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
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
ii
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.
iii
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.
iv
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 v
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 vi
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 vii
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
viii
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
ix
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 onwindow 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
x
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
xi
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
xii
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
xiii
List of Symbols
Latin Symbols
br, bz
C1 ,C2
Cp
D
D
d
d0
E
flimit
hint
h
ℎ!
ℎ!
h0
hmax
kgas
kair
L
P
q
R
Q
𝑅!"#
T
T1, T2
Tsurf
T∞
Tair
Ti
Tcoh
Tliq
Tsol
t
V
Greek Symbols
α
𝜀!""
𝜀! , 𝜀!
𝜀
Description
components of body force
equation constants or coefficients
specific heat
elastic matrix
displacement in sensor calibration equation
size of the gap
initial size of the air gap
elastic modulus
equation factor
interfacial heat transfer coefficient
heat transfer coefficient
radiation heat transfer coefficient
conduction heat transfer coefficient
baseline heat transfer coefficient
maximum heat transfer coefficient
conductivity of gas
conductivity of air
latent heat
pressure
heat flux
thermal resistance
volumetric heat source term
thermal resistance of the interface
temperature
temperatures of the two surfaces at the interface
temperature of the surface
temperature of the coolant or environment
temperature of the ambient environment
initial temperature
mechanical coherency temperature
liquidus temperature
solidus temperature
time
voltage output
Units
Pa
J/kg/K
mm
m
m
Pa
W/m2/K
W/m2/K
W/m2/K
W/m2/K
W/m2/K
W/m2/K
W/K/m
W/K/m
kJ/kg
Pa
W/m2
K·m2/W
W/m3
K·m2/W
°C
°C
°C
°C
°C
°C
°C
°C
°C
s
volts
thermal expansion coefficient
effective emissivity
emissivity of cast and mould
strain
xiv
𝜀! , 𝜀! , 𝜀!
𝜀!", 𝜀!" , 𝜀!! v
ρ
µ
σ
σr , σz , σθ
τij
ur , uz
Components of strain
elastic, inelastic and thermal strains
Poisson’s ratio
density
shear modulus
Stefan-Boltzmann constant
components of stress
shear stress
components of displacement
kg/m3
Pa
Pa
Pa
m
xv
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.
xvi
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.
1
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 micro2
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].
3
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
4
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
5
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.
6
Figure 1.1 A schematic of a low pressure die casting process machine [1]
7
In-board Rim Flange
Out-board Rim Flange
Figure 1.2 Automotive wheel sections and a directional solidification path [5]
8
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:
𝑞=
!! !!!
!
= ℎ 𝑇! − 𝑇! (2. 1)
9
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
10
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 1.55.0×104 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].
11
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
12
conduction and radiation in parallel. Hence, the heat flux though the interface is the sum of the
two heat flows [15]
𝑞 = 𝑞! +𝑞! = ℎ! 𝑇! − 𝑇! + ℎ! 𝑇! − 𝑇! = (ℎ! + ℎ! )(𝑇! − 𝑇! ) =
(!! !!! )
!!"#
(2. 2)
where T1, T2 = temperatures of the two surfaces at the interface between the solid and gas
ℎ! =
ℎ! =
!" !!! !!!!
!! !!!
!!"#
!
𝑅!"# = !
= radiation heat transfer coefficient in the gas
=conduction heat transfer coefficient of the gas
!
! !!!
= thermal resistance of the interface
in which 𝜎 is the Stefan-Boltzmann constant, and 𝜀 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:
𝜀!"" =
1
1
1
𝜀! + 𝜀! − 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.
13
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].
14
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-
15
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:
ρCp 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), ρ 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.
16
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.
17
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 nonlinear 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].
! !
! !"
! !
! !"
!
𝑟𝜎! + !" 𝜏!" −
!
!!
!
+ 𝑏! = 0
𝑟𝜏!" + !" 𝜎! + 𝑏! = 0
(2. 4)
(2. 5)
18
where, σr, σz , σθ are the stresses acting on planes with normals in the r, z and θ directions; the τij
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]:
!!!
𝜀!
𝜀
𝜀 = 𝜀! =
!
𝛾!"
!"
!!!
!"
!!
!!!
!"
!
+
(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:
𝜎!
𝜎
𝜎 = 𝜎! = 𝐷𝜀
!
𝜏!"
(2. 7)
D is the elastic matrix containing elastic modulus E and Poisson’s ratio v. For an isotropic
material [29],
" 1− v,
v,
v,
0
$
v,
0
E
$ v, 1− v,
D=
v, 1− v,
0
(1+ v)(1− 2v) $ v,
$
0,
0,
(1− 2v) / 2
$# 0,
%
'
'
'
'
'&
(2. 8)
To incorporate the effect of thermal strain and plastic behavior, the total strain is given by
𝜀 = 𝜀!" + 𝜀!" + 𝜀!!
(2. 9)
19
where 𝜀!", 𝜀!" , 𝜀!! 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:
εth =
T
α
Ti
T dT
(2.10)
where α 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
20
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].
21
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].
22
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=
kair
A
+
σ T2metal, int +T2mold, int Tmetal, int +Tmold, int
1
1
+
-1
ϵmetal εmold
(2. 12)
23
where A is the width of the air gap (mm), σ is the Stefan-Boltzmann constant, and ε 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=
1
k·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
24
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.
25
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=
1
1
1
+
h0 hair +hr
with hair =
λair
g
(2. 14)
where λair 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
26
(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 =
1
d
kair
+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
27
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.
ℎ = 0.0011𝑃! − 0.112𝑃! + 6.605𝑃 + 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.
28
ℎ!"#$%!$ = 𝛾!
! !!
!
(2. 17)
where P is contact pressure at the interface, H is the Vickers hardness, and γ1 and γ2 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.
29
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]
30
Figure 2.3 A schematic of heat transfer through gas at solid/gas/solid interface
(a)
(b)
(c)
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
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]
35
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
36
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:
37
•
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.
38
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 watercooled 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
39
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
40
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
41
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
42
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
43
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
44
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
45
compared with the model predictions. In this way, one of the mechanical aspects of the thermomechanical 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 −4 T +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
46
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.
47
Figure 4.1 Thermocouple locations in top die
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)
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.
Figure 4.6 KD-1950 high temperature displacement sensor [37]
50
Sensor
Quartz disc
H13 Steel
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]
51
Figure 4.9 KD-1950 calibration data for A356 [39]
52
Gauge block
A356 sample
target
Tube holding
displacement sensor and
quartz disc
25.4 mm
C-Clamp
Figure 4.10 Sensor test set-up
Table 4.1 Sensor test result
Test
1
2
3
4
5
6
7
8
Displacement
(mm)
0.0762
0.1524
0.2718
0.5105
0.7849
1.0465
1.3030
1.5875
Measurement
(mm)
0.076
0.155
0.2332
0.5472
0.8352
1.0742
1.3572
1.5972
Error
(mm)
-0.0002
0.0026
-0.0386
0.0367
0.0503
0.0277
0.0542
0.0097
53
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
54
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
55
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.
56
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.
57
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.
α' =22.598×10-6 +2.387×10-8 T
(5. 1)
In which α’ 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.
58
In ABAQUS, the thermal expansion coefficient, α, is formulated to define the total thermal
expansion from a reference temperature [45]. This α is not the typical physical thermal expansion
coefficient (α’) provided in the literature. α(T) for ABAQUS can be calculated as [46]:
1
α T = T-T
T ' T
α
0 T0
dT
(5. 2)
where α(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=2µ 1+ν
300-T
µ=2.54×104 (1+ 2T
melt
)
(5. 3)
In Eq. 5.3, E is the Young’s modulus in MPa, µ is the shear modulus in MPa, T is the
temperature in K, Tmelt is 614 °C, and ν 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
59
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
60
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).
61
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.
62
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.
63
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.
64
Two types of interfacial heat transfer coefficients have been implemented in the thermomechanical 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: 𝑇 ≥ 614 ℃, ℎ!"# = ℎ!"#
Stage II: 567 ≤ 𝑇 < 614 ℃, ℎ!"# = ℎ!"# ×
!!!"#
!"#!!"#
×(1 − 𝑓!"#"$ ) + 𝑓!"#"$
Stage III: 𝑇 < 567 ℃, ℎ!"# = ℎ!"# ×𝑓!"#"$
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:
𝑆𝑡𝑎𝑔𝑒 𝐼: 𝑇 ≥ 614,
ℎ!"# = ℎ!"#
65
𝑆𝑡𝑎𝑔𝑒 𝐼𝐼: 567 ≤ 𝑇 < 614,
ℎ!"# = ℎ 𝑇, 𝑃 =
ℎ!"# = ℎ 𝑇, 𝑑 =
𝑇 − 567
𝑃
∗ 0.3 + 0.7 ∗ 1 + 0.2 ∗
, 𝑖𝑓 0 < 𝑃 ≤ 5 𝑀𝑃𝑎,
614 − 567
5 ∗ 10!
𝑇 − 567
ℎ!"# ∗
∗ 0.3 + 0.7 ∗ 1 + 0.2 ,
𝑖𝑓 𝑃 > 5 𝑀𝑃𝑎
614 − 567
ℎ!"# ∗
1
(𝑑 − 𝑑! )
1
+
𝑇 − 567
𝑘!"#
ℎ!"# ∗
∗ 0.3 + 0.7
614 − 567
𝑆𝑡𝑎𝑔𝑒 𝐼𝐼𝐼: 𝑇 < 567,
ℎ!"# = ℎ 𝑃 =
ℎ!"# = ℎ 𝑑 =
ℎ!"# = ℎ 𝑇, 𝑑, 𝑃
, 𝑖𝑓 (𝑑 − 𝑑! ) > 0
ℎ!"# = ℎ 𝑑, 𝑃
𝑃
, 𝑖𝑓 0 < 𝑃 ≤ 5 𝑀𝑃𝑎,
5 ∗ 10!
ℎ!"# ∗ 0.7 ∗ 1 + 0.2 ,
𝑖𝑓 𝑃 > 5 𝑀𝑃𝑎
ℎ!"# ∗ 0.7 ∗ 1 + 0.2 ∗
1
(𝑑 − 𝑑! )
1
+
𝑘!"#
ℎ!"# ∗ 0.7
, 𝑖𝑓 (𝑑 − 𝑑! ) > 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 ºC; (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
66
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 ºC) 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.
67
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
68
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:
69
hrad =σε T2surf +T2air Tsurf +Tair
(5. 6)
where hrad is the radiation heat transfer coefficient, σ is the Stefan-Boltzmann constant, ε 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.
70
(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
71
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.
72
a)
b)
73
c)
d)
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 onwindow section
a)
75
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
76
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
a)
b)
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]
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
Table 5.1 Thermal physical properties of A356 [41], H13 [3], Cast iron [42], Tungsten carbide
[43]
Materials
Density
ρ (kg/m3)
Thermal conductivity
T (°C)
k (W/m/K)
Specific heat
T (°C)
Cp (J/kg/K)
A356
2380
25
100
200
300
380
400
500
567
614
700
800
20
200
500
600
800
25
100
200
300
400
500
600
700
20
25
100
200
300
380
400
500
567
614
700
163
165
162
155
153
153
145
134
400* (65.8)
400* (67.9)
400* (70.0)
24.6
26.25
27.3
27.76
28.07
49
48
46
43
42
41
38
35
84.2
880
921
967
1011
1046
1055
1098
1127
1190
1190
23
458.8
200
518.5
400
587.8
600
726.2
700
905.4
Cast iron
7200
25
490
100
510
200
555
300
600
400
640
500
700
600
785
700
1000
Tungsten carbide
15600
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.
H13
7800
83
Table 5. 2 Latent heat of A356 [3]
Temperature range
(°C)
613.2 > T ≥ 610.7
610.7 > T ≥ 588.2
588.2 > T ≥ 567.2
567.2 > T ≥ 563.6
563.6 > T ≥ 533.0
Latent heat
(kJ/kg)
51.02
91.10
48.74
170.36
36.18
Table 5.3 Temperature dependent thermal expansion coefficient of A356; reference temperature
is 25 °C [44]
T
(°C)
25
75
125
175
225
275
325
375
425
475
525
567
568
600
700
Real α
(°C-1)
2.32E-05
2.44E-05
2.56E-05
2.68E-05
2.80E-05
2.92E-05
3.04E-05
3.15E-05
3.27E-05
3.39E-05
3.51E-05
3.61E-05
0
0
0
α for ABAQUS
(°C-1)
2.32E-05
2.38E-05
2.44E-05
2.50E-05
2.56E-05
2.62E-05
2.68E-05
2.74E-05
2.80E-05
2.86E-05
2.92E-05
2.97E-05
2.96E-05
2.80E-05
2.38E-05
84
Table 5.4 Temperature dependent Young’s modulus of A356 [30]
T
(°C)
25
100
200
300
400
500
559
563
567
606
610
614
700
Young's modulus
(Pa)
6.76E+10
6.48E+10
6.10E+10
5.72E+10
5.33E+10
4.95E+10
4.65E+10
4.60E+10
4.50E+10
2.50E+09
1.00E+09
1.00E+08
1.00E+08
Table 5.5 Temperature dependent yield stress of A356 [30]
T
(°C)
34
103
200
249
299
348
399
444.4
494.9
700
Yield stress
(Pa)
1.06E+08
9.95E+07
9.43E+07
9.17E+07
8.75E+07
6.25E+07
4.40E+07
2.70E+07
1.70E+07
1.70E+07
85
Table 5.6 Heat transfer coefficient for various interfaces of the dies and sprues
Top die/top die drum core
Top die/top die drum core (thermal break)
Top die/top die center pin
Top die/top die center pin (thermal break)
Top die drum core/top die center pin
Top die drum core/top die center pin (thermal break)
Time
(s)
0 ≤ t ≤ 210
0 ≤ t ≤ 210
0 ≤ t ≤ 210
0 ≤ t ≤ 210
0 ≤ t ≤ 210
0 ≤ t ≤ 210
Heat transfer
coefficient
(W/m2/K)
2500
1000
1000
250
750
250
Side die/top die
Side die/top die (thermal break)
7 ≤ t < 154.9
7 ≤ t < 154.9
800
250
Side die/side die cooling
Side die/side die cooling (thermal break)
0 ≤ t ≤ 210
0 ≤ t ≤ 210
750
200
Side die cooling/bottom die
Side die cooling/bottom die (thermal break)
0 ≤ t ≤ 154.9
0 ≤ t ≤ 154.9
750
250
Bottom die/upper sprue
Bottom die/upper sprue (thermal break)
Bottom die/lower sprue
0 ≤ t ≤ 210
0 ≤ t ≤ 210
0 ≤ t ≤ 210
500
200
500
Interface
86
Table 5.7 Heat transfer coefficient at the wheel/die interfaces
Interface
Temperature
(°C)
T ≥ 614
567 ≥ T > 614
T < 567
Heat transfer coefficient
(W/m2/K)
3700
15.74 T - 7817
1110
Top die/wheel (thermal break)
T ≥ 614
567 ≥ T > 614
T < 567
500
2.13 T - 1056
150
Bottom die/wheel
T ≥ 614
567 ≥ T > 614
T < 567
3500
14.89 T -7395
1050
Side die/wheel
T ≥ 614
567 ≥ T > 614
T < 567
3250
13.83 T -7192
650
Upper sprue/wheel
T ≥ 614
567 ≥ T > 614
T < 567
3500
14.89 T - 7395
1050
Lower sprue/cast sprue
T ≥ 614
567 ≥ T > 614
T < 567
4000
17.02 T -8451
1200
Top die/wheel
87
Table 5.8 Heat transfer coefficient for the cooling channels (cooling channel locations are shown
in Figure 5.4)
Cooling Channel
Time
(s)
0<t<7
7 ≤ t ≤ 210
HTC
(W/m2/K)
200
2500
Sink temperature
(°C)
50
50
TDDC_CC
0 < t < 67
67 ≤ t ≤ 107
107 < t ≤ 117
117 ≤ t ≤ 210
4
4000
50
4
25
25
25
25
SDC_CC
0<t<7
7 ≤ t ≤ 157
157 < t ≤ 167
167 ≤ t ≤ 210
4
2000
50
4
25
25
25
25
BD_CC1
0 < t < 47
47 ≤ t ≤ 187
187 < t ≤ 197
197 ≤ t ≤ 210
4
200
50
4
25
25
25
25
BD_CC2
0 < t < 77
77 ≤ t ≤ 127
127 < t ≤ 137
137 ≤ t ≤ 210
4
8000
50
4
25
25
25
25
TD_CC
88
Table 5.9 Heat transfer between the die and the environment
Heat transfer Ambient
coefficient
temperature
2
(W/m /K)
(°C)
Surface
emissivity
Surface
Step
Top die
External surfaces
Mating surface with Side die
Internal surface with wheel
step 1-5 20
step 2-5 20
step 4-5 20
100
100
100
0.7
0.7
0.7
Side die
Heater surface
External surfaces
Mating surface with top and bottom die
Internal surface with wheel
step 1-5
step 1-5
step 2-5
step 2-5
475
100
100
100
0.7
0.7
0.7
Bottom die
External surfaces
Mating surface with side die
Internal surface with wheel
step 1-5 20
step 2-5 20
step 2-5 20
100
100
100
0.7
0.7
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)
External surfaces
Internal surface with cast sprue
step 1-5 2000
step 1-5 20
step 2-5 20
700
100
500
0.7
0.7
Upper sprue
Internal surface with wheel
Internal surface with cast sprue
step 2-5 20
step 2-5 20
100
500
0.7
0.7
1500
20
20
20
89
Table 5.10 Non-default convergence tolerance parameters
Convergence criteria
Convergence tolerance parameters for force
Criterion for residual force for a nonlinear problem
Criterion for displacement correction in a nonlinear problem
Criterion for zero displacement relative to characteristic length
Convergence tolerance parameters for heat flux
Criterion for residual heat flux for a nonlinear problem
Criterion for temperature correction in a nonlinear problem
Default
User-defined
0.005
0.01
1.00E-08
0.5
1
0.01
0.005
0.01
0.5
1
90
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
91
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 ºC), 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 inboard rim flange and the side die shown in Figure 6.2. At this location, an air gap forms at the
92
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.
93
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 ºC). The pressure reaches a peak value of 36.1 MPa prior to the completion of wheel
solidification (wheel surface temperature equal to 573 ºC). 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 ºC. 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
94
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
95
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
96
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 thermomechanical 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
97
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.
98
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
99
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
100
(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:
𝑇 < 567 ℃, ℎ!!! = ℎ! + 𝐶! 𝑑!!! − 𝑑! + 𝐶! 𝑃!!! − 𝑃!
101
𝜕ℎ
𝜕𝑑
𝜕𝑃
= 𝐶!
+ 𝐶!
𝜕𝑡
𝜕𝑡
𝜕𝑡
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:
ℎ!"# = ℎ 𝑑 =
!
!
!
!
!! !!"#
ℎ! = ℎ!"# ∙ 𝑓!"#"$ = 0.7ℎ!"#
!!
!"
!!
!"
= !" ∙ !"
!!
𝐶! = !" = −
!
!!
+!
!
!"#
!!
∙!
!
!"#
For the case where contact pressure develops at the interface, assuming:
!
ℎ!"# = ℎ 𝑃 = ℎ! ∗ 1 + 0.2 ∗ !∗!"!
!!
!"
!!
!"
= !" ∙ !"
!!
!.!"
𝐶! = !" = !∗!"!!
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
102
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
103
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.
104
a)
b)
105
c)
d)
106
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.
107
0.0226 m
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
108
0.013 m
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
109
0.011 m
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
110
0.245 m
Figure 6.8 – Location of thermocouples
111
(a)
(b)
112
(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)
113
(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
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
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
a)
b)
116
c)
d)
117
e)
f)
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
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
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
a)
121
b)
Figure 6. 17 – Flowcharts for incrementally calculating the interfacial heat transfer coefficient
based on: a) updating every iteration, or b) updating every increment
122
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.
123
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
124
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
125
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.
126
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. 100111, 2006.
[2]
B. Zhang, S. Cockcroft, D. Maijer, J. Zhu and A. Philion, "Casting Defects in LowPressure 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 Lowpressure 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
127
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,
128
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].
129
[38] "Kaman High Temperature Displacement System User Manual," [Online]. Available:
http://www.kamansensors.com/pdf_files/Kaman_High_Temperature_Sensors_manual_we
b.pdf. [Accessed 10 2013].
[39] Intertechnology, "Displacement System Calibration Record," 2013.
[40] "ABAQUS
6.11
Documentation,"
SIMULIA,
http://abaqus.ethz.ch:2080/v6.11/. [Accessed 10 2013].
[Online].
Available:
[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. 119126 .
[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.
130