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Thermal mechanical analysis of interfacial behavior in aluminum alloy wheel casting process Wei, Xiaodan 2014

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THERMAL MECHANICAL ANALYSIS OF INTERFACIAL BEHAVIOR IN ALUMINUM ALLOY WHEEL CASTING PROCESS by XIAODAN WEI B. Eng., Zhejiang University, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2014 ?Xiaodan Wei, 2014 ii   Abstract The focus of this project is to improve the understanding of the interfacial heat transfer behavior within the Low-Pressure Die Casting (LPDC) process, which is the main manufacturing process for A356 aluminum alloy wheels, and to develop an improved methodology/expression for calculating the heat transfer across the wheel/die interface.  To formulate and assess expressions for the interfacial behavior, a 2D-axisymmetric coupled thermo-mechanical model has been developed in the commercial finite element package, ABAQUS.  The model was capable of predicting the thermal history, deformation and the variation of the air gap and pressure along the wheel/die interface.  The temperature predictions of the coupled thermo-mechanical model were compared with temperature measurements obtained at Canadian Auto Parts Toyota Inc obtained on a production die. A displacement measurement setup using a high temperature eddy current displacement sensor was designed and tested in a lab setting but not employed in a plant trial due timing issues. Initially, the coupled thermo-mechanical model was run with a temperature dependent interfacial heat transfer coefficient to obtain preliminary air gap and pressure behavior at various locations. Comparisons with the thermocouple measurements suggest that the model is able to generally qualitatively, and at some locations quantitatively, predict the temperature changes from the main physical phenomena occurring during the casting process.  The preliminary air gap and pressure predictions were used to develop a temperature, gap size and pressure dependent interfacial heat transfer coefficient based on literature review. The interfacial heat transfer coefficient was implemented in the model, and was found to improve the iii   agreement between the model predictions and measured temperatures, but was prone to numerical convergence issues. A new methodology of incrementally changing the interfacial heat transfer coefficient has been proposed to solve the issues. The methodology was implemented in an EXCEL spreadsheet to test it and the calculated interfacial heat transfer coefficients were found to be continuous, reflecting the effects of air gap and pressure evolution. The methodology and corresponding algorithm should be further developed for use in an ABAQUS model in the future.    iv   Preface The temperature measurements in Chapter 4 were performed by Dr. Carl Reilly, a Research Associate, with assistance from two other colleagues in the research group at the University of British Columbia.  The thermal model mentioned in Chapter 5 used as a starting point to accelerate the development of the coupled thermo-mechanical model in the current work was also developed by Dr. Carl Reilly as part of a larger program. The development of the coupled thermo-mechanical model, investigation of the interfacial behavior and comparisons to the temperature measurements are my work.      v   Table of Contents  Abstract ........................................................................................................................................... ii	 ?Preface ............................................................................................................................................ iv	 ?Table of Contents ............................................................................................................................ v	 ?List of Tables ............................................................................................................................... viii	 ?List of Figures ................................................................................................................................ ix	 ?List of Symbols ............................................................................................................................ xiv	 ?Acknowledgements ...................................................................................................................... xvi	 ?Chapter 1  Introduction ................................................................................................................... 1	 ?1.1 Wheel Casting ....................................................................................................................... 1	 ?1.2 Wheel/die Interfacial Behavior ............................................................................................. 4	 ?Chapter 2  Literature Review .......................................................................................................... 9	 ?2.1 General Description of Interfacial Heat Transfer .................................................................. 9	 ?2.1.1 Principles of Interfacial Heat Transfer ........................................................................... 9	 ?2.1.2 Casting/Die Interface Heat Transfer ............................................................................. 14	 ?2.2 Computational Process Modeling ....................................................................................... 15	 ?2.2.1 Thermal Models ............................................................................................................ 16	 ?2.2.2 Coupled Thermo-Mechanical Models .......................................................................... 18	 ?vi   2.3 Quantification of Casting/Die Interfacial Heat Transfer ..................................................... 22	 ?2.3.1 Incorporating Interfacial Heat Transfer in Models ....................................................... 22	 ?2.3.2 Interfacial Heat Transfer Coefficient ............................................................................ 23	 ?Chapter 3  Scope and Objectives .................................................................................................. 36	 ?3.1 Scope of Research ............................................................................................................... 36	 ?3.2 Objectives of Research Project ........................................................................................... 37	 ?Chapter 4  Experimental Measurements ....................................................................................... 39	 ?4.1 Temperature Measurements ................................................................................................ 40	 ?4.1.1 Experimental Procedures .............................................................................................. 40	 ?4.2 Displacement Measurements ............................................................................................... 42	 ?Chapter 5  Coupled Thermo-mechanical Model Development .................................................... 54	 ?5.1 Model Geometry ................................................................................................................. 55	 ?5.2 Mesh .................................................................................................................................... 56	 ?5.3 Materials Properties ............................................................................................................. 57	 ?5.3.1 Thermophysical Properties ........................................................................................... 57	 ?5.3.2 Mechanical Properties .................................................................................................. 58	 ?5.4 Initial Conditions ................................................................................................................. 60	 ?5.5 Thermal Boundary Conditions ............................................................................................ 62	 ?5.5.1 Interfacial Heat Transfer ............................................................................................... 64	 ?vii   5.5.2 Surface Film Heat Transfer .......................................................................................... 68	 ?5.5.3 Radiation ....................................................................................................................... 69	 ?5.6 Mechanical Boundary Conditions ....................................................................................... 70	 ?5.7 Convergence Criteria ........................................................................................................... 71	 ?Chapter 6 Results and Discussion ................................................................................................. 91	 ?6.1 Results of Model with Temperature Dependent Heat Transfer Coefficient ....................... 91	 ?6.1.1 Distribution of Temperature and Deformation ............................................................. 91	 ?6.1.2 Predictions of Wheel/die Interfacial Behaviour ........................................................... 92	 ?6.2 Results of Model with Temperature, Gap size and Pressure Dependent HTC ................... 97	 ?6.2.1 Implementation of Interfacial Behaviour ..................................................................... 98	 ?6.2.2 Comparisons to Thermocouple Data ............................................................................ 99	 ?6.2.3 Discussion ................................................................................................................... 100	 ?6.3 New Methodology to Calculate Interfacial Heat Transfer Coefficients ........................... 101	 ?Chapter 7  Summary and Conclusions ........................................................................................ 123	 ?7.1 Recommendations for Future Work .................................................................................. 125	 ?References ................................................................................................................................... 127	 ?   viii   List of Tables Table 2.1 Temperature dependent heat transfer coefficient applied in LPDC A356 wheel casting process [3] ..................................................................................................................................... 35 Table 4.1 Sensor test result ........................................................................................................... 53 Table 5.1 Thermal physical properties of A356 [2], H13 [3], Cast iron [4], Tungsten carbide [5]....................................................................................................................................................... 83 Table 5. 2 Latent heat of A356 [3] ................................................................................................ 84 Table 5.3 Temperature dependent thermal expansion coefficient of A356; reference temperature is 25 ?C [7] .................................................................................................................................... 84 Table 5.4 Temperature dependent Young?s modulus of A356 [10] ............................................. 85 Table 5.5 Temperature dependent yield stress of A356 [10] ........................................................ 85 Table 5.6 Heat transfer coefficient for various interfaces of the dies and sprues ......................... 86 Table 5.7 Heat transfer coefficient at the wheel/die interfaces ..................................................... 87 Table 5.8 Heat transfer coefficient for the cooling channels (cooling channel locations are shown in Figure 5.4) ................................................................................................................................. 88 Table 5.9 Heat transfer between the die and the environment ...................................................... 89 Table 5.10 Non-default convergence tolerance parameters .......................................................... 90    ix   List of Figures Figure 1.1 A schematic of a low pressure die casting process machine [1] ................................... 7 Figure 1.2 Automotive wheel sections and a directional solidification path [5] ............................ 8 Figure 2.1 A schematic of the physical contact between two solid materials and the temperature profile through the solids and the contact interface [9] ................................................................ 30 Figure 2.2 Experimental setup of a drop of casting solidifying on a nickel substrate, which is instrumented with thermocouples to analyse the thermal resistance at the casting/die interface. The size of the metal drop is about 10-11 mm in diameter and 4-5 mm in height. [13] [14] ....... 30 Figure 2.3  A schematic of heat transfer through gas at solid/gas/solid interface ........................ 31 Figure 2.4 Three stages of air gap formation at the cast/mold interface, showing (a) Stage I, (b) Stage II, (c) Stage III when a gap forms [8] ................................................................................. 31 Figure 2.5 Elastic modulus versus temperature data used in modeling of an AZ31 Magnesium Billet casting [21] .......................................................................................................................... 32 Figure 2.6 Casting apparatus with Thermocouples and LVDT [31] ............................................ 33 Figure 2.7 A plot of heat transfer coefficient as a function of air gap. The squares depict the heat transfer coefficient calculated from experimental data. Line 1 displays predictions based on the semi-empirical equation, Line 2 shows the heat transfer coefficient estimated from the analytical equation. Line 3 shows the estimated radiation heat transfer coefficient [31] ............................. 34 Figure 2.8 Heat transfer coefficient history derived from the experimental results. Segment ?ab? indicates a rapid drop in HTC due to formation of asperities and intermittent gaps. Segment ?bc? indicates a small increase in HTC because of the expansion of the die. Segment ?cd? depicts a significant decrease in HTC, which is due to the metal-die gap starting at ?c? and continuing to x   increase in size. Vertical dotted line L1 indicates the moment when the air gap is detected by LVDT measurments; L2 indicates the moment when the cast has solidified and a constant HTC is reached. [7] ................................................................................................................................ 35 Figure 4.1 Thermocouple locations in top die .............................................................................. 48 Figure 4.2 Thermocouple locations in side die shown in a section view ..................................... 48 Figure 4.3 Temperature history of TC 1 at steady state (from Cycle 26 to Cycle 32) ................. 49 Figure 4.4 Temperature history of TC 2 at steady state (from Cycle 26 to Cycle 32) ................. 49 Figure 4.5 Cast-in thermocouple locations in the wheel for different cycles, from left to right, Cycle_27, Cycle_29 and Cycle_30. Note that 29_B is outside of the wheel because it was cast in the window section, trapped between top die and bottom die. ..................................................... 50 Figure 4.6 KD-1950 high temperature displacement sensor [37] ................................................. 50 Figure 4.7 Sectioned view of the side die showing the KD-1950 sensor installation .................. 51 Figure 4.8 Left: KD-1950 sensor dimensions; right: KD-1950 mounting [38] ............................ 51 Figure 4.9 KD-1950 calibration data for A356 [39] ..................................................................... 52 Figure 4.10 Sensor test set-up ....................................................................................................... 53 Figure 5.1 a) 427 Wheel geometry; b) front view; c) back view; d) side view ............................ 74 Figure 5.2 A section through half of the 427 wheel, showing the on-spoke section and on-window section ............................................................................................................................. 75 Figure 5.3 a)  A 36? slice of the 427 wheel which has been simplified for modeling by merging the riser blocks in Figure 5.2into one block to allow simplified meshing; b) Wheel and die geometry of 3D thermal model showing die sections. Note that: the heater is not included as a part in the model; a thermal boundary condition is applied to model its effect ............................ 76 xi   Figure 5.4 Wheel and die geometry of the 2D axisymmetric coupled thermo-mechanical model, Note:  TD_CC is top die cooling channel, TDDC_CC is top die drum core cooling channel, SDC_CC is side die core cooling channel, BD_CC1 is bottom die cooling channel 1, and BD_CC2 is bottom die cooling channel 2. ................................................................................... 77 Figure 5.5 a) Mesh of 2D axisymmetric coupled thermo-mechanical model; b) zoomed in view of the finer interface mesh ............................................................................................................ 78 Figure 5.6 Temperature dependent Young?s modulus of A356 [30] ............................................ 79 Figure 5.7 Temperature dependent yield stress of A356 [30] ...................................................... 79 Figure 5.8 Initial temperature distributions in the die and the wheel ........................................... 80 Figure 5.9 Die assembly of 427 wheel .......................................................................................... 81 Figure 5.10 Mechanical boundary conditions applied in the model ............................................. 82 Figure 6.1 ? Temperature contours along with deformation at different time, a) 0 s, b) 53 s, c) 80 s, d) 105 s, e) 154 s. Note: the deformation scale factor is 10. ................................................... 107 Figure 6.2 ? Zoomed-in image of the interface between the in-board rim flange and side die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6.3 .................................................................................................................................... 108 Figure 6.3 ? Variation of gap and temperature with time at the interface between the in-board rim flange and side die ...................................................................................................................... 108 Figure 6.4 ? Zoomed-in image of the interface between the spoke and top die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6.5. .... 109 Figure 6.5 ? Variation of pressure and temperature with time at the interface between the spoke and top die ................................................................................................................................... 109 xii   Figure 6. 6 ? Zoomed-in image of the interface between the rim and side die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6. 7. ... 110 Figure 6. 7 ? Variation of temperature, air gap and pressure with time at the interface between the rim and side die ..................................................................................................................... 110 Figure 6.8 ? Location of thermocouples ..................................................................................... 111 Figure 6.9 ? Comparisons of predicted and measured temperatures at three locations in the upper part of the rim section: (a) TC_1, (b) TC_2, and (c) TC_52 ...................................................... 113 Figure 6.10 ? Comparisons of predicted and measured temperatures at three locations in the lower part of the rim section: (a) TC_4, (b) TC_5, and (c) TC_46 ............................................ 114 Figure 6. 11 ? Variation of interfacial heat transfer coefficient and real size of air gap with time at the interface between the in-board rim flange and side die as shown in Figure 6.2 ............... 115 Figure 6. 12 ? Variation of interfacial heat transfer coefficient and contact pressure with time at the interface between the spoke and top die as shown in Figure 6.4 .......................................... 115 Figure 6. 13 ? Comparisons of the model predictions with thermocouple data ......................... 118 Figure 6. 14 ? Variation of air gap size and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing the interfacial heat transfer coefficient at a location on the side die / wheel interface .................................................................................................. 119 Figure 6. 15 ? Variation of air gap size and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing interfacial heat transfer coefficient at a location on the top die / wheel interface ........................................................................................................ 119 xiii   Figure 6. 16 ? Variation of air gap size, pressure and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing interfacial heat transfer coefficient at a location on the side die cooling core / wheel interface ............................................................... 120 Figure 6. 17 ? Flowcharts for incrementally calculating the interfacial heat transfer coefficient based on: a) updating every iteration, or b) updating every increment ...................................... 122 xiv   List of Symbols Latin Symbols Description Units br, bz  components of body force Pa C1 ,C2  equation constants or coefficients  Cp specific heat J/kg/K D  elastic matrix  D  displacement in sensor calibration equation mm d size of the gap m d0  initial size of the air gap m E elastic modulus Pa flimit  equation factor  hint   interfacial heat transfer coefficient W/m2/K h heat transfer coefficient  W/m2/K ?? radiation heat transfer coefficient W/m2/K ?? conduction heat transfer coefficient W/m2/K h0  baseline heat transfer coefficient  W/m2/K hmax  maximum heat transfer coefficient W/m2/K kgas conductivity of gas W/K/m kair conductivity of air W/K/m L latent heat kJ/kg P pressure Pa q  heat flux  W/m2 R  thermal resistance K?m2/W Q  volumetric heat source term  W/m3 ???  thermal resistance of the interface K?m2/W T  temperature  ?C T1, T2 temperatures of the two surfaces at the interface ?C Tsurf  temperature of the surface ?C T?  temperature of the coolant or environment  ?C Tair  temperature of the ambient environment ?C Ti  initial temperature ?C Tcoh  mechanical coherency temperature ?C Tliq  liquidus temperature ?C Tsol  solidus temperature ?C t  time  s V  voltage output  volts Greek Symbols   ?  thermal expansion coefficient  ???  effective emissivity  ??, ?? emissivity of cast and mould  ? strain  xv   ?? , ?? , ?? Components of strain  ?? ,?? , ??? ? elastic, inelastic and thermal strains  v Poisson?s ratio  ?  density  kg/m3 ?  shear modulus Pa ?  Stefan-Boltzmann constant  ?r, ?z , ??  components of stress Pa ?ij shear stress Pa ur , uz components of displacement m    xvi   Acknowledgements I would like to express my sincere gratitude to my supervisor, Professor Steve Cockcroft, for his guidance, encouragement and support in the completion of my research work. I am extremely grateful for the breakthrough ideas he contributed to the project, and for providing the freedom that let me explore study and life. I am particularly thankful for his advice and unconditional support at the turning point of my life.  I would like to thank my supervisor, Professor Daan Maijer, with whom I have studied computational modeling during the last two and a half years. I appreciate the countless hours of discussion we had in discussing the finite element models. I owe much of my success in finite element models to him. I am also very appreciative of the joy, hope and optimism he always brought. I am grateful to Dr. Carl Reilly, for his research support in models and experiments. I would also like to thank him for the effort he made in revising this thesis.  I would like to thank my colleagues and friends in the casting group. Special thanks to Dr. Matt Roy, for motivating and helping me through the difficulties. To Jianglan Duan, for her help in programming languages. To Sara Moayedinia, for sharing the difficulties and happiness since the first day of graduate school. Those times we spend together will always be in my mind.  Finally, thanks due to my family and friends for their love and support.   1  Chapter 1  Introduction This thesis is part of a larger program funded by Automotive Partnerships Canada (APC) in partnership with Canadian Auto Parts Toyota aimed at developing an advanced water-cooled die design for the production of aluminum alloy (A356) automotive wheels. The work presented in this thesis is focused on heat transfer across the interface between the solidifying wheel and the steel die and constitutes one of the sub-tasks that comprise the larger program. A team-based approach is adopted in the program with the various sub-tasks assigned to individual team members. Care is taken to appropriately identify the individuals that contributed to aspects of this study. 1.1 Wheel Casting The aluminum alloy wheel is a critical automotive component because of its importance to both vehicle safety and aesthetics. The low-pressure die casting process (LPDC) is currently the main manufacturing process for aluminum alloy wheels, owing to its ability to produce wheels with good surface finish and high mechanical performance at a relatively low cost. There are two types of LPDC process: the conventional air-cooled version (compressed air forced through cooling channels) and the water-cooled version (water circulated through cooling channels). The water-cooled LPDC process offers enhanced cooling and solidification rates with reduced cycle time, and produces aluminum alloy wheels with finer microstructure and superior mechanical properties compared to the air-cooled version.  2   A typical low-pressure die-casting machine, shown in Figure 1.1 [1], has a die located above a holding furnace, which is connected via a transfer tube. The die is comprised of a top die, at least two side dies, a bottom die, and a sprue. The dies are typically made of H13 tool steel and cast iron. When a casting cycle starts, the side dies close and the top die moves down, forming the wheel cavity. The holding furnace is then pressurized with air forcing liquid metal (A356 aluminum alloy, Al-7Si-0.3Mg typically at a temperature in the range of 700 to 740 ?C) up the transfer tube and through the sprue into the wheel cavity. The liquid metal is cooled until solidified by transferring heat from the wheel to the die, and from the die to the cooling channels and also from the die to the ambient environment. Once the wheel is solidified, the side dies open and the top die is raised with the wheel attached to it (due to thermal contraction). The wheel is then ejected on to a transfer tray for post-processing [2]. After each cycle the dies are visually inspected before a new cycle starts. Typical cycle times are between three and four minutes for the water-cooled LPDC process. Currently, the competition in the automotive industry is fierce due to increased supply from wheel manufacturers located in Asia and Mexico. In order to remain competitive within the industry, North American wheel manufacturers need to further develop their wheel casting technology to improve wheel quality, reduce weight and reduce cost.  Automotive wheels must satisfy several stringent specifications related to rotational balance, air tightness, surface finish, and mechanical performance. The later includes strength, impact and fatigue performance [3]. In the manufacturing process, defects that diminish wheel quality may occur and lead to wheel rejection. The most common defect in wheels is porosity that forms during the solidification process [2]. There are two types of porosity: macro-porosity and micro-3   porosity. Macro-porosity, also known as shrinkage porosity, occurs when liquid metal is entrapped in solidified metal and there is an inability to feed liquid metal to compensate for the metal shrinkage during solidification. Macro-porosity has a great impact on mechanical properties such as strength, fatigue and impact performance.  Micro-porosity, or hydrogen-based porosity, is porosity of a smaller size (<~300 um) [2] and is mainly caused by the decrease in hydrogen solubility between the liquid and solid aluminum. It can impact on the surface finish and fatigue performance of wheels (if large and present in highly stressed regions of the wheel) [2].  Entrainment of oxides and oxide films within the wheel, another category of defect, can arise due to poor liquid metal and excessive turbulence during filling. Once entrained in the solid, metal oxides and oxide films can act to concentrate stresses leading to crack initiation. They can also act as nucleation sites for hydrogen-based porosity during solidification and can restrict liquid metal feeding leading to shrinkage porosity. Oxide films can be controlled by improving melt quality and using a screen at the top of the sprue to filter out the debris and reduce turbulence during filling.  To control macro-porosity, it is critical to manage the heat flow to achieve directional solidification in the wheel. More specifically, the solidification front should follow a path that ensures continuous liquid metal feeding to the liquid/solid transition zone to compensate the volumetric shrinkage due to solidification. In a wheel casting, the directional solidification path necessary to avoid macro-porosity starts at the top in-board rim flange, progresses down through the rim, along the spokes and ends at the hub [4], as shown in Figure 1.2 [5]. 4   Besides directional solidification, high cooling rates should also be promoted to further reduce casting defects and improve wheel quality. High cooling rates reduce the size of hydrogen-based micro-porosity, and also produce wheels with finer microstructure (secondary dendrite arm spacing). Moreover, higher cooling rates could also reduce the cycle time, and thus improve the production rates.  However, it is challenging to achieve both directional solidification and a fast cooling rate simultaneously, because directional solidification requires some parts of the wheel to remain hot enough to allow liquid metal to feed the liquid/solid two phase zone, which to some extent contradicts the need to ?cool as fast as possible? [4]. To meet this challenge, computational modeling tools are necessary to simulate the LPDC wheel casting process and predict the thermal history accurately.  1.2 Wheel/die Interfacial Behavior In the LPDC wheel casting process, heat transfer across the interface between the wheel and die is important and complex. The resistance evolves with time and strongly influences solidification rates and overall wheel quality. Consequently, it also has a bearing on cooling channel placement, timing (on and off times) and overall process cycle times. During filling, the liquid metal is in good physical contact with the die wall because the liquid metal is able to conform to the solid surface thereby reducing resistance ? defined to be 1/hint, where hint (W/m2/K) is the interfacial heat transfer coefficient. Moreover, the substantial amount of fluid flow during filling reduces the boundary layer at the interface further reducing the resistance (increasing hint). During the period of die filling and while the metal is above the liquidus temperature the heat transfer 5   coefficient between the wheel/die is high (estimated to be as high as 2500 W/m2/K [3]). As the wheel cools and solidification begins at the interface, localized shrinkage occurs at the scale of the developing microstructure leading to reduced contact and a gradual reduction in hint. This arises due to the volumetric shrinkage that occurs during the liquid-to-solid phase transformation, which is in the range of 6 to 8 vol % in aluminum alloys [6]. The increase in resistance occurs over the liquidus to solidus interval and leads to a 20 to 30% reduction in hint [7]. As solidification proceeds deeper into the cast component, shrinkage occurs at the scale of component. In the context of wheel production, this can lead to the wheel either pulling away from the die and the development of a gap or the wheel shrinking on the die and the development of pressure. , At the same time, the die sections are heating up and expanding prior to cooling and contracting as heat is transferred from the die to the cooling channel. Die heating and thermal expansion would tend to reduce any gap that formed or increase the pressure that develops.   The formation of a gap changes the thermal resistance to the heat transfer at the wheel/die interface, and significantly reduces the interfacial heat transfer coefficient. Within the gap, heat is transferred by a combination of conduction, radiation and convection through the air gap. Due to the much lower thermal conductivity of air, the limited space available for convection and the decreasing temperature of the casting surface (which makes radiation heat transfer negligible), the interfacial heat transfer coefficient is much lower, and continues to decrease as the gap size increases. This has a significant effect on the cooling and solidification rate of the wheel, and therefore needs to be quantified accurately. In contrast, the formation of pressure at the interface will tend to increase heat transfer and maintain a relatively small thermal resistance at the 6   interface. Finally, because the die is subject to a temperature increase and a temperature decrease within a casting it is possible that interface can experience pressure and the formation of a gap within a single casting cycle.  The formation and evolution of the air gap and pressure at the casting / die interface makes for a  coupled mathematical problem wherein changes to the heat transfer affect the deformation of the wheel and die and vice versa. This coupling of thermal and mechanical responses of the wheel and die make the wheel/die interfacial behavior complex and difficult to quantify and predict mathematically. In summary, the wheel/die interfacial behavior has a significant effect on the wheel cooling and solidification rates; therefore it is critical to be able to predict the wheel/die interfacial behavior to achieve an accurate prediction of the thermal history. The research presented in this thesis is focused on understanding the wheel/die interfacial behavior. A literature review related to this topic is presented in the following chapter before defining the scope and objectives of the project.    7     Figure 1.1 A schematic of a low pressure die casting process machine [1] 8    Figure 1.2 Automotive wheel sections and a directional solidification path [5]    In-board Rim Flange Out-board Rim Flange 9   Chapter 2  Literature Review A review of literature relevant to the wheel/die interfacial behaviour is presented in this chapter.  Principles of interfacial heat transfer and a general description of interfacial heat transfer in the casting process are introduced first to establish a knowledge foundation. Computational modeling of casting processes that include the characterization and implementation of the interfacial behaviour are then discussed. Various researchers have studied interfacial heat transfer and have described the cast/die interfacial behaviour as a function of temperature and/or gap and/or pressure. The relevant research results in the literature will be discussed.   2.1 General Description of Interfacial Heat Transfer During the LPDC process for the production of aluminum alloy wheels, heat transfer across the cast/die interface happens through one or more of three different forms as the wheel solidifies: liquid/solid contact, solid/solid contact and heat transfer through an air gap. It is helpful to first review the principles of interfacial heat transfer and then investigate the cast/die interfacial heat transfer.  2.1.1 Principles of Interfacial Heat Transfer The heat transfer equation across the interface is generally defined as: ? = ?????? = ? ?? ? ??  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(2. 1) 10   where q is the interfacial heat flux (W/m2), R is the thermal resistance (K?m2/W), (T2 - T1) is the temperature difference across the interface (driving force) (K) and h is the effective heat transfer coefficient (W/m2/K). The various phenomena that affect heat transfer at the interface are generally accommodated by varying the heat transfer coefficient. These phenomena include the thermophysical properties of the casting, mould materials (including mould coatings) and the atmosphere, the volume change associated with the liquid-to-solid phase transformation, the thermal expansion coefficients of the relevant materials and the physical configuration of the casting and mould. In the following section, the thermal resistance that develops under conditions of liquid/solid contact, solid/solid contact and transport through a gas filled gap will be discussed.  2.1.1.1 Liquid/Solid Thermal Resistance During a casting process, a liquid/solid contact interface forms immediately after the liquid metal is poured into a die or mould. The mechanism for heat transfer at the interface under this condition is conduction. An interfacial thermal resistance may exist due to the presence of the oxide-based mould or die coating that is usually applied to protect the die metal though the thermal resistance is usually very small [8]. Surface tension effects may also limit the ability of the liquid metal to fully wet the die coating.  2.1.1.2 Solid/Solid Contact Resistance During the casting process, the first metal to solidify against the die or mould establishes an interface with discontinuous solid/solid contact. Figure 2.1 shows a schematic representation of the solid/solid contact that forms between to materials (A and B) [9]. The surfaces do not 11   perfectly fit together as there are irregularities on each surface that result in intermittent gaps along the interface. The irregularities in the cast surface arise due to solidification whereas those present on the mould wall stem from the machining/die coating. The intermittent gaps increase the thermal contact resistance at the interface, resulting in an abrupt temperature drop at the interface as shown in Figure 2.1. Heat transfer occurs across the solid/solid interface by conduction at contact locations (asperities), conductive heat transfer through the gas or fluid filling the gaps, and radiation between the surfaces [9].  Thermal contact resistance at the interface is sensitive to many factors, such as external pressure (the force pressing the two surfaces together), the fluid (gas) filling the interface gaps, surface roughness and the solid material properties [9]. Fried [10] investigated the effect of external pressure on the contact resistance between metal surfaces in a vacuum. It was found that increasing external pressure reduces the contact resistance significantly. For example, when the external pressure is 100 kPa, the contact resistance between two aluminum surfaces is 1.5-5.0?104 m2 K/W; however, when the external pressure is increased to 10,000 kPa, the contact resistance is reduced to 0.2-0.4?104 m2 K/W [10].  It has also been shown that putting an interfacial fluid or gas with high thermal conductivity at the contact interface reduces the thermal contact resistance compared to the case with air at the interface [9]. For example, under 105 Pa contact pressure, thermal contact resistances for aluminum-aluminum interface with air, helium and hydrogen as the interfacial fluid are 2.75 m2K/W, 1.05 m2K/W and 0.720 m2K/W, respectively [10]. 12   The surface roughness of each surface influences the thermal contact resistance by affecting the area of physical contact at the interface.  If the contact surfaces are smooth and flat, the contact resistance is relatively small; as the surface roughness increases, the physical contact area decreases and the thermal resistance increases. Surface roughness also affects the contact pressure with a given load, and thus further affects the contact resistance [11, 12].  Loulou et al. [13, 14] developed an experiment to investigate interfacial heat transfer during solidification of three pure metals (tin, lead and zinc) on a water-cooled substrate. The experimental set up, shown in Figure 2.2, was based on observing a molten metal drop on the surface of a nickel mould. Thermocouples were embedded in the mould, and the data acquired was used to estimate the contact thermal resistance. Material properties of the metals were used to explain the differences in the estimated interfacial thermal contact resistances between the three different metals and the nickel mould. The results indicated that the higher the latent heat, the lower the contact resistance. It was also noted that the wettability of the metal has an effect on the thermal contact resistance.  2.1.1.3 Heat Transfer through a Gas Filled Gap As solidification progresses, the interface between a casting and die may change from solid/solid contact condition to solids separated by a gas as the casting pulls away from the die and an air gap forms. Assuming the gap is small and convection heat transfer can be neglected, the gas filled gap heat transfer scenario can be simplified to that shown schematically in Figure 2.3. The schematic diagram shows that heat is transferred from one solid to another solid through a gas by 13   conduction and radiation in parallel. Hence, the heat flux though the interface is the sum of the two heat flows [15]  ? = ??+?? = ?? ?? ? ?? + ?? ?? ? ?? = (?? + ??)(?? ? ??) = (?????)? ??         (2. 2) where T1, T2 = temperatures of the two surfaces at the interface between the solid and gas ?? = ? ????????????  = radiation heat transfer coefficient in the gas ?? = ? ???  =conduction heat transfer coefficient of the gas  ??? = ?????? = thermal resistance of the interface in which ? ?is the Stefan-Boltzmann constant, and ? is the effective emissivity of the interface surfaces, kgas is the conductivity of gas, and d is the size of the gap. Note: for two parallel plates the effective emissivity is: ??? = 11?? + 1?? ? 1 In a casting process, the development of the air gap introduces extra thermal resistance to the interface, and changes in the size of the air gap will affect the thermal resistance and the heat transfer significantly.  14   2.1.2 Casting/Die Interface Heat Transfer Heat transfer across the casting/die interface is an important phenomenon in the LPDC process, and has a critical effect on the cooling and solidification rate of the cast component. As introduced in Section 2.1.1, it is influenced by a large number of phenomena that can act individually or in combination with one another.  Interfacial heat transfer in an LPDC process evolves within each casting cycle, but can be characterized by three stages, as shown in Figure 2.4. Stage I is defined as when liquid metal is in contact with the die, which occurs immediately after the liquid metal is poured into the die, and is characterized by low thermal resistance. Stage II starts when the liquid metal cools and solidification begins. Interfacial heat transfer decreases as asperities formed on the surface of the casting contract parallel to the surface resulting in intermittent contact and small gaps typical of a solid/solid contact interface. As noted in Section 2.1.1.2, the thermal resistance in this stage is largely dependent on the contact resistance of the surface asperities. The resistance across the cast/die interface is proportional to the length of the asperity and inversely proportional to the conductivity of the materials. A small portion of the heat is also transferred by convection and radiation through the air pockets. Stage III occurs when the casting surface breaks contact with the die surface due to continued local thermal contraction or deformation caused by phenomena occurring elsewhere in the casting. Heat transfer in this stage is governed by conduction, convection and radiation through the air gap. Because of the much lower conductivity of air, the limited space available for convection, and the relatively low temperature of the casting surface (limiting radiation heat transfer in the aluminium LPDC scenario), the heat transfer coefficient at this stage is much lower than the previous two stages [16]. 15   Note that in the LPDC process used for automotive wheels, not all locations of the casting will experience each stage. For instance, the casting/side die interface on the rim section of a wheel will pull away from the side die to form a gap and experience all the three stages. However, on the other side of the rim at the casting/top die interface, the wheel will contract onto the top die, preventing a gap from forming and stage III will not occur. Under these conditions, the heat transfer coefficient remains constant or goes up if the solid/solid contact pressure increases.   Various phenomena affect the interfacial heat transfer in a casting process. Ho and Pehlke [17] studied the effect of chills on interfacial heat transfer with two types of castings. In the first casting, a copper chill was placed on the top of a cylindrical casting; while in the second casting, the chill was placed at the bottom. A gap formed at the chill / casting interface in the first type of casting; while in the second type of casting, the casting remained in contact with the chill. The effects of different die materials (copper, steel and bonded sand) were also investigated. Castings made using a copper die cooled down the fastest, followed by steel, and then the bonded sand. As expected, these trends are consistent with the thermal conductivities of these three materials. Though copper allows faster cooling, industrial die assemblies are predominantly made of mild steel, cast iron or H13 steel, which have good resistance to erosion and thermal fatigue.   2.2 Computational Process Modeling Computational process models are important tools that can be used to investigate the various phenomena that impact on complex manufacturing processes such as the low-pressure die casting process for A356 aluminum alloy wheels. These models can be used to predict the cast-16   ability of wheels prior to prototyping and mass-production and can be used to limit costly experimental trials. Thus, they can save money and time for industry [3].  2.2.1 Thermal Models A complete model describing the physics involved in the aluminum wheel casting process requires the coupled modeling of fluid flow, heat transfer, phase transformation, and deformation. However, thermal models that only consider heat transfer and approximate the effects of other phenomena are often adopted. These models offer simplicity (easier to achieve convergence) and lower computational cost compared with fully coupled models.  The governing equation describing transient heat conduction solved by these models is presented in the following equation: ?Cp T ?T?t = ? k T ?T +Q                                                     (2.3) where T is the temperature (?C), t is the time (s), k is the thermal conductivity (W/K/m), ? is the density (kg/m3), Cp is the specific heat (J/kg/K), Q is the volumetric heat source term (W/m3) associated with the latent heat of solidification.  One challenge in a thermal model lies in how to incorporate the effect of the liquid to solid phase transformation properly. This often involves the evaluation of temperature dependent materials properties, including conductivity and latent heat. Another big challenge is how to implement thermal boundary conditions, especially the interfacial heat transfer at the wheel/die interface, which is complex and has a great effect on the thermal history. 17   Zhang et al [3] developed a 3-D thermal model of the LPDC process for A356 aluminum alloy wheels. In the work, latent heat is included using a heat source term as in Equation 2.4. Latent heat is released throughout a discrete temperature range in proportion to the evolution in fraction solid [18]. To approximate the effects of natural convection when metal is molten, the thermal conductivity of the metal is augmented at temperatures above the liquidus. The wheel/die interfacial heat transfer is handled in this model by using a temperature dependent heat transfer coefficient to approximate the three-stage resistance behaviour of the interface. The three stages are defined by two temperatures (560 and 540 ?C), which reside near the liquidus and solidus point of the alloy, respectively. When temperature is higher than 560 ?C, the interfacial heat transfer is high and constant; when temperature is between 560 ?C and 540 ?C the interfacial heat transfer coefficient drops linearly to a lower bound; and when temperature is lower than 540 ?C, the interfacial heat transfer coefficient remains constant at the lower bound.  These two temperatures are selected to transition the heat transfer coefficient because they resulted in better agreement between the predictions of the model and experimentally derived data. The temperature dependent heat transfer coefficient does not fully describe the effect of gap and pressure development on interfacial heat transfer. This thermal only model was able to predict the temperature field and the progress of the solidification front during the casting process with reasonable accuracy, and has the advantage of being computationally efficient. However, there are obvious shortcomings with the approach as it is unable to accurately characterize the differences in casting/die interfacial behaviour between the top die, where there is a build up in pressure and no gap, and side die where there is the formation of a gap.  18   2.2.2 Coupled Thermo-Mechanical Models A coupled thermo-mechanical model is needed to describe the effects of deformation on heat transfer and vice versa [19]. Since the thermal resistance of the air gap that forms at the casting / die interface will control the heat transfer rate, a coupled thermo-mechanical model is needed to predict the wheel deformation and air gap evolution.  Many obstacles arise during the computational modeling of thermo-mechanical behaviour in solidification processes such as LPDC. These obstacles include: 1) the treatment of the liquid/solid transition from a constitutive behaviour standpoint; 2) the mechanical constraints needed to prevent rigid body movement without causing over-constraint issues; 3) penetration and over-closure issues at contact interfaces; 4) and highly non-linear coupling between the heat transfer and stress calculations. Extensive effort is required to model the coupled, highly non-linear phenomena to obtain reasonable accuracy [19, 20, 21, 22, 23, 24, 25].  2.2.2.1 Governing Equations for Stress Analysis  Thermal-mechanical analysis involves solving the equilibrium equations, constitutive equations, and compatibility equations, which relate force to stress, stress to strain, and strain to displacement, respectively [26].  The differential equations of equilibrium for the axisymmetric problem in cylindrical coordinates can be derived by conducting a force balance on an elemental volume [27, 28]. ?? ?? ??? + ?? ? ? ? ??? + ?? = 0                                          (2. 4) ?? ?? ?? ? + ?? ?? + ?? = 0                                                (2. 5) 19   where, ?r, ?z , ?? are the stresses acting on planes with normals in the r, z and ? directions; the ?ij are the shear stresses acting on planes with normals i in direction j, and br, bz are the components of the body force acting in the r and z directions.  In order to solve the equilibrium equations, the displacement, strain and its relationship to stress must be considered [27]. The strain vectors are defined in terms of the displacements as below [29]: ? = ???????? =??????????????? + ????                                                    (2. 6) where, ur is the radial displacement and uz is the axial displacement.  Assuming linear elastic behavior, the relationship between stress and strain is: ? = ??????? ? = ??                                                        (2. 7) D is the elastic matrix containing elastic modulus E and Poisson?s ratio v. For an isotropic material [29],  D = E(1+ v)(1? 2v) 1? v, v, v, 0v, 1? v, v, 0v, v, 1? v, 00, 0, 0, (1? 2v) / 2"#$$$$$ %&'''''                            (2. 8) To incorporate the effect of thermal strain and plastic behavior, the total strain is given by ? = ?? + ?? + ???                                                      (2. 9) 20   where ?? ,?? , ??? ?are the elastic, inelastic and thermal strains, respectively. Thermal strains arise due to thermal contraction/expansion that happens during the cooling of the metal in the solid state/heating up of the die. Differences in thermal strain cause non-uniform deformation and the generation of stresses [19]. The thermal strain can be calculated with the following equation:  ?th= ? T dTTTi                                                               (2.10) where ? is thermal expansion coefficient, and Ti is the initial temperature. The temperature history is found by solving the energy equation, including the effects of transient heat conduction, solidification, interfacial heat transfer, etc.  Plastic or inelastic strain occurs when the stress exceeds the flow stress of the material, which dependents critically on temperature and strain-rate and to a lesser extent microstructure. Creep deformation is generally ignored in wheel casting, as the cycle times are relatively short in the range of 3 to 5 minutes, which limits creep strain accumulation. It is difficult to characterize the constitutive behaviour experimentally owing to the broad range of conditions that exist within a wheel and rapidly changing temperature. As a result, it is difficult to predict the stress and strain behaviour of the cast component and die accurately [19].   2.2.2.2 Constitutive Behaviour of A356 Roy et al. [30] characterized the constitutive behavior of A356 aluminum alloy in the as-cast condition with compression tests performed over a range of temperatures (30-500 ?C) and strain rates (~0.1-10 s-1). Empirical expressions were used to fit the flow stress behaviour as a function 21   of temperature and strain rate and develop a comprehensive constitutive equation. The work provides deformation data for A356 at temperatures up to 500 ?C and strain rates of ~0.1-10 s-1.  Relatively little data is available in the literature regarding constitutive behavior at the high temperatures (>500 ?C), low strains, and low strain rates important to casting [19]. Hao et al. [21] reported the high temperature flow stress used in a model of direct chill casting of an AZ31 magnesium alloy: for temperatures above 400 ?C, the yield stress was linearly extrapolated down to 1 MPa at Tcoh (mechanical coherency temperature, also known as mechanical coalescence temperature, which is defined as the temperature where the solidifying material is first able to develop stress, and it is assumed to occur at fraction solid ~0.90 in this literature, 578 ?C for AZ31 magnesium alloy); the yield stress was then linearly increased to 10 MPa at Tliq (liquidus temperature, 635 ?C for AZ31 magnesium alloy). This methodology was implemented in order to minimize plastic strain accumulation at temperatures above Tcoh. The work also reports a framework to define the elastic modulus (lower the elastic modulus by several orders of magnitude as temperature increases within the mushy zone to ensure a low level of stress in the metal above coherency temperature). The estimated Young?s modulus is as shown in Figure 2.5. Drezet and Phillion [22] employed a similar method to vary Young?s Modulus when modeling the deformation in an aluminum alloy AA6063 billet. Young?s modulus was decreased from 10 GPa at Tsol (solidus temperature, 557 ?C for AA6063) to 0.1 GPa at Tcoh (for AA6063, Tcoh = 630 ?C corresponding to a fraction solid of 0.88) and to 0.01 GPa at Tcoh + 5 K. However, the steep drop of Young?s modulus might cause computational convergence problems for complicated models like wheel casting. It may also lead to large elastic strain accumulation, which can create unreasonable stresses after solidification [19]. 22   2.3 Quantification of Casting/Die Interfacial Heat Transfer One of the biggest challenges of coupled thermal-mechanical modeling lies in the quantification of casting/die interfacial heat transfer, which can vary with time, position, temperature, air gap size and contact pressure.  2.3.1 Incorporating Interfacial Heat Transfer in Models Since interfacial heat transfer phenomena, as described in Section 2.1, are complex, the interface is not typically modeled explicitly. Instead, it is generally described using a heat flux boundary condition at the interface. The heat flux, q, across the interface is calculated by:  q = h (T2 ? T1)                                                                       (2.11) where h is the interface heat-transfer coefficient, and (T2 - T1) is the temperature difference across the interface.  Proper determination of h is critical for obtaining accurate simulation results. There are three main mathematical methods described in the literature for estimating h, including (1) analytical, (2) semi-analytical or empirical, (3) numerical techniques. Pure analytical methods are only applicable for simple geometry and constant material properties, and can only provide constant interfacial heat transfer coefficients. Semi-analytical and empirical methods do not rigorously solve the heat transfer equation; instead, they use curve fitting and experimental data to achieve an empirical correlation for h. Numerical methods use optimization methods, such as inverse heat conduction analysis, to iteratively determine h. These numerical approaches are both complex and computationally expensive [17].  23   A variety of numerical descriptions have been used for interfacial heat transfer coefficients in models reported in the literature. These include constant values, functions of time, and/or temperature, and/or functions of interfacial gap size/pressure. These studies show that defining h as a function of temperature works reasonably well [3, 16, 17].  However, when more complex changes occur at the casting/die interface - i.e. gap formation dependent on far field heat transfer - the appropriateness of defining h as a function of casting surface temperature alone becomes questionable. In these cases, h may be defined as a function of time in thermal only analyses. The prediction of gap formation and evolution requires the use of a coupled thermal-mechanical model but allows for both casting surface temperature and contact pressure/gap size (or gap size alone) to be accounted for [16].  2.3.2 Interfacial Heat Transfer Coefficient Researchers have performed experiments to quantify the variation of h with casting surface temperature and contact condition (gap size or pressure) and have suggested several correlations for h based on casting surface temperature and gap size [3, 7, 8, 23, 24, 31]. Trovant and Argyropoulos [8] developed the following analytical expression to estimate the overall heat transfer coefficient for perfectly flat interface: Stage I: h = 1/RT1 Stage II: h= hcontact= f (k, Pcontact, hardness, roughnesssurface) Stage III: h = hcond, gas + hradiation= kairA + ? Tmetal, int2 +Tmold, int2 Tmetal, int+Tmold, int1?metal+ 1?mold-1                (2. 12) 24   where A is the width of the air gap (mm), ? is the Stefan-Boltzmann constant, and ? is the effective emissivity of the interface surfaces. The heat transfer coefficients calculated with the analytical expression for Stage III was compared with experimentally estimated data, but the match was observed to be poor especially when the air gap was small. The difference between experimental and analytical solutions diminished with increasing gap size.    Instead of the analytical expression, Trovant and Argyropoulos [8] proposed a semi-empirical equation characterize the heat transfer coefficient as a function of air gap.  h= 1k?A+r+C                                                         (2. 13) This equation is a function of the air gap size, which was observed to dominate the heat transfer conditions, and uses the constants k, r, and C to lump together the effects of other factors, such as the thermal conductivity of air in the gap, radiation, die roughness and coating, etc. The constants k, r, and C will depend on the particular condition of each casting/die system they are used to analyze. The units of k, A, r, and C are W-1 m2 K mm-1, mm, W-1 mm K, and W m-2 K-1, respectively.  In follow-up research by Argyropoulos and Coates [31], experiments were performed using a unique casting apparatus (Figure 2.6) that enabled the interfacial heat transfer coefficient to be correlated with air gap. The instrumentation used in the apparatus included four thermocouples and two linear variable displacement transducers (LVDT), and allowed the temperature profiles in the die and casting as well as the air gap variation at the casting/die interface to be measured during the solidification process. The heat transfer coefficient at the interface was estimated by inverse heat transfer analysis. The results show that a significant drop in the heat transfer 25   coefficient coincided with air gap formation. The semi-empirical equation was found to provide excellent correlation between the heat transfer coefficient and the air gap size.   Figure 2.7 shows a comparison of the calculated heat transfer coefficient (square symbols) as a function of air gap size from the onset of the air gap formation, based on inverse analysis, to predictions using the semi-empirical equation). The constants in the semi-empirical equation for this data are k = 1.2318 x 10-2 W-1 m2 K mm-1, r = 3.3085 x 10-4 W-1 m2 K, and C = 116.1491 W m-2 K-1. The predicted h represents an excellent fit to the calculated data. Line 2 is the estimate of the heat transfer coefficient using the analytical equation, assuming a perfectly flat interface between the casting and die. Line 3 shows the estimated radiation heat transfer coefficient as a function of the air gap size. It is evident that radiation heat transfer contributes very little to the overall heat transfer coefficient for these castings.  In summary, the expression proposed by Argyropoulos and his colleagues can be used to provide a good estimate of heat transfer coefficient as a function of air gap size for the specific conditions of commercial purity aluminum cast against steel and cast iron dies with specific surface roughness. The applicability of this equation to different casting systems (alloy and die materials) and different geometries has yet to be tested. This equation only describes the heat transfer coefficient after the formation of the air gap. The questions of what value/equation should be used for a heat transfer coefficient before the air gap forms, what happens when air gap does not form and contact pressure increases instead, and how to make a smooth transition through these different stages still remain. 26   Kron et al. [23] proposed an equation to describe the effective heat transfer coefficient at the interface between two types of aluminum alloys (eutectic Al-13%Si and Al-7%Si-0.3%Mg) and a steel mould (0.14% C, 0.35% Si and 1.2% Mn) for a cylindrical casting geometry. The equation is: h= 11h0+ 1hair+hr  with hair= ?airg                                                          (2. 14) where ?air is the thermal conductivity of air, g is the air gap width, hr is the radiation heat transfer coefficient, and h0 is the baseline heat transfer coefficient when no gap exists. The authors indicated that the coefficient h0 is greatly affected by the presence of a coating at the interface.  Its value may be approximated by the ratio of the coating material conductivity over the coating thickness. The radiation term, as described before, can be neglected to further simplify the equation as radiation has a very limited contribution to the overall interfacial heat transfer coefficient. This equation has a theoretical basis, and may also be applied when the air gap is zero - i.e. h = h0. However, it is notable that h0 does not represent the heat transfer coefficient at the beginning of the casting process - i.e. stage I - when the casting and die are in liquid/solid contact. Instead, h0 is the heat transfer coefficient just before air gap opens - i.e. at the end of Stage II, when heat transfer coefficient has already dropped because of solid/solid contact resistance.  In a paper describing a 3-D thermal model of low-pressure die-cast process of A356 aluminum alloy wheels, Zhang et al. [3] reported the use of a temperature dependent heat transfer coefficient for the wheel/die interface. As shown in Table 2.1, when the temperature at the wheel surface is higher than the liquidus temperature (560 ?C), a constant heat transfer coefficient 27   (2000 W/m2/K) was used; as the temperature drops below the liquidus temperature but is higher than a temperature near the solidus temperature (540 ?C), the heat transfer coefficient drops linearly to a minimum value; for surface temperatures below 540 ?C, a constant heat transfer coefficient was applied. Though the definition of this heat transfer coefficient as a function of temperature does not explicitly incorporate the effect of the air gap and/or pressure, it provides a reference for the heat transfer coefficient at the die wheel interface for the LPDC of aluminum wheels. Argyropoulos and Carletti [7] used a casting apparatus similar to Figure 2.6 to determine the time history of the heat transfer coefficient at the casting/die interface. As shown in Figure 2.8,  there is a steep rise in the h to a maximum (point a) over a very short time period after pouring. Then the h decreases rapidly (segment ?ab?) as the liquid metal in contact with the die solidifies and asperities form. Segment ?bc? shows a small increase in h due to the expansion of the die, which enhances contact between the solidified metal and the die. The following segment, ?cd?, indicates that h decreases gradually as the air gap, which forms at ?c?, increases in size. The drop in h prior to air gap formation is ~ 30%.  Koric et al. [24] reported a gap dependent heat transfer coefficient that included a critical gap size, below which h was assumed not to change.  hg=h0 , h? d0 hg= 1dkair+Rc +hrad, d0<d                                                      (2. 15) where h0 is the heat transfer coefficient corresponding to the critical gap size of d0. Setting the heat transfer coefficient to be constant for small gaps was proposed to minimize the effects of 28   minor variations in contact. Applying this method in computational models could improve numerical convergence. As for cases when an air gap does not form and contact pressure develops, there is not much literature about the effect of contact pressure on the metal / die heat transfer coefficient during solidification. Most of the literature discusses an applied pressure in squeeze casting or the roll pressure during metal rolling process. Sekhar et al. [32] investigated the effect of applied pressure on the heat transfer coefficient at the metal / die interface during solidification of Al-Si eutectic alloy against H13 die during the squeeze casting process. They found that the heat transfer coefficient increased with increasing applied pressure (107~20?107 Pa). The heat transfer coefficient increased as much as one order of magnitude, or more, after application of pressure and the overall effect of pressure diminished as the pressure increased.  Ilkhchy et al. [33] also studied the effect of external pressure on the heat transfer coefficient at the interface of A356 aluminum alloy and metal mold. A finite difference heat conduction code was adopted to estimate the interfacial heat transfer coefficient from thermocouple data. A third order expression was proposed to correlate the external pressure and the interfacial heat transfer coefficient.  ? = 0.0011?? ? 0.112?? + 6.605? + 2927.57                                     (2. 16) Fackeldey et al. [34] provided a simplified relation of heat transfer coefficient as a function of contact pressure at the interface, considering the mean hardness of the casting and die. The relation is shown below. 29   ????? = ?? ?? ??                                                (2. 17) where P is contact pressure at the interface, H is the Vickers hardness, and ?1 and ?2 are material parameters depending on the material combinations used in the casting process. Chen et al. [35] studied heat transfer during rolling. It was pointed out that the heat transfer coefficient at the interface between the slab and the roll in the roll gap was strongly dependent on the roll pressure. High pressure improved the contact between the slab and the roll, reducing the thermal resistance. A linear relationship was found to exist between the mean heat transfer coefficient and the mean roll pressure. This relationship can be used to calculate interfacial heat transfer coefficient based on estimation of the rolling load, consequently, the mean roll pressure.  In summary, both theory and empirically based expressions to calculate the interfacial heat transfer coefficients as a function of air gap size have been reported in the literature. The effect of pressure on interfacial heat transfer has also been discussed, though there is less directly applicable work. However, a method to incorporate the combined effects of temperature, air gap and pressure on interfacial heat transfer coefficient for LPDC process of aluminum alloy wheels has not been investigated. Moreover, from the standpoint of numerical convergence the heat transfer coefficient should be made as smooth as possible at the transitions between the different interfacial heat transfer stages to avoid convergence difficulties.  This makes implementing the interfacial heat transfer coefficient in computational modeling more challenging.      30    Figure 2.1 A schematic of the physical contact between two solid materials and the temperature profile through the solids and the contact interface [9]   Figure 2.2 Experimental setup of a drop of casting solidifying on a nickel substrate, which is instrumented with thermocouples to analyse the thermal resistance at the casting/die interface. The size of the metal drop is about 10-11 mm in diameter and 4-5 mm in height. [13, 14]   31     Figure 2.3  A schematic of heat transfer through gas at solid/gas/solid interface   Figure 2.4 Three stages of air gap formation at the cast/mold interface, showing (a) Stage I, (b) Stage II, (c) Stage III when a gap forms [8]   (a) (b) (c) 32    Figure 2.5 Elastic modulus versus temperature data used in modeling of an AZ31 Magnesium Billet casting [21] 33    Figure 2.6 Casting apparatus with Thermocouples and LVDT [31] 34    Figure 2.7 A plot of heat transfer coefficient as a function of air gap. The squares depict the heat transfer coefficient calculated from experimental data. Line 1 displays predictions based on the semi-empirical equation, Line 2 shows the heat transfer coefficient estimated from the analytical equation. Line 3 shows the estimated radiation heat transfer coefficient [31]   35    Figure 2.8 Heat transfer coefficient history derived from the experimental results. Segment ?ab? indicates a rapid drop in HTC due to formation of asperities and intermittent gaps. Segment ?bc? indicates a small increase in HTC because of the expansion of the die. Segment ?cd? depicts a significant decrease in HTC, which is due to the metal-die gap starting at ?c? and continuing to increase in size. Vertical dotted line L1 indicates the moment when the air gap is detected by LVDT measurments; L2 indicates the moment when the cast has solidified and a constant HTC is reached. [7]  Table 2.1 Temperature dependent heat transfer coefficient applied in LPDC A356 wheel casting process [3]    36   Chapter 3  Scope and Objectives 3.1 Scope of Research  The goal of this project is to improve the understanding of, and develop a methodology/expression for calculating, heat transfer across the wheel/die interface during the wheel casting process. In order to formulate and assess expressions for the interfacial behavior, a computational model has been developed. This model is capable of predicting the thermal history and thermally induced deformation of the cast wheel and die during the LPDC process of aluminum alloy wheels. The coupled thermal-mechanical model was developed within ABAQUS, a commercial finite element package, and is capable of predicting the interfacial gap and/or pressure that develops during solidification and cooling of the wheel.  As a first step, the model was fit to experimentally obtained thermocouple data using a straightforward temperature dependent heat transfer coefficient based on previous work appearing in the literature to obtain preliminary gap and pressure behavior at various locations in the casting. This data was then used to develop a preliminary interfacial heat transfer coefficient correlation that was a function of temperature, gap width and pressure based on a literature review. The interfacial heat transfer coefficient was defined and implemented in the model using a user-subroutine. The original approach was found to improve the agreement between the model predictions and thermocouple measurements, but was found to be prone to numerical convergence issues due to large sudden changes in the heat transfer coefficient associated with gap formation and the transition from gap to pressure that appears to occur in some locations of 37   the wheel casting. A new methodology of incrementally changing the interfacial heat transfer coefficient was also proposed to alleviate the convergence issues of the model, which has been demonstrated in an Excel-based spreadsheet. To produce the data required for model validation, plant trials were performed in collaboration with Canadian Auto Parts Toyota Inc. (CAPTIN) at their wheel manufacturing facility in Delta, British Columbia. Two sets of measurements were planned; the first, involved temperature measurements at discreet locations within the wheel and die; and the second, involved using a high accuracy eddy current displacement sensor to characterize the evolution of the air gap at the wheel/die interface. Temperature measurements were completed during a pre-production plant trial performed by other colleagues at UBC. The measured temperatures have been be used for model validation of temperature predictions. Due to timing issues, the displacement measurements have not been conducted with the industrial partner; however, the design of an experiment for this purpose has been completed and provides a reference for future work.  3.2 Objectives of Research Project The primary objective of the project is: ? To develop a better description of the interfacial heat transfer coefficient for use in mathematical casting process models - i.e. to move from the temperature dependent heat transfer coefficient to a temperature, gap width and pressure dependent heat transfer coefficient.  To achieve the primary objective, the following subtasks are necessary: 38   ? Develop a coupled thermal-mechanical model, within ABAQUS, capable of predicting the thermal deformation/distortion of the wheel and die sections during the LPDC process for aluminum wheels.  ? Develop and implement a subroutine within ABAQUS to describe the interfacial heat transfer coefficient as a function of temperature, gap, and pressure.  ? Use measured temperature data to assess the temperature predictions of the model and the effects of the revised interfacial heat transfer description. ? Design a displacement measurement experiment capable of characterizing the gap width at the interface.    39   Chapter 4  Experimental Measurements Experimental measurements are needed to support the development of an improved interfacial heat transfer coefficient description. More specifically, the data will be used to tune and validate the process model. Comparing model predictions with measured data will provide an assessment of whether the model correctly represents the real physical system, provides accurate predictions, and meets its intended requirements/abilities. For this project, a thermo-mechanical model of the LPDC process for wheel production has been developed and experimental measurements were needed to assess the model?s ability to predict the temperature history and the evolution of the air gap at the wheel/die interface.  Two sets of measurements were planned; one involved temperature measurements at discreet locations within the wheel and die structure; the second involved using a high accuracy eddy current temperature displacement sensor to characterize the evolution of the air gap at the wheel/die interface. Initially both sets of measurements were to be performed in collaboration with Canadian Auto Parts Toyota Inc. (CAPTIN) at their wheel manufacturing facility in Delta, British Columbia. Initially, these measurements were to be made on a prototype advanced water-cooled die as part of the larger APC program. Unfortunately, due to delays in fabrication of the prototype die, temperature data from an earlier trial on a production die with a simple 5-spoke geometry had to be used. This precluded the availability of interface displacement data as the eddy current sensor to measure interface displacements was not available at the time of the earlier trial (the first trial with the prototype die without the eddy current sensor was completed 40   on December 14th, 2013). The development and testing of a set-up to test the eddy current sensor was performed and will be discussed in the following sections to provide a reference for future work. It is anticipated that these measurements may be made in future plant trials at CAPTIN.  4.1 Temperature Measurements To provide data to develop and tune the boundary conditions and to validate the model from the perspective of temperature history, thermocouples were inserted at a number of locations within a production die and cast into wheels. Data from the thermocouples was logged over a number of cycles including under steady-state cyclic conditions in order to measure and record the temperature history at discrete locations within the die and wheel. Dr. Carl Reilly, a Research Associate at UBC and member of the APC team, completed the experimental program related to the measurement of temperature, which is described in the following section. 4.1.1 Experimental Procedures Stainless steel sheathed, Type-K thermocouples (1/8th inch) were inserted into holes machined in the die sections (top, side, bottom dies) and into pressure-lock fittings at 40 locations in a die (427 wheel production die). Thermocouples placed close to the casting/die interface were welded into the face of the die at approximately 2 mm below the surface. The locations of the thermocouples placed in the top and sides die are presented in Figure 4.1 and Figure 4.2. Data from the 40 thermocouples together with the signals from 6 pressure sensors (2 for process control) were recorded for 32 casting cycles using a National Instruments CompactRIO DAQ system and LabView software at 5 Hz.  After 25 cycles, nominal steady state operation was 41   achieved, determine based on when the die temperatures at the end of each casting cycle were nearly identical to their values at the start of the cycle.  The thermocouple data collected under steady state operation (from Cycle_26 to Cycle_32) is used for model validation. Figure 4.3 shows the recorded temperature history of thermocouple 1 (TC_1) from Cycle_26 to Cycle_32 (refer to Figure 4.2 for thermocouple location) plotted based on individual cycle times (overlaying each cycle). The temperature data for TC_1 shows the characteristic temperature cycle experienced at this location and that cyclic steady state was achieved (except for a small deviation in Cycle_29). The characteristic shape of the temperature curve can be linked to different stages in the casting process. From 0 to ~30 s, the temperature at this location decreases because the liquid metal takes about 30 seconds to reach the height of TC_1, and while heat transfer within the die and to the surroundings continues. At around 30 s, the temperature starts to increase because the heat from the casting has reached this location in the die. Following a rapid increase as the die draws heat from the wheel, the temperature decreases as the heat is transferred to the coolant and ambient environment. Similar trends are observed in the temperature history of TC_2, as shown in Figure 4.4 and for other thermocouples in the die adjacent to the interface (not shown here) with shifts in the time at which the temperature begins to increase consistent with when the liquid metal reaches the point (height) in the cavity where the thermocouple is located. In addition to measuring the temperature of the die, three 1/16th inch thermocouples were cast into the wheels produced during several casting cycles (Cycle_27, Cycle_28, Cycle_29, Cycle_30) to record the cooling curves at discrete locations within the wheel. The thermocouples 42   that were to be cast-in were placed into the die cavity when top and side dies were open. When the casting cycle started, the top and side dies close, trapping the thermocouples within the die cavity. Before the die opened in each of these cycles, the thermocouples were severed to allow the wheel to be ejected and to carry on down the production line. The final locations of these cast-in thermocouples were determined by transmission X-ray imaging after casting. Figure 4.5 shows the locations of the thermocouples in the wheel for Cycle_27, Cycle_29 and Cycle_30. The wheel from Cycle_28 was thrown out by accident before it was X-rayed; therefore the locations of the cast-in thermocouples in Cycle_28 are unknown. The locations of the cast-in thermocouples were different for each cycle because they must be installed during a 10 s period, into a ~400 ?C die while the die is open, and the positions of the thermocouple tips change when the die closes and the thermocouples are clamped inside. 4.2 Displacement Measurements  Along the casting / die interface, the development of an air gap depends on the relative displacements of the casting and die surfaces. A technique has been developed to characterize the air gap evolution at the casting/die interface to provide data for validating the coupled thermal-mechanical model. Unfortunately, this technique has not been implemented and tested in a production die at the time of writing of this thesis due to factors beyond UBC?s control. A review of the sensor selection, installation plan, and calibration testing is provided here for future reference. There are many techniques available to measure displacement. These may be broadly segregated into two categories: contact and non-contact. A popular type of contact displacement 43   measurement equipment applied in casting experiments is the linear variable displacement transducer (LVDT), which is able to measure large displacement ranges, is insensitive to target material, can measure a localized spot (measure the displacement of a discrete point rather than the average of a surface), and is generally low cost [36]. If a contact measurement system was used to characterize the wheel casting process, the sensor would be in contact with liquid aluminum melt, which has a temperature of 700 ?C, causing the instrument to be sacrificed after each casting cycle. In the laboratory setting, Argyropoulos et al. [7, 31] circumvented this issue by attaching a quartz rod to the end of an LVDT to make contact with the liquid metal. Quartz has the added benefits that it can sustain high temperatures with little shape change (low thermal conductivity and thermal expansion). The quartz rod is replaced for each casting [7, 31]. However, the complexity (die shape and motion) and cyclic nature of the production aluminum wheel casting process negates the use of this approach. A fairly extensive review of the literature failed to identify a suitable means of employing contact methods for the LPDC process. The wheel casting process requires that the displacement sensor be embedded in or attached to the die structure and be operational for several cycles without human interference (replacement of sacrificial parts).  Non-contact measurement technologies such as optical fiber sensors and ultrasonic sensors were considered but rejected due to some of their limitations. For example, optical fiber sensors, which would require glass fibers to be inserted in the die to transmit light, are very sensitive to environmental contaminants. Ultrasonic sensors would have problems measuring the distance to the wheel surface because the signal would be lost through the air gap; moreover, the internal discontinuities in the die (voids, inclusions, and cracks) may disrupt the signal. In addition, no 44   ultrasonic probes that could be identified would be capable of withstanding the required operating temperatures (~ 500 ?C). Another non-contact measurement technology is the inductive eddy current sensor. It can measure the displacement of electronically conductive target without detecting the nonconductive material between the sensor and the target. In operation, an electromagnetic field is generated by the dual coils in the sensor. The electromagnetic field penetrates the front of the sensor and induces eddy currents in the conductive target. When the target displacement changes, the impedance in the coil changes correspondingly. The change of impedance is detected by the signal conditioning electronics and converted to a voltage output signal [37]. Therefore, the eddy current sensors can detect the position and movement of the aluminum wheel surface without direct contact.  The Kaman High Temperature Displacement Sensor is based on inductive eddy current technology, and was selected as a device potentially able to measure the displacement in the aluminum wheel casting process. It provides accurate non-contacting measurement of conductive surface motion in hostile environments [37]. Detailed information can be found in the product datasheet and user manual [37, 38]. The KD?1950 Displacement Sensor was selected because: firstly, with an operating temperature range from -196 ?C to +537 ?C (648 ?C short term), it can withstand the high temperature casting environment (~500 ?C in the die); secondly, its measurement range and resolution (0-3.81 mm and ? 0.0013 mm, respectively) satisfies the wheel/die interface air gap size (0-1 mm, estimated 45   from preliminary modeling results) and the accuracy requirement of the experiment (? 0.05 mm). The KD-1950 displacement sensor is as shown in Figure 4.6. An experimental set-up for the displacement sensor, shown in Figure 4.7, was developed to allow installation in the side die of an operational wheel casting process. As seen in the figure, a through hole would be bored in the side die, and the KD-1950 displacement sensor would be installed near the hot face of the side die behind a quartz disc. Two steel pieces would be used to position the sensor and hold it in the side die. Monitoring the side die / wheel interface was identified as a suitable location based on preliminary modeling results that indicate an air gap forms along on the wheel/side die interface.  A 2 mm thick quartz disc (extremely low thermal conductivity) would be placed between the sensor head and the liquid metal to protect the sensor during casting. As quartz is not electrically conductive, it will not impair the electro-magnetic field of the sensor and would not be detected by the sensor. However, it reduces the available measurement range of the sensor by 2 mm, making the measuring range 0 to 1.81 mm, which is still enough to measure the interfacial gap.    Two steel retaining pieces were specially designed to hold the sensor and more importantly to meet the specifications of the minimum clearance for conductive material around the sensor face. For KD-1950, the minimum clearance is shown in Figure 4.8 along with the sensor dimensions.  In the current application, the sensor will move with the side die, and detect the displacement of the corresponding wheel surface to measure the variation of air gap at the wheel/die interface for each casting cycle. It was originally anticipated that the collected displacement data would be 46   compared with the model predictions. In this way, one of the mechanical aspects of the thermo-mechanical model could have been validated.  The signal conditioning electronics output an analog voltage signal that is linearly proportional to displacement. Figure 4.9 shows the calibration data supplied by KAMAN for the KD?1950 sensor. It is calibrated for a target material of A356 at three reference temperatures, 25 ?C, 350 ?C, and 538 ?C [39]. Based on the calibration data, a temperature dependent equation has been derived to convert the voltage output to displacement output. D = (V ? 2.163?10?4T +1.188?10?1) / (?1.673?10?5 + 4.082?10?1)          (4. 1)           where D is the displacement in mm, V is the voltage output in volts, T is the temperature in degrees Celsius. A simple test, shown in Figure 4.10, was performed to check whether the sensor could detect the displacement of an A356 sample with a quartz disc in between the sensor and the sample, and to assess the accuracy of the equation derived from calibration data. The displacement sensor was installed in a steel tube with a quartz disc in a manner to that shown in Figure 4.7. The tube was fixed to a lab bench. A square piece (~6?6?1 cm3) of A356 was machined from the mid-height of a wheel rim for use as a target sample in the test. The target sample was fixed to an aluminum gauge block with a C-clamp and placed on a horizontal steel plate to ensure that the target surface of the sample was parallel to the face of the sensor. The target sample was placed at known distances from the sensor by inserting different thicknesses of shim stock or gauge blocks in between the quartz disc and the target sample. The thickness of the shim stock and gauge 47   blocks was measured with a micrometer. The voltage output from the sensor was recorded and converted to displacement using Equation 4.1. The testing was undertaken at a temperature of 21.8~22.1 ?C. The quartz disc was measured to be ~2.14 mm in thickness. Table 4.1 summarizes the test results. Based on the results, the accuracy of the sensor in this application was ? 0.05 mm.  This testing proved that KD-1950 displacement sensor could detect the displacement of a room temperature A356 target with a quartz disc in between. In the future, the sensor will be used on a sand casting to test its performance at high temperature. These tests will verify its capabilities to measure the wheel/die air gap evolution in the relevant temperature range and continued attempts will be made to employ it with industrial partners.   48    Figure 4.1 Thermocouple locations in top die   Figure 4.2 Thermocouple locations in side die shown in a section view 49    Figure 4.3 Temperature history of TC 1 at steady state (from Cycle 26 to Cycle 32)  Figure 4.4 Temperature history of TC 2 at steady state (from Cycle 26 to Cycle 32) 50    Figure 4.5 Cast-in thermocouple locations in the wheel for different cycles, from left to right, Cycle_27, Cycle_29 and Cycle_30. Note that 29_B is outside of the wheel because it was cast in the window section, trapped between top die and bottom die.   Figure 4.6 KD-1950 high temperature displacement sensor [37]  51    Figure 4.7 Sectioned view of the side die showing the KD-1950 sensor installation   Figure 4.8 Left: KD-1950 sensor dimensions; right: KD-1950 mounting [38] Sensor H13 Steel Quartz disc 52    Figure 4.9 KD-1950 calibration data for A356 [39]  53    Figure 4.10 Sensor test set-up  Table 4.1 Sensor test result Test Displacement Measurement Error   (mm) (mm) (mm) 1 0.0762 0.076 -0.0002 2 0.1524 0.155 0.0026 3 0.2718 0.2332 -0.0386 4 0.5105 0.5472 0.0367 5 0.7849 0.8352 0.0503 6 1.0465 1.0742 0.0277 7 1.3030 1.3572 0.0542 8 1.5875 1.5972 0.0097    A356 sample target Gauge block Tube holding displacement sensor and quartz disc C-Clamp 25.4 mm 54   Chapter 5  Coupled Thermo-mechanical Model Development A 2D-axisymmetric coupled thermo-mechanical model of LPDC process for A356 aluminum alloy wheels has been developed to predict the thermal history, wheel deformation and the variation of the air gap and pressure along the wheel/die interface. The model has been developed in the commercial finite element package, ABAQUS 6.11 [40].  Compared to a conventional thermal-only model, the coupled thermo-mechanical model considers not only the thermal behaviour but also the mechanical behaviour, including the thermal expansion/contraction of the various materials in the wheel and the die, their stress state and the overall deformation of the wheel and die assembly as a function of time. The deformation behaviour in turn allows for an estimation of the variation in gap or pressure at the wheel/die interface with time, which in turn can be used to predict the heat transfer behaviour at the interface. This two-way coupling between deformation and heat transfer is an important aspect of the wheel casting and critical for the accurate prediction of the thermal history in the process. The coupled thermo-mechanical model employed in this work has been developed based on a thermal-only model developed as part of a larger engineering research and development program on advanced die design. The thermal boundary conditions of the thermal-only model and the thermo-mechanical model are identical except for those describing the die / casting interface. The thermal-only model uses a temperature dependent interfacial heat transfer coefficient, while the coupled thermo-mechanical model employs a temperature, gap size and pressure dependent 55   formulation. The steady state die temperature field predicted by the thermal-only model has been used as the initial condition for the coupled thermo-mechanical model.  In this chapter, details of the coupled thermo-mechanical model development will be discussed, including model geometry, mesh, material properties, initial condition, thermal and mechanical boundary conditions, and convergence criteria. Note: the relationships for the wheel/die interface used in the thermal-only and thermo-mechanical models will both be described. 5.1 Model Geometry The geometry of a current CAPTIN production wheel, referred to as the 427 wheel, was selected for use in the model because of its relatively simple geometry. As shown in Figure 5.1, the 427 is a five-spoke wheel, and has symmetry planes on the planes radially bisecting the spokes and windows. Figure 5.2 is a section through the wheel, showing the cross-section of the on-spoke and on-window symmetry planes of the wheel. Thus the smallest circumferential section that can be analyzed taking advantage of the symmetry condition that exists in the wheel is a 36? section - see Figures 5.3 (a), which shows a 36? section of the wheel, and Figure 5.3 (b), which shows a 36? section of the wheel and the various die components included in the analysis.  Since the coupled thermal-mechanical model is computationally more intensive compared to the thermal only model ? i.e. two degrees of freedom per node compared to one for the thermal model - the model geometry was further simplified from a 3D 36? section to a 2D axisymmetric geometry, based on a plane through the on-spoke section. It was reasoned that the resulting reduction in computational time would allow numerous runs to be completed within a reasonable timeframe allowing a variety of interface boundary condition formulations to be examined. As 56   shown in Figure 5.4, the 2D axisymmetric geometry contains the same die components and cooling channels as the 3D geometry. It should be noted that the 2D axisymmetric geometry ignores the effects of the windows on heat transfer and deformation; since an axisymmetric wheel based on this geometry would have a solid spoke face.  5.2 Mesh The geometry shown in Figure 5.5 was meshed using ICEM, an ANSYS mesh generation software, before being translated by an in-house conversion code to an ABAQUS readable format. The mesh shown in Figure 5.5 a) contains 55137 elements and 60190 nodes. The elements are 4-node axisymmetric bilinear displacement and temperature elements with reduced integration (identified as CAX4RT in ABAQUS). The mesh edge length is finer (0.4 mm) along the edges of each component and courser (4 mm) towards the center as shown in Figure 5.5 b). The fine mesh along the component edges or interfaces was found to be necessary to improve the solution resolution in order to avoid contact convergence issues. The mesh was coarsened at the interior of the components to reduce the number of nodes and computational time. Elements on the surfaces of each component in the model are grouped into element sets and used to define surfaces within ABAQUS. Surfaces of die sections or die and casting sections that are in close proximity to each other were defined as contact pairs with surface-to-surface heat transfer and contact boundary conditions applied.  57   5.3 Materials Properties The wheel is cast from A356 (Al-7Si-0.3Mg) aluminum alloy and the die structure, except the sprue, is fabricated from H13 tool steel. The sprue is comprised of two parts (an upper and lower sprue), where the lower sprue is made of cast iron and the upper sprue is made from tungsten carbide. Both the thermophysical and constitutive properties for each of these materials used in the model were based on values reported in the literature.    5.3.1 Thermophysical Properties The wide temperature range of the casting process makes the use of temperature dependent material properties necessary though it adds non-linearity to the model. The thermophysical properties, which include density, thermal conductivity and specific heat, adopted in the model are listed in Table 5.1 [3, 41, 42, 43].  As the model does not include fluid flow the volume change associated with the liquid-to-solid phase transformation is ignored. A constant value for the density of each material is applied in the model, which is consistent with the density at a fraction solid of 0.9 (567 ?C, also known as the mechanical coherency point). At temperatures above 567 ?C liquid metal feeding is assumed to compensate the volume shrinkage of the metal during the phase transformation. As the model is fully coupled, changes in density are calculated by virtue of the evolving displacements (driven by thermal strain accumulation) within the wheel and die. For A356, the thermal conductivity is artificially increased to 400 W/m/K at temperatures higher than the liquidus (614 ?C) to approximate the effects of natural convection in the liquid [3]. This represents a 6x increase in conductivity relative to the nominal (measured) conductivity of the liquid. 58   The latent heat released during the liquid to solid phase transformation in A356 has been included as a source term (*Latent Heat input command in ABAQUS), and is released throughout a discrete temperature range (533.0 ?C to 613.2 ?C) in proportion to the evolution in fraction solid [18]. The latent heat data reported by Zhang et al. [3] has been adopted in this model. The data is presented in Table 5. 2. The total latent heat released is 397.5 kJ/kg.  5.3.2 Mechanical Properties Mechanical properties including thermal expansion coefficient and the constitutive behaviour of the various materials comprising the wheel and die are critical in determining the strain and stress state within the wheel and die and the resulting deformation as a function of time.  5.3.2.1 Thermal Expansion Coefficient J. F. Hetu et al. [44] reported a temperature dependent thermal expansion coefficient for A356, as described by Equation 5.1. ?'=22.598?10-6+2.387?10-8T                                         (5. 1) In which ?? is the thermal expansion coefficient (?C-1), and T is temperature (?C).  For a temperature lower than the mechanical coherency temperature, Equation 5.1 is adopted for the thermal expansion coefficient ? see discussion related to density in the previous section. For temperatures higher than the mechanical coherency temperature, the thermal expansion coefficient is assumed to be zero. The thermal expansion above the mechanical coherency temperature is expected to be minimal as liquid metal is able to flow to accommodate expansion.  59   In ABAQUS, the thermal expansion coefficient, ?, is formulated to define the total thermal expansion from a reference temperature [45]. This ? is not the typical physical thermal expansion coefficient (??) provided in the literature. ?(T) for ABAQUS can be calculated as [46]: ? T = 1T-T0 ?' TTT0 dT                                                       (5. 2) where ?(T) is the temperature dependent thermal expansion coefficient as defined for ABAQUS, T0 is the reference temperature. The thermal expansion coefficient with a reference temperature of 25 ?C for ABAQUS, calculated with Equation 5.1 and Equation 5.2, is given in Table 5.3. 5.3.2.2 Constitutive Behaviour An expression for the Young?s modulus of A356 at temperatures below 567 ?C (mechanical coherency temperature, fraction solid 0.9), has been reported in literature [30, 47, 48] and was adopted in this model.  E=2? 1+?  ?=2.54?104(1+ 300-T2Tmelt )                                                (5. 3) In Eq. 5.3, E is the Young?s modulus in MPa, ? is the shear modulus in MPa, T is the temperature in K, Tmelt is 614 ?C, and ? is the Poisson?s ratio, which is equal to 0.33.  Above 567 ?C, the Young?s modulus is assumed to decrease linearly from 47.0 GPa to 100 MPa [21, 22] at the liquidus temperature of 614 ?C, and remain constant for temperatures above 614 ?C. The reduction in Young?s modulus at these temperatures is meant to ensure a low level of stress above the coherency temperature. The drawback to this approach is that the rapid change 60   in Young?s modulus, (three-order of magnitudes over the 567~614?C range) can cause convergence issues in the model. To avoid computational issues, the transition at 567 ?C and 614 ?C has been smoothed with additional points - i.e. instead of a steep linear curve over the 567~614 ?C range, two extra points at 559 and 563 ?C were added to achieve a more gradual transition at 567 ?C; similarly, two points at 606 and 610 ?C were also added to alleviate the abrupt transition at 614 ?C. The Young?s modulus data used in the model is summarized in Table 5.4 and Figure 5.6. Elastic-perfectly plastic behavior is assumed in the model formulation - i.e. hardening is not considered. Deformation data from constant strain rate (0.1 s-1) compression tests, published by Roy et al. [30], has been adopted for A356 at temperatures up to 494.9 ?C. For temperatures higher than 494.9 ?C, there is no experimental data available. According to Hao et al [21], the yield stress is supposed to be linearly extrapolated down to 1 MPa at mechanical coherency point, and then linearly increased 10 magnitudes at liquidus temperature to minimize plastic strain accumulation in the semi-solid and liquid state. However, the model had computational convergence issues with this method. To achieve computational convergence, the yield stress is assumed to be constant for temperature higher than 494.9 ?C in the model. The yield stress data applied in the model is presented in Table 5.5 and Figure 5.7. 5.4 Initial Conditions In the LPDC process, dies are pre-heated to approximately 450 ?C before the first casting cycle is run. The temperature of the die evolves during each subsequent cycle until a cyclic steady state 61   condition is reached, which may require up to 25 casting cycles. Cyclic steady state occurs when the die temperature fields at the end of each cycle are equal to those at the start of the cycle. In the thermal-only model, the initial die temperature is defined as 450 ?C throughout the various die sections. In subsequent cycles, the initial die temperature is taken as the die temperature distribution from the end of the previous cycle. An in-house script, written in Perl, is used to repeatedly run the model to simulate cyclic operation. The script stops running the model once cyclic steady state has been reached. In the thermal-only model, the cyclic steady state condition has been defined to occur once the largest temperature difference in the die temperature field at the end of a cycle, compared to the beginning, is less than 5 ?C.  With the current wheel and die geometry, it takes about 15 cycles to reach cyclic steady state. To reduce the number of runs of the thermo-mechanical model, the cyclic steady state die temperature distribution predicted by the thermal only version of the model has been imported as the initial condition for the die in the 2D axisymmetric thermo-mechanical model, as shown in Figure 5.8. Since ABAQUS cannot simulate the filling process, a height dependent initial temperature has been defined for the wheel. The dependence of the initial temperature on height approximates the effects of heat loss from the wheel during filling. The initial temperature of the metal in the sprue is set equal to the temperature of the A356 aluminum alloy melt in the production process (700 ?C). The initial temperature of the wheel then decreases linearly from 700 ?C at the hub region above the sprue to 625 ?C at the top of the wheel (inboard rim flange).  62   5.5 Thermal Boundary Conditions Thermal boundary conditions implemented in the thermal-only model were originally based on values reported in the literature [3] and then refined using a trial-and-error approach to obtain a best fit with thermocouple data. The thermal boundary conditions are split into five steps to describe the conditions active in the LPDC process. Step 1 is the most significant for this work, since it includes the wheel solidification and cooling phases. The total time for a single casting cycle is 210 s.  The five steps and the relevant process events occurring within them are: Step 1 (154.9 s duration): For the first 3 s of the cycle, the casting machine performs interlock checks, with side dies closed and the top die raised (open). During this time, there is no coolant flowing in the die. Heat transfer occurs between the side die and bottom die sections by contact, and between the dies and the ambient environment through convection and radiation. By 7 s the die has closed, the filling starts, and the cooling channels may be activated (cooling channel locations and timing are shown in Figure 5.4 and Table 5.8). Heat transfer mechanisms between the top and side die sections (contact) and active cooling of the die via cooling channels are added. Also at this time, the pressurization of the holding furnace begins and liquid metal moves up the stalk (liquid metal goes up to the bottom of the cast sprue). At 10.4 s, filling of the sprue and die cavity starts and heat transfer between the wheel and the die occurs as a function of the height that the liquid metal has reached. It takes 32 s from start of filling until the inboard rim flange at the top of the wheel is filled according to thermocouple data from a plant trial. During the remaining time (~112s), the wheel cools and solidifies within the die cavity by transferring heat from the wheel to the die, and the die to cooling water and the ambient environment.  63   Step 2 (0.1 s duration): At 154.9s in the casting cycle, the air pressure in holding furnace is released and the liquid metal in the sprue falls back into the reservoir. The side dies pull back (open) and the top die is raised with the wheel attached to it (due to thermal contraction). At this point, heat transfer between the wheel and the bottom and side dies is finished.  Step 3 (8.9 s duration): The wheel remains in contact with the top die while the ejection process occurs. The wheel continues to cool through conduction heat transfer across the wheel/top die interface, and convection and radiation heat transfer at the wheel surfaces exposed to the ambient environment. The side and bottom die surfaces, which were in contact with the wheel, are now exposed to the environment and cool through convection and radiation heat transfer.  Step 4 (0.1 s duration): The wheel is ejected from the top die. At this point in the model, the wheel is removed from the simulation. The top die surfaces that were in contact with the wheel are now exposed to the ambient environment and cooling occurs via convection and radiation heat transfer.  Step 5 (46 s duration): The die continues to cool in the ambient environment with the die cavity open providing opportunity for operators to perform intermittent maintenance. The die temperature distribution at the end of this step is taken as the initial die temperature for the next cycle.  Heat transfer varies among the steps. In general, there are three types of thermal boundary conditions applied in the model: (1) Interfacial heat transfer, (2) Surface film heat transfer (convection), and (3) Radiation.  64   5.5.1 Interfacial Heat Transfer  Interfacial heat transfer is used to simulate the heat transfer at the wheel/die interface, wheel/sprue interface and various interfaces that may occur between the dies and sprue sections. This type of heat transfer can be expressed as:  q=h(T1-T2)                                                         (5. 4) where q is the heat flux at the interface (W/m2), h is the interfacial heat transfer coefficient (W/m2/K), and T1 and T2 are the temperature of the surfaces on either side of the interface.  The heat transfer coefficients used for various interfaces of dies and sprues, summarized in Table 5.6 were based on the expected contact conditions and literature reported values [3]. Note that some of these interfacial heat transfer boundary conditions are inactive at specific points during the casting cycle due to changing die contact conditions.  For the wheel/die interface, the time delay in heat transfer resulting from progressive filling of the die is approximated by activating the casting / die interfacial heat transfer based on the estimated maximum height that the liquid metal would have reached during filling of the die cavity. Filling takes place sequentially, moving from the sprue, to the spoke and finally to the rim, and maximum height of liquid metal is estimated as a linear function of time as filling progresses. When the height of a location in the wheel is higher than the estimated maximum height of the liquid metal, the heat transfer coefficient between the wheel and the die is set equal to zero; when the height of a location is lower than the maximum height, heat transfer between the wheel and die is active. When filling is completed, heat transfer along the entire casting / die interface is active.  65   Two types of interfacial heat transfer coefficients have been implemented in the thermo-mechanical model:  1) Temperature dependent interfacial heat transfer coefficients (the same as used in the thermal- only model) 2) Temperature, gap size and pressure dependent interfacial heat transfer coefficients 1) The temperature dependent interfacial heat transfer is the same as the thermal model, and is based on literature reported values and gradually refined to achieve the best fit with experimental data. The temperature dependent interfacial heat transfer coefficients are defined as: Stage I: ? ? 614 ??, ??? =  ????  Stage II: 567 ? ? < 614 ??, ??? = ??? ? ?? ???? ? ?? ?(1? ???? )+ ????  Stage III: ? < 567 ??, ??? = ??? ?????  where hmax is the heat transfer coefficient when the casting is liquid ? i.e. Stage I, flimit is a factor indicating how much the interfacial heat transfer coefficient drops. During Stage II, h drops linearly from hmax to (hmax?flimit). The values of hmax and flimit are a function of the specific wheel/die interface. The data used to define the temperature dependent interfacial heat transfer coefficients based on this definition is shown in Table 5.7.  2) The temperature, gap size and pressure dependent interfacial heat transfer coefficients used in the thermo-mechanical model are defined based on literature review as:  ????? ??: ?? ? 614, ??? = ???  66   ????? ???: ?567 ? ? < 614, ??? = ? ?, ?, ?  ??? = ? ?, ? = ??? ? ? ? 567614? 567 ? 0.3+ 0.7 ? 1+ 0.2 ? ?5 ? 10? , ?? ?0 < ? ? 5 ????,??? ? ? ? 567614? 567 ? 0.3+ 0.7 ? 1+ 0.2 , ?? ?? > 5 ????  ??? = ? ?, ? = 11??? ? ? ? 567614? 567 ? 0.3+ 0.7 + (? ? ??)??? ? ? ?, ??? ?(? ? ??) > 0 ????? ????: ?? < 567, ??? =  ?? ?, ?  ??? = ? ? = ??? ? 0.7 ? 1+ 0.2 ? ?5 ? 10? , ?? ?0 < ? ? 5 ????,??? ? 0.7 ? 1+ 0.2 , ?? ?? > 5 ????  ??? = ? ? = 11??? ? 0.7 + (? ? ??)???  ? ? ?, ??? ?(? ? ??) > 0 where hmax (W/m/K) is the interfacial heat transfer coefficient when the casting is liquid; T (K) is temperature of the wheel surface; P (Pa) is contact pressure at the interface; kair (W/K/m) is the thermal conductivity of air equal to 0.0515 W/K/m; d (m) is the size of the air gap predicted by the model; d0 (m) is the initial size of the air gap at 614  ?C; (d ? d0) is the corrected size of the air gap. The regions of liquid (or semi-solid) are not explicitly tracked in the thermo-mechanical model and hence are included in the calculation. Instead, the model employs temperature dependent material properties to differentiate between the liquid and solid behaviour. This approach to minimizing the effect of the liquid on the stress calculation results in the thermo-mechanical 67   analysis calculating displacements, stresses and strains in the liquid (and semi-solid regions). Preliminary modeling results showed that an air gap may develop along the casting / die interface prior to the material reaching the mechanical coherency point. This is unrealistic as the liquid metal would flow and conform to the die surface prior to solidification. In addition to being unrealistic in a physical sense, the significant size of the air gap at the transition to a gap size dependent heat transfer coefficient function causes a steep drop of hint and leads to convergence issues.  Attempts to reduce / eliminate air gap formation in the model prior to solidification by changing the material properties were unsuccessful. To accommodate this issue, the initial gap size, d0, at the liquidus temperature (614 ?C) is recorded by the model at each location along the interface and then used to reset the effective gap size used in the heat transfer coefficient function to 0 at the mechanical coherency temperature.  A corrected gap size is calculated as (d - d0) and used to calculate the thermal resistance term for conduction across the air gap: ? ?????? . Also, radiation heat transfer at these interfaces has been neglected according to the literature review. As for contact pressure, a linear correlation between hint and pressure has been assumed up to a maximum pressure. More specifically, when pressure is lower than 5 MPa, hint increases linearly with pressure up to 20%; and above 5 MPa, further increases in pressure do not affect hint and it remains constant. 68   5.5.2 Surface Film Heat Transfer A surface film heat transfer condition is applied to simulate the convective heat transfer on the die in the cooling channels and to the ambient environment consistent with what was used in the original thermal-only model.  The convection heat flux on the die surface is calculated using:  q=h(Tsurf-T?)                                                      (5. 5) where q is the heat flux at the surface film (W/m2), h is the convective heat transfer coefficient (W/m2/K), Tsurf is the temperature of the surface (?C), and T? is the coolant or environment temperature (?C). Table 5.8 shows the heat transfer coefficients used in the cooling channels. There are five cooling channels in total and each is controlled separately.  The heat transfer coefficient varies with both cooling channel location and time. Before the cooling water flow is activated, a heat transfer coefficient representing stagnant air/water cooling (natural convection) conditions is applied (~200 W/m2/K for stagnant water in top die cooling pond, ~4 W/m2/K for stagnant air in the other cooling channels). When the water flow is activated in a cooling channel, the heat transfer coefficient is increased to approximate forced convective heat transfer to water. Although the actual heat flux is a complex function of water and surface temperature, flow rate, geometry, and pressure, constant heat transfer coefficients that range from 2000~8000 W/m2/K were used for the various channels. Note: that side die cooling channel 1 is an air-cooled channel, so the heat transfer coefficient during active cooling is relatively low (~200 W/m2/K). After a programmed time period, coolant flow is terminated by either stopping the water flow (top die cooling pond) or by purging the water/air out of the cooling channel with air (the other 69   cooling channels). In the model, it is assumed that the heat transfer coefficient for flow air purging is 50 W/m2/K. After purging, stagnant air heat transfer conditions are assumed to exist in these channels and the heat transfer coefficient is returned to a lower constant value (~4 W/m2/K). Note: according to a recent plant trial, two-phase flow was observed in the cooling channels during purging, therefore these assumptions may not be accurate.  Table 5.9 summarizes the various surface film boundary conditions applied throughout one cycle. In the case of the die, the surfaces include the external surfaces surrounded by the ambient environment, mating surfaces with other die structures, and internal surfaces. Some of the boundary conditions are not applied for the entire cycle. For example, the internal surfaces of the side dies and bottom die are originally in contact with the wheel, and they are exposed to the environment only after side dies open and top die moves up in Step 2. Similarly, the internal surface of top die is exposed to the environment only after the wheel is ejected in Step 4. Also, the mating surfaces between top die and side die are exposed to the environment only after the side die opens in Step 2. Furthermore, a surface film boundary condition has been used to include the effects of heaters that are installed on the side die. In the case of the sprue, surface film boundary conditions include the base surface on top of a stalk, internal surface with cast sprue, and external surfaces with the ambient environment. For ambient environment boundary conditions, heat transfer is a combination of convective heat transfer and radiative heat transfer.  5.5.3 Radiation Radiation occurs on the die surfaces that are exposed to the ambient environment.  Effective radiation heat transfer coefficients are calculated using the following equation: 70   hrad=?? Tsurf2 +Tair2 Tsurf+Tair                                        (5. 6) where hrad is the radiation heat transfer coefficient, ? is the Stefan-Boltzmann constant, ? is the effective emissivity of the die surface, Tsurf is the surface temperature, and Tair is the environment temperature. The radiation heat transfer conditions applied in the model are summarized in Table 5.9. 5.6 Mechanical Boundary Conditions The 427 LPDC die assembly (Figure 5.9) is a complex assembly of die components that are welded or bolted together and whose major sections are actuated by hydraulic rams. The bottom die is bolted to an H13 steel base plate. The side dies move along a rail built into the base plate and rest on the bottom die when closed. The top die is an assembly of die sections that are raised and lowered as a unit and when closed presses against the side dies.   For an axisymmetric model, the displacements in the x (radial) direction at the centerline are automatically constrained by ABAQUS. Besides this automatic constraint, a number of mechanical boundary conditions, shown in Figure 5.10, are necessary to constrain the major die sections in the model and avoid rigid body motion. There are three types of mechanical boundary conditions used in the model: (1) displacement/rotation boundary conditions, (2) multi-point constraints, and (3) spring constraints.  (1) Displacement/rotation boundary conditions prescribe displacement/rotation in selected degrees of freedom (DOF). Specifying a prescribed displacement magnitude of 0 in DOF 2 on the top and bottom dies stops them from moving vertically.    71   (2) Multi-point constraints (MPC) can be used for connections and joints. The PIN function of the MPC option in ABAQUS provides a pinned joint between two nodes and makes the displacements equal. Using PIN MPC enables the surface nodes of two adjacent parts to be held together to eliminate relative movement of the pinned nodes. A PIN MPC was applied between the side die cooling channel and the side die, between the lower and upper sprues, and between the upper sprue and the bottom die. (3) Spring constraints can model not only actual physical springs but also restraints to prevent body motion. Spring 1 option in ABAQUS is between a node and ground, acting in a fixed direction. Linear spring behaviour is defined by specifying a constant spring stiffness (force per relative displacement). The use of a spring is a less strict method to constrain nodes as it allows some amount of movement. In the model, a spring in DOF 2 (vertical) with a stiffness of 1?107 MPa is applied to the cast in the sprue to restrain it?s motion while allowing it to accommodate contraction in the casting. The top die center pin is also constrained by a spring in DOF 2 with a stiffness of 1?107 MPa to hold the top die components in place.  These mechanical boundary conditions were developed through a trial and error approach using the model. They must be carefully applied in order to properly constrain the parts and avoid over-constraint issues.  5.7 Convergence Criteria The complex boundary conditions, temperature dependent material properties, and coupled thermal mechanical solution procedure result in a model that is highly nonlinear and difficult to 72   achieve a converged solution based on the default convergence criteria of ABAQUS. The default solution control parameters defined in ABAQUS are designed to provide optimal solutions to a wide range of non-linear problems, and in most cases need not to be changed. However, the default control parameters and convergence criteria are very strict by engineering standards [49]. Consequently, the solution control parameters have be adjusted in the model to define less strict convergence criteria, as shown in Table 5.10. Using these parameters in the model allows convergence to be achieved.    73    a)  b) 74    c)  d) Figure 5.1 a) 427 Wheel geometry; b) front view; c) back view; d) side view   75    Figure 5.2 A section through half of the 427 wheel, showing the on-spoke section and on-window section  a)  76     b) Figure 5.3 a)  A 36? slice of the 427 wheel which has been simplified for modeling by merging the riser blocks in Figure 5.2into one block to allow simplified meshing; b) Wheel and die geometry of 3D thermal model showing die sections. Note that: the heater is not included as a part in the model; a thermal boundary condition is applied to model its effect  77     Figure 5.4 Wheel and die geometry of the 2D axisymmetric coupled thermo-mechanical model, Note:  TD_CC is top die cooling channel, TDDC_CC is top die drum core cooling channel, SDC_CC is side die core cooling channel, BD_CC1 is bottom die cooling channel 1, and BD_CC2 is bottom die cooling channel 2.  78    a)  b) Figure 5.5 a) Mesh of 2D axisymmetric coupled thermo-mechanical model; b) zoomed in view of the finer interface mesh 79    Figure 5.6 Temperature dependent Young?s modulus of A356 [30]   Figure 5.7 Temperature dependent yield stress of A356 [30]  80     Figure 5.8 Initial temperature distributions in the die and the wheel    81    Figure 5.9 Die assembly of 427 wheel  82    Figure 5.10 Mechanical boundary conditions applied in the model     83   Table 5.1 Thermal physical properties of A356 [41], H13 [3], Cast iron [42], Tungsten carbide [43] Materials Density Thermal conductivity Specific heat  ? (kg/m3) T (?C) k (W/m/K) T (?C) Cp (J/kg/K)       A356 2380 25 163 25 880   100 165 100 921   200 162 200 967   300 155 300 1011   380 153 380 1046   400 153 400 1055   500 145 500 1098   567 134 567 1127   614 400* (65.8) 614 1190   700 400* (67.9) 700 1190   800 400* (70.0)   H13 7800 20 24.6 23 458.8   200 26.25 200 518.5   500 27.3 400 587.8   600 27.76 600 726.2   800 28.07 700 905.4 Cast iron 7200 25 49 25 490   100 48 100 510   200 46 200 555   300 43 300 600   400 42 400 640   500 41 500 700   600 38 600 785   700 35 700 1000 Tungsten carbide 15600 20 84.2 25 166     100 183     200 196     300 205     400 211     500 217     600 221     700 224 Note: The conductivity values of A356 marked with asterisk (*) have been augmented intentionally to include convection, and the natural values are provided in parenthesis.   84   Table 5. 2 Latent heat of A356 [3] Temperature range Latent heat (?C) (kJ/kg) 613.2 > T ? 610.7 51.02 610.7 > T ? 588.2 91.10 588.2 > T ? 567.2 48.74 567.2 > T ? 563.6 170.36 563.6 > T ? 533.0 36.18  Table 5.3 Temperature dependent thermal expansion coefficient of A356; reference temperature is 25 ?C [44] T Real ? ? for ABAQUS (?C) (?C-1) (?C-1) 25 2.32E-05 2.32E-05 75 2.44E-05 2.38E-05 125 2.56E-05 2.44E-05 175 2.68E-05 2.50E-05 225 2.80E-05 2.56E-05 275 2.92E-05 2.62E-05 325 3.04E-05 2.68E-05 375 3.15E-05 2.74E-05 425 3.27E-05 2.80E-05 475 3.39E-05 2.86E-05 525 3.51E-05 2.92E-05 567 3.61E-05 2.97E-05 568 0 2.96E-05 600 0 2.80E-05 700 0 2.38E-05    85   Table 5.4 Temperature dependent Young?s modulus of A356 [30] T Young's modulus (?C) (Pa) 25 6.76E+10 100 6.48E+10 200 6.10E+10 300 5.72E+10 400 5.33E+10 500 4.95E+10 559 4.65E+10 563 4.60E+10 567 4.50E+10 606 2.50E+09 610 1.00E+09 614 1.00E+08 700 1.00E+08   Table 5.5 Temperature dependent yield stress of A356 [30] T Yield stress (?C) (Pa) 34 1.06E+08 103 9.95E+07 200 9.43E+07 249 9.17E+07 299 8.75E+07 348 6.25E+07 399 4.40E+07 444.4 2.70E+07 494.9 1.70E+07 700 1.70E+07    86   Table 5.6 Heat transfer coefficient for various interfaces of the dies and sprues Interface Time Heat transfer coefficient   (s)  (W/m2/K) Top die/top die drum core 0 ? t ? 210 2500 Top die/top die drum core (thermal break) 0 ? t ? 210 1000 Top die/top die center pin 0 ? t ? 210 1000 Top die/top die center pin (thermal break) 0 ? t ? 210 250 Top die drum core/top die center pin 0 ? t ? 210 750 Top die drum core/top die center pin (thermal break) 0 ? t ? 210 250    Side die/top die 7 ? t < 154.9 800 Side die/top die (thermal break) 7 ? t < 154.9 250    Side die/side die cooling 0 ? t ? 210 750 Side die/side die cooling (thermal break) 0 ? t ? 210 200    Side die cooling/bottom die 0 ? t ? 154.9 750 Side die cooling/bottom die (thermal break) 0 ? t ? 154.9 250    Bottom die/upper sprue 0 ? t ? 210 500 Bottom die/upper sprue (thermal break) 0 ? t ? 210 200 Bottom die/lower sprue 0 ? t ? 210 500    87   Table 5.7 Heat transfer coefficient at the wheel/die interfaces Interface Temperature Heat transfer coefficient   (?C)  (W/m2/K) Top die/wheel T ? 614 3700  567 ? T > 614 15.74 T - 7817  T < 567 1110    Top die/wheel (thermal break) T ? 614 500  567 ? T > 614 2.13 T - 1056  T < 567 150    Bottom die/wheel T ? 614 3500  567 ? T > 614 14.89 T -7395  T < 567 1050       Side die/wheel T ? 614 3250  567 ? T > 614 13.83 T -7192  T < 567 650    Upper sprue/wheel T ? 614 3500  567 ? T > 614 14.89 T - 7395  T < 567 1050    Lower sprue/cast sprue T ? 614 4000  567 ? T > 614 17.02 T -8451  T < 567 1200      88   Table 5.8 Heat transfer coefficient for the cooling channels (cooling channel locations are shown in Figure 5.4) Cooling Channel Time HTC  Sink temperature   (s) (W/m2/K)  (?C) TD_CC 0 < t < 7 200 50  7 ? t ? 210 2500 50     TDDC_CC 0 < t < 67 4 25  67 ? t ? 107 4000 25  107 < t ? 117 50 25  117 ? t ? 210 4 25     SDC_CC 0 < t < 7 4 25  7 ? t ? 157 2000 25  157 < t ? 167 50 25  167 ? t ? 210 4 25     BD_CC1 0 < t < 47 4 25  47 ? t ? 187 200 25  187 < t ? 197 50 25  197 ? t ? 210 4 25     BD_CC2 0 < t < 77 4 25  77 ? t ? 127 8000 25  127 < t ? 137 50 25  137 ? t ? 210 4 25    89   Table 5.9 Heat transfer between the die and the environment Surface Step Heat transfer coefficient Ambient temperature Surface emissivity    (W/m2/K)  (?C)  Top die     External surfaces step 1-5 20 100 0.7 Mating surface with Side die step 2-5 20 100 0.7 Internal surface with wheel step 4-5 20 100 0.7      Side die      Heater surface step 1-5 1500 475 - External surfaces step 1-5 20 100 0.7 Mating surface with top and bottom die step 2-5 20 100 0.7 Internal surface with wheel  step 2-5 20 100 0.7      Bottom die     External surfaces step 1-5 20 100 0.7 Mating surface with side die step 2-5 20 100 0.7 Internal surface with wheel  step 2-5 20 100 0.7      Wheel surfaces (side, bottom) step 2-3 20 100 0.7      Cast sprue (base) step 1 2000 700 -      Lower sprue     Base (on top of stalk) step 1-5 2000 700 - External surfaces step 1-5 20 100 0.7 Internal surface with cast sprue step 2-5 20 500 0.7      Upper sprue     Internal surface with wheel step 2-5 20 100 0.7 Internal surface with cast sprue step 2-5 20 500 0.7    90   Table 5.10 Non-default convergence tolerance parameters Convergence criteria Default User-defined Convergence tolerance parameters for force   Criterion for residual force for a nonlinear problem 0.005 0.5 Criterion for displacement correction in a nonlinear problem 0.01 1 Criterion for zero displacement relative to characteristic length 1.00E-08 0.01    Convergence tolerance parameters for heat flux   Criterion for residual heat flux for a nonlinear problem 0.005 0.5 Criterion for temperature correction in a nonlinear problem 0.01 1    91   Chapter 6 Results and Discussion In this chapter, the baseline results from the coupled thermal-mechanical model with temperature dependent interfacial heat transfer coefficient are presented. To begin, the temperature predictions from the thermal-mechanical model, obtained with the temperature dependent interfacial heat transfer coefficient, are compared to measurements. The predicted air gap size and pressure behavior at various locations are then used in the development of an interfacial heat transfer coefficient that is a function of temperature, gap size and pressure. The temperature, gap size and pressure dependent heat transfer coefficient is implemented in the model, and the modeling results are presented with comparisons to previous model and temperature measurements. Issues and possible solutions to improve this method are discussed. In the end, a new methodology of defining interfacial heat transfer coefficient is proposed.   6.1 Results of Model with Temperature Dependent Heat Transfer Coefficient The model was run with the temperature dependent heat transfer coefficients (refer to Chapter 5 for more details) to obtain preliminary temperature, gap size and pressure predictions at various locations in the wheel and die.  The model was started with steady state temperature conditions from the thermal-only model.  6.1.1 Distribution of Temperature and Deformation The predicted temperature distribution and deformation of the wheel and die at discrete times within a cycle are presented in Figure 6.1. The deformation in the wheel and die are magnified 92   10 times in order to show the gap at the interface more clearly. Initially, at time 0 s (Figure 6.1 a), the simulation starts with the steady state temperature field from the thermal-only model applied to the die and a temperature gradient imposed in the wheel (not visible due to the contour scale). At this time, there are no gaps along the wheel/die interface, except at the inboard rim flange where the die is not completely filled. At 53 s (Figure 6.1 b), solidification has begun in the rim section of the wheel and there is still no significant air gap observed. At 80 s (Figure 6.1 c), as the wheel continues to cool down and solidify, the temperature at some locations in the wheel drops below the mechanical coherency point (567 ?C), and air gaps begin to develop at locations such as the out-board rim flange. In Figure 6.1 d), the prediction for 105 s into the casting cycle, shows various air gaps that continued to develop and are more readily observed. The predicted temperature and deformation at 154 s (just prior to opening the die to eject the wheel) is shown in Figure 6.1 e). As can be seen, in terms of the wheel / die interface, the most extensive air gaps form between the side die and the wheel. Additional gaps form at locations along the top die / wheel, and the bottom die / wheel interfaces but they are not as large as those formed at the side die / wheel interface. A detailed inspection of the air gaps (not presented here) that have developed reveals that their size ranges from 0 to ~0.5 mm and the largest air gap (0.445 mm) forms between the side die and the in-board rim flange of the wheel. At the interface between the spoke and the top die, there is no air gap as the wheel contracts and attaches to top die.  6.1.2 Predictions of Wheel/die Interfacial Behaviour The evolution of the wheel / die interface contact conditions at three locations exhibiting a range of behaviours have been further investigated. The first location is at the interface between the in-board rim flange and the side die shown in Figure 6.2. At this location, an air gap forms at the 93   interface as the wheel contracts towards the top die. The model predictions of the air gap width and temperature (of wheel surface) have been plotted as a function of cycle time in Figure 6.3. For reference, lines indicating the times within the cast cycle when the solidus and liquidus temperatures are reached have been overlaid on the graph. At this location, the model predicts that the gap begins to develop at the beginning of the casting cycle before solidification takes place in the wheel (i.e. the temperature of the wheel surface is still higher than the liquidus temperature). The air gap forms at this time and location due to thermal expansion of the side die when it heats up. Although the wheel is still liquid at this point in the cycle, it does not flow to fill the gap due to limitations in the thermal-mechanical model formulation. The width of the gap is small (7x10-5 m) when the wheel surface temperature cools to the liquidus temperature. After the onset of solidification, the width of the gap decreases to 3x10-5 m.  As the cycle proceeds, the gap size increases gradually to a maximum of 4.2x10-4 m as the wheel continues cool.  The predicted formation of an air gap prior to solidification is inconsistent with the expected behaviour of liquid in the die. At this point in the cycle, the liquid metal should completely fill the die cavity and conform to the die surface. One possible reason for the formation of an air gap prior to solidification in the model predictions is the approximate thermo-mechanical properties used to describe the liquid and solid material behaviour. In particular, the material properties employed for the liquid phase were formulated to limit stress development (low modulus) and prevent plastic deformation from occurring (enhanced flow stress). Other limitations in the model formulation include: 1) pressure is not applied to the liquid at the inlet to push it up against die surfaces; and 2) gravity is not applied as a body force which would result in a downward force and flow out to the die surfaces. 94   The second location where the interface predictions were examined is between the wheel spoke and the top die (refer to Figure 6.4). At this location, an air gap does not form and the casting and the die maintain contact throughout the casting cycle. The evolutions of interface contact pressure and temperature versus time have been plotted in Figure 6.5. The contact pressure begins to increase after the wheel solidification has started (wheel surface temperature equal to 602 ?C).  The pressure reaches a peak value of 36.1 MPa prior to the completion of wheel solidification (wheel surface temperature equal to 573 ?C). Then the pressure drops as the wheel cools. At 127 s, the pressure drops abruptly as the rate of temperature decrease in the wheel increases significantly. One possible explanation of what happens at 127 s is that the end of solidification is reached at this location, ending the release of latent heat. This results in a transition in the cooling rate. The wheel shrinks in volume and away from the top die surface, therefore the contact pressure at this location of the wheel/top die interface drops.  The development of an air gap or contact pressure at the interface is very complicated. It varies with location and time. It has been observed that at certain locations, contact pressure develops initially, but then drops to zero and an air gap develops as the casting cycle proceeds. The third location chosen for examination is at the side die / wheel interface opposite the side die cooling core and is shown in Figure 6. 6. The data presented in Figure 6. 7 shows that the pressure increases and then decreases to zero twice, and then an air gap develops. The first drop in pressure starts when temperature of the wheel is 567 ?C. This temperature corresponds to the mechanical coherency point when the thermal contraction in the wheel starts and the wheel surface pulls away from the die surface. At ~100 s the pressure redevelops due to macroscopic 95   deformation of the wheel and/or die. At ~120 s the pressure drops again due to contraction of the die associated with a water cooling channel and a gap develops.  6.1.3 Temperature Comparisons  Temperature data measured on a 427 (five spoke) wheel was compared with the temperatures predicted by 2-D axisymmetric thermo-mechanical model with thermal-only interfacial heat transfer conditions. The heat transfer conditions in the rim section of the 427 wheel / die are nearest to those approximated by axisymmetric analysis. Therefore, comparisons were limited to locations in this area where thermocouple data was available. The three primary thermocouple locations selected for comparison are shown in Figure 6.8: TC_1 and TC_2 in side die, and TC_52 in top die. The measured and predicted temperatures at these locations are compared in Figure 6.9. Beside the three primary thermocouples, comparison was also performed at three secondary thermocouple locations lower in the rim section. The comparison of measured and predicted temperatures at the secondary locations, TC_4, TC_5 and TC_46 shown in Figure 6.8, are presented in Figure 6.10 (a) (b) (c).  The characteristic shape of the temperature curves at each location is linked to the different events (i.e. filling sequence, cooling channel cycles, and die section opening) that occur during each casting cycle. At the location of TC_1 for example, as shown in Figure 6.9 a), from 0 to ~30 s, the measured temperature decreases because the liquid metal takes ~30 seconds to reach the height of TC_1. During this time, heat transfer occurs within the die and to the surroundings. At ~30 s, the temperature starts to increase rapidly because the liquid metal has reached this location in the die. Following this rapid increase as the die draws heat from the wheel, the 96   temperature decreases as the heat is transferred to the coolant in the cooling channels and to the ambient environment. Similar trends are observed in the temperature history of other thermocouples. In addition, at the locations corresponding to TC_2, TC_4 and TC_5, there is a clear transition in the temperature curve at ~155 s, which corresponds to side dies opening at the end of Step 1.  Figure 6.9 compares the measured and predicted temperatures at the primary locations in the upper (inboard) rim section. For the most part, the difference between the measured and predicted temperatures varies from 10 ~ 50 ?C. At most locations, the model initially underestimates the temperature of the die by 40~50 ?C. This difference decreases as the temperature starts to increase. At locations TC_1 and TC_2, the predicted temperature surpasses the measured value (at ~40 s) for a short period of time, and then remains 10~50 ?C lower than the measured value. For TC_52, the predicted temperature is always lower than the measured. The maximum difference between the predicted and measured temperature is ~40 ?C. The model predicts the time at which the temperature starts to increase accurately, but in the model the temperature starts to drop earlier than in the experimental data.  Figure 6.10 shows the comparison between the predicted and measured temperature at locations lower down in the rim section.  The overall agreement between the measurements and the predictions is generally not as good at these locations relative to the upper rim locations. The predicted temperature at location TC_5 is higher than the measured temperature with a maximum difference of ~60 ?C. This may be because location TC_5 is in close proximity to the spoke and window present in the 427 wheel. In the axisymmetric model, the spoke is a solid disc section, which represents a large thermal mass compared to cyclic spoke / window geometry of 97   the 427 wheel. Thus in the model, the additional thermal mass results in higher temperatures in the die near TC_5 compared to the measured temperatures. At locations TC_4 and TC_46, the results show similar trends to those observed for locations TC_1 and TC_2, except that the maximum temperature difference is larger (~60 ?C).  In summary, the comparison of the predicted and measured temperatures suggests that the model is able to qualitatively predict the temperature variation in the die. The model correctly predicts the temperature changes resulting from the main physical phenomena occurring during a casting cycle. The differences between the measured and predicted temperatures may be caused by a variety of factors.  Based on the literature reviewed in Chapter 2, the development of an air gap or contact pressure at the wheel/die interface will affect interfacial heat transfer. An air gap at the wheel/die interface introduces additional thermal resistance and the interfacial heat transfer coefficient will drop. While the development of contact pressure results in enhanced contact at the wheel/die interface (i.e. reduced thermal resistance) and the interfacial heat transfer coefficient should increase accordingly. The predictions presented in this section showcase how the wheel and die may deform during a casting cycle, but did not include the effects of thermo-mechanical deformation on interfacial heat transfer. A better description of the interfacial heat transfer coefficient as a function of temperature, air gap width and contact pressure is needed to improve the model accuracy.  6.2 Results of Model with Temperature, Gap size and Pressure Dependent HTC The model incorporating temperature, gap size and pressure dependent interfacial heat transfer (refer to Chapter 5 for more details) was used to predict the temperature and deformation of the 98   wheel and die during a single casting cycles. As in the last section, the model was started with an initial condition based on the steady state conditions from the thermal only model. 6.2.1 Implementation of Interfacial Behaviour The variation in interfacial heat transfer coefficient, gap size and pressure during the casting cycle at two locations in the die, are shown in Figure 6. 11 and Figure 6. 12. During initial runs of the model with the gap size dependent interfacial heat transfer coefficient, the early development of the gap - i.e. prior to the start of solidification - resulted in discontinuity in the heat transfer coefficient at the transition from the temperature based formulation to the gap and pressure based formulation. The rapid discontinuous decrease in heat transfer between the wheel and die results in convergence issues and poor comparison with the measured temperature data. A gap size correction, obtained by subtracting the gap size at liquidus temperature (614 ?C) from the predicted gap size, was implemented and used to calculate the interfacial heat transfer coefficient (for values of the corrected gap size less than zero it was set to zero).  As shown in Figure 6. 11, when the temperature is higher than 614 ?C, the interfacial heat transfer coefficient remains constant based on the temperature dependent formulation. When the temperature is between 567 and 614 ?C, the interfacial heat transfer coefficient decreases linearly with temperature, again based on a temperature dependent formulation. For temperatures below 567 ?C, the interfacial heat transfer coefficient switches to a gap-based formulation. For the first 40 s with the gap-based function, the corrected gap is zero and the heat transfer coefficient is constant. At times greater than 70 s, the corrected gap becomes greater than zero and the interfacial heat transfer coefficient decreases as the corrected gap size increases.  99   Figure 6. 12 shows the variation of the interfacial heat transfer coefficient and contact pressure during the casting cycle at a location on the interface between the rim and top die. At this location, the interfacial heat transfer is influenced by the development of a contact pressure. When temperature is higher than 614 ?C, the interfacial heat transfer coefficient is temperature dependent and constant. When temperature is between 567 and 614 ?C, the heat transfer coefficient is a function of both temperature and pressure (refer to Chapter 5 for more details). When the temperature is lower than 567 ?C, the interfacial heat transfer coefficient is dependent on contact pressure. Referring to Chapter 5, page 67, the interface boundary condition has been formulated such that the interfacial heat transfer coefficient is invariant with pressure at pressures above 5 MPa. As a result, the heat transfer coefficient does not vary between approximately 65 s and 120 s. Beyond 120 s, it decreases with decreases in pressure.  6.2.2 Comparisons to Thermocouple Data The predicted temperatures from the model with temperature, gap size and pressure dependent interfacial heat transfer coefficients implemented as described above are compared to the measured temperatures in Figure 6. 13. To highlight the effects of the interfacial boundary conditions, the temperature predictions from the model with the temperature dependent interfacial heat transfer coefficients have also been plotted in Figure 6.13. As can be seen from the plot, the two model predictions are different. By using the temperature, gap size and pressure dependent interfacial heat transfer coefficients, a better fit to the measured data at locations corresponding to TC_52, TC_46 and TC_5 are achieved. At location TC_52, for example, the prediction matches the thermocouple data between 50~80 s before diverging at 100   times later than ~80 s much more accurately compared with the thermal-only interfacial formulation. In another example, at location TC_5, the shape of the temperature curve matches the thermocouple data much better. At location TC_46, the maximum temperature difference is reduced from ~60 to 40 ?C and for TC_1, TC_2, and TC_4, the general shape of the temperature curve is better than for the temperature based formulation. At TC_1, TC_2, and TC_4, the temperature predictions show a rapid increase when the first heat from the wheel reaches them. Following this increase there is a slow decrease followed by a plateau and then another gradual decrease. These trends are qualitatively more like those seen in the measured temperatures, however, the initial increase is too rapid.  6.2.3 Discussion The predicted development of an air gap before solidification has started is inconsistent with the behaviour expected in the process. Thus, the technique of correcting the gap size has been used to deal with this issue to compensate for the inaccuracy of the model. It is important to note that it is not being suggested this technique represents the physics of the problem.  To properly tackle this problem, the flow of the liquid metal should be modelled with CFD simulation, using software like CFX or ABAQUS/CFD, and coupled to a thermal-mechanical model of the solid domain. This type of simulation represents a fluid-structure interaction problem which is not within the scope of this project but may be possible as part of future work. The temperature, gap size and pressure dependent interfacial heat transfer coefficient has added the effects of air gap and contact pressure, but it is still inadequate in several ways. Firstly, the temperature that the air gap is zeroed at should be near to the mechanical coherency temperature 101   (567 ?C) instead of the liquidus temperature (614 ?C). Effort has been spent to implement this modification, but model runs did not finish due to convergence issues. Secondly, the transition between Stage II and Stage III interfacial behaviours is not smooth due to the sudden change of equations based on a temperature criterion and abrupt evolution of the interfacial air gap/pressure. Finally, the baseline heat transfer coefficient parameters should be adjusted to achieve better fit with the measured temperatures. For example, the maximum and minimum interfacial heat transfer coefficient could be changed to reduce the temperature discrepancies in comparison with the temperature measurements. Also, the temperature ranges defining the stages of interfacial heat transfer could be adjusted ? i.e. Stage II may not occur in 567~614 ?C. These types of changes should be considered when improving the interfacial heat transfer description in the future.  6.3 New Methodology to Calculate Interfacial Heat Transfer Coefficients The new methodology to calculate the interfacial heat transfer coefficients is intended to make the transition between Stage II and Stage III behaviours continuous.  The main idea is to limit the change between two connecting increments by storing the heat transfer coefficient of last increment and updating it based on the current increment conditions, but limiting the maximum change in an increment. This methodology would only be applied to Stage III interface behaviour (including the transition from Stage II and Stage III). The interfacial heat transfer coefficient in Stage III would be calculated as:   ? < 567 ??, ???? = ?? + ?? ???? ? ?? + ?? ???? ? ??  102   ???? = ?? ???? + ?? ????  where hi+1 and hi are the interfacial heat transfer coefficients at increment i+1 and i, respectively,   di+1 and di are the air gap sizes, Pi+1 and Pi are the contact pressures, and C1 and C2 are two constants or coefficients depending on the interfacial behaviour and can be estimated as: For the case where an air gap develops at the interface, assuming:  ??? = ? ? = ????? ?? ??  ?   ?? = ??? ? ???? = 0.7???   ??? = ??? ? ? ?   ?? = ??? = ? ??? + ?? ?? ?? ? ?? ??  For the case where contact pressure develops at the interface, assuming:   ??? = ? ? = ?? ? 1+ 0.2 ? ??? ? ?   ??? = ??? ? ? ?   ?? = ??? = ?. ? ??? ? ? This methodology has been implemented in EXCEL to test it using the temperature, gap size, and contact pressure predict by the thermo-mechanical model. The calculated interfacial heat transfer coefficients at the three example locations discussed in Section 6.1.2 are plotted in 103   Figure 6. 14, Figure 6. 15, and Figure 6. 16. As can be seen in Figure 6. 15 and Figure 6. 16, the calculated interfacial heat transfer coefficients are smooth and continuous between stages. However, in Figure 6. 14, the transition between Stage II and Stage III is somewhat abrupt, and may lead to numerical convergence issues. The calculated interfacial heat transfer coefficients reflect the change of air gap size/pressure ? i.e. the heat transfer coefficient decreases when air gap size increases or pressure decreases, and increases when air gap size decreases or pressure increases. An advantage of this method is that it uses the rate of change of the air gap size/pressure to calculate the interfacial heat transfer coefficient increment instead of the absolute value of the air gap size/pressure, and thus it avoids the effects of inaccuracies in the prediction of air gap size/pressure (induced by not explicitly tracking the liquid to solid phase change and the regions of liquid / solid in the thermo-mechanical model). Moreover, the constants/coefficients C1 and C2 can be estimated by taking the derivative of the analytical expression first and then gradually modified by trial-and-error to achieve the best fit.  To implement the new methodology in ABAQUS, an algorithm (Fortran code) would need to be developed within an ABAQUS subroutine. In order to calculate the incremental changes, the values of the interfacial heat transfer coefficient, gap size and pressure must be stored in a common block data structure. It is important to note that for every time increment, ABAQUS performs several iterations to achieve equilibrium. Thus, it is possible to update the heat transfer coefficient either every iteration or every increment. Figure 6. 17 shows a flow chart for the algorithm based on incrementally changing the interfacial heat transfer coefficient every iteration, and Figure 6. 17 b) is a flow chart for the algorithm based on updating the interfacial heat transfer coefficient at the start of a new increment only. Though the latter approach is more 104   realistic, it is also more complicated and computationally expensive.  The implementations of these algorithms are deemed to be beyond the scope of this thesis and thus should strongly be considered for future work in this area.   105    a)  b) 106    c)  d) 107    e) Figure 6.1 ? Temperature contours along with deformation at different time, a) 0 s, b) 53 s, c) 80 s, d) 105 s, e) 154 s. Note: the deformation scale factor is 10. 108    Figure 6.2 ? Zoomed-in image of the interface between the in-board rim flange and side die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6.3  Figure 6.3 ? Variation of gap and temperature with time at the interface between the in-board rim flange and side die 0.0226 m  109    Figure 6.4 ? Zoomed-in image of the interface between the spoke and top die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6.5.  Figure 6.5 ? Variation of pressure and temperature with time at the interface between the spoke and top die 0.013 m  110    Figure 6. 6 ? Zoomed-in image of the interface between the rim and side die (the deformation scale factor is 1). The red dot indicates the location for model results plotted in Figure 6. 7.  Figure 6. 7 ? Variation of temperature, air gap and pressure with time at the interface between the rim and side die 0.011 m 111      Figure 6.8 ? Location of thermocouples   0.245 m 112     (a)  (b) 113    (c) Figure 6.9 ? Comparisons of predicted and measured temperatures at three locations in the upper part of the rim section: (a) TC_1, (b) TC_2, and (c) TC_52  (a) 114    (b)  (c) Figure 6.10 ? Comparisons of predicted and measured temperatures at three locations in the lower part of the rim section: (a) TC_4, (b) TC_5, and (c) TC_46 115    Figure 6. 11 ? Variation of interfacial heat transfer coefficient and real size of air gap with time at the interface between the in-board rim flange and side die as shown in Figure 6.2  Figure 6. 12 ? Variation of interfacial heat transfer coefficient and contact pressure with time at the interface between the spoke and top die as shown in Figure 6.4 116     a)  b) 117    c)  d) 118    e)  f) Figure 6. 13 ? Comparisons of the model predictions with thermocouple data 119    Figure 6. 14 ? Variation of air gap size and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing the interfacial heat transfer coefficient at a location on the side die / wheel interface  Figure 6. 15 ? Variation of air gap size and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing interfacial heat transfer coefficient at a location on the top die / wheel interface 120     Figure 6. 16 ? Variation of air gap size, pressure and the interfacial heat transfer coefficient calculated by the methodology of incrementally changing interfacial heat transfer coefficient at a location on the side die cooling core / wheel interface  121    a) 122    b) Figure 6. 17 ? Flowcharts for incrementally calculating the interfacial heat transfer coefficient based on: a) updating every iteration, or b) updating every increment   123   Chapter 7  Summary and Conclusions This project has been focused on improving the understanding and developing methods for describing heat transfer across the wheel/die interface during the aluminum alloy wheel casting process. In particular, temperature dependent expressions for describing interfacial heat transfer were extended to be temperature, gap-size and pressure dependent. To formulate and assess expressions for the interfacial behavior, a computational model was developed within ABAQUS, a commercial finite element analysis package. A 2D axisymmetric thermo-mechanical model, capable of predicting the temperature history, and the interface gap width and pressure that develops during solidification and cooling of the wheel, was developed.  Data measured on an industrial low-pressure die-casting machine, configured to produce a 427 wheel, was used to validate the 3-D thermal-only process model developed as part of a larger program focused on design of an advanced water-cooled die. The plant trial to measure the temperature history at various locations in a die was performed by colleagues within the casting research group at UBC. Unfortunately, a second plant trial to measure the development of the air gap in the die was not conducted due to delays in design and fabrication of the die that was to be used to conduct the displacement measurements (the die to be used to measure interface displacements was to be based on a prototype of the advanced water-cooled die). However, as part of the program described in this thesis, an eddy current displacement sensor was purchased, tested and calibrated for use in future work. In addition, the modifications needed for the side-die to accommodate the eddy current sensor were also designed. 124   Initially, a temperature dependent heat transfer coefficient based on previous published work was used to develop the coupled thermo-mechanical model and to obtain preliminary gap and pressure behavior at various locations in the casting process. The temperature predictions of this model were compared to the measured temperatures obtained from the 427 die, and the results indicate that the model is able to correctly predict the effects of the main physical phenomena occurring during a casting cycle. The model was then re-formulated to include a temperature, air gap size and pressure dependent interfacial heat transfer coefficient. The expression for the interfacial heat transfer coefficient was formulated from expressions reported in the literature and tuned using the preliminary model data. The interfacial heat transfer coefficient was defined and implemented in the model using a user-subroutine that included a formulation to correct the gap size for premature accumulation of displacement occurring while the cast metal was above the liquidus temperature. The use of this expression was found to improve the agreement between the model predictions and measured temperatures, but was found to be prone to numerical convergence issues due to sudden changes in the calculated heat transfer coefficient arising between the various stages defined in the model to describe the evolution in heat transfer with the evolving interface temperature, air gap and pressure.  To solve the convergence issues, a new methodology of incrementally updating the interfacial heat transfer coefficient was proposed. The rationale behind this methodology is to limit the change between two calculation increments by storing the heat transfer coefficient of last increment and updating it based on the new temperature, gap size, and pressure conditions. To ensure smooth transitions, the maximum change in an increment is also limited. The 125   methodology was implemented in EXCEL to test it and the calculated interfacial heat transfer coefficients were smooth and continuous between stages and reflect the change of air gap size and pressure. An algorithm to apply the methodology in ABAQUS was developed, but has not been implement at this time.  7.1 Recommendations for Future Work The results of the coupled thermo-mechanical model illustrate that significantly more accurate predictions can be made with a thermal-mechanical model capable of describing the dependence of wheel/die interface heat transfer on the evolution of the gap and/or pressure at the interface.  Preliminary implementation of a gap/pressure dependent formulation for interfacial heat transfer indicates that a new methodology of incrementally changing the interfacial heat transfer coefficient is necessary to resolve convergence issues. The new methodology should be implemented in ABAQUS within the GAPCON subroutine and fit to the experimental data using the tunable parameters available in the formulation. Also, the effect of contact pressure on the interfacial heat transfer coefficient in the LPDC process should be further investigated to obtain a more accurate equation of interfacial heat transfer coefficient.  During the development of the coupled thermo-mechanical model, convergence difficulties were an on-going challenge. More effort should be devoted to understanding and solving the convergence issues. Resolving these convergence issues will reduce development times for new models, and allow this type of modeling to be an efficient tool for research and industrial applications. Additionally, resolving the convergence issues should allow a transition from 2D axisymmetric models to 3D symmetric slice models. 3D models are necessary to accurately 126   describe the wheel / die geometry, and would provide additional insight into the interface behavior.   Finally, a trial to characterize interface displacement with the high temperature eddy current displacement sensor should be conducted. This measurement will provide data necessary to validate the mechanical aspect of the coupled thermo-mechanical model and will provide further insight into the interface behavior.         127   References [1]  A. Miller and D. Maijer, "Investigation of erosive-corrosive wear in the low pressure die casting of aluminum A356," Materials Science and Engineering A, no. 435-436, pp. 100-111, 2006.  [2]  B. Zhang, S. Cockcroft, D. Maijer, J. Zhu and A. Philion, "Casting Defects in Low-Pressure Die-Cast Aluminum Alloy Wheels," JOM, pp. 36-43, 2005.  [3]  B. Zhang, D. Maijer and S. Cockcroft, "Development of a 3-D Thermal Model of the Low-pressure Die-cast (LPDC) Process of A356 Aluminum Alloy Wheels," Materials Science and Engineering A, no. 464, pp. 295-305, 2007.  [4]  S. Cockcroft, D. Maijer and A. Phillion, "APC Project Full Proposal," 2010.  [5]  N. Velluvakkandi, in Developing and Effective Die Cooling Technique For Casting Solidification, Master thesis, Auckland University of Technology, 2009, p. 10. [6]  D. G. Eskin, in Physical Metallurgy of Direct Chill Casting of Aluminum Alloys, CRC Press, 2008, p. 186. [7]  S. A. Argyropoulos and H. Carletti, "Comparisons of the Effects of Air and Helium on Heat Transfer at the Metal-Mold Interface," Metallurgical and Materials Transactions B, vol. 39B, pp. 457-468, 2008.  [8]  M. Trovant and S. Argyropoulos, "Finding Boundary Conditions: A Coupling Strategy for the Modeling of Metal Casting Processes: Part I. Experimental Study and Correlation Development," Metallurgical and Materials Transactions B, vol. 31B, pp. 75-86, 2000.  [9]  F. Kreith and M. S. Bohn, in Principles of Heat Transfer, 2001, pp. 11-14. [10]  E. Fried, "Thermal Conduction Contribution to Heat Transfer at Contacts," Thermal Conductivity, Academic press, vol. 2, 1969.  [11]  M. Bahrami, J. Culham, M. Yovanovich and G. Schneider, "Thermal Contact Resistance of Nonconforming Rough Surfaces, Part 1: Contact Mechanics Model," Journal of Thermalphysics and Heat Transfer, vol. 18, no. 2, pp. 209-217, 2004.  [12]  M. Bahrami, J. Culham, M. Yovanovich and G. Schneider, "Thermal Contact Resistance of Nonconforming Rough Surfaces, Part 2: Thermal Model," Journal of Thermalphysics and Heat Transfer, vol. 18, no. 2, pp. 218-227, 2004.  [13]  T. Loulou, E. Artyukhin and J. Bardon, "Estimation of Thermal Contact Resistance During the First Stages of Metal Solidification Process: I ? experimental principle and 128   modelisation," International Journal of Heat and Mass Transfer, no. 42, pp. 2119-2127, 1999.  [14]  T. Loulou, E. Artyukhin and J. Bardon, "Estimation of Thermal Contact Resistance During the First Stages of Metal Solidification Process: II ? experimental setup and results," International Journal of Heat and Mass Transfer, no. 42, pp. 2129-2142, 1999.  [15]  F. Kreith and M. S. Bohn, in Principles of Heat Transfer, 2001, pp. 38-39. [16]  P. Kobryn, "Heat-Transfer Interface Effects for Solidification Process," ASM Handbook, vol. 22A: Fundamentals of Modeling for Metals Processing, pp. 144-151, 2009.  [17]  K. Ho and R. D. Pehlke, "Metal-Mold Interfacial Heat Transfer," Metallurgical Transactions B, vol. 16B, pp. 585-594, 1985.  [18]  S. Thompson, S. Cockcroft and M. A. Wells, "Advanced Light Metals Casting Development: Solidification of Aluminum Alloy A356," Mater. Sci. Tech., vol. 20, pp. 194-200, 2004.  [19]  B. G. Thomas, "Issues in Thermal-Mechanical Modeling of Casting Processes," Iron and Steel Institute of Japan International, vol. 35, pp. 737-743, 1995.  [20]  C. Li and B. Thomas, "Thermomechanical Finite-Element Model of Shell Behavior in Continuous Casting of Steel," Metallurgical and Materials Transactions B, vol. 35B, pp. 1151-1172, 2004.  [21]  H. Hao, D. Maijer, M. Wells, A. Philion and S. Cockcroft, "Modeling of the Stress-Strain Behavior and Hot Tearing during Direct Chill Casting of an AZ31 Magnesium Billet," Metallurgical and Materials Transactions A, vol. 41A, pp. 2067-2077, 2010.  [22]  J. M. Drezet and A. Phillion, "As-Cast Residual Stresses in an Aluminum Alloy AA6063 Billet: Neutron Diffraction Measurements and Finite Element Modeling," Metallurgical and Materials Transactions A, vol. 41A, pp. 3396-3404, 2010.  [23]  J. Kron, M. Bellet, A. Ludwig, B. Pustal, J. Wendt and H. Fredriksson, "A Comparison of Numerical Simulation Models for Predicting Temperature in Solidification Analysis with Reference to Air Gap Formation," Int. J. Cast Metals Research, vol. 17, pp. 295-310, 2004.  [24]  S. Koric, L. C. Hibbeler and B. Thomas, "Explicit coupled thermo-mechanical finite element model of steel solidification," International Journal for Numerical Methods in Engineering, no. 78, pp. 1-31, 2009.  [25]  K. S. and B. Thomas, "Efficient Thermal-Mechanical Model for Solidification Processes," International Journal for Numerical Methods in Engineering, no. 66, pp. 1955-1989, 129   2006.  [26]  D. M. Maijer, Mathematical Modeling of Microstructure and Residual Stress Evolution In Cast Iron Calender Rolls, Phd Thesis, University of British Columbia, 1999.  [27]  S. Cockcroft, "5.2 Development of Mathematical Model of Stress Generation," in Thermal Stress Analysis of Fused-Cast Monofrax-S Refractories, Phd Thesis, University of British Columbia, 1991, pp. 82-90. [28]  "Advanced Finite Elment Methods, University of Colorado at Boulder," [Online]. Available: http://www.colorado.edu/engineering/cas/courses.d/AFEM.d/. [Accessed 9th January 2014]. [29]  O. Zienkiewicz and R. Taylor, "Axisymmetric Stress Analysis," in The Finite Element Method, McGRAW-HILL Book Company Europe, 1989, pp. 72-88. [30]  M. Roy, D. Maijer and L. L. Dancoine, "Constitutive Behavior of As-cast A356," Materials Science and Engineering A , no. 548, pp. 195-205, 2012.  [31]  B. Coates and S. A. Argyropoulos, "The Effects of Surface Roughness and Metal Temperature on the Heat-Transfer Coefficient at the Metal-Mold Interface," Metallurgical and Materials Transactions B, vol. 38B, pp. 243-255, 2007.  [32]  J. Sekhar, G. Abbaschian and R. Mehrabian, "Effect of Pressure on Metal-Die Heat Transfer Coefficient during Solidification," Materials Science and Engineering, no. 40, pp. 105-110, 1979.  [33]  A. F. Ilkhchy, M. Jabbari and P. Davami, "Effect of Pressure on Heat Transfer at the Metal/Mold Interface of A356 Aluminum Alloy," International Communications in Heat and Mass Transfer, no. 39, pp. 705-712, 2012.  [34]  M. Fackeldey, A. Ludwig and P. Sahm, "Coupled Modelling of the Solidification Process Predicting Temperatures, Stresses and Microstructures," Computational Materials Sicence, no. 7, pp. 194-199, 1996.  [35]  W. Chen, I. Samarasekera and E. Hawbolt, "Fundamental Phenomena Governing Heat Transfer During Rolling," Metallurgical Transactions A, vol. 24A, pp. 1307-1320, 1993.  [36]  "Inductive Technology Handbook," Kaman Measuring and Memory Systems, 2012.  [37]  "Kaman Extreme Environment Sensor Data Sheet," [Online]. Available: http://www.kamansensors.com/html_pages/Extreme_Environment.html. [Accessed 10 2013]. 130   [38]  "Kaman High Temperature Displacement System User Manual," [Online]. Available: http://www.kamansensors.com/pdf_files/Kaman_High_Temperature_Sensors_manual_web.pdf. [Accessed 10 2013]. [39]  Intertechnology, "Displacement System Calibration Record," 2013.  [40]  "ABAQUS 6.11 Documentation," SIMULIA, [Online]. Available: http://abaqus.ethz.ch:2080/v6.11/. [Accessed 10 2013]. [41]  K. Mills, "Al-LM25," in Recommended Values of Thermalphysical Properties for Selected Commercial Alloys , Cambridge, Woodhead Publishing Limited, 2002, pp. 43-49. [42]  K. Mills, "Grey Cast Iron," in Recommended Values of Thermalphysical Properties for Selected Commercial Alloys, Cambridge, Woodhead Publishing Limited, 2002, pp. 119-126 . [43]  J. F. Shackelford and W. Alexander, CRC Materials Science and Engineering Handbook, CRC Press, 2000.  [44]  J. F. Hetu, D. M. Gao, K. Kabanemi, S. Bergeron, K. T. Nguyen and C. Loong, "Numerical Modeling of Casting Processes," Advanced Performance Materials, no. 5, pp. 65-82, 1998.  [45]  SIMULIA, "Thermal expansion, Section 25.1.2 of the Abaqus Analysis User's Manual," in Abaqus 6.11 Documentation.  [46]  C. Estey, Thermal Mechanical Analysis of Wheel Deformation Induced From Quenching, Thesis of Master degree, University of British Columbia, 2004.  [47]  H. Frost and M. Ashby, "Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics," Pergamon Press, 1982.  [48]  M. J. Roy and D. Maijer, "Modeling of As-cast A356 for Coupled Explicit Finite Analysis," Light Metals, TMS (The Mineranls, Metals & Materials Society), pp. 377-382, 2012.  [49]  SIMULIA, "7.2 Analysis convergence controls, Abaqus Analysis User's Manual," in Abaqus 6.11 Documentation.     

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