哈尔滨工业大学

哈尔滨工业大学 EMBR Effect of Ultra-Strong Transverse Magnetic Field on the Liquid Flow in Directionally Solidifying Shaped Castings 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: damingxu@hit.edu.cn HIT–MSE

哈尔滨工业大学 INTRODUCTION • 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 the3rd classof Static Magnetic applications is toemploy 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). HIT–MSE

哈尔滨工业大学 • With the advance in developments of super-conductor magnetic techniques of lower costs, ultra-strong static magnetic fields, up toover 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: • Can exhibit intuitivelythe EM-solidification transport behaviors in an opaque metallic fluid/solidification system; • 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; • Costs less, etc. HIT–MSE

哈尔滨工业大学 • 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 presentation is to show some numerical simulation results for the influences of transverse static magnetic fields, of different strengths from 0T toan ultra-high magneticup 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 bygravityandvolume variationsincluding thesolidification shrinkage.(Several special solidification magnetohydrodynamics (MHD) and transport phenomena in the applied static magnetic fields are presented) . HIT–MSE

哈尔滨工业大学 A Brief Introduction to:used EM-Solidification Model and the Numerical Solution Methods HIT–MSE

哈尔滨工业大学 Blade-like Casting AC-EM Induction Coil HEATING ZONE HEATING ZONE g S N COOLING ZONE COOLING ZONE Bottom Cooler Withdrawal Rod v0 v0 z x y Physical Model for EM-Directional Solidification Static Magnetic Field N HIT–MSE

哈尔滨工业大学 • Solidification heat energy transfer [(cP)mT]/t + (fLLcPLVT) = [mT] + 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 + (fLLVCL) = [DL(fLLCL) + DS(fSSCS)](3) where: Cm t = SCSfSt + fSSCSt + LCLfLt + fLLCLt(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) HIT–MSE

哈尔滨工业大学 • Solidification mass conservation: m/t = (fLLV) (6) where, mt SfSt + fSSt + LfLt + fLLt (7) • Momentum transfer for bulk/interdendritic liquid flow: (fLLV)/t + [(fLLV)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= fLLg + FL (9) the Lorentz force acting on the moving liquid phase during solidification: FL = fL(E + VB)B = fL{JGB + [(VB)B B2V]} (10) HIT–MSE

哈尔滨工业大学 Direct current induction coils Magnetic conduction iron Air domain SHELL MOLD Directionally solidifying CASTING Heating Zone, TH Coordinate system for computation g EM-field analysis boundaries z Cooling Zone, TC o y [Unit：mm] Bottom COOLER Withdrawal velocity,Vo A 2D configuration and dimensions of the directional solidification of alloy shaped castings in a transverse static magnetic field (right half). HIT–MSE

哈尔滨工业大学 Boundary Conditions(Refers to the Solidification System Configuration) • (1) at z=0 and y[0,W2]: T/z=(A/LA1)(TTCooler), C/z=0, Vy=0 and Vz=0 • (2) at z=H1and y[W1,W2]: T/z=(A/LA3)(TTM), C/z=0, Vy=0 and Vz=0 • (3) at z=H2and y[W1,W2]: T/z=(A/LA5)(TTM), C/z=0, Vy=0 and Vz=0 • (4) at z=H3and 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=W2and z[0, H1]: T/y=(A/LA2)(TTM), C/y=0, Vy=0 and Vz=0 • (7)at y=W1and z[H1, H2]: T/y=(A/LA4)(TTM), C/y=0, Vy=0 and Vz=0 • (8) at y=W2and z[H2, H3]: T/y=(A/LA6)(TTM), 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, in the present modeling; the top feeding rate, Vfd = {(Wj,k)i+1/[ti+1(fLj,KoAj)i+1] (for fLj,Ko>fLc); 0. (for fLj,KofLc)}, where, (Wj,k)i+1 and (fLj,KoAj)i+1/2 represent the total liquid-volume contraction over the casting domain in a time interval ti+1 and the effective liquid area on the casting 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. HIT–MSE

哈尔滨工业大学 FVM for Solidification Heat, Mass & Momentum Transfer Simulation c) a) b) FEM for ANSYS 8.0 Magnetic Fields Analysis Both FEM(for EM analysis)and FVM/FDM(for solidification transport simulation)are used 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) HIT–MSE

哈尔滨工业大学 Staggered Mesh for Calculating Vz [j, k-1] [j, k-1] Main FDM Mesh [j, k-1/2] [j+1/2, k] Staggered Mesh for Calculating Vy [j+1, k] [j+1, k] [j, k] [j-1, k] [j-1, k] [j, k] [j+1/2, k+1/2] [j, k+1] [j, k+1] Numerical Solution Methods to the Equations for Alloy Solidification Transport Processes Main FVM Mesh [j,k] for Computations of the Scalar Variables T、fS、CL&P Staggered Meshes [j+1/2, k] and [j, k-1/2] for Computations of the Vectorial Variables Vy&VZ HIT–MSE

哈尔滨工业大学 Level-rule model Scheil model HIT–MSE • Iterative Computations for the Nonlinear and Strong TfSCLCoupling Available for BOTH Single-Phase and Eutectic Solidification Portions (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 ANY Solid Back-Diffusion

哈尔滨工业大学 SIMPLE Scheme widely used for Fluid Transport Systems by Subas V. Patankar Direct-SIMPLE Scheme by present Authors (No Iteration NeededC. Efficiency；Available both for PV Coupling in Pure Fluid Systems and Alloy Solidification Systems) Let P= P* + P，take an initial estimation：Po = P* Let: ti+1 times pressure=ti times pressure + pressure change in the time interval ti+1, i.e. Pi+1 = Pi + Pi+1 (1) Let: ti+1 times 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 VYi+1 = VY* + VY (2)/(3) Calculate the approximate velosity: VX* & VY* based on the up-to-date pressure Pn and Momentum Equations Using ti times pressure field: Pi and ti times V-field: Vxi and VYi, calculate the approxi. V-field: VX* and VY* from momentum eqs. Using VX*& VY*, V-correction eqs. and mass-conserv. eq. to calculate the approximate pressure:P and Pn+1 = Pn + P Using the obtained ti+1 times T, fS and CL fields, and the approxi. V-field: VX*/VY*, solve the algebraical eq. set for Pi+1 which is derived from the mass conservation eq. and the V-correction eqs. Using VX* , VY*, P&V-corr. eqs. to calculate approxi. V: VXn+1&VYn+1; calculate other fields that influence P&V If Pn+1, VXn+1 and VYn+1 converge? Using VX*、VY*&Pi+1 and the V-corr. eqs., calculate the accurate ti+1 times pressure and velosity fieds:Pi+1&Vi+1from eqs.(1)-(3). Pn = Pn+1 No Yes End of P-V coupling calculation End of P-V coupling calculation (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.) (From: Subas V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, 1980) HIT–MSE • Numerical Computations of the Nonlinear and Strong PV Coupling

哈尔滨工业大学 FEM mesh configuration for the directional solidification system of alloy shaped castings under a applied transverse static magnetic field HIT–MSE

哈尔滨工业大学 Numerical Simulation Results HIT–MSE

哈尔滨工业大学 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 0.066T). HIT–MSE

哈尔滨工业大学 Has an almost uniformly transverse magnetic distribution. 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). HIT–MSE

哈尔滨工业大学 11 T Comparison of the Vector Distributions of Applied Transverse Static Magnetic of Different Strengths in the Blade-Like Casting 0.66 T Have the Same Vector Distribution Form ! HIT–MSE

哈尔滨工业大学 (a) 0 At 500 At 1000 At 2500 At 10000 At Solidification Transport Processes of a Blade-like Al-4.5%Cu Casting under Transverse Static Magnetic Fields of different Load Strengths (in At) 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. HIT–MSE

哈尔滨工业大学 (a) 0 At500 At1000 At2500 At10000 At Solidification Transport Processes of a Blade-like In718-4.85%Nb Casting under Transverse Static Magnetic Fields 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.. HIT–MSE

哈尔滨工业大学 (b) 0 At500 At1000 At2500 At10000 At Solidification Transport Processes of a Blade-like In718-4.85%Nb Casting under Transverse Static Magnetic Fields 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.. HIT–MSE

哈尔滨工业大学 (c) 0 At 500 At 1000 At 2500 At 10000 At Solidification Transport Processes of a Blade-like In718-4.85%Nb Casting under Transverse Static Magnetic Fields 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.. HIT–MSE

哈尔滨工业大学 0 At 500 At 1000 At 2500 At 10000 At Channel Segregations in the Solidified Blade-like In718-4.85%Nb Casting under Transverse Static Magnetic Fields of different Load Strengths (in At) 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) HIT–MSE

哈尔滨工业大学 (a) (b) (c) (d) Solidification Transport Processes of a Blade-like (TiAl)-55Al%at. Casting under Transverse Static Magnetic Fields of different Load Strengths (in T) 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) HIT–MSE

哈尔滨工业大学 (a) (b) (c) (d) Solidification Transport Processes of a Blade-like (TiAl)-55Al%at. Casting under Transverse Static Magnetic Fields of different Load Strengths (in T) 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) HIT–MSE

哈尔滨工业大学 a steep pressure-gradient established ! (a) (b) (c) (d) Solidification Transport Processes of a Blade-like (TiAl)-55Al%at. Casting under Transverse Static Magnetic Fields of different Load Strengths (in T) 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 0.66 T magnetic field applied) HIT–MSE

哈尔滨工业大学 (a) (b) (c) (d) Solidification Transport Processes of a Blade-like (TiAl)-55Al%at. Casting under Transverse Static Magnetic Fields of different Load Strengths (in T) 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) HIT–MSE

哈尔滨工业大学 a very steep pressure-gradient established ! (a) (b) (c) (d) Solidification Transport Processes of a Blade-like (TiAl)-55Al%at. Casting under Transverse Static Magnetic Fields of different Load Strengths (in T) 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) HIT–MSE

哈尔滨工业大学 Animation(S): V&fS Evolution in a Blade-like (TiAl)-55at.%Al Casting directionally Solidifying under Transverse Static Magnetic Fields of Different Strengths 0T 0.0165T11T (transverse static magnetic fields) HIT–MSE

哈尔滨工业大学 0.066 T The transverse distributions of solid volume fraction at a height of ~10 mm under different static magnetic fields, at about 21.0s HIT–MSE

哈尔滨工业大学 The transverse distributions of liquid concentration in mushy zone at a height of ~10 mmunder different static magnetic fields, at about 21.0s HIT–MSE

哈尔滨工业大学 The values of VLmax，Vmmax and PL/z according to the numerical results shown in Figs.1-5, 11 and 12 under Transverse Static Magnetic Fields of Different Strengths HIT–MSE

哈尔滨工业大学 |B|  |∂P/∂z|! HIT–MSE

哈尔滨工业大学 Summary • 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 staticmagnetic 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. HIT–MSE

哈尔滨工业大学 Thank you HIT–MSE

哈尔滨工业大学 HIT–MSE

哈尔滨工业大学 A Symplified Analytical Model • From the Momentum Transport Equation for Liquid Flow: (fLLV)/t + [(fLLV)V] = [(fLV)] (fLP)  (fL2/K)V + fLLg +fL{JGB + [(VB)BB2V]}(8)-(10) For the present upwards directional solidification case under a strong transverse static magnetic field, the Momentum Transfer Equation reduces to: (fLLVZ)/t + [(fLLVZ)VZ]/z = [ (fLVZ)/z]/z (fLP)/z  (fL2/K)VZ+ fLLg |B|2VZ  0  0  0 ↗↗↗ 0 ↗ In this case, VZ→0, and for the bulk liquid region theMomentum Transfer Equation further reduces to: (fLP)/z  fLLg  |B|2VZ HIT–MSE

哈尔滨工业大学 Electromagnetic solidification model • A 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 solid-liquid interfaces 5) Newtonian and laminar liquid flow present; 6) the model alloy is a binary system. HIT–MSE

哈尔滨工业大学 • Maxwells 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+fLVB)(Ohm Law for the Moving Metallic Melt) (17) HIT–MSE

哈尔滨工业大学 EM-Induced Joule Heat Fluid Flow Field Electromagnetic Field Heat and Mass Transfer S/L Interface Morphology Structure and Properties Lorentz Force Gravity • 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. EM-Induced Joule Heat Lorentz Force HIT–MSE

哈尔滨工业大学 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) = 2.310-2 [W/mmC] • Solid specific heat:  0.67832 – 6.432810-6T + 2.5417 10-13T3 , (25C < T  660C) cPS(T) =  0.68286 + 1.364410-5T , (660C < T  882C) [J/gC]  0.65193 + 1.051310-5T , (882C < T  1490C) • Liquid specific heat: cPL(T,CL) = 0.86074 , (1490C < T  2727C) [J/gC] • Solid density of the alloy:S(T,CL) = 3.810-3 , [g/mm3] • Liquid density of the alloy: L(T,CL) = 3.63210-3 – 2.010-7(T – T liq.) – 9.3210-6(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 + 23.01678CL – 0.20279CL2 , (CL[54.86, 74.61at.%Al]) [C] • Partition coefficient: k(CL/Al) = 69.24438-4.03153CL+8.89410-2CL2-8.7003110-4CL3 + 3.1859510-6CL4 (CL[54.86, 74.61at.%Al]) [-] • Liquid diffusion coefficient: DL/Al(TC) = 510-3[mm2/s] • Solid diffusion coefficient:DS/Al(TC) = 0.11exp[-14504/(T+273.15)] [mm2/s] • Dynamic viscosity coefficient:(T) = 3.510-3 [Pas] (= [Ns/m2] = [g/mms])  4.010-4fL3.0 , (fL0.7088) • Permeability coefficient: K =  1.6318fL27.155 ,(0.7088fL0.99999) [mm2]  1.01018 , (fL0.99999) • Primary dendrite spacing: d1 = 1.410-1 [mm] • Secondary dendrite arm spacing: d2 = 3.010-2 [mm] HIT–MSE

哈尔滨工业大学 (b) 0 At 500 At 1000 At 2500 At 10000 At Solidification Transport Processes of a Blade-like Al-4.5%Cu Casting under Transverse Static Magnetic Fields of different Load Strengths (in At) 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. HIT–MSE

哈尔滨工业大学 (d) 0 At 500 At 1000 At 2500 At 10000 At Solidification Transport Processes of a Blade-like Al-4.5%Cu Casting under Transverse Static Magnetic Fields of different Load Strengths (in At) 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. HIT–MSE

哈尔滨工业大学 Model 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) HIT–MSE

哈尔滨工业大学

哈尔滨工业大学

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