperatures were recorded at 2Hz using a Data Acquisitionsystem (daq) connected to a computer running the LabVIEW software 2 .After extracting the castings from the mould, they were cut into thinner sections(shown in Figure 4.6 to be polished and analyzed. The cross-section was assumedto be representative of the whole casting since the casting is axisymmetric. Eachsection was mounted in epoxy 3 to aid polishing. Samples were hand polished with240, 320, 400, 600 and 1800 grit Si-C paper. This was followed by two secondarypolishing steps with 6 and 1 µm diamond paste. The surface of each casting cross-section was then mapped using Scanning Electron Microscopy (sem) imaging. sem2National Instruments LabVIEW3System Three Cold Cure72CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSanalyses were conducted on a FEI Quanta 650 scanning electron microscope with aBurker XFlash 6|30 detector. A total of 3000 and 12000 images at 100X magnificationwere taken from the castings produced from the small mould and the large mould,respectively. It should be noted that, based on the assumption of axisymmetry, onlyhalf of the polished surface was imaged in this process. A python code was then usedto automatically stitch the images together to create a high resolution map of halfof the polished surface. This image was then mirrored to create a full section andanalyzed using the method explained in chapter 3. Nevertheless, in order to assess thesymmetry assumption, a full section of the casting with natural cooling was mappedand analyzed.Figure 4.6: Sectioning of dumbbell-shaped castings forpolishing and analysis.4.2 ResultsThis section presents the results obtained from the casting and analysis of the outlinedin section 4.1. For each casting, first the thermocouple results are shown, followed bya full resolution montage of the section of the casting. Results of the AMD analysis73CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGS(AMD curves) are then shown to acquire an optimum mesh size for final analysis.Finally, using the optimum mesh size the silicon segregation contour plot is shown.4.2.1 Casting with natural coolingThe recorded thermocouple data for the dumbbell-shaped casting with natural cool-ing is presented in Figure 4.7. In the first 20 seconds, temperature drops rapidlybecause of the high heat transfer to the cold mould. As the mould heats up andprimary solidification begins, the temperatures decrease with a slower rate until theyreach the eutectic temperature. Depending on the cooling rate, the time at whicheach thermocouple reaches eutectic temperature is different. For instance, TC0, dueto its proximity to the bottom plate, exhibits a larger cooling rate than the otherthermocouples.Figure 4.7: Temperature data recorded for thedumbbell-shaped casting with natural cooling.During eutectic solidification, the rate of latent heat release increase which mani-fests as a plateau region in the thermocouple curves. Looking at the first 20 secondsof the solidification, TC2 cools faster than TC0 which indicates a non-directionalsolidification in this time interval. Afterwards the measured temperatures indicate74CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSthat solidification is directional from bottom to top. However, compositional analysisis needed to verify this finding. It should be noted that the TC1 data was discardeddue to thermocouple failure.The high resolution montage image of the microstructure in the dumbbell-shapedcasting with natural cooling is presented in Figure 4.8. It should be noted that inthis mapping 2cm from top and bottom of the original section was not imaged. Thisimage has been segmented into three phases; silicon is shown in gray, α-aluminum isshown in white and pores are shown in black. Some large-scale shrinkage porosity isobserved near the top of the casting (area circled in red in Figure 4.8).75CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.8: High resolution montage for thedumbbell-shaped casting with naturalcooling. Red circle indices the shrinkageporosity.76CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.9 shows the results of the segmentation process. As can be seen, theeutectic phase (circled in red) is segmented into its constituents. This was achieveddue to the resolution of the images captured using the sem. It should be noted that,the noise in the segmented image is due to micropores visible in the sem image.(a) (b)Figure 4.9: Image segmentation for microstructure images from naturalcooling setup. (a) original image and (b) segmented image. Redcircles indicate the eutectic phase.The amd analysis methodology, described in section 3.1.5, was applied to the im-age montage to calculate the optimal mesh size for visualization of macrosegregation.Figure 4.10 shows the amd graph for this image. A global minimum of 1390 pixels(∼4mm) is visible from the graph. This mesh size was then used with the meshingalgorithm to complete the macro-segregation visualization. Figure 4.11 shows thegenerated contour image of Si composition and a plot of the Si composition along thecenterline of the casting. It should be noted that the nominal composition in thiscase is 7wt%Si because the alloy used has slightly less silicon compared to the othercases.The contour plot shows a deplete region along the centerline of the joint section,and more enriched regions on the sides. Considering the Si composition along thecentreline (refer to Figure 4.11b), the Si composition in the bottom section of the cast-77CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.10: amd curve for the dumbbell-shapedcasting with natural cooling.(a) (b)Figure 4.11: a) Contour image of Si composition on cross-section andb) plot of Si composition along the centreline of thedumbbell-shaped casting with natural cooling.78CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSing decreases with distance from the bottom. In the joint section, the compositionincreases with distance from the bottom to slightly above the nominal compositionbefore decreasing to below the nominal composition. Top section of the casting ex-hibits a highly deplete region. The enriched region at the bottom and the depleteregion at the middle of the casting indicate that in fact the solidification was notdirectional.The contour plot shows fairly symmetric results along the axis. Due to this sym-metry, as discussed in the methodology section, only half of the cross-section will besegmented and analyzed for the next castings.4.2.2 Casting with insulated jointThe temperature history for the dumbbell-shaped casting with the insulated jointsection is shown in Figure 4.12. As can be seen, the temperature of the casting after600s has decreased to 420◦C compared to 405◦C for natural cooling. Also temperatureat location TC2 for the case of insulated casting drops less rapidly compared to naturalcooling configuration.Figure 4.12: Temperature data recorded for thedumbbell-shaped casting with insulated joint.The microstructure of the insulated casting is shown in Figure 4.13. The high79CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSresolution montage has been segmented into three sections similar to the previouscase (i.e. pores, Si and α-Al). Shrinkage porosity, visible in the top section andat the bottom of the joint section of the casting marked, has been circled in red.The shrinkage porosity at the bottom of the joint section indicates that liquid metalwas encapsulated suggesting that solidification was not directional. The depth ofthe shrinkage porosity in the top section also indicates encapsulation. The differencein the appearance between two shrinkage porosity regions may be due to differingextents of solidification in the areas at the time of encapsulation.Figure 4.13: High resolution montage for thedumbbell-shaped casting with insulatedjoint. Red circles indicate shrinkageporosity.80CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.14 shows the results of the segmentation procedure for the casting withinsulated joint. As can be seen, similar to the previous case, the eutectic is segmentedinto primary and silicon phase.(a) (b)Figure 4.14: Image segmentation for microstructure images from thecasting with insulated joint. (a) original image and (b)segmented image. Red circles indicate the eutectic phase.Figure 4.15 shows the AMD curve for the image montage from this casting. Theminimum lies at 1400 pixels (∼ 4mm).Figure 4.16 shows the generated macrosegregation contour plot and the plot of Sicomposition along the centerline of the casting. The contour plot shows an enrichedregion in the top part of bottom volume and a deplete region in the bottom part ofthe top volume. Note that unlike the case with natural cooling, there is very littleradial gradient in composition in this case. Considering the plot of Si compositionalong the centerline, Si increases from 6wt% at the bottom to a maximum of 9.7wt%in the transition to the joint. The Si composition then decreases until the top of thejoint section, where a slight increase occurs through the transition to the top volume.Similar to the previous case, there is a deplete region at the top of the casting.81CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.15: amd curve for the dumbbell-shapedcasting with insulated joint.(a) (b)Figure 4.16: Measured macrosegregation in thedumbbell-shaped casting with insulatedneck.82CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGS4.2.3 Casting with forced cooling on the jointFigure 4.17 shows the temperatures measured in the dumbbell-shaped casting withforced cooling on the joint. The addition of cooling on the joint results in rapid coolingin this section and has lead to reduced solidification time throughout the casting (i.e.joint as well as top and bottom sections).Figure 4.17: Temperatures data recorded from thedumbbell-shaped casting with forced cooling on thejoint.The high resolution montage, segmented into three sections, of the casting isshown in Figure 4.18. Significant shrinkage porosity (circled in red) is visible in thetop section and in the transition from the joint to the bottom of section of the casting.Similar to the previous cases, the shrinkage porosity in the transition from the jointto the bottom section results from liquid encapsulation at that region. This confirmsthat solidification was not directional and is expected in this case since the joint wasactively cooled. Fast cooling in the joint section would result in early encapsulation ofthe liquid and consequently, is the reason for the larger shrinkage porosity comparedto the insulated case.83CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.18: High resolution montage for thedumbbell-shaped casting with forcedcooling on the joint. Red circles indicatethe shrinkage porosity.84CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.19 shows the result of the segmentation process taken from the section ofthis casting. Similar to the previous cases, the eutectic, circled in red, are segmentedinto primary and silicon phase, due to the resolution of the image.(a) (b)Figure 4.19: Image segmentation for microstructure images from thecasting with forced cooling on the joint. (a) original image and(b) segmented image. Red circles indicate the eutectic phase.Figure 4.20 shows the results of the AMD analysis applied to the casting withforced cooling on the joint. It can be seen that minimum point in this case lies at1310 pixels (∼3.5 mm).Figure 4.21 shows the generated macrosegregation contour plot and the plot of Sicomposition along the centerline of the casting with forced cooling on the joint. Thecontour plot shows a high degree of depletion in the top part of the joint and, bottompart of the top section. Moreover, a highly enriched region is visible in the top partof the bottom volume. Looking at the centerline, from the bottom of the castingthe composition increases to a maximum of 14wt%. Since this value is not realistic,the maximum value was cutoff to 12.6wt%, which is the eutectic composition. Thecomposition then starts decreasing until the top of the joint section, where a slightincrease can be seen. Similar to the previous cases above the enriched region, a depleteregion can be seen.85CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.20: amd curve for the dumbbell-shapedcasting with forced cooling on the joint.(a) (b)Figure 4.21: a) Contour image of Si compositionon cross-section and b) plot of Sicomposition along the centreline of thedumbbell-shaped casting with forcedcooling on the joint.86CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGS4.2.4 Large castingFigure 4.22 shows the temperature from the thermocouples of the large dumbbell-shaped casting. It can been seen that the solidification time has significantly increaseddue to the size of the casting. The temperature history suggests that the solidificationconditions were directional and that there was little to no liquid encapsulation in thecasting.Figure 4.22: Thermocouple data from the largedumbbell-shaped casting.Figure 4.23 shows the high-resolution montage of the section of the large dumbbell-shaped casting. Due to the large size of this section, it had to be cut into three separatesections to be polished and imaged. The acquired images were then used to createthis montage. It can be seen that, unlike the previous cases, there is no shrinkageporosity visible in the cross-section of casting. However, there seems to be a higheramount of hydrogen porosity throughout the casting.87CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.23: High resolution montage for thelarge dumbbell-shaped casting.88CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.24 shows additional higher resolution imaging of the segmentation processfrom the section of this casting. Similar to the previous cases, the eutectic has beensegmented into primary and silicon phase.(a) (b)Figure 4.24: Image segmentation for microstructure images from thelarge casting. (a) original image and (b) segmented image. Redcircles indicate the eutectic phase.Figure 4.25 shows the AMD results for the large casting. It can be seen that aminimum point for this case occurs at 2520 pixels (∼7.5 mm).Figure 4.26a illustrates the macrosegregation contour plot from the image analysis.Figure 4.26b shows the plot of Si composition along the centerline of this casting. Thecontour plot shows a degree of enrichment in the top section of the bottom volume.Moreover, a deplete region is visible extending from the middle part of the joint tothe bottom part of the top volume. Looking at the centerline, from the bottom ofthe casting the composition increases to a maximum of 7.9wt% Si. It then startsdecreasing until the middle part of the top volume, where a slight increase can beseen.89CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSFigure 4.25: amd curve for the largedumbbell-shaped casting.(a) (b)Figure 4.26: a) Contour image of Si compositionon cross-section and b) plot of Sicomposition along the centreline of thelarge dumbbell-shaped casting.90CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGS4.3 DiscussionA metric is defined to measure macrosegregation in each casting based on compositionvariation. Macrosegregation value (the defined metric) facilitates the comparison ofdifferent cases where macrosegregation is present. The macrosegregation values werecalculated by subtracting the minimum composition from maximum composition onthe centreline of each casting and dividing by the nominal composition.Table 4.1: Macrosegregation value for each casting.Casting Condition Macrosegregation valueCasting with natural cooling 0.29Casting with insulated joint 0.62Casting with the forced cooling on the joint 1.17Large casting 0.16Overall, comparing the composition variation along the centerline, the segregationis the lowest in the large casting followed by the casting with natural cooling andcasting with insulated joint. The amount of segregation is the highest on the castingwith forced cooling on the joint. In order to better understand this one should lookinto the the solidification behaviour of these castings.For the case of the casting with natural cooling, there is similar cooling on thewalls of each section. Therefore, some degree of encapsulation is expected in thevicinity of the section with smaller volume (i.e. the joint section). This dependson the cooling rate from the bottom plate, the lower the cooling rate the higherthe chance of encapsulation. In this case, the high resolution image (Figure 4.8),shows no encapsulation in the joint section. However, the contour image shown inFigure 4.12, illustrates a non-uniform segregation along the centerline. This indicatesa slight degree of encapsulation, which did not turn into shrinkage porosity. Furtheranalyzing Figure 4.8, there can be seen a shrinkage porosity in the top section of thetop volume. This porosity is in fact present in all the other castings, since the top91CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGSsurface of the casting, which is open to air, solidifies first and encapsulates the liquidbelow.The goal in the casting with insulated joint section was to delay the solidificationin the joint section. In this case the cooling rate on the sides of the bottom and thetop volumes are higher than the cooling rate on the joint section. Depending on thecooling rate from the bottom plate, this might result in solidification of the top andbottom volumes prior to the solidification of the joint section hence an encapsulationin the middle part of the joint section. Figure 4.13 shows a shrinkage porosity in themiddle of the joint section, which confirms this solidification rationale. From Figure4.16, it can be seen that the silicon variation follows the cooling procedure explainedpreviously. There is enrichment in the region at the bottom of the joint section anda deplete region at the top of the joint section.In the casting with forced cooling on the joint, the cooling rate on the joint sectionhas been increased significantly. This results in solidification of the joint section priorto the solidification of the bottom volume. This then results in a large region ofencapsulation, ranging from the mid point of the bottom volume to the transition tothe joint section, which is prominent in Figure 4.18. Figure 4.21 shows that there isa high degree of enrichment in this area. In order to compensate for shrinkage, thecasting pulled the liquid from the top part of the joint section, which resulted in adeplete region in this section.The cooling conditions for the large casting are similar to the casting with naturalcooling. Therefore, as in the smaller casting with natural cooling, there is no visibleshrinkage porosity in the high resolution image (Figure 4.23). Nonetheless, the ex-pected silicon distribution is different due to the size of the casting. The enrichmentis higher in the bottom of the joint section compared to the casting with naturalcooling.Comparing the AMD curves shows that as the measured silicon distribution be-92CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGScomes more complex, the resulting AMD curves become more noisy and, the mini-mum AMD value increases. For instance comparing the centerline compositions ofthe large casting with the casting with forced cooling on the joint, the silicon massfraction varies from 6.7%wt to 7.9%wt for the large casting. Whereas, this variationis between 4%wt and 12.6%wt for the case of forced cooling. This results in a muchnoisier AMD curve. Furthermore, the minimum AMD for large casting is 0.009 wherethe minimum AMD for the casting with forced cooling is 0.011.4.4 SummaryA series of dumbbell-shaped castings with different sizes and cooling conditions weredeveloped to study the effects of macrosegregation. These castings were then sec-tioned, polished and analyzed for silicon macrosegregation using the method discussedin chapter 3. The following is a short summary of the results:• The casting with natural cooling resulted in no shrinkage porosity. However,analyzing the segregation map, it was concluded that there was some degree ofencapsulation present in the bottom part of the joint section.• The casting with an insulated joint section resulted in a shrinkage pore in themiddle of the joint section. The segregation map for this casting showed a highdegree of enrichment in the vicinity of this pore.• The casting with forced cooling on the joint resulted in a shrinkage porosityin the bottom of the joint section which was accompanied by a high degree ofenrichment.• The large casting similar to the casting with natural cooling resulted in noshrinkage porosity. However, the segregation was more exaggerated comparedto the casting with natural cooling.93CHAPTER 4. MACROSEGREGATION IN SHAPE CASTINGS• It was observed that the segregation was the highest on the casting with forcedcooling rate on the joint and the lowest on the casting with natural cooling.94Chapter 5Modeling of Macrosegregation inShape Castings1A mathematical model has been developed to better understand the various physicalphenomena leading to macrosegregation in castings. The numerical simulation givesadditional insight into how macrosegregation occurs by enriched compensatory flowsand has been applied to the experimental casting produced for this study.5.1 Model descriptionIn this section, a mathematical model based on mass, energy, momentum and speciesconservation equations is introduced. After introducing the governing equations, thegeometry and the meshing technique are discussed. This is followed by the bound-ary condition formulation and material properties. Finally, the solution technique isintroduced.5.1.1 Governing equationsThe model uses an enthalpy-porosity method which considers the casting to be asingle fluid phase (i.e. continuum assumption) where the material properties of the1Portions of this chapter have been published in:• Khadivinassab H., Fan P., Reilly C., Yao L., Maijer D. M., Cockcroft S. L., Phillion A.B.,“Study of the macro-scale solute redistribution due to liquid metal feeding during thesolidification of A356”, Light Metals Production, Processing and Applications Symposium,The 53rd Annual Conference of Metallurgists, (2014)95CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSfluid are a function of temperature. The governing equations for this formulation aresummarized in this section.5.1.1.1 Mass balanceThe continuity equation in single-phase form is:∂ρ∂t+∇ · (ρu) = 0 (5.1)where ρ is the temperature dependent density and u is the superficial velocity givenby the velocity in the liquid multiplied by the fraction liquid, this assumes that thevelocity of the solid is zero. It should be noted that the mushy zone in this caseis considered to be a porous media where the liquid metal flows through the poresdefined by the solid microstructure. The microstructure is not tracked explicitly,instead the liquid fraction is used as an indicator of porosity level in the solidifyingmaterial.5.1.1.2 Energy balanceThe enthalpy balance is solved to predict the heat flow in the domain. The energyequation in enthalpy form is:∂ρh∂t+∇ · (ρuh) = ∇ · (k∇T ) + Se (5.2)where h is the sensible enthalpy in the system, k is the thermal conductivity and Seis a source term added to the energy equation. The equation considers heat flow byadvection, ∇· (ρuh), and heat flow by diffusion,∇· (k∇T ). The release of latent heatduring solidification is included through the source term Se which has 2 terms linkedto the evolution of the fraction liquid.96CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSSe = −∂(ρ∆H)∂t−∇ · (ρu∆H) (5.3)where ∆H is the incremental latent heat which is defined by overall heat releasedmultiplied by fraction liquid. The final form of the enthalpy equation is:∂ρH∂t+∇ · (ρuH) = ∇ · (k∇T ) (5.4)where H is the total enthalpy defined in equation 5.5.H = href +∫ TTrefCpdT + βLf (5.5)where href is a reference enthalpy, Tref is a reference temperature, Lf is the latentheat of fusion and Cp is the specific heat.5.1.1.3 Momentum balanceTo predict fluid flow during solidification, the transient momentum equations must besolved. This equation incorporates the effects of advection, buoyancy and microstruc-ture evolution, which are explained in more detail in further paragraphs. Equation5.6 gives the general form of the momentum equations.∂ρu∂t+ (u · ∇)ρu = −∇P +∇(µ∇ · u)−∇× (µ∇× u) + ρg+ F (5.6)where µ is the viscosity, g is the gravitational acceleration vector, P is the liquidpressure and F is a source term. The term (u ·∇)ρu represents the convective effects,and the terms ∇(µ∇ · u) − ∇ × (µ∇ × u) collectively represent the diffusive effectswhere viscosity acts to diffuse momentum. The term ρg incorporates the effect ofgravity on fluid motion.The source term is used to incorporate the resistance of the evolving microstruc-97CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSture that acts to dampen flow in the mushy region. The mushy zone is assumed tobe a porous media where liquid flows through the solid, porous microstructure. Asnoted earlier, the geometry of the solid microstructure is not predicted. The sourceterm is added to the momentum equation based on Darcy’s law (Equation 5.7).F = µK−1u (5.7)where K is the permeability tensor. If the structure of the mushy region is assumedto be isotropic, the permeability tensor reduces to KI where K is a scalar function ofthe fraction liquid and of the morphology of the mush. K can be then approximatedbased on the Carman-Kozeny relation [11].K =das2180β3(1− β)2 (5.8)where β is the fraction liquid. It should be noted that, in the original formulation ofthe Carman-Kozeny equation, the structure of mushy zone was assumed to consistof packed solid spheres with fluid flowing between them. It is a common procedurein solidification models to relate the average diameter of the solid spheres to thedas [11]. Combining equations 5.6 to 5.8, the final form of the momentum equationcan be written as:∂ρu∂t+ (u · ∇)ρu = −∇P +∇(µ∇ ·u)−∇× (µ∇×u) + ρg+ 180µdas2(1− β)2β3u (5.9)5.1.1.4 Solute balanceSimilar to the other transport phenomena, species transport can be described byperforming a species balance considering advective and diffusive contributions. Inorder to develop the single-phase species transport equation, one should first look at98CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSthe two-phase species transport equation (Equation 5.10):∂(βρlCl + (1− β)ρsCs)∂t+∇·(βρlClul+(1−β)ρsCsus) = ∇·(βρlDl∇Cl+(1−β)ρsDs∇Cs)(5.10)where C is the concentration of solute and D is the diffusion coefficient. Subscriptss and l represent the solid and liquid phases, respectively. In order to convert thistwo-phase model into a one-phase model a relationship must be constructed to linkthe composition of liquid to the composition of solid. This can be done using theScheil equation. To allow the Scheil approximation to be used, the solidificationbehaviour was based on the Al-Si binary alloy. It should be noted that by applyingthe Scheil approximation, this assumes no diffusion in the solid and infinite diffusionin the liquid. Therefore, the concentration of solute in the solid phase will not changewith time. By considering equilibrium at the solid-liquid interface, the differencein composition in the liquid and solid at the interface is described by the partitioncoefficient, κ. Substituting Cs with κCl, and, adding ∂(ρCl)∂t to both sides of theequation, the final form of the solute transport equation is arrived at:∂(ρCl)∂t+∇ · (ρClu) = ∇ · (D+∇Cl)− ∂((1− β)ρCl)∂t− κCl∂((1− β)ρ)∂t(5.11)where D+ is given by ρβDl. It should be noted that in this equation velocity of solidphase was considered zero as there is no solid phase movement.5.1.1.5 Fraction liquidOlder solidification simulation methods use fixed temperature/solute dependent liquidfraction formulation, where the user should manually input the liquid fraction. In this99CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSsimulation an automated technique is used to update the liquid fraction as a functionof the interface temperature.The simplest method to calculate the fraction liquid is to use a temperature ratio.From the phase diagram of a binary alloy, when the temperature T is greater thanthe solidus temperature and smaller than the liquidus temperature, one can write:β =T − TsolTliq − Tsol (5.12)Employing this formulation results in instability in the numerical solution pro-cedure [78]. Moreover, this formulation does not represent the liquid fraction accu-rately, as it is only taking into consideration the temperatures. Voller et al. suggesteda method to calculate fraction liquid based on the Scheil equation which proved tobe stable [78]. The other advantage of this method is that it can be applied to afixed grid. This formulation is based on an iterative method which used the followingequations:βn+1 = βn − λ ap(T − T∗)∆tρV L− ap∆tLf ∂T ∗∂β(5.13)where λ is a relaxation factor, ap is coefficient in discretization equation, ∆t is time-step, V is the cell volume and T ∗ is the interface temperature given by Scheil equationas follows:T ∗ = Tmelt − (Tmelt − Tliq)β(Tsol−TliqTmelt−Tsol) (5.14)where Tmelt is the melting point of the pure substance. It should be noted that Tsoland Tliq are dependent upon the changes in local composition, and can be accessedusing equations 5.15 and 5.16.Tsol = Tmelt +MCSi/κSi (5.15)100CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSTliq = Tmelt +MCSi (5.16)where M is the slope of the liquidus surface.5.1.2 GeometryTo reduce the computational requirements for the model, the dumbbell casting wasassumed to be symmetric about the centerline, allowing a 2D axisymmetric domainto be adopted. The geometry, mesh and boundary conditions used in the modelare shown in Figure 5.1. The mesh is generated using quadrilateral elements oflength 3.5mm for the regular mould and 7.5mm for the large casting. It should benoted that using a very small mesh size resulted in increased computation time andincreased chance of divergence. increasing the mesh size on the other hand, resultedin divergence and increased levels of inaccuracies. Both the geometry and the meshwere generated using Ansys Workbench.Figure 5.1: Geometry, mesh and boundary conditions ofmodel.101CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS5.1.3 Boundary conditionsAs can be seen from Figure 5.1 there are three types of boundary conditions in thissimulation: Wall type, Opening type and Axis type boundary conditions. The Walltype boundaries include bottom Heat Transfer Coefficient (htc), side htc and jointhtc. The following is the description of each of the boundary conditions.5.1.3.1 Wall boundary conditionsWall type boundary condition assume a no-slip condition where velocity of the fluidat the wall is zero. Heat transfer through the wall is specified based on htcs.In order for the model to accurately reflect the heat transfer present in the dumb-bell castings for the three different cooling conditions, the htcs applied to the sidewall of the casting were adjusted so that the predicted temperatures matched themeasured temperature from the experiment. This was a complex task because eachhtc was customizable for each section (i.e. bottom side, joint side, and top side) andfor the different cooling conditions (i.e. natural convection, forced convection, andinsulation). To begin with, the htcs of the side wall sections of the mould were firstapproximated by a heat transfer resistance approximation. As the model does notinclude geometry/mesh representing the copper mould and the region surroundingthe casting, an estimate of the overall htc between the casting outer surface and theambient environment was needed. Despite this calculation, the values do not repre-sent the initial heat transfer between the mould and the melt. This was accountedfor by adding a large htc value for the first few seconds of the solidification.As shown in Figure 5.2a, an interface exists between the casting surface and theinner surface of the mould. An Interfacial Heat Transfer Coefficient (ihtc) is definedbetween these two sections. It is assumed that when the metal is liquid, there isperfect contact (i.e. a high ihtc) between the casting and the mould. However, asthe metal solidifies, the contact degrades as small local gaps form along the interface102CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS(depicted in Figure 5.2a). Therefore, the value of the ihtc changes as a function oftemperature. There have been various studies published on ihtc evaluation duringcasting. Trovant et al. recommend an ihtc of 500 W/m2K for the temperaturescolder than solidus temperature and 2300 W/m2K for the temperatures hotter thansolidus for solidification of A356 in a copper mould [79].(a) (b)Figure 5.2: (a) Depiction of a part of the castingsystem with relevant temperature points,(b) Respective resistance model for heattransfer calculation.Heat transfer can be modeled after electrical circuits, where heat flux acts ascurrent, temperature acts as voltage and heat transfer coefficient acts as resistiveimpedance (reciprocal of resistance). Similar to an electric circuit, one can find theequivalent htc by adding the individual htcs. For the htcs connected in parallel,the equivalent htc is calculated by adding the individual heat transfer coefficients.However, for the components connected in series, the equivalent htc is calculated bytaking the reciprocal of the summation of the reciprocals of the individual htcs.The calculation of an equivalent resistance for heat conduction in the coppermould is straight forward and can be determined by dividing the width of the sectionby the conductivity of copper. Estimating the htc between the outer surface of the103CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSmould and the ambient environment, however, is a difficult task, thus a number ofassumptions have been made. First, it is assumed that heat flows out of the mouldby two parallel mechanisms: convection (natural or forced) and radiation. In Figure5.2b, these mechanisms are illustrated using two parallel resistors.The htc for natural convection (laminar flow) can be calculated using the Churchill-Chu correlation given in Equation 5.17 [80].hconv =kL(0.825 +0.387Ra1/6(1 + (0.492/Pr)9/16)8/27)2(5.17)where L is the characteristic length, Pr is the Prandtl number and Ra is the Rayleighnumber associated with the characteristic length. It should be noted that this equa-tion is valid for Rayleigh numbers above 109.The htc for forced convection conditions may be calculated using Churchill-Bernstein correlation given in Equation 5.18 [80].hconv =kL(0.3 +0.62Re1/2Pr1/3(1 + (0.4/Pr)2/3)1/4(1 + (Re/282000)5/8)4/5)(5.18)where Re is the Reynolds number. It should be noted that this equation is valid whenPrRe > 0.2.In order to calculate accurate htcs using these equations, the temperature de-pendent material properties for air must be used. As illustrated in Figure 5.2a, aboundary layer is expected to develop on the mould wall and the material propertiesof air are assumed to be dependent upon the temperature in the boundary layer. Thistemperature is calculated by averaging the ambient temperature and the temperatureof the outer surface of the mould.104CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSTo estimate the radiative heat transfer contributions to cooling on the side wall, aradiative htc was calculated. It was assumed that the hot mould is radiating energyto its cooler surroundings (ambient environment in our case). The following equationcan be used to calculate the radiative heat transfer coefficient:hrad = εσ(Tside + T∞)(T 2side + T2∞) (5.19)where ε is the emissivity and σ is the Stefan-Boltzmann constant.Combining the resistances defined, the effective heat transfer coefficient was de-termined and illustrated in Figure 5.3 for both natural and forced convection. Theeffective htcs are only dependent upon the temperature of the casting surface andtherefore can be used directly in the simulations.(a) (b)Figure 5.3: Effective heat transfer coefficient dueto (a) natural and (b) forced convection ina copper mould.As mentioned previously, these htcs do not take into account the initial heattransfer between the mould and the melt. In the simulation, the copper mould as-sumed to initially absorb a large amount of heat, with htc of 1200 W/m2K. Thehtc then gradually reduces to the steady state value over a duration of 5 seconds.The side htc in the castings for all three cases were based on the effective htc105CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSvalues for natural cooling. The joint htc was consistent with the different conditions.For the case of natural and forced cooling, the joint htc was considered to be thecalculated effective htc, respectively. For the insulated case, the effective htc washeld constant at 15 W/m2K. It should be noted that this value was determinedthrough trial and error.The bottom htc was assumed to have a behaviour similar to the side htc. How-ever, since it has a thicker cross-section and is in contact with a support, the effectivehtc was initially assigned to be 2000 W/m2K and it gradually decreases over 10seconds to the nominal value of 20 W/m2K. It should be noted that these valueswere chosen by trial and error during a fitting exercise.5.1.3.2 Opening Boundary ConditionDue to changes in density in the casting (shrinkage due to the phase change andoverall thermal contraction), a boundary condition was needed so that mass staysconserved. It should be noted that if the configuration does not include an OpeningBoundary Condition, continuity equation will diverge due to changes in density. Anopening boundary condition was applied at the top of the casting domain to give themodel the ability to draw material in to the domain to compensate for the shrinkage.Although this seems far from what happens in the real-life case, the parameters ofthe opening boundary condition can be adjusted to achieve better accuracy, whileconserving mass.The Opening Boundary Condition, applied to the top of the casting domain,enables material to be drawn into the domain with predefined temperature and com-position. In this study, the temperature and composition of the material entering thedomain at each time step was set to be equal to the temperature and compositionof the metal at the boundary elements. This results in a zero gradient in temper-ature and composition at the boundary. It should be noted that a more accurate106CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSrepresentation of temperature would be to draw in a different material, such as air,with a temperature equal to the surrounding air temperature during the experimentand with appropriate properties so that the heat flux between the casting and theair would better represent the experiment. This implementation was explored duringmodel development but it was eventually discarded as it resulted in divergence.It is expected that zero temperature gradient would result in underestimation ofhtc at the top section, hence a higher temperature is expected at the top section.5.1.3.3 Axis boundary conditionThe axis boundary condition is used to implement the axisymmetric formulation.This means that at a particular radius from the axis and a particular height, eachflow variable has the same value on different angles.5.1.4 Material propertiesThe thermo-physical properties for A356 used in the model are summarized in Table5.1 and Figure 5.4.Table 5.1: Thermo-physical properties used in themathematical model.Properties Value Referencesk Figure 5.4a [81]Cp Figure 5.4b [81]Lf 395000 J/kg [82]ρ Figure 5.4c [81]µ Figure 5.4d [81]κSi 0.13 [83]The values for thermal conductivity linearly decrease in solid zone from 163 to144 W/m.K. The value of thermal conductivity for liquid zone, is an enhanced valueof 400 W/m.K in order to account for the convection in liquid zone. The valuesin the mushy zone are interpolated linearly based on temperature. Specific heat107CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS(a) Temperature dependantthermal conductivity.(b) Temperature dependantspecific heat.(c) Temperature dependantdensity.(d) Temperature dependantviscosity.Figure 5.4: Temperature dependant material properties.exhibits values of 880 and 1190 J/kg.K for the solid and liquid phases. The valuesin the mushy zone are interpolated linearly as function of temperature. Density wasassumed to have a constant value of 2578 kg/m3 in the solid region. In the liquidzone it linearly decreases from 2406 to 2325 kg/m3. Density in the mushy zone wasapproximated using three points as shown in Figure 5.4c. Viscosity in the liquid zonewas set to 0.001 Pa.s. In the solid region it was set to a very large value (1000 Pa.s)to constrict the flow. The values in the mushy zone were interpolated as shown inFigure 5.4d.108CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS5.1.5 Solution procedureA coupled thermal - fluid flow - composition model, based on the equations, boundaryconditions, and material properties described in the previous sections, was developedusing the commercial Computational Fluid Dynamics (cfd) software package FLU-ENT to predict solute (silicon) segregation during solidification of the dumbbell cast-ings. The model incorporates the relevant solidification phenomena including mushyzone resistance (Darcy flow damping), latent heat evolution, buoyancy, and siliconpartitioning during the phase transformation. An incremental approach was taken todevelop the coupled thermal - fluid flow - composition model, where the complexityof the model was increased in stages.Four cases were simulated based on the four experiments explained in the previouschapter: natural cooling, insulated, forced cooling and natural cooling of a largecasting. Testing and validation of the model was performed in two stages in thisstudy. Initially, the temperature data from the simulation was compared againstthe thermocouple data from the experiment and the heat transfer coefficients wereadjusted until the predicted temperature history matched the measured temperatures.In the second stage, the macrosegregation results from the simulation were comparedand validated against the results from the image processing technique.5.2 Results5.2.1 Casting with natural coolingFigure 5.5 shows the comparison of thermocouple data from experiment and simula-tion for the dumbbell-shaped casting with natural cooling conditions. Results showa good fit in thermocouple data between the experiment and simulation. To achievethis fit, the heat transfer coefficient were manipulated as explained in section 5.1.3.1.The only inconsistency seems to be the top thermocouple (TC4). This is the result109CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSof complications arising from the top opening boundary condition description. InFLUENT, it is not possible to define an htc for the opening boundary condition. Asdiscussed in section 5.1.3.2, this results in overestimation of temperature.Figure 5.5: Thermocouple data from experiment andsimulation for the dumbbell-shaped casting withnatural cooling conditions.Temperature contours from the dumbbell-shaped casting with natural cooling con-ditions are shown in Figure 5.6. It should be noted that solidification progress wascalculated based on the solidification time. As can be seen, the solidification at firstis not directional (temperature exhibits an increasing-decreasing-increasing profile)but it becomes directional as the htcs decrease (temperature is increasing uniformlyfrom bottom to top).Figure 5.7 shows the liquid encapsulation plots from the simulation. These plotswere generated based on the liquid fraction of 0.5. Since the liquid metal flow beyondthis liquid fraction is negligible, the liquid metal feeding to encapsulated areas iscut. It is suggested from the results that the two areas shown are prone to shrinkageporosity since liquid is encapsulated and there is no feeding.110CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.6: Temperature contours from thedumbbell-shaped casting with natural coolingconditions.Figure 5.7: Liquid encapsulation results from simulation forthe dumbbell-shaped casting with natural coolingconditions.111CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.8 shows the silicon segregation in the dumbbell-shaped casting with nat-ural cooling conditions. A silicon-enriched region occurs in the vicinity of the en-capsulated region shown in Figure 5.7. A silicon-deplete region can be seen at thebottom of the top volume, where it is attached to the neck.Figure 5.8: Simulated silicon segregation in thedumbbell-shaped casting with natural coolingconditions.Figure 5.9 shows comparison of measured and predicted silicon mass fraction alongthe centerline of the dumbbell-shaped casting cooled by natural convection (refer toFigure 5.8). Moving up the centerline from the bottom of the casting, the predictedSi content increases rapidly until a maximum is reached in the bottom volume. Con-tinuing along the centerline, the silicon concentration then decreases until the topvolume is reached. In the top volume, a slight increase is observed. Finally, at thetop of the casting, a deplete region is predicted. The measured Si composition ex-hibits similar trends to the predicted in terms of where the composition increasesand decreases, however, the magnitude of the maximum composition predicted in the112CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.9: Silicon mass fraction along the centerline fromthe experiment and the simulation for thedumbbell-shaped casting with natural coolingconditions.bottom volume is much higher than measured.5.2.2 Casting with insulated jointFigure 5.10 shows a comparison of the measured and predicted temperatures forthe insulated dumbbell-shaped casting. Similar to the previous case there is someinaccuracy in predicting the top thermocouple, but overall the predicted temperaturesmatch the measured temperatures at most of the locations in the casting. Figure 5.11shows the temperature contours for the insulated case. The temperature contoursshow an encapsulated area at 10% solidified configuration. However, after this pointthe solidification exhibit a directional nature.Figure 5.12 shows the liquid encapsulation plots from the simulation. Resultsshow that two areas are prone to shrinkage porosity since liquid is encapsulated andthere is no feeding.Figure 5.13 shows the silicon segregation in the insulated dumbbell-shaped casting.113CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.10: Thermocouple data from experiment andsimulation for the insulated dumbbell-shapedcasting.Figure 5.11: Temperature contours from the insulateddumbbell-shaped casting.114CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.12: Liquid encapsulation results from simulationfor the insulated dumbbell-shaped casting.A silicon-enriched region is observable in the vicinity of the encapsulated region shownin Figure 5.12. A silicon-deplete region can be seen at the bottom of the top volume,where it is attached to the neck. These results are similar to the observations fromthe casting with natural cooling conditions. Nevertheless, the variation of silicon inthe latter case is higher.Figure 5.14 shows comparison of silicon mass fraction from the experiment andthe simulation along the centerline shown in Figure 5.13. The results show goodagreement. However, the simulated scenario shows higher silicon concentration in theenriched region comparing to the experimental results.5.2.3 Casting with forced cooling on the jointFigure 5.15 shows the comparison of the measured and predicted temperature his-tory for the dumbbell-shaped casting with forced cooling. Similar to the previouscases there is some inaccuracy in the predictions for the top thermocouple, but thecomparison is good at the other thermocouple locations in the casting. Figure 5.16115CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.13: Simulated silicon segregation in the insulateddumbbell-shaped casting.Figure 5.14: Silicon mass fraction along the centerline fromthe experiment and the simulation for the insulateddumbbell-shaped casting.116CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSshows the temperature contours for the case with forced cooling. As expected, Thetemperature gradients developed in this case are much larger than of previous twocases.Figure 5.15: Thermocouple data from experiment andsimulation for the dumbbell-shaped casting withforced cooling.Figure 5.17 shows the liquid encapsulation plots from the simulation. Similar tothe previous cases there are two areas prone to shrinkage porosity. It should be notedthat the location of the encapsulated region in the bottom of the joint section is lowerone on the previous two cases.Figure 5.18 shows the silicon segregation in the dumbbell-shaped casting withforced cooling. A silicon-enriched region is observable in the vicinity of the encapsu-lated region shown in Figure 5.17. This region is much larger than the previous cases.Similar to the previous cases, a silicon-deplete region can be seen at the bottom ofthe top volume, where it is attached to the neck. The variation of silicon throughoutthe casting is higher than the previous cases.Figure 5.19 shows comparison of the measured and predicted silicon mass fraction117CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.16: Temperature contours from thedumbbell-shaped casting with forced cooling.Figure 5.17: Liquid encapsulation results from simulationfor the dumbbell-shaped casting with forced cooling.118CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.18: Simulated silicon segregation in thedumbbell-shaped casting with forced cooling.Figure 5.19: Silicon mass fraction along the centerline fromthe experiment and the simulation for thedumbbell-shaped casting with forced cooling.119CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSalong the centerline of the dumbbell-shaped casting with forced cooling (refer toFigure 5.18). The predictions show higher silicon concentration in the enriched regioncompared to the experimental results. The area of discrepancy, similar to the previouscases, lies in the vicinity of the encapsulated region at the bottom of the joint section.5.2.4 Large castingFigure 5.20 shows a comparison of the measured and predicted temperature historyfor the large dumbbell-shaped casting. Similar to the previous cases there is someinaccuracy in predicting the top thermocouple, but the temperature predictions at theremaining thermocouple locations are accurate. Figure 5.21 shows the temperaturecontours for the large casting. At 10% solidified, the figure shows the onset of liquidencapsulation . However, similar to the previous cases the solidification exhibits adirectional nature.Figure 5.20: Thermocouple data from experiment andsimulation for the large dumbbell-shaped casting.Figure 5.22 shows the liquid encapsulation plots from the simulation. Similar tothe previous cases there are two areas prone to shrinkage porosity. The encapsulatedregions are similar in position compared to the previous cases.120CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.21: Temperature contours from the largedumbbell-shaped casting.Figure 5.22: Liquid encapsulation results from simulationfor the large dumbbell-shaped casting.121CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.23 shows the silicon segregation in the dumbbell-shaped casting withforced cooling. A silicon-enriched region is observed in the vicinity of the encapsulatedregion shown in Figure 5.22. The silicon content variation is smaller compared to othercases presented previously.Figure 5.23: Simulated silicon segregation in the largedumbbell-shaped casting.Figure 5.24 shows a comparison of measured and predicted silicon mass fractionalong the centerline of the large dumbbell-shaped casting (refer to Figure 5.18). Ascan be seen, both measured and predicted Si composition values start from a lowvalue of silicon mass fraction and move to a maximum at around 10cm from thebottom of the casting. Afterwards a gradual decrease is evident from the graph forboth cases. The composition then starts increasing starting from the bottom of thetop volume until the middle of this volume at around 21cm from the bottom.122CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.24: Silicon mass fraction along the centerline fromthe experiment and the simulation for the largedumbbell-shaped casting.5.3 Discrepancy in the enriched regionBased on the results presented in the previous section, the prediction of liquid encap-sulation in / near the joint section seems to correspond to over-prediction of the siliconcomposition in this area. This is especially evident in the forced cooling case, wherethe model over-predicts the composition in the transition from the bottom volume tothe joint section by approximately 40%. The predicted value at this region also raisesquestions because it increases well above the eutectic composition. It seems that thisover-prediction is also related to the cooling rate as well. For cases with low coolingrates, i.e. the casting with insulated joint and large casting, the over-prediction isminimal and the results match the experimental cases. Nonetheless, for the castingsproduced with natural cooling and forced cooling conditions, where the cooling ratesare higher, the simulation and experimental results exhibit substantial discrepancies.To further analyze this discrepancy, predicted liquid encapsulation and compo-sition results for the casting with forced cooling on the joint, where the observeddiscrepancy is the highest, were plotted side by side at different instances during so-123CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSlidification. The time instances were based on the onset of encapsulation (t = 12s),the end of solidification (t = 150s) and several intermediate values shown in Figure5.25.It can be seen from Figure 5.25 that liquid encapsulation in the bottom of thejoint region occurs at time t = 12s. It is expected that since the liquid metal feedingto this region is restricted at this time, there should be no increase in the siliconconcentration. However, the enriched region keeps growing in the next instances.Figure 5.26 shows the Si mass fraction along the centerline for these instances.The results from Figure 5.26 show that the composition keeps increasing in theencapsulated region even after the eutectic composition is reached which is physicallyunrealistic. This suggests that the simulation is encountering a numerical instabilityin this region. This may be influenced by the high heat transfer rates in the castingswith natural cooling and forced cooling on the joint. However, since the htcs areset to match the heat transfer rates of the experiments, any changes in htcs wouldresult in deviation of the thermal history.It is a common practice in cfd to utilize relaxation factors to overcome instabilityand divergence problems [84]. Relaxation factors control the stability and convergencerate of the iterative process. Under relaxation increases the stability while over relax-ation increases the rate of convergence [84]. Equation 5.20 shows the implementationof this concept.xk+1 = αxcalc + (1− α)xk (5.20)where x is a variable that is being solved for by a differential equation, superscriptsk+ 1 and k are the iteration steps, and xcalc is the current solution of the variable forthe differential equation. α is considered to be an under-relaxation factor if 0 < α < 1and an over-relaxation factor if 1 < α < 2.124CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS(a) (b)(c) (d)(e) (f)Figure 5.25: Development of over-prediction of enrichment in thebottom of the joint region for the casting with forced on thejoint.125CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.26: Silicon mass fraction along the centerline forinstances presented in Figure 5.25.In order to resolve the prediction instability leading to over-prediction of thesilicon mass fraction, the relaxation factor for the species transport equation in Fluentwas progressively decreased. A relaxation factor of 0.2 was found to eliminate theinstability and the over-prediction of concentration in the vicinity of the encapsulatedregion. Figure 5.27 shows the silicon mass fraction on the centerline of the casting inaforementioned time instances.Figure 5.28 shows the predicted silicon mass fraction variation for the casting withforced cooling on the joint using an relaxation factor of 0.2. It can be seen that notonly is the maximum composition value now below the eutectic composition, but alsothe size of the enriched region has decreased.Figure 5.29 shows the composition comparison of the simulated case using a relax-ation factor and the experiment for the casting with forced cooling on the joint. It canbe seen that the under-relaxation helped the model to better match the experimentalresult.126CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.27: Si mass fraction along the centerline forunder-relaxed model in several instances.Figure 5.28: Simulated silicon segregation in thedumbbell-shaped casting with forced cooling on thejoint with an under-relaxation factor.127CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.29: Measured and predicted silicon mass fractionalong the centerline of the casting with forcedcooling on the joint with an under-relaxation factor.The same under-relaxation factor was applied to the casting with natural coolingcondition. Figure 5.30 shows the silicon composition variation from the casting withnatural cooling simulated using an under-relaxation factor for species equation. Itcan be seen that the highly enriched region visible in Figure 5.8 at the bottom of thejoint section is less enriched.Figure 5.31 shows the composition comparison of the simulated case with relax-ation factor and the experiment for the casting with natural cooling. It can be seenthat, similar to the previous case, the under-relaxation helped the model to bettermatch the experimental result.5.4 Yield strength predictionTo complete the analysis of the casting, the expression developed for flow stress inChapter 2 has been applied to predict the yield strength variation in the each ofthe castings. In order for accomplishing this task, das predictions, as well as the Si128CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSFigure 5.30: Simulated silicon segregation in thedumbbell-shaped casting with natural cooling withan under-relaxation factor.Figure 5.31: Measured and predicted silicon mass fractionalong the centerline of the casting with naturalcooling with an under-relaxation factor.129CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGScomposition, were needed. Shabani and Mazahery studied the effects of cooling rateon microstructure of A356 [49]. Figure 5.32, borrowed from this study, shows thecooling rate vs. das for A356.Figure 5.32: das vs. cooling rate for A356 [49].Equation 5.21 shows the power law type model that was fit to this data.das = 40.71×R−0.33 (5.21)The cooling rate, R, at each location in the casting is defined as the liquidustemperature, Tliq, minus the solidus temperature, Tsol, divided by the solidificationtime, tsolidification, (Equation 5.22).R =Tliq − Tsoltsolidification(5.22)Combining these equations with the modified Ludwig-Holloman expression withparameters n, K and 0 as a function of das, Si content and T6 state, yield strengthwas predicted through a post-processing operation. The results are shown in Figure130CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS5.33. It can also be seen from the results that variation of yield strength is the highestfor the case of the casting with forced cooling on the joint and the lowest for the largecasting.Figure 5.34 shows the yield strength variation in these casting if T6 heat treatmentwas applied. The results show a similar trend as the ac condition. However, thevariation of yield strength in the T6 condition is increased by a factor of 4. Thisshows that in an already segregated sample, T6 heat treatment results in a muchlarger variation in mechanical properties.5.5 DiscussionMacrosegregation values for the simulated cases are summarized in Table 5.2. Thesevalues were calculated based on the formula introduced in section 4.3. As can beseen, the large casting has the lowest segregation followed by the casting with naturalcooling and casting with insulated joint. The casting with forced cooling on the jointhas the largest degree of segregation. This trend is shown in both the measured andpredicted results. It can be seen that the relative error in the values for the largecasting, the casting with insulated joint and the casting with forced cooling on thejoint are within 10%. However, the error for the casting with natural cooling is 40%.This difference is also evident from the Figure 5.31, where there is a visible differencein the enriched region at the bottom of the joint section.Table 5.2: Comparison of predicted and measuredmacrosegregation values for each casting condition.Casting Condition Predicted MeasuredCasting with natural cooling 0.41 0.29Casting with insulated joint 0.65 0.62Casting with the forced cooling on the joint 1.04 1.17Large casting 0.15 0.16Comparing the encapsulation results from Figures 5.7, 5.12, 5.17 and 5.22 with131CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS(a) (b)(c) (d)Figure 5.33: Predicted yield strength for the simulated castings accondition. (a) for the casting with natural cooling, (b) for thecasting with insulated joint, (c) for the casting with forcedcooling on the joint and (d) for the large casting.132CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGS(a) (b)(c) (d)Figure 5.34: Predicted yield strength for the simulated castings T6condition. (a) for the casting with natural cooling, (b) for thecasting with insulated joint, (c) for the casting with forcedcooling on the joint and (d) for the large casting.133CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGShigh resolution images from Figures 4.8, 4.13, 4.18 and 4.23, respectively, showsthat in all four cases the model was able to predict the shrinkage porosity at the topsection of the casting. The model also correctly predicted the location of the shrinkageporosity in the joint section of the casting with forced cooling and insulation at thejoint. For the case of the casting with natural cooling and the large casting, despitethe model predicted encapsulation at the joint section, there is no shrinkage porosityvisible in the experimental results.The results from Figure 5.33 show that yield strength is highly dependent onthe silicon composition, more so than das. Overall, the yield strength results suggestthat macrosegregation is a defect that needs to be controlled in shape casting designs,especially where strength uniformity is important.5.6 SummaryA complex mathematical model was developed to simulate macrosegregation in dumbbell-shaped castings. Four models were developed based on the cooling condition and size,which are discussed in detail in the previous chapter. Results can be summarized asfollows:• Temperature curves from the thermocouple location show good agreement withthe experimental results. However, the top thermocouple in the simulation over-predicted temperature. This was due the opening boundary condition assignedto the top boundary in the model.• The simulated model was able to correctly predict all the shrinkage porositiesthrough encapsulated liquid analysis, expect for the cases of the casting withnatural cooling and the large casting where it incorrectly predicted encapsula-tion in the bottom of the joint section.• The model initially over-predicted the values of composition up to 40% near134CHAPTER 5. MODELING OF MACROSEGREGATION IN SHAPE CASTINGSthe encapsulated areas (i.e. bottom of the joint section). This issue was foundto be a stability related problem, which was then resolved implementing anunder-relaxation factor for the species model in the simulation.• The predicted yield strength results show that yield strength variation is moresignificantly influenced by the variation of silicon composition rather than das135Chapter 6Summary and ConclusionsThis work has studied the macrosegregation of silicon in shape cast, aluminum alloyA356. First, the effects of silicon composition, das and the heat treatment state onthe flow stress of Al-Si-Mg alloys were investigated. In order to do this, a series ofplate castings were produced from Al-Si-Mg alloys where the Si content was varied andthe constitutive behaviour was characterized. The results were then used to establishan empirical expression correlating das, silicon content and the heat-treated statewith flow stress based on a modified Hollomon equation.In the second portion of the project, an image processing method was developedto characterize the macrosegregation occurring on cross-sections of shape castings.The method utilizes a combination of image segmentation, pixel-to-pixel analysis andtessellation techniques to construct a quantitative map of the solute distribution onlarge samples. The accuracy and validity of the method were assessed through aseries of artificially designed micrographs.In the third portion, a series of dumbbell-shaped castings were produced to pro-mote and/or limit the effects macrosegregation. This was done by controlling thecooling on the joint section of a dumbbell-shaped casting mould. Four castings wereproduced using this technique, three small castings with natural cooling, force aircooling and insulation on the joint section and one large casting with natural cool-ing. The temperature history of the casting during solidification was recorded usingK-type thermocouples. To determine the segregation, the castings were sectioned,polished and analyzed using the developed image processing technique.136CHAPTER 6. SUMMARY AND CONCLUSIONSIn the final portion of the project, a numerical model was developed to simulate themacrosegregation in the aforementioned castings. Silicon species segregation duringsolidification was calculated assuming the Scheil approximation, and was coupled withmacro-scale transport incorporating the resistance of the mushy zone and feeding flow.The model has been implemented within the commercial CFD software, FLUENT,which simultaneously solves the thermal, fluid flow fields and species segregationon the macro-scale. Finally, the constitutive behaviour expression developed in thefirst phase of the project was applied to predict the yield strength variation in thedumbbell-shaped castings based on the simulation data.6.1 ConclusionsThe findings of this PhD research project can be summarized as below:• An equation was developed to characterize the constitutive behaviour of hy-poeutectic Al-Si-Mg alloys. Analyzing this equation it was concluded that,apart from das, silicon composition variation plays a crucial role in the flowstress behaviour of the material. Furthermore, sensitivity analysis results showthat the maximum yield strength occurs when das is low and silicon composi-tion is high.• A technique was developed to accurately characterize the spatial variation ofmacrosegregation in eutectic alloys. This method utilizes a tessellation tech-nique to mesh the image, then calculates the area fraction of the desired phasein each element. The current method uses a continuous map of micrographsto calculate a segregation map which results in more accurate results, whereprevious methods used a discrete number of samples in order to do so.• Analyzing the dumbbell-shaped castings it has been found that the segregationis the lowest in the large casting and the highest where forced cooling was137CHAPTER 6. SUMMARY AND CONCLUSIONSapplied to the joint section.• Analyzing the simulation results, it was observed that the numerical model over-predicted the segregation in the transition to / bottom of the joint section, whereliquid encapsulation was predicted to occur. This result is caused by instabilityin the species model, and was resolved by applying an under-relaxation factor.Similar to the experimental results, it was predicted that segregation in thelarge casting is the lowest.6.2 Future workThe current study considers a wide range of issues related to macrosegregation inA356 aluminum alloy shape castings. The following items have been identified toextend the current work:• In describing constitutive behaviour, the effects of temperature and strain rateshould be included. This is necessary for developing a comprehensive constitu-tive equation and to allow the deformation of a casting that has experiencedmacrosegregation during solidification to be predicted.• The most challenging section for the image analysis technique was developinga segmentation method to segment the micrographs into desired phases. Animprovement for this would be to include a more sophisticated segmentationtechnique. This can be done by training an object detection algorithm to classifythe phases. This will allow the technique to be used for wider range of alloysto help improve the performance of the alloy.• In order to characterize macrosegregation more accurately, castings with a widevariety of sizes and cooling rates should be used. This will help to build a morecomprehensive database of different solidification conditions and corresponding138CHAPTER 6. SUMMARY AND CONCLUSIONSmacrosegregation patterns, which in turn allows to improve the accuracy of themethod.• In the current simulation, the top opening boundary condition does not accu-rately represent the conditions present in the castings. In order to improve thisa user-defined boundary condition should be developed in Fluent that allowsthe implementation of htcs onto the opening boundary condition.• A series of macro-hardness tests could be implemented to verify the yieldstrength variation results from the simulation. This would further verify boththe flow stress model and the developed numerical model.139Bibliography[1] B. Zhang, D. M. Maijer, and S. L. Cockcroft. Development of a 3-D thermalmodel of the low-pressure die-cast (LPDC) process of A356 aluminum alloywheels. Materials Science and Engineering A, 464(1-2):295–305, 2007. 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Heat transfer over the circumference of the heatedcylinder in transverse flow. Tech. Rep. NACA-TM-1050. National AdvisoryCommittee for Aeronautics. (Cited on pages 69 and 70.)147AppendicesAppendix A: Term Sensitivity of the Flow StressModelA sensitivity analysis has been conducted to assess the relative significance of eachterm in equation 2.1 for n, K and 0. Relative significance in this case is definedas the percentage of contribution of each term in the equation and is calculated bydividing the magnitude of each term by the sum of magnitudes of all the terms. Thefive terms which were studied are; a, b×t6 , (t6+1)×c×CSi, (t6+1)×d×das−1/2and (t6+ 1)× e×CSi×das−1/2. These terms are referred to with roman numerals Ito V, respectively. The analysis was conducted over a combination of cases with lowand high das values (i.e. 20 and 50µm) and low and high Si contents (i.e. 4 and 10wt%) in T6 and ac conditions. The results are shown in Tables 6.1 to 6.3.Table 6.1: Relative significance of terms I to V on parameter n invarious conditions. Note that the values are in percentages.148BIBLIOGRAPHYn (ac) low das high daslow CSiI II III IV V I II III IV V53.8 0.0 7.6 29.7 9.0 68.4 0.0 9.6 16.9 5.1high CSi 29.5 0.0 24.8 16.2 29.5 39.4 0.0 33.2 9.7 17.6n (T6) low das high daslow CSiI II III IV V I II III IV V25.4 31.0 7.1 28.0 8.5 31.8 38.8 8.9 15.7 4.7high CSi 14.3 17.4 24.1 15.7 28.5 18.9 23.1 31.8 9.3 16.9Table 6.2: Relative significance of terms I to V on parameter K invarious conditions. Note that the values are in percentages.K (ac) low das high daslow CSiI II III IV V I II III IV V10.4 0.0 13.3 63.6 12.7 18.0 0.0 23.0 49.2 9.8high CSi 4.5 0.0 34.7 27.7 33.2 6.8 0.0 52.2 18.6 22.4K (T6) low das high daslow CSiI II III IV V I II III IV V4.0 26.9 10.2 49.0 9.8 5.9 39.9 15.2 32.5 6.5high CSi 2.0 13.4 30.7 24.5 29.4 2.8 19.1 43.7 15.6 18.7Table 6.3: Relative significance of terms I to V on parameter 0 invarious conditions. Note that the values are in percentages.0 (ac) low das high daslow CSiI II III IV V I II III IV V53.4 0.0 9.1 31.8 5.7 67.3 0.0 11.5 18.0 3.2high CSi 30.7 0.0 31.5 18.3 19.5 38.8 0.0 39.8 10.3 11.0149BIBLIOGRAPHY0 (T6) low das high daslow CSiI II III IV V I II III IV V35.9 1.4 12.3 42.8 7.6 49.8 1.9 17.0 26.5 4.7high CSi 18.0 0.7 36.9 21.5 22.9 23.9 0.9 48.9 12.7 13.6It can be seen from the sensitivity results that the parameters are more sensitiveto changes in Si content rather than das. This suggests a similar behaviour for theflow stress. Furthermore, applying the T6 heat treatment to the alloy seems to havea small effect on the contributions of terms III and V – the terms dependent onSi content. This is expected since applying a T6 heat treatment does not changethe silicon content and its contribution to the flow stress. Furthermore, applyinga T6 heat treatment activates term II, which is inactive in the as-cast condition.This results in a contribution from term II and consequently a decrease in relativesignificance of the other terms. Overall, the contribution of all the parameters aresignificant in the equation.150BIBLIOGRAPHYAppendix B: Area Fraction to Volume FractionConversionFrom stereological point of view there are only a few bulk microstructural parametersthat can be accessed from analyzing a 2D surface, where volume fraction is one ofthem [74]. According to Kaplan [74], volume fraction is equal to area fraction if thecross-sectional plane is randomly positioned.In order to test this method, a series of synthetic microstructures were gener-ated using Blender1. All the synthetic microstructures were confined to cube witha similar volume, and the microstructure itself was modelled using randomly placedellipsoids with random sizes and aspect ratios. Figure 6.1 shows a sample syntheticmicrostructure generated using this method.Figure 6.1: Sample synthetic microstructure generated usingBlender.1Blender 2.69 - a 3D modelling and rendering package151BIBLIOGRAPHYA randomly positioned plane was then used to cut the sample and acquire theresulting cross-sectional projection. Figure 6.2 shows a sample cross-sectional imageacquired from a synthetic microstructure.(a) (b) (c)Figure 6.2: Sample cross-sectional images acquired fromsynthetic microstructures.Volume fraction data, calculated within Blender, and the cross-section images werethen written into files. This was carried out for five hundred iterations. A Pythoncode was then used to calculate the area fraction of the ellipsoid cross-sections andcompare it to the volume fraction calculated from Blender.Figure 6.3 shows the error distribution plot for the area fraction to volume fractionconversion, attained from the ellipsoid analysis. The initial observation suggests thatthe data is normally distributed. Therefore, a Gaussian distribution was fitted ontothe error data.The fitted Gaussian distribution has a mean of 0.00 and a standard deviation of0.11, which suggests that the majority of the data lies within 10% error. In otherwords, area fraction value, calculated using a random cross-sectional plane, is morelikely to be within 10% error. This suggests that area fraction and volume fractioncan be used interchangeably with a reasonable error margin. This concept combinedwith the volume fraction to mass fraction conversion is used in various sections toconvert area fraction to mass fraction.152BIBLIOGRAPHYFigure 6.3: Error distribution plot for area fraction tovolume fraction conversion analysis.153

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