Riser Feeding Evaluation Method for Metal Castings Using

Riser Feeding Evaluation Method for Metal Castings

Using Numerical Analysis

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Nadiah Ahmad

Graduate Program in Industrial and Systems Engineering

The Ohio State University

2015

Dissertation Committee:

Dr. Jerald Brevick, Advisor

Dr. Theodore Allen

Dr. Jose Castro

Nadiah Ahmad

2015

Abstract

One of the design aspects that continues to create a challenge for casting designers is the optimum design of casting feeders (risers). As liquid metal solidifies, the metal shrinks and forms cavities inside the casting. In order to avoid shrinkage cavities, risers are added to the casting shape to supply additional molten metal when shrinkage occurs during solidification. The shrinkage cavities in the casting are compensated by controlling the cooling rate to promote directional solidification. This control can be achieved by designing the casting such that the cooling begins at the sections that are farthest away from the risers and ends at the risers. Therefore, the risers will solidify last and feed the casting with the molten metal. As a result, the shrinkage cavities formed during solidification are in the risers which are later removed from the casting.

Since casting designers have to usually go through iterative processes of validating the casting designs which are very costly due to expensive simulation processes or manual trials and errors on actual casting processes, this study investigates more efficient methods that will help casting designers utilize their casting experiences systematically to develop good initial casting designs. The objective is to reduce the casting design method iterations; therefore, reducing the cost involved in that design

ii processes. The aim of this research aims at finding a method that can help casting designers design effective risers used in sand casting process of aluminum-silicon alloys by utilizing the analysis of solidification simulation.

The analysis focuses on studying the significance of pressure distribution of the liquid metal at the early stage of casting solidification, when heat transfer and convective fluid flow are taken into account in the solidification simulation. The mathematical model of casting solidification was solved using the finite volume method

(FVM). This study focuses to improve our understanding of the feeding behavior in aluminum-silicon alloys and the effective feeding by considering the pressure gradient distribution of the molten metal at casting dendrite coherency point. For this study, we will identify the relationship between feeding efficiency, shrinkage behavior and how the change in riser size affects the pressure gradient in the casting. This understanding will be used to help in the design of effective risers.

iii

Acknowledgments

All praises and thanks to Allah, the Almighty for His blessings that have allowed me to succeed in achieving my goal of obtaining a doctoral degree.

Special gratitude and profound appreciation to my advisor, Dr. Jerald Brevick for his invaluable insights, advices, encouragements, supports and understanding throughout my research work. I would also like to express my gratitude and great appreciation to my dissertation committee members, Dr. Theodore Allen and Dr. Jose

Castro for their advices, reviews and recommendations in helping me progressing well toward completing my dissertation.

I would like to sincerely thank the Government of Malaysia, through Universiti

Teknikal Malaysia Melaka and The Ministry of Education for providing scholarship opportunity to fund my graduate studies.

Finally, I am deeply grateful to my spouse, Radin Zaid Radin Umar and my family for their continuous supports and encouragements during my study. Special thanks as well for everyone who had helped me in any ways during my reseach conduct.

Vita

2002 – 2003 ...... American Degree Foundation Program,

Universiti Teknologi MARA, Malaysia

2003 – 2007 ...... B.S. Industrial Engineering,

University of Wisconsin – Madison, WI

2007 – 2009 ...... Tutor,

Universiti Teknikal Malaysia Melaka

2009 – 2010 ...... M.S. Industrial Systems Engineering,

The Ohio State University

2011 to 2015 ...... Graduate Teaching Associate, Department

of Integrated Systems Engineering,

The Ohio State University

Fields of Study

Major Field: Industrial and Systems Engineering.

Manufacturing Systems and Processes.

Casting Solidification Analysis and Optimization. v

Table of Content

Abstract ...... ii

Acknowledgments...... iv

Vita ...... v

List of Tables ...... ix

List of Figures ...... x

Chapter 1: Introduction ...... 1

1.1 Metal Casting Overview ...... 1

1.2 Research Problem ...... 5

1.3 Research Objective and Outline ...... 8

Chapter 2: Literature Review ...... 11

2.1 Riser Design Approaches ...... 12

2.1.1 Geometric Reasoning Method ...... 12

2.1.2 Numerical Modeling Techniques ...... 17

2.2 Solidification Modeling ...... 21

2.3 Feeding Mechanisms ...... 24

2.4 Summary ...... 27

Chapter 3: Numerical Modeling of Solidification ...... 31

3.1 Solidification Model ...... 32

3.1.1 Governing Equations ...... 33

3.1.1.1 Momentum Source and Sink ...... 35

3.1.1.2 Energy Source Term ...... 37

3.2 Boundary Conditions ...... 38

3.2.1 Temperature ...... 38

3.2.2 Velocity ...... 39

3.3 Finite Volume Method ...... 40

3.3.1 Solution Algorithm for Pressure and Velocity Calculation ...... 42

3.3.1.1 Predictor ...... 44

3.3.1.2 Corrector Step ...... 44

3.3.2 Numerical Schemes...... 46

3.3.3 Pressure Gradient Calculation...... 46

Chapter 4: The Analysis of the Casting Solidification Behavior Using Convective Fluid

Flow ...... 49

4.1 Casting Designs and Input Parameters ...... 52

4.2 Thermal Analysis and Experimental Results ...... 55

vii

4.3 Flow Analysis of the Casting Models ...... 59

4.4 Pressure Analysis ...... 74

4.5 Evaluation of Feeding Efficiency Using Convective Fluid Flow Analysis ...... 79

4.6 Sensitivity Analysis of the Constant C Value in the Solidification Simulation. .... 88

4.6 Evaluation of Feeding Effectiveness on Long Freezing Range Alloy: Application

to AlSi7Mg Casting ...... 93

Chapter 5: Summary ...... 105

5.1 Contributions ...... 106

5.2 Future Recommendations ...... 108

References ...... 110

Appendix A: OpenFOAM Programming Codes ...... 123

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List of Tables

Table 3.1. Numerical schemes used in OpenFOAM solver...... 46

Table 4.1. Riser designs used in simulations...... 53

Table 4.2. AlSI9Cu alloy elements’ composition...... 54

Table 4.3. Themophysical properties of AlSi9Cu used in the simulation...... 54

Table 4.4. Silica sand mold properties used...... 55

Table 4.5. Minimum and maximum velocities of the castings at coherency...... 68

Table 4.6. The modulus difference between the casting and riser and casting yield for each model...... 79

Table 4.7. The depreciation rate of the pressure gradient after coherency...... 87

Table 4.8. The depreciation rate of the pressure gradient after coherency with (a) constant C=103and (b) constant C = 107...... 92

Table 4.9. AlSi7Mg alloy elements' composition…………..…………………………………………….94

Table 4.10. Themophysical properties of AlSi7Mg used in the simulation...... 94

Table 4.11. Riser designs used in simulations...... 95

Table 4.12. The modulus difference between the casting and riser and the casting yield for each model...... 95

Table 4.13. The depreciation rate of the pressure gradient after coherency...... 96

List of Figures

Figure 1.1. Typical terminologies in sand casting (“Sand casting”, 2006)...... 3

Figure 1.2. Types of metal contractions in metal casting (Campbell, 2004 ) ...... 5

Figure 1.3. Classification of shrinkage defects (Stefanescu, 2005)...... 7

Figure 2.1. Illustration of feeding mechanism during solidification of alloy (adapted from

Dahle and St. John, 1998)...... 27

Figure 3.1: A generalized 2D control volume (Jana et al., 2007)...... 41

Figure 3.2: PIMPLE solution algorithm...... 43

Figure 4.1. Multi-steps casting geometry with a riser...... 53

Figure 4.2. Shrinkage porosities predictions using Niyama Criterion...... 57

Figure 4.3. Casting of (a) Model 1, (b) Model 2, (c) Model 3 and (d) Model 4...... 58

Figure 4.4. Velocity fields of Model 1 at (a) 10 s, (b) 20 s and (c) 31 s...... 60

Figure 4.5. Velocity fields of Model 2 at (a) 10 s and (b) 20 s...... 61

Figure 4.6. Velocity fields of Model 2 at (a) 30 s and (b) 40 s...... 62

Figure 4.7. Velocity fields of Model 3 at (a) 10 s and (b) 20 s...... 63

Figure 4.8. Velocity fields of Model 3 at (a) 30 s and (b) 50 s...... 64

Figure 4.9. Velocity fields of Model 4 at (a) 10 s and (b) 20 s...... 65

Figure 4.10. Velocity fields of Model 4 at (a) 30 s and (c) 60 s...... 66

Figure 4.11. Velocity fields at casting coherency for (a) Model 1 and (b) Model 2...... 69

Figure 4.12. Velocity fields at casting coherency for (a) Model 3 and (b) Model 4...... 70

Figure 4.13. Plot of temperature over time...... 71

Figure 4.14. Scaled velocity vectors when the casting sections reached coherency...... 73

Figure 4.15. Pressure distribution of Model 1 at (a) 1 s, (b) 5 s, (c) 10 s and (d) 30 s time intervals...... 76

Figure 4.16. Pressure gradient profiles of Model 1 at (a) 5 s, (b) 10 s, (c) 20 s and (d) 30 s.

...... 78

Figure 4.17. Pressure gradient distributions with liquidus and coherency temperature isotherms when all of area inside the casting sections had reached coherency temperature; (a) Model 1, (b) Model 2, (c) Model 3 and (d) Model 4...... 83

Figure 4.18. Plot of pressure gradient over time at a point between riser and casting. . 84

Figure 4.19. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting...... 85

Figure 4.20. Regression applied on the pressure gradient over time; (a) Model 3and (b)

Model 4 ...... 86

Figure 4.21. Plot of pressure gradient over time at a point between riser and casting with constant (a) C = 103 and (b) C = 107...... 90

Figure 4.22. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting with (a) constant C=103and (b) constant C = 107...... 91

Figure 4.23. Pressure gradient distribution when all of the casting section reached coherency for (a) Model 1, (b) Model 2, and (d) Model 3...... 97

Figure 4.24. Plot of pressure gradient over time at a point between riser and casting. . 98

Figure 4.25. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting...... 99

Figure 4.26. Regression applied on the pressure gradient over time; (a) Model 2 and (b)

Model 3 ...... 100

Figure 4.27. (a) Model 2 and (b) Model 3 AlSI7Mg castings...... 102

Figure 4.28. Plot of temperature over time...... 103

Figure 4.29. Framework for riser design and optimization...... 104

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Chapter 1

Introduction

Designing a casting system which minimizes defects from occurring in a metal casting, such as shrinkage defects, has been a subject of active research in industry and academia. One of the design aspects that remains a challenge for casting designers is the efficient design of casting risers. The purpose of this chapter is to give an overview about metal casting and its challenges, as well as the objectives of this study pertaining to casting riser design.

1.1 Metal Casting Overview

Metal casting is among the oldest methods of shaping materials, dating back to approximately 4000 B.C when copper was cast for making various objects. The metal casting process involves pouring or injecting molten metal into a mold cavity, which is in the shape of the part to be manufactured. The molten metal is then allowed to cool and solidify, before the metal can be removed from the mold. Due to its versatility and

1 economic nature, metal casting has been one of the important near net-shape manufacturing technologies.

To date, there have been several casting techniques developed, each with its own characteristics and applications. The traditional method of metal casting is the sand casting. This method is still important, as more than 70% of metal casting is performed using sand casting (Degarmo et al., 2003). Sand casting, which uses sand as the mold material, is also the least expensive method as compared to other casting techniques.

According to Kalpakjian (1995), approximately 15 million tons of metals are cast annually by this technique in the United States alone.

In sand casting processes, a pattern is used to form a mold cavity in the sand mixture contained in a flask (See figure 1.1). The pattern is usually made of wood, foam, plastic, wax or metal; and the sand is packed around the pattern to make the mold. For a simple casting design, a single solid pattern is use. As for a more complex design, a split pattern is used which consists of the upper part, called cope and the lower part, called drag. The core is an insert placed in the mold to create holes or interior surface of the casting. Figure 1.1 shows an example of a pattern shape and the terminologies used in sand casting. As molten metal is poured into the pouring cup, it will travel through a gating system which control the flow of the metal, down the sprue. A runner is a channel that allows the molten metal to flow into the casting mold cavity. The molten metal will first reach the gate that acts as inlet into the mold cavity.

The sand casting method that uses a polystyrene foam to make the pattern is called lost-foam casting (LFC) method, or evaporative-pattern casting (EPC) method, since the foam will evaporate as the molten metal is poured into the foam-filled mold.

The advantages of LFC method are that, it requires no cores and it only requires a single solid pattern as the foam is easy to manipulate even for complex geometries. The LFC method differs from the conventional sand casting in that, it uses unbonded dry sand for molding, making it easier to maintain than the bonded sand molding; and eliminates the damaging effect of the steam from the bonded sand. LFC method has gained its popularity within the last four decades especially in making the aluminum-based automative parts (Wang et al., 1993).

Figure 1.1. Typical terminologies in sand casting (“Sand casting”, 2006).

When a part design is received from a client or design engineer, the tool builder will start designing the sand casting tooling such as pattern, cores and etc. Designing and building casting tooling requires a deep knowledge in casting processes and foundry operations. There are many considerations that need to be taken into account. These considerations include on how to achieve the specified shape of the casting, how to flow the molten metal and feed during solidification of the metal, as well as the shrinkage of the casting during solidification and cooling to room temperature.

Figure 1.2 displays three types of contractions or shrinkage that occur in metal casting. Liquid contraction occurs as the liquid metal temperature decrease from the pouring temperature to liquidus temperature which signifies the start of the metal solidification. The solidification contraction begins at the liquidus temperature until the metal reaches solidus temperature at which the metal solidification ends. As the metal temperature decreases further to room temperature, solid contraction occurs.

Figure 1.2. Types of metal contractions in metal casting (Campbell, 2004 )

1.2 Research Problem

In metal casting, one primary issue that arises is the formation of shrinkage defects during solidification. As liquid metal solidifies, the metal shrinks and forms cavities inside the casting. The cavities are normally formed at the last area where the liquid melt solidifies. When designing a casting, it is important to design the casting

5 system that allows feed metal to compensate for the shrinkage of the casting during its solidification.

Shrinkage defects can be classfied into two types: open shrinkage defects and closed shrinkage defects. Open shrinkage defects that are exposed to the atmosphere, are caused by metal contraction while cooling in liquid state and during solidification.

Closed shrinkage defects occur inside the casting and are also called porosities. The shrinkage porosities correspond to contraction and nucleation of pores in the mushy region. They can be in the form of macroporosities and microporosities. Figure 1.3 illustrates the classification of the various types of shrinkage defects explained above.

To overcome the defects caused by solidification shrinkage, risers are usually employed along with chills or paddings, if necessary. Risers, which play a key role in feeding system, are added to the casting shapes to supply additional metal when shrinkage occurs during solidification. Uninterrupted flow of molten metal to feed the solidifying regions can produce sound castings. The shrinkage in the casting is compensated by controlling the cooling rate to promote directional solidification

(Kalpakjian, 1995). This control can be achieved by the mold and casting design such that the cooling begins at the sections that are furthest away from the risers and end at the risers. Therefore, the risers will solidify last and feed the molten metal to the region with developing casting cavities. As a result, the shrinkage cavities formed during solidification are in the risers which later are cut off from the casting (Johns, 1987).

Figure 1.3. Classification of shrinkage defects (Stefanescu, 2005).

In order to prevent shrinkage defects, there have been studies in literatures for effective feeding. These studies also involve the determination of feeding distance of a riser. Feeding distance indicates how far in a casting a riser can supply compensating liquid melt. Effective risers are able to provide enough continuous feed flow into the castings without causing waste in material due to their excessive sizes. If the risers are too small, they will not be able to provide uninterrupted feed flow into the castings. The design of effective risers is very important to reduce the production cost, especially for low volume production, since repetitive casting trials will increase the casting cost. This cost includes the metal cost and the removal cost of the risers. Riser effectiveness can be influenced by its dimension, shape, location on casting, and connection type to the casting (Lewis, 1983). The designers commonly determine the riser designs based on

7 knowledge and experiences in casting gained through trial-and-error procedures and some fundamental feeding rules.

There are also several software packages that can simulate and evaluate the casting design characteristics, providing inputs on design improvements to achieve the best casting results. Therefore, time and cost spent on the design processes can be reduced as compared to when they are done manually. This software can predict the filling and solidification processes in order to identify potential defects that could occur in the casting (Tan et al., 1998). The designers will therefore, redesign and reevaluate the casting designs. Even though current technology allows the identification of the potential shrinkage areas such as through the simulation of the hot spot formation, there is still difficulty in assuring that the riser placements will provide uninterrupted feeding flows, and will reach the targeted hot spot locations due to unknown limited feeding distance. Consequently, the design engineer can still spend considerable time to develop a satisfactory design. The aim of this research is therefore to address this issue, and find design methods that can assist casting engineers in designing good castings with minimized design iterations.

1.3 Research Objective and Outline

Since casting simulation has become an important tool in the design of castings, there is need of more efficient methods that exploit the results from the simulation.

The objective of this study is to use the heat transfer and convective fluid flow simulation of casting solidification to evaluate the design of the risers. Most of previous

8 research related to the design of risers has only utilized the heat transfer models. The current study seeks to show the advantages and importance of convective fluid flow model in providing solutions for designing effective risers. This is based on the hypothesis that the numerical heat transfer and convective fluid flow simulation results together can provide a more accurate prediction of riser effectiveness. Simulating the convective fluid flow has been known to be computationally expensive. For that reason, current study also attemps to investigate if the results at the early stage of the solidification can lead to identification of feeding problems in hope that, full solidification simulation involving convective fluid flow can be avoided. The mathematical models of casting solidification will be solved using the finite volume method (FVM).

Feeding has been known to take place in the direction of the maximum temperature gradient along the interface between the solid and liquid metal during solidification (Sutaria and Ravi, 2012). In this study, we aim to use the simulation of casting solidification to analyze the effect of riser characteristics on feeding effectiveness using the analysis of pressure distribution inside the casting; as we know that the liquid flows in the path of positive pressure gradient (Campbell, 2004).

We intend that this study will improve our understanding of the feeding behavior in alloys for effective feeding system design. For this study, we will discover the relationship between the pressure distribution, shrinkage behavior and how the change in riser size affects the pressure distribution, by taking into account the development of dendrite coherency during the casting solidification. This understanding will be used to

9 improve the design method of effective feeding. The experimental study for validation was conducted using the LFC sand casting process for aluminum-silicon alloys.

Chapter 2 of this thesis will describe the literature review related to this study. An overview related to riser calculation methods and studies pertaining to feeding distance will be presented in this chapter. Later, the chapter provides an overview of studies on casting optimization methods available in the literature. This chapter will also cover the studies on solidification modeling and simulation approaches as well as an overview of casting feeding mechanisms.

Chapter 3 contains the description of the casting solidification models used in this study. The governing conservation equations and the sources terms used are mentioned in this chapter. The finite volume method used to solve the discretized equations is also explained. The summary of the pressure-velocity coupling solution algorithm is shown and was implemented in OpenFOAM software. Chapter 4 presents the analysis of the casting solidification simulation. The analysis of the convective fluid flow and pressure gradient is described for AlSi9Cu, using a “step” casting design. A method to evaluate the effectiveness of riser is then explained. Further application to AlSi7Mg alloys is also presented. Chapter 5 provides the summary and future recommendations of this research.

Chapter 2

Literature Review

There are various factors that influence the failure of risers in feeding during solidification to make sound castings. Among the main causes are the riser sizes being too small in volume or too low in height, improper riser locations and isolated hotspots in the castings which are not connected to the riser feeding channels. In addition, the failure is caused by the feeding channels being too narrow or too far from the casting sections as influenced by the shapes and sizes of the castings (Zhou, 2005). The following sections review some of the techniques available in the literature to design effective risers which act as additional metal reservoirs to compensate the metal shrinkage in order to produce sound castings or castings that are free from shrinkage defects.

There has been research that developed and documented casting design principles, guidelines and simple rules (Campbell, 1991-2004; Bralla, 1998; Upadhya and

Paul, 2005). These guidelines and rules are used by engineers to design good castings

11 which include series of decisions made to achieve defect-free castings at low cost. The decision making includes the riser designs (number, size, shape, and location), the interface conditions, and mold conditions. Ransing and Sood (2005) published a comprehensive review paper on optimization techniques in casting process. There are two streams of research exist in the work of optimization of casting process. They are geometric reasoning techniques and numerical modeling methods. In the context of riser design optimization, the objective is to create strategies to determine shape and process parameters that produce shrinkage-defects-free castings while maximizing casting yield. Casting yield is the ratio of the casting volume to the volume of both the casting and riser.

2.1 Riser Design Approaches

2.1.1 Geometric Reasoning Method

There are two main factors that determine the requirements for risers in order to obtain a sound casting. The first one is the thermal properties which require that the risers must solidify after the casting has solidified. The second factor is the solidification characteristics that the risers must be able to provide enough feed metal to avoid shrinkage cavity or porosity being formed during the solidification (Flinn, 1963;

Merchant, 1959). The most common rule used to evaluate riser design is based on the

Chvorinov’s rule which gives the solidification time for any mass of molten metal. The earliest study by Chvorinov (1940) stated that the time for the metal to remain in a liquid form is proportional to the squared of the ratio of the metal volume (V) to its

12 surface area (A). According to Chvorinov’s rule, the solidification time of the riser must be greater than the solidification time of the casting.

Chvorinov’s rule leads to the following relation:

(2.1)

The ratio of is then called a freezing ratio where and represent the riser and casting respectively. Gaindhar et al. (1988) used nonlinear optimization to find the optimum dimension, i.e., the riser volume for various shapes of risers. By applying Chvorinov’s Rule, the problem is solved as geometric programming problem.

Caine (1952) introduced riser calculation method that is based on the

Chvorinov’s rule. Caine based his method on indirect use of V/A calculation. Caine established charts based on empirical studies that relate the V/A ratio of casting to V/A ratio of riser. From the chart, the limit for shrinkage and soundness for various casting shapes and feeding methods can be obtained. The following is the relationship found by

Caine (1952):

(2.2) where

solidification rate relative measure

Caine method works for simple and moderately complex shapes. This method later was simplified by Bishop et al. (1955) through the use of a shape factor (SF). This shape factor replaces the tedious calculation of the freezing ratio in the Caine’s curve.

The study showed that the V/A ratio can simply be replaced with a simple relationship between the length (L), width (W) and thickness (H). This relationship is SF= (L+W)/H.

The Bishop method was proven in practice to provide approximately similar riser dimensions for simple shapes and also was demonstrated to more complex shapes

(Flinn, 1963). Both Caine and Bishop et al. relationships however are only valid when the ratio of the riser’s height to its diameter falls between 0.5 and 1 and were studied on steel alloy.

Merchant (1959) did further analysis on the relationship studied in Caine and

Bishop et al. and obtain a linear relationship as follows:

(2.3)

14 where m and c ‘ are constants for slope and intercept respectively. For steel, these values are m=0.081 and c’=0.041.

Ruddle as cited in (Lewis, 1983) utilized the Chvorinov relationship and defined a casting modulus (M) as follows:

(2.4)

Therefore, as the modulus increases, the time to solidify also increases. Through empirical study, he showed that feed metal will be sufficient if the modulus of the riser is greater than that of the casting by twenty percent. This modulus principle is used to check any given dimension of riser if it meets the basic requirements. It however, does not give optimal dimension but requires various possible dimensions to be validated.

Many optimization techniques of risers in literature are based on modulus principle such as seen in Ravi (2009), Zhao and Jing (2003) and Jacob et al. (2004).

So far the use of modulus principle has a limitation in that, it only accounts for the solidification time requirement which is not sufficient to ensure sound castings. In addition, casting shapes having the same modulus had been shown to have different solidification characteristics (Hansen and Sahm, 1988). Casting geometries can affect thermal gradients inside the castings. Even changing the process parameters such as the mold material and pouring temperature can show unpredictable results on the solidification time. Besides, solidification mode also differs with alloys. Aluminum alloys

15 for example, are less likely to form a sturdy solid outer shell especially for alloys having wide freezing range and are more quickly filled with dendrites (Sigworth and Wang,

1993). Finding reliable relationship between process parameters and casting shapes is consequently difficult.

There are studies that utilize feeding distance as a measure to evaluate how well well a riser can feed the casting during the solidification (“feedability”). Feeding distance (FD) is the maximum distance that a riser can feed a metal such that, within this distance the casting section is relatively free of shrinkage defect (SFSA, 2001). The distance from the edge of the riser to the farthest point in the casting section that the riser feeds, give the measure for the feeding distance. The most detailed work on feeding distance was first done by Naval Research Labaratory on steel castings (Bishop et al.,1950-51). The rules developed for feeding distance are based on the empirical casting trials on simple shapes mainly plates, bars and cylinders.

Feeding distance is geometrically influenced by the section thickness which affects the cooling rate of the section (SFSA, 2001). Consequently, the feeding distances established for simple shapes (plate, bar and cylinder) are commonly a function of casting section thickness. The shape of the casting will also influence the temperature gradient. This temperature change per unit length during solidification controls the feeding path of the feed metal and therefore, influences the feeding distance of the casting. The use of chill added on the casting increases the temperature gradient which as a result, increases the feeding distance. On that accounts, different shapes produce

16 different feeding distances. These studies were based on the theory that when the solidification gradient toward the end of the freezing drops below some minimum value, shrinkage cavities would form in that location of the casting (Ou et al., 2005).

More recent studies on feeding distance are from Ou et. al on alloy steel and more extensive literature review on feeding distance can be found in the their work (Ou et al,2002-05). Ou et al. developed the feeding rules that are less conservative than the rules developed by previous studies. The rules were developed using numerical heat transfer simulations for both high and low alloy steel grades. In their studies, the feeding distance was determined after taking into account factors that can affect feeding distance such as sand mold material, alloy composition, superheat and the use of chills. The feeding distance for a top-risered casting is calculated as follow:

(2.5)

Nonetheless, the above equation is only accurate for up to W/H=15. Beyond that ratio, FD/H ratio will have a constant value of 9.0. Most studies in feeding distance however, are limited to steel casting and simple shapes.

2.1.2 Numerical Modeling Techniques

There has also been considerable numerical research in optimizing the casting process design. However, the optimization particularly in the field of riser design, has received less attention in spite of its significance (Campbell, 1991). Most of these works obtain the optimal riser design, i.e., the riser design that resulted in a sound casting, 17 systematically through numerical modeling and simulation. Hence, the determination of the last region to solidify, called hotspot, and the ability of the riser to feed this region laid the foundation for the traditional riser design techniques. In designing the riser, the hotspot is required to be in the riser, in order to ensure the riser remains molten to feed the casting.

Numerical simulation of casting solidification is one of the techniques used to quantify and predict the behavior of the solidifying metal. The results of the simulation such as the temperature distribution throughout the casting, will be used to facilitate feeding evaluation and optimization including the prediction of the shrinkage locations.

One of the advantages of numerical modeling is that, it allows how the changing of the process parameters affects the solidification characteristics, be known. This method is timely more expensive but usually results in a more accurate prediction of solidification pattern. Numerical method involves an approximation to the mathematical analysis of the solidification problem. Consequently, its application can encompass more practical problems as compared to the exact mathematical analysis which only applicable to the most simple solidification problems. In foundries, numerical simulation can be used to evaluate the effectiveness of a casting design including the riser design.

There are several works that integrate numerical optimization methods

(Morthland et al., 1995; Tortorelli et al., 1994). The riser design was formulated as a shape optimization problem. The designer provides the initial design and the program will evaluate the design to supply information to the numerical optimization. Lewis

(1983) formulated the problem with the objective to minimize the cost associated with

18 the riser design by selecting the best design from various sets of risers and chills design combinations provided by the casting engineer. For each combination chosen, the algorithm determines the optimal riser dimension and position so as to obtain a shrinkage-defect-free casting. The method uses modified Benders’ Decomposition to solve the problem and solidification simulation to evaluate the chosen design.

Nevertheless, the method required extensive amount of CPU time in order to gain adequate simulation iterations.

Morthland et al. (1995) and Tortorelli et al. (1994) used gradient based minimization method with sensitivity analysis for the solidification parameters. The objective function is formulated to be the riser volume. While the previous works depend on the initial riser design from the engineer, Tavakoli and Davami (2008) developed a method that automate the riser design. In this method, the designer only provides the casting geometry without a riser, as an input. The riser design will be initialized by the program and optimized using shape and topology optimization method. This method designs the biggest riser as its initial design and presented the smallest ones during its iterations. This method however, does not work well when the casting requires multiple risers with similar size or for more complex castings.

Tavakoli and Davami (2008) formulated the feeding design as volume constrained topology optimization and is solved with the finite element analysis. In

(Ransing and Sood, 2005) they added finite difference analysis of the solidification process with evolutionary topology optimization and claimed their method to be

19 efficient, easy to implement and define the initial design. In (Tavakoli and Davami,

2007), they formulated the problem as additive evolutionary topology optimization method using a riser growth technique, in which the dimension of the riser iteratively grows until the desired result is achieved.

There has been limited integrated CAD system developed for designing risers.

Liao et al. (2011) integrated 3D models to simulate castings and design risers with the use of CAD system. The authors used perimetrischen quotient model that is based on modulus method to determine the dimension of the risers. The method was verified on a simple casting design and showed good casting results. However, the performance of the system on a more complex casting is not known.

Jian and Tao (2003) developed an integrated CAD/CAE system for casting process using concurrent engineering method that uses input from CAD module to design the casting system. The riser design is determined using the modulus method. The casting geometry is split into segments and for each segment, the riser modulus and riser neck modulus must be equal to the transfer modulus. Jacob et al. (2004) used CAD and genetic algorithm for determining the riser design. CAD platform is used to divide casting into segments and calculate the casting modulus. The inputs were feeded into the genetic algorithm which finds the riser dimensions that optimize the yield for each segment.

The key to practical simulation package is to determine the most important factors in designing the casting process (Ravi, 2008). Ravi (2009) developed computer-

20 aided casting design and simulation package called AutoCast that can help design the riser dimensions based on the modulus principle after the user has determined the location of the risers. Kor etc. (2009) integrated numerical optimization technique with a commercial simulation software, MAGMASOFT. The method uses genetic algorithm to solve multi-objective optimization problem for finding optimum gating and riser design.

Baha’l (2004) studied the use of simulation package called FLUENT that is based on computational fluid dynamic in designing good risers. Various designs were simulated and the solidification simulation results provide input for the riser design.

2.2 Solidification Modeling

There are two common categories of methods for solving the solidification problem involving convection. The first one is based on a multi-phase model while the second group is based on a single-phase approach or the mixture model. In the multi- phase or multi-domain approach, the momentum and energy equations are solved separately for each domain. As a consequence, a continuous update of the interface position that defines the domain regions is required at each time iteration. See review in

Samarskii (1993).

In the single-phase approach or also called a homogenous model, a set of momentum and energy equations are solved for the entire physical domain.

Consequently, the interface position is not explicitly computed and the energy balance is automatically conserved at the interface. This model is based on the fixed-grid approach, one of its special cases is called enthalpy-porosity formulation or porous

21 media formulation. This model is widely used for phase-change problem but is usually used when the thermophysical properties of different domains are assumed equal

(Belhamadia et al.,2012). In the enthalpy-porosity model, the total enthalpy is solved as the dependent variable for the energy equation instead of the temperature. The sensitive enthalpy h varies by temperature T and is handled using the relationship of

The specific heat is assumed constant.

In molten alloy or impure substance, the freezing takes place over a range of temperature and therefore, is called non-isothermal phase change. This temperature range produces a mushy region that must be known its transport properties in modeling. Voller and Prakash (1987) were the first to develop enthalpy-porosity models that deal with the mushy region. The mushy region in the molten metal is simulated by using the Darcy’s source term in the momentum equation. This approach uses the assumption that the thermal properties are constant with temperature and phase. In

Voller et al. (1989), a set of assumptions are identified that allow this approach applicable to a binary alloy system. The release of the latent heat during the phase change is accounted for using the source term in the energy equation.

Belhamadia et al. (2012) developed a modified enthalpy porosity model that takes into account, the different properties of thermophysical data related to each phase or domain present in the melt. Vakhrushev (2010) includes the turbulence effect into the model through the use of effective viscosity and effective conductivity in the momentum and energy equation respectively.

There are studies that solve the temperature instead of the total enthalpy in the energy equation. This approach is classified as the temperature-based fixed-grid method and uses a specific way of handling the heat coefficient or the source term in the energy equations. See also Beckermann and Viskanta (1988) and Voller and Prakash (1989).

Zeng and Faghri (1994) used this method to simulate the mushy region by combining an effective heat coefficient and heat source term handling in the energy equation to represent the evolution of the latent heat of fusion during the phase change.

There are three typical numerical methods for solving the solidification problem involving phase transformation from liquid to solid. There are fixed grid, variable grid and transformed grid methods (Hong, 2004). In the fixed grid method, the grids of the computational domain are uniform and orthogonal. The grids also remain unchanged throughout the calculation iterations. The evolution of the latent heat is incorporated through a suitable source term in the governing equation. The location of the moving interface between solid and liquid is not explicitly known but can be interpolated through the calculated temperature distributions.

On the other hand, for the variable grid method, one of the grids for the space and time domains is not fixed but to be determined so that the moving interface always remains at a grid point. This method requires energy balance equation at the interface to account for the release of the latent heat of fusion and therefore the interface position can directly be known. The front tracking method is one such technique that is based on the variable grid method.

The transformed grid method is an adaptive grid method that can be used for problems with complex geometries, in which the irregular region is mapped into regular region in the computational domain. The grids are generated for every time iteration causing a considerable use of the computational time. Fixed grid method is among the most commonly used techniques to solve phase change problem in solidification of alloys due to the occurrence of mushy zone, which causes other methods used to solve a single discreet phase change are not applicable. Current study employed a fixed frid method using the mixture model to simulate the solidification process.

2.3 Feeding Mechanisms

There are five distinct feeding mechanisms exist during alloy solidification as suggested by Campbell (1969) and further rheologically explained by Gourlay and Dahle

(2005). As solidification begins simultaneously in the castings and risers, the liquid flows from the risers to feed the contraction in the castings. The feeding flow at the beginning of the solidification is called liquid feeding through viscous flow mechanism. The driving forces involved in the liquid feeding are ones due to the volume change as a result of temperature changes, and pressure differential induced by the solidification shrinkage.

However, the pressure difference during the liquid feeding is often considered negligible

(ASM Handbook, 2008).

Shrinkage induced flow occurs when the nuclei started to form in the melt and this flow is called mass feeding. Mass feeding dominates from the liquidus temperature,

to coherency temperature during which the material behaves as liquid or slurry solid

24 grains. The material exerts little resistance to feeding during this stage as the dendrites are independent and the pressure difference required for mass feeding is also small.

Mass feeding diminishes at packing temperature, at which the network of dendrites is well packed.

The next stage of feeding is termed interdendritic feeding during which winding channels have formed between dendrites after the dendrites impose on one another.

The temperature at which this dendrite mesh formed is termed coherency temperature,

. The size and morphology of dendrites can influence . Materials that develop larger grains or irregular shapes tend to reach coherency earlier. At this point the

coherency solid fraction, begin to exchange stresses and interact mechanically.

Some liquid can flow through these channels to feed the shrinkage occurred towards the end of the mushy zone through interdendritic feeding.

At solid fraction slightly higher than , dendrites can slide, roll and reorient as a result of network collapse for grain rearrangement (Metz and Flemmings, 1970).

Through movements of dendrites in reduced scale, mass feeding can continue to occur

after coherency at solid fraction slightly higher than . However, as primary dendrite arms growth decreases and secondary dendrite arm growth dominates, the dendrite network formed tend to resist fluid flow thus can result in inadequate feeding. As the solid fraction increases, the pressure gradient required for interdendritic flow increases causing amplified difficulty in interdendritic feeding. Isolated mushy zones surrounded

25 by solids can also form when the fraction liquid is so small hence, promoting shrinkage porosities.

Burst feeding occurs as a result of dendritic network collapse by the reason of increased stresses in solids to the extent exceeding the network strength. Burst feeding can occur at both macroscopic and microscopic levels. At macroscopic burst feeding, the region of lower solid fraction can block off the region of higher solid fraction creating a barrier to interdendritic flow. The barrier breaks down when the pressure in the liquid is reduced such that it causes the stress in solid fraction to exceed the network strength.

At microscopic burst feeding, through similar mechanism, the ruptured dendrite arms feed the isolated liquid region between the dendrite arms.

As a liquid region is further isolated from the supply of feed melt, the mush can deform inward to release the hydrostatic tension as the isolated liquid continues to freeze. This is called solid feeding as high solid fraction mush becomes viscoplastic. After all solid have formed at solidus temperature, , solid contraction starts to occur. Figure

2.1 illustrates the five feeding mechanism as described above.

Figure 2.1. Illustration of feeding mechanism during solidification of alloy (adapted from Dahle and St. John, 1998).

2.4 Summary

Even though some works have been done to help build an optimum casting design that is free from shrinkage defect, most of these proposed methods still require the designer to provide an initial design for evaluation and optimization. In order to save computational cost and reduce the time to design the casting, it is important that the designer develops a good initial design (Jensen et al., 1995). Moreover, the most common numerical simulations of casting solidification from above studies are performed through heat transfer calculations using numerical approaches such as finite

27 difference method (FDM), finite element method (FEM) and finite volume method

(FVM).

According to Pehlke (2002), the earliest study found to solve the heat transfer problem of casting solidification using computer simulation is by a German researcher in

1962 utilizing the FDM technique. The heat transfer problem for solidification is normally referred to as phase change or moving boundary problem. Earlier studies focus on solving the heat transfer caused by heat conduction without regard to important convection factors in solidification. Campbell (1998) warned that the results of such models should be dealt vigilantly. Several review literatures on solidification modeling can be found in Voller et al. (1990), Campbell (1991), Pehkle (2002), and

Vijayaram et al. (2006).

In metal solidification, fluid flow also plays a major role which creates the effect of heat and mass convection. Diffusion in addition, naturally occurs alongside convection and therefore these two terms are often handled together. Natural convection which occurs due to the temperature difference in the liquid phase plays a role by slowing down the solidification process but accelerate the melting process.

Therefore, consideration of convective fluid flow effect requires the coupling of energy equation and the Navier-stokes equation. Studies considering the effect of natural convection in phase change problems started to appear among researchers towards late

1970s. Zhang et al. (1997) conducted a study on the effect of natural convection during solidification based on heat transfer around a horizontal tube.

Fluid flow due to natural convection in casting predominantly occurs within part of the melt with a higher temperature due to non-equilibrium conditions that prevail during solidification (Prabhakar, 1993). Consequently, natural convection is more significant in the riser section which is designed to solidify later than the casting.

Convection within the melt influences solidification at both microscopic and macroscopic levels. It can change the shapes of isotherms and reduces thermal gradients in the melt. Convection hence causes the top regions to solidify slower than the bottom regions. As convection promotes mixing, it is accountable for nonuniform redistribution of alloy constituents and the grain sizes, orientation and distribution

(Prescott and Incropera, 1996). Many castings problems can be explained by convection as it can results in poor prediction of temperature distribution and feeding effectiveness

(Campbell, 1998).

Most studies that include fluid flow focus on predicting the location of the shrinkage porosities of a casting thus, requires extensive computational cost. Fluid flow analysis in commercial casting software however, only available for mold filling simulation due to its expensive computational cost. Current study focuses on evaluating riser designs using both the heat transfer and convective fluid flow analysis, by taking into account the convection and diffusion factors of the molten metal without the need to run full solidification simulation of a casting. This model is useful to analyze the feeding behavior as the casting developes dendrite coherency, since the metal still predominantly behaves as liquid. The effectiveness of the riser to produce a sound casting was evaluated by running the simulation for only the initial stage of the

29 solidification until the casting developed dendrite coherency. Consequently, the computational cost related to the convective fluid flow analysis can be reduced.

This study employed a new approach to evaluate the effectiveness of the riser design, which is particularly based on the analysis of the pressure gradient characteristic. Due to unavailability of the traditional sand casting system, the results of current study were compared to experimental castings using the LFC casting method. As most studies on risering techniques were validated using traditional sand casting system on steel alloys, this study provides insight into effective feeding applicable to the relatively new LFC casting method using aluminum alloys.

Chapter 3

Numerical Modeling of Solidification

In this chapter, a three dimensional solidification model used in this study for metal casting involving natural convection will be described. Fluid flow caused by convection is an important phenomenon in many casting processes as it will influence the shape of the solid-liquid interface and the temperature distribution in the metal casting. The following section will describe the general governing equations of heat, momentum and mass conservations for the casting problem. These mathematical models are based on the mixture model as discussed previously in chapter 2. Section 3.2 illustrates the numerical method of finite volume approach to solve the governing equations. The finite volume discretization of the governing equations is shown based on a fixed grid solution method. Section 3.3 will discuss the specific treatment to the latent heat of solidification and the source terms used in the governing equations. The model of this study is implemented for the casting solidification process in OpenFOAM adapted from meltFoam solver established by Rosler and Br ggemenn (2011).

3.1 Solidification Model

In this dissertation study, the solidification modeling of liquid metal for casting process involves two phases, i.e. liquid and solid phases. This study employed the mixture-based model and therefore, the two phases are implicitly captured in the formulation of the heat and mass conservation equations (Rosler and Br ggemenn,

2011, Brent et al., 1988). The equations solved will regard the liquid and solid phases as one single phase. In literature this model is called the single-phase formulation. The gas phase was not considered in this study even though it can present in some castings. The assumptions for the model used are as follows:

a) The flow in the liquid is laminar.

b) The liquid metal is Newtonian ,i.e., the shear stress is directly proportional to the

deformation rate of the fluid.

c) The fluid metal is incompressible. The variation in density within the flow is

neglected.

d) Boussinesq approximation is valid for modeling the buoyancy effects, i.e. the

fluid density is assumed constant (temperature-independent). In most fluid, the

density varies due to the temperature difference and this variation in density

creates buoyancy force that influences the flow motion in liquid. Boussinesq

model assumes the variation in densities with temperature is negligible.

e) A continuous error function approach approximates the liquid fraction as a

function of temperature through a direct substitution into the energy

conservation equation.

f) The mold is instantaneously filled with molten metal with a given pouring

temperature and the liquid is initially stagnant once pouring is complete.

g) For simplification, the thermophysical properties of the metal are assumed

constant and equal in the liquid and solid regions.

h) The pressure values calculated by the simulation program were not absolute but

relative values.

3.1.1 Governing Equations

The employed single phase model is based on the following continuum equations of mass, momentum and energy conservation. The mass conservation can be written as

(3.1) where V is the velocity vector.

The conservation law of momentum dictates that the difference between the rates of momentum in and out of the control volume must equal to the rate of momentum accumulation in that control volume. In other words, the change in momentum must equals to the sum of all active forces. The forces include the shear stress, pressure variations and body force due to gravitational acceleration. The momentum equation is described as follows:

(3.2)

P is pressure, is the density and represents the momentum source term. The second term on the left side of the equation is known as the convective term while the third term on the left side of the equation is called the diffusion or viscous term.

The concept of energy conservation is based on the first law of thermodynamic.

The law of energy conservation is expressed as the heat added to the control volume minus the heat released from the volume must equal to the energy increase in the volume. The energy conservation equation that will calculate the temperature and solid fraction is

(3.3)

In the above equation, is the specific heat, k is thermal conductivity, and is the energy source term. The momentum and energy source terms will be discussed further in subsections 3.1.2 and 3.1.3 respectively.

In the mold, phase change does not occur and conduction is the only heat transfer mechanism considered. Therefore, the energy conservation in the mold is as follows:

(3.4)

3.1.1.1 Momentum Source and Sink

The momentum source consists two terms as described follows

(3.5) where

(3.6)

The term is the Darcy-type source term used to model the effect of solidification on the momentum, which describes the flow in the mushy region as a function of the liquid fraction of the solidifying metal. K is a Carman-Kozeny term that denotes the permeability of solid structure formed during solidification which embodies the resistance of the solidifying structure to the feeding flow and is also called the porosity function (Miehe, 2014). is a constant with a large value. The source term is derived on the assumption that the solid and the mushy region behaves like a porous medium allowing the liquid to flow through (Voller et al., 1990). In casting solidification,

Darcy flow model is important when the fraction solid present in the mush is significant to form dendrite coherency (rigid dendritic network or structure). It assumes the solid phase does not move and is fixed to the numerical grid. In the early stage of

35 solidification, this term becomes insignificant as solidification is governed by eiquiaxed or eutectic solidification.

When in liquid, the Darcy term turns zero as the liquid fraction, is unity and causes the velocity to be determined by other terms of the momentum equation. In other words, the momentum equation represents the actual fluid velocity. On the contrary, as the volume turns to solid, the term becomes zero and causes the Darcy term to act as a prevailing momentum sink in the solid. This as a result, forces the velocity to be zero. In order to prevent division by zero, a small constant, is used in the denominator.

In current study, since the permeability is difficult to measure, we assumed that the constant C that is a characteristic of the mushy region morphology is similar for aluminum alloys. This assumption allows us to quantitatively evaluate how the thermophysical properties of the alloys affect the feeding behavior of the riser.

The fraction of liquid, is computed based on the following equation

(3.7)

where is the melting temperature determined by the arithmetic mean between and (Rösler and Brüggemann, 2011). and are the liquidus and solidus temperature respectively.

The second term, is the buoyancy source term based on the Boussinesq model for approximating the natural convection in the fluid. This term represents the buoyancy force that occurs from the density difference due to the temperature variation in the solidifying metal. The buoyancy force is approximated by the Boussinesq approximation when the density of the fluid is treated as constant except in the gravitational term. The density difference, is estimated as follows:

(3.8)

The buoyancy force is in the direction of the gravity vector denoted by which is in the positive -direction. is the volumetric thermal expansion coefficient and is the liquidus temperature. The for the molten metal was determined using the following equation

(3.9) where is the percent change of the densities between the densities at the solidus and liquidus temperatures (Nareder et al., 2013).

3.1.1.2 Energy Source Term

The latent heat of fusion which is the change in enthalpy due to the phase change from a liquid to a solid, is taken into account using the energy source term, .

During the solidification between the liquidus and the solidus temperatures, the latent heat is released. The source term used is described as

(3.10)

H is the latent heat of fusion. The derivation of the above source term can be seen in

Rösler and Brüggemann (2011). The source term is derived by directly substituting the continuous liquid fraction function decribed previously into the energy conservation equation. According to Rösler and Brüggemann (2011), iterative solution procedure to update the liquid fraction is necessary when coupling the liquid fraction function with energy conservation equation. The continuous entalphy function unnecessitates updating the solution to the energy conservation equation in every iteration.

3.2 Boundary Conditions

3.2.1 Temperature

Initially at time, ,

and

where is the room temperature, is the pouring temperature, is the sand mold temperature and is the metal temperature.

Across the metal casting and mold, the temperatures are coupled using the solidWallMixedTemperatureCoupled wall function adapted in OpenFOAM. The heat transfer mechanism calculated by the function is conduction only with the assumption of negligible thermal contact resistant between the mold and the metal interface. The heat flux continuity between the metal and sand mold is described by the following equation

(3.11)

where and are the metal and sand thermal conductivities respectively.

At the mold wall, the boundary condition is set using a convective heat transfer coefficient between the mold and the environment. The prescribed heat transfer coefficient is set using the GroovyBC library with OpenFOAM. The boundary condition as normal to the mold surface is set based on the following expression:

(3.12) where n is normal vector to the boundary surface and is the heat transfer coefficient from the sand mold to the environment.

3.2.2 Velocity

A non-slip condition is assumed at the boundary. Therefore, the boundary condition at the wall of the metal casting is zero velocity. Because of zero velocity condition at the boundary and the buoyant flow, the pressure boundary condition is set as bouyantPressure which will calculate the normal gradient appropriately by taking into account the difference in densities from Boussinesq approximation.

3.3 Finite Volume Method

In the FVM approach, the computational domain is divided into discrete control volumes as the computational grids (see Figure 3.1). The finite difference equations are derived by applying the integral form of the conservation equations into the control volume (CV) allowing the fluid properties, such as mass and momentum be conserved at the discrete level. Most spatial derivative terms are then transformed into integrals over the surfaces A bounding the cell volume using Gauss’s theorem as shown in

Equation 3.13 (Greenshields, 2015).

(3.13)

This process of discretization will produce a system of linear equations through appropriate schemes that will be solved to gain the solution of the variables.

Figure 3.1: A generalized 2D control volume (Jana et al., 2007).

The following is the general form of the discretized equation based on the FVM approach that sums up the fluxes going through all the faces of the CV. The algebraic equations show the link between the dependant variable values and the neighboring values.

(3.14)

The central coefficient is derived from the neighboring nodes, (nb=E, W, N, S), and transient terms while the coefficient only contains the contribution from the neighboring nodes. The term represents the source terms.

The set of linear algebraic equations can be solved with direct method or iterative method. Some instances of direct method are Gaussian elimination and

Cramer’s rule matrix inversion. Iterative method uses a simple algorithm that is applied repeatedly until convergence. Such examples of this method are Jacobi and Gauss-

Siedel point-by-point iterative techniques. Iterative method use less computing time but convergence of the solution is not guaranteed (Versteeg and Malalasekera, 2007).

3.3.1 Solution Algorithm for Pressure and Velocity Calculation

This subsection will discuss the solution method for the pressure and velocity coupling used in this thesis. The method adopted is called PIMPLE algorithm which is a combination of Semi-Implicit Method for Pressure Linked Equations (SIMPLE) and

Transient Pressure Implicit with Splitting of Operator (PISO) algorithms which is adapted for iterative solution procedure (Versteeg and Malalasekera, 2007). In PISO, the solution to pressure-velocity coupling involves one momentum predictor step and two pressure corrector steps in every time iteration. Therefore, PISO is similar to SIMPLE algorithm introduced by Patankar (Patankar, 1980) except with an additional corrector loop.

PIMPLE however, needs only one corrector step but the pressure-velocity coupling was solved twice in every time iteration. PISO method strictly requires very small time step nonetheless, PIMPLE method allows bigger time step without compromising stability making the computational cost less expensive. The following figure summarizes the procedure in PIMPLE algorithm to find the solution values for the pressure and velocity in each time iteration:

Start

Solve the discretized momentum equation to predict velocity.

Solve all other discretized equations.

Calculate pressure from the predicted momentum and correct velocity

Resolve discretized equations from the new pressure and corrected velocity.

Recalculate pressure from the predicted momentum and correct velocity.

No Convergence?

Yes

Stop

Figure 3.2: PIMPLE solution algorithm.

3.3.1.1 Predictor

The equation below is derived from the linearization of the conservation equations. The convective term of the continuum equation is linearized by using the convective velocity from the previous time step n. The discretized equations of the momentum equation can be generalized in matrix-vector form as follows:

(3.15)

is the coefficient matrix of the solution vector while is the source terms. Note that the matrix can change depending on the factors incorporated into the momentum conservation equation. The matrix can be separated into a diagonal matrix and off-diagonal matrix .The left hand side of the above equation can be therefore rewritten as shown below:

(3.16)

The pressure values are taken from the previous iteration . The above form of equation is implemented and solved in OpenFOAM using a matrix solver that predicts the value of the velocity so as to represent the calculation of the velocity predictor step in PIMPLE.

3.3.1.2 Corrector Step

The value of will not satisfy the continuity conservation until the pressure is corrected giving Let first consider the velocity corrector step. The velocity corrector step can be applied through the following matrix-vector-form descritized equation.

(3.17)

Equation (3.10) above can be rearrange to calculate the first corrected velocity

as in the following

(3.18) where .

By taking the divergence of (3.11) and considering that , equation (3.18) becomes

(3.19) which is known as the Poisson equation to calculate the first corrected pressure A more detail derivation of the above equations can be referred to (Oliviera and Issa,

2001).The example of the above implementation in OpenFOAM is shown in

BuoyantBoussinesqPimpleFoam tutorial case.

The above solution method was implemented in the CFD software, OpenFOAM as in the Appendix A. The chtMultiRegion solver was adapted to simulate the solidification of the casting process. This solver allows the conjugate heat transfer between the casting and the mold to be simulated. In order to define the boundary conditions for the casting mold and for mold-metal interface, groovyBC dictionary and solid-fluid wall coupling were used respectively.

3.3.2 Numerical Schemes

The numerical schemes for discretizing the mathematical terms in the equations were set in the fvSchemes dictionary in the system directory. The following schemes were specified in this study:

Mathematical terms Schemes

Gradient Gauss Linear

Time derivatives Euler

Divergence, Gauss upwind

Laplacian Gauss Linear corrected

Surface normal gradient Corrected

Table 3.1. Numerical schemes used in OpenFOAM solver.

3.3.3 Pressure Gradient Calculation.

In current study, the pressure gradient distribution was analyzed to help understands the feeding behavior during the casting solidification. In the numerical model used in OpenFOAM, the pressure gradient in the momentum equation 3.2 described in section 3.1 is related to the motion of the flow.

Since Boussinesq approximation was used to model natural convection, the density was assumed constant in the liquid and caused the hydrostatic pressure at a location to always remain the same. Therefore, OpenFOAM excluded the hydrostatic pressure to account only the dynamic pressure that is responsible for driving the flow in the fluid. Consequently, the flow behavior can be analyzed through the change in the pressure related to the liquid motion.

One of the field values calculated by OpenFOAM is the kinematic pressure value which is represented by the term p_rgh to differentiate from the total pressure field available in OpenFOAM as denoted by the term p. The kinematic pressure is the pressure divided by the density of the fluid. To investigate the flow behavior during the casting solidification, the kinematic pressure gradient was calculated as post-processing by invoking the fvc::grad function in OpenFOAM. The code was adapted from the foamCalcFunctions available in OpenFOAM library which is used for post-processing field data. If the pressure is denoted by p, then the gradient of p is a vector field as follows:

(3.22)

In this study, the fvc::grad function in OpenFOAM calculates the gradient of a field by invoking Gauss Integration as the numerical scheme for solving the discretized equations which was specified using the gradScheme keyword in an input file as

47 previously presented in Table 3.1. The program codes for solving the solidification models can be found in Appendix A.

Chapter 4

The Analysis of the Casting Solidification Behavior

Using Convective Fluid Flow

This chapter describes the results and analysis of the solidification simulation on sand casting cases as described in the following sections. The analysis focuses on studying the behavior of the solidifying castings at the early stage of the solidification process when the metal melt still dominantly behaved as liquid inside the castings. The objective of this analysis was to investigate if the early results have any influence on the shrinkage defects behavior of the castings. In this study, we investigated the rheological behavior of the solidifying liquid metal at the early stage of casting solidification, when heat transfer and fluid flow with the effect of natural convection are taken into account in the simulation models. The goal of this study was to improve our understanding of the feeding behavior in aluminum-silicon alloys and develop strategies to design effective risers by considering the convective flow behavior of the metal.

Many aluminum alloys are the materials that pose a great challenge in designing the risers to feed the shrinkage during the solidification process in castings. The application of risering techniques which are mostly based on thermal principle analysis as discussed in chapter 2, for aluminum alloys can be limited in used since not much information is available for these alloys pertaining to effective feeding or the same feeding criteria cannot be generalized to other castings with different thermal conditions. There is no non-conservative relationship that exists between the shapes or sizes of the risers and the soundness of the castings for aluminum alloys.

The most popular Niyama criterion available in many simulation packages, that is based on thermal gradient to predict shrinkage location which works relatively well with materials such as steel, was shown to be less effective with aluminum alloys (Liotti

& Previtali, 2006, Jakumeit et al., 2012). This challenge is as a result of their mushy nature during solidification and due to their high percentage of volumetric contraction of the aluminum element which is approximately 7 percent (Campbell, 1991).

Consequently, most aluminum alloys need literally large risers to feed the shrinking casting sections and casting yields of 25 to 45 percents based on weight poured become relatively common in casting practices (Heine et al., 1967).

Dahle and St. John (1998) described that the mushy regions during the solidification of metal casting which can be divided into three regions, comprise of different mechanical and feeding characteristics. The presence of these regions is related to the way the mush response to shear. Stress and pressure gradient present in

50 the mushy region that are responsible for driving the liquid motion, can influence the formation of porosity defect. Dahle and St. John also claimed that the combined understanding of the casting solidification in terms of mushy zone development and its rheological behavior with flow induced by the casting and its feeding process, has the potential to explain the defects occurred in castings. They also pointed out that the development of dendrites at the effective liquidus temperature characterizes the solidification process of most aluminum alloys.

The focus of this section is to show the influence of the convective fluid flow during the liquid and mass feeding of the castings on the solidification patterns. Most studies on risering methods are based on thermal principle analysis. The fluid flow was omitted due to ability of thermal analysis to give good results for some alloys, and high computational requirement for fluid flow analysis. In order to reduce the need for high computational time for using fluid flow analysis and due to difficulty to quantify interdendritic feeding, only the liquid and mass feeding stages were analyzed in current study particularly until the casting section has reached coherency.

As described in the earlier chapter, the mass feeding dominates from the liquidus temperature to coherency temperature and the material behaves as liquid or slurry solid grains before it diminishes at packing temperature. The pressure difference required for mass feeding is also small. A study by Dahle and Arnberg (1997) on aluminum alloys showed that the material has not developed any strength prior to dendrite coherency indicating that the dendrites are floating freely and are independent

51 in the mushy zone. Therefore, the models used are relevant to analyze the solidification pattern during which the metal still behaves as liquid.

The information gained in this study were used to determine how the changes in the convective fluid flow behavior affect the effectiveness of the riser in feeding the casting. The analysis was done on multi-steps casting using AlSi9Cu aluminum alloy.

AlSi9Cu has dendrite coherency temperature at about 849.15K (576 Celsius) (Backerud et. al, 1990). The material properties and the geometry of the castings used in this study are illustrated in Section 4.1.

4.1 Casting Designs and Input Parameters

The geometry of the casting and the riser used in the study is shown in Figure

4.1. The simulations were run on the part design for four types of risers. Table 4.1 summarizes the designs of the risers used in the simulations.

Figure 4.1. Multi-steps casting geometry with a riser.

Model Riser diameter, Riser Height, Neck dimension, d (mm) h (mm) (dxh) mm 1 55 50 - 2 60 100 - 3 80 100 - 4 100 100 55 x 10

Table 4.1. Riser designs used in simulations.

The simulations for the above casting models were carried at pouring temperature of 973.15 K ( 700oC ). Table 4.2 and 4.3 show the standard material

53 compositions and the parameters for the thermophysical properties of the alloy used in the numerical simulations respectively as taken from Magmasoft database.

Element Si Fe Cu Mn Mg Zn Ti Fe/Mn Composition % 7.5-9.5 1.0 3.0-4.0 0.5 0.10 2.9 0.35 2.0

Table 4.2. AlSI9Cu alloy elements’ composition.

Properties Values Density , 2570 kg/m3 Thermal conductivity, k 70 W/m K Latent heat of fusion, 479.234 kJ/kg

Specific heat capacity, 1000 J/kg K

Solidus temperature, 752.15 K

Liquidus Temperature, 851.15 K Volumetric Thermal expansion 3.26 x 10-4 /K coefficient, Dynamic viscosity, 4.5 x 10-7 kg/ m s

Table 4.3. Themophysical properties of AlSi9Cu used in the simulation.

The mold of the sand casting was of dry silica sand with initial temperature of about 300K. The dimension of the mold used is 280 mm x 140 mm x 260 mm. The interfacial heat transfer coefficient (IHTC) between the mold and atmosphere used was

11.2 W/ m2K. The properties of sand mold used in the simulations are shown in Table

4.4.

Properties Values Density , 1520 Kg/m3 Thermal conductivity, k 0.6 W/m K Specific heat capacity, 1170 J/kg K Table 4.4. Silica sand mold properties used.

The meshes of 2,040,000 cells were generated using the SnappyHexMesh utility in OpenFOAM that requires CAD drawing of the part in standard Stereolitography (STL) format. The time step used in the simulations was 0.5 second. The experimental castings of the models were carried out using the LFC sand casting.

4.2 Thermal Analysis and Experimental Results

Magmasoft software package was used to run the heat transfer simulation of the casting models studied. Magmasoft only produces results based on the temperature analysis but not the convective fluid flow analysis. Figure 4.2 portrays the shrinkage porosities predictions based on the Niyama Criterion, which is derived from the temperature gradient and cooling rate of the metal. The figure shows that all of the models except model 1 could potentially produce sound castings. The experimental castings, on the other hand, showed that Model 1-3 produced unsound castings while

Model 4 casting was free from shrinkage defect thus is considered sound (see Figure

4.3). From this results, it is evident that a thermal-based criterion can lead to inaccurate prediction of casting performance.

57 Model 1 Model 2

Model 3 Model 4 Figure 4.2. Shrinkage porosities predictions using Niyama Criterion.

(a) (b)

Figure 4.3. Casting of (a) Model 1, (b) Model 2, (c) Model 3 and (d) Model 4.

4.3 Flow Analysis of the Casting Models

This section illustrates the flow patterns from the numerical simulations of the four multi-steps castings. Figures 4.4-4.10 show the velocity profiles taken at various time steps for each casting model. The results show that as the riser size increases the effect of natural convection or buoyancy also increases. Since natural convection promotes mixing, it can be seen that the largest riser had more circular motion of liquid flow compared to other sizes of risers. The mixing helps smooth out the temperature gradient thus causing the riser and the casting to maintain a higher temperature for a longer period of time. The mixing effect is mainly influenced by the temperature difference and the thermal expansion coefficient, specified in the Boussinesq equation for buoyancy term in the momentum conservation equation.

As the temperature decreases, the effect of buoyancy also decreases creating a more smooth upward flow channel at the middle of the riser. From Fgures 4.6, 4.8 and

4.10 in part (b), the circular flows which occurred at a later time become more uniform inside the riser as a result of higher solid fraction formed, thus reducing the fluidity of the molten metal. At this point the relative flow inside the riser is lower than 0.01 m/s.

(a)

(b)

(c)

Figure 4.4. Velocity fields of Model 1 at (a) 10 s, (b) 20 s and (c) 31 s.

(a)

(b)

Figure 4.5. Velocity fields of Model 2 at (a) 10 s and (b) 20 s.

(a)

(b)

Figure 4.6. Velocity fields of Model 2 at (a) 30 s and (b) 40 s.

(a)

(b)

Figure 4.7. Velocity fields of Model 3 at (a) 10 s and (b) 20 s.

(a)

(b)

Figure 4.8. Velocity fields of Model 3 at (a) 30 s and (b) 50 s.

(a)

(b)

Figure 4.9. Velocity fields of Model 4 at (a) 10 s and (b) 20 s.

(c)

(d)

Figure 4.10. Velocity fields of Model 4 at (a) 30 s and (c) 60 s.

Figures 4.11-4.12 portray the velocity fields at the time when the entire casting sections had reached coherency. In addition, Table 4.5 contains the minimum and the maximum velocities of each casting model when the casting section reached coherency.

Between the liquidus and the coherency temperatures, the flow rates were in the range between the flow with order of magnitude 10-4 and the flow with order of magnitude

10-3. For making comparisons, the liquidus flow rate is defined as the flow relatively higher than 0.0001 m/s which is approximately the minimum flow rate occurs above the liquidus temperature inside the risers, when the casting sections reached coherency.

The coherency flow rate is defined as the flow relatively higher than 0.00001 m/s as this flow rate approximately occurs as the minimum above the coherency temperature.

From Table 4.5, Model 1 part had the maximum velocity only slightly higher than the minimum coherency flow rate while other models had maximum flow higher than the minimum liquidus flow rate. From Figures 4.11-4.12, it can also be seen that all models except Model 1 had the risers dominated by the liquidus flow rate. Figure 4.12 also shows that Model 4 still had a significant buoyancy effect in the riser as the flow in the riser was less smooth than the flow in other risers. Model 3 and 4 indicate that the liquid supply is still abundant in the riser for the casting sections. Model 3 however had liquidus flow rate at a higher region from the casting sections. Model 3 and 4 had liquidus flow rate close to the regions between the riser and the casting section. By comparing with the experimental results, it can be concluded that the casting sections that is disconnected from the region with liquidus flow rate in the riser at casting coherency, will be prone to shrinkage defect.

Model Min velocity Maximum velocity 1 2.39e-17 7.79e-5 2 2.48e-13 .00773 3 1.15e-13 .0208 4 2.69e-13 .0328

Table 4.5. Minimum and maximum velocities of the castings at coherency.

From this analysis, it is found that the size of the riser has significant effect on the flow in the riser when the casting section reached coherency. Inefficient riser will produce a very small flow rate at the region between the riser and the casting therefore, increases the difficulty to feed the casting with feed metal. Small flow rate suggests that the pressure gradient that drives the flow of the fluid is also small. In conclusion, it is important to design a riser that is still entirely dominated by liquidus flow rate at the time when the casting has reached coherency to ensure continuous feeding from the riser.

The plot of the cooling curve at the beginning of the solidification for the location studied is shown in Figure 4.13. From the plot, it can be seen that the bigger the riser the higher the temperature at a certain period in time. However, for the sound casting as of Model 4, the temperature initially reduced faster compared to that of

Model 3 casting before it maintained at higher temperature than that of Model 3 at later time. This observation could be due to the higher convection effect in Model 4 which was responsible for advecting higher volume of hot liquid upward, reducing the

68 local temperature at the bottom faster as colder liquid glides down the casting

(Campbell, 1998).

(a)

(b)

Figure 4.11. Velocity fields at casting coherency for (a) Model 1 and (b) Model 2.

(a)

(b)

Figure 4.12. Velocity fields at casting coherency for (a) Model 3 and (b) Model 4.

980

960

940

920

Model 1 900 Model 2

880 Model 3 Temperature (K) Temperature Model 4 860

840

820 0 20 40 60 80 100 120 Time

Figure 4.13. Plot of temperature over time.

Figure 4.14 demonstrates the flow field by the scaled vectors at coherency. In

OpenFOAM, vectors are drawn as it is, therefore a vector of magnitude 1 m/s will be drawn 1 meter long. In order to fit the castings without significantly affecting the flow patterns, the vectors were scaled by 0.5 of its original magnitude value. From Figure

4.14, Model 1 casting had no apparent flow at coherency while Model 2 flow was only apparent at the top of the riser suggesting that the flow at the bottom of the riser in bulk motion had disappeared or became stagnant at macro level. Model 3 still had significant flow occurring in most part of the riser except the very bottom of the riser.

Only Model 4 showed a presence of significant flow of metal in bulk motion at the region between the riser and the casting section at casting coherency.

The above results suggest that the appearance of apparent flow field inside the riser is critical in determining a continuous feed flow from the riser when the casting section has reached coherency. When the casting section is disconnected from the apparent flow field, the casting will be prone to shrinkage porosities. For that regards, it is important for a good riser to maintain a distinct flow of the metal inside the entire riser at the time the casting section reaches coherency.

AM AM pos sim plic des Model 2 pro pro Mu soli pre Model 1 the the the the the

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tin tio

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ed ed ati ati mi

en oa Th

ad na ng ap an ng Pr

co co co ra

re re

ix ix

m m

A A

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Model 3 Model 4

Figure 4.14. Scaled velocity vectors when the casting sections reached coherency. 73

4.4 Pressure Analysis

This section analyzes the pressure profiles calculated by the solver in

OpenFOAM. The pressure gradient as described in the momentum equation is what driving the flow at the beginning of the alloy solidification. This pressure difference is induced by the dynamic of the fluid motion. The pressure discrepancy in the castings was only observed prior to the start of the solidification before it diminished as the liquid melt started to solidify. This was due to the pressure difference governed by the flow of the metal in the casting due to thermal or natural convection. As previously discussed, during the mass feeding stage the alloy still behaves like a liquid. After that the pressure gradient inside the dendritic mush is derived according to the Darcy’s Law for the interdendritic porous medium as follows (Voller et. al, 1987):

(4.1)

Even though the pressure gradient in the liquid metal is relatively small, the following analysis investigates the pressure profiles driven by the liquid flow until the flow in the casting reaches dendrite coherency or the formation of dendritic network.

Figure 4.15 illustrates the relative kinematic pressure profiles for Model 1 casting at different time intervals until the pressure had turned uniform across all part geometry.

Initially there were significant pressure differences in the casting due to higher motion

74 of the liquid metal. The higher motion of the liquid melt at the beginning of the solidification created lower pressure regions inside the casting. As the velocity in the casting decreased, the liquid motion became more restricted and thus reducing the pressure gradient across the casting. As the fraction solid increased, the pressure inside the casting also increased.

Figure 4.16 portrays the kinematic pressure gradient distribution inside the casting at various time intervals until the pressure gradient diminished or became significantly very low. As we analyzed the gradient profile and plot the isotherms that represent the liquidus temperature, (colored in white) and coherency temperature,

(colored in red), it can be seen that the isotherm closely bounded the region that exhibits a significant presence of pressure gradient followed by the isotherm. Since the pressure gradient calculated is related to the force due to flow, considerable pressure gradient was more apparent inside the region filled with liquid metal before the temperature reached the liquidus point, i.e, before the solidification started. From the figure, it can also be seen that the pressure gradient dropped significantly once the solidification started, i.e, after the melt reached .

As the coherency isotherm moved towards the region outside of the casting section, i.e, into the riser, the pressure gradient decreased significantly creating a larger region of a very low pressure gradient at the bottom within the liquidus isotherm loop.

The gap between the isotherm and isotherm also widen as the isotherms leaved the casting section from the lower part of the riser. The following section

76 (a) (b)

Figure 4.15. Pressure distribution of Model 1 at (a) 1 s, (b) 5 s, (c) 10 s and (d) 30 s time intervals.

76 will show how the sizes of the risers influence the pressure gradient distribution in relation to the dendrite contour line. Based on this information, the feedability of the riser design will be evaluated.

(a) (b)

Figure 4.16. Pressure gradient profiles of Model 1 at (a) 5 s, (b) 10 s, (c) 20 s and (d) 30 s.

4.5 Evaluation of Feeding Efficiency Using Convective Fluid Flow

Analysis

This section discusses how the analysis of the pressure gradient can help determine the design of the riser that produce porosity-free casting. The casting models as described in section 4.1 have the modulus difference between the riser and casting from 26% to 346% as illustrated in Table 4.6. Based on Table 4.6, it can be seen that the sound casting from Model 4 has low casting yield of 26 percents. Therefore, it is not surprising that the part casting required large riser with volume exceeding the volume of the casting to eliminate the shrinkage porosities from the casting section.

Parts (%) Casting Yield,Y (%) Model 1 26 69 Model 2 114 49 Model 3 227 35 Model 4 346 26

Table 4.6. The modulus difference between the casting and riser and casting yield for

each model.

Figure 4.17 displays the pressure gradient distributions for all models at the time when the coherency had reached the entire casting section. The isotherms correspond to the liquidus and coherency temperatures are also shown with red and white lines

79 respectively. As the coherency contour line exited the casting from the lower region of the riser, it can be seen that the gap between the coherency isotherm and liquidus isotherm at the bottom of the risers widen for the unsound castings. The smaller the riser, the wider the gap was. The model of sound casting is shown to maintain the narrow gap between the two isotherms as depicted in Figure 4.16 (d).

The pressure gradient at the lower region of the riser also reduced significantly as the entire casting section reached coherency. The smaller the riser the much lower the pressure gradient became. The smallest riser as depicted in Figure 4.16(a) showed an overall very low pressure gradient compared to others with no apparent pressure gradient differential as the liquidus isotherm had vanished. The sound casting on the other hand showed a significant presence of pressure gradient at the bottom of the riser above the liquidus isotherm. From above findings, it is evident that understanding the convective fluid flow behavior during the solidification can help us understand the feedability of the riser.

In order to clearly see the influence of the flow behavior on the feedability of the riser, the plot of the pressure gradient over time taken at a middle point between the riser and the casting section was made as in Figure 4.18. The plot illustrates the differences in the magnitudes of the pressure gradients as the riser sizes varied. The maximum pressure gradient increased as the riser size increased corresponding to the increase in the liquid momentum. After the pressure gradient reached its maximum, it decreased gradually until it turned down to zero.

Figure 4.19 is the close-up plot of the pressure gradients over time within the range they turned to zero. The figure portrays how the pressure gradient diminished as the size of the riser increased. The red marking on each of the lines representing a different riser size, corresponds to the time when the casting point reached coherency at temperature . Even though the magnitudes of the pressure gradients are relatively small that often considered negligible, it can be clearly seen that the pressure gradients of the first two models of the castings which had smaller risers, diminished to zero immediately after coherency. Model 3 casting showed the pressure gradient gradually depreciated over some time interval before it became zero. Model 4 casting which produced sound casting had a slower rate of pressure gradient depreciation over a significantly longer time period. The pressure gradient declining to zero as it is related to the motion of the fluid could signify the end the mass feeding so that interdendritic feeding dominates the casting after coherency. The longer the mass feeding become available after the casting section reaches coherency, the more liquid can be fed into the casting. Therefore, the size of the riser could influence how the feeding mechanisms characterize the solidification pattern of the casting.

In order to quantify the rate of the pressure gradient depreciation, a simple linear regression method was applied to the data from the point of coherency until it became zero (see Figure 4.20). The regression analysis was only performed on Model 3 and 4 since only these two models showed gradual decline of the pressure gradients over significant time ranges. Table 4.7 summarizes the rate of the pressure gradient depreciation based on the slope of the regression line. The time range from coherency

81 to zero is also shown in the table. The table shows that Model 4 had the decline rate approximately twice less than that of Model 3 over a time range about double than that of Model 3. This suggests that a sound casting requires more time for the liquid flow to occur from the riser after the casting section reaches coherency. As a conclusion, a good riser must allow continuous liquid feeding after the casting reaches coherency over sufficient time interval.

As the pressure gradient is commonly perceive as negligible during mass feeding, this study shows that the pressure gradient can have significant influence on the effectiveness of the riser and the soundness of the casting. The pressure gradient can portray the penetration degree of the convective flow into the coherent mushy region.

By taking into account the rheological behavior of the mushy region, it can therefore give more accurate prediction about the soundness of the casting for a given riser design.

83 (a) (b)

Figure 4.18. Plot of pressure gradient over time at a point between riser and casting.

Figure 4.19. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting.

0.0012

0.001

2 0.0008

0.0006

0.0004 PressureGradient(m/s)

0.0002

0 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 Time (s)

(a)

0.0009

0.0008

2 0.0007 0.0006 0.0005 0.0004 0.0003

Pressure Gradient (m/s)GradientPressure 0.0002 0.0001 0 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 Time (s)

(b) Figure 4.20. Regression applied on the pressure gradient over time; (a) Model 3and (b) Model 4

Model Model 1 Model 2 Model 3 -.0000895 11 s Model 4 -.0000331 22 s

Table 4.7. The depreciation rate of the pressure gradient after coherency.

4.6 Sensitivity Analysis of the Constant C Value in the Solidification

Simulation.

The constant C in the momentum equation is the permeability constant that depend on the morphology of the mushy region. The choice of constant C can influence the shape of the mushy region. The constant influences the penetration degree into the mushy region by the convection field (Brent et.al, 1988). Since this value is hard to determine accurately in practice, it is normally assumed to be constant. It is interesting to know how the values of C affect to feedability of the risers on the studied geometry.

Therefore, simulations were run by varying the values of constant C in the momentum equation. The values and were chosen based on the values recommended by

Guthrie and Tavares (1998) and Zeng et. al (2009). as well as found in the Van Tol

(1998).

Figure 4.21 illustrates the plot of the pressure gradients over time for the studied casting designs while Figure 4.22 shows the close-up of the plot over time from the coherency point until the pressure gradient diminished to zero. As can be seen, the constant C affects the convective flow behavior of the molten metal as portrayed by the shape of the curves in the plot. Nonetheless, changing the value of constant C did not change the feedability patterns among the riser designs studied. Even though lower value of constant C increases the time interval over which the pressure gradient diminished for all of the riser designs due to increase in convective flow, the pattern for the sound casting is still distinguishable from the unsound castings. As in Figure 4.22,

88 the sound casting always took almost or twice the time for the decline of the pressure gradient after casting coherency, than that of the intermediate design of which the decline took over some time range but the casting was unsound. The depreciation rates from the slopes of the linear regression analysis differed by one order of magnitude between the sound and unsound castings for both C values as shown in Table 4.8 which is more significant than the difference showed by C value of 106 . As these two values of constant C do not change the result of the riser effectiveness, the value of constant C of

106 as was used in initial analysis is therefore a reasonable value to use in the solidification simulation of the AlSi9Cu alloy.

(a)

(b)

Figure 4.21. Plot of pressure gradient over time at a point between riser and casting with constant (a) C = 103 and (b) C = 107.

(a)

(b)

Figure 4.22. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting with (a) constant C=103and (b) constant C = 107.

Model Model 1 -.000142 18 s Model 2 -.000142 27 s Model 3 -.000211 28 s Model 4 -.0000157 50 s

(a)

Model Model 1 Model 2 Model 3 -.000170 11 s Model 4 -.0000609 23 s

(b)

Table 4.8. The depreciation rate of the pressure gradient after coherency with (a)

constant C=103and (b) constant C = 107.

4.6 Evaluation of Feeding Effectiveness on Long Freezing Range Alloy:

Application to AlSi7Mg Casting

This section describes the use of the evaluation technique for feeding effectiveness of the riser applied to AlSi7Mg alloy. This alloy has a lower range of silicon content from 6.5 to 7.5 percent compared to the previous alloy which has the range from 7.5 to 9.5 percent. Studies have shown that silicon content in aluminum alloys can influence the grain morphology and the fraction solid at coherency (B ckerud et al.,

1990). The higher the silicon content is, the more dendritic the grain morphology is and the less the fraction solid present at coherency. AlSi7Mg alloy has coherency temperature at about 4 degree Celsius below its liquidus temperature. Thus the larger gap between the and corresponds to the lower silicon content which results in the alloy reaches coherency at a higher fraction solid. AlSi7Mg is also known as a long freezing range alloy partly due to its magnesium content (Kaufman, 2000).

The convective fluid flow analysis was carried out on the same steps geometry from the previous section. The following data of the thermophysical properties were used in the simulations (see Table 4.10). The pouring temperature was 1008.15 K ( 735

OC ) to maintain the same range between the liquidus and the pouring temperatures as was done for AlSi9Cu alloy. Since this alloy has a higher volumetric contraction percentage of approximately 5.35 percent, it is expected that the part will require a larger riser to produce a sound casting for the similar step geometry. For the purpose of

93 the study analysis, the simulations were run for three models of risers with the following dimensions as described in Table 4.11.

Element Si Fe Cu Mn Mg Zn Ti Composition % 6.5-7.5 0.5 0.25 0.35 0.25- 0.35 0.25 0.45

Table 4.9. AlSi7Mg alloy elements’ composition.

Properties Values Density , 2430 kg/m3 Thermal conductivity, k 68 W/m K Latent heat of fusion, 430.518 kJ/kg

Specific heat capacity, 1168 J/kg K

Solidus temperature, 815.15 K

Liquidus Temperature, 886.15 K Volumetric Thermal expansion 7.53 x 10-4 /K coefficient, Dynamic viscosity, 4.5 x 10-7 kg/ m s

Table 4.10. Themophysical properties of AlSi7Mg used in the simulation.

Model Riser diameter, d Riser Height, h (mm) Neck dimension (dxh) (mm) mm 1 80 100 55 x 10 2 100 100 55 x 10 3 120 90 55 x 10

Table 4.11. Riser designs used in simulations.

Parts (%) Casting Yield,Y (%) Model 1 227 35 Model 2 346 26 Model 3 420 21

Table 4.12. The modulus difference between the casting and riser and the casting yield

for each model.

Figure 4.23 displays the pressure gradient distributions as the casting sections reached coherency. From the figure, it is evident that Model 1 would not produce a sound casting. To further evaluate the feedability of Model 2 and Model 3, the plot of the pressure gradient over time was constructed as in Figure 4.24-4.25. Following similar trend from previous alloy, the plot in Figure 4.25 shows that Model 1 and Model

2 were not going to be sound castings as the pressure gradients depreciated immediately or at high rate after coherency. Model 3 portrayed a slower rate of depreciation over a wider time interval before reaching zero. Regression analysis in

Table 4.13 shows that the depreciation rate of Model 3 is almost twice less than the rate

95 of Model 2 and the time interval of the pressure gradient decline after coherency was more than twice of Model 2. Based on these results, Model 3 was expected to be a sound casting.

Model Model 1 0 Model 2 -.0000681 7 Model 3 -.0000446 15

Table 4.13. The depreciation rate of the pressure gradient after coherency.

The experimental castings showed that only Model 3 make a sound casting hence is in agreement with the result above (see Figure 4.27). Model 2 which was a sound casting with AlSi9Cu alloy, showed scattered small porosities present throughout the riser and the region in the casting below the riser. Other than small scattered internal porosities seen, note that this alloy also had a surface sink shrinkage at the top of the riser which is a common form of shrinkage defect seen on this alloy (Reis et al.,2008). Model 3 were found to have no internal porosities like in model 2 except the surface sink at the top of the riser. Thermal analysis also showed consistent result as previously found with AlSi9Cu alloy. In Figure 4.28, the temperature curve for Model 3 became lower than that of Model 1 and 2 initially and later maintained a higher temperature than those two curves.

(a)

(b)

(c)

Figure 4.23. Pressure gradient distribution when all of the casting section reached coherency for (a) Model 1, (b) Model 2, and (d) Model 3.

Figure 4.24. Plot of pressure gradient over time at a point between riser and casting.

Figure 4.25. Close-up plot of pressure gradient as it depreciated over time to zero at a point between riser and casting.

0.0006

0.0005

0.0004

0.0003

0.0002 PressureGradient(m/s) 0.0001

0 37 39 41 43 45 47 49 51 53 55 57 Time (s)

(a)

0.0007

0.0006

0.0005 2

0.0004

0.0003

0.0002

PressureGradient(m/s) 0.0001

0 37 39 41 43 45 47 49 51 53 55 57 -0.0001 Time (s)

(b)

Figure 4.26. Regression applied on the pressure gradient over time; (a) Model 2 and (b) Model 3

100

The results of the analysis on the long-freezing-range alloy, AlSi7Mg are consistent with the results on the short-freezing-range alloy, AlSI9Cu. Even though the solidification characteristics of long freezing range alloys are distinct from short freezing range alloys, the analysis of pressure gradient seems promising in determining the feedability of a riser and its effectiveness. It is therefore evident that influence of the convective fluid flow on the solidification and shrinkage behavior is important in understanding the feedability of the risers. Figure 4.29 present the framework for the implementation of the analysis method in designing optimal risers.

101

(a)

(b)

Figure 4.27. (a) Model 2 and (b) Model 3 AlSI7Mg castings.

102

980 Model 1 Model 2 Model 3 960

940

920 Temperature (K) Temperature 900

880

860 0 20 40 60 80 100 Time (s)

Figure 4.28. Plot of temperature over time.

103

Figure 4.29. Framework for riser design and optimization.

104

Chapter 5

Summary

The main objective of this research is to investigate the influence of the convective fluid flow on the solidification behavior of castings using analysis of solidification simulation and develop method of riser effectiveness evaluation based on the convective fluid flow analysis. Solidification model using heat transfer and convective fluid flow with natural convection was implemented in an open-source CFD program called OpenFOAM thus any interested parties can easily implement the method. The simulations were carried on multi-steps castings with various riser designs for AlSi9Cu and AlSi7Mg alloys. During the analysis, the rheological state of the mushy region was taken into account particularly when the casting section developed dendrite coherency.

105

5.1 Contributions

As point out by Dahle and St. John (1998), porosity formation can be influenced by the development of the mushy region and the ways it responses to shear stress.

Taking adavantage of the knowledge of the interdendritic coherency development during casting solidification, this study shows how the characteristics of pressure gradient induced by the motion of the convective flow can signify the feeding effectiveness of the risers into the castings. It is also found that the behavior of the flow during the early stage of the solidification particularly during the interdendritic structure development is essentially useful in identifying any feeding difficulty based on the riser characteristics. The analysis presented in this study can therefore, help detects the feeding problem early on hereby increasing the chance being right the first time. The findings thus prove the claim by Dahle and St. John (1998) that dendrites development after liquidus temperature characterizes the solidification process of most aluminum alloys.

Campbell (1998) stated that sufficient pressure gradient must be present for feeding to occur in castings. Effective risers show significant presence of pressure gradient between the risers and the castings after dendrite coherency as shown by the analysis on the aluminum-silicon alloys. As a result, the analysis using thermal and convective fluid flow models proves to be important to understand the feeding of the risers and help determine the best risers that produce sound castings. As most studies using thermal analysis focus on the short freezing range alloy, the same current technique was also found to work on the long freezing range alloy as shown by the

106 analysis on AlSi7Mg. Such outcome could be due to better accuracy gained from including convective fluid flow analysis. Even so, current technique only requires partial simulation of the solidification process making it reasonable for practical application in foundries.

Current study has successfully shown the application of the OpenFOAM simulation tool to help in improving the design of castings. Even though casting simulation can significantly help foundries in reducing the cost and time spent in the design process thus becoming more competitive, the cost of the simulation packages that are available commercially can be very expensive. This can be a disadvantage to small scale foundries especially in low income countries. Studies using OpenFOAM, an open-source software that is available for free, allow the small enterprises gain the benefits from the advance of the computing technology in the context of casting simulation.

While the experimental study was conducted on the lost foam casting, the numerical models used are relevant to any solidification process in general and therefore, applicable to other forms of sand casting process including the investment casting. However, the models used are based on some simplifying assumptions that could affect the accuracy of the results. In spite of that, the use of OpenFOAM gives the user the flexibility to exploit the models to tailor for specific applications. In order to expand the validity and applicability of the techniques presented in this study, the following section discusses some future work that could be undertaken.

107

5.2 Future Recommendations

The followings are the recommendations for further research related to the topic of this study.

a) Perform analysis on complex geometries and casting with multiple top risers.

Complex geometries can be divided into several section, each with its own

feeding system.

b) Since different materials have different thermophysical properties and

solidification characteristics, it is interesting to compare the results of the

following analyses:

a. Do feedability analysis of risers using the same techniques on different

aluminum alloys and other alloys such as steel, copper alloys and etc.

b. Determine and compare the characteristic of the pressure gradient

decline rates that correspond to sound castings for various aluminum

alloys.

c. Do analysis using different casting process such as green sand casting and

permanent mold casting.

c) For more accurate solidification prediction, use solidification model that consider

variable densities in solid and liquid and other thermophysical properties.

d) Do analysis using solidification model with different solid fraction function such

as linear function or Scheil model.

108 e) Develop and implement a stopping criterion into the simulation program that

allows sufficient time for the pressure gradient decline after the casting section

reaches coherency.

109

References

ASM Handbook (2008). Volume 15 Casting. Materials Park: ASM International.

B ckerud, L., Chai, G., & Tamminen, J. (1990). Solidification characteristics of aluminum

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Appendix A

OpenFOAM Programming Codes

This section contains the programming codes for the solver and the post- processing application for calculating the pressure gradient. The solver, named solidFoam, is adapted from the OpenFOAM chtMultiRegion solver for simulating conjugate heat transfer in multi regions. The following summarizes the structure of the solver for simulating the solidification of the metal,represented by the fluid region; and heat transfer inside the casting mold, represented by the solid region. The fluid simulation is adapted from the meltFoam solver as available from the forum at www.cfd-online.com.

123 solidFoam solver

solidFoam.C

readPIMPLEcontrols.H

regionProperties

regionProperties.H

regionProperties.C

fluid

createFluidFields.H

createFluidMeshes.H

icoContinuityErrors.H

icoCourantNo.H

icoCourantNo.C

icoMultiRegionCourantNo.H

initContinuityErrs.H

readFluidMultiRegionPIMPLEControls.H

readTransportProperties.H

setRegionFluidFields.H

storeOldFluidFields.H

readFluidMultiRegionPIMPLEControls.H

Ueqn.H

hEqn.H

pEqn.H

solid

createSolidFields.H

createSolidMeshes.H

readSolidTimeControls.H

setRegionSolidFields.H

solidRegionDiffNo.H

solidRegionDiffNo.C

solidRegionDiffusionNo.H

readSolidMultiRegionPIMPLEControls.H

solveSolid.H

Make

files

options

include

setInitialMultiRegionDeltaT.H setMultiRegionDeltaT.H 124

A.1 SolidFoam.C ------#include "fvCFD.H" #include "fixedGradientFvPatchFields.H" #include "regionProperties.H" #include "solidRegionDiffNo.H" #include "mathematicalConstants.H" #include "icoCourantNo.H"

// * * * * * * * * * * * * * * * * * ** * * * * * * // int main(int argc, char *argv[]) { #include "setRootCase.H" #include "createTime.H"

regionProperties rp(runTime);

#include "createFluidMeshes.H" #include "createSolidMeshes.H" #include "createFluidFields.H" #include "createSolidFields.H" #include "initContinuityErrs.H" #include "readTimeControls.H" #include "readSolidTimeControls.H" #include "icoMultiRegionCourantNo.H" #include "solidRegionDiffusionNo.H" #include "setInitialMultiRegionDeltaT.H"

while (runTime.run()) { #include "readTimeControls.H" #include "readSolidTimeControls.H" #include "readPIMPLEControls.H" #include "icoMultiRegionCourantNo.H"//added #include "solidRegionDiffusionNo.H" #include "setMultiRegionDeltaT.H"

runTime++;

Info<< "Time = " << runTime.timeName() << nl << endl;

if (nOuterCorr != 1) { forAll(fluidRegions, i) { #include "setRegionFluidFields.H" #include "storeOldFluidFields.H" } }

125

// --- PIMPLE loop for (int oCorr=0; oCorr

forAll(solidRegions, i) { Info<< "\nSolving for solid region " << solidRegions[i].name() << endl; #include "setRegionSolidFields.H" #include "readSolidMultiRegionPIMPLE Controls.H" #include "solveSolid.H" } }

runTime.write();

Info<<"ExecutionTime = "<

Info<< "End\n" << endl;

return 0; }

// *********************************************** //

A.2 readPIMPLEcontrols.H ------// Construct the fvSolution for the runTime. fvSolution solutionDict(runTime); const dictionary& pimple = solutionDict.subDict("PIMPLE"); const int nOuterCorr =

126

pimple.lookupOrDefault("nOuterCorrectors", 1);

A.3 regionProperties.H ------//Simple class to hold region information for coupled region simulations //SourceFiles regionProperties.C

#ifndef regionProperties_H #define regionProperties_H

#include "IOdictionary.H" #include "Time.H" namespace Foam {

/*------*\ Class regionProperties Declaration \*------*/ class regionProperties : public IOdictionary { // Private data

//- List of the fluid region names List fluidRegionNames_;

//- List of the solid region names List solidRegionNames_;

// Private Member Functions

//- Disallow default bitwise copy construct regionProperties(const regionProperties&);

//- Disallow default bitwise assignment void operator=(const regionProperties&);

public:

// Constructors

//- Construct from components regionProperties(const Time& runTime);

127

//- Destructor ~regionProperties();

// Member Functions

// Access

//- Return const reference to the list of fluid region names const List& fluidRegionNames() const;

//- Return const reference to the list of solid region names const List& solidRegionNames() const; };

} // End namespace Foam

#endif

A.4 regionProperties.C ------#include "regionProperties.H"

// * * * * * * * * * * Constructors * * * * * * * * * * * * * //

Foam::regionProperties::regionProperties(const Time& runTime) : IOdictionary ( IOobject ( "regionProperties", runTime.time().constant(), runTime.db(), IOobject::MUST_READ, IOobject::NO_WRITE ) ), fluidRegionNames_(lookup("fluidRegionNames")), solidRegionNames_(lookup("solidRegionNames")) {}

// * * * * * * * * * * Destructor * * * * * * * * * * * * * * //

Foam::regionProperties::~regionProperties()

128

{}

// * * * * * * * * * Member Functions * * * * * * * * * * * * // const Foam::List& Foam::regionProperties::fluidRegionNames() const { return fluidRegionNames_; }

const Foam::List& Foam::regionProperties::solidRegionNames() const { return solidRegionNames_; }

A.5 fluid/solveFluid.H ------if (finalIter) { mesh.data::add("finalIteration", true); }

#include "UEqn.H" #include "hEqn.H" // PISO loop for (int corr=0; corr

A.6 fluid/UEqn.H ------// Solve the momentum equation

tmp UEqn ( fvm::ddt(U) + fvm::div(phi, U) - fvm::laplacian(nu, U) + fvm::SuSp(DC, U) );

129

UEqn().relax();

if (momentumPredictor) { solve ( UEqn == fvc::reconstruct ( ( - ghf*fvc::snGrad(rhok) - fvc::snGrad(p_rgh) )*mesh.magSf() ) ); }

A.7 fluid/hEqn.H ------// Solving the energy equation

{ fvScalarMatrix TEqn ( fvm::ddt(cp, T) + fvm::div(phi*fvc::interpolate(cp), T) + hs*4.0*exp(-pow(4.0*(T-Tmelt)/(Tl- Ts),2))/Foam::sqrt(constant::mathematical::pi)/(Tl-Ts) *fvm::ddt(T) + hs*4.0*exp(-pow(4.0*(T-Tmelt)/(Tl-Ts),2))/ Foam::sqrt(constant::mathematical::pi)/(Tl-Ts)*(U & fvc::grad(T)) - fvm::laplacian(lamda/rho, T) );

TEqn.relax(); TEqn.solve();

alpha = 0.5*Foam::erf(4.0*(T-Tmelt)/(Tl-Ts))+scalar(0.5);

cp = alpha*cpL+(1.0-alpha)*cpS; lamda = alpha*lamdaL+(1.0-alpha)*lamdaS; nu = alpha*nuL+(1.0-alpha)*nuS; rhok = 1.0 - (beta*(T - Tl)); DC = DCl*Foam::pow(1.0-alpha,2)/(Foam::pow(alpha,3)+DCs); }

130

A.8 fluid/pEqn.H ------{ volScalarField rAU("rAU", 1.0/UEqn().A()); surfaceScalarField rAUf("(1|A(U))", fvc::interpolate(rAU)); p_rgh.boundaryField().updateCoeffs(); U = rAU*UEqn().H(); phi = (fvc::interpolate(U) & mesh.Sf()) + fvc::ddtPhiCorr(rAU, U, phi);

surfaceScalarField buoyancyPhi(rAUf*ghf*fvc::snGrad(rhok) *mesh.magSf()); phi -= buoyancyPhi;

for (int nonOrth=0; nonOrth<=nNonOrthCorr; nonOrth++) { fvScalarMatrix p_rghEqn ( fvm::laplacian(rAUf, p_rgh) == fvc::div(phi) );

p_rghEqn.setReference(pRefCell, getRefCellValue(p_rgh, pRefCell));

p_rghEqn.solve(mesh.solver(p_rgh.select (pimple.finalInnerIter())));

p_rghEqn.solve ( mesh.solver ( p_rgh.select ( ( oCorr == nOuterCorr-1 && corr == nCorr-1 && nonOrth == nNonOrthCorr ) ) ) );

if (nonOrth == nNonOrthCorr) { // Calculate the conservative fluxes phi -= p_rghEqn.flux();

// Explicitly relax pressure for momentum corrector p_rgh.relax();

131

// Correct the momentum source with the pressure gradient flux calculated from the relaxed pressure U -= rAU*fvc::reconstruct((buoyancyPhi + p_rghEqn.flux())/rAUf); U.correctBoundaryConditions(); } }

#include "icoContinuityErrors.H"

p = p_rgh + rhok*gh;

if (p_rgh.needReference()) { p += dimensionedScalar ( "p", p.dimensions(), pRefValue - getRefCellValue(p, pRefCell) ); p_rgh = p - rhok*gh; } }

A.9 solid/solveSolid.H ------if (finalIter) { mesh.data::add("finalIteration", true); }

{ for (int nonOrth=0; nonOrth<=nNonOrthCorr; nonOrth++) { tmp TEqn ( fvm::ddt(rho*cp, T) - fvm::laplacian(K, T) ); TEqn().relax(); TEqn().solve(); }

Info<< "Min/max T:" << min(T) << ' ' << max(T) << endl; } if (finalIter) { mesh.data::remove("finalIteration"); }

132

A.10 solidWallMixedTemperatureCoupled.H ------//Mixed boundary condition for temperature, to be used by the conjugate heat transfer solver.

//SourceFiles solidWallMixedTemperatureCoupledFvPatchScalarField.C

#ifndef solidWallMixedTemperatureCoupled_H #define solidWallMixedTemperatureCoupled_H

#include "mixedFvPatchFields.H" namespace Foam {

/*------*\ Class solidWallMixedTemperatureCoupledFvPatchScalarField Declaration \*------*/ class solidWallMixedTemperatureCoupled : public mixedFvPatchScalarField { // Private data

//- Name of field on the neighbour region const word neighbourFieldName_;

//- Name of thermal conductivity field const word KName_; public:

//- Runtime type information TypeName("solidWallMixedTemperatureCoupled");

// Constructors

//- Construct from patch and internal field solidWallMixedTemperatureCoupled ( const fvPatch&, const DimensionedField& );

//- Construct from patch, internal field and dictionary solidWallMixedTemperatureCoupled (

133

const fvPatch&, const DimensionedField&, const dictionary& );

//- Construct by mapping given // solidWallMixedTemperatureCoupledFvPatchScalarField onto a new patch solidWallMixedTemperatureCoupled ( const solidWallMixedTemperatureCoupled&, const fvPatch&, const DimensionedField&, const fvPatchFieldMapper& );

//- Construct and return a clone virtual tmp clone() const { return tmp ( new solidWallMixedTemperatureCoupled(*this) ); }

//- Construct as copy setting internal field reference solidWallMixedTemperatureCoupled ( const solidWallMixedTemperatureCoupled&, const DimensionedField& );

//- Construct and return a clone setting internal field reference virtual tmp clone ( const DimensionedField& iF ) const { return tmp ( new solidWallMixedTemperatureCoupled ( *this, iF ) ); }

// Member functions

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//- Get corresponding K field const fvPatchScalarField& K() const;

//- Update the coefficients associated with the patch field virtual void updateCoeffs();

//- Write virtual void write(Ostream&) const; };

} // End namespace Foam

#endif

A.11 solidWallMixedTemperatureCoupled.C ------#include "solidWallMixedTemperatureCoupled.H" #include "addToRunTimeSelectionTable.H" #include "fvPatchFieldMapper.H" #include "volFields.H" #include "mappedPatchBase.H" #include "regionProperties.H"

// * * * * * * * * * Constructors * * * * * * * * * * * * * * //

Foam::solidWallMixedTemperatureCoupled:: solidWallMixedTemperatureCoupled ( const fvPatch& p, const DimensionedField& iF ) : mixedFvPatchScalarField(p, iF), neighbourFieldName_("undefined-neighbourFieldName"), KName_("undefined-K") { this->refValue() = 0.0; this->refGrad() = 0.0; this->valueFraction() = 1.0; }

Foam::solidWallMixedTemperatureCoupled:: solidWallMixedTemperatureCoupled ( const solidWallMixedTemperatureCoupled& ptf, const fvPatch& p, const DimensionedField& iF,

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const fvPatchFieldMapper& mapper ) : mixedFvPatchScalarField(ptf, p, iF, mapper), neighbourFieldName_(ptf.neighbourFieldName_), KName_(ptf.KName_) {}

Foam::solidWallMixedTemperatureCoupled:: solidWallMixedTemperatureCoupled ( const fvPatch& p, const DimensionedField& iF, const dictionary& dict ) : mixedFvPatchScalarField(p, iF), neighbourFieldName_(dict.lookup("neighbourFieldName")), KName_(dict.lookup("K")) { if (!isA(this->patch().patch())) { FatalErrorIn ( "solidWallMixedTemperatureCoupled::" "solidWallMixedTemperatureCoupled\n" "(\n" " const fvPatch& p,\n" " const DimensionedField& iF,\n" " const dictionary& dict\n" ")\n" ) << "\n patch type '" << p.type() << "' not type '" << mappedPatchBase::typeName << "'" << "\n for patch " << p.name() << " of field " << dimensionedInternalField().name() << " in file " << dimensionedInternalField(). objectPath() << exit(FatalError); }

fvPatchScalarField::operator=(scalarField("value", dict, p.size()));

if (dict.found("refValue")) { // Full restart refValue() = scalarField("refValue", dict, p.size()); refGrad() = scalarField("refGradient", dict, p.size()); valueFraction() = scalarField("valueFraction", dict, p.size()); } 136

else { // Start from user entered data. Assume fixedValue. refValue() = *this; refGrad() = 0.0; valueFraction() = 1.0; } }

Foam::solidWallMixedTemperatureCoupled:: solidWallMixedTemperatureCoupled ( const solidWallMixedTemperatureCoupled& wtcsf, const DimensionedField& iF ) : mixedFvPatchScalarField(wtcsf, iF), neighbourFieldName_(wtcsf.neighbourFieldName_), KName_(wtcsf.KName_) {}

// * * * * * * * * * Destructor * * * * * * * * * * * * * * * //

// * * * * * * * * Member Functions * * * * * * * * * * * * // const Foam::fvPatchScalarField& Foam::solidWallMixedTemperatureCoupled::K() const { return this->patch().lookupPatchField(KName_); }

void Foam::solidWallMixedTemperatureCoupled::updateCoeffs() { if (updated()) { return; }

// Get the coupling information from the mappedPatchBase const mappedPatchBase& mpp = refCast ( patch().patch() ); const polyMesh& nbrMesh = mpp.sampleMesh(); const fvPatch& nbrPatch = refCast ( nbrMesh 137

).boundary()[mpp.samplePolyPatch().index()];

// Force recalculation of mapping and schedule const mapDistribute& distMap = mpp.map();

tmp intFld = patchInternalField();

const solidWallMixedTemperatureCoupled& nbrField = refCast ( nbrPatch.lookupPatchField ( neighbourFieldName_ ) );

// Swap to obtain full local values of neighbour internal field scalarField nbrIntFld = nbrField.patchInternalField(); mapDistribute::distribute ( Pstream::defaultCommsType, distMap.schedule(), distMap.constructSize(), distMap.subMap(), // what to send distMap.constructMap(), // what to receive nbrIntFld );

// Swap to obtain full local values of neighbour K*delta scalarField nbrKDelta = nbrField.K()*nbrPatch.deltaCoeffs(); mapDistribute::distribute ( Pstream::defaultCommsType, distMap.schedule(), distMap.constructSize(), distMap.subMap(), // what to send distMap.constructMap(), // what to receive nbrKDelta );

tmp myKDelta = K()*patch().deltaCoeffs();

// Both sides agree on // - temperature : (myKDelta*fld + nbrKDelta*nbrFld)/ (myKDelta+nbrKDelta) // - gradient : (temperature-fld)*delta // We've got a degree of freedom in how to implement this in a mixed bc. // (what gradient, what fixedValue and mixing coefficient) 138

// Two reasonable choices: // 1. specify above temperature on one side (preferentially the high side) // and above gradient on the other. So this will switch between pure // fixedvalue and pure fixedgradient // 2. specify gradient and temperature such that the equations are the // same on both sides. This leads to the choice of // - refGradient = zero gradient // - refValue = neighbour value // - mixFraction = nbrKDelta / (nbrKDelta + myKDelta())

this->refValue() = nbrIntFld;

this->refGrad() = 0.0;

this->valueFraction() = nbrKDelta / (nbrKDelta + myKDelta());

mixedFvPatchScalarField::updateCoeffs();

if (debug) { scalar Q = gSum(K()*patch().magSf()*snGrad());

Info<< patch().boundaryMesh().mesh().name() << ':' << patch().name() << ':' << this->dimensionedInternalField().name() << " <- " << nbrMesh.name() << ':' << nbrPatch.name() << ':' << this->dimensionedInternalField().name() << " :" << " heat[W]:" << Q << " walltemperature " << " min:" << gMin(*this) << " max:" << gMax(*this) << " avg:" << gAverage(*this) << endl; } }

void Foam::solidWallMixedTemperatureCoupled::write ( Ostream& os ) const { mixedFvPatchScalarField::write(os); os.writeKeyword("neighbourFieldName")<< neighbourFieldName_ << token::END_STATEMENT << nl; os.writeKeyword("K") << KName_ << token::END_STATEMENT << nl; 139

} namespace Foam { makePatchTypeField ( fvPatchScalarField, solidWallMixedTemperatureCoupled );

} // End namespace Foam

A.12 Pgradient/PGradient.C ------//Calculates the gradient for field p_rgh in each timestep as post-processing

# include "fvCFD.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // int main(int argc, char *argv[]) { # include "addRegionOption.H" # include "setRootCase.H" # include "createTime.H" # include "createNamedMesh.H" //# include "createMesh.H" # include "createFields.H"

Info<< "\nCalculating solution pressure gradient\n" << endl;

instantList timeDirs = timeSelector::select0(runTime, args);

forAll(timeDirs, timeI) { runTime.setTime(timeDirs[timeI], timeI);

Info<< "Time = " << runTime.timeName();

//Info<< "Reading field p_rgh\n" << endl;

//Reading field

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volScalarField p_rgh ( IOobject ( "p_rgh", runTime.timeName(), mesh, IOobject::MUST_READ, IOobject::AUTO_WRITE ), mesh );

Info<< "; max-min p_rgh: " << max(p_rgh).value() << " " << min(p_rgh).value() << endl;

tmp tmp=fvc::grad(p_rgh); PGrad=tmp();

//runTime.write(); PGrad.write(); }

Info<< "End\n" << endl;

return(0); }

A.13 PGradient/createFields.H ------volVectorField PGrad ( IOobject ( "PGrad", runTime.timeName(), mesh, IOobject::MUST_READ, IOobject::AUTO_WRITE ), mesh );

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