Numerical prediction of the incremental melting and solidification process

Journal of University of Science and Technology BeijingMaterialsVolume 10, Number 4, August 2003, Page 50Numerical prediction of the incremental melting and solidification processJun Wang, Chengchang Jia, and Sheng YinMaterials Science and Engineering School, University of Science and Technology Beijing, Beiing 100083, ChinaAbstract: A mathematical formulation is applied to represent the phenomena in the incremental melting and solidification process(IMSP)，and the temperature and electromagnetic fields and the depth of steel liquid phase are calculated by a finite difference tech-alues are in good agreement with theobservations. Inaccordance with the calculated values for different kinds of materials and different size of molds, the technological parameter of theIMS process such as the power supply and the descending speed rate can be determined.Key words: incremental melting and solidification process (IMSP); numerical prediction; electromagnetic field; temperature field;technological parameter1 Introductionfield inside the mold, which cause the induction cur-rent in the raw materials. The equations governing theIMSP (incremental melting and solidification proc-electromagnetic fields in the raw materials are Max-ess) is a new field of materials process. It importswell's equations [6], which ignore the electric fluxsome merits of other processing methods and inter-density D because it is very small.crosses with investment casting， continuous castingaBand powder metallurgy process. In addition, it canVxH=J;VxE=-;V.B=0;B = μH.make fine metallurgy combine with preparation engi-neering of materials. Its operation principle can be de-and Ohm's law .scribed as follows: the materials are melt by the in-J= oE.duction heating in the mold and the continual granulesOwing to the fact that the exciting current is sinu-or powders of conducting materials are fed to themold. The heating zone is limited to a small part ofsoidal, the following equations can be obtained ac-mold. The incremental melting and solidificationcording to the above equations:process proceeds from the bottom to the top of theV'B = σμ2Bviz. V2B = σujoB(1mold by changing the relative position between the)tmold and the heating apparatus, which achieved theBecause the induction coil is circular and the shap-aim of shaping [1-4].IMSP can be used to produce metal matrix compo-ing body is axisymmetric in the cylindrical coordinatesystem, the magnetic flux density (B) owns the fol-sites and functionally gradient materials (FGM) withlowing properties: .continuous variation of components and properties [2,5]. The influence of the technological parameter ofaB,IMSP to the material properties had been researcheda=0; a=0; Bg=0.[3]. This paper shows a mathematical formulation ap-So the following equations in B are obtained for theplied to represent the phenomena in the incrementalr component:melting and solidification process (IMSP), and thetemperature and electromagnetic fields and the depth0B,、 10B， 2B,--B, =orjioB,(2)of steel liquid phase calculated by a finite differencer ortechnique using the control volume method.the z component2 Theoretic analysis02B。1 aB,中国煤化工(32.1 The calculation of electromagnetic fieldOr2 r OrYHCNMHGThe induction coil gives rise to a traveling magneticB, and B: in equations (2) and (3) are treated asCorresponding author: Jun Wang, E-mail: wbb34@hotmail.com.J. Wang et al, Numerical prediction of the incremental melting and solidification process51complex numbers, so the four differential equationscordance with equation (7). In addition, the magneticare used to calculate B instead of the above two equa-flux density on the symmetry axis owns the followingtions.properties.Because there is only a single-turn induction coilaBzand the induction current is distributed mainly in the :B,=0; =0.minute range of the shapingbody close to the coil, themagnetic flux density B over the boundary can be2.2 The calculation of temperature fieldcaculated through Biot-Savart law:Supposing the steel liquid is an incompressiblefluid, the equations of motion for cylindrical geometry(4)are as follows.4π J1 r3Continuity equation:where dl is the line element of induction coil, and itsdirection is the same as the current of coil; r the vector1. a(r,)+ 0V=0(8)whose direction is that dl points to the dot P whichr arneed to be solved (figure 1) and whose quantity is thedistance between the dl and the dot P; I the electricMomentum equation:current of induction coil.(a) r component:induction.2( av,8v,1 a(... ov ).P[ ota;” 8z=ror'lener))v,)]_180v,Ler .I'lefrOz小rar*" ormold.body(9r(b) θcomponent:( ovo，av。Figure 1 Diagrammatic sketch for computation.orH4er ar +I=Imeioazr'ler0v。). wo.yf.e_ AemVo_ PY.Vo + F。Gzrarr2where Im is the amplitude of the electric current.(10)In the cylindrical coordinate system, the coordinateof the dot M on the coil (figure 2) is (R,0, Z) and the(C) z component:dot P on the boundary is (p, 0, Z). According to equa-[1 av2)tion (4), the magnetic flux density B can be calculated.(8= 8z)r8refar)Radial direction (the x component):18"Her 2z2(42021。0v:)+B,=A R(Z-Z)j ^[ρ2 + R2- 2pRcos(0 - 01)+_可(11)(Z - Z1)2]-3/2 cos(θ - 0l)dθl(5)Energy equation:Axial direction (the Z component):(aTaT. k aT\.Bz= r{"[ρ2 +R2 - 2pRcos(0 - 01)+留)F品(C, 2r)+4(Z - ZI)2 ]-3/2[R - pcos(0 - θl)]d0l(6)okoT+.9Angular direction (the θ component):z(C,orJ+ C,(12)Bo=" R(Z- zl)[。" [p2 + R2 - 2pRcos(0 -01)+Since flow is assumed to be turbulent. the effective4元viscosity Per in中国煤化工ists of lami-(Z - ZI)2 ]-3/2sin(0l- 0)dOl7)nar viscosity μ;YHCNMHG.Theturbu-It is obvious that the θ component B。is zero in ac-lence model used in this study is the well known K-ε.52J. Univ. Sci. Technol. Bejjing, Vol.10, No.4, Aug 2003model [7].According to equations (5)-(7), it's obvious that theK equation:following equations are right.「0K. 6(v,K)、 8(v2K)]_ | 1 8(. μer 8KB,=0, oB._B.)θ a0at)r2:_ror(' σK OrSo, J can be witten as follows:0(.leraKPV,ε.a'σx的z+G- pε(13)1 ,B，_ aB.J:=0;J,=O; Jo=-(-oε equation:In accordance with the Joule-Lenz's law, the input「Oε、 6(v,&)， 8(v2e)]_ | 1 .(. Hloitpower (q) shows as follows:ar0r(σ。aq=J2/σ.a(. Ler BεPV.ε + εThe boundary for temperature field can be treatedaz(' σ。0zPr-+ =(CG-C2P) (14)as heat convection and radiation through the methodof additional source term. The latent heat can be treat-whereed through the method of temperature recovery [8].G=μ(台)-(倍)(](倍.az3 Computation procedureThe electromagnetic, flow and temperature fields(15)were calculated by a finite difference technique usingμer= u+ μ(16)the control volume method. These three fields' calcu-从=CpK2(17)lations were performed using the code written by theεauthors. A simplified flow sheet of the computationalThe values of the constants used in equations (13)-procedure is shown in figure 2.(17) are C=0.09, C=1.44, C=1.92, σ=1.0, σ= 1.3( Beg[7].The boundary conditions may be written as follows.Input Source data(parameter on shaping,(i) Symmetry axis:mold, power and material)vo=0ov,_ Ovaθ~ aθDividelattices(i) Free surface:ovo_ av,=0ectromagnatic feld Iv_= 0;Dz 8zon boundarv,(ii) Wall:Compute theelectromagnetic ficld,v,=0;vo=0; v=0.the current densityIncreasing the calculatingelectromagnetic forcgSince the electromagnetic field is varying sinusoi-length according to thedescending rate of molddal with time, it is necessary to represent the electro-magnetic force in equations (9), (10) and (11) in atemperature fieldtime averaged form. The components of the time av-eraged electromagnetic force are given byCompute the \flow andemperatuure fieldF,=-Re(JoB:'-J.B)，Fo=;Re(J.B; -J,B?),F:=- Re(J,B; -JoB;).No_height satisfied?where Re is the complex real part and the asterisk isthe conjugate complex. The induction current density中国煤化工J can be calculated by the Maxwell's equation( ProcessMYHCNMHGVxH=J,B= μH.Figure 2 Flow sheet of the computational procedure..J. Wang et al, Numerical prediction of the incremental melting and solidification process534 Computed resultsdescending speed rate of 5 mm/min, which closely re-sembles the computed result.4.1 Temperature fieldThe shaping body materials is 65Mn steel and themold materials is Al2O3. The inner diameter of the30 tmold is 25 mm and the thickness of the mold is 2.5mm. The heat capacity at constant pressure, the den-昌20Computed resultssity and the thermal conductivity of shaping body(V=5 mm/min)materials vary according to the temperature, which are10 t-- Computed results(V=i0 mm/min)shown in reference [9]. Figure 3 is the computed tem-士. Experiment resultsperature on the given conditions even as the movingdistance of induction coil is 40 mm.10 12 14P/kWsymmetry axisFigure 4 The height of melting (h) vs. the power supply (P)40a)2028(V.- -the descending speed rate). .32182819782078**Figure 5 is the height of melting vs. the descending1778昌24十speed rate. According to this figure, the height ofmelting reduces with the descending speed rate in-心1(creasing, and the melting procedure stops when the1678descending speed rate reaches a certain value at a1628 15781428given power supply.15281478_ : 1378 .0 2.50 5.00 7.50 10.00 12.50 15.0040「- - Computedr /mm士Experiment! symmetry axiswall40[(6E2032 1778_ 18280t唱24-172812市* 16 20“1-4678--.1628v。/ (mmmin~)15781528-1228 ，Figure 5 The height of melting (h) vs. the descending148128283723 I1178speed rate (V) ( the power supply is7 kW).2.50 5.00 7.50 10.00 12.50 15.005 Conclusions .field (temperature unit: K; powerA mathematical formulation is applied to representsupply: 7 kW), descending speed rate: (a) 5; (b) 10 mm/min.the phenomena in the incremental melting and solidi-4.2 The comparison of computed results with ex-fication process (IMSP), and the temperature andperimentelectromagnetic fields and the depth of steel liquidFigure 4 is the height of melting on the given pow-phase are calculated by a finite difference techniqueer supply. With the power supply increasing, theusing the control volume method. The result showsheight of melting increases and tends to a definitethat the predicted values are in good agreement withvalue. When the power supply is 15 kW, the height ofthe observations. In accordance with the calculatedmelting reaches the definite value of 30 mm, and atvalues for different kinds of materials and differentthis time the descending speed rate rarely affects thesize of molds, the technological parameter of the IMSheight of melting. In order to melt the raw materials atprocess such as the power supply and the descendingthe given value of descending speed rate, the powerspeed rate can be determined.supply must reach a certain value. And with the de-Referencesscending speed rate increasing, the power supply val-中国煤化工ues must increase correspondingly. The dotted line[1] c.C. Jia, Inc[R], Postdoctor arroscaui Npui al uiliversity of Science.MHCN M H Gation Processwith the triangle mark is the experiment result at the.54J. Univ. Sci. Technol. Bejjing, Vol.10, No.4, Aug 2003and Technology Beijing, 1996.μ: The magnetic permeability, Hm-';[2] C.C. Jia, T. Lin, Z.M. Guo, H.B. Tian, and Nobuyoshi Sa-σ. The electric conductivity, S:m-';saki, Making gradient materials by the incremental melt-ing and solidification process [], ACTA Metall. Sin.t: The time, s; .35( 199), No.2, p.190.[3] C.C. Jia, Z.M. Guo, T Lin, and H.B. Tian, A research onJ: The unit of imaginary number;the parameters in incremental melting and solidificationo: The angular frequency, rads |;process [], ACTA Metall. Sin, 35(1999), No.2, p.187.[4] C.C. Jia and Nobuyoshi Sasaki, Incremental melting anI: The electric current of induction coil, A;solidification molding process [], J. Univ. Sci. Technol.Bejing, 18(1996), No.5, p.436.Im: The amplitude of the current, A;[5] J. Wang, C.C. Jia, and S. Yin, Making metal matrix com-T: The temperature, K;posites and functionally gradient materials by the incre-mental melting and solidification process, [in] 13 Inter-J: The induction current density, A.m ;national Conference on Composite Materials [C], (2001),p.225.v: The velocity, m-s |;[6] Nathan Ida and Joao P. A. Bastos, Electromagnetics andp: The density, kg.m 3;Calculation of Fields [M], Springer-Verlag, New York,Berlin Heidelberg, 1997.P: The pressure, Pa;[7] W.Q. Tao, Numerical Heat Transfer [M], Xi an Jiaotongr: The radial cylindrical coordinate;university, Xi'an, 1995.[8] Z. Liu, Z.J. Wu, J.Z. Wu, and Y. Zhang, Numerical Simu-z: The axial cylindrical coordinate;lation of Heat Treatment Process [M], Science publishingcompany, Beijing, 1996.0. The azimuthal cylindrical coordinate;[9] Q. Yang and Z. Zhang, Numerical Simulation of Metalq: The input power, W;Solidification and Casting Process [M], Zhejiang Univer-sity, Hangzhou, China, 1996.V: Hamiton operator;NomenclatureK: The turbulent kinetic energy, m2.s 2;ε. The turbulent kinetic energy dissipation rate, m2.B: The magnetic flux density, T;H: The magnetic field intensity, A.m ); .Note: The subscript r denotes the radial componentE: The electric field intensity, V.m-;of the vector, and z denotes the axial component, andθ denotes the azimuthal component.D: The electric flux density, C.m-2;中国煤化工MHCNMH G.