EMBR Effect of Ultra-Strong Transverse Magnetic Field on the Liquid Flow in Directionally...

EMBR Effect of Ultra-Strong Transverse EMBR Effect of Ultra-Strong Transverse Magnetic Field on the Liquid Flow in Magnetic Field on the Liquid Flow in

Directionally Solidifying Shaped CastingsDirectionally Solidifying Shaped Castings

哈尔滨工业大学哈尔滨工业大学

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Daming Xu1, Yunfeng Bai1, Jingjie Guo1 and Hengzhi Fu1,2

School of Materials and Engineering, 1)Harbin Institute of Technology, 2)Northwestern Polytechnological University

Contact E-mail: [email protected]

INTRODUCTIONINTRODUCTION• DC or static magnetic fields are widely used as important

applications to electromagnetic processing of materials (EPM):1) for electromagnetic (EM) purifications of metallic alloy melts

(nonmetallic inclusions separation by electromagnetic Archimedes force);

2) for solidification structures refining/control of ingots/castings by EM vibrations (e.g. with a static magnetic field + AC current);

3) applied to continuous casting of various alloy slabs, including bimetallic slabs (for reduction of inclusions entrapment and improvement of the

slab surface quality), to crystal growth or liquid metal heat energy transport systems (for a suppressed or desired stable flow control) etc.

• The key technique in the 3rd class of Static Magnetic applications is to employ the electromagnetic brake (EMBR) effect to suppress the alloy melt convections in the liquid regions (induced heavily by the inlet melt fluxes or/and gravity etc).


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• With the advance in developments of super-conductor magnetic techniques of lower costs, ultra-strong static magnetic fields, up to over 10T, have become available for practical uses in materials science and engineering researches, taking advantage of the possible effects of the ultra-strong Lorentz or/and Magnetization forces on the various processed materials.

• Computer modeling is a useful method to investigate the heat and mass transfer and fluid flow behaviors in various materials solidification processes under different EM fields, due to the advantages of:

1) Can exhibit intuitively the EM-solidification transport behaviors in an opaque metallic fluid/solidification system;

2) Can test an EM-solidification transport problem under a condition that can not be or is difficult to be practically realized, e.g. under a magnetic field of strength >20T so far;

3) Costs less, etc.


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• A continuum model for solidification transport processes in alloy ingots/castings and the related numerical solution algorithms, proposed by the authors, are recently extended to an EM solidification case, and shows to be successful and available for the numerical simulations of solidification heat, mass and momentum transport processes under a harmonic or a static magnetic field.

• The aim of this presentationThe aim of this presentation is to show some numerical simulation results for the influences of transverse static magnetic fields, of different strengths from 0T to an ultra-high magnetic up to 25T, on the liquid flow behaviors, as well as heat and mass transports in upwards directional solidification processes of blade-like castings (Al-4.5wt.%Cu, pseudo-binary In718 base-

4.85wt.%Nb and (TiAl)-55at.%Al), in which the liquid flow are driven both by gravity and volume variations including the solidification shrinkage. (Several special solidification magnetohydrodynamics (MHD) and transport phenomena in the applied static magnetic fields are presented) .


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A Brief Introduction to:A Brief Introduction to: used EM-Solidification Model and used EM-Solidification Model and

the Numerical Solution Methodsthe Numerical Solution Methods

Physical Model for EM-Directional SolidificationPhysical Model for EM-Directional Solidification


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N

g

z

x y

AC-EM Induction Coil HHEEAATTIINNGG

ZZOONNEE

CCOOOOLLIINNGG

ZZOONNEE

SS

v0

Withdrawal Rod

v0

NN

HHEEAATTIINNGG

ZZOONNEE

CCOOOOLLIINNGG

ZZOONNEE

Static Magnetic FieldBlade-like Casting

Bottom Cooler


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• Solidification heat energy transfer

[(cP)mT]/t + (fLLcPLVT) = [mT] + Sh(fS/t) + qJ (1)

where, the electromagnetically inducted Joule heat is given by

qJ＝ JGE = JG2/ (2)

• Solidification solute mass transfer

(C)m/t + (fLLVCL) = [DL(fLLCL) + DS(fSSCS)] (3)

where:

Cm t = SCSfSt + fSSCSt + LCLfLt + fLLCLt (4)

= /(1 + ), = (DS(T)/Rf)A2N and = k(1 + )fS/fL2

• L-S phase-change characteristic function for a specific alloy

TLiq = TLiq(CL) (5)

• Solidification mass conservation:

m/t = (fLLV) (6)

where, mt SfSt + fSS

t + LfLt + fLLt (7)

• Momentum transfer for bulk/interdendritic liquid flow:

(fLLV)/t + [(fLLV)V] = [(fLV)] (fLP) (fL2/K)V + FB (8)

For the present modeling, the body force term induced by external fields includes

the gravity and Lorentz force:

FB = fLLg + FL (9)

the Lorentz force acting on the moving liquid phase during solidification:

FL = fL(E + VB)B = fL{JGB + [(VB)B B2V]} (10)


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[Unit： mm]

g

Heating Z

oneH

eating Zone

,, T THH

Cooling

Cooling

Zone

Zone

, , TTCC

A 2D configuration and dimensions of the directional solidification of alloy shaped castings in a transverse static magnetic field (right half).

Direct current induction coils

Magnetic conduction iron

Air domain

SHELL MOLDSHELL MOLD

Bottom COOLERCOOLER

Directionally solidifying CASTINGCASTING

EM-field analysis boundaries

Coordinate system for computation

o y

z

Withdrawal velocity, Vo

Boundary ConditionsBoundary Conditions (Refers to the Solidification System Configuration)(Refers to the Solidification System Configuration)

• (1) at z=0 and y[0,W2]: T/z=(A/LA1)(TTCooler), C/z=0, Vy=0 and Vz=0

• (2) at z=H1 and y[W1,W2]: T/z=(A/LA3)(TTM), C/z=0, Vy=0 and Vz=0

• (3) at z=H2 and y[W1,W2]: T/z=(A/LA5)(TTM), C/z=0, Vy=0 and Vz=0

• (4) at z=H3 and y[0,W2]: T/z=qConve.+qRadia., C/z=0, PL=0, Vy/z=0 and Vz=Vfd

• (5) at y=0 and z[0, H3]: T/y=0, C/y=0, Vy=0 and Vz/y=0

• (6) at y=W2 and z[0, H1]: T/y=(A/LA2)(TTM), C/y=0, Vy=0 and Vz=0

• (7) at y=W1 and z[H1, H2]: T/y=(A/LA4)(TTM), C/y=0, Vy=0 and Vz=0

• (8) at y=W2 and z[H2, H3]: T/y=(A/LA6)(TTM), C/y=0, Vy=0 and Vz=0

• where, the air gap thicknesses, LAi, i=1, 2, …, 6, take constant values between 0.05~0.15 mm, i

n the present modeling; the top feeding rate, Vfd = {(Wj,k)i+1/[ti+1(fLj,KoAj)i+1] (for fLj,Ko>fL

c); 0. (for fLj,KofLc)}, where, (Wj,k)i+1 and (fLj,KoAj)i+1/2 represent the total liquid-volume cont

raction over the casting domain in a time interval ti+1 and the effective liquid area on the casti

ng top at the time ti+1/2, respectively. In the present modeling, the critical liquid-volume fraction

that allows a free top feeding takes, fLc = 0.35.


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Both Both FEMFEM ((for EM analysisfor EM analysis)) and and FVM/FDMFVM/FDM ((for for

solidification transport simulationsolidification transport simulation)) are used are used


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a) A schematic FEM mesh pattern for a casting domain; b) A schematic FVM mesh pattern for the same casting domain; c) Overlap of a) and b)

a)

b)

c)

FEM FEM for ANSYS 8.0 Magnetic Fields Analysis

FVM FVM for Solidification Heat, Mass & Momentum Transfer Simulation

Numerical Solution Methods to the Equations for Numerical Solution Methods to the Equations for Alloy Solidification Transport ProcessesAlloy Solidification Transport Processes


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Staggered Meshes [j+1/2, k] and [j, k-1/2] for Computations of the Vectorial Variables Vy&VZ

Main FVM Mesh [j,k] for Computations of the Scalar Variables T、 fS、 CL&P

[j, k-1]

[j, k+1]

[j-1, k] [j+1, k][j, k]

Main FDM Mesh

[j, k]

[j, k+1]

[j+1, k]

[j, k-1]

[j-1, k]

[j, k-1/2]

[j+1/2, k]

[j+1/2, k+1/2]

Staggered Mesh for Calculating Vz

Staggered Mesh for Calculating Vy


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• Iterative Computations for the Nonlinear and

Strong TfSCLCoupling

(From: Daming Xu and Qingchun Li, “ Numerical Method for Solution of Strongly Coupled Binary Aalloy Solidification Problems”, Numerical Heat Transfer, Part A, 1991, Vol.20, 181-201.)

Available for ANYANY Solid Back-Diffusion

Available for BOTHBOTH Single-Phase and Eutectic Solidification Portions

Level-rule model

Scheil model


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SIMPLESIMPLE Scheme widely used for Fluid Transport Systems by Subas V. Patankar

Direct-SIMPLEDirect-SIMPLE Scheme by present Authors(No Iteration NeededC. Efficiency； Available both for PV

Coupling in Pure Fluid Systems and Alloy Solidification Systems)

• Numerical Computations of the Nonlinear and Strong PV Coupling

(From: Subas V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, 1980)

Let P = P* + P， take an initial estimation： Po = P*

Calculate the approximate velosity: VX* & VY* based on the up-to-date pressure Pn and Momentum Equations

Using VX*& VY*, V-correction eqs. and mass-conserv. eq. to calculate the approximate pressure: P and Pn+1 = Pn + P

Using VX* , VY*, P&V-corr. eqs. to calculate approxi. V: V

Xn+1&VY

n+1; calculate other fields that influence P&V

If Pn+1,VX

n+1 and VYn+1

converge?

End of P-V coupling calculation

Yes

NoPn = Pn+1

(From: Daming Xu and Qingchun Li, “ Gravity- and Solidification-Shrinkage-Induced Liquid Flow in a Horizontally Solidified Alloy Ingot”, Numerical Heat Transfer, Part A, 1991, Vol.20, 203-221.)

Let: ti+1 times pressure=ti times pressure + pressure change in the time interval ti+1, i.e. Pi+1 = Pi + Pi+1 (1)Let: ti+1 times accurate V-field=an approximate V-field + the V-corrections induced by the pressure change in time ti+1: Pi+1, i.e

. VXi+1 = VX* + VX and VY

i+1 = VY* + VY (2)/(3)

Using ti times pressure field: Pi and ti times V-field: Vxi and VY

i,

calculate the approxi. V-field: VX* and VY* from momentum eqs.

Using the obtained ti+1 times T, fS and CL fields, and the approxi. V

-field: VX*/VY*, solve the algebraical eq. set for Pi+1 which is de

rived from the mass conservation eq. and the V-correction eqs.

Using VX*、 VY*&Pi+1 and the V-corr. eqs., calculate the accurat

e ti+1 times pressure and velosity fieds: Pi+1&V i+1 from eqs.(1)-(3).

End of P-V coupling calculation

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FEM mesh configuration for the directional solidification system of alloy shaped castings under a applied transverse static magnetic field


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Numerical Simulation Numerical Simulation ResultsResults


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Magnetic flux lines of the applied transverse static magnetic in the EM-DS system, calculated and output from ANSYS 8.0 (Having an averaged value of 0.066 T).


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Vectors of the Applied Transverse Static Magnetic in the Blade-like Casting, calculated and output from ANSYS 8.0 (Having an averaged value

of 0.66 T).

Has an almost Has an almost uniformly uniformly transverse transverse magnetic magnetic

distribution.distribution.


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Comparison of the Comparison of the Vector Distributions of Vector Distributions of

Applied Transverse Applied Transverse Static Magnetic of Static Magnetic of

Different Strengths in Different Strengths in the Blade-Like Castingthe Blade-Like Casting

0.66 T

11 THave the Have the

Same Vector Same Vector Distribution Distribution

Form !Form !

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Solidification Transport Processes of a Blade-like Solidification Transport Processes of a Blade-like Al-4.5%CuAl-4.5%Cu Casting under Casting under Transverse Static Magnetic FieldsTransverse Static Magnetic Fields of different Load Strengths (in At) of different Load Strengths (in At)


0 At 500 At 1000 At 2500 At 10000 At

(a)

Comparisons of liquid flow and directional solidification behaviors of Al-4.5%Cu shaped casting under static magnetic fields inducted by different current-turns at t=21.0 sec: (a) Velocity vectors of liquid phases; (b) Contours of relative pressure in the liquid; (c) Contours of solid volume fraction; (d) Contours of temperature.

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Solidification Transport Processes of a Blade-like Solidification Transport Processes of a Blade-like In718-4.85%NbIn718-4.85%Nb Casting Casting under under Transverse Static Magnetic FieldsTransverse Static Magnetic Fields of different Load Strengths (in At) of different Load Strengths (in At)


Influences of transverse static magnetic strengths on the liquid flow behaviors and temperature /concentration distributions in directionally solidified In718 shaped castings at t=21.0 sec.: (a) Velocity vectors of liquid phases; (b) Contours of temperature; (c) Contours of liquid concentration, Nb%wt..

0 At 500 At 1000 At 2500 At 10000 At

(a)

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0 At 500 At 1000 At 2500 At 10000 At

(b)

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0 At 500 At 1000 At 2500 At 10000 At

(c)

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Channel Segregations in the Solidified Blade-like Channel Segregations in the Solidified Blade-like In718-4.85%NbIn718-4.85%Nb Casting Casting under under Transverse Static Magnetic FieldsTransverse Static Magnetic Fields of different Load Strengths (in of different Load Strengths (in AtAt))


Influences of the different transverse static magnetic strengths on the macro-composition distribution in the directionally solidified In718 shaped castings at a later stage and on the freezing time to arrive at the same solidification stage. (Under the same solidification condition as the previous Figure)

0 At 500 At 1000 At 2500 At 10000 At

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Solidification Transport Processes of a Blade-like Solidification Transport Processes of a Blade-like (TiAl)-55Al%at.(TiAl)-55Al%at. Casting Casting under under Transverse Static Magnetic FieldsTransverse Static Magnetic Fields of different Load Strengths (in of different Load Strengths (in TT))


The distributions of: (a) Velocity vectors of liquid phases; (b) Contours of relative pressure in the liquid; (c) Contours of temperature; (d) Contours of liquid concentration at t=21.0 sec.(No magnetic field (0.0 T) applied)

(a) (b) (c) (d)

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The distributions of: (a) Velocity vectors of liquid phases; (b) Contours of relative pressure in the liquid; (c) Contours of temperature; (d) Contours of liquid concentration at t=21.0 sec.(A 0.0033 T magnetic field applied)

(a) (b) (c) (d)

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(a) (b) (c) (d)

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(a) (b) (c) (d)

a steep pressa steep pressure-gradient ure-gradient established !established !

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(a) (b) (c) (d)



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The distributions of: (a) Velocity vectors of liquid phases; (b) Contours of relative pressure in the liquid; (c) Contours of temperature; (d) Contours of liquid concentration at t=21.0 sec.(A 11 T magnetic field applied)

(a) (b) (c) (d)



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The distributions of: (a) Velocity vectors of liquid phases; (b) Contours of relative pressure in the liquid; (c) Contours of temperature; (d) Contours of liquid concentration at t=21.0 sec.(A 25 T magnetic field applied)

(a) (b) (c) (d)

a very steep a very steep pressure-grapressure-gradient establisdient establis

hed !hed !


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0T 0.0165T0T 0.0165T 11T (transverse static magnetic fields)11T (transverse static magnetic fields)

Animation(S) : VV&f&fSS Evolution in a Blade-like Evolution in a Blade-like (TiAl)-55at.%Al Casting directionally (TiAl)-55at.%Al Casting directionally Solidifying under Transverse Static Magnetic Fields of Different StrengthsSolidifying under Transverse Static Magnetic Fields of Different Strengths


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The transverse distributions of solid volume fraction at a height of ~10 mm under different static magnetic fields, at about 21.0s

0.066 T0.066 T


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The transverse distributions of liquid concentration in mushy zone at a height of ~10 mm under different static magnetic fields, at about 21.0s

The values of The values of VVLLmaxmax ，， VVmmmaxmax and and PPLL//z according to the numz according to the numerical results shown in Figs.1-5, 11 and 12 under Transverse Staterical results shown in Figs.1-5, 11 and 12 under Transverse Stat

ic Magnetic Fields of Different Strengthsic Magnetic Fields of Different Strengths


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Strength of Static Magnetic Fields Applied

B ， [T] VLmax , [mm/s] Vmmax , [mm/s]

Averaged Vertical Pressure Gradients in Bulk Liquid Region PL/z , [Pa/mm]

Notes

0.00 [T], (only in g) 14.7932 0.00439009 0.054778

0.0033 [T], (500At) 9.65907 0.00463883 0.053011 Max. PL/z (upper

half) =-0.1096

0.0066 [T], (1000At) 3.94108 0.00477937 0.050000

0.0165 [T], (2500At) 1.46466 0.00507709 0.043053

0.033 [T], (5000At) 0.51531 0.070038 0.035417

0.066 [T], (10kAt) 0.155329 0.104446 0.00 PL/z fluctuations max=-0.026

0.66 [T], (100kAt) 0.15063 0.14224 5.7051 WhenB 0.066 [T], convections can be basically suppressed, only downwards feeding exists, i.

e. VLz0。

6.6 [T], (1MAt) 0.15188 0.15234 473.20

11 [T], (1.6667MAt) 0.15169 0.14812 1377.2

25 [T], (3.788MAt) 0.15166 0.16213 6793.8


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|B| |∂P/∂z| !

SummarySummary哈尔滨工业大学哈尔滨工业大学

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• The used continuum model and numerical solution methods still show available and efficient for the numerical simulation of solidification transport processes under a static magnetic field, which could be very strong at least up to 25T;

• According to the present computational results, the natural convections in the bulk-liquid region can be effectively suppressed under a relatively weak static magnetic(~0.1T). However, the solidification-shrinkage-driven liquid flow, though much smaller than the bulk liquid convection by order(s), can not be suppressed by a very strong static magnetic field even up to 25T.

• Under an ultra-strong transverse static magnetic field, a small vertical velocity may cause a very steep pressure-gradient to establish in the bulk liquid region.


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Thank youThank you


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• From the Momentum Transport Equation for Liquid Flow:

(fLLV)/t + [(fLLV)V] = [(fLV)] (fLP) (fL2/K)V

+ fLLg + fL{JGB + [(VB)B B2V]} (8)-(10)

For the present upwards directional solidification case under a strong transverse static magnetic field, the Momentum Transfer Equation reduces to:

(fLLVZ)/t + [(fLLVZ)VZ]/z = [ (fLVZ)/z]/z (fLP)/z

(fL2/K)VZ + fLLg |B|2VZ

0 0 0 ↗ ↗ ↗ 0 ↗

In this case, VZ → 0, and for the bulk liquid region the Momentum Transfer Equation further reduces to:

(fLP)/z fLLg |B|2VZ

A Symplified Analytical ModelA Symplified Analytical Model

Electromagnetic solidification modelElectromagnetic solidification model

• A Continuum ModelA Continuum Model (based on Mixture Average Theory) with the following assumptions:

1) the external forces involved in a solidifying system are gravity and Lorentz force;

2) no pores will occur, i.e. the geometric continuity, fL+fS=1, holds for any region in the casting/ingot domain;

3) the solid phase is macroscopically static during solidification;4) local thermodynamic equilibrium holds at the microscopic soli

d-liquid interfaces5) Newtonian and laminar liquid flow present; 6) the model alloy is a binary system.


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• Maxwells equations

×H = J+D/ t (Law of Maxwell-Ampere) (11)

×E = -B/ t (EM-Induction Law of Faraday) (12)

·D = e (Law of Gauss) (13)

·B = 0 (Continuity of Magnetic Flux) (14)

• Constitutive equations for the involved EM-medium materials

D = E (Constitutive Relationship of Electric Properties) (15)

B = H (Constitutive Relationship of Magnetic Properties) (16)

J = (E+fLVB) (Ohm Law for the Moving Metallic Melt) (17)


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• In such EM materials processing applications, the EM-induced solidification heat/mass transfer, and especially the melt flow behaviors can be highly enhanced (or suppressed in the case of a static magnetic field applied), through the Lorentz forces and Joule heats, and might be in a much more complex pattern.


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Ele

ctro

mag

netic

Fie

ld

Flu

id F

low

Fie

ld

Hea

t and

Mas

s T

rans

fer

S/L

Int

erfa

ce M

orph

olog

y

Str

uctu

re a

nd P

rope

rtie

s

EM-Induced Joule Heat

GravityLorentz Force

EM-Induced Joule Heat

Lorentz Force


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The physical properties used for the present computer modeling of solidification transport phenomena of (TiAl)-Al%at. pseudo-binary alloy

• Thermal conductivity: S(T) = L(T) = 2.310-2 [W/mmC]

• Solid specific heat: 0.67832 – 6.432810-6T + 2.5417 10-13T3 , (25C < T 660C)

cPS(T) = 0.68286 + 1.364410-5T , (660C < T 882C) [J/gC]

0.65193 + 1.051310-5T , (882C < T 1490C)

• Liquid specific heat: cPL(T,CL) = 0.86074 , (1490C < T 2727C) [J/gC]

• Solid density of the alloy: S(T,CL) = 3.810-3 , [g/mm3]

• Liquid density of the alloy:

L(T,CL) = 3.63210-3 – 2.010-7(T – T liq.) – 9.3210-6(CL – CO) [g/mm3]

• Latent heat of fusion: h(CS) = 435.4 [J/g]

• Liquidus of the (TiAl)-Al%at. alloys:

Tliq(CL/Al) = 802.19745 + 23.01678CL – 0.20279CL2 , (CL[54.86, 74.61at.%Al]) [C]

• Partition coefficient:

k(CL/Al) = 69.24438-4.03153CL+8.89410-2CL2-8.7003110-4CL

3 + 3.1859510-6CL4

(CL[54.86, 74.61at.%Al]) [-]

• Liquid diffusion coefficient: DL/Al(TC) = 510-3 [mm2/s]

• Solid diffusion coefficient: DS/Al(TC) = 0.11exp[-14504/(T+273.15)] [mm2/s]

• Dynamic viscosity coefficient: (T) = 3.510-3 [Pas] (= [Ns/m2] = [g/mms])

4.010-4fL3.0 , (fL0.7088)

• Permeability coefficient: K = 1.6318fL27.155 , (0.7088fL0.99999) [mm2]

1.01018 , (fL0.99999)

• Primary dendrite spacing: d1 = 1.410-1 [mm]

• Secondary dendrite arm spacing: d2 = 3.010-2 [mm]

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0 At 500 At 1000 At 2500 At 10000 At

(b)

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0 At 500 At 1000 At 2500 At 10000 At

(d)

Model Numerical Solution StrategyModel Numerical Solution Strategy

• Electromagnetic Fields (B and J vector fields etc)

Solved by ANSYS 8.0 in a FEM-Scheme;

• Solidification Transport Phenomena Equations (scalar T, fS, CL, P and vector V fields)

Solved by a FDM-Scheme-based

Computer Simulation Code

(developed by the present authors)


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Numerical Solution Methods to the Equations for Numerical Solution Methods to the Equations for Alloy Solidification Transport ProcessesAlloy Solidification Transport Processes

• Calculations of the Nonlinear and Strong Pressure Velocity (PV) Coupling in a Solidification Transport Process of Alloy Casting/Ingot Takes Most Portion (>90%) of the Entire Computational Efforts for the STP Numerical Simulation;

• Therefore, it is of Great Importance to Adopt a Algorithm of High Computational Efficiency for a Practical STP-Based Computer Simulation for the Solidification Processes of Alloy Castings/Ingots.


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Table I Initial and technological parameters for the present Table I Initial and technological parameters for the present example simulations with example simulations with 3 different kinds of Alloys3 different kinds of Alloys


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Alloys IN718 base-4.85wt.%Nb (TiAl)-55at.%Al Al-4.5wt.%Cu

Pouringtemperature, TP

1450 oC 1500 oC 700 oC

Initial moldtemperature, TM

1500 oC 1550 oC 700 oC

Heating zonetemperature, Th

1600 oC 1600 oC 950 oC

Cooling zonetemperature, TC

45 oC 45 oC 25 oC

Bottom coolertemperature, TBC

45 oC 45 oC 25 oC

Withdrawalvelocity, V0

0.15 mm/sec 0.15 mm/sec 0.15 mm/sec

EMBR Effect of Ultra-Strong Transverse Magnetic Field on the Liquid Flow in Directionally...

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