Casting: An Analytical Approach (Engineering Materials and Processes) (Engineering Materials and Processes)

Engineering Materials and Processes

Series Editor Professor Brian Derby, Professor of Materials Science Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK

Other titles published in this series: Fusion Bonding of Polymer Composites C. Ageorges and L. Ye Composite Materials D.D.L. Chung Titanium G. Lütjering and J.C. Williams Corrosion of Metals H. Kaesche Corrosion and Protection E. Bardal Intelligent Macromolecules for Smart Devices L. Dai Microstructure of Steels and Cast Irons M. Durand-Charre Phase Diagrams and Heterogeneous Equilibria B. Predel, M. Hoch and M. Pool Computational Mechanics of Composite Materials ´ski M. Kamin Gallium Nitride Processing for Electronics, Sensors and Spintronics S.J. Pearton, C.R. Abernathy and F. Ren Materials for Information Technology E. Zschech, C. Whelan and T. Mikolajick Fuel Cell Technology N. Sammes Computational Quantum Mechanics for Materials Engineers L. Vitos Publication due August 2007

Alexandre Reikher and Michael R. Barkhudarov

Casting: An Analytical Approach

123

Alexandre Reikher, PhD Albany Chicago Co. 8200 100th St. Pleasant Prairie, WI 53158 USA

Michael R. Barkhudarov, PhD Flow Science, Inc. 683A Harkle Road Santa Fe, NM 87505 USA

British Library Cataloguing in Publication Data Reikher, Alexandre Casting : an analytical approach. - (Engineering materials and processes) 1. Founding I. Title II. Barkhudarov, Michael R. 671.2 ISBN-13: 9781846288494 Library of Congress Control Number: 2007928128 Engineering Materials and Processes ISSN 1619-0181 ISBN 978-1-84628-849-4 e-ISBN 978-1-84628-850-0

Printed on acid-free paper

© Springer-Verlag London Limited 2007 FAVOR™ and FLOW-3D® are trademarks and registered trademarks of Flow Science Inc., 683 Harkle Rd. Ste A, Santa Fe, NM 87505, USA. http://www.flow3d.com MATLAB® is a registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA. http://www.mathworks.com Visual Basic® is a registered trademark of Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399, USA. http://www.microsoft.com The software disk accompanying this book and all material contained on it is supplied without any warranty of any kind. The publisher accepts no liability for personal injury incurred through use or misuse of the disk. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. 987654321 Springer Science+Business Media springer.com

Michael Barkhudarov: I dedicate this work to my family, Natasha, Sophia and Philip, and to my dear parents

Alexandre Reikher: I dedicate this work To my dear parents To my wife Marianna And to my children Daniel and Alison

Preface

This book is the result of 40 years of the combined authors’ experience in mechanical and fluid dynamics engineering. It gives an overview of product and process development from the analytical standpoint. This book has not been intended to revolutionize the casting industry. The principals of fluid dynamics and static mechanics were largely developed in the nineteenth century, but process development still largely remains a trial and error method. This book is intended to underline the principals of strength of materials and fluid dynamics that are the foundation of the casting product and process development. This book has been written as a resource and design tool for product and process engineers and designers who work with aluminium castings. It combines many aspects of product and process development, which include the basic principals of static mechanics and fluid dynamics as well as completely developed applications allowing solving problems at every stage of the development process. This book has five main parts: (1) overview of casting processes, (2) fluid dynamics, (3) strength of materials, (4) sand casting, permanent mould, and die casting process development, and (5) quality control. The unique feature of this book is a combination of real life problems, which product and process engineers face every day, with user-friendly applications written in MATLAB® and Visual Basic. The comprehensive unit conversion calculator as well as examples in the area of strength of materials is intended to serve as a reference for practicing engineers as well as for students who are beginning to study mechanical engineering. It cab also be used by process engineers, who contribute greatly to product design. Completely developed process design applications will help process engineers to take full advantage of the power that computers and software bring to engineering. Product engineers can also get inside of process development procedures as well as the governing equations that are used in casting process development. Examples in this book are solved with the help of MATLAB® functions, Visual Basic applications as well as a general purpose CFD code FLOW-3D®. All MATLAB® functions, art work and Visual Basic applications are developed by the

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Preface

authors. Proper references are given for commonly available equations and theories. Considerable effort has been made to avoid errors. The authors would appreciate reader’s comments and suggestions for any corrections and improvements to this book. Alexandre Reikher, Milwaukee, Wisconsin, USA, April 2007 Michael Barkhudarov, Los Alamos, New Mexico, USA, April 2007

Contents

1 Casting of Light Metals .................................................................................... 1 1.1 Casting Processes....................................................................................... 1 1.2 Sand Casting ............................................................................................. 1 1.2.1 Gating System ................................................................................ 2 1.2.2 Risers and Chills............................................................................. 2 1.3 Permanent Mould ...................................................................................... 3 1.3.1 Gravity Casting............................................................................... 4 1.3.2 Low-pressure Permanent Mould Casting ....................................... 4 1.3.3 Counterpressure Casting ................................................................. 4 1.4 Die Casting ................................................................................................ 5 1.4.1 Die-cast Process ............................................................................. 5 1.4.2 Die-cast Dies ................................................................................. 8 1.4.3 Runner System ............................................................................... 8 1.4.4 Cavity of Die-cast Die ................................................................. 11 1.4.5 Air Ventilation System ................................................................ 11 1.4.6 Ventilation Blocks ....................................................................... 12 2 Introduction to Fluid Dynamics .................................................................... 13 2.1 Basic Concepts ....................................................................................... 13 2.1.1 Pressure ...................................................................................... 13 2.1.2 Viscosity ...................................................................................... 14 2.1.3 Temperature and Enthalpy ........................................................... 16 2.2 Equations of Motion ................................................................................ 18 2.3 Boundary Conditions ............................................................................... 19 2.3.1 Velocity Boundary Conditions at Walls ....................................... 20 2.3.2 Thermal Boundary Conditions at Walls ....................................... 20 2.3.3 Free Surface Boundary Conditions............................................... 21 2.4 Useful Dimensionless Numbers............................................................... 23 2.4.1 Definitions .................................................................................... 23 2.4.2 The Reynolds number................................................................... 24 2.4.3 The Weber Number ...................................................................... 24

x

Contents

2.5 2.6

2.7

2.4.4 The Bond Number ........................................................................ 25 2.4.5 The Froude Number...................................................................... 25 The Bernoulli Equation............................................................................ 26 Compressible Flow .................................................................................. 26 2.6.1 Equation of State .......................................................................... 27 2.6.2 Equations of Motion ..................................................................... 28 2.6.3 Specific Heats............................................................................... 29 2.6.4 Adiabatic Processes ...................................................................... 31 2.6.5 Speed of Sound............................................................................. 31 2.6.6 Mach Number............................................................................... 33 2.6.7 The Bernoulli Equation for Gases ................................................ 33 Computational Fluid Dynamics ............................................................... 35 2.7.1 Computational Mesh..................................................................... 35 2.7.2 Numerical approximations ........................................................... 36 2.7.3 Representation of Geometry ......................................................... 38 2.7.4 Free Surface Tracking .................................................................. 39 2.7.5 Summary ...................................................................................... 41

3 Part Design ...................................................................................................... 43 3.1 Design of Light Metal Castings ............................................................... 43 3.2 Static Analysis ....................................................................................... 43 3.2.1 Moment of an Area....................................................................... 44 3.2.2 Moment of Inertia of an Area ....................................................... 44 3.3 Loading ................................................................................................. 46 3.4 Stress Components................................................................................... 47 3.4.1 Linear Stress ................................................................................. 48 3.4.2 Plane Stress................................................................................... 50 3.4.3 Mohr’s Circle................................................................................ 51 3.5 Hooke’s Law............................................................................................ 55 3.6 Saint-Venant’s Principle .......................................................................... 56 3.7 Criteria of Failure..................................................................................... 57 3.7.1 Ductile Material Failure Theory .................................................. 58 3.7.2 Brittle Material Failure Theory.................................................... 58 3.8 Beam Analysis ........................................................................................ 58 3.8.1 Free Body Diagram ...................................................................... 67 3.8.2 Reactions ...................................................................................... 67 3.9 Buckling ....................................................................................... 68 3.10 Bending Stresses ...................................................................................... 71 3.10.1 Normal Stresses in Bending ......................................................... 72 3.10.2 Shear Stress .................................................................................. 74 3.11 Stresses in Cylinders................................................................................ 76 3.12 Press-fit Analysis ..................................................................................... 82 3.13 Thermal Stresses ...................................................................................... 86 3.14 Torque...................................................................................................... 87 3.15 Stress Concentration ................................................................................ 93

Contents

xi

3.16 Fatigue ...................................................................................................... 94 3.17 MATLAB® Functions Used in Chapter 3 ................................................. 96 4 Process Design ................................................................................................. 97 4.1 Evaluation of Dimensionless Numbers.................................................... 97 4.1.1 Gravity Pour ................................................................................. 98 4.1.2 High-pressure Die Casting: Slow Shot ......................................... 98 4.1.3 High-pressure Die Casting: Fast Shot........................................... 99 4.2 Flow in Viscous Boundary Layer .......................................................... 100 4.3 The Bernoulli Equation.......................................................................... 102 4.3.1 Stagnation, Dynamic and Total Pressure.................................... 102 4.3.2 Gravity Controlled Flow............................................................. 103 4.3.3 Flow in the Runner System ........................................................ 104 4.3.4 Filling Rate ................................................................................. 106 4.4 Flow in a Shot Cylinder ......................................................................... 108 4.5 Gas Ventilation System ......................................................................... 116 4.6 Ventilation Blocks ................................................................................. 125 4.7 Little’s Formula ..................................................................................... 126 4.8 Poisson Process and the Exponential Distribution................................. 127 4.9 Cooling ................................................................................................. 132 4.9.1 Lumped-temperature Model ....................................................... 132 4.9.2 Heat Flow into a Semi-infinite Medium ..................................... 138 4.10 CFD Simulations.................................................................................... 141 4.10.1 High-pressure Die Casting.......................................................... 142 4.10.2 Gravity Sand Casting.................................................................. 152 5 Quality Control ............................................................................................. 157 5.1 Basic Concepts of Quality Control ........................................................ 157 5.2 Definition of Quality.............................................................................. 158 5.3 Definition of Control ............................................................................. 158 5.4 Statistical Process Control ..................................................................... 159 5.5 Tabular Summarization of Data............................................................. 160 5.6 Numerical Data Summarization............................................................. 160 5.6.1 Normal Probability Distribution ................................................. 161 5.6.2 Poisson Probability Distribution................................................. 162 Appendix ............................................................................................................ 165 A.1 Unit Conversions ................................................................................... 165 A.2 Prefixes ............................................................................................... 171 References ............................................................................................................ 173 Index .................................................................................................................... 175

1 Casting of Light Metals

1.1 Casting Processes There are several casting methods used to produce light metal parts. The most widely used are x x x

Sand casting Permanent mould casting Die casting

Usually economic considerations are the driving force in deciding which casting process can be used. The sand casting process requires the least amount of up-front investment in tooling. But parts cannot be produced with close tolerances and minimum machine stock. It will require extra machining operations, which will drive part price cost up. Permanent mould requires up-front investing in tooling. But parts can be cast with much closer tolerances and reduced machine stock. Due to intensive cooling, parts can be produced with much shorter cycle time, compared with sand casting. The die-cast process requires a large up-front investment in tooling. Due to high pressure used during the die-cast process, parts can be produced with close tolerances and minimum machine stock.

1.2 Sand Casting Sand casting is the oldest way to produce near net shape parts. Sand casting moulds (Figure 1.1) are made using green or chemically bonded sand. Green sand moulds use either a mixture of natural sand and clay or synthetic sands. A typical sand casting mould has a gating system, risers and chills.

2

Casting: An Analytical Approach

1.2.1 Gating System The primary function of the gating system is to allow metal to flow into the cavity. Good design of the runner and gating system should use basic principles of liquid flow. It should allow filling the cavity at the slowest possible fill rate to avoid air entrainment and to ensure complete fill of the thinnest areas of the casting. A typical gating system includes x x x x

A pouring basin A sprue A runner In-gates

To avoid shrink porosity, heavy sections have to be supplied with metal during solidification. That is why it is a good practice to gate into thicker sections of the part. The basic principles of gating system design are as follows: 1. 2. 3. 4. 5. 6.

The ratio of the pouring basin to sprue diameter should allow keeping the gating system full during fill time. The gating system should allow directional solidification of the casting. In gating system design, sharp turns and sudden changes in cross section should be avoided. The gating system should allow maximum flow rate at minimum velocity. This will help to avoid turbulence in metal flow and reduce gas entrainment. If a casting has multiple combinations of thick and thin sections, several sprues and gates have to be used. This also applies to large castings to reduce fill time and avoid filling the cavity with overheated metal. Avoid gating directly into the core or into the wall. It will produce splashing and result in gas porosity.

1.2.2 Risers and Chills During solidification metal density increases and the volume of the casting is decreasing. This causes a decrease in internal pressure. Differences in casting internal pressure force metal to flow from hotter sections that are still in a liquid state to cooler shrinking sections. At some point, metal reaches the state called critical solid fraction. This is state at which metal can’t flow any more to compensate for decreasing volume. At this point, shrink porosity is formed. Risers are placed in the last areas to solidify to continue to fit metal into the thicker, slow solidifying sections of the casting. Some features of the casting are isolated by a thinner section that prevents feeding metal during solidification. An example is a boss attached to a thinner wall. In this case, chills are used to accelerate solidification and minimize or eliminate formation of shrink porosity. In complex castings, multiple risers and chills can be used to force directional solidification.

Casting of Light Metals

3

Figure 1.1. Sand casting mould

1.3 Permanent Mould Permanent mould casting referred to a method of casting in which mould is not destroyed during extraction of the casting. Permanent moulds are capable of producing large number of the same casting. Castings produced in permanent moulds have finer grain structure and superior mechanical properties compared with sand castings. Castings also have almost no gas porosity, major defect of the die-castings. Permanent mould has next major components 1. 2. 3. 4.

Gating system, which direct metal into the cavity at selected rate. Feeding system, which feed metal to thicker areas of the part during solidification period. Chills, which complement feeding system by cooling thicker areas of the part. Venting system, which allows gases to escape during cavity fill process.

In general, permanent mould operationally very similar to a sand casting. It employs gravity as a feeding method. In order to ensure proper fill of the casting sufficient head has to be provided. Position of the gating system, feeders, risers, and chill has to allow direction solidification, starting from the areas of the casting away from the gate and moving into the direction of the gates and feeders. Incorrectly designed and positioned gating system will result in short fill and

4

Casting: An Analytical Approach

shrink porosity. Mistakes in design of the feeding system and chills will result in excessive shrink porosity, or longer dwell time. Incorrectly placed and sized ventilation channels will result in excessive gas porosity in the casting. There are three major processes are currently used to produce castings in permanent moulds x x x

Gravity casting Low-pressure castings Counterpressure casting

1.3.1 Gravity Casting Gravity casting is a basic casting process that uses gravity to fill the cavity of the mould. This process can be used for simply shaped parts that are not going to be used in high stress or leak free applications. 1.3.2 Low-pressure Permanent Mould Casting Low-pressure permanent mould casting is a process that uses pressure to feed metal in to the cavity. Castings produced by this method have higher density and lower gas and shrink porosity. Molten metal is fed from the bottom of the cavity through the riser tube under some pressure (0.5 – 0.8 Bar). Advantages of this method are 1. 2. 3. 4. 5.

The process can be easily automated, which allows control of metal velocity, reduces the turbulence of the metal flow and minimizes air entrainment. A hermetically sealed furnace minimizes metal oxidation and avoids unwanted inclusions. Metal is fed from the bottom of the bath which allows feeding cleaner metal into the cavity of the mould. Directional solidification to the riser allows feeding metal until the casting is completely solidified. This reduces the amount of shrink porosity. This method allows producing quality casting with thinner walls.

1.3.3 Counterpressure Casting Counterpressure casting is a method that uses low pressure to feed metal into the cavity from the bottom of the mould, similar to the low-pressure permanent mould casting method, as well as pressurized cavity. As the cavity is filled with metal, the pressure constantly increases which suppresses hydrogen precipitation. Counterpressure permanent mould casting method allows achieving the highest mechanical properties in a casting. The pressurized cavity eliminates shrink porosity without using risers.

Casting of Light Metals

5

1.4 Die Casting The earliest examples of die casting by pressure injection, as opposed to casting by gravity pressure, occurred in the mid – 1800s. A patent was awarded to Sturges in 1849 for the first manually operated machine for casting printing type. The process was limited to printer’s type for the next 20 years, but development of other shapes began to increase toward the end of the century. By 1892, commercial applications included parts for phonographs and cash registers, and mass production of many types of parts began in the early 1900s. The first die-casting alloys were various compositions of tin and lead, but their use declined with the introduction of zinc and aluminium alloys in 1914. Magnesium and copper alloys quickly followed, and by the 1930s, many of the modern alloys still in use today became available. The diecasting process has evolved from the original low-pressure injection method to techniques including high-pressure casting at forces exceeding 4500 pounds per square inch squeeze casting and semisolid die casting. These modern processes are capable of producing high integrity, near net-shape castings with excellent surface finishes. Alloys of aluminium, copper, magnesium, and zinc are most commonly used for casting. x x x x

Aluminium is a lightweight material exhibits good dimensional stability, mechanical properties, machinability, thermal and electrical conductivity. Copper alloy is a material with high strength and hardness. It has high mechanical properties, dimensional stability, and wear resistance. Magnesium is the lightest cast alloy. It about 4 times lighter than steel and 1.5 times lighter than aluminium. It has a better strength to weight ratio than some steel, iron and aluminium alloys. Zinc is the easiest alloy to cast. It can be used to produce castings with 0.5 mm wall thickness.

1.4.1 Die-cast Process High-pressure die casting is used for a wide range of applications in all major industries. Advantages of die castings are 1. 2. 3. 4. 5. 6. 7.

High mechanical properties in combination with light weight. High thermal conductivity. Good machinability. High resistance to corrosion. Parts can be produced with no or a limited amount of machining. Parts can be cast with close dimensional tolerances. Low scrap rate.

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Casting: An Analytical Approach

Die casting is a precision manufacturing process in which molten metal is injected at high pressure and velocity into a permanent metal mould. There are two basic die-casting processes: 1. 2.

The hot chamber process. The cold chamber process.

In a hot chamber diecast machine (Figure 1.2), a metal injection system is immersed in the molten metal. Advantages of hot chamber die-cast process are 1. 2.

Cycle time kept to a minimum. Molten metal must travel only a short distance, which ensures minimum temperature loss during cycle time.

The hotchamber process can be used only for alloys with a low melting point (lead, zinc). Alloys with a higher melting point will cause degradation of the metal injection system. The hot-chamber die cast process has these steps 1. 2. 3. 4. 5.

Hydraulic cylinder applies pressure on plunger (Figure 1.2). Plunger pushes metal from the sleeve through the gating system into the cavity (Figure 1.3a). High pressure is maintained during solidification process . After solidification is complete, the die opens (Figure 1.3b). The part is ejected from the cavity (Figure 1.3c).

Figure 1.2. Hot chamber die cast machine

Casting of Light Metals

a

7

b

c Figure 1.3. Steps in the hot-chamber die-cast process: a. plunger pushes metal from the sleeve through the gating system into the cavity; b. after solidification process is complete, the die opens; c. the part is ejected from the cavity.

The cold chamber die-casting process is used for alloys with a high melting point (aluminium, brass). In a cold chamber die-casting machine (Figure 1.4), the metal is in contact with the machine injection system for only a short period of time. A typical process consists of several steps (Figure 1.5): 1. 2. 3. 4. 5. 6.

Molten metal is ladled into the shot sleeve (Figure 1.5a). Hydraulic cylinder applies pressure on plunger (Figure 1.5b). Plunger pushes metal from the sleeve through the gating system into the cavity (Figure 1.5c). High pressure is maintained during solidification process (Figure 1.5d). After solidification is complete, the die opens (Figure 1.5e). The part is ejected from the cavity (Figure 1.5f).

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Casting: An Analytical Approach

Figure 1.4. Cold chamber die-cast machine

1.4.2 Die-cast Dies Die-cast dies (Figure 1.6) are made from alloy tool steel and must withstand multiple cooling-heating cycles. A die-cast die usually consists of two halves, the stationary (cover) side and the ejector side. The cover side of the die is mounted to a fixed platen on the die-cast machine and the ejector side is mounted to a movable platen. The cover and ejector halves are separated at the parting line to allow for removal of the casting. The molten metal enters the die through the shot hole in the cover half of the mould. The runner system of the die allows metal to flow from the shot hole into the cavity. An ejector mechanism is mounted on the ejector side of the die. Its purpose is to push the casting out of the cavity. There are several ventilation channels cut on the parting line of the die to allow air to escape. These channels are either open to the atmosphere or connected to a vacuum system. The cooling system of the die-casting die allows excess energy to be removed to cool the casting below solidification temperature. 1.4.3 Runner System The runner system of the die-cast die delivers metal from the shut cylinder of the die-cast machine into the cavity of the die with minimum losses. As in every hydraulic system, head losses in the runner system are the product of major and minor losses. hL

hMajor  hMinor .

(1.1)

Major losses that occur due to friction between metal flowing through the runner system and cavity walls can’t be avoided. That is why calculations of the runner system have to account for these losses.

Casting of Light Metals

a

b

c

d

e

f

9

Figure 1.5. Steps in the cold chamber die-cast process: a. molten metal is ladled into the shot sleeve; b. hydraulic cylinder applies pressure on plunger; c. plunger pushes metal from the sleeve through the gating system into the cavity; d. high pressure is maintained during solidification; e. after solidification is complete, the die opens; f. the part is ejected from the cavity.

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Casting: An Analytical Approach

Figure 1.6. Die-cast die

Major losses can be described by the Darcy-Weisbach equation:1 hMajor

f

l V2 . D 2g

(1.2)

where f = coefficient of friction L = length of the flow pass D = equivalent diameter of the runner system V = velocity of the metal through the gate g = acceleration of gravity. Minor head losses that are results of dissipation of kinetic energy can be minimized or completely avoided (Figure 1.7).

1

Equation developed as a variation of the empirical equation developed by Gaspard de Prony.

Casting of Light Metals

11

Figure 1.6. Flow pattern for the sharp and rounded edge runner

1.4.4 Cavity of Die-cast Die After the first stage of the die cast process is complete and metal has been delivered to the cavity, the flow pattern through the cavity has to be designed to fill the cavity in the most efficient way. Die-cast dies use forced cooling method where water, oil or steam flows through the cooling channels. The location and effectivness of the die-cast die cooling system are limiting factors in calculating cavity fill time. Longer fill time can result in metal reaching point where an increase in metal density can significantly influence fluidity of the metal, which will result in a nonfill condition. In the die design stage a lot of consideration has to be given to a location of the gates, because they will define the flow pattern of the metal through the cavity. After runner and gate size and location have been defined, the cooling system has to be designed to allow for uniform cavity temperature. 1.4.5 Air Ventilation System The air ventilation system is a part of the die-cast die. After the die-cast die is closed and ready for the next cycle, the shot cylinder, gate system, cavity, and ventilation system are filled with air. As the molten metal is forced through the gating system into the cavity, it displaces air. Air flows to the outside or into a vacuum tank through the ventilation channels. Correctly placed and sized ventilation channels allow most of the air to escape from the cavity. Air compressibility, coefficient of friction, temperature changes in the air, and the shape of the ventilation channel all have to be taken into account to prevent pressure loss and shock wave formation in the airflow stream. Gases entrapped in the cavity form internal porosity. They are a major defect in the high-pressure die-casting process. Gas porosity can originate from many sources. A properly designed ventilation system can minimize the amount of gas entrapped in the cavity.

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Casting: An Analytical Approach

1.4.6 Ventilation Blocks Properly positioned ventilation channels are connected to the last area of the diecast die cavity to be filled. To prevent the molten metal from escaping through the ventilation channels, ventilation blocks are used. Ventilation blocks are used for both conventional and vacuum ventilation systems. When metal flows through the ventilation blocks due to the thermal exchange between the blocks and the molten metal, its temperature is reduced until it reaches its critical point of rigidity. There are two different types of ventilation blocks - valve - valveless. Ventilation blocks with valves have larger cross-sectional areas in the gas evacuation channel. The disadvantage of these types of ventilation blocks is a mechanical system that shuts-off the ventilation channel as soon as metal reaches the valve. Periodic failure, constant maintenance requirements and low reliability of the mechanical system limit the application of this type of ventilation system.

2 Introduction to Fluid Dynamics

2.1 Basic Concepts The behaviour of metals during filling and solidification can be described by applying the laws of fluid dynamics. In the liquid state, metals behave largely like common liquids such as water or liquefied natural gas. Molecules in liquids do not form a rigid crystalloid structure and, therefore, can move easily relative to each other. This behaviour distinguishes fluids in general from solid materials. At the same time, these molecules are packed sufficiently close to each other to experience strong forces of mutual attraction that make it hard to pull a piece of liquid apart. For the same reason, it is also hard to compress a liquid to a smaller volume. Therefore, liquids, unlike gases, can be treated as essentially incompressible materials, a property that greatly simplifies the governing equations.2 Fluid flow behaviour is characterized by density, pressure, temperature and velocity. Density, U, is the amount of mass, represented by molecules, in a unit volume of fluid. The incompressibility property implies that density stays constant during flow. In other words, no matter how a fluid is stretched, sheared or pressed, the number of molecules in a fixed volume stays more or less constant, even though some molecules may have moved out of it and others have entered it in their place. 2.1.1 Pressure The resistance of fluid to compression is characterized by pressure. Huge pressures must be applied to compress a fluid by as little as 1% of its original volume. In

2

In this chapter, when referring to metals we will use the terms liquid and fluid. Although the term fluid generally includes both incompressible liquids and compressible gases, we will primarily mean the former unless specifically clarified otherwise.

14

Casting: An Analytical Approach

most situations, even in high-pressure die casting, the pressures are not sufficient to change the fluid volume noticeably. Pressure is one of the main parameters that control the flow of fluid. It is related to the rate at which molecules transfer the momentum of their random microscopic motion to their neighbours through collisions. Since this random motion occurs in all directions, pressure at a point in the fluid also acts in all directions. But when molecules in one part of the fluid transfer more momentum to the molecules in an adjacent region than they receive in return, a macroscopic force arises between these fluid regions. This force can be described as the pressure gradient, which is the difference in pressures at two locations in the fluid, divided by the distance between those points. When pressure P is a function of the threedimensional coordinates x, y and z, then at every point in the fluid, the pressure gradient is a vector, defined as

’P

§ wP wP wP · ¨¨ ¸¸ . , , © wx wy wz ¹

(2.1)

In the absence of other forces, flow is initiated in the direction of the pressure variation from high values to low, that is, in the direction opposite to the direction of the pressure gradient, as shown in Figure 2.1.

Figure 2.1. Iso-lines of pressure (isobars) showing the distribution of pressure and the direction of the pressure gradient

2.1.2 Viscosity

As with any moving objects, the motion of most fluids experiences additional forces due to friction. Frictional forces also arise from the collisions of molecules in a moving fluid with molecules in the slower adjacent fluid regions. These collisions result in a net transfer of momentum from the faster flow regions to the

Introduction to Fluid Dynamics

15

slower ones, giving rise to frictional force. This force acts in the direction opposite the faster flow, dissipating its energy and generating heat, similar to the effect of taxes on the flow of capital. For example, if two streams of fluid are moving at different speeds parallel and next to each another, the faster moving stream will gradually slow down and the slower one will accelerate. As a result, the boundary between the two layers will widen with time leading to the development of the viscous boundary layer, the region where flow transitions from the velocity in one fluid layer to the velocity in the other. It is noteworthy that, as the two fluid layers exchange momentum, the overall kinetic energy in the flow decreases.

Figure 2.2. Viscous frictional force acting between two streams of fluid

The frictional properties of a fluid are conveniently described with a single variable called the dynamic viscosity coefficient P, and these forces are called viscous forces or stresses. Fluids with larger dynamic viscosity coefficients generate higher viscous forces than less viscous fluids in the same flow conditions. Additionally, larger differences in velocities result in a higher rate of transfer of the momentum from the faster moving fluid to the slower and hence in more viscous friction. Finally, the distance between two fluid regions also plays a role: the closer they are, the faster the transfer of momentum occurs. According to these observations, the viscous frictional force Ffr acting between the two streams of fluid in Figure 2.2 can be estimated as Ffr

AP

U1  U 2 . d

(2.2)

where A is the contact area between the streams, U1 and U2 the average velocities in the two streams and d the distance between them. In this form, Ffr is the force acting on the fluid moving with the velocity U2. For the fluid stream with the velocity U1 the sign of the force is the opposite. In a differential form, the righthand side of Equation 2.2 can be expressed for a unit contact area as

16

Casting: An Analytical Approach

Ffr A

P

wU . wz

(2.3)

where z is the coordinate axis normal to the direction of flow. The expression on the right-hand side of Equation 2.3 is called shear stress. Note that viscous stresses are reduced to zero in a uniform flow since the righthand side of Equation 2.3 vanishes when there is no velocity variation. However, it is hard to achieve uniform flow in practical situations where a fluid is typically confined by the walls of the channel or the container. Fluid molecules collide with these walls and bounce back. The surface of a typical material has roughness that far exceeds the size of a fluid molecule. Even a super finished metal surface at best has a roughness in excess of tens of nanometres. This is still more than a hundred times bigger than a fluid molecule. Other cutting and finishing techniques produce roughness in the range from 100 to 50,000 nm (0.1 to 50 Pm). So for a fluid molecule hitting a wall, its surface looks like the Black Forest to a football. After several collisions, it is very likely to lose all information about where it was coming from before it hit the surface. The usually irregular shape of the molecules and atoms only accelerates the “loss of memory.” When a fluid molecule returns into the flow after interacting with the wall, its original momentum component normal to the wall may be retained, but the direction of the tangential component is completely random. In macroscopic terms this behaviour is expressed in the form of the no-slip boundary condition. It means that the fluid velocity component tangential to the surface of a wall boundary is equal to zero. The no-slip boundary condition means that friction, or viscous shear stress, is always present in a flow near walls. In addition to the pressure gradient, it is one of the main factors controlling flow. It leads to the development of viscous boundary layers, in which flow transitions from zero velocity at the surface to the flow in the bulk. Moreover, the relatively large size of the surface roughness may produce more flow loss than can be suggested just by its interaction with the individual fluid molecules. Large clusters of these molecules can be deflected, redirected and trapped by the small bumps and pits on the surface that make up the surface roughness. This may contribute to the development of turbulence in the flow. Turbulence can be described as a form of flow instability, when random oscillatory motion develops in the otherwise ordered mean fluid flow. This random motion occurs on much larger time and length scales than the molecular motion, but its effect is similar. It accelerates the transfer of momentum between different parts of the fluid and, therefore, results in more friction. 2.1.3 Temperature and Enthalpy

The thermal state of a fluid is usually represented by temperature, T, which is a measure of and proportional to the kinetic energy of the chaotic motion of its molecules. Fluid specific thermal energy¸ I, is proportional to the temperature I

CVT ,

(2.4)

Introduction to Fluid Dynamics

17

with the coefficient of proportionality CV, called the specific heat at constant volume. It is equal to the amount of heat that is needed to raise the temperature of a unit mass of fluid by 1o. The subscript ‘V’ means that the volume of fluid would be kept constant during such a procedure. This clarification is necessary for a compressible gas, which, if allowed to expand upon heating, would require more energy to raise its temperature. For incompressible fluids, this distinction is not very important. As a result, the value of the specific heat at constant volume is very close to that of the specific heat at constant pressure, Cp. For obvious reasons, it is easier the measure Cp for metals by simply keeping the specimen at atmospheric pressure during measurement, whereas for a gas placed in a fixed container, it is easier to measure CV. CP is used to calculate another useful quantity called enthalpy, E, C P T  L(1  f S ) .

E

(2.5)

The second term on the right-hand side accounts for the release of thermal energy during solidification. Fluid molecules in the liquid phase have more freedom to move than in the solid state where they are locked in a crystalloid structure. As the metal cools and passes from the liquid state to the solid, the excess energy is released in the form of latent heat. The solid fraction, fS, is the mass fraction of the solidified phase in a given amount of metal. Upon cooling, its value changes from 0.0 in the pure liquid to 1.0 in the pure solid phase allowing for the latent heat release in Equation 2.5. One of the mechanisms for the exchange of thermal energy within fluids is thermal conduction. As molecules collide with each other, they transfer momentum, which is responsible for pressure and viscous forces, and also the kinetic energy of their chaotic motion. Consequently, any temperature variations in a thermally insulated volume of fluid would disappear over time, resulting in a uniform temperature distribution. The rate of heat exchange by conduction is described by the thermal conduction coefficient, k. The heat flux q between two regions of fluid at temperatures T1 and T2 separated by the distance d is then calculated as q

k

T2  T1 , d

(2.6)

wT . wx

(2.7)

or in differential form,

q

k

Equation 2.7 is the Fourier law stating that the heat flux by thermal conduction is linearly proportional to the temperature gradient [Holman, 1976]. Note that the form of Equation 2.7 is similar to that of Equation 2.3 for the viscous dissipation of momentum.

18

Casting: An Analytical Approach

2.2 Equations of Motion Pressure gradients and viscous stresses are the main internal forces present in fluids. External forces can include gravity and electro-magnetic forces. According to Newton’s second law, the sum of all these forces results in a net acceleration of the fluid, which is inversely proportional to its mass, or density. This can be expressed in the form of the Navier-Stokes equations, which for an incompressible viscous fluid can be written in the following form [Batchelor, 1967] wu wu wu wu  u  v  w wt wx wy wz



wv wv wv wv  u  v  w wt wx wy wz



ww ww ww ww  u  v  w wt wx wy wz

P § w 2u 1 wP w 2u w 2u · ¸  Gx  ˜ ¨¨ 2   U wx U © wx wy 2 wz 2 ¸¹

w 2v w 2v · P § w 2v 1 wP (2.8) ¸  Gy  ˜ ¨¨ 2   wy 2 wz 2 ¸¹ U wy U © wx P § w2w w2w w2w · 1 wP ¸  Gz .   ˜¨   U wz U ¨© wx 2 wy 2 wz 2 ¸¹

Here u, v and w are the three components of the fluid velocity vector U at any point in the flow, and G = (Gx, Gy, Gz) is the external force, which we will assume here consists only of gravity. The left-hand side of Equation 2.8 represents the components of fluid acceleration, the components of the pressure gradient, viscous stresses and gravity are summed up on the right-hand side. These forces are divided by fluid density U, therefore, the same forces would produce a higher acceleration for a lighter fluid. The ratio of the dynamic viscosity coefficient and density is often called the kinematic viscosity coefficient Q PU. Mass conservation is another important law governing the motion of fluids. It states that mass cannot be created or lost and is expressed through the continuity equation. For incompressible fluids, this equation reduces to the condition of zero divergence of the velocity vector divU

wu wv ww   wx wy wz

0.

(2.9)

and simply means that for any amount of fluid entering a given volume from one side, exactly the same amount must leave on the other side. When heat transfer and solidification are of interest, then additional equations are needed to track the evolution of temperature and the solid fraction. This is done in the energy conservation equation, which, similar to the mass conservation one, says that energy is not lost or created. As for the equation of motion, the energy transport equation is simplified by the assumption of incompressibility. Written in terms of enthalpy, defined in Equation 2.5, it has the following form:

Introduction to Fluid Dynamics

wE wE wE wE u v  w wt wx wy wz

w 2T w 2T k § w 2T ¨ 2   U ¨© wx wy 2 wz 2

· ¸. ¸ ¹

19

(2.10)

The left-hand side of Equation 2.10 constitutes the rate of change of enthalpy, and the right-hand side describes thermal conduction. An appropriate relationship between solid fraction and temperature must also be devised to complete the model. Equations 2.8 – 2.10 constitute the basic set of equations describing the evolution of an incompressible fluid such as metal. It can be applied to a wide range of flow problems, from ocean currents to MEMS, from external to internal flows, steady-state or transient. Metal casting, of course, is one of the areas where the rules of fluid dynamics can be used. When turbulence is present, conventional turbulence models seek to enhance viscous mixing and dissipation in the flow by evaluating the turbulent dynamic viscosity coefficient and using it in Equation 2.8 in place of the molecular value [Batchelor, 1967]. The left-hand sides of Equations 2.8 and 2.10 have similar forms and describe the transport of the quantities shown in the partial derivatives (u, v, w in Equation 2.8, and E in Equation 2.10). The leading term is called the temporal derivative. It is the rate of change of a quantity at a given point in the flow. For instance, wE / wt could be evaluated by inserting a thermocouple into the flow and then using its readings and Equation 2.5. The rest of the terms on the left-hand sides of these equations are convective terms. They are responsible for carrying fluid quantities with the flow and are characteristic of continuum mechanics when a particle of fluid moves, another particle comes in its place bringing with it its unique properties such as temperature and velocity. Diffusion is another means of transport in fluids. The diffusion of thermal energy is described by the thermal conduction terms on the right-hand side of Equation 2.10. The diffusion of momentum is represented by the terms in parentheses on the right-hand side of Equation 2.8. In incompressible fluids, as well as in solids, pressure can actually be negative because the intermolecular forces in these materials include the forces of attraction that are responsible for keeping the molecules close together. Pressure in Equation 2.8 can be relative, or gauge pressure. For example, it can be set relative to one atmosphere, in which case the normal pressure will be equal to zero. This is possible for incompressible materials because pressure in the equations of motion is present only in the gradient operand, therefore, adding or subtracting a constant does not change the flow dynamics.

2.3 Boundary Conditions Equations 2.8 – 2.10 are usually solved in a finite domain that has external and internal boundaries. Therefore, proper descriptions of these boundaries, or boundary conditions, are needed to find the flow solution. In addition to material properties, boundary conditions distinguish low-pressure from high-pressure die

20

Casting: An Analytical Approach

casting or lost foam casting from gravity pour. Boundary conditions, therefore, play an important role in determining the solution, and it is worth saying a few words about them here. 2.3.1 Velocity Boundary Conditions at Walls

There are two flow boundary conditions at the walls bounding the flow. Since fluid cannot penetrate solid obstacles, the component of the velocity normal to the wall must be equal to zero:

U˜n

u ˜ nx  v ˜ n y  w ˜ nz

0,

(2.11)

where n(nx,ny,nz) is the unit length vector normal to the wall surface. The second boundary condition enforces the no-slip condition, that is, the velocity component tangential to the wall must also be equal to zero: UIJ

0.

(2.12)

Combined together, Equations 2.11 and 2.12 simply state that flow velocity at the wall is equal to zero. It is useful, however, to define the two conditions separately since Equation 2.12 is not necessary when an inviscid flow approximation is used (i.e., when viscous stresses are small and can be neglected in Equation 2.8). 2.3.2 Thermal Boundary Conditions at Walls A boundary condition at walls is also needed for Equation 2.10 for enthalpy. This is typically done by defining a heat flux, q, at the interface between fluid (metal) and wall (mould) as follows: q

h ˜ (Tfluid  Twall )

(2.13)

with the heat transfer coefficient, h, representing the thermal properties of the interface itself. Factors like surface roughness, coating or lubrication affect the value of h. The wall boundary condition given by Equation 2.13 can be replaced by the one that directly specifies the heat flux, possibly as a function of time, q

q0 (t ) .

(2.14)

Equation 2.14 is useful when modeling exothermic sleeves or water-cooled mould surfaces.

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21

2.3.3 Free Surface Boundary Conditions Free surface is a special type of boundary; it moves with the liquid. The influence of air on flow can usually be ignored because air is much lighter than most liquids, especially metals. The fact that free surface is a boundary between a liquid and the ambient air is expressed in the so-called kinematic boundary condition, stating that the velocity of the free surface, Ub, is equal to the velocity of the liquid: Ub

U(t , x, y, z ) .

(2.15)

This obvious condition is nevertheless necessary to include free surface properly in the flow model. Equation 2.15 ensures that liquid and free surface do not get separated. The lightness of the ambient air in comparison with liquid gives rise to the dynamic boundary conditions at a free surface. The first one states that fluid pressure at a free surface, P0, is equal to the air pressure, Pa. P0

Pa .

(2.16)

Moreover, if we ignore the variation of pressure in the air due to gravity, then Pa is constant along a contiguous section of the free surface. This does not necessarily mean that it is constant in time, however. For example, during filling, the air may not be able to escape quickly enough, causing the air pressure in the cavity to increase, thus making Pa a function of time. Moreover, multiple air pockets will generally have as many different pressures, each serving as the boundary condition for the segment of metal surface bounding the respective air pocket. Surface tension forces at a free surface can also be taken into account. A liquid molecule located at the free surface interacts with the liquid molecules on one side of the interface and with the adjacent air molecules on the other side. The asymmetry of the inter molecular forces gives rise to a macroscopic force, which is proportional to the curvature of the interface. This force is typically expressed in terms of the surface tension pressure, Ps, which is a product of the surface tension coefficient, V, and the interface curvature, N, Ps

VN

V (’n) ,

(2.17)

where ’ ˜ n wnx wx  wn y wy  wnz wz is the divergence of the unit outward normal vector of the surface ( Figure 2.3). Liquid metals have the highest surface tension coefficients among liquids, with mercury leading the pack. Additionally, the buildup of a surface film due to the oxidation of metal in contact with air adds to the molecular forces at a free surface [Campbell, 1991].

3

The surface tension coefficient is not so much a property of the fluid as of the interface between two media, such as aluminium and air.

22

Casting: An Analytical Approach

Figure 2.3. Surface tension pressure acting on an element of free surface

Surface tension is an important force when the free surface curvature is large as, for example, in small droplets in an atomized flow common in high-pressure die casting. Equation 2.16 then needs to be modified to include the surface tension force. Pa  Ps .

P0

(2.18)

The second dynamic boundary condition at a free surface is derived from the assertion that viscous friction between fluid and air is negligibly small or, using Equation 2.3, wU wn

0.

(2.19)

where the derivative of the fluid velocity near a free surface is taken in the direction normal to the surface. Thermal boundary conditions at the free surface during casting are often assumed to be adiabatic, i.e., for simplicity heat losses to the air are neglected in comparison with the heat fluxes inside metal and at mould walls. However, more realistic relationships, similar to that given by Equation 2.13, can also be used. For example, radiative heat losses, qR, which maybe important for high temperature alloys, can be computed as qR

4 H ˜ ] ˜ (Tfluid  Tair4 ) ,

(2.20)

where H is the emissivity of the surface (H < 1), ]=5.5604˜10-8 kg s-3 K-4 the StefanBoltzmann constant, and temperature is expressed in the absolute units of degrees

Introduction to Fluid Dynamics

23

Kelvin, K. Due to the power of four on the right-hand side of Equation 2.20, the radiative heat flux grows quickly with an increase in surface temperature. For example, the pouring temperature of steels, 1700 – 1800 K, is around twice that of a die-cast aluminium alloy and, therefore, with similar emissivity coefficients, the radiative heat loss from the surface of the steel is about sixteen times larger.

2.4 Useful Dimensionless Numbers Equations 2.8 – 2.10, together with the appropriate boundary conditions, describe a very wide range of flows. It is often useful to estimate the relative importance of various terms in these equations and thus determine the most significant aspects of the physical behaviour of the fluid in a given situation. This, in turn, may enable simplification of the equations before one proceeds with the solution. As a minimum, it would be useful to understand what type of flow to expect. A set of dimensionless numbers, each representing an estimate of the ratio of a pair of forces, can be conveniently employed for that purpose. These numbers are derived from the dimensionless form of the equations of motion. This form, in turn, is obtained by scaling the equations by the characteristic values of length and velocity. As their name suggests, for a given flow each dimensionless number has the same value, irrespective of the units system employed to evaluate it. 2.4.1 Definitions The commonly used dimensionless numbers are Reynolds number: Re

UUd P

Weber number:

UU 2l V

Bond number:

Froude number:

We

Bo

Fr

UGl 2 V

U Gh

fluid inertia . viscous forces

(2.21)

fluid inertia

.

(2.22)

surface tenstion

gravity surface

.

(2.23)

tension

fluid inertia . gravity

(2.24)

24

Casting: An Analytical Approach

Here U is the characteristic velocity, d, l and h denote the appropriate characteristic lengths and G is gravity. 2.4.2 The Reynolds Number For the Reynolds number U is the average variation of the velocity in the flow between its minimum and maximum values, and d is the distance over which this variation occurs. According to Equation 2.12, in a typical filling, U can be defined as the difference between the velocity at the walls, which is zero, and in the bulk of the flow, or as the average metal velocity. Then d becomes half of the minimum wall thickness or half of the channel width. The Reynolds number is one of the most important parameters characterizing fluid flow. When its value is small, Re < 1, then flow is dominated by viscous forces. For very small values of Re, the convective terms in Equation 2.8 can be neglected in comparison with viscous dissipation of the momentum, reducing it to the so-called creeping, or Stokes, flow approximation. As is shown in the next section, the Reynolds number in metal flow in most castings is much greater than one, indicating that, at least during filling, viscosity plays a secondary role to fluid inertia. With the increase in the speed of the flow, it transitions from laminar to turbulent due to the development of flow instabilities initiated by spatial variations in fluid velocity. The transition begins at Re | 2000 and turns into a fully turbulent flow when Re exceeds 10,000. Only in extremely carefully controlled flow experiments can the laminar regime be extended to Re up to 20,000. Fully developed turbulence enhances the dissipation of fluid momentum, in addition and significantly beyond the dissipation due to the molecular viscosity, even though a large value of the Reynolds number may suggest that viscous forces are not important in the flow. 2.4.3 The Weber Number In Equation 2.22 for the Weber number, U characterizes the average variation in fluid velocity near a free surface. To be more precise, it is the velocity component normal to the free surface that is of the interest here. Due to the no-slip boundary condition at walls and Equation 2.15, we can say the U is the average velocity of the free surface. As with the Reynolds number, the distance l then becomes the minimum width of the flow channel. During filling, internal fluid forces can cause distortion of the metal surface, sometimes called surface turbulence [Campbell, 1991], that would lead to folding of the surface, additional oxidation and other undesirable effects. The process can be visualized by imagining a submerged jet of metal directed at an area of the free surface. Its energy will create a bulge on the initially undisturbed surface. The Weber number can be used to determine if the surface tension forces can prevent the rupture of the surface film and restore its shape. The velocity U and distance l in Equation 2.22 in this case relate to the jet velocity and size of the bulge, respectively. If We < 1, then we can hope that the energy of the flow will be contained within the confines of the existing free surface. If We > 1, as is the case

Introduction to Fluid Dynamics

25

in most filling scenarios, then the folding and entrainment of the surface oxide film and possibly air cannot be avoided. It has also been observed experimentally that a free surface breaks up into small droplets when the Weber number exceeds the critical value of around 60 [Manzello and Yang 2003]. 2.4.4 The Bond Number The Bond number is another measure of the relative importance of surface tension. This time it is compared to gravity, which is useful to determine if a free surface will stay flat or bulge. The natural tendency of the surface tension forces is to bend the initially horizontal free surface to reach a constant curvature at its every point, and in the absence of other forces it will do just that. Gravity in this case acts in the opposite direction trying to flatten it. When gravity is strong and the surface’s horizontal extent l is large, that is Bo > 1, a free surface is likely to stay flat and undisturbed by the surface tension forces as in a glass of water or a metal pouring cup. If the size of the container is gradually reduced, then at some point the value of the Bond number will drop below unity and the shape of the free surface will be determined more by the surface tension than by gravity. This can be observed inside a half-filled (transparent) drinking straw or when placing a small droplet of water on a dry surface. 2.4.5 The Froude Number The Froude number is often employed to estimate the importance of such as surface waves in open-channel flows, like rivers and canals. It is also useful to look at the waves in the horizontal runners in gravity pour castings and shot sleeves in high-pressure die casting. In all these cases, the waves are driven by gravity. The variable h in Equation 2.24 is the average depth of the fluid. When Fr is much smaller than one, Fr << 1, surface waves are much faster than the main flow, U. Such flow is called sub-critical. In the time it takes for the fluid to move the length of the container, the waves will pass in both directions multiple times, dissipate their energy and, therefore, can be deemed unimportant for the overall configuration of the flow. In the case of large values of the Froude number, Fr > 1, the flow is faster than the surface waves, or super critical. Any such waves are quickly swept away by the flow toward the boundaries of the flow domain. The fact that these waves can move in only one direction may result in their accumulation at the downstream walls. This, in turn, produces a buildup of fluid near the walls and eventually develops into a hydraulic jump, a narrow area in the flow in which the fluid transitions from the high velocity upstream to the low velocity downstream of the jump. The transition of the flow from one side of a hydraulic jump to the other is also characterized by an abrupt change in pressure, fluid depth and, of course, turbulence. The latter often results in excessive entrainment of air into the bulk of the fluid at the transition point, which is highly undesirable during mould filling.

26

Casting: An Analytical Approach

2.5 The Bernoulli Equation Once of the most commonly used solutions of the general fluid motion equations is the Bernoulli equation. It can be derived from Equations 2.8 and 2.9 when the flow is steady and inviscid, and can be expressed in the following form P 

1 UU 2  Ugh 2

C,

(2.25)

where g is the magnitude of the gravity vector and h is the height above a reference point. C is an abitrary parameter that is constant along any streamline. It can be evaluated by using pressure and velocity at a single point along the streamline P 

1 UU 2  Ugh 2

P1 

1 UU12  Ugh1 . 2

(2.26)

Stagnation, Dynamic and Total Pressure If the variation in fluid elevation h is small or gravity forces are negligible compared to pressure and inertia, like in high pressure die casting or in air, then Equation 2.26 can be reduced to P 

1 UU 2 2

P1 

1 UU12 . 2

(2.27)

As fluid accelerates along a streamline, pressure drops so that the sum on the lefthand side of Equation 2.27 stays constant. The maximum value of pressure occurs at the point where velocity is zero, or at the stagnation point. This pressure is called stagnation pressure. The term 1/2UU2 is the dynamic pressure, as opposed to the static pressure represented by P. The sum of static and dynamic pressures in Equation 2.27 is termed the total pressure. The Bernoulli equation in the form of Equation 2.27 led to the development of the theory of the airfoil [Abbott, 1959]. The difference between the static pressures on the lower and upper surfaces of an airplane wing creates the lift necessary to keep the plane in the air.

2.6 Compressible Flow Strictly speaking, all fluids are somewhat compressible. In other words, if external pressure is applied to a fluid volume, the latter will decrease in size. Among other things, compressibility of materials enables the propagation of acoustic waves. For most liquids, however, this change is negligible, even if the pressure is large. Fluids for which the compressibility effect is significant are called gases. The average distance between molecules in a gas is large, much larger than the size of

Introduction to Fluid Dynamics

27

the molecules themselves. This allows them to move freely in space, interacting with other molecules mostly through collisions. Unlike liquids, gases occupy all available space bound by solid or liquid surfaces as, for example, propane in a steel tank or an air bubble inside liquid metal. 2.6.1 Equation of State If a gas is not too dense and sufficiently hot, then two things can be said about its molecules. First, they interact with each other mostly through collisions, with only two molecules participating in any collision. Second, the kinetic energy of the molecules comes primarily from their translational motion. That is, molecules of a gas can be closely approximated by small, elastic, identical spherical balls moving around and colliding with each other in a chaotic manner. Such a fluid is called an ideal or perfect gas [Sedov, 1972]. The variables that define a thermodynamic state of a gas are pressure, density and temperature. For an ideal gas, they are related to each other through the equation of state: P

RUT ,

(2.28)

where R=8.3144 J mol– 1 K–-1 is the universal gas constant. One important result of this equation is that the thermodynamic state of an ideal gas is defined by just two parameters: density and temperature, pressure and temperature or pressure and density. Equation 2.28 is a very common equation of state that has been successfully applied to many real gases. In general, molecules in a real gas are far from spherical, or elastic, or even of the same size. Therefore, their rotational and oscillatory motions contribute to the total kinetic energy and are also exchanged during collisions. Moreover, if the gas is dense and cold, interactions between a pair of molecules cannot be described as simple collisions. In this case, the exchange of energy and momentum between molecules occurs over longer distances and times and with multiple molecules interacting at the same time. All these factors result in the behaviour that deviates from Equation 2.28. However, it is only significant at near cryogenic temperatures or very high pressures. For most gases, they are negligible in a wide range of temperatures and pressures. Air is an example of a compressible multi component real gas that can be described by Equation 2.28 with good accuracy. When modelling gas flow, the absolute values of pressure and temperature must be used. Degrees Kelvin or Rankine should be used for temperature. Unlike incompressible fluids, gauge pressure is not used for gases because pressure is present in the equation of state. The use of the absolute scale for these parameters is important for Equation 2.28 to be valid. A pressure of one atmosphere is 1.013 106 dyne/cm2 in CGS units or 1.013 105 N/m2 in SI units. Pressure, temperature and density for gases are always positive.

28

Casting: An Analytical Approach

2.6.2 Equations of Motion In general, the density of a gas can vary in time and space. The continuity equation that we wrote for incompressible fluids, Equation 2.9, is not valid in this case. It must be replaced by the full transport equation for density wU wUu wUv wUw    wt wx wy wz

0.

(2.29)

The full form of the specific thermal energy transport equation for gases. w ( UI ) w ( UI ) w ( UI ) w ( UI ) u  v  w wt wx wy wz § w 2T w 2T w 2T k ¨¨ 2   2 wy wz 2 © wx

· § wu wv ww · ¸  P¨¨ ¸.   ¸ x y w w wz ¸¹ © ¹

(2.30)

The last term on the right-hand side is the work term associated with the compression and expansion of the gas. It is equal to zero for incompressible fluids. Equation 2.30 manifests the first law of thermodynamics described in Section 2.6.3 below. Solution of the flow equations for liquids, Equations 2.8 and 2.9, is not coupled to the energy equation since neither density nor pressure depend directly on temperature, so that, generally, the solution of the energy transport equation, Equation 2.10, for liquids is optional. This is no longer true for gases. Both pressure and density depend on temperature through the equation of state. Therefore, the thermal energy transport equation above must always be included in the solution for gas flow. Compared to the momentum equations for incompressible fluids, Equation 2.8, the viscous terms in the momentum equations for gases include extra terms associated with compression:

Introduction to Fluid Dynamics

29

wu wu wu wu  u  v  w wt wx wy wz w 2u w 2u ·¸ 1 wP P §¨ w 2u P w  ˜    ˜ ¨ ¸ 2 2 2 U wx U U wx wy wz ¹ © wx wv wv wv wv  u  v  w wt wx wy wz



§ wu wv ww · ¨¨ ¸  Gx   wy wz ¸¹ © wx

w 2v w 2v ·¸ wv ww · 1 wP P §¨ w 2v P w § wu ¨¨ ¸  Gy   ˜    ˜   ¸ 2 2 w w w wz ¸¹ y x y U wy U ¨ wx 2 U © w w y z © ¹ ww ww ww ww  u  v  w wt wx wy wz 

1 wP P §¨ w 2 w w 2 w w 2w ·¸ P w  ˜    ˜ ¸ U wz U ¨ wx 2 U wz wy 2 wz 2 ¹ ©

(2.31)

§ wu wv ww · ¨¨ ¸  Gz .   wy wz ¸¹ © wx

The second term in parentheses on the right-hand side contains velocity divergence and represents the viscous force associated with the compression and expansion of the gas. According to Equation 2.8, it is equal to zero for incompressible fluids. 2.6.3 Specific Heats As mentioned in Section 2.1.3, specific heats at constant volume, CV, and at constant pressure, CP, differ significantly from each other for a gas. Because, when held at constant pressure, the gas expands upon heating. A part of the thermal energy goes into the work against the external pressure, leaving less energy for the actual heating of the gas. Consequently, more thermal energy is required to raise the gas temperature by 1o than when the gas volume is kept constant, and, therefore, CP is larger than CV. The difference between CP and CV is constant and identical for all ideal gases. It can be calculated from Equation 2.28 and the first law of thermodynamics. The latter states that the change in the total thermal energy of a gas, MdI, is equal to the amount of heat added to it, q, minus the amount of work done by the gas, W, MdI

q W,

(2.32)

where M is the total mass of the gas (see Figure 2.4). The work done by an expanding or contracting gas is the product of the gas pressure and the change in its volume

W

PdV .

(2.33)

30

Casting: An Analytical Approach

Figure 2.4. Heating of a gas volume at constant pressure P, during which its temperature and density change and the gas expands by volume dV, producing work PdV

Since V

M

U

,

(2.34)

then for the change in volume, dV V



dU

U

.

(2.35)

From Equation 2.28, it follows that when a gas is heated at a constant pressure, the corresponding changes in its density and temperature are related to each other: TdU  UdT

0,

(2.36)

and the heat flux in Equation 2.32 is by definition, q

MC P dT .

(2.37)

C v dT .

(2.38)

while from Equation 2.4, dI

Now combining Equations 2.32 – 2.38 yields

Introduction to Fluid Dynamics

CP  CV

31

(2.39)

R

Equation 2.39 states that the difference between the specific heats is the same for all gases that fit the ideal gas model. 2.6.4 Adiabatic Processes An adiabatic process is a process during which no heat is added or subtracted from the system, i.e., q = 0 in Equation 2.32. Using Equations 2.32, 2.35 – 2.39, it can be derived that in such a process the change in pressure is related to the change in density according to C P dU . CV U

dP P

(2.40)

After combining Equations 2.35 and 2.40, the equation of state, exrepssed in terms of volume and pressure, can be written in the following form J

P

§V · P0 ¨¨ ¸¸ , © V0 ¹

J

CP , CV

(2.41)

with pressure P0 and volume V0 expressing the state of the gas at a certain point in time. Note that, according to Equation 2.39, J> 1 for ideal gases. According to Equation 2.41, the thermodynamic state of gas during an adiabatic process is defined by only one parameter. This parameter can be either volume (or density), or pressure, or temperature. Given the simplicity of the equation of state, the adiabatic gas model is a useful approximation to gas flows where heat fluxes are small compared to other factors affecting the energy. This is often true when the process takes a relatively short time, for example, in supersonic flows. A Bernoulli-type solution can also be derived for processes governed by Equation 2.41 (see Section 2.6.7 below). 2.6.5 Speed of Sound In an acoustic, or sound wave, the material undergoes small, localized compressions and expansions. These changes in density result in corresponding changes in pressure and temperature. In turn, the variations in pressure create a force that causes these perturbations to propagate through the medium, that is, to actually behave like a wave. The rate at which acoustic waves propagate is called the speed of sound and is a property of the material. Note that, even though the sound speed in most solid and fluid materials is relatively large (of the order of several hundred metres per second), the actual displacement of the medium in an acoustic wave is small because of the small amplitude of the fluctuations in it. In other words, there is no transport of mass, energy and momentum associated with acoustic waves.

32

Casting: An Analytical Approach

Heat transfer due to conduction in a sound wave is also negligible because temperature gradients are small and also because the rate of heat transfer due to conduction is usually small compared to the speed of sound. This means that the propagation of sound waves is an adiabatic process. For small velocities in a sound wave, the viscous effects on its propagation are also negligible. The speed of sound, a, can be easily derived from Equations of motion 2.29 and 2.31 in the following way. Let’s assume for simplicity that the gas is initially at rest, in with a uniform pressure, temperature and density and that a single acoustic wave propagates in the x direction. As the wave moves, it introduces perturbations in the gas. Since all flow perturbations in such a wave are small, we will ignore all terms in these equations that are second order and higher with respect to these parameters. A change of density dU produced by the passing wave results in a corresponding change in pressure dP, which in turn causes a change in the velocity du. The latter two are related to each other through the momentum Equation 2.31 du



dt dP , U dx

(2.42)

where dx is the distance travelled by the wave in the time dt, that is, dx . dt

a

(2.43)

According to the mass conservation Equation 2.29, du and dU are also interrelated

dU

dtU

du . dx

(2.44)

Now substituting Equation 2.44 in Equation 2.42 yields a2

dP dU

,

(2.45)

where the derivative on the right-hand side is taken with the condition of adiabaticity. With the help of Equations 2.28, 2.34 and 2.41, this derivative can be evaluated as a

JP U

JRT .

(2.46)

Introduction to Fluid Dynamics

33

Equation 2.46 shows that in an ideal gas the speed of sound is a function solely of temperature. At higher temperatures, the molecules are more energetic and, therefore, are more capable of transmitting local changes in pressure and density to the adjacent gas volume in the end, resulting in an increase in the speed of sound. 2.6.6 Mach Number Sound waves provide the means of transmitting information about flow conditions in different parts of the gas. When the speed of the flow exceeds the speed of sound, that is, the flow travels faster than information about it, then shock waves can arise. Typically, a shock wave separates two regions of the gas with principally different flow conditions. On the upstream side, the flow is supersonic, and on the downstream side, it becomes subsonic. The transition occurs on a very small length scale which can be estimated as the distance travelled by a gas molecule between two successive collisions with other molecules. That is to say that the thickness of a shock wave can be as small as 0.03 micron. The flow of gas is characterized by the Mach number which is the ratio between the flow speed and the speed of sound M

U . a

(2.47)

Obviously, M > 1 in a supersonic flow and is less then one in the subsonic. When the Mach number if less than about 0.1, the compressibility effects can usually be neglected. It is possible then to model the gas as an incompressible liquid. Shock waves may occur when a supersonic flow meets with geometric obstructions, such as a flying airplane, or near sudden changes in the flow path, such as at an orifice or at the entrance to a vent. 2.6.7 The Bernoulli Equation for Gases The equations of motion 2.29 and 2.31 can be integrated for an adiabtic process to yields a solution similar to the Bernoulli equation for the incompressible fluids, Equation 2.25. Gravity is usually omitted for gases because its effect is small compared to pressure forces. The result is § J ·§ P · 1 2 ¨¨ ¸¸¨¨ ¸¸  U © J  1 ¹© U ¹ 2

C.

(2.48)

With the help of Equations 2.28 and 2.41, Equation 2.48 can be rendered in several other useful forms, for example,

34

Casting: An Analytical Approach

§ J ·§ P0 ¨¨ ¸¸¨¨ © J  1 ¹© U 0

·§ P ¸¨ ¸¨ P ¹© 0

· ¸ ¸ ¹

J 1 J



1 2 U 2

C,

(2.49)

and CPT 

1 2 U 2

C.

(2.50)

where the constant C is the same for all three equations. It can be determined by calculating the left-hand side at some point along the streamline. For example, it can be defined by the flow parameters at the stagnation point, that is, where U=0:

C

§ J · P0 ¨¨ ¸¸ © J  1¹ U0

C P T0 ,

(2.51)

where the subscript ‘0’ indicates the values of the respective parameters at the stagnation point. P0 and T0 are also called the total pressure and temperature, respectively.

Maximum Speed The Bernoulli equation can be used to calculate the maximum speed, Umax, that can be achieved in a steady-state adiabatic gas flow. Since J!1¸ then, according to Equation 2.49, the maximum speed is obtained at a point where pressure is equal to zero, or U max

2J P0 J  1 U0

2CPT0 .

(2.51)

Note that the maximum velocity is only a function of the gas stagnation temperature. Flow with close to adiabatic and steady-state conditions can be obtained by letting a gas escape from a large container (or cavity) through a small hole. If the container is sufficiently large compared to the hole, then we can assume that the conditions inside are close to stagnation. The maximum velocity can then be achieved by placing the container in vacuum. For air at room temperature of 300 Kelvin and CP = 1000 J/Kg K, the maximum velocity comes to about 775 m/sec.

Introduction to Fluid Dynamics

35

2.7 Computational Fluid Dynamics In this and other sections, we have presented full systems of equations of motion for fluids, together with an array of approximate approaches to solving them. These approximations have the benefit of offering analytical dependencies for various flow parameters in simple situations and are useful tools in a design process. At the same time, they provide only limited information and often cannot be successfully applied to general, transient three-dimensional flows with turbulence, heat transfer and phase change. In this case, numerical methods must be employed to solve the fluid flow equations. The science (and often art) of developing numerical approximations to the differential and integral equations of fluid motion is called computational fluid dynamics [Roache, 1985]. Fluid motion is described with non linear, transient, coupled, second-order differential equations. A numerical solution of these equations involves approximating its various terms with algebraic expressions. The resulting equations are then solved to yield an approximate solution to the original problem. The process is called simulation. 2.7.1 Computational Mesh Typically, a numerical model starts with a computational mesh, or grid. It consists of a number of interconnected elements of various shapes, e.g., tetrahedrals or bricks. These elements subdivide the physical space into small volumes where at least one node is associated with each such volume. The nodes are used to store values of the unknowns, such as pressure, temperature and velocity. The mesh is effectively the numerical space that replaces the original physical one. It provides the means for defining the flow parameters at discrete locations, setting boundary conditions and, of course, for developing numerical approximations of the fluid motion equations. The mesh discretizes the physical space. Each fluid parameter is represented in a mesh by an array of values at discrete points. Since the actual physical parameters vary continuously in space, a mesh with fine spacing between nodes provides a better representation of reality than a coarse one. We arrive then at a fundamental property of a numerical approximation: any valid numerical approximation approaches the original equations as the grid spacing is reduced to zero. If an approximation does not satisfy this condition, then it is incorrect. Reducing the grid spacing, or refining the mesh, for the same physical space results in more elements and nodes and, therefore, increases the size of the numerical model. But apart from the physical reality of fluid flow and heat transfer, there is also the reality of design cycles, computer hardware and deadlines, which combine in forcing the simulation engineers to choose a reasonable size for the mesh. Reaching a compromise between satisfying these constraints and obtaining accurate solutions is a balancing act that is no lesser an art than the CFD model development itself.

36

Casting: An Analytical Approach

Figure 2.5. Regular two-dimensional staggered computational mesh. The cell indexing i and j are shown for a two-dimensional case

The regular two-dimensional grid shown in Figure 2.5 is called staggered. All scalar quantities, such as pressure and temperature, are calculated and stored at the centre of each rectangular cell, and velocity vectors are assigned to the respective faces of the cell for better stability of the numerical algorithm. The full velocity vector can be reconstructed at the centre of the cell by simply averaging its components from the four faces. Each element of such grid has five nodes one for the scalar quantities and four for the respective velocity components. Each cell shares velocity nodes with its immediate neighbours. Grids like that one shown in Figure 2.5 are very easy to generate and store because of their regular, or structured, nature. A non uniform grid spacing adds flexibility when meshing complex flow domains. The computational cells are numbered in a consecutive manner using three indices: i in the x direction, j in the y direction and k in the z direction. This way each cell in a three-dimensional mesh can be identified by a unique address (i, j, k), similar to coordinates of a point in the physical space. Structured rectangular grids carry additional benefits of the relative ease of the development of numerical methods, transparency of the latter with respect to their relationship to the original physical problem and, finally, accuracy and stability of the numerical solutions. The oldest numerical algorithms based on the finite difference and finite volume approaches have been developed on such meshes and are still widely in use. 2.7.2 Numerical Approximations The finite difference (FD) method is based on the properties of the Taylor expansion and on the straighforward application of the definition of derivatives. It

Introduction to Fluid Dynamics

37

is the oldest of the methods applied to obtain numerical solutions to differential equations and the first application is considered to have been developed by Euler in 1768. The idea of the finite difference method is quite simple. For a function u(x), the derivative at a point x can be approximated as wu u ( x  'x)  u ( x) | . wx 'x

(2.51)

The spatial increment 'x can be now selected as the distance between two adjacent mesh nodes, u(x+'x) and u(x) are the values of the function at these nodes. Then Equation 2.51 can be rewritten in the form that is more common in CFD: u( xi )  u( xi  1 ) wu | wx xi  xi  1

ui  ui 1 . xi  xi  1

(2.52)

Here we use a subscript i to indicate the location on the x axis where the values of u and x are taken. Thus we obtain an approximation of the derivative of the function u(x) on a structured computational grid. From the definition of derivative, the accuracy of the approximation given by Equation 2.52 improves with smaller grid spacing. A similar approach is taken to approximate all first and second-order spatial derivatives in the fluid equations of motion, Equations 2.8 – 2.10 for incompressible fluids and Equations 2.29 – 2.31 for gases. The approximation of the second-order derivatives, which are present in the diffusive terms, typically requires at least three nodes. A time step, 't, is used to divide time into discrete increments so that the temporal derivatives can also be approximated: wu u (t  't )  u (t ) | wt 't

uin  1  uin , tn 1  tn

(2.53)

where superscript n refers to the order of the time points at which the values are taken. Once all derivatives are replaced with their respective finite difference approximations, the differential equations are replaced by a set of algebraic equations. As with the original equations of motion, proper boundary conditions must also be defined for the finite difference equations. These, for example, include the no-slip and heat transfer conditions at the interface between the fluid and the surrounding walls. The unknowns in that system of algebraic equations are the values of the flow parameters velocity, pressure and temperature at the mesh nodes. For a unique solution of this system to exist, the number of algebraic equations must match the number of nodes. Therefore, the size of a numerical model increases with the increase in mesh resolution. For example, for a three-dimensional mesh with a

38

Casting: An Analytical Approach

hundred nodes in each coordinate direction, a million equations must be solved. Moreover, the solution must be found at each incremental time point tn, starting at the beginning, t = 0, and finishing at a predefined moment tN = T, multiplying accordingly the number of operations required to find the complete solution. The actual procedure for the solution of each equation in the numerical model is usually quite simple. It is the huge number of the elementary operations that is so daunting for a human brain and where the electronic brain power is really needed. CFD simulations in this book use a combination of the finite difference and finite volume approaches to the solution of the fluid flow equations on computational grids similar to that shown in Figure 2.5. Unlike the finite difference method, the finite volume (FV) method uses the integral form of conservation laws. For every elementary control volume, fluxes of the conserved quantity (e.g., mass) are calculated at its boundaries. The net flux then translates into the change in that quantity inside the control volume. One of the strengths of the finite volume method is its conservation property adherence to the physical conservations laws that naturally arises from the method’s definition. For structured rectangular meshes, the look of the formal expressions for the control volume approximations is often indistinguishable from those of the finite difference method. 2.7.3 Representation of Geometry As mentioned above, finite difference methodology has been in use for many decades. In modern times, it is difficult to accurately describe the complex geometry within the framework of the traditional finite difference method. A rectangular cells is either fully open or fully blocked by the mould resulting in a familiar stair-step representation of the curved surfaces. Such a limitation makes the definition of accurate boundary conditions problematic. For example, the zigzag shape of the interface introduces unphysical additional flow losses due to friction. Also the surface area of a zigzag surface is larger than that of the original smooth surface, and the difference does not improve with mesh refinement. An innovative technique called Fractional Area Volume Obstacle Representation (FAVORTM) has been developed to remedy this problem [Hirt, 1985]. In this method, a geometric surface can cut through a rectangular mesh cell dividing it into blocked and open portions, as shown in Figure 2.6. The ratio of the open volume in a cell to its total volume is called the fractional volume. The intersections of the surface with the faces of the cell (six in three dimensions) are computed and stored as fractional areas, which are the ratios of the open area to the total area at respective cell faces. Similarly to the velocity vectors, the fractional areas are computed and stored at the staggered, cell-face locations. The complete geometry in every computational cell is thus converted into fractional volumes and areas. With adequate mesh resolution, the reconstruction of the original geometry from those fractional quantities is possible with a high degree of reliability.

Introduction to Fluid Dynamics

39

Figure 2.6. Geometric representation (shaded area) in the finite difference mesh using fractional area and volumes

The area and volume fraction are subsequently incorporated into the numerical equations and are also used to define boundary conditions. The resulting model provides significantly better accuracy in the numerical solution than the traditional stair-step approach. 2.7.4 Free Surface Tracking Free surface exists in most metals flows and, certainly, during filling. It is challenging to model a free surface in any computational environment because flow parameters and materials properties, such as density, velocity and pressure, experience a discontinuity at it. Moreover, the motion of the free surface is the result of a combination of dynamic and kinematic flow conditions. Proper account of these conditions is critical to accurate modelling of free surfaces. One of the commonly used methods to model free surfaces is the Volume-offluid (VOF) method [Hirt, 1981; Rider, 1998]. It consists of three main components: the definition of the volume-of-fluid function F, a method to solve the transport equation for F and the proper boundary conditions at the free surface.

Volume-of-fluid Function The VOF function F is defined equal to one in the fluid and to zero outside. Averaged over a cell volume, it becomes the fractional volume of fluid, that is, the amount of fluid in the cell divided by the cell’s total volume. Thus F=1.0 in a cell that is full of fluid and F=0.0 in an empty cell. A cell with free surface would have a value of F in the range between zero and one as shown in Figure 2.7.

40

Casting: An Analytical Approach

Figure 2.7. Illustration of the calculation of the Volume-of-fluid function in selected cells. The shaded area represents fluid

The VOF Equation The VOF function can be interpreted as a kind of a tracer ink added to a fluid. Then it it should be carried through space by the fluid. This consideration leads to a transport equation for the volume-of-fluid function: wF wFu wFv wFw    wt wx wy wz

0.

(2.54)

Equation 2.54 is similar to Equation 2.29 for gas density. It is often called the kinematic equation because there are no forces present. It just states that the values of F move around according to the velocity field, like smoke in the air. This applies to the free surface itself making Equation 2.54 automatically include the kinematic free surface boundary condition given by Equation 2.15.

Tracking Free Surface If conventional computational methods described earlier in this section are applied to the VOF equation the result will most likely be unsatisfactory because the free surface represents a boundary between the values of F in the fluid and outside. The derivative of F across the free surface is effectively infinite since its value changes from one to zero on an infinitely small length scale, that is, across the “thickness” of the free surface. Therefore, any attempt at approximating this variation with the finite difference method, e.g., using Equation 2.52, is bound to be very inaccurate. The most viable way to solve Equation 2.54 numerically is to use the geometric method, where the shape and location of the free surface within a computational cell are reconstructed using the values of the VOF function in its vicinity before computing the fluxes of the fluid volume at the cell faces. For example, if we know that fluid is filling the left portion of a cell, as shown in Figure 2.8, it will be some time before it crosses into the neighbour on the right.

Introduction to Fluid Dynamics

41

Figure 2.8. The calculation of fluid volume fluxes using the geometric method. The shaded area represents fluid, while vectors represent the direction of fluid motion. The cell in the middle must fill completely before fluid can cross into its neighbour on the right

Typically, the segment of the free surface within a computational cell is represented in three dimensions in a piecewise linear fashion, that is, with a section of a plane. Then its slope can be computed from the gradient of the VOF function, and its location is pinned by the value of F inside the cell. Then the respective amounts of fluid moving in or out of the cell at each of its six faces can be evaluated. This particular approach is sometimes called piecewise linear interface calculation or the PLIC method.

Free Surface Boundary Conditions The components of the VOF method described so far can be applied to both twophase (metal and air) and one-phase (metal and void) flows. In the former case, the space outside the F = 1.0 region is filled with air, and flow equations are solved for both fluids. In the one-phase case, the inertia of the air is neglected and the F=0.0 space is effectively empty, void of mass, represented only by uniform pressure and temperature. This approach has the advantage of not wasting CPU time on modelling air since in most cases the details of its motion are unimportant for the motion of the much heavier metal. However, in this case, free surface becomes one of the fluid external boundaries and requires the definition of dynamic boundary conditions. These boundary conditions are contained in Equations 2.16 - 2.20. A proper definition of the boundary conditions is important for accurate capture of free surface dynamics. 2.7.5. Summary The finite difference/finite volume approach to numerically solving the equations of fluid motion, combined with the FAVORTM and VOF methods provide the basis of the CFD calculations for incompressible and compressible flows in this book. The numerical solutions also contain turbulence, heat transfer and solidification. These and other numerical models are part of a general-purpose CFD code FLOW3D® [FLOW-3D, 2006]. This code has been used through this book to validate analytical models and to provide transient, three-dimensional solutions that are beyond those simplified approaches.

3 Part Design

3.1 Design of Light Metal Castings The advantages of light metal parts have secured them a place in modern machinebuilding industries. The ability of casting processes to produce near net-shape parts explains their popularity as a leading manufacturing method. All aspects of the engineering process, including the product, tools, process design, and engineering must be considered as a whole to achieve the best possible product. The product engineer must keep in mind that an abrupt change in the cross-sectional area of the part, thick walls, or small fillets will reduce the mechanical properties of the part. The tool designer must also understand how the die design will affect the performance of the part. An incorrectly chosen location for the ejector pins, parting line, or the water line’s layout can cause premature failure of the part. Metal temperature, process variation, and the amount of moisture in the die will also influence part performance. In this chapter, product design will be the main topic of discussion. It will help die casters understand the thought processes of the product designer. The development of Visual Basic applications and fundamentals of the static analysis of the mechanical system will help the die-cast engineer to understand product functionality, suggest modifications, and find the process that best complements the final product.

3.2 Static Analysis Static analysis is the subject of engineering mechanics that studies stationary solid bodies. In most real-life problems, bodies are subjected to complex loading. Finding stresses using analytical equations is extremely difficult. Complex cases are solved by breaking down complex loading cases into the sum of simple linear states. To simplify analytical solutions, the following assumptions must be implemented:

44

Casting: An Analytical Approach

1. The beam has a uniform cross section. 2. The beam satisfies Hooke’s law, which states that stress in the material is linearly proportional to strain. 3. The beam is of a homogeneous material with the same modulus of elasticity in compression and tension. 4. Beam deformation is significantly smaller than its size. Loads and reactions remain normal in relation to the axis of the beam. 5. The maximum stress does not exceed proportional limits. 6. The cross section of the beam after deformation remains flat and perpendicular to the longitudinal plane of symmetry. 7. No unbalanced forces are acting on the beam. Let us define the major geometric characteristics of a beam’s cross section. 3.2.1 Moment of an Area The moment of an area of an element can be determined by multiplying each element of the area dA by its distance y from the axis 0x of axis 0y. Therefore, ydA ; dS y

dS x

xdA .

(3.1)

The sum of the entire moment of an area of an element will give us the moment of the area: Sx

³ ydA ; S

y

A

³ xdA .

(3.2)

A

3.2.2 Moment of Inertia of an Area The moment of inertia of an area is the integral of the products, obtained by multiplying each element of an area by the square of its distance from the axis. Jx

2

³ y dA ; A

Jy

2

³ x dA . A

Moments of inertia of common shapes are given in Table 3.1.

(3.3)

Part Design 45 Table 3.1. Moments of inertia

Moment of inertia 1

2

3

Ix

Iy

h4 12

Ix

Iy

2 3 h t 3

Ix Iy

4

Ix Iy

5

6

ah 3 12 ha 3 12 th 3 § a · ¨ 3  1¸ 6 © h ¹ 3 ta § h · ¨ 3  1¸ 6 © a ¹

Ix

1 4

§ ah 3 Sd 4 · ¨ ¸ ¨ 3  16 ¸ © ¹

Iy

1 4

§ ha 3 Sd 4 · ¨ ¸  ¨ 3 16 ¸¹ ©

Ix Iy

th 3 a  H 3  h3 12 12 ht 3 a3 H  h  12 12





46

Casting: An Analytical Approach Table 3.1. Moments of inertia (continued)

7

t8

1 3

Iy

h 4  h  a 12

Ix Iy

9

10

3 ª 4 a· º § 4 «2b  2b  a  a¨ h  2b  ¸ » 2 ¹ »¼ © «¬

Ix

4

1 3 3 t h  a  ba 3  b  t a  t 3 1 3 3 t b  c  hc 3  h  t c  t 3

> >

Ix

Iy

Ix

Iy

@ @

Sd 4 64

S D 4  d 4 64

3.3 Loading Loading is the result of the rigid body’s contact with external components. Loading can be a force, torque, or a shearing traction. In a rigid homogeneous body, stress is always proportional to a load. There are two types of forces that are studied in engineering mechanics 1.

2.

External force Distributed force Concentrated force Dynamic force Impulse force Internal force

An external force is the result of interaction with other bodies. The most important question that engineering analysis can answer is how bodies behave under an external load. Three of Newton’s laws describe the behaviour of solid bodies under an external load:

Part Design 47

x x x

Newton’s first law: If external equally balanced forces are applied to a body, it will remain at rest or continue in motion at a constant velocity. Newton’s second law: The acceleration of a body is proportional to the resultant force in the direction of the resultant force. Newton’s third law: For every external force acting on a body, there is an equal reaction opposite in direction.

3.4 Stress Components An external load applied to a body causes stress inside the body. Stress is the result of the interactions between the elements of the body caused by the external loading applied to the body.

Figure 3.1. Solid body

Imagine that every object consists of an infinite number of small elements “A” (Figure 3.1). Under stress, these elements attempt to move closer together (compression stress) or away from each other (tensile stress). These types of stresses are called normal stresses. Stresses that cause elements to slide one against another are called shear stresses. If stresses on the side of any given element are pure compression or tension, the sides of this element are called principal planes. Stresses that act on the principal plane in a direction perpendicular to them are called principal stresses. Stresses acting on an element can be 1. Linear. When stresses in two principal directions are zero, the stress is considered linear or uniaxial. 2. Biaxial.

48

Casting: An Analytical Approach

When stress in one principal direction is zero, the stress is considered biaxial or plane. 3. Triaxial. When none of the three principal stresses equals zero, the stress is considered triaxial.

Figure 3.2. Single element

3.4.1 Linear Stress Linear stresses are often found in straight bars of uniform cross section. Let us assume that the bar (Figure 3.3) loaded on both ends is under uniformly distributed stresses. Uniformly distributed stress can be calculated as

V

P , A

(3.4)

where P is the force applied at the end of the bar and A the cross-sectional area of the bar. Stresses will be the same at any cross section parallel to the end of a bar. Shear stresses are equal to zero. To calculate stress at any cross-sectional area of a bar, let’s consider the section at angle Į to the direction of stress ı (Figure 3.4).

Part Design 49

Figure 3.3. Bar under uniformly distributed stresses

Figure 3.4. Bar section at angle Į to the direction of stress ı

The cross-sectional area of the bar is defined as AD

cos D . A

(3.5)

The total stress in the bar can be calculated as

V total

P AD

P cos D A

V cos D .

(3.6)

50

Casting: An Analytical Approach

Normal and shear stresses in any cross-sectional area can be calculated as

VD

V Total cos D

WD

V Total sin D

V cos 2 D .

V 2

sin 2D .

(3.7)

(3.8)

Normal stresses are considered positive when directed outwardly. Shear stresses are positive when they are clockwise. 3.4.2 Plane Stress An oblique plane at a random location is cut by a plane element as shown on Figure 3.5.

Figure 3.5. Stresses on an oblique plane

Two mutually perpendicular stress components ıx and ıy are acting on the side of the element. Normal and shear stresses on the edge of AĮ can be found by summing all of the forces caused by the stresses in the element.

V DX

Vx  Vy 2

V DY



Vx  Vy

Vx  Vy 2

2



cos 2D  W xy sin 2D

Vx  Vy 2

cos 2D  W xy sin 2D .

(3.9)

(3.10)

Part Design 51

Vx  Vy

W DXY

2

sin 2D  W xy cos 2D .

(3.11)

Summing both sides of Equation 3.9 yields

V X  VY .

V DX  V DY

(3.12)

Equation 3.11 shows that the sum of normal stresses in perpendicular directions are the same in any cross section of a body and are equal to the sum of principal stresses in the same direction. There are two ways to determine stresses in a plane element. Analytically, stresses can be determined using Equations 3.9 and 3.11. Differentiating Equations 3.9 and 3.10 with respect to Į, setting a derivative equal to zero yields the following results: 2W XY . V X  VY

tan 2D

(3.13)

In a similar manner, by differentiating Equation 3.11, we obtain tan 2D



V X  VY . 2W XY

(3.14)

Differentiating Equations 3.10 and 3.11 with respect to Į yields

V 1, 2

V X  VY 2

2

r

§ V X  VY · 2 ¨ ¸  W XY , 2 © ¹

(3.15)

2

W 1, 2

§ V  VY · 2 r ¨ X ¸  W XY . 2 © ¹

(3.16)

Graphically, stresses can be determined using Mohr’s circle.

3.4.3 Mohr’s Circle Otto Mohr introduced the graphical method for determining principal and shear stresses in 1882. Mohr’s circle is an effective way to visualize a stress state at any given point. Mohr’s circle shown in Figure 3.6, completely describes the stresses in the plane element in Figure 3.7. Normal stresses are plotted on the abscissa, and shear stresses are plotted on the ordinate. By convention, tensile stresses are positive and are plotted on the right side of the origin; compressive stresses are negative and are plotted on the left side of the origin. Shear stresses acting in a

52

Casting: An Analytical Approach

clockwise direction are positive, and stresses acting in a counter clockwise direction are negative. As shown in Figure 3.6, normal stresses V 1 and V 2 are plotted on the abscissa.

Figure 3.6. Mohr’s circle

Part Design 53

Figure 3.7. Stresses in a plane element

Equation 3.10 can be rewritten as V  Vy Vx  Vy (3.17) V DX  x cos 2D  W xy sin 2D . 2 2 1 into Equation Substituting the basic geometric equation cos 2 2D  sin 2 2D 3.17 yields 2

V  VY · § 2 ¨ V Dx  X ¸  W XY 2 © ¹ The radius of the Mohr’s circle is

2

§ V X  VY · 2 ¨ ¸  W XY . 2 © ¹

(3.18)

2

§ V X  VY · 2 (3.19) ¨ ¸  W XY 2 © ¹ The distance of the Mohr’s circle from the origin can be determined as V 2  V1 V Arg . (3.20) 2 From the centre “O” of the circle, we can draw a line at the 2Į angle. The intersection of this line with the circle will be at point “A.” Coordinates of point A will be V X and W XY . Coordinates of point B will be V Y and W YX Two problems can be solved using the graphical method:

R

1. Stresses acting parallel to the main coordinate system become known. Stresses in the direction inclined at angle Į to the main coordinate system must be determined.

54

Casting: An Analytical Approach

2. Stresses acting at an angle to a main coordinate system become known. Stresses parallel to the main coordinate system must be calculated. Example 3.1

An aluminium bar of a uniform cross section is suspended vertically. A uniform symmetrical load is applied at the bottom of the bar. A crosssectional element is subjected to stress plane conditions.

V x = – 60 mPa

V y = 115 mPa

W xy = 25 mPa Determine the principal stress and shear stress. We will solve this example using MATLAB® function Mohrdirect. ............................. Calculated values: .............................

Sigma1 = 118.501 mPa with angle 8.1301o Sigam2 = í 63.5014 mPa with angle 81.8699o Taymax = 91.0014 mPa with angle 37.0273o and 127.027o. Average normal stress SigmaAver = 27.5 mPa.

Part Design 55 Example 3.1 (continued)

3.5 Hooke’s Law In 1660, Robert Hooke discovered the law of elasticity, which describes the mathematical relations between stress and strain. When a straight bar is subjected to an axial tensile load, it elongates. Changes in the length of bars under a tensile load are called stretch. The elongation of an elastic body per unit of length is called strain. Bernoulli’s hypothesis states that, if the cross section of a bar was flat before deformation, it will stay flat after deformation, and will move along the axis of the bar in the direction of the applied force.

'l l

H

const ,

(3.21)

where 'l = total elongation of the bar and l = length of the bar. According to Hooke’s law, the relation between stress and strain can be described a

H

V E

,

(3.22)

where E = modulus of elasticity (Young's modulus). The modulus of elasticity is a constant characteristic of a material. Force applied to the end of the bar can be defined as P

³ VdA , A

where A is the cross-sectional area of the bar.

(3.23)

56

Casting: An Analytical Approach

Using Equation 3.22, we can obtain

V

HE = const.

(3.24)

P . A

(3.25)

Then Equation 3.23 will yield

V

3.6 Saint-Venant’s Principle To simplify the definition of the stressed-boundary conditions of a body, SaintVenant’s principle must be applied. Saint-Venant’s principle states that if forces acting on an elastic body are replaced by a statically equivalent system of forces, acting on the same relatively small area of the body, stresses in the parts of the body at a significant distance from the area of loading do not undergo any noticeable changes. This principle is shown on Figure 3.8.

Figure 3.8. Saint-Venant’s principle

Elongation can be expressed as

H

P . EA

(3.26)

Part Design 57

Lateral strain can be found as

H Lateral

'$ . $

(3.27)

Now we can define Poisson’s ratio, which relates the lateral to the axial strain for an axially loaded body:

P

H Lateral . H

(3.28)

3.7 Criteria of Failure The most important outcome of engineering calculations is determining of whether or not the structure will fail under known stress conditions. Engineering materials can be divided into two categories: 1. Ductile materials 2. Brittle materials An ultimate elongation of 5% is the dividing point between this two material types. Yield or permanent deformation is the criterion used to predict the failure of ductile materials. Rupture is the criterion used to predict the failure of brittle materials. The major characteristics of both types of materials are described in Table 3.2. Table 3.2. Characteristics of ductile and brittle materials

Ductile materials

Brittle materials

Exhibit considerable elongation

Exhibit little ultimate elongation

Yield point, yield strength clearly defined

Yield point is not clearly defined

Ultimate tensile and compressive strength are approximately the same

Stronger in compression than in tension

58

Casting: An Analytical Approach

3.7.1 Ductile Material Failure Theory There are three commonly used failure theories for ductile material:

1. 2. 3.

The maximum stress theory, which states that failure occurs whenever maximum normal stress equals or exceeds maximum tensile strength. The maximum shear stress theory, which states that failure occurs when the maximum shear strength reaches one-half of the yield strength. The maximum distortion energy theory, which states that failure occurs when von Mises stress reaches the yield strength of the material.

3.7.2 Brittle Material Failure Theory There are three commonly used failure theories for brittle material: 1. 2.

The maximum normal stress theory, which states that failure occurs when the ultimate strength is reached. The Coulomb-Mohr theory, which states that failure occurs when the stress condition satisfies the following equation:

V V1  3 t 1. V UT V UC 3.

(3.29)

A modified Mohr’s theory, which states that failure occurs when Equation 3.29 is satisfied, except in the fourth quadrant, where V 1 is in tension and

V 2 is in compression. At this point, the material is stronger than the Coulomb-Mohr theory would have predicted. 3.8 Beam Analysis Recent studies have shown that Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642) made the first attempts to develop the beam theory; however, it was not until the late 19th century that Leonard Euler and Daniel Bernoulli developed the elementary beam theory, which for the first time gave engineers a way to calculate a beam’s deflection and loading capacity in a relatively quick and easy way. This theory was first tested during the construction of the Eiffel Tower (1887–1889) and the Ferris Wheel (1893). Euler-Bernoulli’s equation for displacement is f ( x)

d2 dx 2

§ d2y· ¨ EI ¸, ¨ dx 2 ¸¹ ©

(3.30)

Part Design 59

where E = Young’s modulus I = area moment of inertia of the beam’s cross section y = displacement The beam under consideration is of uniform cross section and isentropic material. Based on these assumptions, E and I are constant. Then the Euler-Bernoulli equation can be written as EI

d4 w( x) dx 4

f ( x) , 0 d x d L .

(3.31)

The right side of the Equation 3.31 is a forcing function:

f ( x)

q( x)  f 0G ( x  x f )  W

d G ( x  xW ) , dx

where q(x) = Distributed external load f0 = Pointwise force applied at xf IJ = Torque applied at xIJ į = Dirac4 delta function. Table 3.3. Static beam boundary conditions

1

4

Dirac delta function was introduced by British physicist Paul Diarc.

(3.32)

60

Casting: An Analytical Approach Table 3.3. Static beam boundary conditions (continued)

Moment 0<x
M

P

Rotation slope a=0

( L  a) x L

T

a<x
M

T

PL2 16 EJ

Shear Force 0<x
G P( L  a) 2 2  u L2  L  a x  x 3 6 EJL

^>

@

Q

`

P

Q

G

P

½ P( L  a ) ­ L u® x  a 3  x 3  L2  ( L  a) 2 x ¾ 6 EJL ¯ ( L  b) ¿

>

@

2

Moment 0<x
qL2 2

Rotation slope x=0

§x x2 · ¨  2¸ ¨L L ¸¹ ©

T

G

3 4 qL4 ª x §x· §x· º «  2¨ ¸  ¨ ¸ » 24 EJ ¬« L ©L¹ © L ¹ ¼»



qL3 24 EJ

x=L

T Displacement 0<x
L  a L

a<x
a<x
M

PL2 16 EJ

a=L

a P ( L  x) L

Displacement 0<x




qL3 24 EJ

Shear Force 0<x
Q

§1 · q¨ L  x ¸ 2 © ¹

a L

Part Design 61 Table 3.3. Static beam boundary conditions (continued)

3

Moment 0< x
M

Rotation slope x=0

ª§ a b· x x2 º  ¸u  qa 2 «¨ » L¹ a 2a 2 ¼ ¬© 2 L

a<x
T

qa 2 L § a · ¨1  ¸ 6 EJ © 2L ¹

2





qa 2 L 12 EJ

· ¸ ¸ ¹

x=L 2

M

qa § x· ¨1  ¸ L¹ 2 ©

Displacement 0<x
G



qa 3 L 24 EJ

2 ª § º a · x «4¨1  » ¸ 2L ¹ a « © » » u« a · § 3 « » ¨  b¸x x4 » « ¹ ©2  4  « a 2 L2 a 3 L »¼ ¬

a<x
G

2 ª § º a · x «4¨1  ¸ » 2L ¹ a « © » « » a § · 3 « »  b x ¨ ¸ qa 3 L « » ©2 ¹   4 » 2 2 24 EJ « a L « » 4 « x4  x  a » « 3  » a3L » « a L « » ¬ ¼

T

§ a2 ¨1  ¨ 2 L2 ©

Shear Force 0<x
Q

§ a2 · ab   x ¸¸ q¨¨ L © 2L ¹

a<x
Q



qa 2 2L

62

Casting: An Analytical Approach Table 3.3. Static beam boundary conditions (continued)

4

Moment 0<x
M

qL2 6

Rotation slope x=0

§x x3 · ¨  3¸ ¨L L ¸¹ ©

T

Displacement 0<x
5



8qL3 360 EJ

Shear Force 0<x


7qL3 360 EJ

x=L

T

G



qL 360 EJ

3

5

§ x x x · ¨ 7  10 3  3 5 ¸ ¨ L L L ¸¹ ©

Q

§L 2 x2 · ¸ q¨¨  L ¸¹ ©6

Part Design 63 Table 3.3. Static beam boundary conditions (continued)

Moment 0
Rotation slope x=0

Displacement 0<x
Shear Force

G



T

M 0 L2 § x· ¨1  ¸ 2 EJ © L¹

2

M 0L EJ

Q=0

6

Moment 0<x
M

Rotation slope x =0

 Px

T

Displacement 0<x
G



PL3 6 EJ

PL2 2 EJ

Shear Force 0<x
Q

P

7

Moment 0<x
M



Rotation slope x=0

qx 2 2

T

Displacement 0<x
G



qL4 24 EJ

qL3 6 EJ

Shear Force 0<x
Q

qx

64

Casting: An Analytical Approach Table 3.3. Static beam boundary conditions (continued)

8

Moment 0<x
M



a<x
M

Rotation slope x =0

b

qx 2 2

T

a· §  qa¨ x  ¸ 2¹ ©

Displacement 0<x
b

Shear Force 0<x
Q

L  a 3

G

ª º b «3  4 3 » L » « « b4 » « 4  4 » 4 qL « L »  24 EJ « § b 3 ·» «u ¨¨1  3 ¸¸» L ¹» « © « 4 » «u x  x » «¬ L L4 »¼

a<x
G

L  a qL3 § b3 · ¨1  3 ¸ 6 EJ ¨© L ¸¹

ª b3 º «3  4 3 » L » « « b4 » « 4  4 » « L » 4 qL « § b 3 ·» «u ¨1  3 ¸¸ »  24 EJ « ¨© L ¹» « 4 » «u x  x » « L L4 » « 4 » «  x  a » «¬ »¼ L4

 qx

a<x
Q

 qa

Part Design 65 Table 3.3. Static beam boundary conditions (continued)

9

Moment 0<x
M



Rotation slope x=0

qx 6L

T

Displacement 0<x
G



qL4 120 EJ

qL3 6 EJ

Shear Force 0<x
§ x x5 · ¨4  5  5 ¸ ¨ L L ¸¹ ©

Q

qx 2 2L

Let’s derive practical equations that will allow us to estimate beam deflection for different beam types and loadings. 1. Simply supported beam with a pointwise load. The beam shown in Figure 3.9 is simply supported with pointwise load.

Figure 3.9. Simply supported beam with a pointwise load

E = modulus of elasticity I = moment of inertia of the section EI = bending stiffness L = length of the beam

66

Casting: An Analytical Approach

The static response of the beam includes [Yang, 2006] Shear force: Q( x)

EI

d3 w( x) . dx 4

(3.33)

M ( x)

EI

d2 w( x) . dx 2

(3.34)

Bending moment:

Rotation slope:

T ( x)

d w( x) . dx

(3.35)

Example 3.2 Simply supported beam subjected to a pointwise force P=10 N at xf = 0.5L length of the beam is 100 mm. Find the equation of the elastic curve. Let’s solve this problem using MATLAB® function beamstr. ............................. Initial Values: ............................. Case N 1 Table 3.3 Length of beam "L" = 100 mm Force "P" = 10 N "a" = 25 mm Modulus of elasticity "E" = 71000 MPa Moment of inertia "J" = 1645 mm4 ............................. Calculated values: ............................. Moment 187.5 N mm Maximum displacement Delta = 0.00119207 mm Location of maximum displacement x = 38.1881 mm

Part Design 67

3.8.1 Free Body Diagram In real-life problems, analyzing complex structures is often necessary. To simplify the analysis, a small portion of the system must be isolated. All possible interactions between the isolated segment and the rest of the system are replaced by forces, moments and reactions. If the system is in equilibrium, applied external loads must be in equilibrium as well. When forces are applied to this segment, it is called a free body diagram.

3.8.2 Reactions When external forces or moments are applied to a solid body, they exert reactions in the supports. For a solid body under external loads to remain in equilibrium, the sum of external loads and reactions must be equal to zero. For static analysis, the beam is replaced with its centre axis. All forces and reactions are applied to the central axis of the beam. The types of possible beam supports are shown in Table 3.4. Table 3.4. Static beam supports

Type of support Roller

Pin

Fixed

The types of beams are shown in Table 3.5.

68

Casting: An Analytical Approach Table 3.5. Support types

Types of beams Simply supported

Overhanging beam

Cantilever beam

3.9 Buckling When an axial compression load is applied to a long column, it retains a straight form equivalent to a certain value of the external force. At some point, when the load reaches a critical value, the column buckles sideways. In the 18th century, Swiss mathematician, Leonhard Euler, developed a formula that allows calculating the critical load for long axially loaded columns: PCR

S 2 EI

kl 2

,

(3.36)

where E = modulus of elasticity, I = area moment of inertia, l = unsupported length of a column, k = a constant whose value depends on the conditions of the end support of the column.

Part Design 69 Table 3.6. End restraint conditions and effective length

Restriction conditions

k

1

2

2

0.69

3

0.5

4

1

5

y/L 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

k 2 1.9 1.7 1.6 1.5 1.35 1.23 1.1 1

70

Casting: An Analytical Approach Table 3.6 End restraint conditions and effective length (continued)

Restriction conditions 6

k

y/L

k

0.1

2

0.2

1.8

0.3

1.7

0.4

1.6

0.5

1.4

0.6

1.3

0.7

1.1

0.8

0.98

0.9

0.76

Example 3.3

Initial conditions An aluminium bar has dimensions: A = 15 mm b = 26 mm L = 200 mm Is subjected to an axial load P. The modulus of elasticity is 71 000 mPa.

Part Design 71 Example 3.3 (continued)

1. Calculate the cross-sectional area of the beam: A = 15 u 26 = 390 mm2 2. The moment of inertia of the cross-section of the beam: 26 u 15 3 = 7312.5 mm4 12

I

3. Using MATLAB® function buckling, we can calculate the critical load and critical stress on the bar: ............................. Initial values: ............................ Moment of inertia I = 7312.5 mm4 Modulus of elasticity E = 71 000 mPa Length of beam L = 200 mm Area of beam cross-section A = 390 mm2 End restriction coefficient k = 2 ............................. Calculated values: ............................. Critical load = 32026.1 N Critical Stress = 82.1182 mPa

3.10 Bending Stresses To develop relations for bending stresses, the next assumptions must be made: 1. 2. 3. 4. 5. 6. 7. 8.

The material is isentropic and homogeneous. The beam has an axis of symmetry in the plane of deformation. The cross section of the beam remains plane after deformation. The beam is subjected to pure bending. The material of the beam obeys Hooke’s law. The beam is straight before an external load is applied. The beam has a constant cross section. Relations between the dimensions of the beam are such that the beam is going to fail by bending.

72

Casting: An Analytical Approach

3.10.1 Normal Stresses in Bending To simplify the development of the calculation procedure for beams subjected to a bending moment, let us examine the case of pure bending. In pure bending, only moment around the “x” axis acts on the beam. The condition of equilibrium of a beam under pure bending is MX

³ VydA .

(3.37)

A

A beam under pure bending is shown in Figure 3.10. When the bending moment acts in a positive direction, the upper zone of the beam is under compression stresses, and the lower zone of the beam is under tensile stresses. As shown in Figure 3.11, the bending stress is changing from maximum compression in the upper zone of the beam to maximum tensile in the lower zone of the beam. Stress distribution in the cross section of the beam is passing through a neutral zone, where it is equal to zero.

Figure 3.10. Beam under pure bending

The strain in a beam can be defined as the ratio of elongation of fibres divided by its original length.

H

dl . dy

(3.38)

The strain of fibres at distance z from a neutral axis with an initial length dy can be found:

H

( R  z )dI  dy dy

( R  z )dI  RdI RdI

z . R

(3.39)

Part Design 73

Figure 3.11. Bending stresses in a beam

From Hooke’s law, shown in Equation 3.22, normal stress is

V

EH .

(3.40)

Substitute the strain value from Equation 3.40 into Equation. 3.41.

V

E z. R

(3.41)

M . EJ X

(3.42)

Now Equation 3.33 can be rearranged as 1 R

If Equations 3.41 and 3.42 are combined, the following is obtained:

V

Mz . JX

(3.43)

Equation 3.43 shows that stress and strain are proportional, and maximum stress varies from zero on the neutral axis to the maximum value when the distance from the neutral axis is equal to zmax.

74

Casting: An Analytical Approach

3.10.2 Shear Stress Pure bending, as discussed in the previous chapter, is very rare in practical application. Pure bending in the case of the beam deformation was selected to simplify the development of the governing equations. Stress distributions throughout the cross section of a beam remains the same whether or not shear stresses are present. Relations between shear force and the bending moment are dM . dy

T

(3.44)

To visualize the shear stress distribution better in the beam’s cross section, we will cut out a section of the beam, as shown on Figure 3.12. Both the bending moment and shear force act on the section of the beam. Because of the shear force acting on the beam, the bending moment is changing along the length of the beam. If we refer to the bending moment at the near section as M, the bending moment on the far end of the beam will be equal to M + dM. Moment M produces normal stress ı. Moment M + dM will produce normal stress ı. The resultant stress in the far end section is greater than that in the near side section. To prevent the section of the beam from sliding in an íx direction, it must be balanced by a shear force. The balanced equation of the beam will be in form

N1  FS

N2 ,

(3.45)

where

Wbdx .

FS

(3.46)

The normal force acting on the beam’s cross section is equal to the area of the beam’s cross section times the normal stress. By designating dF as the beam’s cross section, the normal force acting on the near side section is ıdF. The force acting on the beam’s farside is ıdF. Now the force acting on the entire crosssectional area of the beam is

N1

³ VdF F

N2

"

³ V dF ³ F

³ F

My dF I

( M  dM ) y dF I

M S, I

M  dM S. I

(3.47)

(3.48)

Part Design 75

a

c

b

Figure 3.12. Shear stress distribution in a beam. a. Simply supported beeam under distributed load; b. changes in the bending moment along the length of the beam; c. stresses generated by the bending moment.

Next, using Equations 3.46–3.48 and solving Equation 3.45 for shear stress results in

W

TS . bI

(3.49)

where S = first moment of the area of the vertical face about the neutral axis b = width of the section at parallel distance y from the neutral axis I = second moment of inertia of the entire cross section about the neutral axis.

76

Casting: An Analytical Approach

3.11 Stresses in Cylinders In practical applications, cylindrical parts are often subjected to tangential and radial stresses. Examples can be passages that carry fluids in high-pressure, pressfit assemblies. Let us examine the cylinder shown on Figure 3.13. The cylinder has an inside radius “r” and an outside radius “R.” The cylinder is subjected to external pressure p1 and internal pressure p2. To calculate radial and tangential stresses in the cross section of the cylinder, we will assume that longitudinal elongation is constant around the circumference of the cylinder. This means that if the cross-sectional area of the cylinder was flat before an external force was applied, it will remain flat under stress as well.

Figure 3.13. Cylinder under internal and external stress

Figure 3.14. Stresses in the element of a cylinder

Part Design 77

Let’s cut out the element in Figure 3.14. There are radial ır and tangential ıt stresses acting on the element. Since the element remains in equilibrium, we can write

¦X

0;

¦Y

0.

(3.50)

Referring to Figure 3.14, the element is symmetrical relative to the X axis. This means that all forces projected on the Y axis in positive and negative directions cancel each other. The conditions of equilibrium in the X direction will be

¦X

V r rdG  (V r  dV r )(r  dr )dG  2(V t dr sin

dG ) 2

0 . (3.51)

With a certain degree of accuracy, the condition of equilibrium in the X direction is r

dV r  Vr  Vt dr

0.

(3.52)

To solve Equation 3.52, we need to find the radial and tangential stresses. Let’s denote the elongation of the surface with radius r as er, the elongation of the cylindrical surface with radius R as eR and the length of the element as l. Strain in the radial direction is

Hr

der . l

(3.53)

RdG  rdG , rdG

(3.54)

The tangential strain is

Ht

where R = r + l. Substituting R in Equation 3.54 will yield

Ht

(r  l )dG  rdG rdG

l . r

(3.55)

Hooke’s law for radial and tangential stresses will be

Vr Vt

E (H r  PH t ), 1  P2 E (H t  PH r ). 1  P2

(3.56)

78

Casting: An Analytical Approach

Putting the values of radial and tangential strain from Equations 3.53 and 3.55 into 3.56 will yield

E § der e ·  P r ¸, 2 ¨ r¹ 1  P © dr E § er de · ¨  P r ¸. dr ¹ 1  P2 © r

Vr Vt

(3.57)

Using the values of radial and tangential stresses from Equation 3.57 in Equation 3.52 will result in a differential equation:

d 2 er dr

2



e 1 de r  r2 r dr r

d ª 1 d er r º dr «¬ r dr »¼

0,

0.

(3.58)

(3.59)

The general solution of Equation 3.59 can be found by integrating it twice: er

A1r  A2

1 . r

(3.60)

A1 and A2 are constants of integration. They can be derived from boundary conditions. By combining equations 3.60 and 3.57, the following are obtained:

Vr Vt

E ª 1  P A1  1 2 P A2 º». 2 « r 1 P ¬ ¼ E ª 1  P A1  1 2 P A2 º». r 1  P 2 «¬ ¼

In our case, the boundary conditions are as follows:

Vr

 p1 , V R

 p2

Using the boundary conditions in Equation 3.61 yields the following:

(3.61)

Part Design 79

E ª 1  P A1  1 2 P A2 º», 2 « 1 P ¬ r ¼ E ª 1  P A1  1 2 P A2 º». r 1  P 2 «¬ ¼

 p1  p2

(3.62)

From Equation 3.62, we can derive constants of integration: 1  P r 2 p1  R 2 p2 , E R2  r 2 1  P R 2 r 2  p1  p2 . E R2  r 2

A1 A2

(3.63)

Importing the constant of integration into Equations 3.60–3.63 will yield er

1  P r 2 p1  R 2 p 2 1  P r 2 R 2  p1  p 2 1 r .  in E E rin R2  r 2 R2  r 2

(3.64)

This equation was developed by French mathematician, Gabriel Lamé (1795– 1870).

Vr

r 2 p1  R 2 p2 r 2 R 2  p1  p2 1 .  2 2 R  r R2  r 2 rin

(3.65)

Vr

r 2 p1  R 2 p2 r 2 R 2  p1  p2 1 .  2 2 R  r R2  r 2 rin

(3.66)

Summarizing Equations 3.65–3.66, we can see that Vr  Vt const ,

(3.67)

P

(V r  V t ) const . (3.68) E Equation 3.64 proves that cross sections of a cylinder remain plane after radial deformation. For the special case when outside pressure p2 = 0,

H



Vr

§ r2 R2 · ¨1  2 ¸ p , 2 ¨ R r © r ¸¹

(3.69)

Vr

r2 R2  r 2

(3.70)

2

§ R2 · ¨1  2 ¸ p , ¨ r ¸¹ ©

80

Casting: An Analytical Approach

er

1 P r2 p 1  P r 2R2 p 1 r  , E R2  r 2 E R 2  r 2 rin

V r_max

(V r ) r

 p,

V t_max

V t r

1  c2 p, 1  c2

(3.71)

(3.72)

where c

r . R

Equation 3.68 is a special case when R o f and c o 0 :

V r r V t r V Total

 p,

(3.73)

p,

V 1  V 3 d >V @ .

(3.74)

V t r V r r

(3.75)

A special case is when c o 0 :

V1 V3

p,  p.

Then Equation 3.70 will be 2 p d >V @, p d

>V @ .

(3.76)

2

Equation 3.76 shows that the maximum pressure that can be applied to the inside of a cylinder is a function of the material, not the thickness of the cylinder wall. An increase in the wall thickness of the cylinder will not necessarily increase the maximum allowable inside pressure. Let’s develop an equation for the case when the cylinder is loaded only by external pressure only (p1 = 0 and p2 = p). Equations 3.64 – 3.66 will be

Part Design 81

er

1  P R2 p 1  P r 2 R2 p 1 r ,  in E R2  r 2 E R 2  r 2 rin

(3.77)

Vr



§ R2 r2 · ¸p , ¨ 1  R 2  r 2 ¨© rin2 ¸¹

(3.78)

Vt



§ R2 r2 · ¸p . ¨ 1  R 2  r 2 ¨© rin2 ¸¹

(3.79)

As we can see from Equations 3.78 and 3.79, both stresses are compression.

Vt ! Vr . On the inside surface of the cylinder,

V r r V t r er

0, 2 p, 1  c2 r 2  p. E 1  c2 

(3.80)

On the outside surface of the cylinder,

V r R V t R e r R

 p, 1  c2 p, 1  c2 · R §1  c2 ¨  Q ¸¸ p.  2 ¨ E ©1  c ¹ 

(3.81)

The stress distribution throughout the cylinder wall shows that maximum tangential stress is on the inside wall.

82

Casting: An Analytical Approach

3.12 Press-fit Analysis Press-fit assemblies are commonly used for many applications. A press or shrinkfit assembly eliminates the need for fasteners and simplifies the assembly operation in applications where liquids or gases must be stored or transported under high pressure. Let us examine the stress distribution in cases where two cylindrical parts are assembled on another. After assembly the inside cylinder will be under compression stress and the outside cylinder will be under tensile stress. Figure 3.15 shows the stress distribution in the thick wall cylinder as well as the press-fit assembly of two cylinders. The calculation procedure of the press-fit assembly starts with the definition of the contact pressure between the outside surface of the inside cylinder and the inside surface of the outside cylinder. The difference in diameters between the outside surface of the inner member and the inside surface of the outer member is called radial interference. Radial interference is the amount of radial deformation that assembled members experience:

G in  G out

G Total ,

(3.82)

where G

in

= radial deformation of the inner member

G out = radial deformation of the outer member. rin and c2 R Equation 3.81 will be

Let’s denote c1

R rout

G in G out

§ 1  c12 · ¨ ¸ ¨ 1  c 2  P in ¸, 1 © ¹ 2 § · Rp 1  c 2 ¨  P out ¸¸. 2 ¨ E out © 1  c 2 ¹

Rp E in

(3.83)

Part Design 83

Figure 3.15. Stress distribution in the press-fit assembly of two cylinders

Substituting the values of c1 and c2 into Equation 3.83 yield:

G in

G out

· § r2 ¸ ¨ 1  in 2 · Rp ¨ Rp § R 2  rin2 R  P ¸ ¨  P in ¸¸, in 2 2 2 ¸ ¨ ¨ E in E in © R  rin r ¹ ¸¸ ¨¨ 1  in 2 R ¹ © 2 · § R ¸ ¨1  2 2 · rout  R2 ¸ Rp ¨ Rp § rout ¨ P  P out ¸¸.  out ¸ ¨ 2 2 2 ¨ E out ¨ E out © rout  R R ¹ ¸ ¸ ¨1  r 2 out ¹ ©

(3.84)

Now use Equation 3.78 for total deformation:

G Total

Rp E in

§ R 2  rin2 · Rp ¨ ¸ ¨ R 2  r 2  P in ¸  E out in © ¹

2 § rout ·  R2 ¨ ¸ ¨ r 2  R 2  P out ¸ . © out ¹

Based on Equation 3.81, the interfacial pressure is

(3.85)

84

Casting: An Analytical Approach

p

G 2

2 in

2 § R  r rout P P  R2 R¨¨  in   out 2 2 2 2 E ( R r ) E E ( r R ) Eout   in in out out © in

· ¸ ¸ ¹

.

(3.86)

In the special case when the inner and outer cylinders are made of the same materials, Equation 3.82 can be simplified: p

G 2  rin2 ) R2 2 R 2 (rout 2  R2 ) E ( R 2  rin2 )(rout

2  R2 ) GE ( R 2  rin2 )(rout

2R3

2 (rout  rin2 )

.

(3.87)

As shown in Equation 3.77, the radial stresses in the inner and outer members are V R  p . When the interfacial pressure is known, Equation 3.81 can be used to determine the stress distribution in the inner and outer cylinders: R 2  rin2 V T _ in , p 2 R  rin2 (3.88) 2 rout  R2 V T _ out , p 2 rout  R 2 where V T_in = tangential stress in the inner cylinder

V T_out = tangential stress in the outer cylinder.

Example 3.4

Part Design 85

The steel ring is press-fit into the aluminium housing. Calculate the hoop stress in the aluminium housing. Properties and dimensions of aluminium housing: Outside diameter of aluminium housing (D3) = 100 mm Modulus of elasticity = 71 000 mPa Poisson ratio = 0.33 Maximum yield strength of the aluminium housing is 160 mPa Properties and dimensions of steel ring: Nominal outside diameter of the steel ring (D2) = 88 mm Inside diameter of the steel ring (D1) = 84 mm Modulus of elasticity = 206 800 mPa Poisson ratio = 0.29 Maximum yield strength of the steel ring is 285 mPa Diametral interference = 0.2 mm Safety factor for strength calculations = 2 ............................. Initial values: ............................. Outside diameter of Outer housing D3 = 100 mm Nominal outside diameter of the inside ring D2 = 88 mm Inside diameter of the steel ring D1 = 84 mm Diametral interference delta = 0.2 mm Modulus of elasticity of outer housing Eouter = 71000 mPa Modulus of elasticity of inner ring Einner = 206800 mPa Poisson ratio of outer housing Pouter = 0.33 Poisson ratio of inner ring Pinner = 0.29 Maximum yield strength of outer housing Youter = 160 mPa Maximum yield strength of inner ring Yinner = 285 mPa Safety factor = 2 ............................. Calculated values: ............................. Interfacial pressure = 10.4232 mPa Hoop stress = 86.86 mPa Maximum allowable stress = 80 mPa Maximum allowable diametral interference = 0.184205 mm

86

Casting: An Analytical Approach

3.13 Thermal Stresses When the temperature of an elastic body is changed, the body will experience changes in volume, area and length. The magnitude of the dimensional changes will depend on the coefficient of thermal expansion. If the object temperature changes uniformly, normal strain is

H th

D 't ,

(3.89)

where

D = coefficient of thermal expansion 't = temperature change.

There is no difference in determining stresses, whether a temperature change or an applied load causes the strain. According to Hooke’s law, if the constrained object experiences thermal expansion, the stress is

V th

Example 3.5

H th E .

(3.90)

Part Design 87

An aluminium beam of rectangular cross section is held rigidly at two ends. The modulus of elasticity of the aluminium E = 71 GPa, the coefficient of thermal expansion is 21.6e—6 1/0C. The dimensions of the beam are A = 25 mm b = 30 mm L = 300 mm. Initially the beam is at 250C. Find the cross-sectional area, strain, stress, and axial force in the beam after its temperature increases to 800C. We are going to use MATLAB® function for thermal stress. It will yield the results shown below. ............................. Initial values: ............................. Beam width a = 25 mm Beam height b = 30 mm Beam length L = 300 mm Modulus of elasticity E = 71 GPa Initial temperature Ti = 25 C Final temperature Tfinal = 80 C Coefficient of thermal expansion = 2.16e – 005 1/oC ............................. Calculated values: ............................. Change in area = 751.783 mm2 Stress in the beam = 84.348 MPa Axial force in the beam = 63411.4 N

3.14 Torque The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. Round is the most commonly used cross section for the bars subjected to pure torsion. Figure 3.16 shows the bar of a round cross section subjected to a torque moment MT. The torque angle depends on its distance from the rigidly held end of the bar and the magnitude of the torque moment. For further analysis, we will make the following assumptions:

88

Casting: An Analytical Approach

1. 2. 3. 4.

The distance between cross sections of the bar does not change. The cross sections of the bar that were flat before the load was applied will remain flat. The material obeys Hooke’s law. The magnitude of the torque moment is in the range of the linear relations between the value of the moment and the angle of twist.

Figure 3.16. Beam under torque

Based on the assumptions above, any small element separated from the bar under torsional load (Figure 3.16 ), is under pure torque. If the bar is in static equilibrium, the torque moment and shear stress relations can be described by

MT

³ bdAW

b

,

(3.91)

A

where b = distance from the centre of the bar to an element dA = area of the element W b = shear stress at distance b from the centre of the element. From Equation 3.91, we can conclude that the torque moment is equal to zero at the centre of the bar and has a maximum value at the outer surface. Let us separate the portion of the bar with length dl (Figure 3.17).

Part Design 89

Figure 3.17. Element of the bar under torque.

Due to the relatively small angle of rotation, the shearing strain is tgD | D

bc ac

rdD ’ dl

(3.92)

where dD

= the angle of twist (ș). dl The relation between rotational shearing strain and the twist angle is

D

rT ,

(3.93)

Substituting the rotation shearing strain by the shear stress and using Hooke’s law

W

GTr .

(3.94)

Where G = modulus of rigidity. Based on the previous made assumption that cross sections remain flat after a torque moment is applied to the bar, we can find the shear stress at any of point in the section:

Wb Solving Equations 3.94 and 3.91 yields

GTb .

(3.95)

90

Casting: An Analytical Approach

GT b 2 dA

MT

GTJ ,

³

(3.96)

A

where J = the polar second moment of area. Now we can derive the equation for the angle of twist: MT . GJ

T

(3.97)

Now, the angle of twist definition from Equation 3.92 is used to solve Equation 3.90. MTb . J

Wb

(3.98)

When b = r, the maximum shear stress will be

W max

MT r . J

(3.99)

Equation 3.94 can be used only for circular sections. For solid round sections

Sd 4

J

32

.

(3.100)

For a hollow round section,

J

S 32

D

4



 d4 .

(3.101)

The maximum shear stress for a round section is defined as

W max

16 M T . Sd 3

(3.102)

For a hollow section,

W max

16 M T . d4 · 3§ SD ¨¨1  4 ¸¸ D ¹ ©

(3.103)

Part Design 91

Let’s denote maximum shear strength as >W @ ; then W max d >W @ . From 3.101, we find that d t

3

16 M T

S >W @

.

(3.104)

From Equation 3.98, if d 4 D 4 , it is known that 16 M

D t 3

§ d4 · S ¨¨1  4 ¸¸>W @ D ¹ ©

.

(3.105)

With the strength requirements, it is sometimes necessary to determine the maximum allowable twist angle:

MT d >T @ . GJ

T max

(3.106)

The maximum diameter of a solid round bar is d t

4

32M T . SG>T @

(3.107)

The outside diameter of a hollow bar is D t

32 M T § d4 · S ¨¨1  4 ¸¸G>T @ D ¹ ©

.

(3.108)

92

Casting: An Analytical Approach

Example 3.6

An aluminium shaft is subjected to torque. The shear modulus is 71 000 MPa, length is 100 mm and diameter is 20 mm. Find the torque required to twist the shaft 30. ............................. Initial values: ............................. Polar moment of inertia J = 7853.98 mm4 Shear modulus G = 71000 MPa Diameter of shaft D = 20 mm Length of shaft L = 100 mm Twist angle 30 ............................. Calculated values: ............................. Torque T = 291976 N mm The same shaft subjected to 500 000 N mm torque. Find the maximum stress in the shaft. ............................. Calculated values: ............................. Maximum stress = 636.62 MPa Twist angle = 5.13741 Deg

Part Design 93

Example 3.7

A thin-walled shell is subjected to torque of 500 MPa. Wall thickness is 6 mm. Area enclosed to wall centreline is 265 000 mm2 Find tensional shear strength and average shear flow. ............................. Initial Values: ............................. Wall thickness t = 6 mm Area Enclosed to the centreline of shell A = 265 000 mm^2 Torque T = 500 N mm ............................. Calculated values: ............................. Torsional shear stress = 0.000157233 MPa Shear flow = 0.000943396 N/mm

3.15 Stress Concentration Development of the previously described static equations was based on the assumption that objects have a constant cross section and are of homogeneous

94

Casting: An Analytical Approach

material properties. In practical applications, imagining an object without any discontinuity in the cross section is difficult. This discontinuity can be product design or manufacturing related.

x x x x x x x x x

Examples of product design discontinuities are abrupt changes in cross-sectional area, holes, sharp corners, and changes in shape. Examples of discontinuities caused by the manufacturing process are machine tool marks, ejector pin marks, tear marks in skin (high pressure die casting), and heat sinks.

Such discontinuities are called stress risers. The area around the stress riser is called the area of stress concentration.

3.16 Fatigue To determine the strength of material, loads are applied gradually to allow sufficient time to develop the strain. This type of load is called a static load. In many practical cases, numerically repeated loads are applied to a part. It was noticed that materials could fail under repeated loads even if the stress is less than the maximum allowable. This type of failure is called fatigue. To determine the strength of material under repeated loads, a test specimen is subjected to a specific repeated or varying load until failure. The second test is then performed with less load than previously. The process is repeated until the desired number of cycles is reached. All results are plotted on an S-N curve (Fatigue strength-number of stress cycles). The stress under which the S-N curve becomes flat is called the endurance limit. Note that light metals do not have an endurance limit. The fatigue limit for these materials is determined based on the number of cycles they are subjected to before fatigue failure occurs. Several empirical equations for the endurance limit have been suggested. The most commonly used is the Goodman diagram. Figure 3.18 shows an example of the Goodman diagram developed in 1890. Each loading cycle stress varies from the maximum V max value to a minimum V min value.

Part Design 95

Figure 3.18. Goodman diagram

The mean stress is

Vm

1 V max  V min , 2

(3.109)

Va

1 V max  V min . 2

(3.110)

and the alternating stress is

In Equations 3.104 and 3.105, the additions and subtractions are algebraic. The Goodman diagram is plotted on a rectangular coordinate system, where V m is measured horizontally and V a is measured on the vertical axis. As seen on the Goodman diagram, when V m equals zero, V a is equal to an endurance limit for the fully reversed stress, and when V a equals zero, V m is the ultimate tensile strength. A more conservative approach was suggested by Soderberg in 1930. He suggested that when V a equals zero, V m is equal to the yield strength, instead of the ultimate tensile strength. Goodman stated that the ordinate of the points on the line AB represents the maximum allowable alternating stress V a that can be applied with the corresponding mean stress V m . Any points above the AB line represent the stress conditions that will eventually cause failure and any points bellow the AB line represent stress conditions with a margin of safety, more or less. Goodman’s diagram has been found to correlate positively with experimental results for steel and light metals.

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Casting: An Analytical Approach

3.17 MATLAB® Functions Used in Chapter 3 MATLAB® function minertia mohrdirect beamstr buckling bressfit thermal torque torquethin

Used Table 3.1 Example 3.1 Example 3.2; Table 3.3 Example 3.3 Example 3.4 Example 3.5 Example 3.6 and 3.7 Example 3.8

4 Process Design

4.1 Evaluation of Dimensionless Numbers Let’s estimate the dimensionless numbers defined in Section 2.3 for an aluminium alloy in a gravity pour and high-pressure die casting. For the metal we will assume the density U= 2,500 kg/m3, surface tension coefficient V= 0.9 N/m and dynamic viscosity coefficient P= 0.0015 Pa s [Campbell, 1991]. For the gravity casting, we will take that the metal is poured down a sprue of height H=0.25 m, at the bottom of which it gains the velocity of about U 2GH § 2.4 m/s. We can expect that velocities of that magnitude will prevail in the horizontal runner and will be somewhat smaller inside the cavity. For the characteristic length, we will pick the runner width of 0.02 m. A typical high-pressure die-casting process has a range of velocities and length scales because of the slow and fast shot stages, and also due to the small size of the gates compared to the runner and probably the average wall thickness in the die cavity. For the slow shot stage, we will use the velocity of the plunger, Us=0.2 m/s, as the approximation of the average metal velocity in the short sleeve of radius 0.3 m. We will assume that the shot sleeve is initially filled with liquid metal to 50%. For the fast shot, the plunger velocity of Uf=2 m/s will be used in the shot sleeve, the gate velocity of Ug=40.0 m/s and the gate width d = 0.002 m. The values of the dimensionless numbers that correspond to these parameters are shown in Table 4.1.

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Casting: An Analytical Approach

Table 4.1. Estimates of the dimensionless numbers for typical gravity and high-pressure die casting

Re We Bo Fr

Gravity pour

Slow shot (shot sleeve)

40 000 160 2.7 7.7

100 000 33 2 450 0.12

Fast shot Shot sleeve

Gate

1 000 000 3 300 2 450 1.2

67 000 4 400 0.03 400

From the values of the Reynolds number it is clear that the flow is turbulent in all cases. The highest level of turbulence may be expected in the shot sleeve due to its relatively large size. The variations in the values of the remaining parameters warrant a more detailed discussion.

4.1.1 Gravity Pour The Weber number of 160 in the gravity pour example sufficiently exceeds the critical value of 60 to expect a breakup of the metal surface and splashing in the sprue and runner. There is enough kinetic energy in the flow to overcome surface tension and tear it into individual droplets as the liquid stream shears and turns in the runner and impinges on the walls. The Bond number of 2.7 indicates that gravity may be able to stabilize the surface, but only where the flow is horizontal. Surface tension is still sufficiently strong to produce a noticeable curvature of the free surface. When the metal surface is not horizontal, gravity has either a neutral effect or may actually contribute to its breakup. Since the Froude number is equal to 7.7, the flow in the horizontal runner is supercritical. If the runner remains only partially filled for a while, then the formation of a hydraulic jump and additional surface turbulence is likely where the flow reaches the dead end. If this is inevitable, then measures need to be taken to decrease the intensity and duration of these jumps by filling the runner earlier or to force it where it does the least harm.

4.1.2 High-pressure Die Casting: Slow Shot The high values of the Weber and Bond numbers mean that surface tension by far is not an important factor for flow in the shot sleeve. Since the surface is mostly horizontal, gravity would flatten out any perturbations of the free surface due to surface tension. The Froude number of 0.12 is quite low, so surface waves will travel faster than the plunger. If there is no motion in the metal initially, then after the plunger begins to move, one can reasonably expect a single wave travelling ahead of it.5 This wave will typically be shallow, without breaking and overturning of metal. As 5

In actual situations there maybe significant motion in the metal prior to the movement of the plunger because of the residual flow left from the shot sleeve filling stage.

Process Design 99

the plunger moves forward, the level of metal rises and the crest of the wave will at some point start interacting with the ceiling of the shot sleeve. By this time, the wave is likely to have travelled the length of the shot sleeve several times, and it is hard to predict its exact location without a more detailed analysis. Ideally, the remaining air should be expelled into the runner and not trapped in the sleeve by the wave.

4.1.3 High-pressure Die Casting: Fast Shot If there is any free surface still remaining in the shot sleeve at the start of the fast shot, then surface tension is even less relevant here than during the slow shot. The Froude number of 1.2 means that the plunger is just fast enough to prevent a wave separation from its tip. In reality, the flow and wave formation are complicated by the cylindrical shape of the sleeve, the changing level of metal and by the plunger acceleration from slow to fast stages. To avoid air entrainment, the amount of the remaining air in the sleeve at the beginning of the fast shot should be minimized. The very high value of the Weber number at the gate, many times over the critical value, points to the very unstable nature of the free surface. Since the Bond number is small, gravity does not play any stabilizing role. The huge kinetic energy, combined with large perturbations of the flow at the gates due to friction and surface roughness, results in flow atomization. Soon after the metal emerges from the gates into the open space of the die cavity, it breaks up into a stream of small droplets. The rate and degree of atomization cannot be derived using only the dimensionless numbers, but we can be sure that it does occur. For example, the free surface in such flow is perturbed by small-scale subsurface turbulent eddies, and if these eddies have sufficient energy, then further surface rupture would occur. The large value of the Froude number confirms that gravity is not going to play a significant role in the flow at the gates. Once past the gates, the motion of the metal is controlled mostly by its kinetic energy, momentum and the geometry of the cavity. It takes only milliseconds for a small piece of metal to cover the distance between a gate and the far end of the cavity. In that time, gravity is only going to alter its velocity by a tiny fraction. For the travel time of say 10 ms, gravity adds about 0.1 m/s, which is negligible compared to the gate velocity of 40 m/s.6 The dimensionless numbers discussed in this section give a useful insight into some general aspects of flow. But they are not sufficient to obtain details of the interaction between different forces in the fluid, turbulence, waves and instabilities, temporal and spatial variations of the flow parameters. For this purpose in the rest of this chapter, we will discuss several simple approximate analytical solutions of Equations 2.8 and 2.9 that help to gain further understanding of metal flow in casting. 6

By dividing the Bond number by the Weber number, we get another dimensionless 2

relationship, Gl U , between gravity and inertia. The ratio is small, less than 0.01, for flow in high-pressure die casting.

100

Casting: An Analytical Approach

4.2 Flow in Viscous Boundary Layer Due to friction and the no-slip boundary condition at the walls of the runnners and mould, a viscous boundary layer develops in the flow. Usually the thickness of the boundary layer is thin compared with the overall size of the geometry. Unlike the bulk of the flow where gravity and pressure are its main driving forces, flow in the boundary layer is dominated by viscous forces. These considerations allowed Ludwig Prandtl to find a set of approximate solutions for laminar flow in the boundary layer [Prandtl, 1928]. Consider a one-directional, uniform flow of viscous liquid over a semi-infinite thin plate as shown in Figure 4.1. A viscous boundary layer starts developing downstream from the tip of the plate. Fluid velocity transitions within the boundary layer from zero velocity at the surface of the plate to the velocity U0 in the main flow. The thickness of the layer, G, increases with distance x along the plate according to

G

D

Px , UU 0

(4.1)

where D is a constant dependent on the definition of the transition point between the boundary layer and the rest of the flow. For example, if we define this transtion as the point where velocity is equal to 99.5% of U0, then D|5.2. Equation 4.1 shows that the thickness of the viscous boundary layers is not constant, but grows as the boundary layer gradually develops in the flow. The growth rate is proportional to the square root of the distance along the plate. Although Equation 4.1 was derived for a relatively simple, steady-state flow configuration, it has been successfully applied to much more general cases of transient three-dimensional flows and curved boundaries. Moreover, this solution holds up well for Reynolds number values of up to 400 000. We can estimate G in Equation 4.1 using examples in Section 2.3.6. For gravity casting, assuming that the length of the runner is L=0.25 m and flow velocity U0 = 2.4 m/s, the maximum thickness of the boundary layer comes to about 1.3 mm. For high-pressure die casting, with the same runner length and velocity of U0 = 5 m/s, it is even smaller, G§ 0.9 mm. In the gate, the thickness of the boundary layer is even smaller. All these estimates indicate that bulk flow occupies most of the available cross section of the runner system. Although the boundary layer in the two examples is thin, this is not necessarily to say that friction at the wall is negligible. In fact, the thinner the boundary layer, the larger the gradient of the velocity across it and thus the shear stress. Shear stress, W at any point on the surface of the plate is

W

§ wu · ¸¸ © wy ¹ y

P ¨¨

0.332 0

UPU 03 x

.

(4.2)

Process Design 101

Figure 4.1. Viscous boundary layer in flow over a horizontal plate

The magnitude of the shear stress decreases with distance x along the plate. The total friction force R acting on one side of a plate of length L and width b is R

b

³

L

0

Wdx

0.664b UPLU 03 .

(4.3)

Note that the force is proportional to the 3/2 power of the bulk flow velocty. If we stretch the model a bit, then we can probably say that the force in Equation 4.3 would also be acting on the metal in a runner of length L and cross-sectional perimeter b. For a high-pressure die casting runner of length L = 0.25 m, width b = 0.15 m and U0 = 5 m/s, the total force due to friction at the walls of the runner is about 1 N, which is far below the pressure force driving the flow. This means that flow in the runner can be approximated by the inviscid, or ideal, flow model with a good degree of accuracy.7 In most real world mould filling flow is turbulent. Turbulence usually originates within the viscous boundary layer and spreads into the bulk of the flow. The boundary layer itself becomes turbulent with a very thin laminar sub layer close to the wall. Turbulence results in widening of the viscous boundary layer and

7

The viscosity coefficient P is set equal to zero in Equation 2.8 for the inviscid flow model.

102

Casting: An Analytical Approach

in steepening of the velocity gradients in it. Consequently, the viscous friction forces increase. Nevertheless, it is useful to see how the inviscid flow model can be applied to describe certain aspects of liquid metal flow.

4.3 The Bernoulli Equation Once of the most commonly used solutions of the general fluid motion equations is the Bernoulli equation. It can be derived from Equations 2.8 and 2.9 when the flow is steady and inviscid and can be expressed in the following for: P 

1 UU 2  Ugh 2

C,

(4.4)

where g is the magnitude of the gravity vector and h is the height above a reference point. C is an abitrary parameter that is constant along any streamline. It can be evaluated by using pressure and velocity at a single point along the streamline: P 

1 UU 2  Ugh 2

P1 

1 UU12  Ugh1 2

(4.5)

4.3.1 Stagnation, Dynamic and Total Pressure If the variation in fluid elevation h is small or gravity forces are negligible compared to pressure and inertia, as in high-pressure die casting or in air, then Equation 4.5 can be reduced to P 

1 UU 2 2

P1 

1 UU12 2

(4.6)

As fluid accelerates along a streamline, pressure drops so that the sum on the lefthand side of Equation 4.6 stays constant. The maximum value of pressure occurs at the point where velocity is zero, or at the stagnation point. This pressure is called stagnation pressure. The term 1/2UU2 is the dynamic pressure, as opposed to the static pressure represented by P. The sum of static and dynamic pressures in Equation 4.6 is termed the total pressure. The Bernoulli equation in the form of Equation 4.6 led to the development of the theory of the airfoil [Abbott, 1959]. The difference between the static pressures on the lower and upper surfaces of an airplane wing creates the lift necessary to keep the plane in the air.

Process Design 103

4.3.2 Gravity Controlled Flow For gravity casting, gravity is important and cannot be neglected. When a streamline is located on the free surface, then, according to Equation 2.16, pressure along it is constant and, therefore, Equation 4.4 can be reduced to 1 2 U  gh 2

1 2 U1  gh1 . 2

(4.7)

Figure 4.2. Examples of flows where the free surface is also part of a streamline

Figure 4.2 shows three examples of such flow. In the first, metal is poured out of a ladle, in the second, it flows over a weir and in the third case, it flows out of a small hole in a container. In the latter case, the streamline is not located completely on the free surface, but its beginning and ending points are. In each of these three cases, flow originates in the area where the metal velocity is small compared to the velocity in the jet, where the flow is the fastest. If we select that quiescent area to evaluate the right-hand side of Equation 4.7 and assume U1§0, then for the velocity elsewhere along the streamline, we get

U

2 g (h1  h)

2 g'h .

(4.8)

This is the classic expression for the velocity in a free flowing fluid, which is named after Torricelli. When pouring from a ladle held 'h = 0.2 m above the pouring basin, the velocity at which the jet impinges on the pool of metal in the basin is about 2 m/s, irrespecive of the alloy in the ladle.

104

Casting: An Analytical Approach

4.3.3 Flow in the Runner System Once the metal is in the runner system, the Bernoulli principle, Equation 4.5, can be applied to calculate he velocity of metal at various points in the flow. Let us assume that flow is inviscid and that all air has been purged, that is, the runner is completely filled by metal. If flow originates from the same reservoir, e.g., a pouring basin or a shot sleeve, then the right-hand side of Equation 4.5 is the same for all streamlines within the runner, (Figure 4.3).

Figure 4.3. Streamlines during gravity (left) and high pressure die casting filling

Let us also assume that the values of h1, U1 and P1 are known and C1 is the value of the expression on the right-hand side of Equation 4.5. Then at any point in the flow inside the runner system, P 

1 UU 2  Ugh 2

C1 .

(4.9)

At the same time, due to the incompressibiity of metal, the flow rate, Q, at any cross section must be the same: UA

Q,

(4.10)

where A is the cross-sectional area. If the runner splits into multiple channels, then the left-hand side of Equation 4.10 is the sum over all channels.

Process Design 105

When the metal emerges from the runner system into the casting cavity, the metal pressure at the gates is equal to the initial pressure of the air in the cavity, Pa. Then according to Equation 4.9, the initial velocity of the metal at a gate, Ug, is P · §C 2¨¨ 1  gh  a ¸¸ . U U ¹ ©

Ug

(4.11)

Equation 4.11 says that if there are multiple gates, the initial velocity of metal at these gates will differ only because of the variations in their heights relative to each other and does not depend on the size of each gate. For high-pressure die casting, where gravity effects on the flow in the runner system can be ignored, Equation 4.11 becomes Ug

§C P · 2¨¨ 1  a ¸¸ U U¹ ©

(4.12)

meaning that metal emerges at all gates at the same speed, irrespective of the size of the gate. This does not mean that the size of a gate is not important. If one of the gates is made larger, then the flow at each gate will come out more slowly, yet the initial metal velocity at all gates will be equal. These conclusions hold for the initial gate velocities. As the cavity fills up, gate velocities will vary in less predictable ways because of the back-pressure buildup. Another assumption implicitly used in calculating the initial gate velocities was that metal arrives at all the gates at the same time. If it does not, then we cannot use the same value of C1 for all gates. Note that in none of the examples, where we have applied the Bernoulli equation, is flow really steady-state. But temporal variations of the flow are small and, as before, we hope that our approximations are valid to a reasonable degree. Let us now look at the variations of pressure in a runner system as a function of flow rate and cross-sectional area. Substituting Equation 2.34 in Equation 2.33 and solving for pressure yields P

C1  Ugh 

UQ 2A

.

(4.13)

According to Equation 4.13, pressure will be the lowest in the areas with the smallest cross section. In sand castings, if pressure drops below one atmosphere, air aspiration through the porous mould may occur, causing porosity in the final casting. In a die casting, where the mould is impervious to air, low pressure in the runner system is less likely to cause defects in the casting unless it is near a partition line through which air can seep. But if pressure drops below the pressure of the dissolved gasses in the metal, then cavitation may occur, when these gases, mainly air and hydrogen, evolve into bubbles. The appearance and violent collapse

106

Casting: An Analytical Approach

of these bubbles is usually accompanied by large momentary pulses of pressure that can damage the surface of the die. After ignoring the gravity term, Equation 4.13 can be rewritten as P

C1 

UQ 2A

.

(4.14)

The flow rate Q in high-pressure die casting is primarily defined by the preprogrammed motion of the plunger in the shot sleeve and less by the geometry of the runner system. If Q is sufficiently large and A small, then the pressure can drop to the level that may initiate cavitation. On the other hand, if the cross-sectional area of the runner decreases monotonically from the metal entry point at the shot sleeve end to the gate, then such low pressures are unlikely to occur.

4.3.4 Filling Rate Another useful application of the steady-state Bernoulli solution to an unsteady flow is for gravity filling with bottom gating shown in Figure 4.4. Let’s write Equation 2.29 for two points on a streamline beginning in the pouring basin and ending inside the cavity. The first point is located at the top of the pouring basin, and the second point R is somewhere inside the runner with the cross-sectional area AR. We will assume that the level of metal in the basin is fixed at the height h0 above point R and the metal velocity there is close to zero. The ambient air pressures above the basin and inside the cavity are the same, equal to the atmospheric pressure, Pa. Finally, we will assume that the free surface inside the cavity is more or less horizontal and h is its time-dependent height above the gate. Even though the flow is, of course, time-dependent, once metal enters the mould cavity, the rate of change of pressure and velocity is relatively small compared to the variations of pressure due to gravity. This is primarily due to the dissipation of the flow energy in the mould. So we can still assume that the Bernoulli equation holds to a good degree of accuracy at every time during filling. The Bernoulli equation for a point in the pouring basin and point R on a streamline is Pa  Ugh0

PR 

UU R2 2

,

(4.15)

where subscript R refers to the values at point R. The rate of change of the level of metal in the mould cavity can be expressed as a function of the flow rate and the average horizontal cross-sectional area in the cavity, A: dh dt

Q . A

(4.16)

Pressure PR in Equation 4.15 will vary in time as the level of metal in the cavity increases and can be approximated as

Process Design 107

PR0  Ugh .

PR

(4.17)

where PR0 is the value of the pressure at point R at the moment when metal enters the mould cavity. Finally, velocity UR can be expressed using the flow rate and the area AR: Q . AR

UR

(4.18)

Substituting Equations 4.15, 4.17 and 4.18 in Equation 4.16 and solving the resulting differential equation for h yields the following expression for h h

h0 

1 2g

2 ­° AR ª º ½° E E 2 gh g ( t t )     ® 0 0 » ¾, « A ¬ ¼ °¿ °¯

(4.19)

where E 2 ( P a  PR0 ) U and t0 is the time of metal entry into the mould cavity. The value of PR0 depends on variations in the cross-sectional area in the runner and gates between point R and the cavity. For example, if the area is constant and equal to AR, then PR0 can be estimated as Ugh1, where h1 is the height of the bottom of the mould cavity above point R (see Figure 4.4). From Equations 4.16 and 4.19, one can find the flow rate as a linear function of time: Q

A ª º AR « E  2 gh0  R g (t  t 0 )» . A ¬ ¼

(4.20)

From Equation 4.19, one can also calculate the time required to fill a mould. If hmax is the elevation of the top of the mould cavity, then the fill time, tfill is given by t fill

1 A g AR

> E  2gh

0



@

E  2 g (h0  hmax ) .

(4.21)

Given the values of the average horizonal cross-sectional area A of the casting and

E, Equation 4.21 gives the fastest possible fill time. Viscous and other flow losses

in the runner and gate system increase the fill time. Nevertheless, these equations are useful in providing rough estimates of the flow parameters, short of performing detailed numerical or experimental studies.

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Casting: An Analytical Approach

Figure 4.4. Derivation of the filling rate as a function of time

4.4 Flow in a Shot Cylinder The speed of the plunger in a horizontal shot cylinder must be carefully controlled to avoid unnecessary entrainment of air in the metal. If the plunger moves too fast, it creates large overturning waves on the surface of the metal that faciliate turbulent mixing of the air into the bulk metal. A plunger moving too slowly results in waves travelling along the length of the shot sleeve in both directions, preventing proper expulsion of air into the die cavity. In either case, the outcome is excessive porosity in the final casting. In an ideal situation, the slope of the metal free surface should be directed away from the plunger everywhere along the length of the cylinder and at the same time, should not be too steep. The dynamics of waves in a horizontal shot sleeve can be analyzed by drawing an analogy with the flow in an open channel. From the start, we will approximate a cylindrical shot sleeve by a channel of rectangular cross section filled with liquid metal to depth h0. This simplification of the shape of the cylinder is justified for initial fill fractions of 40-60% [Lopez et al, 2003] and allows for some useful solutions. For a shallow wave travelling along the free surface due to gravity g, the speed of the wave, c0, is given by c0

gh0 .

(4.22)

Equation 4.22 is valid for waves that are long and shallow compared to the mean depth of the fluid. Note that the wave speed is independent of the properies of the metal. If the speed of the plunger is too slow, these waves will travel a distance

Process Design 109

equal to several times the length of the sleeve, reflecting off the moving plunger and the opposite end of the sleeve, before the transition to the fast shot.

Figure 4.5. Schematic illustration of a propagating surface wave when the plunger is moving slowly (upper image) and the hydrauic jump forming ahead of the fast moving plunger

As the plunger accelerates, it first catches up, then overtakes the waves, that is, the flow becomes supercritical. As a result, metal piles up to the top of the sleeve in front of the plunger, creating a flow condition called hydraulic jump. The hydraulic jump moves ahead of the plunger and is similar to a shock wave in a gas where flow undergoes a sharp transition through its relatively thin front. Since waves cannot overtake it, metal in front of the jump “does not know” about its violent approach and continues to move slowly. As soon as the hydraulic jump engulfs a volume of metal ahead of it, the metal is quickly accelerated to a much higher speed. Figure 4.5 schematically illustrates the two flow zones in a short sleeve separated by a hydraulic jump. If the plunger is fast enough, then the metal will pile up to the top of the sleeve. If we neglect the relatively slow speed of the metal in front of the jump, then the speed of this front, D, can be estimated from the balance of mass as D

Up 1H

,

(4.23)

where Up is the plunger velocity and H is the fill fraction of the sleeve ahead of the front [Garber, 1982]. Equation 4.23 shows that the hydraulic jump always moves faster than the plunger and that, just like the wave speed in Equation 4.22, its speed is independent of metal properties. Equations 4.22 – 4.23 provide some guidance to what the plunger speed can be during the slow shot stage. A more detailed analysis is possible by modelling the flow of metal in a rectangular shot sleeve of length L and height H using the shallow water approximation [Lopez et al, 2000]. In this approximation, the flow in the vertical direction is neglected in comparison with the horizontal velocity component. The flow is modeled in two dimensions, with the x axis directed in the direction of motion of the plunger, and the z axis pointing upwards. If viscous forces are omitted, then the flow has only one velocity component, u, along the length of the channel. Pressure at every point in the flow is then hydrostatic:

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Casting: An Analytical Approach

P

P0  Ug (h  z ) ,

(4.24)

where h(x,t) is the height of the fluid at point x and time t, as shown in Figure 4.6.

Figure 4.6. Schematic representation of the flow in a shot sleeve and the coordinate system

With these assumptions about the flow Equations 2.8 and 2.9 reduce to w (u  2c) w (u  2c)  (u  c) wt wx w (u  2c) w (u  2c)  (u  c) wt wx

0,

(4.25) 0,

where c ( x, t )

gh .

(4.26)

The plunger speed in the positive x direction is given by dX/t = X’(t), where X(t) defines the position of the plunger at time t>0. At the surface of the plunger, u ( X (t ), t )

X ' (t )

(4.27)

At all other walls of the channel, including the end at x=L, the normal velocity component is equal to zero. The initial conditions at t=0 are X ( 0)

0,

X ' (0) u ( x,0)

0, 0,

h( x,0)

h0 .

(4.28)

Equation 4.25 defines two sets of waves travelling at the respective speeds of u+c and u-c along the metal surface. The quantity u+2c is conserved in the first set of waves, and u – 2c is conserved in the second set. Combined with the boundary

Process Design 111

and initial conditions, Equations 4.27 and 4.28, this yields the following solution for a wave that separates from the surface of the plunger at time t = tp 3 ª º X (t p )  «c 0  X ' (t p )» (t  t p ), 2 ¬ ¼ u ( x, t ) X ' (t p ), x(t )

(4.29)

2

h ( x, t )

1 ª 1 º gh0  X ' (t p )» . « g ¬ 2 ¼

Figure 4.7. Geometric representation of the slow shot soluion, Equation 4.29. The thick solid line represents the position of the plunger X(t) as any given time, with its slope defining the plunger velocity X’(t). The waves on the metal surface, or charactersitics, are represented by thin solid lines. The tangent of the slope - of a characteristic is equal to the speed of the wave. As the plunger accelerates, the speed of the waves originating at its surface, and hence the slope -, increases. After time tc, the speed of the plunger becomes constant, so all charactersitics that are created after this time have a constant slope. When two characteristics intersect each other at a point P, the metal surface slope becomes vertical causing overturning of the waves. The horizontal dashed line repesents the end of the shot cylinder at x=L.

As the plunger moves along the length of the channel, it sends waves forward. Each wave represents a small segment of the metal free surface and the column of metal directly below it (Figure 4.6). Flow parameters in each such wave are constant and depend only on the time of separation from the plunger, tp. According to Equation 4.29, once a wave detaches from the plunger, it travels ahead of it at the constant speed of u c

X ' (t p ) 

gh

gh0 

3 X ' (t p ) . 2

(4.30)

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Casting: An Analytical Approach

The metal depth, h, and velocity, u, are constant in the wave. They both increase with the speed of the plunger. Therefore, to maintain a monotonic profile of the metal surface away from the plunger, the latter must not decelerate, that is, its acceleration must not be negative at any time:

X ' ' (t ) t 0 .

(4.31)

If the plunger accelerates, then each successive wave will move faster than waves generated earlier. This will lead to the steepening of the surface slope as the waves travel further down the channel, and can result in overturning of the waves. The solution given by Equation 4.29 can be illustrated geometrically. Figure 4.7 depicts the motion of the plunger and of the waves on a graph with time plotted along the horizontal axis and distance along the vertical one. The thick line represents the function X(t). The thin lines represent travelling waves. Since their respective speeds are constant, these lines are straight and are called characteristics. The slope of each line is related to the wave speed given by Equation 4.30: tan(T )

u  c,

(4.32)

where Tis the angle between the characteristic and the horizontal axis. The characteristic that originates at the (0,0) point corresponds to the wave in a quiescent pool of metal with its speed given by Equation 4.22. As the plunger accelerates, the slope of the characteristics increases, therefore, they intersect each other at some point down their paths. The intersection point of a pair of characteristics is where the faster wave catches up with the slower one. Having two solutions at the same location can be interpreted as the slope of the free surface becoming vertical, at which point the wave is likely to overturn and trap air. However, if the intersection occurs beyond the end of the shot sleeve at x = L, then we can say that the two waves never meet and the surface slope does not reach the vertical.

Figure 4.8. Illustration for calculating the slope of the metal’s free surface

Process Design 113

Let us analyze the evolution of the surface slope between two waves generated at the plunger at close instances, t2 > t1 (see Figure 4.8). The slope is given by tan(D )



dh dx



h1  h2 . x1  x2

(4.33)

Using expression for x and h in Equation 4.29, we can rewrite the right-hand side as follows: (4.34)

tan(D )

2

2

1 1 ª º ª º c0  X ' (t1 )»  «c 0  X ' (t 2 )» « 2 2 1 ¬ ¼ ¬ ¼  g 3 3 ª º ª º X (t1 )  X (t 2 )  «c 0  X ' (t1 )» (t  t1 )  «c 0  X ' (t 2 )» (t  t 2 ) 2 2 ¬ ¼ ¬ ¼

If 't=t2-t1 is small, then we can expand X(t2) into a Taylor series around X(t1): X (t 2 ) X ' (t 2 )

1 X ' ' (t1 ) ˜ 't 2 ,... 2 X ' (t1 )  X ' ' (t1 ) ˜ 't  ....

X (t1 )  X ' (t1 ) ˜ 't 

(4.35)

Substitution of Equation 4.35 in Equation 4.34 and omitting higher-order terms with respect to 't yields

tan(D )

1  g

1 ª º «c 0  2 X ' (t1 )» X ' ' (t1 ) ¬ ¼ . 1 3 c0  X ' (t1 )  X ' ' (t1 )(t  t1 ) 2 2

(4.36)

Equation 4.36 gives the expression for the free surface slope as a function of time t. Note that if the plunger moves at a constant speed, i.e. X’’(t1)=0, then the right-hand side of Equation 4.36 becomes zero and the slope of the free surface is horizontal. If the plunger accelerates, then the denominator on the right-hand side of Equation 4.36 decreases and the slope grows with time. When the denominator turns zero, the slope becomes vertical. The maximum slope, Dmax, is achieved when the wave reaches the end of the shot sleeve at t=tL. This time can be computed from

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Casting: An Analytical Approach

tL

t1 

L  X (t1 ) . 3 c0  X ' (t1 ) 2

(4.37)

Replacing t in Equation 4.36 with tL and rearranging terms yields X D' ' (t1 ) max

1 3 ª ºª º «c 0  2 X ' (t1 )» «c 0  2 X ' (t1 )» tan(D max ) ¬ ¼¬ ¼ . 1 ª 1 3 ºª º ' ( ) ' ( ) tan( ) ( ) c  X t c  X t  D  L  X t 0 1 »« 0 1 » max 1 2 2 g «¬ ¼¬ ¼

(4.38)

Equation 4.38 represents the final and most useful form of the solution for the flow of metal in a shot sleeve. It can now be used to calculate the velocity of the plunger to maintain a certain slope of the metal surface during the slow shot stage. For example, if Dmax is set equal to 10°, then the plunger acceleration given by Equation 4.38 ensures that the slope of 10° is not exceeded anywhere and anytime during the motion of the plunger. Equations 4.31 and 4.38 can be combined to give a range of values for the plunger acceleration at any give time 0 d X ' ' (t ) d X D' ' (t ) . max

(4.39)

Two things are achieved when the plunger acceleration stays within this range. The slope of the metal surface is directed away from the plunger and toward the opposite end of the shot cylinder. Second, the slope will not exceed the angle defined by Dmax at anytime during the slow shot process. To obtain the solutions for X(t) and X’(t), Equation 4.38 needs to be integrated numerically with respect to t1 using the initial conditions for X and X’ given by Equation 4.28. Figure 4.9 shows numerical solutions for the plunger position, X(t), acceleration, X’’(t), and speed, X’(t) (the latter is shown as a function of both time and distance along the channel length) for several values of Dmax. The integration was done for a shot cylinder of length L = 0.7 m and height of H = 0.1 m and an initial fill fraction of 40%, i.e., h0 = 0.04 m. Note that the plunger motion is slower for smaller values of Dmax. It takes the plunger 1.66 seconds to get to the end of the shot sleeve for the most conservative case considered with Dmax = 5o; for Dmax = 90o, the time is 0.83 seconds. The difference between these two extreme cases is just 0.83 seconds. However, these times will be longer if we add an additional constraint of the plunger velocity not to exceed the critical velocity at which the metal surface reaches the ceiling of the channel at h = H [Garber, 1982]. The critical velocity of the plunger can be derived from the solution for h(t,x) given by Equation 4.29 [Tszeng and Chu, 1994]:

Process Design 115

X cr'



2 gH 



gh0 .

(4.40)

and is shown in Figure 4.9 by the horizontal dashed line. For the selected parameters of the shot sleeve, X’cr = 0.73 m/s. Even for Dmax = 5o, the plunger velocity reaches the critical value after it moved just over 60% of the channel length at tc = 1.35 s. For steeper surface slopes, the critical velocity is reached at earlier times, for example, for Dmax = 90o tc = 0.58 s and the plunger position is 22% of L.

Figure 4.9. Solutions of Equation 4.38 for the plunger position a., acceleration b., velocity c. and velocity as a function of distance along the length of the shot channel (d), at different maximum surface slopes Dmax: 1 – 90o, 2 – 60o, 3 – 45o, 4 – 30o, 5 – 15o and 6 – 5o. The horizontal dashed lines in plots c and d represent the critical plunger velocity given by Equation 4.40.

When the plunger reaches the critical velocity, the metal surface comes in contact with the ceiling of the shot cylinder. Beyond this point, the solution given by Equation 4.29 becomes invalid. It can also be argued that if the plunger continues to accelerate, then the potential for creating an overturning wave increases since all the energy of the flow is now redirected forward by the walls and ceiling of the channel. It is usually recommended to keep the plunger velocity below the critical value. The CFD simulation of the slow shot process, presented in Section 4.10.1, provides more details of the flow before and after the critical velocity is reached. It is often assumed that the overturning of the metal surface that causes air entrainment occurs when the wave profile becomes vertical, that is, Dmax = 90o. In reality, the breaking of the wave surface may happen at more moderate angles, as

116

Casting: An Analytical Approach

can be seen while observing ocean waves. Equation 4.38 allows the engineer to define any maximum permitted wave slope, obtaining a sufficient safety margin to avoid any air entrainment. Being able to define a safety margin for the surface slope is also important because we made some simplifications in arriving at Equation 4.38, such as replacing the cylindrical channel with a rectangular one. Obviously, the curved walls of the shot cylinder will exacerbate the potential for wave overturning as the metal level rises. Besides, the critical velocity is attained faster in a cylindrical channel than in a rectangular one of the same width and, therefore, an extra safety margin must be used in an estimating of the critical velocity by Equation 4.40. Equation 4.36 can be used to obtain the slope Dmin of the metal surface right at the plunger. This can be done by setting t = t1. Then it reduces to a very simple form tan(D min )



X ' ' (t1 ) . g

(4.41)

Equation 4.41 gives the initial surface slope for a wave detaching from the plunger at time t = t1; it is a function of only the plunger’s acceleration and not its position or even velocity. As the wave propagates along the length of the channel, it steepens reaching the maximum slope, Dmax, at the end of the channel at x = L, given by Equations 4.36 and 4.37. One can easily get carried away in preventing air entrainment by defining a very small value of Dmax in Equation 4.38. This would lead to a very slow plunger motion. Of course, in real life situations, the requirement of minimum air entrainment must be combined with other criteria that control the quality of the casting, such as a specific filling rate and minimal early solidification in the shot sleeve and runner system.

4.5 Gas Ventilation System The size of the cross-sectional area of the ventilation system has to be based on the volume of the gas to be evacuated and the parameter of the die casting process. They will define how much time is available to evacuate the air from the cavity of the die-cast die. The gas ventilation system has to be calculated as a converging– diverging nozzle. For subsonic flow in the air stream, the static pressure just inside the ventilation channel exit will be equal to the pressure in the downstream region. There are two reasons that explain that equality. There is no internal mechanism that can produce the difference in static pressure between the inlet and outlet of the ventilation channel. For subsonic flow, the pressures in the inlet and outlet are equal. Any disturbances in the outlet of the ventilation channel can be propagated back to the inlet through the entire flow field. A continuous increase in the upstream pressure will cause an increase in the mass flow as well as in velocity of

Process Design 117

the air until the local Mach number at the exit reaches unity. One-dimensional compressible flow theory can be applied to a ventilation channel. The effect of viscous friction in this model is neglected. The flow can be assumed to be isentropic when no shocks are present. The critical pressure ratio at which the flow becomes sonic can be determined [Munson, Young and Okiishi 2006]: J

§ p2 · ¨¨ ¸¸ © p1 ¹ CR

§ 2 · J 1 ¨¨ ¸¸ . © J  1¹

(4.42)

Using isentropic theory, the relation between mass flow rate m and crosssectional area A can be described as m p1 A

RT1

J

J 1 ª º § p2 · J » § 2 ·« ¸¸ «1  ¨¨ ¸¸ » , ¨¨ © J  1 ¹« © p1 ¹ » ¬ ¼

1/ J

§ p2 · ¨¨ ¸¸ © p1 ¹

(4.43)

J

§ 2 ·J 1 p ¸¸ for ¨¨ d 2  1 , and p1 © J  1¹ J 1

m p1 A

RT1

J

§ 2 · 2(J  1) ¸¸ ¨¨ , © J  1¹

(4.44)

J

§ 2 ·J 1 p ¸¸ . for 2  ¨¨ p1 © J  1¹

The mass flow through the ventilation channel depends on the ratio p2/p1 of downstream to upstream pressure when the flow is subsonic. If the pressure ratio is equal to or less than (p2/p1)CR, then the velocity at the throat becomes sonic. There is no convenient rule to apply define mass flow when the ratio of downstream to upstream pressure is more than critical. The general rule for obtaining mass flow through the ventilation channel is 1. If the ratio of downstream to upstream pressure is less than the critical mass flow, it can be determined using Equation 4.43. 2. If the ratio of downstream to upstream pressure is less than the critical mass flow, it can be calculated using Equations 4.43 and 4.44. The lesser of the two values is used. Let us determine how the flow in the throat section of the nozzle develops. As is shown in Figure 4.10: 1. The shock wave curves upstream.

118

Casting: An Analytical Approach

2.

The shock first originates upstream of the M =1 line of the supersonic zone.

Now let us look at the actual amount of mass flow compared with that one determined from one-dimensional theory. The actual flow, even disregarding friction, is as much as 5% less than that predicted by one-dimensional theory. The choked flow is about 0.7% less than that predicted by one-dimensional theory. These numbers will vary with the shape of the nozzle [Shapiro, 1953, V1]. Steady-state flow can be used for most engineering calculations. During the high pressure die-cast process, a sudden flow (within 10–15 m/s) changes from slow (20–30 m/s) to fast (150–200 m/s). In this case, unsteady flow has to be considered to describe the behaviour of the gas as it moves through the ventilation system.

Figure 4.10. Shock wave formation in a divergent nozzle

To avoid mathematical difficulties, it will be necessary to make a number of assumptions: 1. The flow will be considered geometrically one-dimensional, implying that all fluid properties are uniform over each cross section of the passage, and that changes in cross-sectional area take place very slowly. 2. The viscosity and thermal conductivity of the gas will be neglected. That means that all parts of the gas are related through the isentropic relations unless the shock of changing strength appears. 3. It will be assumed that the equation of state is that of a perfect gas. 4. The gravity effects are negligible. 5. The fluid can be treated as a continuum. Flow characteristics have to be established based on the normal shock relations for a perfect gas. The air ventilation system of the die-cast die is a diverging–converging nozzle with an abrupt decrease in cross-sectional area. Due to limitation of the measurement equipment in being unable to measure temperature, pressure, and

Process Design 119

humidity inside of the cavity of the die, measurements of the gas exiting die have to be used for further analysis. To evaluate the pressure and temperature inside the cavity of the die-cast die, equations that describe changes in the thermodynamic state due to abrupt changes in the cross-sectional area of the ventilation channel have to be derived. Basic thermodynamic equations for the conservation of energy, mass and conservation of momentum are used to define correlations between area contraction and variations in pressure, density and temperature. The Mach number is used as an initial parameter. The equation of conservation of energy can be written as V12 § J · p1 ¸¸  ¨¨ 2 © J  1 ¹ U1

V 22 § J · p2 ¸¸ ,  ¨¨ 2 © J  1¹ U2

where V = velocity of gas J = ratio of specific heat at constant pressure to specific heat at constant volume p = pressure, ȡ = density. Subscripts 1 and 2 indicate quantities before and after an abrupt contraction. Constant acceleration of gravity:

³

dp



U

1 2 V  gz 2

C.

(4.45)

For a perfect gas,

URT .

p

For an ideal gas in isentropic flow, p

C.

UJ From here,

U

p1 / J C 1 / J .

So, Equation 4.45 can be rewritten as C1 / J

³p

1 / J

dp 

1 2 V  gz 2

const .

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Casting: An Analytical Approach

The constant C can be evaluated at any point on a streamline: C1 / J

p11 / J

C1 / J

p12 / J

U1

,

or

U2

.

By integrating pressure terms between points 1 and 2: C

1/J

³p

1 / J

dp

C

1/J

J 1 J 1 · § J ·§¨ J ¸¸ p 2 ¨¨  p1 J ¸ ¸ © J  1 ¹¨© ¹

p § J ·§ p 2 ¸¸¨¨ ¨¨  1 U1 © J  1 ¹© U 2

· ¸¸ . ¹

Now Equation 4.45 can be rewritten: V2 § J · p1 ¨¨ ¸¸  1  gz1 2 © J  1 ¹ U1

V2 § J · p2 ¨¨ ¸¸  2  gz 2 . 2 © J  1¹ U2

For one-dimensional flow z1=z2 and Equation 4.45 can be recast as V2 § J · p1 ¸¸ ¨¨  1 2 © J  1 ¹ U1

V2 § J · p2 ¸¸ ¨¨  2 . 2 © J  1¹ U2

(4.46)

U 2 A2V2 .

(4.47)

The continuity equation is

U1 A1V1

The equation of conservation of momentum is

U 2 A2V22  U1 A1V12

 A2 ( p2  p1 ) .

(4.48)

Equation 4.45 can be rearranged as

Jp § J  1· 2 ¨ ¸V1  1 U1 © 2 ¹

2

Jp § J  1 ·§ V 2 · 2 ¨ ¸¨¨ ¸¸ V1  2 . U2 © 2 ¹© V1 ¹

The speed of sound can be determined using the equation:

(4.49)

Process Design 121

§ wp · ¸¸ . ¨¨ © wU ¹ s

c

(4.50)

Subscript s is used to designate that the partial differentiation occurs at constant entropy. Equation 4.50 suggests that the speed of sound can be calculated by determining the partial derivatives of pressure with respect to density at constant entropy. For an isentropic ideal gas, c2

§ wp · ¸¸ ¨¨ © wU ¹ s

CJU J  1

p

JU J  1

UJ

J

p

U

.

(4.51)

When Equation 4.51 is substituted in Equation 4.49, 2

Jp § J  1 ·§ V 2 · 2 ¨ ¸¨¨ ¸¸ V1  2 U2 © 2 ¹© V1 ¹

§ J  1· 2 2 ¨ ¸V1  c1 © 2 ¹

or 2

2 § J  1 · V1 ¨ ¸ 2 1 © 2 ¹ c1

§ J  1 ·§ V 2 ¨ ¸¨¨ © 2 ¹© V1

§ J  1· 2 ¨ ¸M 1  1 © 2 ¹

p U § J  1 ·§ V 2 · 2 ¨ ¸¨¨ ¸¸ M 1  2 1 . V 2 p1 U 2 © ¹© 1 ¹

· V12 p U ¸¸  2 1 , 2 p1 U 2 ¹ c1

and finally 2

(4.52)

From the equation of conservation of momentum, p2 p1

1J

§ A1 V · M 12 ¨¨1  2 ¸¸ A2 V1 ¹ ©

1J

§ U A · A1 M 12 ¨¨1  1 1 ¸¸ A2 U 2 A2 ¹ ©

(4.53)

And from the equation of continuity, V2 V1

A1 U1 . A2 U 2

(4.54)

By using the perfect gas equation of state, the temperature ratio can also be computed:

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Casting: An Analytical Approach

p2 / p1 U 2 / U1

T2 T1

(4.55)

Now, using Equation 4.54, § M2 · ¨ ¸ ¨M ¸ © 1¹

2

2

V2 / c2 2 V1 / c1 2

§ V2 · p1 U 2 ¨ ¸ ¨V ¸ p U © 1¹ 2 1

 A1 /

2

A2

(4.56)

p2 U 2 p1 U1

and, therefore, A1 M1 A2

M2

p2 U 2 p1 U1

.

When Equations 4.53 and 4.56 are substituted into Equation 4.52, § J  1· 2 ¨ ¸M 1  1 © 2 ¹ § J  1 ·§ U 1 · ¸¸ ¨ ¸¨¨ © 2 ¹© U 2 ¹

2

§ A1 ¨¨ © A2

2

2

(4.57)

2

· §A · §U · U ¸¸ M 12  1 1  JM 12  J ¨¨ 1 ¸¸ M 12 ¨¨ 1 ¸¸ . U2 ¹ © A2 ¹ © U2 ¹





which can be recast into a quadratic equation for the density ratio: § U1 · ¨ ¸ ¨U ¸ © 2¹

2

2

§ A1 · J  1 U1 ¨ ¸ M 12  ¨A ¸ 2 U2 © 2¹

§ A · M 2 (J  1)  2 ¨1  JM 12 1 ¸  1 ¨ 2 A2 ¸¹ ©

0 . (4.58)

Equation 4.58 yields the following solution:

U1 U2



1  JM 12  A1 A2

 A1

(4.58)

2

A2 M 12 J  1

2J  A1 / A2 M 12 (1  A1 / A2 )  1  2( A1 / A2 ) 2 M 12  ( A1 / A2 ) 2 M 14 ( A1 / A2 ) 2 M 12 (J  1)

.

To obtain the equation for the pressure ratio, the result of the Equation 4.58 must be applied to the inverted form of Equation 4.53:

Process Design 123

p1 p2

1J 1  JM 12  A1 A2

(4.59)

 J 2J  A1 / A2 (1  A1 / A2 )  1  2( A1 / A2 ) 2 M 12  ( A1 / A2 ) 2 M 14

.

Differentiation of Equation 4.59 with respect to A1/A2 yields the expression for the area ratio that gives the maximum pressure difference:

§ A1 ¨¨ © A2

· ¸¸ ¹ CR

J 1

M 12  1 2 . 2(J  1)  M 12

J 

(4.60)

Equations 4.56 – 4.60 establish relationships between pressure, density, area and Mach number.

Figure 4.11. Relationship between air pressure ratio and area ratio of a ventilation channel

The curves shown in Figure 4.11 were derived by using Equation 4.60. After the curves shown on Figure 4.11 were computed, the maximum pressure differential was found by using Equation 4.60 and the MATLAB® code. In prior discussions, it was established that transient flow can occur when the Mach number reaches 0.9. Using the curves in Figure 4.7 the maximum pressure differential found for Mach number 0.9 was p1/p2 = 0.72. This value corresponds to the area differential = 0.6. As shown before, during die-cast operation, 50% of the

124

Casting: An Analytical Approach

gas ventilation system can be plugged up. To account for the reduction in the area of the ventilation system and still maintain maximum possible gas mass flow, the area differential variation was taken as 0.3. Using Equation 4.60 with a Mach number equal to 0.9 and the area differential equal to 0.3, the calculated pressure differential was 0.78. A general criterion for calculating the gas evacuation from a die cast die is established. Now two separate procedures can be outlined. 1. Calculation of the conventional gas evacuation system. 2. Calculation of the vacuum assist gas ventilation system. Calculation of the conventional gas evacuation system. 1. The volume of the die-cast part has to be calculated. 2. The fill time has to be calculated. The fill time is the time necessary to fill the cavity of the die cast die with aluminium. 3. Using the density of steam at standard atmospheric conditions 1.23 kG/m3, the cavity fill time and the volume of the part, the mass flow of the gas can be calculated. As shown above, mass flow that takes into account two-dimensional gas flow is about 5% less than that calculated using one-dimensional theory. To account for this difference, mass flow has to be increased by 5%. Ideally, all the gas has to be evacuated from cavity of the die cast-die. 4. Now Equation 4.43 can be rewritten to calculate the cross-sectional area of the gas evacuation system: m p1

A 1/J

§ p2 · ¨ ¸ ¨p ¸ © 1¹

RT

J

J 1 ª º § p2 · J » 2 « 1  ¨¨ ¸¸ » J 1« © p1 ¹ » «¬ ¼

.

(4.61)

where J= 1.4 p2 = 0.78 p1 p2 = 101 kPa p1 = 101/0.78 = 129.5 kPa R = 287 J/kg K T = 317K. 4. The maximum length of the ventilation channel for a calculated cross section can be determined from Equation 4.44 [Munson, Young and Okiishi, 2006]:

Process Design 125

L max

D 4f

­°1  M 2 J  1 ª (J  1) M 2 º ½° ln « .  ® 2 2 »¾ 2J °¯ JM ¬ 2  (J  1) M ¼ °¿

(4.62)

If a ventilation channel has to be longer, then the cross-sectional area has to be increased.

4.6 Ventilation Blocks Properly positioned ventilation channels are connected to the last area of the diecast die cavity to be filled. Ventilation blocks are used to prevent the molten metal from escaping through the ventilation channels and are used for both conventional and vacuum ventilation systems. When metal flows through the ventilation blocks due to the thermal exchange between the blocks and the molten metal, its temperature is reduced until it reaches the critical point of rigidity. There are two different types of the ventilation blocks: - valve - valveless. A valve system uses a valve shutoff to prevent metal from flowing outside of the die-cast die. Valveless systems rely on the metal to solidify before it reaches the end of the block. A vacuum assist gas ventilation system consists of a die-cast die, vacuum tank, vacuum pump, gate, and runner system that connect the cavity of the die-cast die with the vacuum tank. During the cycle, gas is forced from the cavity of the diecast die into the vacuum tank. To maintain pressure in the vacuum tank within specified values, the vacuum pump is used to evacuate the gas into the atmosphere. The number of die-cast machines that can be connected to the same vacuum system is limited by the volumetric pumping speed of the vacuum pump. Several die-cast machines can be connected to the same vacuum tank. There are two types of problems that must be solved: 1. Calculation of the maximum number of machines that can be connected to the same vacuum tank based on gas pump productivity, the volume of gas flowing into the tank and total process time. 2. Calculation of the size of the vacuum tank and volumetric pumping speed of the vacuum pump based on the number of die-cast machines connected to the same vacuum assist gas ventilation system. Queueing theory can be used to solve both problems. Before proceeding any further, the fundamentals of queueing theory must be defined.

126

Casting: An Analytical Approach

4.7 Little’s Formula John D. Little developed one of the fundamental relations of queueing theory. He related the steady-state mean system size to steady-state average customer waiting times. Letting Tq represent the time the customer (transaction) spends waiting in the queue prior to entering service and T represent the total time a customer spends in the system, Tq  S ,

T

(4.63)

where S – the service time. T, Tq, S are random variables. Two often used measures of system performance with respect to customers are Wq E[Tq ] and W E[T ] , the mean waiting time in the queue and the mean waiting time in the system. Little’s formulas are L

OW

(4.64)

Lq

OW q

(4.65)

where

O = average rate of customers entering the queueing system W = the mean waiting time in the system Wq = the mean waiting time in the queue.

4.8 Poisson Process and the Exponential Distribution The most common stochastic queueing models assume that interarrival times and service times obey an exponential distribution or, equivalently, that the arrival rate and service follow a Poisson distribution. The general formula for a Poisson probability distribution with mean Ot is pn (t )

( O t ) n  Ot e . n!

(4.66)

Thus, if we consider the random variable defined as the number of arrivals to a queueing system by time t, this random variable has the Poisson distribution given by Equation 4.66 with mean of Ot arrivals, or a mean arrival rate of O [Gross, 1998].

Process Design 127

We can calculate the vacuum system where several die-cast machines are connected to the same vacuum tank. Every die-cast machine can start the cycle according to a Poisson distribution with mean N/min (where N = the number of machines connected to a vacuum system). After every cycle, gas from the die-cast die is forced into the vacuum tank. When the pressure in the tank reaches the specified limit, the vacuum pump evacuates gas into the atmosphere. The amount of time it takes the pump to evacuate gas from an upper to a lower limit of pressure in the vacuum tank is considered a server’s service time. The criterion used in choosing the maximum number of die-cast machines that can be connected to the same vacuum system will be a portion of the pump’s busy time (it should be less than 1). A simple Markovian birth-death queueing model can be employed to estimate optimal size and to ensure steady-state operating conditions for the vacuum assist ventilation system. The amount of gas that flows from the die-cast die into the vacuum tank is the sum of 1. The amount of gas in the shot sleeve. V shs

D ss2 * S * L * (1  Pf ) , 4

(4.67)

where D = diameter of the shot sleeve. L = length of the shot sleeve. Pf = percent of fill of the shot sleeve 2. The amount of gas in cavity, gates, and overflows ( Vc ) The total volume of gas is V Total

Vshs  Vc

(4.68)

As molten metal fills the cavity of the die-cast die, it forces gas into the vacuum tank. Pressure rises in the vacuum tank with every shot. When the pressure in the vacuum tank exceeds the specified limit, the vacuum pump pumps air out of the tank. Gas flows into the vacuum tank at a very high velocity. The average fill time is 0.055-0.075 s. The vacuum pump operates much more slowly. For the vacuum system to operate within the specified parameters, the amount of gas that flows into the vacuum tank and the volume of gas that is extracted from the tank must be in balance. It can be assumed that die-cast machines cycle according to a Poisson distribution with a mean rate of N/min (N = the number of machines under consideration). We will consider this system a single server, since there is only one vacuum pump. The mean time to complete the service is the amount of time it takes the pump to evacuate gas from the vacuum tank. The assumption will be that the service time of the vacuum pump is exponential.

128

Casting: An Analytical Approach Example 4.1

Pump volumetric speed is 22 SCFM Average amount of gas flow into the tank (each shot) = 0.274 ft 3 Average cycle time = 0.41min Number of machines under consideration = 10 Nominal pressure in the vacuum tank = 2.5 PSI Maximum acceptable pressure rise in the vacuum tank = 0.25 PSI The MATLAB® code to calculate the maximum number of die-cast machines that can be connected to the same vacuum system. The number of machines that can be connected to the vacuum system will be based on the pump’s busy time. For the vacuum pump to keep up with the amount of gas entering the vacuum tank, the pump’s busy time must be less than 1. Ps=input('Volumetric pumping speed (ft^3/min)= '); Nm=input('Number of die cast machines under consideration= '); As=input('Average cycle time (min)= '); Pv=input('Part volume (ft^3)= '); Np=input('Nominal pressure in vacuum tank (PSI)= '); Pt=input('Maximum acceptable pressure rise(PSI)= '); n=linspace(1,Nm,Nm); lambda=n./As; %Arrival rate (arrival/unit of time) MeanSerTime=(Pv.*lambda)./(Ps); %Mean time to complete service MeanInterTime=1./lambda; %Mean interarrival time mu=1./MeanSerTime; %Service rate (# served/unit of time)

k=0;

Process Design 129 Example 4.1 (continued)

for i=1:length(lambda); Sb(i)=lambda(i).*MeanSerTime(i); if Sb(i)<1; %Fraction of time that server is busy (Busy time must be < 1) Pi(i)=1-Sb(i); %fraction of time server is idle Lq(i)=(Sb(i).^2)./Pi(i); %Expected queue size L(i)=Sb(i)./Pi(i); %Expected system size L_q(i)=Lq(i)./(1-Pi(i)-Sb(i).*Pi(i)); %Expected non empty queue size Wq(i)=Sb(i)./(mu(i)-lambda(i)); %Expected waiting time in the queue W(i)=Wq(i)+MeanSerTime(i); %Expected waiting time in the system k=k+1; else; Sb(i)=Sb(k); Pi(i)=1-Sb(i); %fraction of time server is idle Lq(i)=(Sb(i).^2)./Pi(i); %Expected queue size L(i)=Sb(i)./Pi(i); %Expected system size L_q(i)=Lq(i)./(1-Pi(i)-Sb(i).*Pi(i)); %Expected non empty queue size Wq(i)=Sb(i)./(mu(k)-lambda(k)); %Expected waiting time in the queue

130

Casting: An Analytical Approach Example 4.1 (continued)

W(i)=Wq(i)+MeanSerTime(k); %Expected waiting time in the system end; end; j=linspace(1,k,k); V=1.2.*(Pv.*j)./((((Np+Pt)/Np)^(1/1.4))-1); %Size of the vacuum tank (ft^3) subplot(2,2,1); plot(lambda,Sb); grid; title('Fraction of time pump is busy', 'colour', 'green'); xlabel('Number of die cast machines'); ylabel('Fraction of time'); subplot(2,2,2);plot(lambda,Lq); grid; title('Expected queue size', 'colour', 'green'); xlabel('Number of die cast machines'); ylabel('Queue size'); subplot(2,2,3);plot(lambda,W); grid; title('Expected waiting time in the system', 'colour', 'green'); xlabel('Number of die cast machines'); ylabel('Waiting time'); subplot(2,2,4);plot(lambda,Wq); grid; title('Expected queue size', 'colour', 'green'); xlabel('Number of die cast machines'); ylabel('Expected waiting time in the queue');

Results of calculations are summarized in Figure 4.12 and Table 4.2.

Process Design 131 Table 4.2. Tabulated results for Example 4.1

Number of die cast machines L L_q Lq Sb Wq W V ft 3

1

2

0.0716 1.072 0.0048 0.067 0.002 0.029 4.207

0.3646 1.3645 0.097 0.267 0.02 0.075 8.41

3 1.51 2.507 0.906 0.601 0.123 0.206 12.62

As shown in Figure 4.12, the maximum number of machines connected to a specified vacuum system, based on the pump’s busy time is equal to 3.

Figure 4.12. Results of calculations. The number of machines that can be connected to the vacuum system.

To proceed in solving the second problem, the measure of traffic congestion must be defined. Traffic congestion is:

C

where

P = mean time to complete service.

O , P

(4.69)

132

Casting: An Analytical Approach

When traffic congestion exceeds 1, the queue continues growing. Since the goal is to achieve steady-state conditions, traffic congestion should never exceed 1.

4.9 Cooling Two analytical solutions for the heat transfer process are presented in this section. The first describes a simple cooling process when temperature is assumed uniform throughout the cooling body, that is, the lumped-temperature approach is used. In the second solution, we look at one-dimensional, transient solution for heat transfer from a fixed-temperature boundary into a semi-infinite medium.

4.9.1 Lumped-temperature Model When two objects at different temperatures are put in contact over a period of time, their average temperatures will gradually draw closer to each other due to heat exchange. During this process the colder object acquires internal energy from the hotter one. The energy exchange occurs by collisions between the molecules at the contact surface. By definition temperature is the measure of the average kinetic energy of chaotic molecular motion. As a body gains or loses kinetic energy, its temperature increases or decreases, respectively. In our case, when energy is transferred from the hot casting to cold water in the cooling system, assuming that there is no heat lost into the environment, the conservation of energy principle implies that the exact amount of energy lost by one object must be gained by the other. Let us denote the amount of heat gain or loss by the objects as Q. Then the conservation principle can be described as QGAIN

QLOSS .

(4.70)

Although the internal energy of an object is directly proportional to its mass, it does not necessarily mean that the two objects of the same mass and temperature have the same amount of internal energy. If they are made of different materials, then their specific heats are generally different. The specific heat is defined as the quantity of heat required to raise the temperature of 1 gram of a substance by 1°C. If T1 and T2 are the initial temperatures of the two objects, and T0 is the terminal temperature after a prolonged contact, then the respective amounts of heat lost or gained by the objects are Q1

M 1C1 (T0  T1 ),

Q2

M 2 C 2 (T0  T2 ),

(4.71)

where M denotes mass and C specific heat, with the indices referring to the two objects. According to Equation 4.70, Q1 is equal to -Q2, a condition that allows us to calculate the final temperature:

Process Design 133

M 1C1T1  M 2C2T2 . M 1C1  M 2C2

T0

(4.72)

Equation 2.72 gives the steady-state solution for the heat transfer problem of two bodies exchanging heat, which is achieved after a very long (theoreticallyinfinite) period of time. To find the transient solution for temperature, we can simplify the problem to have only a single body cooling to the constanttemperature environment. Let us consider a casting of mass M, specific heat C and initial temperature, TIN, cooling to the air at a constant temperature TENV. We will employ here Newton’s law of cooling, which states that the rate of heat loss of a body is proportional to the difference in temperature between the body and its surroundings. It can be expressed as MC

dT dt

hAT  TENV ,

(4.73)

where h is constant called the heat transfer coefficient and A is the surface area of the casting. Integration of Equation 4.73 using the variable separation method yields T

TENV  TIN  TENV ˜ e  Įt .

(4.74)

To obtain Equation 4.74, we took into account that at t = 0, T = TIN, and we introduced a constant

D

hA . MC

(4.75)

Equation 4.74 shows the evolution of the casting average temperature with time. It starts at the initial temperature and approaches the temperature of the environment at a rate defined by the constant D Moreover, the rate is higher for larger surface area and heat transfer coefficient or for smaller mass and specific heat. The transient solution for cooling can also be used to evaluate the time it takes to cool the casting to a certain temperature TFIN. Rearranging terms in Equation 4.74 gives the desired formula t FIN



1

D

§T  TENV ˜ ln¨¨ FIN  T © IN TENV

· ¸. ¸ ¹

(4.76)

According to this expression, it will take an infinite amount of time for the temperature to become equal to the surrounding temperature TENV because the argument of the logarithm becomes zero at that temperature. However, it takes a

134

Casting: An Analytical Approach

finite period of time to cool to a temperature above TENV, no matter how close to TENV that temperature may be. For smaller values of D the wait is longer. After the casting is extracted from die-cast die, the last stage of the process is to cool it to room temperature. There are passive and active ways to cool a casting. A casting can be left in the open area to be cooled by the air, or it can be quenched in a tank of water. When two objects at different temperature are kept together over the certain period of time, eventually they will reach the same temperature. This process is called heat exchange. The energy that is transferred from one object to another is called internal energy. In our case when energy is being transferred from the casting into the water and we can assume that there is no heat lost into the environment, then the conservation of energy principle implies that the energy lost by one object must be gained by the other. By definition, temperature is the measure of the average kinetic energy of molecular motion. As a body gains or loses kinetic energy, its temperature will increase or decrease. Although the internal energy lost or gained by an object is directly proportional to its mass, it does not means that two objects of the same mass and temperature have the same amount of internal energy. Temperature reflects only the kinetic energy portion of the internal energy, so an object with a greater fraction of its internal energy in the form of potential energy will have a greater internal energy at a given temperature. This property is reflected in the quantity called the specific heat. The specific heat is defined as the quantity of heat required to raise the temperature of 1 gram of a substance by 1°C. Now we can write an equation that allows us to define the quantity of heat lost or gained by a body: Q

MC (T  T0 ) .

(4.77)

To calculate the time required to cool casting to a room temperature we will use Newton’s law of cooling, which states that rate of the heat loss of the body is proportional to the difference in temperature between the body and its surroundings. It can be expressed as dT dt

k (TC  TENV ) ,

(4.78)

where TC = temperature of the body TENV = temperature of the environment k = constant. Separating variables in Equation 4.78 yields dT TC  TENV

kdt .

(4.79)

Process Design 135

Integrating Equation 4.79, we obtain ln(TC  TENV )

 kt  ln C ,

(4.80)

kt ,

(4.81)

ln(TC  TENV )  ln C

ln

TC  TENV C

TC  TENV C

TC  TENV

 kt ,

(4.82)

e  kt ,

(4.83)

Ce  kt .

(4.84)

To complete the solution of Equation 4.84, we need to apply boundary conditions. Let us denote the temperature of the body at t = 0 as Tin. Then Equation 4.84 becomes

TIN

T (t

C

0)

C  TENV ,

TIN  TENV

(4.85)

(4.86)

Now Equation 4.84 becomes

TC  T ENV

(T IN  T ENV )e  kt

(4.87)

Now let us develop the equation of the temperature change during the cooling process.

136

Casting: An Analytical Approach

Example 4.2

An aluminium casting weighing 20 kg is extracted from the die-cast die and placed into water. The casting is extracted from the die at 315°C. The quench tank contain 1000 kg of water. The water in the tank is at 37°C. The specific heat of aluminium is 0.23 kcal/kg C. After the aluminium casting is quenched into the water tank, using Equation 2.47, we can write for the casting, Qout = 0.23 x 20 x [15 –T(t)C], where T(t)C = temperature of the casting, changing during cooling. Now we can write a similar equation for the water, Qin = 1 x 1000 x [T(t)w – 37] where T(t)w = temperature of the water changing over the same period of time. According to Equation 2.70 Qin = Qout 0.23x20x[315-T(t)c]=1000x[T(t)w - 37] Solving for T(t)w: T(t)w = 38.449-0.0046T(t)c Now using Newton’s law of cooling, Equation 4.73, we obtain dT dt  k (T (t ) c  T (t ) w )

k (T (t ) c  38.449  0.0046T (t ) c )

k (1.0046T (t ) c  38.449) 1.0046T (t )C k  38.449k .

We can rewrite it as linear differential equation: dT  1.0046T (t ) c k 38.449k . dt The integrating factor is given as 1.0046 kdt e³ e1.0046 kt . Then (T (t )C e1.0046 kt )' Integrating,

38.449ke1.0046 kt .

Process Design 137 Example 4.2 (continued)

T (t ) C e 1.0046 kt

38.449e 1.0046 kt  c,

T (t ) C 38.449  ce  1.0046 kt . Using initial conditions at T(t=0)c = 315°C 315=38.449 + c c = 276.551. T(t)C = 38.449 + 276.551e– 1.0046kt To find k, we apply boundary condition TC(t=30)= 40.5: 40.5 = 38.449 + 276.551e–30.138k. e–30.138k = 0.0074. 30.138k = ln134.84. k=0.163. TC(t) = 38.449 + 276.551e–0.1637t.

Figure 4.13. Cooling curve for an aluminium casting quenched in water

Example 4.3

Air cooled casting example. Casting extracted from the die at 188°C. After it placed in the room at 25°C for 15 min, the temperature of the casting is 104°C. Draw the cooling curve. Employing Newton’s law of cooling: dT k >TC (t )  25@ . dt

138

Casting: An Analytical Approach

Example 4.3 (continued)

To solve this differential equation, we need to separate variables: dT kdt . T  25 Integrating, we obtain ln T  25 kt  c T ce  kt  25 . We have two unknowns c and k. To solve this equation, we are going to use the initial conditions: At t = 0, T(t)=188°C, therefore, c=163 and T (t ) 163e  kt  25 . Now to find k we are going to use the second condition: At t=15, T(t)=104°C, therefore, k=0.048 and T (t ) 163e 0.048t  25 .

Figure 4.14. Cooling curve of air cooled aluminium casting

4.9.2 Heat Flow into a Semi-infinite Medium Let us consider the well-known problem of a one-dimensional flow of heat from a boundary into a semi-infinite medium. The boundary is at a fixed ambient temperature Tb. In this case, the temperature in the medium is a function of the time and distance from the boundary. If we choose the coordinate system so that the boundary is at x = 0 and the x axis is normal to it, as shown in Figure 4.15, then the governing equation is

Process Design 139

wT wt

D

w 2T , wx 2

(4.88)

where D= k/(UC) is the thermal diffusion coefficient. Equation 4.88 is solved with the following initial and boundary conditions: 0: x 0:

t

T (0, x) q (t ,0)

Ti ,

(4.89)

h>Tb  T (t ,0)@,

where h is the constant heat transfer coefficient, q is the heat flux at the boundary and Ti is the uniform initial temperature.

Figure 4.15. One-dimensional heat flow from a fixed-temperature boundary into a semiinfinite medium

The exact solution to Equations 4.88 and 4.89 is [Holman, 1976] T (t , x)  Ti Tb  Ti § x 1  erf ¨¨ © 2 Dt

§ x · § hx h 2 Dt · ª h Dt ¨ ¸  exp¨ ¸  ¨ k  k 2 ¸ «1  erf ¨ ¸ k © ¹ ¬« ¹ © 2 Dt

where erf(z) is the error function:

·º ¸», ¸» ¹¼

(4.90)

140

Casting: An Analytical Approach

erf ( z )

2

S

z

˜

³e

 t2

dt .

(4.91)

0

The error function is not an elementary function. Its values can be calculated either numerically or using an approximation with elementary functions. The temperature at the boundary, x=0, is T (t ,0)  Ti Tb  Ti

§ h Dt · º § h 2 Dt · ª ¸» . 1  exp¨¨ 2 ¸¸ «1  erf ¨ ¨ k ¸» © k ¹ «¬ © ¹¼

(4.92)

In reality, of course, it is rare, if possible at all, to have a situation when the medium can be considered semi-infinite. There is usually another boundary at the opposite end of the domain. However, it is often possible to ignore that boundary for a certain period of time, before information from it travels far enough to affect the solution. With that in mind, some useful information can be obtained from Equation 4.90. For example, we can evaluate temperature changes in different parts of the die or mould. Also, the thickness of the thermal boundary layer at a given time can be obtained. These estimates provide information on the temporal and spatial scales of the thermal fluxes during the casting process.

Figure 4.16. Evolution of the thermal boundary layer in an H-13 die and a sand mould

We choose here to define the thermal boundary layer, x = xb, as the region adjacent to the boundary where the temperature changes by more than 1% of the initial difference Tb – Ti: 0 d x d xb :

T (t , x)  Ti ! 0.01 . Tb  Ti

(4.93)

Process Design 141

Figure 4.16 shows the growth of the thermal boundary layer with time for an H-13 steel die and a sand mould. The properties of the materials and the heat transfer coefficients used in the calculations are givein in Tables 4.3 and 4.4. It is clear that the heat penetrates into the die much faster than into the sand mould. For example, it takes about 27 milliseconds for the heat to move 1 mm into the die (which is comparable with a typical fill time), while it is more than 413 milliseconds for the sand mould. The times for the 10 mm penetration are 1.5 sec and 25 sec, respectively. Another interesting estimate can be made for the temperature at the very surface. The die surface temperature increases by 1% of Tb – Ti in just 0.03 milliseconds. For the sand mould, the time is 0.33 milliseconds. These estimates are used in the following chapter to determine the importance of heat transfer during filling, allowing for some simplifications of the computational models.

4.10 CFD Simulations In this section, we present the results of several simulations using computational fluid dynamics. These simulations illustrate that the real world behaviour of metal during casting processes is more complex sometimes a lot more complex than an analytical solution may suggest. It is important, therefore, not to lose sight of such reality when making the simplicifications necessary to obtain the exact solutions. We consider here two casting processes high-pressure die casting and gravity sand casting. The alloy material is the same in both cases, aluminium alloy A356. The die material is H13 steel and the sand mould is made of silica sand. All material properties and heat transfer coefficients used in the simulations are listed in Tables 4.3 – 4.5. The aspects of the die casting process considered in the simulations are flow in the shot sleeve during the slow shot stage, thermal die cycling, filling with subsequent solidification, and air flow through a vent. For sand casting, we model the filling and the subsequent solidification of the metal. Table 4.3. Thermophysical properties of the materials used in the simulations A356 alloy Liquid density, kg/m3 Solid density, kg/m3 Viscosity, Pa sec Liquid specific heat, J/kg C Solid specific heat, J/kg C Liquid thermal conductivity, W/m C Solid thermal conductivity, W/m C Latent heat, J/kg Solidus temperature, C Liquidus temperature, C Critical solid fraction Coherency solid faction

2420 2570 0.00119 1194 1265 86.9 185 429000 555 611 0.68 0.23

142

Casting: An Analytical Approach

Table 4.3. Thermophysical properties of the materials used in the simulations (continued) H-13 die steel Density*specific heat product, J/m3 C Thermal conductivity, W/m C Silica Sand Density*specific heat product, J/m3 C Thermal conductivity, W/m C

5.106 29.7 1.7.106 0.61

Table 4.4. Heat transfer coefficients

Interface Metal/die Metal/sand mould Die/cooling channels

Heat transfer coefficient, W/m2 C 20000 500 5000

Figure 4.17. The complete geometry of the high pressure die casting model with the shot sleeve, runner system, single cavity and cooling channels

4.10.1 High-pressure Die Casting The complete geometry of the shot sleeve, the runner and gating system, the cavity, an overflow and the cooling channels is shown in Figure 4.17. It is a single cavity die for casting an automotive engine cover. From the standpoint of an engineer, the production of a casting is a continuous process from placing the metal into the ladle and pouring it into the shot sleeve to

Process Design 143

pushing the liquid into the cavity with the plunger, to solidification and, finally, to the opening of the die and retrieving the piece. However, for CFD modelling, it is beneficial to divide the process into stages that are easier to model separately from the whole process. Here we choose the following stages: i. ii. iii. iv. v.

the flow in the shot cylinder during the slow shot stage; thermal die cycling; filling; solidification after filling; flow of air through a vent.

Each stage is characterized by certain time and length scales, boundary conditions and the dominant physical phenomena. Identifying these specific features of each stage in turn allows the modeller to concentrate on the aspects of the casting process that are critical for the quality of the part, e.g., air entrainment in the shot sleeve, unbalanced flow in the gates, die overheating and so on. The appropriate computational methods and algorithms can then be applied to obtain accurate and efficient numerical solutions. Finally, the ability to use data from one simulation as the initial or boundary conditions for another links them into a solution for the complete process, continuous in time and space.

Figure 4.18. Three-dimensional snapshots of the metal in the shot cylinder during the slow shot stage

Flow in the Shot Cylinder The main purpose of this simulation is to test the validity of the analytical solution given by Equations 4.29, 4.38 and 4.40, which was derived for an inviscid fluid pushed by a plunger along a rectangular channel. In the simulation, more realistic conditions of viscous flow and a circular channel cross section are used. Heat transfer and solidification are not included in the model based on the assumption that solidification in the shot sleeve is minimal, if any, and does not significantly affect the flow.

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Casting: An Analytical Approach

The length of the channel is L = 0.7 m, the same as that used to obtain solutions in Figure 4.9. The shot diameter D = 0.1 m, and the initial fluid depth is h0 = 0.04 m. The velocity of the plunger is defined as a function of time using the solution for the maximum slope of the metal surface of Dmax=5o, given by Equation 4.38 (curve 6 in Figure 4.9c). The results are shown in Figures 4.18 and 4.19.

Figure 4.19. Two-dimensional snapshots of the metal in the shot cylinder during the slow shot stage. Contour lines indicate the variation in the magnitude of the horisontal velocity.

There are several aspects of the numerical solution that match the analytical solution quite well. The slope of the wave stays largely within the 5o limit. The circular nature of the channel does not seem to affect much the profile of the free surface in the transverse direction. The critical point, at which the metal surfaces touch the top of the channel, is reached at t = 1.37 s, very close to where curve 6 crosses the critical velocity in Figure 4.9c. The velocity of the plunger at that time is 0.725 m/s, close to the value of 0.73 m/s given by Equation 4.40 for the rectangular channel. Finally, the first wave arrives at the end of the shot sleeve at t

Process Design 145

= 1.15 s, the theory predicts 1.12 s, based on the wave speed in undisturbed flow given by Equation 4.26. Given the three-dimensional nature of the simulation and the fact that the flow is viscous, the agreement with the analytical solution is quit remarkable. It is important to note that we used the same initial depth, h0, of metal for the round shot sleeve in the simulation and for the rectangular one in the analytical solution. In our particular example, with h0 = 0.04 m, the initial fill fraction in the circular cylinder comes to 37.4%, as opposed to 40% for the rectangular channel. If we used the same initial fill fraction instead, the results of the two solutions would not be so close. This is not to say that there are no differences in the two solutions. The velocity contours shown in the plane of symmetry in Figure 4.19 indicate that the main assumption used to obtain the analytical solution that all flow variables vary only in the horizontal direction does not quite hold. First, a viscous boundary layer develops at the bottom of the shot sleeve. Second, the numerical results show that flow near the free surface moves faster than the bulk of the metal below it, resulting in a sort of a surge wave. The slope of the metal surface in this wave is about one and a half to two times larger than 5o. It reaches the end of the channel at around 1.3 sec and then reflects back. As a result, air may be entrained in the last stages of the process unless, for example, the reflected surge wave is redirected into the runner system. Thermal Die Cycling In practice, a die is used for many thousands of shots. The heat is constantly added with each shot through the injection of liquid metal and extracted by the cooling channels and, to a lesser extent, by sparaying the die inner surface with lubricant when the die is open. After many cycles, this leads to a complex distribution of temperature throughout the die. The die cycling simulation serves two purposes. First, we want to analyze the thermal balance within the die to ensure that the cooling system is sufficient to prevent the die from overheating and hence to prolong its life. Second, the die cycling simulation provides a realistic temperature distribution within the die that can be used for filling simulation. The actual filling analysis is usually ignored during a thermal die cycling simulation. This is justified by the short duration of the filling several milliseconds compared to the length of each cycle several tens of seconds. At the beginning of each cycle, the metal is instantaneously placed into the cavity at the initial uniform temperature of 650°C, and the simulation, therefore, is reduced to a purely thermal problem. To describe the changing conditions during a single cycle, the temperature and heat transfer coefficient at the inner surface of the die are vary with time, as shown in Table 4.5. Each cycle is divided into four segments the first corresponds to the closed die with the metal inside. The remaining three segments describe the open die.

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Casting: An Analytical Approach

Table 4.5. Conditions at the die inner surface during one thermal cycle

Cycle segments Die closed Die open to air Spraying Die open to air

Temperature °C (initial) 650 30 50 30

Heat transfer coefficient, W/m2 C 20000 10 3000 10

Duratio n, sec 25 5 10 15

The cooling channels are included in the model since they provide the main means of removing heat from the die. They are filled with water at the uniform, constant temperature of 40°C. The initial uniform temperature of the die is guessed at 205°C and the simulation is run for twenty five cycles. Each cycle lasts 55 s, and the total simulation time is 1375 s. Figure 4.20 shows the variation in the total thermal energy of the die as a function of time. The thermal cycles are clearly visible, as well as a slow approach to a steady-state solution when the die temperature is the same at the beginning of each cycle.

Figure 4.20. Total die thermal energy as a function of time during twenty five thermal die cycles

Filling Filling is one of the most important stages in die casting that defines the quality of the final part. The filling sequence, pressures and velocities that develop in the flow, temperature distribution and solidification, air and oxide film entrainment, efficienty of overflows and vents are some of the important results provided by a numerical simulation. In the present simulation, the metal is pushed into the cavity by a plunger with variable speed. The maximum plunger velocity is 3.81 m/s, corresponding to a gate

Process Design 147

velocity of 40 m/s. The total time of the simulation, including the slow shot stage, is 1.775 s. The fast shot stage takes 0.175 s. The average Reynolds number during the fast shot stage is over 200,000; therefore, the flow is highly turbulent. The k-H turbulence model is used in the model to account for this. Heat transfer and solidification are also included so that thermal losses and any early solidification can be evaluated. The initial temperature of the die is taken from the end of the thermal die cycling simulation. The cooling channels are not included in the simulation because the penetration distance of the heat from the metal into the die is significantly smaller than the distance between the cavity surface and the cooling channels. This heat penetration distance can be estimated from Figure 4.16; for a fill time of 175 ms, it is about 3 mm, well short of reaching the cooling channels. According to Equation 4.92, the die surface temperature rises during filling by almost half of the difference between the initial die temperature and the temperature of the metal, or, in our case, about 75°C. Given the thermal boundary layer thickness of only 3 mm, the temperature gradients in the die are quite steep, 25°C per millimetre of the die material. Nevertheless, we assume that the die temperature is constant during the simulation, using the temperature distribution obtained at the end of the die cycling simulation. This may appear as a stretch in simplifying the model, but it provides a significant benefit for the speed of the simulation. The air in the cavity is treated as an adiabatic compressible gas. The volume and pressure of each air pocket are computed using the equation of state given by Equation 2.41. For air, J= 1.4. No venting is included in the model, making it the worst-case scenario in terms of the amount of entrained air. Figure 4.21 shows selected snapshots of the filling sequence. The metal reaches the gates more or less simultaneously, but once inside the cavity, the metal front undergoes severe deformations and breakup. As a result, multiple small air pockets, each with its own pressure, are created. These bubbles continue to breakup and coalesce throughout filling. The images just before the end of filling show the last places to be filled, which can help to place vents and overflows. The amount and location of entrained air and oxide film are shown in Figure 4.22. As expected, these are mostly pushed into the last areas to be filled. Ideally, the metal contaminated most with entrained oxide film and air should be caught by the overflows. The overflow cavity at the centre of the casting appears to be functioning properly in this respect.

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Casting: An Analytical Approach

Figure 4.21. Simulated filling sequence for high-pressure die casting

Process Design 149

Figure 4.22. Oxide film (left) and entrained air distribution at the end of high-pressure die casting filling. The metal with top 45% of oxide concentration is highlighted with a white colour. For the entrained air, the white colour shows the metal with top 25% of air entrained during filling.

Solidification Finally, a solidification simulation is done using the solution from the end of filling as the initial conditions. As in the die cycling case, we negelect the residual flow of metal during solidification for faster compuation. This can be justified by the assertion that most of the heat is extracted through heat transfer and conduction rather than convection. The main purpose of the solidification run is to analyze the solidification sequence and porosity formation and to obtain an estimate of the solidification time, which in turn can tell when it is safe to open the die. In this case we cannot ignore the evolution of temperature within the die, so the full energy equation is solved within its domain. The predicted solidification time of the part is 5.13 s. The biscuit fully solidifies at 17.1 s, therefore, the duration of the closed stage of 25 s used in the die cycling simulation is adequate. The maximum porosity of 2.5% is predicted around the thicker central section of the part, near the overflow located at the central opening. Air Flow in a Vent In this brief study of air flow through a vent, we consider two types of vents. One has a constant rectangular cross section of 6 mm u 21.66 mm, and the other has a variable cross section, where the vent starts at the same cross-sectional area as the first vent and then quickly expands into a large area of 10 mm u 21.66 mm. The geometries of the two vents are shown in Figure 4.23. The flow of air at room temperature is generated by a pressure difference between the inlet and outlet of the flow region. The pressure drop is varied continuously with time across a wide range of values, simulating different vacuum conditions. The variation occurs gradually so that at any given time the flow is close to steady state.

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Casting: An Analytical Approach

Figure 4.23. Geometry of the vents: constant area (left) and variable area geometry (right)

Figure 4.24. Calculated mass flow rates of air through two types of vent as a function of the ratio of the inlet and outlet pressures. The dashed line represents the constant area vent, and the solid line represents the vent with a variable cross sectional area.

Process Design 151

Figure 4.24 shows the mass flow rates obtained as a function of the ratio of the inlet and outlet pressures. In each case, the mass flow rate initially increases with the pressure ratio until it reaches a maximum; beyond that the mass flow rate does not increase as the outlet pressure continues to drop. At this critical point, the flow in the vent becomes supersonic. The flow patterns and the areas of supersonic flows are shown in Figure 4.25. For the constant area vent, the pressure ratio at the critical point is about 3.0; for the variable geometry vent, it is about 3.5. The maximum flow rate is larger in the vent with variable geometry. It is interesting to note that the sudden change in the cross-sectional area results in flow separation in narrowest area, which is reflected in the oscillations of the mass flow rate for that vent in Figure 4.24. If the transition from small to larger area were smooth, these flow instabilities might be avoided all together.

Figure 4.25. Flow patterns at the maximum flow rate for the constant area (left) and variable geometry vents. Dark shading indicates the areas of supersonic flow

These results indicate that for simple vent geometry, there is a limit on how much air can be evacuated through it, no matter what vacuum conditions are maintained outside the die. A larger mass flow rate can be achieved if the jet of air

152

Casting: An Analytical Approach

emerging from the die cavity into the vent is allowed to expand. The expansion must be done to prevent flow separation and instabilities in the vent channel.

4.10.2 Gravity Sand Casting The complete geometry of sand casting, together with the, risers, sprue and runner system, is shown in Figure 4.26. The casting is a simplified version of an automatic transmission cover.

Figure 4.26. Geometry of gravity sand casting

Two simulations were run for this case a filling and the subsequent solidification. The A356 alloy and silica sand mould properties used in the model are listed in Tables 4.3 and 4.4. Filling In the filling simulation, the metal is assumed to be poured from the constant height of 0.1 m above the top of the sprue at temperature of 750ºC. The initial mould temperature is 30ºC. With these conditions, the model predicted filing time of 5.25 s. Figure 4.27 shows selected snapshots of the filling sequence. After the initial splashing in the sprue and runner system, the flow stabilizes and progresses smoothly through the remaining part of the casting.

Process Design 153

Figure 4.27. Simulated filling sequence for gravity casting

The distribution of the entrained oxides and air at the end of filling are shown in Figure 4.28. As with high-pressure die-casting filling (Figure 4.22), the areas of the highest concentrations of oxides and air in the metal are somewhat similar and correlate with the filling sequence. Given enough time, air maybe able to escape through the porous mould back into the atmosphere, but oxides are likely to stay within the metal and potentially result in leakage or structural defects.

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Casting: An Analytical Approach

Figure 4.28. Oxide film (left) and entrained air distribution at the end of gravity casting filling. The metal with top 65% of oxide concentration is highlighted with a white colour. For the entrained air, the white colour shows the metal with top 15% of air entrained during filling.

Figure 4.29. Simulated microporosity distribution at the end of solidification of the gravity sand casting

Solidification As with die-casting, the solidification simulation was a continuation of the filling run with no flow but using the metal and mould temperatures as the initial conditions. The predicted solidification time is 4 minutes. The maximum predicted microporosity is 2.3%. Figure 4.29 shows the distribution of microporosity through a section of the casting. It appears that it is mostly concentrated near the inner

Process Design 155

surface of the casting and that there is enough thickness in the metal for the porosity not to cause any leakage problems. There is also some porosity in the thicker sections of the casting and runner system, but overall the feeding system appears to do its job well.

5 Quality Control

5.1 Basic Concepts of Quality Control The meaning of the term quality has developed over time. There are seven distinctive interpretations [www.wikipedia.com]: 1. 2. 3. 4. 5. 6.

The degree to which a set of inherent characteristics fulfils requirements as ISO-900. Conformance to specifications. The difficulty with this is that the specifications may not be what the customer wants.8 Fitness for use. Fitness is defined by the customer.9 A two-dimensional model of quality. The quality has two dimensions: “must-be quality’’ and “attractive quality”.10 Value to some person. (Gerald Marvin Weinber).11 Costs go down and productivity goes up, as improvement of quality is accomplished by better management of design, engineering, testing and

8

Philip B. “Phil” Crosby, (June 18, 1926–August 18, 2001) was a businessman and author who contributed to management theory and quality management practices. 9

Joseph Moses Juran, (born December 24, 1904), is an industrial engineer and

philanthropist. Juran is known as a business and industrial quality “guru”. 10

Professor Noriaki Kano is the developer of a program whose simple ranking scheme distinguishes between essential and differentiating attributes related to concepts of customer quality. This system is referred to as a Kano model. 11

Gerald Marvin Weinberg (“Jerry”) is an author and teacher of the psychology and anthropology of computer software development.

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Casting: An Analytical Approach

7. 8.

9.

by improvement of processes. Better quality at lower price has a chance to capture a market. Cutting costs without improvement of quality is futile.12 The loss a product imposes on society after it is shipped. Taguchi's definition of quality is based on a more comprehensive view of the production system.13 Energy quality, associated with both the energy engineering of industrial systems and the qualitative differences in the trophic levels of an ecosystem. Energy quality is the contrast between different forms of energy, the different trophic levels in ecological systems and the propensity of energy to convert from one form to another. One key distinction is that there are two common applications of the term Quality as a form of activity or function within a business. One is Quality Assurance which is the “prevention of defects,” such as the deployment of a Quality Management System and preventative activities like FMEA. The other is Quality Control which is the “detection of defects,” most commonly associated with testing which takes place within a Quality Management System and is typically referred to as verification and validation.

5.2 Definition of Quality The Merriam-Webster definition of quality of an object is “peculiar and essential character, degree of excellence, superiority in kind.” Although the definition of quality is important in general terms, it doesn’t give specific criteria to define the quality of the final product. A product produced by different manufacturing processes as well as for various applications will require its own set of criteria. Throughout history people struggled to find a universal definition of quality in a single set of criteria to define product quality.

5.3 Definition of Control Control is a process that employs a set of procedures to ensure that defined quality standards are met on a consistent basis. A process is said to be controlled, if through the use of past experience, we can predict expected variation of the process in the future.

12 William Edwards Deming (October 14, 1900 December 20, 1993) was an American statistician, college professor, author, lecturer, and consultant.

13

Gen'ichi Taguchi (born January 1, 1924 in Takamachi, Japan) is an engineer and statistician. From the 1950s onwards, Taguchi developed a methodology for applying statistics to improve the quality of manufactured goods. Taguchi methods have been controversial among many conventional Western statisticians unfamiliar with the Taguchi methodology.

Quality Control

159

5.4 Statistical Process Control Statistics is a mathematical science pertaining to the collection, analysis, interpretation and presentation of data. The mathematical methods of statistics emerged from probability theory, which can be dated to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Probability is the measure that describes the chances that an event will occur. Statistical process control is the application of statistical methods to identify and control the special cause of variation in a process. Statistical process controlled was first thought of and developed by Walter A. Shewhart.14 Statistical Process Control (SPC) is the equivalent of a histogram plotted on its side over time. Every new point is statistically compared with previous points as well as with the distribution as a whole to assess likely considerations of process control. Forms with zones and rules are created and used to simplify plotting, monitoring and decision making at the operator level. SPC separates special cause from common cause variation in a process at the confidence level built into the rules being followed. In the foundation of statistical variation and the necessity of statistical control is the basic understanding that, in real life, no single process can have two absolutely identical outcomes. Statistics as a science that takes all possible statistical variations of process outcomes into account and helps to predict future behaviour of the process by analyzing the history of the data. Statistical data can be summarized and presented in several forms: graphical, tabular, and numerical. The main concern of statistics is collecting, analyzing and interpreting data. The ability of the organization to predict and control any variation in the manufacturing process has a profound effect on the quality of the final product. But it is easier said than done. The die-cast process has between 20 – 25 physical characteristics that affect the quality of the casting. Control of the process should encompass not only possible variations between these parameters but also understanding of the relations between parameters under specific boundary conditions. In the casting process, physical parameters from one cycle have very limited influence on the next. Statistical analysis can help to discover simple patterns or patterns of relations between measured parameters and can be used to predict product quality. Summary: - Variation is part of reality. Controlling manufacturing process means understanding possible variations of the process as well as relations between physical parameters that influence this process. - The proper use of statistical tools can enable engineers to predict the quality of the product based on past experience. Adjustment of the process can prevent production of defective product before it actually occurs.

14

Walter Andrew Shewhart (March 18, 1891 March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control.

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Casting: An Analytical Approach

5.5 Tabular Summarization of Data Tabular data summarization includes Collecting data Sorting data in specific order Summarizing results Presenting data. Table 5.1 shows raw data of impact pressure collected over a period of time. Table 5.1. Impact pressure collected overperiod of 8 hours

Time 7:00 7:20 7:40 8:00 8:20 8:40 9:00 9:20 9:40 10:00 10:20 10:40

Impact pressure, mPa 13.2 13.7 14.3 14 13.4 14.4 13.6 14.2 13.9 14.1 13.9 13.6

Time 11:00 11:20 11:40 12:00 12:20 12:40 1:00 1:20 1:40 2:00 2:20 2:40 3:00

Impact pressure, mPa 13.4 13.6 14.1 13.9 14.2 14.1 13.9 13.7 14.2 14.3 13.9 14.0 14.1

5.6 Numerical Data Summarization Collected data can be summarized numerically by calculating its central tendency and data dispersion. The central tendency is a measure of the middle value of the data set. There are many measurements of central tendency. The most common is the simple average: N

_

X

¦X i 1

N

i

,

(5.1)

where _

X = arithmetic mean

Xi = individual variable n = number of variables in a data set. The most useful measurement of the data dispersion is the standard deviation. It can be found by using the formula,

Quality Control

N

¦ (X

_

 X)

i 1

V

161

.

N

(5.2)

There are several methods for estimating probability that variables with certain characteristics will occur in a population. Normal probability disttribution (Gaussian15 distribution) Poisson distribution16 Exponential distribution Binominal distribution.

5.6.1 Normal Probability Distribution One example of a normal probabilty distribution is the probability density function: y

1

V 2S

e  X  P

2

/ 2V 2

,

(5.3)

where e = 2.718 ʌ = 3.14 ȝ = Population mean ı = Population standard deviation. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. A graph of the probability density function is a bell curve (see Figure 5.1).

15

Named after Carl Friedrich Gauss, (April 30 1777 – February 23 1855), a German

mathematician. 16

The distribution was discovered by Siméon-Denis Poisson (1781–1840) and published,

together with his probability theory, in 1838 in his work "Research on the Probability of Judgments in Criminal and Civil Matters.l”

162

Casting: An Analytical Approach

Figure 5.1. Curve of normal distribution

The characteristics of the normal distribution are: The density function is symmetrical about its mean value. The mean is also its mode and median. Of the area under the curve, 68.25% is within one standard deviations of the mean. Of the area under the curve, 95.47% is within two standard deviations of the mean. Of the area under the curve, 99.73% is within three standard deviations of the mean.

5.6.2 Poisson Probability Distribution The Poisson distribution describes the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and are independent of the time since the last event. The probability distribution can be described as: p(t )

(O t ) n e  Ot , n!

where Ȝ = expected number of occurrences that occur during the given interval t = time interval n = number of occurrences e = the base of the natural logarithm (e = 2.718...).

(5.4)

Quality Control

163

Example 5.1 Now when data is collected in tabular form we can analyze it numerically. Let’s suppose that data follow Poisson distribution. 1. Find the mean of the Poisson distribution: c

347.7 25

13.908 .

2. Find standard deviation: 2

N

V

¦ (X

i

 c)

i 1

0.30 .

N

3. Find the upper and lower control limit: U L

c  3V c  3V

13.908  3 u 0.3 13.908  3 u 0.3

14.808. 13.008.

4. All values fall within upper and lower control limits. We can conclude That a process is in control with the upper limit of 14.808 and the lower limit of 13.008. 5. Now we can analyze the central tendency and calculate the limit: 5.1 c ± 1ı = 13.908 ± 0.3 = 14.208, 13.608 5.2 c ± 2ı = 13.908 ± 2 u 0.3 = 14.508, 13.308 5.3 c ± 3ı = 13.908 ± 3 u 0.3 = 14.808, 13.008 6. Data can be summarized in tabular form (see Table 5.2)

Limit

Table 5.2. Summary of results Percent of data

Theoretical Percent of data

c ± 1ı

64%

68.3%

c ± 2ı

80%

95.5%

c ± 3ı

100%

99.7%

Appendix

A.1 Units Conversion Acceleration ft/min

2

ft/min 1

2

2

2

ft/s 0.00028

in/s 0.0033

m/min 0.305

2

2

2

m/s 0.85E-4

mm/s 0.0847

ft/s

2

3600

1

12

1097.28

0.3048

304.8

in/s

2

300

0.083

1

91.44

0.0254

25.4

3.28

0.0009

0.0109

1

0.3E-3

0.278

3.281

39.37

3600

1

1000

0.0033

0.0394

3.6

0.001

1

m/min m/s

11811.02

2

mm/s

2

2

11.8

Angle rad deg min s grad Area 2

yd ft2 in2 m2 cm2

rad 1 0.017 0.00029 0.48E-05 0.0157 yd2 1 0.1 0.00077 1.196 0.0001196

deg 57.296 1 0.0167 0.000278 0.9 ft2 9 1 0.0069 10.76 0.0011

min 3437.747 60 1 0.0167 53.9998

in2 1296 144 1 1550.0031 0.155

m2 0.836 0.093 0.6E-3 1 0.0001

s 206264.81 3600 60 1 3239.987 cm2 8361.2736 929.0304 6.4516 10000 1

grad 63.66 1.11 0.018 0.00031 1 mm2 836127.3 92903.04 645.16 1000000 100

166

Appendix

Area (continued) yd2 2 mm 0.119E05 Density

ft2 0.11E-4

m2 0.1E-5

cm2 0.01

mm2 1

lb/in3 0.58E-3 1 0.36E-4 0.036 0.0186

kg/m3 16.02 27679.9 1 1000 515.38

g/cm3 0.016 27.68 0.001 1 0.52

slug/ft3 0.031 53.71 0.0019 1.94 1

Energy and work Btu Btu 1 J 0.95E-3 erg 0.95E-10 cal 0.0039 dyne.cm 0.948E-10

J 1055 1 0.1E-6 4.184 0.1E-06

erg 0.1056E+11 0.1E+08 1 0.418E+08 1

cal 252.16 0.239 0.24E-7 1 0.239E-07

dyne.cm 0.105E+11 0.1E+08 1 0.418E+08 1

Enthalpy (mass basis) Btu/lb Btu/lb 1 J/kg 0.000429 erg/g 0.43E-07 cal/g 1.799 cal/lb 0.00396

J/kg 2326 1 0.0001 4184 9.224

erg/g 0.23E+08 10000 1 .418E+08 92241.4

cal/g 0.556 0.000239 0.239E-07 1 0.0022

cal/lb 252.1644 0.108411 0.108E-4 453.592 1

3

lb/ft lb/in3 kg/m3 g/cm3 slug/ft3

lb/ft3 1 1728 0.062 62.43 32.17

in2 0.00155

Enthalpy (volume basis) Btu/ft J/m

3

3

3

3

3

erg/cm 372589.4

cal/cm 0.0089

0.268E-4

1

0.1E-05

10

0.239E-6

1

0.1E+08

0.239

1

0.239E-7

0.418E+8

1

26.839

1000000

erg/cm

0.268E-05

0.1

cal/cm

3

112.295

0.418E+7

0.1E-06 4.184

Entropy Btu/lb R 1

J/kg.K

J/g.K

cal/g.K

4186.8

4.1868

1.000669

0.00023 0.239

1 1000

0.001 1

0.000239 0.239005

0

0

Btu/lb R J/kg.K J/g.K

3

J/cm 0.0372

3

J/cm

3

3

J/m 37258.94

Btu/ft 1

Appendix

167

Entropy (continued) 0

Btu/lb R 0.999

cal/g.K

Flow rate (mass basis) lb/min lb/min 1 lb/s 60 kg/min 2.2046 kg/s 132.277

J/kg.K

J/g.K

cal/g.K

4184

4.184

1

lb/s 0.0167 1 0.0368 2.2046

kg/min 0.454 27.2155 1 60

kg/s 0.00756 0.454 0.0167 1

Flow rate (volume basis) 3

3

3

3

3

ft /min 1

ft /s 0.0167

m /min 0.028

m /s 0.47E-3

3

60

1

1.699

0.028

3

35.315

0.588

1

0.01667

3

2118.88

35.315

60

1

ft /min ft /s m /min m /s Force

lbf 1 0.2248 0.000224 2.2046 0.225E-05

lbf N mN kgf dyne

N 4.448 1 0.001 9.8067 0.1E-4

mN 4448.222 1000 1 9806.65 0.01

kgf 0.454 0.102 0.102E-3 1 0.102E-05

dyne 444822.2 100000 100 980665 1

Frequency 1/min 1 60 60

1/min 1/s Hz

1/s 0.01667 1 1

Hz 0.01667 1 1

Heat Flux 2

Btu/(s.ft ) 1

W/cm 1.135

0.881 2

2

2

Btu/(s.ft ) W/cm

2

kcal/(h.m ) cal/(s.cm )

2

2

2

kcal/(h.m ) 9771.39

cal/(s.cm ) 0.271

1

8604.2065

0.239

0.000102

0.000116

1

0.0000278

3.684

4.184

36000

1

168

Appendix

Heat Transfer Coefficient Btu/(s.ft

2 0

W/(m

2 0

W/(in

2 0

cal/s.cm

F)

Btu/(s.ft 1

2 0

F)

2 0

W/(m K) 20441.748

W/(in 7.327

2 0

F)

cal/s.cm 0.488

2 0

K)

0.489E-4

1

0.0003584

0.239E-4

F)

0.136

2790.00558

1

0.0667

2.04679

41840

14.996

1

Btu/h 1 3600 3.412 14.276

Btu/s 0.278E-3 1 0.000947 0.00396

J/s 0.293 1055.0558 1 4.184

cal/s 0.07 252.164 0.239 1

cm 30.48 2.54 100 1 0.1

mm 304.8 25.4 1000 10 1

2 0

C

C

Heat Rate Btu/h Btu/s J/s cal/s Length ft 1 0.083 3.281 0.0328 0.00328

ft in m cm mm

in 12 1 39.3701 0.3937 0.03937

m 0.3048 0.0254 1 0.01 0.001

Mass lb 1 2.2046 0.0022 2000 2240 2204.62

lb kg g US t UK t metric t

kg 0.4536 1 0.001 907.18 1016.047 1000

g 453.592 1000 1 907184.7 0.1E+07 1000000

US t 0.0005 0.0011 0.11E-05 1 1.12 1.1023

UK t 0.000446 0.000984 0.98E-06 0.893 1 0.984

metric t 0.000453 0.001 0.1E-05 0.9072 1.016 1

Moment of Inertia lb.in

lb.in 1

2

2

2

2

2

kg.mm 292.639

US t.ft 0.347E-05

UK t.ft 0.31E-05

Metric.t.m 0.293E-06

0.118E-07

0.106E-07

0.1E-08

0.003417

1

US ton.ft

2

288000

0.84E+08

1

0.893

0.08428

UK ton.ft

2

322560

0.94E+08

1.12

1

0.094

Metrict.m

2

0.34E+07

0.1E+10

11.865

10.594

1

kg.mm

2

2

Appendix

169

Momentum lb.ft/h 1 3600 300 7.233 26038.8 0.26

lb.ft/h lb.ft/s lb.in/s kg.m/h kg.m/s g.cm/s

lb.ft/s 0.28E-3 1 0.08 0.00201 7.233 0.72E-4

lb.in/s 0.0033 12 1 0.024 86.7962 0.87E-3

kg.m/h 0.138 497.718 41.4765 1 3600 0.036

kg.m/s 0.38E-4 0.138 0.01152 0.28E-3 1 0.1E-4

g.cm/s 3.840 13825.5 1152.12 27.778 100000 1

Power Btu/h W kcal/h cal/s hp (Metric)

Btu/h 1 3.412 3.96567 14.2764 2509.627

W 0.2931 1 1.16 4.184 735.499

kcal/h 0.2522 0.86 1 3.6 632.838

cal/s 0.0701 0.239 0.2778 1 175.789

hp (Metric) 0.000398 0.001359 0.00158 0.005689 1

Pa

dynes/cm

atm

bar

1

6894.757

68947.573

0.068

0.0689

0.000145

1

10

0.987E-05

0.00001

0.0000145

0.1

1

0.987E-06

0.1E-05

14.696 14.504

101325 100000

0.1E+07 1000000

1 0.9869

1.01325 1

MPa

Pa

0.006895

6894.757

N/mm 0.006895

dynes/cm 68947.573

145.0377 0.000145 145.0377

1 0.1E-05 1

1000000 1 1000000

1 0.1E-05 1

0.1E+08 10 0.1E+08

0.145E-4

0.1E-06

0.1

0.1E-06

1

N/m

J/m 175.1268 1 1

Pressure lbf/in lbf/in

2

psi

Pa dynes/cm

2

atm bar

2

psi

2

Stress 2

lbf/in psi MPa Pa N/mm

2

dynes/cm

2

lbf/in 1

2

Surface Tension lbf/in lbf/in N/m J/m

2

1 0.0057 0.0057

psi

175.1268 1 1

2

2

2

erg/cm 175126.84 1000 1000

2

dyne/cm 175126.84 1000 1000

170

Appendix

Surface Tension (continued) lbf/in N/m 2

erg/cm dyne/cm

2

0.57E-05

0.001

J/m 0.001

0.571E-05

0.001

0.001

Temperature F F 1 C 1.8*C+32 R R-459.69 K K*1.8-465.088

C (F-32)/1.8 1 (R-491.69)/1.8 K-273.16

R F+459.69 1.8*C+491.69 1 1.8*K

erg/cm 1 1

dyne/cm

2

1 1

K 273.16+(F-32)/1.8 C+273.16 273.16+(R-491.69)/1.8 1

Thermal Conductivity Btu/(h.ft

20

F/in)

Btu/(h.ft 1

20

F/in)

20

W/(in F/in) 0.00204

J/(s.m 0.144

20

W/(in

20

F/in)

491.348

1

70.866

J/(s.m

20

C/m)

6.933

0.0141

1

C/m)

Torque lbf.ft 1 0.0833 0.7376 0.00738 0.00074 0.74E-7

lbf.in 12 1 8.8507 0.0885 0.00885 0.88E-6

N.m 1.356 0.113 1 0.01 0.001 0.1E-06

N.cm 135.581 11.299 100 1 0.1 0.1E-04

N.mm 1355.82 112.98 1000 10 1 0.0001

dyne.cm 0.14E+8 0.11E+7 0.1E+08 100000 10000 1

Velocity (Angular) RPM RPM 1 RPS 60 rad/min 0.159 rad/s 9.549 deg/min 0.00278 deg/s 0.167

RPS 0.0167 1 0.00265 0.1592 0.46E-4 0.00277

rad/min 6.283 376.991 1 60 0.0174 1.047

rad/s 0.105 6.283 0.01667 1 0.00029 0.0174

deg/min 360 21600 57.296 3437.74 1 60

deg/s 6 360 0.955 57.2957 0.01667 1

Velocity (Linear) ft/min ft/min 1 in/s 5 in/min 0.0833 m/s 196.85 cm/s 1.9685

in/s 0.2 1 0.0167 39.37 0.3937

in/min 12 60 1 2362.20 23.622

m/s 0.00508 0.0254 0.00042 1 0.01

cm/s 0.508 2.54 0.0423 100 1

mm/s 5.08 25.4 0.423 1000 10

lbf.ft lbf.in N.m N.cm N.mm dyne.cm

Appendix

Velocity (Linear) (continued) ft/min in/s mm/s 0.1968 0.0394 Viscosity (absolute) lb/(ft.s) lb/(ft.s) slugs/ft.s Pa.s dyne.s/cm kg/(m.s)

2

in/min 2.362

m/s 0.001

cm/s 0.1

slugs/(ft.s)

Pa.s

1 32.174 0.672 0.0672

0.031 1 0.0208 0.002088

1.488 47.8802 1 0.1

dyne.s/cm 14.882 478.802 10 1

0.672

0.0209

1

10

mm/s 1 kg/(m.s)

2

1.4882 47.8802 1 0.1 1

Viscosity (kinematic) 2

2

2

2

2

ft /s 1

in /s 144

cm /s 929.0304

mm /s 92903.04

2

0.006944

1

6.4516

645.16

0.001076

0.155

1

100

0.00001076

0.00155

0.01

1

ft /s in /s 2

cm /s 2

mm /s Volume 3

ft

3

ft 1

in

3

0.00058

m

3

mm liter

3

3

3

3

Prefix Tera Giga Mega Kilo Hecto Deka Deci Centi Milli Micro

liter

in 1728

m 0.02832

mm 0.283E+8

28.317

1

0.1E-4

16387.1

0.0163

35.31467

61023.744

1

0.1E+10

1000

0.353E-07

0.00006

0.1E-08

1

0.1E-05

0.03531

61.0237

0.001

1000000

1

A.2 Prefixes Symbol T G M k h da D C M Ȃ

Value 1012 109 106 103 102 10 10-1 10-2 10-3 10-6

171

References

Abbott I.H., Von Doenhoff A.E. Theory of wing sections. Dover, New York, 1959. Ammen C.W. The complete handbook of sand casting. McGraw-Hill, New York, 1976. Ammen C.W. Metalcasting. McGraw-Hill, New York, 2000. Batchelor G.K. An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967. Campbell J. Castings. Butterworth-Heinemann, Oxford, 1991. FLOW-3D® Manual, Flow Science, Inc., Santa Fe, http://www .flow3d.com, 2006. Garber L.W. Theoretical analysis and experimental observation of air entrapment during cold chamber filling. Die Casting Engineer, May/June 1982. Gryna F.M.Quality planning and analysis. McGraw-Hill, New York, 2001. Henderson F.M. Open channel flow. The Macmillan, New York, 1966. Hirt C.W., Nichols B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Computational Phys. 39, 201 – 225, 1981. Hirt C.W., Sicilian J.M. A porosity technique for the definition of the obstacles in rectangular cell meshes. Proc. Fourth Int. Conf. on Ship Hydrodynamics. National Academy of Science, Washington, DC, 212 – 235, September 1985. Holman J.P. Heat Transfer. McGraw-Hill, New York, 1976. Lopez J., Faura F., Hernandes J., Gomez P. On the critical plunger speed and threedimensional effects in high-pressure die casting injection chambers. Manufacturing Sci. 125, 529 – 537, August 2003. Lopez J., Hernandez J., Faura F., Trapaga G. Shot sleeve wave dynamics in the slow phase of die casting injection. ASME, Fluid Eng. 122, 349 – 356, June 2000. Manzello S.L., Yang J.C. The influence of liquid pool temperature on the critical impact Weber number for splashing. Phys Fluids, 15, 257 – 260, 2003. Munson B.R., Young D.F. Fundamentals of fluid mechanics. John Wiley, New York, 2006. Prandtl L. Motion of fluids with very little viscosity. NACA, Washington, 1928. Rider W.J., Kother D.B. Reconstructing Volume Tracking. J. Computational Phys. 141, 112 – 152, 1998. Roache P.J. Computational fluids dynamics. Hermosa, Albuquerque, 1985. Sedov L.I. A course in continuum mechanics, Vols. 1-4. Wolters-Noordhoff, Groningen, 1972. Tszeng T.C., Chu Y.L. A study of wave formation in shot sleeve of a die casting machine. ASME, Eng. 116, 175 – 182, May 1994. Yang B. Stress, strain, and structural dynamics. Elsevier, New-York, 2005. Young W.C. Roark’s formulas for stress and strain. McGraw-Hill, New York, 1989. Wikipedia, http://www.wikipedia.org.

Index

A acceleration, body, 47 fluid, 18 gravity, 10, 119 plunger, 99, 111, 114 units, 164 aluminium, alloy, 5, 97, 141 air cooled, 138 bar, 54, 70 beam, 87 casting, 136 housing, 85 properties, 141 shaft, 92 water quenched, 137 axial force, 87 B beam, 44, 58, 72, 87 Bernoulli equation, 26, 102, 105 for gases, 33 Bond number, 23, 25, 98 boundary conditions, dynamic, 21, 41 free surface, 21, 41 kinematic, 21 no-slip, 16 stressed, 56 thermal, 20 wall, 20

brittle material, 57 buckling, 68, 71, 96 C cold chamber, 6, 8 compressible flow, 26, 117 computational fluid dynamics (CFD), 35, 141 concentration of, oxides and entrained air, 149, 153 stress, 94 continuity equation, 18 for gases, 28, 120 D Darcy-Weisbach equation, 10 density, 13, 18, 27, 97, 141 units, 165 displacement, 58, 60 ductile material, 57 E elasticity, 55 energy, conservation of, 18, 119 internal, 132 kinetic, 15, 132 quality, 158 thermal, 17, 28, 146 units, 165 enthalpy, 17 units, 165

176

Index

entropy, 121 units, 165 equation of state, 27, 31, 118 Euler-Bernoulli equation, 58 exponential distribution, 126, 161 F fast shot, 97, 99, 147 fatigue, 95 fill time, 107, 147 finite volume, 36, 38 finite difference, 36, 41 first law of thermodynamics, 28 flow, atomization, 22, 99 fluid, 13 gravity-controlled, 102 heat, 138 isentropic, 119 losses, 38, 107 mass, 117, 124, 150 of air, 149 rate, 2, 104, 151, 166 steady-state, 100, 118 subsonic, 33, 116 sub-critical, 25 supercritical, 108 supersonic, 33 turbulent, 16, 19, 24, 98, 101 with separation, 151 force, external, 18, 46 frictional, 15 internal, 18, 46 pressure, 2, 14, 101 shear, 60, 66, 74 surface tension, 22 viscous, 29 units, 166 Fourier law, 17 fractional area-volume obstacle representation (FAVOR), 38 Froude number, 23, 25, 38

G gas porosity, 2, 11 gating, 2, 7, 105, 143 H heat transfer coefficient, 20, 133, 139, 146, 167 Hooke’s law, 44, 55, 73, 86, 90 hot chamber, 6 hydraulic jump, 25, 108 L load, axial, 55, 68 compression, 68 critical, 68, 71 distributed, 75 pointwise, 65 static, 95 tensile, 55 torsional, 89 loading, 46 M Mach number, 33, 117, 123 microporosity, 154 modulus of elasticity, 44, 55, 65 Mohr’s circle, 51 moment, bending, 66, 72 of inertia, 44 torque, 88 units, 167 momentum, conservation of, 119 diffusion, 19 dissipation, 17, 24 units, 168 N Navier-Stokes equaions, 18 for gases, 29 Newton’s laws of motion, 46 Newton’s law of cooling, 133

Index

O overflows, 127, 143, 147 P permanent mould, 3 plunger, 6, 97, 108, 115, 143 Poisson probability distribution, 162 Poisson process, 126, 161 Poisson ratio, 57 pressure, 13, 27, 39, 100, 109, 117, 147 dynamic, 26,102 gradient, 14, 18, 21 stagnation, 26, 102 total, 26, 34, 102 units, 168 R Reynolds number, 23, 98, 147 runner, 2, 10, 109 S Saint-Venant’s principal, 56 shot sleeve, 7, 25, 97, 108, 127, 143 simulation, 35, 115, 141 slow shot, 97, 111, 143 solidification, 2, 18, 41, 141, 149, 154 specific heat, 17, 29, 119, 132, 142 speed of sound, 31, 33, 121 statistical process control, 159 strain, 44, 55, 72 sress, biaxial, 48 bending, 71 components, 47 compression, 47, 72 concentration, 94 linear, 47 normal, 47, 51, 72 plane, 50 principal, 47 risers, 94 shear, 16, 47, 50, 58, 74

177

stress, tensile, 47, 51, 72 thermal, 86 triaxial, 48 viscous, 15, 18 uniaxial, 47 units, 168 von Mises, 58 surface tension coefficient, 21, 97 surface waves, 25, 98 T temperature, 13, 16, 27, 41, 86, 132 units, 169 thermal conduction coefficient, 17 units, 169 thermal diffusion coefficient, 139 thermal expansion coefficient, 86 units, 169 torque, 46, 59, 88 units, 169 turbulence, 2, 16, 24, 41, 98, 147 V velocity, 10, 13, 20, 97 of plunger, 109, 115, 144 units, 169 ventilation, 11,116, 125 viscosity, 14 dynamic, 15 kinematic, 18 turbulent, 19 units, 170 viscous boundary layer, 15, 100, 145 Volume-of-Fluid (VOF), 39 W Weber number, 23, 24, 98 Y Young’s modulus, 55