Mathematical Model for Electroslag Remelting Process

Available online at www.sciencedirect.comScienceDirectJOURNAL OF IRON AND STEEL RESEARCH ,INTERNATIONAL 2007 ,14(5):07-12 ,30Mathematical Model for Electroslag Remelting ProcessDONG Yan-wuJIANG Zhou-hua'，LI Zheng-bang2( 1. School of Materials and Metallurgy , Northeasterm University , Shenyang 110004 , Liaoning , China ;2. Institute for Metallurgical Technology , Central Iron and Steel Research Institute , Beijing 100081 , China )Abstract :A mathematical model , including electromagnetic field equation , fluid flow equation , and temperature field equa-tion , was established for the simulation of the electroslag remelting process. The distribution of temperature field was ob-tained by solving this model. The relationship between the local solidification time and the interdendritic spacing during theingot solidification process was established , which has been regarded as a criterion for the evaluation of the quality of crystal-lization. For a crucible of 950 mm in diameter , the local solidification time is more than 1 h at the center of the ingot with thelongest interdendritic spacing , whereas it is the shortest at the edge of the ingot according to the calculated results. Themodel can be used to understand the ESR process and to predict the ingot quality.Key words : electroslag ; remelting ; mathematical model ; interdendritic spacing ; local solidification timeSymbol ListC ,C2一-Constants in K-ε model ;T一-Temperature ;Cj一-Dissipation rate constant ;-Time ,s ;cp一Speific heat ;v。一-Solidification rate of ingot ,( m' h-' );d一-Interdendritie spacing ,m ;v,，v:一-Velocity of r and z direction ,( m' h-' );G--Generation term for turbulence kinetic energy ;-Axial coordinate ,m ;g一-Cravitational acceleration ,( m' s-2 );zs ,吃一-Position of solidus and liquidus ,m ;-Magnetie field intensity ,( A. m-' );β→- -Coffcient of thermal expansion of slag ;H-- Complex amplitude of H ;-Vorticity ;- Stream function ;H, ,H。,H:一-Magnetic field intensity inr ,0 ,and z direc-K一-Kinetic energy of turbulence ;tion ,respectively ;-Dissipation rate of turbulence energy ;hsw一-Overall heat transfer coeffcient between slag and-Density ,(kg" m^ 3 );cooling water,( W. m-2. K-' );σ一Elctrical conductivity ,( 0+. m~' );hw ; 一-Heat transfer cofficient of other position ,( W. m-2-Angular frequency of current ,( rad. s-' );-Viscosity of slag ,( Pa. s~ ' );1- - Reference value of current ,A ;-Magnetic permeability ,( H m-' );J一-Current intensity ,( A. m-' );μeu一-Efctive viscosity ,( Pa s-' );J一-Complex amplitude of J ;Turbulent viscosity ;SubscriptJ,- -Complex conjugation ofJ ;-Electrode ;h ,的一-Cofficient , depending on steel grade ;k.e一Fffective thermal conductivity ,( W. m-'. K~');中国煤化工Q,--Rate of heat extraction from slag by metal droplets ;nYHCNMHG.-Radiant heat loss ,J ;-Liquld slagR-- -Radius ,m;r-- - -Radial coordinate ,m ;Foundation Item :ltem Sponsored by Weaponry Pre-Research Fund ( 514120203040QT0901 )Biography :两亦数据vu( 1978- ) , Male , Doctor Candidate ; E-mail : yanwu_ dong@ 163. com ; Revised Date : November 29 ,2006.8Jourmal of Iron and Steel Research , InternationalVol. 14The ingot structure is a main objective to be con-1 Mathematical Modeltrolled for electroslag remelting process ，which deter-mines the properties of the final steel products. A modelTo calculate the temperature fields in the system ,for describing the ingot thermal field established in pre-first , the Maxwell's equations to compute the electromag-vious study 1 was not concerned with the thermal field ofnetic force field and the turbulent Navier-Stokes equa-electrode , liquid slag , etc. However , all these thermaltions to calculate the fluid flow should be solved. Thesefields exist in the system , which should be considered inequations influence each other. The physical conceptthe model. As it is known , the electrode remeltingused in the development of the mathematical model ofprocess is a very complex system , involving electromag-the process is sketched in Fig. 1.netic fields , fluid flow phenomena , heat transfer , etc.which should be also considered in the simulation. The1.1Transport equation for magnetic fields andproper staterment of the problem requires the definition ofboundary conditionsboth the appropriate fluid flow equations and electromag-The magneto-hydrodynamic form of Maxwell's equa-netic field equations. Ultimately these equations shouldtions should be solved to calculate the electromagneticbe coupled with a heat-balance equation. In several earli-force field. The transport equation for the magnetic fielder studies--sI ，the turbulent Navier- Stokes equationstakes the fllowing form at cylindrical symmetry-245781 :and thermal transfer equations were presented through1 o( dHo). 22 H。the statement of Maxwell' s equations , which described=joauo H,( 1)r ar)r ))z2the velocity fields , electromagnetic fields , and tempera-ture fields. In present study ,a model involving electro-wherej= / -1 , and the boundary conditions for Eqn.magnetic fields , fluid flow , and thermal fields was estab-( 1 ) are expressed using the following physical con-lished and temperature fields in the system was particu-straints :larly emphasized as evaluation index of the model accu-aH=0(0≤r≤R。,z=0);racy. Ballantyne' 61 showed that the ingot structure is)zcontrolled by the local solidification time( LST ) ratherthan by the pool profile. LST can never be obtained byH=2πR.(r=R。,0≤z≤z );measurement , but it can be achieved by calculation.1However ,till now ,there is less model that can be used( R.≤r≤R ,z=Z );2πrto calculate the LST. Moreover , LST in ingot can beused to calculate the interdendritic spacing , which is a0= 2πR,(r=Rm ,么≤z≤z);criterion to assess an ingot quality. The purpose of thestudy is to establish predictive relationships among cur-1(aH。Ho)_1(0H。H。rent input , pool profiles , LST , and interdendritic spac-σ。dring for ESR process.( r=R。,2