A NUMERICAL MODEL OF MHD SEPARATION OF INCLUSIONS FROM MOLTEN ALUMINUM

knnifer Ye Sheng

A thesis submitted in conformity with the requirements

for the degree of Master of Appüed Science

Graduate Department of Metailurgy and Materials Science

University of Toronto

O Copyright by Ye Sheng (1997)

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To My Parents

SEP-TION

Master of Applied Science, 1997

Ye Sheng

Graduate Department of Metailurgy and Materiais Science

University of Toronto

ABSTRACT

An electromagnetic device is proposed for the separation of inclusions kom Aluminum. The

device utilizes a magnetic structure that genmtes a current within the Bow regime. The

Electromagnetic force field exerts a pressure upon the inclusions which propels them towards a

wall where they accumulate and are prevented fiom re-entering the stream.

The thesis presents two mathematical models which are used to reveal the fundamental

mechanimis of inclusion removal £kom a liquid metal stream. The fmt of these is a simplified

mode1 which assumes a total independence between magnetic and flow fields. The second is a

f i t e element mode1 for fiuid flow which is used to c o b the overall features of the simplified

approac h.

The main results of the thesis show that electromagnetic effects can be used for the removal of

inclusions fkom a liquid metal stream. However, the magnitudes of excitation cments are high,

and the geometry of the device fiindamentaily affects the efficiency of filtration.

ACKNOCVLEDGMENTS

I would like to thank Professor G. Bendzsak for his invaluable advice and supenision

throughout my stay at the University, without whose presence my development as both a

student and an individual would not have progressed as rapidly.

Appreciation is especially to Michael Dawson and Warren Adolphe for their helpfulness

and encouragement. It is also extended to Dr. D. Guo for his invaluable advice and the

helpfulness of Chns Achong, Nicoleta Western and Sameer Kochar.

TABLE OF CONTENTS

ii ABSTRACT üi ACKNOWLEDGMENTS iv TABLE OF CONTENTS

vi NOMENCLATURE ix LIST OF FIGURES xi LIST OF TABLES

CFUPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Fundamentai Principle of Liquid Metal MHD 3 1.3 A review of Relevant Literature 4 1.4 Thesis Outline 5

CHAPTER 2. FUNDAMENTAL PIUNCIPLES OF INCLUSION SEPARATION

2.1 hpurities in Aluminum Production 2.2 Conventional Separation Methods

- 2.3 Magnetohydrodynamic (MHD) separation 2.3.1 Introduction 2.3.2 Proposed physical sysrem

2.4 Force Analysis 2.4.1 Electromagnetic body force generation 2.4.2 î l e impact of EM- body forces on particks

2.5 Analysis of Forces Acting on a Particle 2.5.1 fnertia l forces 7.5.2 Electromagnetic body force 2.5.3 Magnetic force for smnII scaLe, weukly magnetic

particles 2.5.4 Stokes drag force 2.5.5 Buoyancy force

2.6 Order of Magnitude Estunations for Force Cornponents

CHAPTER 3. ANALYSIS OF THE MAGNETIC FIELD 26 3.1 A DC Electromagnetic Separation Model 26 3.2 Magnetic Fields 27

CHAPTER 4.

CHAPTER 5.

APPROXIMATE CHARACTERISTICS OF A MHD SEPARATION METHOD

4.1 Equation of Paxticle Motion 4.2 Predictions of Particle Traces

4.2.2 Cases considered 4.2.2 Case # I - good separation behavior 4.2.3 Case #2 - puor separa tion behavior

4.3 Separation Efficiency

NIJMEXICAL MODELING OF ELECTROMAGNETIC SEPARATION

5.1 introduction 5.2 Flow Mode1

5.2.1 Essential assumptions 5.2.2 FIow equations 5.2.3 Boundary conditions

5.3 Finite Element Approach 5.3.1 Element and mesh 5.3.2 Expansions of Y, , and P

5.4 Results and Discussion 5.4.1 A valid case for the cornputer code 5.4.2 Flow,pressure, E.M. current and E.M.

force PeI& 5.5 Particle Traces

CFIAPTER 6. CONCLUSIONS AND RECOMMENDATIONS 6.1 Conclusions 6.2 Recornmendations for future work

REFERENCES 81

APPENDIX 1 Steps in Derivation of The Finite Element 85

Approac h I I Attached Prog.rams 96

NOMENCLATURE:

radius of the inclusion particle

vector potentid

area

magnetic flux density

maximum magnetic flux density

intensity of magnetization

diameter

electrical potentiai

electromagnetic force acting on a particle

Stokes force acting on a particle

force

acceleration of gravity

strength of the magnetic field between the granules at the point of particle location

magnetic intensity

Hartman number

eIectncal current

current density

dyoamic shape factor

attachment distance

mass

pressure

production rate

radial position

radius of the tube

residual function

Re Reynolds number

Rem magnetic Reynolds number

s area

t time

U particle velocity

v volume

V liquid metal flow velocity

V, initial velocity

v, volume of the particle

W, weight function

z axial position

cylindrical coordinate direction

electrical conductivity

density

kinematic viscosity

absolute permeability of free space (vacuum)

relative permeability

magnetic susceptibility

separation efficiency

dynamic viscosity

velocity shape fûnction

y pressure shape function

0 finite e1ement domain

buoyancy force

electromagnetic force

fluid flow

inertiai force

maximum velocity

pariicle

radial direction

Stokes drag force

axiai direction

magnetic force for small-scale weakly magnetic particles

LIST OF FIGURES

FIGURE 2.1

FIGURE 2.2

FIGURE 2.3

FIGURE 3.1

FIGURE 3.2

FIGURE 3.3

FIGURE 3.4

FIGURE 3.5

FIGURE 3.6

FIGURE 3.7

FIGURE 3 -8

FIGURE 3.9

The physical mode1 of E.M. separation

Diagram illustrating generation of induced currents

A schematic of the forces on a particle

Mechanical flowsheet of separation system

Vector potential produced by current filament of

volume distribution [20]

Circular ~g of charge [20] 3 t

Arrangement of the copper coils 32

Solenoid with 4 layers and 10 turns 33

The arrangement of solenoids around the molten metal tube 3 4

Vector plot of the magnetic flux density for one turn coi1 35

The distribution of axial magnetic density for one-turn and 36

four-layer solenoid

The distribution of radiai magnetic density for one-him 37

and four-layer solenoid

FIGURE 3.10 Magnetic field along the tube (half) 38

FIGURE 3.1 1 The radial magnetic field for four-layer and ten-nim solenoid 39

FIGURE 3.12 The axial magnetic fieid for four-layer and ten-tum solenoid 40

FIGURE 4.1

FIGURE 4.2

FIGURE 4.3

FIGURE 4.4

FIGURE 4.5

FIGURE 5.1

FIGURE 5.2

FIGURE 5.3

FIGURE 5.4

FIGURE 5.5

FIGURE 5.6

FIGURE 5.7

F r G W 5.8

FIGURE 5.9

FIGURE 5.10

The traces of inclusions estimated by Runge-Kutta

method (case #1)

The traces of inclusions estimated by Runge-Kutta

method (case #2)

The separation efficiency vs. Haman nurnber

The separation efficiency vs. maximum magnetic field

The range of inclusions separation in Aluminum

The simplified schematic of the separator

Typical FEM. noddelement configuration

The mesh of the solution region

Nodal points for the local element

Calculated axial velocity distribution of the flow without

E.M. force

The absolute erron of calculated velocity of the flow

without E.M. force

Vector plot of velocity field with E.M. force

Contour plot of current density along the tube (half)

Vector plot of E.M. force field along the tube (half)

The typical particle traces at different entrances with low

separation efficacy

FIGURE 5.1 1 The typical particle traces at different entrances with high

separation efficiency

LIST OF TABLES

TABLE 2.1 Inclusion distribution in Aluminurn production

TABLE 2.2 The order of the forces acting on a A40, particle

TABLE 2.3 The order of the forces acting on a S i 0 particle

TABLE 4.1 The ranges of the parameters in simulation

TABLE 4,2(a) Simulation parameters for r, = 2 cm

TABLE 4.2(b) Simulation parameters for r, = 4 cm

TABLE 4.2(c) Simulation parameters for r,, = 4 cm

TABLE 5.1 Boundary conditions of the tube

TABLE 5.2 The cornparison of E.M. separation emciency for

the numerical and andytical results

INTRODUCTION

1.1 Background

Final product usen have placed an ever increasing burden on manufacturers to

institute strict quality control at every step of their processes. in numerous instances,

product quality problems have been traced to impurities within the metal received fiom

the primary producers. In order to maintain their estabiished shares of the market place,

primary metal producers must meet, or exceed, the demands of ever tighter specifications

regarding acceptable levels of impurities within their products. Thus great efforts are

expended in every sector of the metais industry to increase product cleanliness. One

method for the reduction of inclusions is the filtering of Liquid metals prior to casting.

This is a particularly useful technique for the aluminum industry during the production of

metal destined to be used in very thin sheets.

It will be noted later in this thesis that every technique used for the reduction of

inclusions in a unit volume of liquid aluminum has one or more serious difficulties. One

of the generic reasons for the problems lies in the indirect manner by which forces are

placed upon an impurity. These Vary fiom slight density differences between the

aluminurn and the inclusion to chance encounten with bubbles injected into the liquid

metal. To the best knowledge of the author, there is no successful commercial filtering

device for liquid aluminum which explicitly places forces with sufficient magnitudes

upon inclusions to drive them fiom the metal. This is due to the inability of the devices

to generate controlled amounts of body forces within the liquid rnetal on every single

inclusion to propel it, within a short period of time, to a surface where it collects and is

subsequently removed.

Electromagnetic filtration promises to be a method which addresses the above

problem in the most direct manner. Due to its fundamentai ability to generate body

forces within liquid metais, techniques of liquid metal magnetohydrodynamics (MHD)

can be employed to bear immediately upon inclusions and to drive them in a desired

direction. Unlike others, it is a direct technique, but udomuiately, its potential to date

has not received the serious attention it deserves. The purpose of this thesis is to explore

some of its key theoretical features through exploration of a new device, and establish

directions for other researchers which may lead them to eventual success.

1.2 Fundamental Principle of Liquid Metal MHD

The basic objective of liquid metal MHD is the utilization of body forces, which

are generated by the interaction of a current and a magnetic field, to perforrn a desired

rnetallurgical effect. The objective could be any one of a number, including the

hansportation, confinement, filtering or levitation of liquid metals. While the purpose

may Vary, the fundamental mechanisms are similar, that is, an electric current and an

orthogonal magnetic field produce a force field within electrically conducting metallic

bodies. When the metal is liquid, motion takes place which can be purely rotational,

purely transitionai or a combination of the two dependhg on the nature of the force

field.[l]

The sources of electrical quantities, such as currents and magnetic fields can be

either extemal or intemal to the body. That is, eleceical current can be injected into the

metal or induced by motion of the liquid. Magnetic fields can be genemted either by a

current carrying conductor, i.e a coil, or by the current within the conductor itself. The

phenornenon of interest is produced by interaction between the field and a current. When

this occurs, a force is generated in a direction that is orthogonai to both. This is a true

body force, analogous to gravitation , and can drive a fiow of iiquid metai or produce

motion of entrapped particles across flow streamlines.

It should be reaiized that the generated body forces cause motions of the iiquid

metal which in tum produce elecaical currents which create their own magnetic fields.

Thus, the equations of magnetics (Maxwell) and fluid mechanics (Navier-Stokes )

become intemüngled in a nonlinear manner. As a result, analytical solutions to realistic

MHD problems are difficult, if not impossible to obtain, thus numencal solutions are the

normal methods used to obtain insights into physical events.

1.3 A Review of Relevant Literature

The body of published literature on the principles of electromagnetic filtration of

inclusions ffom liquid metais is not very large. Some of the eariy work has been

conducted in laboratones located in Riga, Latvia and reported in the journal of

Magnetohydrodynamics by Kirko [2] and Galfgat[3]. A device has been descnbed [4]

which employs a magnetic field due a single-tu. coil around a cylindrical tube which

cames liquid metal. The induced currents within the metal interact with the magnetic

field to produce a body force upon the inclusions. The observable effects fiom ihis

device are small at the magnitudes of' magnetic flux levels which can be obtained by a

single turn coil carrying reasonable values of DC current.

Shilova[S] described a conceptuai device to produce body forces which are

generated by the interaction of a DC current with its own magnetic field. This is

commoniy referred to as the pinch efZect and has been recently applied to fi!tration by

Brimacombe [6 ] in a tube having rectangular cross section. As it will be presented in

this thesis, large currents are required to produce reasonable effects upon the inclusions

trapped within the metai flow. This cm easily lead to b o t . stability problems and

secondary fiows which become detrimentai to filtration.

Marty and Aiernany [7] presented a theoretical description of a device employing

obstacles imrnersed in a Bow of liquid metal which carried electical current. The

distortions produced in the electrical field causes rotational body forces which act directly

upon the inclusions to sweep them fiom the flow. The method is completely analogous

to phenornena observed in channel induction h a c e s where the inclusions are driven

fkom the Stream to accumulate along the walls. This phenomenon is responsible for the

fiequent blocking of the channel during the melting of rnetals having high concentrations

of oxides andor debris.

1.4 Thesis Outline

This thesis consists of six chapters. Chapter 2 provides a brief description of the

removal of inclusions through the filtering of liquid metal. The work concentrates

exclusively on Aluminum. The theoretical principles of electromagnetic separation of

inclusions are discussed.

Chapter 3 discusses a proposed DC electrornagnetic(E.M.) separation method.

The magnitude and distribution of the magnetic fields are estimated.

Chapter 4 presents a simplitied mode1 of the proposed E.M. separation. The liquid

metal flow is assumed to be parabolic, thus assuming that the magnetic field does not

change the flow pattern. An estimation of the inclusion traces in the flow and E.M.

separation efficiency are therefor approximated The information obtained is suitable for

design purposes and to explore the fiindamental features of E.M. filtration.

Chapter 5 describes a finite element method used for the numerical simulation of

the flow . It eliminates the simplifications of the previous chapten. It is shown that

accurate simulation is necessary if good estimates for separation efficiency are to be

realized.

Chapter 6 surnmarizes the findings of the study and makes recommendations

regarding future work.

FUMIAMENTAI, PRINCIPLES OF INCLUSION SEPARATION

There are a number of conventional methods which can be used for inclusion

separation in metallurgical processing. Each of these methods have a number of

disadvantages for different kinds of impurities which are entrained within the liquid

metal. In this chapter, the basic principles of electromagnetic separation of inclusions

are introduced and discussed.

2.1 Impurities in Aluminum Production

The main source of the impurities are associated with the feed alumina and

cathode carbon of the reduction celis. Quality improvement of Aluminum and its alloys

is tightly coupled with srneking, casting and the efficiency of the treatment process

used

to eliminate the solid non-metallic inclusions. Therefore, pnor to a study of their

removal, it is necessary to review the nature, sources and mechanisms of formation of

these non-metallic inclusions.[8]

Previous research[8] showed that solid non-metallic inclusions have a non-

uniform distribution in the h a 1 product. These inclusions occur as either isolated

particles or different agglomerate phases which are composed of oxides, nitrides,

carbides, fluondes, chiorides, sulfides, silicates, aluminates and their various

combinations. The metal oxides which are retained as non-metallic inclusions are

typically Al,O,, SiO2, MgO, Cao, or their combinaîions. Other composite inclusions

are sodium silicates, potassium silicates and calcium aluminates.[8] Typical inclusion

contents for some of the above mentioned inclusion species are listed below in Table

2.1.

Table 2.1 Inclusion distribution in Aluminum production

The majority of these inclusions, which have densities close to that of Al, are dispersed

through the entire mass of the liquid metal.

2.2 Conventional Separation Methods

In industry, the separation of non-metallic inclusions nom Alurninum has been

perfomed , or is currently perfomed, by one of the following methods[9] :

1) Removal by sedirnentation.

2) Removal by Botation.

3) Removal with the aid of flues.

These methods have achieved various degrees of separation efficiencies for non-

metallics. These methods, however, are subject to one or more of the following

problems :

A minimum effective size of non-metailic inclusion which rnay be efficiently removed.

Lo w effective density difference behveen particle and melt, thus inhibiting separation.

Entrainment of water vapor thus causing the formation of new non- metallic oxide inclusions and hydrogen dissolution into the melt.

Yield losses.

Presently, researc h is being perfomed to de fine and enhance the understanding

of the specinc chernical mechanisms which are involved in some of the above

processes. It is, however, the main purpose of this work to investigate the potential

viabiiity of developing an MHD technique to separate non-rnetallic inclusions. or

enhance their separation fiom liquid Alminum.

2.3 Magnetohydrodynamic (MHD) Separation

2.3.1 introduction

Magnetohydrodynamics (MHD) is a science concemed with the study of the

motion of electrically conducting fluids under the infiuence of electromagnetic body

forces[lO]. It is the study of mutual interactions of electromagnetic fields and the

motion of conductive fluids. In the past few decades, a nurnber of applications for

MHD concepts have been explored for metallurgical purposes. Research has found

applications in stirring[l 11, levitation, melting[l2], thermal protection, flow control,

pumping[lî] and fiee surface shape controi 1141. In spite of extensive research

activities, to date, MHD separation has found little application in industry.

The key principle of MHD separation is the production of electromagnetic body

forces on inclusion particles which drive them to separate fkom the liquid metal. Several

MHD separation methods have been studied theoretically as well as experimentally [ t 1,

121. One method uses the interaction between the electromagnetic force which is

generated by the magnetic field produced by the curent. and the cunent itself. This is

known as a "pinch" force as it is directed radially inward, effectively squeezing or

"pinching" the liquid metal Stream which carries the current[15]. Other methods,

t O

such as using an AC solenoid to produce the desired electromagnetic forces, have been

also studied. [16]

2.3.2 Proposed physical system.

In the method proposed in this thesis, a magnetic field is produced by a solenoid

carrying DC current. As Iiquid metal flows through the rnagnetic field, an induced

current is produced by the fluid motion through the rnagnetic field. The interaction

behlreen the magnetic field and the induced current generates electromagnetic forces

which act upon the Iiquid metal. The inclusions expenence an electrornagnetic body

force which is in the opposite direction to the electromagnetic force on the metd. This

body force may cause particle motion towards the container walls where the non-

metailic inclusions accumulate.

For the purposes of confining the flow in the following analysis, an enclosed

tubular flow circuit is postulated, and henceforth will be assurned. A rnagnetic field

dong this tube can be generated in a number of ways. It is proposed that the magnetic

field be generated by a solenoid wound around the tube. It is energized by a DC current

which effectively sets up an excitation field. The details of the generation of this field

and the solenoid are discussed in sections 3.3 and 3.4.

A simple diagram of this mode1 is shown on the following page:

! solenoid

Figure 2.1 The Physical Mode1 of E.M. Separation

tn the following sections, the analysis of forces acting on the inclusions will be

discussed and their magnitude will be estimated.

2.4 Force Analysis

2.4.1. EIecîrornagnetic body force generolion

As stated previously, when iiquid metal flows through the magnetic field, it

interacts with it to induce currents. These, in tum interact wîth the magnetic field and

create the electromagnetic body forces which change the flow pattern. In general, flow

' velocity ( fi), current density (7) and the electromagnetic force ( F) form a coupled set

of relationships consisting of the following two parts :

a) One is the imposed field o Ë due to the extemal voltage

b) The other is n((O x P ) and is induced by motion of the fluid, through the

magnetic field.

The equation for the current density is[20]:

In this thesis, the conduction term(o É ) can be ignored since there is no imposed field

across the liquid rnetal. There is, however, a closed path within the fiuid where the

induced currents can circulate, as shown in Figure 2.2.

In general, the electromagnetic@.W.) body force is obtained by [20]:

Substitution of equation (2.1) into (2.2) gives:

Fe = c r ( V x E ) x ~

The problem is assurned to be symmetrical and can be described in a cylindrical

coordinate system (Figure 2.1). Thus, the angular components of both velocity and

magnetic filed are zero, (that is, V, = B, =O). The current density component of

electromagnetic force is caicdated by:

The induced current density caused by rnoving liquid metal through the

magnetic field is in the 8 direction only. Three possibilities can be distinguished for the

current density, and its effect with respect to flow :

a) Br V, > B, Vr ( Opposes flow)

b) B, V, = B, V, ( No effect upon fiow)

C) Br V, < B, Vr (Aids flow)

The interaction of the current density with the magnetic fields creates

electromagnetic forces in both the radial and axial directions. These E.M. body forces

can be described as follows:

where :

Fr = JOB$ = o(B,B,V, - B,~v, )F

F, = O

F, = -J,B,Z = o(B,'v, - B,B,V, )Z

therefore :

Fe = GB, (BrVz - B, Vr)C+~Br(B,Vr -BrVz)2

= o(B,V, - B, V,)(B,P - BrZ)

The radial E.M. force, Fr, is responsible for the separation process. The axial

E.M. force, &, is responsible for either the acceleration or deceleration of the liquid

metal. The magnitudes of these electrornagnetic body forces clearly depend on both

the electromagnetic and velocity fields. Chapter 3 presents the calculations for magnetic

field. The radial and axial veiocities will be obtained fkom the solution of two

dimensional steady state Navier-Stokes equations in Chapter 5.

2.4.2. The impact of E.M. body forces on inclusion pariides

In order to anaiyze the motion of inclusions in liquid metals, it is necessary to

explore the eEect of each force component acting on a particle. The analysis of forces

acting on particles belongs to "particle dynamics", whîch is a branch of general

mechanics deaihg with relative motion between a particle and its surrounding fluid.

The following anaiysis assumes that motion of a single particle in an infinitely

large volume of a fluid. A low particle concentration assumption ensures that the

dynamics of each particle c m be studied individually. Thus, inter-particle dynamics is

not considered in this thesis.

Prior to anaiysis of each force component in detail, a number of hypotheses are

made to simplify the problem. These are:

a) The inclusions concemed are smdl enough to allow an approximation of

their velocities to be that of the bulk flow.

b) The inclusions are sphencal.

c) The inclusions do not interact with one other.

d) The Liquid metal fîow is assumed to be laminar.

The particle movement in a flow of liquid metal under the influence of an

electromagnetic field is described by a set of vector differential equations which are

derived fiom the balance of al1 the forces acting on the particle[lï]:

f l + F q + F , + < + F , = 0 (2.6)

where: F , = inertial force

Ë, = electromagnetic body force

F, = magnetic force for smali-scaie weakly magnetic particles

F, = Stokes drag force

F, = Buoyancy force

The forces acting on the inclusion are shown in the fiee body diagram in Figure 2.3 :

Figure 2.3 A schematic of the forces on a particle

Each force component is now considered texm by term and its effect is quantitatively

explored in Section 2.5. An approximate order of magnitude analysis is conducted to

justify the assumptions in this thesis.

2.5 Analysis of Forces Acting on a Particle

2.5.1 Inertral forces

The distributions of both flow field and electmmagnetic field are nonuniform.

Thus, a particle moves with different velocities at different positions within the flow

field. The inertial force acting upon an inclusion is given by:

F, = qv,dU/dt

where : 4 = density of the particle (kg/m')

V, = volume of the particle (m3)

Ü = velocities of the particle (m/s)

t = time (s)

Usually, the inertial force is srnaller than other forces due to the very small mass

of the particle. This means that it contributes negligibly to particle movement.

Therefore, it is neglected in calculating the particle traces within the flow field.

2.5.2 Eteeîtomagnetic body force

The electromagnetic body force is produced when the liquid metal flows

through a magnetic field. A theoretical equation of the electromagnetic force on a

sphencal particle having an electrical conductivity different fkom that of the liquid

metal, was derived by Leenov and Kolin (O).

where: Ê, = the electromagnetic force acting on a unit volume of liquid metal .

q , q = the.electrica1 conductivity of fluid and particle, respectively.

vp = the volume of the particle.

In this study, only non-conducting inclusions are considered (%=O). Thus, the force

( F ,J becomes:

EP =-R a 3 ( . 7 x B)

where: a = the radius of the particle

2.5.3 Magnetic force for smaf f scale, weakiy magnetic particies

The equation for the force upon a weakly magnetic particle in an external

electromagnetic field is :

F, = POX v p q q

where: p, = ftee space permeabiiity

x = magnetic susceptibiiity

= zero for this thesis,

= strength of the extemal magnetic field.

For a non-magnetic medium, such as Aiuminum and its non-metallic

inclusions, the magnetic susceptibility, x , is zero. Hence, the terni F, does not appear

in this thesis.

2.5.4 Stokes drag force

Due to the viscosity of the liquid metal, any relative motion beween the particle

and its smounding fiuid wili produce a drag force. A Reynolds number, useful for the

characterization of particle motion, is given by:

~ e = @ , I%ÜI pJ/p (2.1 1)

where: p, = the density of the Liquid metal

p = the mo lecular viscosity of the liquid metal.

The drag force is calculated fiom the Stokes equation:

F, = 3x~,p(Ü - O)

where: D, = diameter of the particle.

Ü = velocity of the particle

Y = velocity of the liquid metal flow

2.5.5 Buoyrrnq force

Particles having a different density fiom that of the liquid rnetal will experience

a positive or negative buoyancy force. This force is described by:

F b = (P& v,g

where : g = the acceelration of gravity.

Because of the very short distance and short time in which particles move inside

the proposed filtration system, the buoyancy force can be neglected.

2.6 Order of Magnitude Estimations for Force Components

It is seen fkom equation (2.9) that the force which acts on the particle is opposite

to the electromagnetic force which acts on the liquid metal.

The radial and axial E.M. force components are:

rdirection: Fe, ,=-kvpxFr=-kvpa ( B , B , V , - B ~ ~ V ~ ) ? (2.14)

zdirection: F , = - ~ ~ , ~ F , = ~ ~ , ~ ( B , ~ v , - B ~ B , V , ) ~ (2+15)

where : k = dynarnic shape factor for the sphencal particles (k=3/4)

F, = radial electromagnetic force acting on a particle

F,, = axial electromagnetic forces acting on a particle

v, = volume of the particle

For a first order approximation, the flow of liquid metal in a tube is assumed to

be undisturbed by an electromagnetic field. Thus, the flow is fully developed in the

tube and the electromagnetic force has a negligible effect upon the distribution of the

Bow field. The fùlly developed velocity distribution is given by[18]:

and

where :

v,, = 2yo

V, = mean velocity of the liquid metal flow

V,, = maximum velocity of the liquid flow

Substitution of Equation (2.16) and (2.17) into (2.14) and (2.15), the force equations

gives the radial and axial force as:

3 z direction: Fe, = -n a x <r -Br V,, { [ l - w 2 ] }

where: V,, = the maximum velocity of the liquid metal flow.

Thus, once the magnetic field is known, the distribution of electromagnetic

body forces on the particle in the flow field can be estimated.

It is assurned that liquid Aluminum flows through a cylindrical tube having a

diameter of 4 cm and a length of 30 cm (see Figure 2.1). The magnetic field is

generated by a DC current flowing through the solenoid and is located in the region

where the tlow is fùlly developed. When typical values of axial and radial magnetic

fields are considered, a simple calculation gives approximate magnitudes for the forces

acting on an inclusion.

The magnetic field calculation is presented in detail in Chapter 3. For the

present purposes, cypical values for the magnetic field density in the radial and axial

directions, (Br and Bz), dong with the flow velociv within the charnel, will be used to

estimate the magnitudes of the force components.

An order of magnitude estimate is now included for the forces discussed in

section 2.42. This estimation is used to explore their effects upon the motion of

inclusions. The results of the estimation are listed in Table 2.2 and 2.3 for the

fo llowing conditions:

at position: r = 0.525 mm and z = 100 mm

me 1 : N20, inclusions (p = 3500 kg/m3)

Operating conditions :V- = 0.02 m/s; Br = -0.0 1724 w/m2' B, = 0.09667 w/m2

Table 2.2 The order of magnitude of the forces acting on a Ai@, particle

1 O p

renet; SiO, inclusion (p = 2650 kg/m3)

(The operating parameters are the same as in Case 1.)

I

r 2

Table 2.3 The order of magnitude of the forces acting on a SiOz particle

Inertial Force (N)

*.-.-

y - O

Electromagnetic Force (N)

*

sx 10-l4

Stokes Force (N)

- & ; -SX 10-10

Buoyancy Force (N)

: 3 : O

An examination of the force magnitudes in Tables 2.2 and 2.3 reveals the

fo llowing:

a) The separation effect is improved if the density of inclusions is less than or

equal to that of the liquid metal. This effect is illustrated by Case 2, Table

2.3, in which the test inclusion is SiOL .

b) It can be seen that the buoyancy force is very small and is in the same

direction as the electromagnetic force. This allows for the buoyancy force

to be neglected in future calculations. However, it is important io note that

if the density of the particles is greater than that of the liquid metai, the

effect of the gravity force will be significant and its effect cannot be

neglected (see Table 2.2). in this case, the density of AI,O, is much greater

than that of Al, thus the buoyancy force is larger than the other forces and

opposite to the electromagnetic force.

The above analysis supports the conclusions of reference [19], that for the

removal of nonconducting, non-metallic particles fkom liquid metals, the density of the

particle should be in the order of I/2 to 2/3 of that for the metal.

It can be concluded that the effects of Stokes forces always oppose the

separation process. For decreasing inclusion diameten, the magnitude of the Stokes

drag force approaches that of the electromagnetic force. This renders the removal of

very mal1 hclusions f?om liquid metal difficult. Therefore, to achieve this goal, a very

strong magnetic field is necessary.

ANALYSIS OF THE MAGNETIC F'TELD

3.1 A DC Electromagnetic Separation Model

A flowsheet for a DC electrornagnetic particle separator is shown in Figure 3.1.

Molten aiuminum, which is assumed to contain uniformiy distributed inclusions, is

poured from a holding fumace through the E.M. separator, which is connected to a DC

current power supply. To create the magnetic field, a large amount of current is passed

through the solenoid, thus water cooling is required. The inclusions will be collected on

the tube waU as the molten aluminum flows through the proposed device.

electrode soIenolb-/

water O Figure 3.1 Mechanical fiowsheet of separation system

in general, the velocity Y and magnetic density B have vector components in

radial, axial and angular directions. Hcwever, these vector fields need only be described

by the r and z components due to the axial symrnetry of the solenoid and thus lead to a

slmplified two dimensional problem.

3.2. Magnetic Fields

Maxwell's equations descnbe the relationships between the current and the

magnetostatic field, and according to[20]

c u r l ~ = ~ x R = f AI^') (3.1)

V . B = O (3 -2)

where: 8 = the magnetic field intensity (Mm)

J = the current density (A/mZ)

B = the flux density (W/m2)

Equation (3.1) is known as Ampere's law. In the radial coordinate system, it can be

expressed as:

The magnetic flux density vector B is related to through:

B=~ioli .R (Tl

where: = absolute permeability of free space (vacuum) = 47cx 1 05

= the relative permeability(for aluminum equals 1)

Thus,

B = p, ,H

r u d a = -

From equation (3.1) and (3.3), the reiationship between B and 7 is obtained:

~ x B = ~ , j (3 *4)

Since the flw h s of vector B must be ciosed , it is possible to associate it with a

vector potential A. Hence the solution of electromagnetic equations are simplified such

that:

a a a - - 1 = j (A) âr a az m2

V X A = B

v.A=o

The combination of (3.4) and (3.6) gives[20]:

Equation 3.7 can be re-written, by using standard vector identities, as:

V X V X A = V ( V ~ A ) - ( V * V ) A = ~ , J

Combining equations (3.5) and (3.8) gives:

The vector potential A is a mathematical entity, hence is not a rneasurable

physical quantity. Equation (3.9) links each component of A to a source term 7 . This

Ieads to a simplification of Maxwell's equations allowing the magnetic quantities to be

determined &om various source currents 5 .

In many applications, it is possible to define the volume elements dv as thin

filaments parallei to the direction of the current density 7 , namely dv = dSeds.

where:

a) dS andds point in the directionof j

b) dS dehes a surface element normal to 7,

c) ds is the filament length. (See figure 3.2)

Figure 3.2 Vector potential produced by a cment filament [20]

The vector A can be obtained by integrating equation (3.9) over the control volume,

resulting in the Biot-Savard relationship:

If the cross section of the conductor is small compared with distance r nom the

point of observation P (see Figure 3.3), equation 3.10 can be re-written as:

In the present problem, each turn of the solenoid can be assumed to be an ideal circular

current ring. Each turn is then subdivided into several small rings. When the ring has a

mail cross section, equation (3.8) can be used to calculate the magnetic vector B due to

a single coil. Equation (3.1 1) can be wrinen as[22]:

where: u, . unit vector in the r direction

u, . unit vector in the 4 direction

-. . .*

Figure 3.3 Cucular ring of cunent [20]

The b t component, in the radial direction is canceled by symmetry, and the second

component, in the u2 direction, is tangentid to the cucle. Evaluation of the integral, given

by equation 3.12 leads to[20];

The magnetic field vector B is obtained by differentiating equation (3.13) in a cylindrical

coordinate system[22]:

where: a = the radius of the coi1

K(k) and E(k) are the complete eliiptic integrals of the fint and second kind [21]

respectively, wi't the modulus:

4ra k =

(r + a)' + z'

The magnetic field of the solenoid was detemined by the superposition of a

series of fields due to single coils displaced with respect to each other. Their spacing and

radii are detemined by the coil geometry required for filtering.

3.3 Design of The Solenoid

3.3. 1 Soienoid Dimensions

----------------*---________________________--- ( m a Po-. na

Figure 3.4 Arrangement of the copper coils

Figure 3.4 shows the copper coils used for construction of the solenoid. A hollow

square copper tube was chosen, with a cross sectional area of 100 mm' , and a

thickness of Imm. In order to separate the inclusions fiom the liquid metal, a strong

magnetic field is required The magnitude of the magnetic field can be increased by

increasing the number of tums and layen used in the solenoid. The increase in the

number of layers leads to larger radii which, in tum, produce progressiveiy srnailer

increases in the magnitude of flux density. An increase in the number of tums, however,

leads to linearly increasing the magnitude of the field. The magnetic field of the coi1 in

Figure 15 is created by the superposition of fields produced by a solenoid consisting of

four layen of ten tums each (see Figure 3.5).

Subdivisions

Layer 4

Layer 3

Layer 2

Layer 1

Figure 3.5 Solenoid 4 layen, 10 t u s each. For magnetic field calculations

each tum is subdivided into ten sub rings.

3.4 Distribution of The Magnetic Field

The magnetic field distribution is very important for MHD separation. It directly

affects the magnitude and the direction of the E.M. force, which also affects the motion of

the particles.

electrical ' insutation thermal

insutation layer

Figure 3.6 The arrangement of solenoids around the tube containing molten metal

Figure 3.6 shows the arrangement of the solenoids around the molten metal tube.

The magnetic field for one tum expressed in vector form is s h o w in Figure 3.7. The

plane of the coi1 is at the position z = O and the length arrows indicates the magnitude of

the magnetic flux density , which decreases with increasing of the distance fkom the

position z=0. For example, when PI > 0.02 m , 6 is much smaller than at the z = O plane.

The maximum value of @ is found to be closest to the coil, i.e. at z = O and r = r,, where

r, is the radius of the flow tube. It indicates that the strongest electromagnetic force will

be in the same position as the maximum magnetic field

The magnitude of is plotted by its two components, Br and B, in Figures 3.8 and 3.9.

These two graphs indicate that magnetic flux density in z direction is always positive and

aimost does not change with r (O S B, < 0.0 11 9, while the radial magnetic flux density

is negative when z < O and positive when z > O and zero at z = O ((B,( S 0.0035 T). Aiso,

B,increases with increasing radial position and is zero at r = 0.

axial distance (m)

[mj

Figure 3.8 The distribution of axial magnetic density for one-turn and four-layer soleno id

Figure 3.9 The distribution of radial magnetic density for one-tum and four-layer solenoid

The rnagnetic field for four-Iayer and ten-turn solenoid, represented by the Figure 3.10,

appears to be similar to the one of a single coil, except that the magnitude of is much

larger, as well as the zone of zero magnetic field. Figure 3.1 1 and 3.12 quantitatively

show the distributions of B, and Br respectively. B, almost does not change with r and

reaches the maximum (B, = 0.181 T) at in the rniddle of the solenoid. Br reaches the

minimum @, 2 -0.0303 T) at z = 0.1 rn and the maximum (Br a 0.0303 T) at z = 0.2 m.

In practice, the liquid metal flows in a tube along z direction and it is supposed that v,

equals to zero. When the moving liquid metal is coupled with the magnetic field, the

curent density Io in 8 direction is produced, and so does the magnetic force in r and z

directions. For example, at z = 0.1 m, the r component of magnetic body force, $ is

negative from equation (2.4). Because of the viscous force being balanced by the

magnetic force (if neglecting the effects of convection and pressure gradient), it will

cause liquid metai to flow in r direction. This r direction of the flow makes the liquid

metal to carry small non-conductive particles towards the boundary of the tube. This is

the principle of the MHD separaiion method in removing nonmetallic inclusions fiom

liquid metai. In the next chapter, a more detailed discussion of the process will be

provided.

-

-

O O. 1 0.2

axial dis tance (m)

l1;17001

Figure 3.1 1 The radial magnetic field for four-layer and ten-turn solenoid

0.1 0.2 axlal distance (m)

Figure 3.12 The axial magnetic field for four-Iayer and ten-tum solenoid

APPROXIlMATE CEUMCTEMSTICS OF

A MEED SEPARATION METHOD

Chagter 2 detennined that the dynamics of inclusions in the liquid metai can be

predicted if the flow and magnetic fields are known. The velocity distribution of the flow

will be changed by the application of the magnetic field because of the coupling between

the equarions of electmmagnetics and hydrodynamics. The analytical solution of the

coupled equations exist only for few special cases, consequently numencal techniques

are fiequendy applied Numericd methods by their nature give local solutions, and the

exploration of overall trends requires considerable amounts of work Thus, prior to

numericd shuiation of a magneto hydrodynamic pro b lem, appmximate estimation of

the global behavior of the motions of inclusions is highly desirable. The purpose of this

chapter is to present an analysis of a set of ordinary diEerential equations which can be

used for the study of approximate particle dynamics. The numencal results obtained by a

fourth-order Runge-Kutta method show the approximate behavior of the system, and

allows the estimation of separation efficiency.

4.1 Equation of Particle Motion

It was s h o w in Chapter 2 that the Stokes drag force, and electromagnetic body

forces are responsible for the removal of inclusions fkom liquid metals. Buoyancy force is

particularly effective for particle sizes in excess of 300 microns and when their densities

are vastly different î?om the liquid metal. When the liquid metai flows in the horizontal

direction, the eEect of the buoyancy force upon the particle is particularly effective

because it is directed towards the wall.

This thesis studies the dynamics of inclusions with diameten less than 300

microns which move in a horizontal liquid rnetal flow. The density differences between

the liquid and the solid phases are assumed to be negligible. Consequently, the motion of

the inclusions only depends upon the nun of the electromagnetic and Stokes forces. From

Newton's Law, the motion of a particle obeys the equation:

where: fer= the radial component of the electromagnetic force

fs, = the radial component of the Stokes force.

rn = mass of the particle

t =time

Under the assumption of a fully developed laminar flow, equations for Fe, and fs, have

been obtained in Chapter 2. These are:

3 fer =-na oB,B,V,,

fs, = 6 x pa(U, - V,)

where: Ur = the radial velocity of liquid Alurninum flow

V, = the radiai velocity of the particle

In order to shpli@ the relationships, LT, can be assumed to be negligibly small (LI,

=O) . This simplification will be removed by the numencd solution of the Navier-Stokes

equations as presented in Chapter 5.

Thus, equation (4.1) becomes:

acceleration tenn becomes:

Finaliy, equation (4.4) is written as:

n a 3 0 ~ , ~ , U , where: fe(r) =

m

m = the mass of the particle

v, = the volume of the particle

Solution of the nonlinear ordinary differential Equation (4.5) gives the particle

path within the liquid metai.

4.2 Predictions of Particle Traces

Equation (4.5) is solved for inclusion consisting of M2O3 or SQ, with

correspondhg densities of 3500 kg/m3 and 2600 kg/m3. The parameters of the simulation

magnitude of the electrical current, 1.

radius of the tube, r,,.

initial veiocity of the liquid rnetal fiow, V,.

particle radius, a.

Elec~cal current 1 and radius of the tube r, wiii affect the magnitude of magnetic

density B. It is known from Chapter 3 that the magnetic density in the tube increases with

electrical cment and decreases with the tube radius. It is also known from Chapter 2 that

a higher velocity of the liquid metal flow would produce a higher induced curent density

and increase the magnitude of the electmmagnetic body force.

Table 4.1 lis& the parameters used in the simulation. The particle traces are solved

with different radii.

Table 4.1 The ranges of the parameters in simulation

4.2.1 Cases Considered

The calculation of the particle traces were completed by a variation of the

following paramet ers

a) tube radius.

b) coi1 curent.

c) metal velocity.

d) particle size.

It should be noted that special attention was paid to the selection of particle size.

The range of sizes were chosen such that the effectiveness of the separation could be

illustrated. Tables 4.2 (a), (b) and (c) describe the trials completed using a tube radii of

2.0,4.0 and 6.0 cm, respectively.

Legend :

Moderate separation behavior Q1 - 99%) x improving decending down table Excellent separaiion behavior (1 00%)

Table 4.2 (a) Simulation parameters for r, = 2 cm

Table 4.2 @) Simulation Parameters for r, = 4 cm

Table 4.2 (c) Simulation Parameters for r, = 6 cm

4.2.2 Case #I- Good Separution Behavior

Figure 4.1 shows typical particle paths of the inclusions that display good

separation behavior. The simulation cases that resulted in good separation behavior are

marked in the Tables with an x. The particles arrive at the inlet of the tube (left side) at

various radial distances fiom the center and experience an electromagnetic body force

toward the tube wall. As mentioned earlier, this force results fkom the combination of the

radiai magnetic field and the induced curent in the theta direction. As shown in figure

4.1 some particle traces intenect with the tube wdl. At these points the particle can be

considered to be trapped and removed fiom the liquid metal. If the particles does not hit

the wall, they Boat along with the liquid metal flow out of the separator.

Figure 4.1 indicates that the particles experience a larger electromagnetic body

force if they are closer to the tube wall. This occun because the absolute value of Br

increases with increasing radius r. The larger the radial magnetic density (Br), the greater

the electromagnetic body force and the greater the probability of good separation

behavior.

direction of the flow: b

t

solenoid

2 4

dirmnsioniess axial distance

Figure 4.1 The traces of inclusions estimated by Runge-Kutta method(case 1)

4.2.3 Case #2 Pour Separutiun Behovior

Figure 4.2 shows that particles entering the tube at certain radii will not reach the

wall under the given simulation parametea. These simulation results that retumed poor

separation behavior are marked with an x. This occurs because the electrornagnetic body

force is not large enough to push such small particle sizes to the tube wall. in these cases,

the inclusion's radius is always below 100pm. While increasing the size of particle

radius, more and more particles, at different entry radii can be removed under the

simulation conditions.

sole no id I

l O l 1 I I P 1 I O 2 4 6

dimnsionless axial distance

Figure 4.2 The traces of inclusions estimated by Runge-Kutta method(case 2)

4.3 Separation Emciency

There are a number of parameten which affect the removal of inclusions nom

aluminum. These can be described by the aid of the Reynolds (Re), and the Hartman (Ha)

numbers. The Reynolds number characterizes the fiow behavior, while the Hartman

number relates the parameters of the electromagnetic system to particle size. The

Reynolds number is defined by:

where: D = the diameter of the channel

V, = the mean velocity of the liquid metal flow

p = density of the liquid AIuminum

p = rnolecular viscosity of the liquid metal

Since the density and viscosity of liquid metal are considered constant in this

study, the Reynolds number reflects changes in the magnitudes of flow velocity and

channel diameter.

The magnetic Hartman number for an inclusion is defined as:

where: Bo = maximum magnetic flux density;

a = electrical conductivity

D, = diarneter of the inclusion

Since the electrical conductivity and viscosity are both constant, the Hartman

number represents the product of strength of the magnetic field and diameter of the

inclusion.

The effectiveness of the separation process can be characterized by its efficiency.

In this thesis, the separation efficiency is defined as the ratio of the number particles

removed fiom the system to the total number of particles which entered. The separation

efficiency is proportional to the hNo areas as:

where: Ae, = the area through which aii particles are removed at the exit

Ae, = the total cross sectionai area at the entry

Figure 4.3 The separation efficiency vs. Hartman nurnber

The effects of the Hartman number and Reynolds number on the separation

efficiency are shown in Figure 4.3. With increasing Ha and Re, the separation efficiency

increases because:

1) higher magnitudes of magnetic flux density produces a stronger

electromagnetic body force which acts on a larger surface area.

(i.e. Ha number increases)

higher flow rate increases the magnitude of the induced c u e n t density,

hence the electrornagnetic body force.

(i.e. Re nurnber increases)

O 0.5 1 1.5 2 25 3

Maximum Magnatic Fiold 60 (T)

Figure 4.4 The separation efficiency vs. maximum magetic field

Figure 4.4 shows the effact of the maximum magnetic flux density on t

separation efficiency as funchon of different flow rates. At smaller values of Bo, the flow

rate has a very strong effect on the separation efficiency. But, when the magnitude of Bo

is doubled, the separation efficiency approaches unity and the effkct of the Bow rate is

The effect of the magnetic density on the separation efficiency can be divided into

three zones as shown in Figtne 4.5. These c h t e r i s t i c zones c m be described as:

a) Zone I: The particle wodd not be removed in this area.

b) Zone II: The sepamion efficiency inmases h m O to 1 with i n m b g

the size of the particle (a) or maximum magnetic density (BJ.

c) Zone DI: The partîcle would be removed M y h m any enny radii.

Figure 4.5 illustrates the individuai effect of particle size and magnetic density on

the separation efficiency. This figure dso shows the requirements and limitations of a

E.M. particle separator. Usually, a MHD sepamtion system is Iimited by the magnitude of

magnetic field that can be created. If its magnitude is high, it will result in a large

excitation current which leads to increased power consurnption in the coi1 and will

increase the cooling requirement. Thus, a practical current limit determines both the size

of inclusions which can be removed, and the efficiency of the separation.

7 4 iower Iimit for Re= 1350

\ \ \ \

Figura 4-5 The range of inclusions separation in Aluminum

l 2 3 a z E u

W E R I C A L MORELING OF ELECTROUGNE TIC SEPARA TION

5.1 Introduction

Two general approaches can be used to analyze the effect of electromagnetic

forces on the inciusions in a Liquid metal. The fint is experimental whereby a scaled

down version of the process is used to obtain useful design data. The second is through a

numerical rnodel.

Computational techniques are powerfil for the initiai estimation of system

behavior. An added advantage is that body forces can be incorporated easily into the

computationai h m e work. However, the drawback consists in the modeling method

itself. Because the flow process is complex and cornputer resources are limited,

simplifications in the analysis m u t be made. These, if are not chosen properly, can lead

to significant errors. Thus, numerical models mut be validated and calibnted against

both analytical and experimental results.

In the previous chapters, the magnetic field in the tube was described analytically

and estimations were obtained for the separation efficiency in liquid Aluminum. The

purpose of this chapter is the formulation of a more accurate mathematical mode1 to

describe the separation process. The goveming equations of the flow are obtained through

the enforcement of the conservation of m a s and momentun in a cylindrical, axi-

symmetric coordinate system. As these equations cannot be solved analytically, a

numencal altemate is presented. The numerical approximation of the governing equations

is formulated by Finite Element Method.

5.2 Flow Mode1

As discussed in Chapter 4, when liquid metal flows under the effect of a magnetic

field, radial and axial electromagnetic (E.M.) body forces are produced within the liquid

metal. The flow can be treated as being in steady state and is two dimensionai. Since the

Reynolds number of the liquid metal flow is kept lower than 2000, it is laminar. In order

to study the movement of the inclusions in the liquid metal, the flow field and E.M forces

acting on the fluid must be described mathematically. The two dimensional mass

conservation, and Navier-Stokes equations which include an E.M. force source term,

descnbe the flow subject to specified boundary conditions.

5.2.1. Essentiai assumptions

In solving the momentum and continuity equations, two essential açsumptions are

made:

(a) A turbulent flow causes rnixing which is detrimental to separation. Consequently, the

fiow is considered to be laminar with Re ~2000.

(b) The flow is treated as both incompressible and Newtonian.

5.2.2 FIo w equations

The two dimensional Navier-Stokes equations in cylindncal coordinates are given

by[22] :

d P 2 rcomponent: p b ~ , = - - + p V Vr+fer

3r

where : fer and fez are the electromagnetic body forces in the radial and axial directions,

which were derived in Chapter 3 and are re-stated as :

In order to obtain general results, the governing equations can be cast into their non-

dimensional steady state form by the following normalization:

The normalization o f the Navier-Stokes equations produces a dimensionless nurnber,

called the Reynolds Number Re:

Re = (2 p ro Vo)/p Reynolds Number

u = @P Dynamic Viscosity

The dimensionless goveming equations take the foilowing forms (for simplicicity, the *

symbo l which represents dimensioniess variables has been removed):

r component :

z component :

5.2.3 Boundary conditions

solenoid

t wall A B

Figure 5.1 The simplüied schematic of the separator

The boundary conditions are s h o w in Figure 5.1. It is noted that due to symmetry, only

one half of the channel is shidied. Since it has been assumed that the flow is fully

developed pnor to entering the magnetic field, the inlet axial velocity can be described by

a parabolic velocity prome. The radial velocity is zero. The radiai and axial components

of the velocities at the wail are also zero due to the non-slip condition. The radial velocity

along the tube axis is zero as well due to symmetry. The outlet velocity boundary is a

Neumann type, with zero axial velocity gradient.

Table 5.1 Boundary conditions for the tube

5.3 Finite Eiement Approach

The finite element method is a numerical analysis technique for obtaining

approximate solutions to a wide variety of engineering problems. There are other

approximate numencal analysis methods, such as finite difference etc., which have

evolved over the years. The h i t e difference scheme is a cornrnonly used method and has

many applications in the solution of engineering problems. A finite difference mode1 of a

problem gives a pointWise approximation. The finite element method, however, provides

a piecewise approximation to the governing equations. The basic premise of the finite

element method is that a solution region of space c m be represented analytically, or be

approximated by an assemblage of discrete elements [23]. Since these elements can be

put together in a variety of ways, they can be used to represent exceedingly complex

shapes.

In a continuum pmblem, a field variable, such as velocity or pressure, assumes an

infinite number of values because it is a function of al1 points within the body. In effect,

the value of a variable at a given point is connected to al1 points within the geometry. in

the finite element method, approximations are made by subdividing a region into a set of

finite regions over which the solution is approximated by an interpolating function. The

approxirnating, or interpolation functions, are defined in terms of the values of the field

variables at specified points, called nodes. These usually lie on element boundaries where

adjacent elements are connected. For interior points within the element, the nodal values

and the interpolation functions, completely define the behavior of the field.

Consequently, the nodal values of the field variable become the unknowns, which cari be

subsequently solved by a number of algebraic methods.

element 1

boundary nodes

interior nodes

element 2

Figure 5.2 Typical FEM noddelement configuration

Typical configuration of two elements within a larger structure are shown in

Figure 5.2. Element I is defhed by nodes 1,2, 3 , 4 and 8, whîle element 2 is denoted by

nodes 3, 4, 5, 6, and 7. Note that nodes 3 and 4 are common to both elements, thus

linking the variables in both.

The solution of a continuum problem by the finite element method follows the

steps listed below:

1 . Discretization of the continuum.

The h t step is the division of the continuum or solution region into elements. A

variety of element shapes such as triangular, quadrilateral, etc. may be used. Depending

on the complexity of the geometry, different element shapes may be employed in the

same solution region. For cylindrical flow field in this study, a quadrilateral shape

element was used.

2. Selection of the interpolationfunctions.

Nodes are assigned to each element and then the interpolation function is chosen

to represent the variation of the field over the element. Biquadratic interpolation

£Ùnctions(see Appendix i ) are usually selected for the velocity due to their ease of

integration and differentiation. The numencal solution of the Navier-Stokes equations are

made difficult by the absence of an explicit relationship for pressure. Thus, special

procedures are used for its approximation which will be discussed in Section 5.3.2 below.

3. Element assemblage.

The overall system of equations is determined by a network of elements. Al1 the

element properties are assembled by combining the matrix equations which express the

behavior of the entire system.

4. Imposition of the boundary conditions.

The "real worlà" is coupled into the problem by incorporating the relationships

existing at the geometrical boundaries. Thus, known physical behavior of al1 variables at

the boundary determine the interior solution.

5. Solution of the system equations.

The assembly process results in a set of simultaneous algebraic equations which

must be solved to obtain values for the nodal variables. When the algebraic problern is

linear, and the number of unlaiowns are lirnited to a few hundred, standard direct

reduction methods are adequate for solution. Should the relationships become non-linear,

or the number of nodes exceed a reasonable value, an iterative solution m u t be

employed.

In the following sub-sections, some issues in application of the finite element

method are presented..

Figure 5.3 The mesh of the solution region

5.3.1 Element and mesh

For incompressible flow, the element should satisfj the requirernent for mass

conservation. in practice it will be satisfied if the element has a mid-side velocity nodal

point. It is the nodal point which occurs on the side of the element away from the

corner[24]. In 1-D problems, this means that at least 3 nodal points are required. The

extension to 2-D is to use biquadratic shape functions, which are products of quadratic

shape fimctions in the r and z direction[25].

For the present pmblem, the solution region is divided into 150 elements for the

mesh as shown in Figure 5.3. A 9-node quacirilateral element is considered in this case, as

shown in figure 5.4. Eight corner nodes are placed along the element boundaries, and one

center node is placed in the middle.

Figure 5.4 Nodal points for the local element

5.3.2 Expansions of V, V: and P

The unknowns of radial velocity V, axial velocity V, and pressure P Î n equations

(5.6) to (5.8) are expanded in terms of their respective interpolation functions. Although it

is not strictly necessary, the same interpolation functions are used to represent both V,

and V,. This reflects the fact that V, and V, will Iikely need to be resolved to the sarne

degree of accuracy[26]. The accuracy required for the pressure is different fiom that for

the velocities. The pressure interpolation functions are different from the velocity shape

func tions[25].

where : N = total number of nodes for V, or V,

Np = total number of nodes for pressure WNp)

V, = axial velocity for each nodal point in the

computational domain

Vn = radial velocity for each nodal point in the

computational domain

+&z) = velocity shape function

yi(r,z) = pressure shape function

Pi = pressure for each nodal point in the computational

do main

For incompressible Bows, elements should satis@ the continuity equation, that is mass

must be conserved element-wise. The details of denvations of the necessary relationships

for the finite element method used in this thesis are described in references [27] to [31].

These are, for the sake of convenience, described in Appendix 1. The main features of the

method are :

(a) the use of a weighted residual method to derive the matrix equations for

individual elements,

@) the inclusion of the pressure term in the Navier-Stokes equations to eliminate

the need for explicit relationships for pressure P,

(c) the use of a mixed scheme for tirne marching solution of the problem,

(d) the controi of the stability and accuracy.

5.4 Results and Discussion

The purpose of this section is to describe the results obtained by the theoretical

method presented in section 5.1 through 5.4. The cornputer code (see Appendix II)

developed for the analysis is first verified by cornparison of its results with an analytical

solution. Then, the flow field is described in the presence of electrornagnetic body forces.

The results for separation are presented in tems of non-dimensional parameters and

separation efficiency.

5.4.1 A validation case for the computer code

The anaiyticai solution given by equations (4.2 and 4.3) and the results of the

h i t e element method analysis are shown in Figure 5.5. The error between the two

methods is given by :

I E uW I = ( ~ u l . l * ~ * a l y * . l - ~ W " l . ~ U o . r k i l j

The absolute errors in the radial and axial directions, as a function of radius, are shown in

Figure 5.6. It is seen that the maximum error for the axial velocity is 0.0023 based on

Vm=0.02(m/s) and is 2.2e-8 in the radial direction. The results show that the numericai

solution is very accurate and is expected to produce good results for the addition of

electromagnetic body forces.

Figrire 5.5 Cdculatcd axial velacity distribution of the flow without E.M. force

Figure 5.6 The absolute errors of caiculated velocity of the flow without E.M. force

5.4.2 Flow, pressure, E M. Currenr and E. M. force fields

The following field variable distributions along the tube are for the following

operating conditions shown below:

I=2000A ro=5cm L=30cm a=170pm Vo=O.O1mls

A typical liquid metal velocity distribution with the presence of an E.M. field is shown in

Figure 5.7. Tt is apparent that the magnetic body forces significantly change the nature of

the flow field. There exist two peaks and one valley. The arrows in Figure 5.7 show the

direction of the velocity and the lengtb of the arrow shows the magnitude of the velocity.

The solid iines are the flow Stream lines starting at different radial distances at the inlet on

the 1eR hand side. By comparing Figure 5.7 and the magnetic field plotted in Figure 3.3,

dimensionless axial distance

Figure 5.7 Vector Plot of Velocitv Field With E.M. Force

dimensionless axial distance

Figure 5.8 Contour Plot of Current Density Along The Tube (half)

it is found that the radial flow velocity changes at the exact location where the magnetic

flux density reaches a maximum.

The shape of the flow distribution cm be undentood fiom an analysis of the force

patterns w i t b the flow. Liquid Alurninum can be treated as an incompressible fluid. If

there is no electromagnetic force acting on the liquid metal, and the flow is fully

developed, the axial velocity distribution across the tube is known to have a parabolic

profile. The radial velocity is identicdly zero. Depending on the direction of the radial

electrornagnetic force, the liquid metal is deflected, either towards the wall or the center

thereby assuming a radial velocity component.

Figure 5.8 shows a typical distribution of the elechical current density in the filter.

As descnbed in Chapter 2, there are three regions, conespouding to positive, negative and

zero values of J. The maximum values of current densities are close to the wall of the

tube to increase the magnitudes of the body forces which trap inclusions against the tube

wall.

Figure 5.9 shows a typical distribution of the electromagnetic force vectors in the

tube. As discussed in Chapter 4, the electromagnetic body force is in the opposite

direction to the electromagnetic force. Since the E.M force changes the flow pattern, the

flow pattern in Figure 5.7 is consistent with the direction and magnitude of the E.M force

in Figure 5.9. It is known that the radial E.M body force direction is responsible for

inclusion removal. However, there is a larger positive radial EM force region near the

outlet of the solenoid shown in Figure 5.9, which rnoves the inclusions away nom the

away nom the wall of the tube and renders the separation of inclusions difficult. This can

be seen nom the particle traces presented in the next section. For the present proposed

filtration method, this negative effect is an inherent feature.

5.5 Padcle Traces

Figures 5.10 and 5.1 1 shows typical traces of particles starting from different

initial radial positions at the channel entrance. The chosen examples correspond to a low

and hi& separation efficiency respectively. For the cornparison, the operating conditions

are the same as those chosen in Chapter 4. These are :

(a) For Figure 5.10 :

I=3000A r0=5cm L=30cm a = 100 pm Vo=O.O1 m/s

(b) for Figure 5.11 :

I=3000A ro=5cm L=30cm a=130pm Vo=O.O1 mls

The electromagnetic body force in the radial direction leads to the separation of

the inclusions fkom the flow. The electromagnetic body force is stronger near the tube

wdl than it is near the center üne. It is seen from Figure 5.1 1 that the closer a particle

enters to the flow center he, the more difficult it is to be removed fkom the flow Stream.

This is due to low values of the magnetic force and high values of Stokes drag force

along the center of the tube.

Figure 5.10 The typical particle traces at different entrances having

low separation efficiency

dimonsionleu axia l distance

Figure 5.1 1 The typical particle traces at different entrances

having high separation efnciency

By comparing the numerical results with the analytical solution of the particle

traces, it is found that they have similar patterns are found for the particle path in the

liquid metal flow. For the same operating conditions, the numencal solution has a lower

separation efficiency than that of the analytical solution ( see Table 5.2). This is attributed

to the assumptiom made in the analytical caiculation, which neglects the interaction

between the magnetic field and the fluid flow.

L

Operation Conditions

Table 5.2 The cornparison of E.M. separation efficiency

( anaiytical)

for the numerical and analytical results

Separation Efficiency %

( numerical )

Separation Efficiency %

CONCLUSIONS AND REC0MMEM)ATIONS

6.1 Conclusions

The purpose of this thesis was the investigation of electromagnetic fikration of

Aluminum . The results of the study are summarized as follow:

1) ElectromagneticaUy generated body forces can be applied directly upon inclusions

to remove them from a stream of molten Alutninum.

2) The efficiency of separation is a fûnction of not only the magnitudes of

electromagnetic and flow quantities, but also of the geometry of the filter design.

3) Filter efficiency depends strongly upon the strength of the magnetic field. in

order to achieve high efficiencies magnetic Belds in excess of 2.0 T are required.

4) As a consequence of the above conclusion, the magnitude of excitation curent is

in excess of 6000 Amperes which would Iead to serious thermal design problerns for the

proposed device.

5 ) The region of active zone for filtering in the present device is too narrow and

results in inadequate removal of inclusions fkom regions immediately adjacent to the

centerline.

6) Approxirnate analytical solutions to the problem provide reasonable estimates for

device behavior and ean be usefbl for design purposes alone. The approximations allow

quick assessrnent of the overall behavior of the device and establishes the fundamental

relationships between the electromagnetic parameters, particle sue and weight.

7) Detailed numerical solutions are required to explore the full interdependency of

the fiow and the magnetic field. Detailed analysis is necessary since hdamental

changes occur in the flow patterns which significantly reduce the efficiency of separation.

6.2 Recommendations for Future Work

The present work shows that future work is required along the following lines:

1) Design and conduct experiments to measure and confirm llow patterns and

particle movements.

2) A different flow configuration should be investigated to extend the zone of active

filtering and achieve a higher filter efficiency.

3) The magnetic structure must be modified to elimlnate the "dead zones" inherent

along the axis of the present design.

4) Higher values of magnetic flux densities must be generated either by the use of

superconducting coils, flux concentrators or both.

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Vol. 18, No.4, pp. 426-429, Oct-Dec, 1982

2. Kirko, " Separation Effects in A let Flowing Through A Solenoid ",

Magnetohydrodynamics, No.3, pp275-279, 1976

3. Galfgat, " Effect of Crossed Electric and Magnetic Fields on The Interaction of Two

Solid Spheres in Conducting Liquid ", Magnetohydrodynamics, No.2, pp 142- 144,

Apnl-lune, 1975

4. 1. Pactinen, N.Saluja, LSzekely and l.Kirtley, " Experimental and Cornputationai

Investigation of Rotary Electromagnetic Stirring in Woods Metal System ", ISIJ

Jhtemational, Vo1.34, N0.9, pp. 707-714, 1994

5. E.I. Shilova, " Removal of Nonconducting Irnpurities From Liquid Metals in The Self

Magnetic Field of An Electric Current ", Magnetohydrodynamics, No.2, pp. 142- 144,

April-lune, 1975

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Inclusion Particles From Liquid Steel, ", Journal of Iron and Steel Society of lapan,

2995

7. P. M q and A. Aiemany, " Theoretical and Experimental Aspects of

Electromagnetic Separation ", Magnetohydrodynamics, pp. 245-259, 1965

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The Help of An Alternathg Electromagnetic Field " Magnetohydrodynamics, No.3,

pp. 125-130, 1985

9. K. Grjotheim and B.I. Welch, " Aluminum Smelter Technology " University of

Romania Press, pp. 246-250, 1987

10. H. K.Moffatt, " Some Problems in The Magnetohydrodynamics of Liquid Metals ",

ZAMM 58, T65-T71,1978

11. A.F. Kolemichenko, B.A. Yushchenko, A.D. Podoltsev and lu. 1. Micengendler, "

Electromagnetic Moulds for Steels ", Magnetohydrodynamics in Process Metallurgy,

A hiblication of The Minerals, Metais & Materials Society, pp. 223-229, 1991

12. Sagardia R.. Sergio, Electromagnetic Levitation Melting of Large Conductive Loads,

Doctrate Thesis, Dept. of Electncal Engineering, University of Toronto, 197 1

13. Stanley V.Marshal1, Gabriel G. Skitek, " Electmmagnetic Concepa and

Appficationsbb, Kluwer Academic Publishea, pp. 78- 123, 1987

14. Shuzo Oshima and Ryuichiro Yamane, " Shape-Control of Liquid Metal Free

Surfaces by Means of a Static Magnetic Field ", Magnetohydrodynamics in Process

Metallurgy, A Publication of The Minerals, Metals & Materials Society, pp. 25 1-259,

1991

15. Shoji, S. Keith, " Application of Pinch Force to The Separation of Inclusion Particles

From Liquid Steel ", ISU International, Vol. 34, No. 9, pp. 772-73 1, 1995

16. S. V. Marshall, G.G. Skitek, " Eiectromagnetic Concepts and Applications ",Kluwer

Acadenic Publishers, pp.20 1-203, 1990

17. N. 1. Bolonov, V.M. Dobrychenko, 1. L. Povkh, " Effect of Crossed Electric and

Magnetic Fields on Motion of a Two-Phase Flow ", Magnetohydrodynamics, No.2,

pp. 146-148, 1972

18. Robert A. Granger, " Fluid Mechanies ", Cambridge University Press, pp145-340,

1985

19. A. V. Sandulyak and V.L. Dakhnenko, " Unique Features in Magnetic Deposition by

Filtration ", Magnetohydrodynamics, Vol. 25, No.2, pp. 123-127, 1975

20. Ernst Weber, " Electromagnetic nieory ", Dover hiblication, hc., pp.50-123, 1985

2 1. William H. Press Brian P. Flannery, " Numefieal Recipes ", Cambridge Univenity

Press, pp. 547-554, 1978

22. J.F. Douglas, J.M. Gasiorek, I.A. Swaffied, " Fhid Mechanies ", McGraw-Hill Book

Company, pp. 145-267, 1995

23. Roger Peyret, Thomas D. Taylor, " Computational Methods for Fluid Flow ", CRC

Press, Inc., pp.57-87, 1983

24. Tasos C. Papanastasious, " Applied Fluid Mechanics ", McGraw-Hill Book

Company, pp. 97-167,1994

25. C.Ross Ethier, " Coune Notes for Computational Fluid Mechanics and Heat Transter

", University of Toronto, 1994

26. Baker, A.J., " Finite Element Compu fa tional Fluid Mcchanics ", pp234-26 7,

Springer-Verlag Berlin Heidelberg New York, 1983

27. Ames, " Numerical Methods for Partial Dijjierentiol Equations ", Oxford University

Press, pp78- 145, 1977

28. Anderson, Tannehill, and P;Pletcher, " Computational Fluid Mechanics and Keat

Transfer ", Press Syndicate of The University of Cambridge, pp. 123-26, 1984

29. Cuvelier, Segal and Van Steenhoven, " Finite Element Methoch and Navier-Stokes

Equations ", Oxford University Press, pp 245-367, 1986

30. Fletcher, C M , " Cornpufational Techniques for Fluid Dynamics ", John Wiley &

Sons, Inc., pp. 89-135, 1990

31. Strang & Fix, " An Anaiysis of The Finite Element Method ", Arizona State

University Press,.pp.67-89, 1973

3 2. R. Moreau? " Magnetohydrodynamicî. Fluid Mechania and Irs Applications",

Kluwer Academic Publishers, pp.23 0-245, 1990

33. Chuen-Yen Chow, " Drag of a Sphere of Arbitrary Conductivity in Curent-Canying

Fluid ", The Physics of Fluids, Vol. 12, No. 1 1, pp. 23 17-2322, 1970

Appendh 1 Steps in Derivation of The Finite Element Approach

[2834]

estep 1>

This step is to expand the uhowns q v and p in terms of their respective

shape(basis) functions. The same shape fiinctions are used to represent both u and v. This

reflects the fact that u and v wili be resolved to the same degree of accuracy. The pressure

shape functions will be different nom the velocity shape functions.

where : N = nodes for u and v

Np = nodes for p (N+Np)

S t e p D

This step is to multiply the radial and axial momentun and continuity equations

by the appropriate weight functions, which for the Galerkin method are the same as the

shape bctions. The r and z momentum equations are multiplied by the 4, in the

Galerkin procedure. By default the continuity equation is the equation for the pressure,

thus is multiplied by the y,.

Focusing on the z-momentun equation, the relationship become:

a 2 u +-+--J a 2 u m u --(v*br*bz-"*br2) CJ r, a z 2 r a r pv ,

Weighted residual method:

whereR is the computational domain. The third and forth terms in this integral are then

integrated by parts and transfo& by -g Gauss's theorem.[28] Specifically,

The z component eq. can be written as :

bz - u. br' ) 4

A similar procedure for the r-momentum equation gives:

The continuity equation is :

This step is to insert the expansions for u,v, and p into these expressions.

Focusing on the z-momentum equation term by term[29]:

Thus, we can set: *

<step4>-

there does not exist pressure term in continuity equation, a zero block on the diagonal of

the constant matrix of the h o w n vector is found. This means that the system will have

to be pivoted for a matrix with a large bandwidth[31]. In fact, the matrix is essentially

full. This is due to the pressure terms appearing in momentum equations.

To overcome ihis problem, the penalty fûnction is introduced to add a pressure

terni in the continuity equation. In practice the penalty fùnction is used with only certain

types of finite elements, the so-called Crouzeix-Raviat type elements. These allow the full

efficiency of the penalty function approach to be used.

An element which works very well in conjunction with the penalty function is the so-

called modified enriched Q~+-Q, Crouzek-Raviait element[30].

where

-t means enriched,

Q --quacidateral,

2--2nd order velocity shape function and

1 - 1 st order pressure shape fùnction.

For real finite element simulations, the elements are rarely, if ever, simple rectangles (in

2D) or cubes (in 3D). uistead, they have some more complex shape. We can take account

of this difficulty by definhg a map between the complex (physical) shape and a simpler

(re ference) shape. A 9-nodded quacirilateral element will be considered in this case[fO].

< step 5> 5

There are 3 pressure shape functiions, which are

Y dx,y)=l

w~(x,Y)==x~

w3 (X,Y)T-Y~

where (%,y9) is the location of the centroidal node in the element. The unusual feature of

the C-R element is that al1 of the pressure shape functions have the centroid (x%y9) on

their local elernent. This leads to some major simplifications.

The global velocity vector and pressure vector are reordered as follows:

where {Y} is a vector of u,v velocities at non-centroida1 nodes;

{p, } = a vector of centroid pressure;

{v,} = a vector of y v velbcities at centroidal nodes;

{p,) = a vector of centroidal pressure derivatives.

Mer rearranging the no-penalized continuity equation and momennim equations and

spiitting the matrix to 1 1,12,2 122, it can be shown that[3 11:

{YS } = [RI ]Pl where

nie extremely important f e u e of [R,] is that it can be formed elementaily. This means

that the centroidal velocity of any element can be cornputeci h m non-centroid velocities

of that element. 90

< steP 6> c in order to eliminate pressure terni fiom mornentum equation and make it appear

in the wntinuity equation , a penalty function is introduced by modifying the continuity

equation to:

V .v+&p=O

w h m E is a small parameter. The m r which is di ue to E can be sh own for the velocity

field to be of the order O(&''). For ~a lod to 1oa9, this is acceptably small. Mer

discretization it gives[28]:

In the penalty fiinction approach with C-R elements, only the first M equations are

penafized, the equations resuiting fiom mdtiplying continuity by y, at the M element

cmtroids. This corresponds to sening [Il2!] and [Dn] 4. It is simple to show that p,J

=û[28]. And that

@, ,), = Area of element rn

Thus the penalized continuity equation becornes:

Filally, the pressure term in the momenhim equation can be eliminated as:

where [M] and [SI [30]can be written in a generai matrix [A] and [A] can be divided into

1 1,12,2 1 and 22 partitions:

and S can be computed elementaily and then assembled by direct stifniess method. Pl 1-1 In this way the number of unknowns per element is reduce by 4. Since this elimination

occurs before global assernbly, the resulting in global matrices which are mialler and

therefore easier for LU decornposition in the solution phase.

Equation (4.21) involves only the noncentroidal velocities { Y ) . Once these have been

determineci, the centroidal pressures {p,} can be detemiined bom equation (4.20).

Because of the nature of the C-R element, the matrix [L, , IT [D , , ]-' [L, , ] CM be formeci

elemently and thea inserted into th global matrix b y direct assemb ly[3 11.

7' 3ïa=mdw

The good choice of time-stepping marching scheme is the mixed scheme, which

uses 3rd-Adam Bashword(3rd-AB) explicit treamient for convective terms and Euler

Backward PB) implicit treatment for other tenns[32]. This is an attractive approach,

since the expücit 3rd-AB explicit is a good method for handiing the nonlinear convective

tenns while the difiùsive terms are effectively computed with the highiy stable implicit

EB scheme.

The initiai velocity profile for the flow velocity field is computed from analytical solution

of the fully developed laminar flow. 3rd order Adames-Bashforth method requires u

values at the n-1 and n-2 levels which is not possible at the beginning of the time

marching process. Consequently, the solution should be started with Euler-Forward

treatment, which then tramfers over to 2nd-AB, finally shifts to 3rd-AB method. This

approach may be unstable only for the first two time step and m o r in the solution does

not grow or propagate.

The boundary conditions c m be directly inserted in the right hand vector and adjusted at

the left ma& in equation 5.2 1.

The stability of this scheme is detennined by the explicit steps and by analogy with the

result for f i t e difference we have

0.723 a' h,, At I ---

2 ReU,

where : &, = the smallest element side in the mesh,

U,, = the maximum velocity found in the computational domain.

APPENI.DE II Programs

programe solenoidtubeflow implicit double precision(a-h,o-z) parameter(epsi-1 .d-7,dt=û.02) parameter(nnc=52 1 ,npc=67 1 ,ng=l 042,nt=l34î,ne= 150,nbc=205) dimension uv(nt)

call readrnesh(uv) cd1 timarch(dt,epsi,uv)

stop end

c**********************~~emarching*l******+*******.*. subrouthe tirnarch(dt,epsi,uv) implicit double precision(a-h,L,o-z) parameter(nnc=52 1 ,npc=67 1 ,ng=1042,nH 342,ne=150,nbc=205) cornmonimeshlx(npc),y(npc),index(9,ne) common/mag/a,bx(npc),br(npc),sigma,bO,aa,eta dimension uv(nt),v(8O758),agm(ng7ng),rû(2, 16,ne),r1(2,16,ne),

ibc(nbc),b(ng),ur(l l,6 l),uz(11,6l),rr(l1,6 l),zz(l l,6 L), & dl l(ne),L 1 l(l,16),p(ne)7aj(ll,61),fk(l 1,6 l),fi(l 1,6 1)

open(1 ,filelvelf pm') open(3, file='pres. pm')

open(l1 ,file='currf.pm')

cal1 arnatrix(dt,epsiyagmyv,ibc,rO,r 1 ,b,d 1 1 ,L 1 1) call b-iubksb(v, b) call centrd(uv,b,rO,dl l,L 1 1 ,epsi,p) cd1 bmatrix(dt,agxn,v,ibc,rO,r 1 ,UV)

enddo enddo

do i=l , l l do j=l,6l if(rnod(i,2).eq.O.and.mod0',2).eq.O) then ur(iJ)=ûS*(ur(ij+l)+ur(i ~ 4 ) ) uz(iJ)=û.5*(uz(iJ+ l)+uz(i j- 1)) aj(i j)=O.S*(aj(ij+l)+aj(i j- 1)) fifi j)=O.S*(s(i j+l)+s(i j- 1)) f2(iJ)=O.5*(fz(iJ+l)+fz(i fi)) Mid)-MiJ- 1) endif write(1,l O) z(iJ),n(iJ),uz(iJ),ur(iJ) wite(11 1 O) ~(iJ),rr(iJ),aj(iJ),e(iJ),fi(i j )

enddo enddo

r e m end

co~odma@~bx(npc),b~npc),sigma, bO,aa,eta dimension uv(nt),dl ta(5) open(2,file='rnesh.dat1) open(5, file='rindex.dat') open(8, file='datat) open(9,fi1e='rnag.dat1)

read(5,*) n-elernent do i=l ,-lement

read(5, *) k,(index(j ,k) j=1,9) enddo

do i=1,5 dlta(i)=(y(2*i+ 1)-y(2*i- 1 ))/6.dO

enddo

do i=nnc,npc- 1,5 do j=1,5

y(i+j)=y(i+j)+dlta(j) enddo

enddo

do i=l,npc uv(2*i- 1)=2.d0*(1 .-y(i)**2) uv(2 * i)=O .O

enddo

read(8,*) sigrna,rrû,den,vO,bO,aa,eta read(9,*) n-total do i=L ,n-total read(9,*) num, bx(num), b r(num)

enddo a=sigma*rrû/den/vO

return end

c ******f a and b mate*******************

subroutine amaûix(dt,epsi,agm,v,ibc,rO,r l ,b,dl 1,111) implicit double precision(a-h,L,o-z)

integer i,m,n,offset,point parameter(nnc=521 ,npc=67 1,118- 1 O42,nPL 3 4 2 50,nbc=205) parameter(niocal= l 6 ~ ~ 3 . 6 d l ,alfa=7.4d 1 ) dimension ag(ng,ng),L 1 1(1,16),rO(2,16,ne),Le(3,l a), & amhat(l6,16),agm(ng,ng),shat(l6,16),uv(nt), & Lme(3,18),Lm 1 1 ( l,16),r 1(2,16,ne),d 1 1 (ne) dimension se(l8,18),arne(18,18),ibc(nbc),v(80758),b common/mes h/x(npc),y(npc),index(9,ne) common /matpuam/ m,n,offset

c calculate gauss reference call gaussref

c initial aglobal do i= 1 ,ng do j=l,ng

ag(i j)=û.dO agm(i j)=O.dO

enddo enddo

c assemble global ma& do k=l ,ne call gaussion(k)

c Calculate elemental values of L 1 1 (continuity) and Lml l (momentum)

c calculate matrix rû do j=L,l6 rû(1 j,k)=-Le(2 j)/Le(2,17) rO(2 j,k)-Le(3 j)/Le(3,18) ri (1 j,k)=Lme(2 j)lLme(2,17) r 1 (2j,k)=Lme(3 j)/Lme(3,18)

enddo

c calculate -lement and m-elexnent do i=1,9 do j=l,9

cal1 integ(k,i j,uv,b l ,b2,b3,b4,2) cal1 integ(k,i j,uv,b3,b4,b5,b6,6)

call integ(k,i j,uv,bS,b6,b7,b8,7)

call integ(k,i j,uv,b l,b2,b3,b4,4) ame(i*2-lj*2-1)=b 1 *alfalre/dt ame(i*2j*2)=b 1 *alfdre/dt ame(i82-1 j*2)=û.d0 arne(i*2 j*2-l)=O.dO

enddo enddo

c calculate s h a t and m-hat do i=1,16 do j=1,l6

shat(i j)=û.dO amhat(i j)=û.dO do m=1,2 do 1=1,2

enddo shat(i j)=shat(i ,j)+se(i j) amhat(i j)=amhat(i j)+arne(ij)

enddo enddo

c Calculate the whole elemental matrix do i=1,16 doj=1,16

shat(i ,j)=shat(i j)+Ll l(l,i)*Ll l(1 j)/(epsi*dl 1 (k)) enddo

enddo

do i=l ,nbc read(6, *)nJnode,in, bc

ibc(i)=2*n-bnodein-2 b(i bc(i))=û.dO

if(in.eq. 1) then nb=2 * o n o d e - 1

else nb=2*nJnode endif

do j=l ,ng ag(nb j)=û.dO

enddo ag(nb,nb)=l .dO

enddo do i=l, i l

b(2*i- l)=2.dO2(1 .do-y(i)**2) enddo

j l=i do i=l,n

do j=j 1,i v@oint (i j))=ag(i j) v@oint(i,i))=ag(j,i)

enddo if(i.ge.40) then j l=j l+ l

endif enddo

r e m end

c ********Calculate B matrix*******44*******************4***

subrouthe bmatrix(dt,agm,v,ibc,rû,r 1 ,UV) irnplicit double precision(a-40-2) para1neter(nnc=52 l~pn471~~1042~~1342,n~lSOpbc=205) dimension ane(18,18),anhat(16,16)J0(2,16,ne),rl(2,16,ne) dimension b(702),agm(ng,ng),ibc(nbc),v(80758),

c calculate N matrix do i=l,ng do j=l ,ng

an(i j ) 4 . enddo

enddo

do k=l,ne call gaussion@)

do m=1,9 do n=1,9 b l=O.dO b2=0.d0 b3=0.d0 b44.dO

do i=1,3 do j=1,3

dfl =o.do d£2=0.d0 df34.dO df4--0.d0

do 1=1,9 dfl=dfi+uv(index(l,k)*2- l)*fai(i j ,n)* f a i (

& *(dfdk(ij,i)*dkdx(i j)+dfde(i j,l)*dedx(i j)) dfî-dn+uv(index(l,k)*2- 1)" fai(i j,n)* fai(i j,m)

& *(dfdk(iJ,l)*dkdy(i j)+dfde(i j ,l)*dedy(i j)) df3-dE3+uv(index(l,k) *2)* fai(i j ,n) * fai(iJ ,m)

& *(dfdk(ij,l)*dkdx(i j)+dfde(i&l)*dedx(i j)) df4=df4+uv((index(lTk)*2)*fai(i&n)*fai(ij,m)

& *(dfdk(i j,l)*dkdy (i j)+dfde(i j,l)*dedy (i j)) enddo

bl=bl+dfl*ajcob(i j)*w(i)*w(j)*yl tol(i j ) b2=bZ+df2*ajcob(i j)*w(i)*w(jj*y 1 tol(i j ) b3=b3+rntajcob(i j)*w(i)*w(j)*yl tol(ij) b4=b4+dfPajcob(ij)*w(i)*w(j)*yl tol(i j )

enddo enddo

ane(2*m- 1,2*n- l)=b 1 ane(2*m- 1,2*n)=b2 ane(2*m92*n-1)=b3 ane(2*m72*n)=b4

enddo enddo

c calculate Nhat do i=1,16

do j=I,l6 anhat(i j)=û.dO

do 1=1,2 do m=1,2 anhat(i j)=anhat(i j)+rû(l,i,k)*ane(l+ 16,m+ 16)*rû(m j,k) enddo

anhat(i j)=anhat(i j)+ane(i,l+ l6)*rû(l j,k) & +rû(l,i,k)*ane(l+l6 j)

enddo anhat(i j)=anhat(i j)+ane(i j)

enddo enddo

c assembly gloable rnatrix do i=1,16 do j=l,l6

i f(mod(i,2).eq.O) then ngi=index((i+l)L?,k)*2

else ngî=index((i+ l)/2,k)*2- 1

endif if(mod(i72).eq.0) then ngj=index((j+1)/2&)*2 else ngj=index((j+ 1)/2&)*2- 1 endif

an(n@,ngi)=an(npi,ngi)+anhat(i j) enddo enddo

do i=l ,ng do j= 1 ,ng

if (jt.eq. 1) then b(i)=b(i)+(agm(iJ)-an(i j))*uv(i)

elseif (jt.eq.2) then b(i)=b(i)+(agm(i j)- l.SdO*an(i j))*uv(j)

& +OS*anl(iJ)*uvlü) else

b(i)=b(i)+(agm(i j)-23 .dO*an(i j)/l2.d0)*uv(j) & +4.dOtan2(i j)*uvZ(j)/3 .do-5.dO*an 1 (i j)*uvl @Il 2.dO endif

enddo enddo

do i=l ,nbc b(ibc(i))=O.dO enddo

do i=l, I1 b(2*i- 1)=2.d0*(1 -y(i)**2) enddo

c ************ solve velocity by backward and fonvard ************* cal1 b-lubksb(v,b)

do i=l,nt if0t.q. 1) then

UV 1 (i)=uv(i) elseif(j t.eq.2) then

uv2 (i-v(i) else UV l (i)=uv2 (i) uv2(ihv(i)

enàif enddo

else an 1 (i j)=an2 (i J) d(i j)=an(i j)

endif enddo

enddo enddo

retum end

c *********Salve centroid veloic~******************** subroutine centrd(uv,b,rO,d 1 1 ,Ll 1 ,epsi,p) impiicit double precision (a-h,L,o-z) pararneter(nnc=52 1 ,npc=67 1 ,ngdO42,nt=l342,ne=l5O,nbc=2OS) dimension uv(nt),uc 1 (2*ne),uc2(2 *ne),r0(2,16,ne), b(ng), & d 1 l(ne),Ll l(l,l6),p(ne) common/meshlx(npc),y(npc),index(9 ,ne)

do i=l ,ng uv(i)=b(i)

enddo

do k=l ,ne ucl(2*k-l)=O. uc2(2*k-1)4. uc l(2*k)=û. uc2(2*k)=û.

pl*. do i=I,8

uc 1(2*k-I)=uc 1(2*k-l)+r0(lY2*i-1 ,k)*b(index(i&)*2-1) uc2(2*k- l)=uc2(2*k-l)+r0(1,2*i.k)*b(index(i,k)*2) uc1(2*kmc 1(2*k)+d)(2,2*i-l,k)*b(index(i,k)*2-1) uc2(2*k)=uc2(2*k)+d)(2,2*i,k)*b(index(i,k)*2)

p l=p 1-1 1(1,2*i-l)*b(index(i,k)*2-1)+ & L11(1,2*i)*b(index(i,k)*2))/dll(k)/epsi enddo

uv(2*index(9,k)- l)=ud(2'k-l)+uc2(2*k-l) uv(2*index(9,k))=uc 1 (2* k)+uc2(2*k) PO 1 enddo

return end

aa=û.dO do i= 1 ,nt

ifüt.eq. 1) then aa=aa+(uv(i)-UV 1 (i))* *2

else aa=aa+(uv(i)-uv2(i))* *2 endif

enddo

enddo

return end

subroutine gaussref implicit double precision(a-h,k,o-z) dimension ksi(3),dphi(2,9),phi(9) common/cob/w(3),ajcob(3,3) commonlgaureE/fai(3,3,9),dfdk(3,3,9),~Ede(3,3,9)

return end

subroutine gaussion(nee) implicit double precision(a- h,o-z) parameter(nnc=52 1 ,npc=67 1 ,ng=1 04î,nt=l342,ne=lSO,nbc=2OS) common/cob/w(3),ajcob(3,3) commodgauref7fai(3,3,9),dfdk(3,3,9),dfde(3,3,9) common/Etiins/dkdx(3,3),dkdy(3,3),dedx(3,3),dedy(3,3) cornmon/xytol/xto1(3,3),yt0I(3,3),y 1 tol(3,3) common/mesh/x(npc),y(npc),index(9,ne)

do b 1 , 9 dxâk==dxdk+x(index(k,nee)) *dfdk(i j,k) dxde=ckd~x(index(k,nee)) *dfde(i j, k) dydk=-dydk+y(hdex(k,nee))*dfdk(iJ,k) dyde=dyde+y(index&nee)) *dfde(i j ,k) xtoi(iJ)~ol(iJ)+(x(index(k,nee))-x(hk@,me)))* fai(i j,k) ytol(ij)==ol(i j)+(y(index(k,nee))-y(index(9,ee)))*f~(ijpk)

y1 tol(iJ)==l toi(iJ)+y(index(k,nee))*fai(i j enddo

ajcob(i j)=dxdk*dyde-dxde*dydk dkdx(iJ)=dyde/ajco b(i j) dedx(i J)=-dydklajcob(i j) dkdy(iJ)=dxde/ajco b(i j ) dedy (iJ)=dxdk/ajcob(iJ)

enddo enddo

retum end

subroutine integ(nee7m,n,uv7b 1 ,b2,b3 ,b4jj) implicit double precision(a-h,o-z) parameter(nn~52 1 ,npc=67 1 ,ng=I 042,nt=1342,ne= 150,nbc=205) dimension uv(nt) common/cob/w(3),ajcob(3,3) comrnon/ga~ref7fai(3,3,9),dfdk(3,3,9),dfde(3~3,9) common/xytoVxtol(3,3),yto1(3,3),y 1 tol(3,3)

do i=1,3 do j=lJ

if (jj.eq. 1) then call func l(m,n,i jydfl ,dfî,dB) elseif üj.eq.2) then call func2(m,n,i j,dfl) elseif Üj.eq.3) then dfl=l .dO dQ=L .dO endif

if (jj .eqA) then dfl+ai(i j,n)* fai(i j,m) endif

iQj .eq.6) then cal1 func4(nee7rn,n,i j,dfi &2) elseifQj.eq.7) then caii func5(nee,m,n,ij7dfl ,dQ,dD ,df4) endif

retum end

c This is a code hgment which can be incorporated into a subroutine c to evaluate shape fictions and derivatives. The numbering of shape c fictions is as in the course notes for MEC 12 10. C

c You will need to declare a double precision array of size 9 for phi, c and a double precion 2D array of size 2,9 for phi derivatives. C

c NOTATION: here r is used in place of xi, and s is used in place of c eta C

c Ross Ethier C********************************************************************

subroutine con(r,s,phi,dphi) implicit double precision(a-h,L,o-z) dimension phi(9),dphi(2,9)

* calculate phi phi(1) = r*s*(r-1 .dO)*(s- 1 .dO)/4.dO phi(2) = s*(s-1 .dO)*(l .do-Pr)/Z.dO phi(3) = r*s*(rtt .dO)*(s-l .dO)/4.dO phi(4) = r*(r+l .dO)*(l .do-s*s)/2.d0 phi(5) = r*s*(r+l .dO)*(s+l .d0)/4.d0 phi(6) = s4(s+l .dO)*(l .do-r*r)/2.d0 phi(7) = r*s*(r-1 .dO)*(s+l .dO)M.dO phi(8) = r*(r-1 .dO)*(l .do-s*s)/2.d0 phi(9) = (1 .do-r'r)*(l .do-s*s)

* caicuiate dphi/dr dphi(1,l) = s*(s-1 .d0)*(2.dO*rœ1 .dO)/4.dO dphi(l,2) = -r*s*(s-1 .do) dphî(l,3) = s*(s-1 .dO)*(Z.dO*r+l .dO)M.dO dphi(l,4) = (1 .do-s*s)*(2.dO*W .d0)/2.d0 dphi(l,5) = s*(s+l .d0)*(2.d0**1 .dO)/4.dO dphi(1,o) = -Ps*(s+l.dO) dphi(l,7) = s*(s+l .d0)*(2.d0*r01 .d0)/4.d0 dphi(l,8) = (1 .do-s*s)*(2.dO*r-1 .dO)/Z.dO dphi(l,9) = -2. do*?( 1 .do-s*s)

* calculate dphifds dphi(2,l) = r*(r-1 .dO)*(2.dO*s-1 .dO)M.dO dp hi(2,2) = (1 .do-r*r)*(2.dO*s-1 .d0)/2 .do dphi(2,3) = r*(r+l .dO)*(2.dO*s-1 .d0)/4.d0 dphi(2,4) = -iLs*(r+l .do) dphi(2,S) = ?(rH .d0)*(2.d04s+l .d0)/4.d0

r e m end

~************~~l~~~efun~ti~~l********************t**~**** subrouthe huic 1 (n,m,i j,dfl ,dfZ7df3) implicit double precision(a-h,L,o-z) common/gaure~fai(3,3,9),dfdk(3,3,9),dfde(3,3,9) common/trans/dkdx(3,3),dkdy(3,3),dedx(3,3),dedy(3,3) common/xytoVxtol(3,3),yto1(3,3),y l toI(3J)

iE(n.eq. 1) then dfl=(dfdk(iJ,m)*dkdx(i j)+dfde(i j,m)*dedx(i,j)) df2=(dfdk(i j,m)*dkdy(i j)+dfd e(i j,m)*dedy(i j)+

& fai(iJ,m)/yltoi(iJ)) dB=(dfdk(iJ,m)*dkdy(iJ)+dfde(i j,m)*dedy(i j))

endif

if (n.eq.2) then dfl=xtol(i j)*(dfdk(i j,m)*dkdx(i j )+d fde ( i ( i j)) df2=xto l(i j) *(dfdk(i j ,m)*dkdy(i j)+d fde(i j ( i i ) +

& fai(iJ,m)/yltol(iJ)) dB=xto l(i j)*(df&(i j ,m)*dkdy(i J)+dfde(iJ ,m)*dedy(iJ))

endif

if (n.eq.3) then dfi -ytol(ij)*(dfdk(iJm)*dkdx(i j)+dfde(iJ,m)*dedx(iJ)) dn=ytol(i j)*(dfdk(iJ,m)*dkdy(i J)+dfde(i j ( i J ) +

& fai(i j,m)/y 1 tol(i j)) df3=ytoi(ij)*(dfdk(i jsn)*dkdy (i j ) + d f d e ( i m )

endif

return end

c ****************CaiCdate function2************f ******Ir*** subrouthe func2(m7n,i j,dQ implicit double precision(a-h,L,o-z) common~gaure£tfai(3,3,9),dfdk(3,3,9),dfde(3,3,9) common/transldkdx(3,3),dkdy(3,3),dedx(3,3),dedy(3,3)

dfk==dfdk(i j,m)*dkdx(i j)+dfde(i J ,m) *dedx(i J)) & *(dfdk(i j,n)*dkdx(i j)+dfde(i j,n)*dedx(i j))

dfi-=dfdk(i j,m)*dkdy(i j )+dfde(i j,m)*dedy (i j)) & *(dfdk(i,j,n)*dkdy(i j)+dfde(i j,n)*dedy(i j))

dE--&+de

return end

c ***** ****CalcUIate fiuiction4******* *********** **************** subrouthe huic4(nee,m,n,i j,dfl ,da) irnplicit double precision (a-h,o-z) parameter(nnc=52 1 ,npc=67 1 ,ng=l O42,nel 34î,ne=lSO,nbc=2OS) common/gaureUfai(3,3,9),dfdk(3,3,9),dfde(3,3,9) cornmon/trans/dkdx(3,3),dkdy(3,3),dedx(3,3),dedy(3,3) co"non/mesh~x(npc),y(npc),index(9,ne) comrnonlxytoVxto1(3,3),yt01(3,3),y1 tol(3,3)

dfl =fai(i j,m)* fai(i j,n)/y l tol(i j)/y 1 tol(i j ) df2-fai(i j,m)*(df&(i j,n)*dkdy(i j )

& +dfde(i j,n)*dedy(iJ))/y 1 io l(i j )

r e m end

c **********Calculate function5(source ternis)**** **************+ subroutine func5(nee,m,n,i j,dfl ,df2,df3,df4) irnplicit double precision (a-h,o-z) parameter(mc=52 l,npc=671,ng=1042,n~1342,ne=150pbc=205) common/gaureDTai(3,3,9),dfiik(3,3,9),dfde(3,3,9) comrnon/trans/dkdx(3,3),dkdy(3,3),dedx(3,3),dedy(3,3) common/mesh/x(npc),y(npc),index(9,ne) common/rnagia, bx(npc),br(npc),sigma,bO,aa,e ta

return end

c BANDED MATRlX SOLUTION ROUTINES

c This file contains the following functions: c (i) b-Iubksb: Forwardlback solves a banded system c (ii) b-ludcmp: LU decomposes a banded system c (iii) point : provides location in 1D storage vector c of the i j th entry of a banded matrk.

c Notation: c N: size of system matrix c m: half-bandwidth of system ma& c v: storage vector for coefficients of systern matrix c b: right hand side of system to be solved

c Note that al1 floating point quantities are defined double c precision. C

c Usage: c Youmust: c (i) Declare m, N, and offset as integers and create a c common block called matjaram containhg these variables c (ii) Specify m and N in the main routine; c (iii) Compute offset in the main routine as: offset = m*N - m*(m+1)/2 . c (iv) Have aiiocated memory for v and b.

c Prograrnming notes: c 1. Note that hinction point contains an if-else bath which is used for c emor checking. Since point is calied many times, and if-else constructs c are slow, you might want to delete out the error checking c for production nuis.

c Ross Ethier c Mechanical Engineering c University of Toronto

c Forwardmack solves the system Ax = b, where A has already c been LU decomposed. The answer is retunied in b. A is c n by n, banded, and stord in vector v. The original contents c of b are overwritten.

double precision v(*), b(*), sum integer i j, m, n, point comrnon /matparam/ m,n,offset

c forward solve

do 100 i = l,n sum = b(i) if (i.ne. 1) then do 20 j=maxO(l ,Lm), i-1 sum = sum -v@oint(ij))*bu)

20 continue endif b(i) = s u m

LOO continue

c backsolve

do 200 i =n, 1, -1 sum = b(i) i@.kn) then

do 120 j = i+l, minO(n, i+m) sum = sum -v@oint(ij))*bÿ)

120 continue endif b(i) = sum/v(point(i,i))

200 continue

end C************************************************************** c LU decomposition routine for banded matrices of sire N c by N, and bandwidth m. The entries of the rnatrix are c packed in the storage vector v. The original entries of c v are ovexwritten. C*************************************************t************

subroutine b-Iudcmp(v)

double precision v(*), surn integer m, n&j, ilower, iupper, Wower, point cornmon /matgarard rn,n,o ffset

C

do 100j=l,n ilower = maxO(1, j-m) iupper = minO(n, j+m) do 30 i= ilower, j

klower = rnaxO(1 ,Lm j-m) sum = v@oint(iJ)) do 20 k = klower, i-1 sum = sum - v@oint(i,k))*v@oint&j))

20 continue v(point(i j)) = sum

30 continue do 50 i = j+l, iupper

klower = maxO(1, Lm, j-m) sum = v@oint(iJ)) do 40 k = klower, j-1

s u m = sum - v@oint(i,k))*v@oint&j)) 40 continue

v@oint(ij)) = sum/v(point(j j)) 50 continue 100 continue

retum end

c Function which retunis an address in the vector v for a c given [il [j] location in the original matrix. v is the c vector containing the packed version of a banded maaix. C**************************************************************

function point(ij, integer i, j, m, point, offset, n, delta common /matparam/ -offset

delta = i-j if (iabs(delta).gt.m.or.i.ie.O.or.i.gt.n.o~

+ j.gt.n) then write (6,*) ' Fatal error: i j outside mattix band: i J =',

+ i, j stop else

point = offset + i - delta*(2*n-iabs(delta)+1)/2 endif return end

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