Electroslag remelting (ESR) is duly an important process for the production of high quality special steels and superalloys. Conventional ESR research has long been known as trial and error approach, which is excessively expensive and timeconsuming, due to the complex process mechanism involving interactions of multiple physical fields, simultaneous phase transformations and chemical reactions. As the alternative way of study, ESR numerical simulation has been profoundly developed. Till now, systematically formulated model could demonstrate so many aspects of the process including electromagnetic field, fluid flow, heat and mass transfer, electrode melting, ingot solidification, slag/metal interface phenomenon, solidification structure parameters, ingot elements distribution, etc. There is a trend of multiscale combined simulation, trying to bridge the gap between macro and microscopes, thus could realize the control of solidified structure. Numerical modeling and simulation of ESR process have been widely accepted for its superiority of low cost, high speed, flexible adaptability and systematic results. Through combination of simulation and experiment, the ESR R&D process can be significantly promoted. Further, with the newly developed control technology supported by theoretical models, high precision and perfect quality control are expected to achieve. In this work, a mathematical model and the calculating code for the simulation of practical ESR process were developed based on multiphysics coupling calculation. The model considers many features of the process including the heat of the dropping liquid metal from the electrode, the naturally formed melt pool and the growing of the ingot, the cooling shrinkage of the solidified ingot away from the mould boundary, the changed slag skull thickness along the ingot growing direction, the matching between melt rate and input melting parameters, the specific boundary conditions, etc. The model covers physics of electromagnetic field, fluid flow, heat transfer, and melting and solidification during the remelting process, giving the characteristic information about distributions of temperature and liquid phase volume fraction, shape and size of melt pool and mushy zone, etc. highly concerned with the process control. Using history of temperature distributions and evolution, the model can compute various solidification parameters closely related to the ingot quality. The model realizes predictions for the unknown ESR process with steady state mode calculation and also analysis in transient mode of the whole ESR process from the melting start point of electrode to the end of cooling stage of ingot within the mould. Electromagnetic fields and steady and transient process simulations were carried out and discussed here for the practical IN718 alloy ESR process. The simulated melt pool profile and its depth size approximate to the experimental result of the ingot dissection analysis, and the predicted secondary dendrite spacing distribution coincides with the pictures of dendrite structure analysis fairly well. The model could be applied to the process analysis and optimization, and provide important technical support for the R&D of new product and technology.
Qing LI,Zixing WANG,Shuyuan XIE. Research on the Development of Mathematical Model of the Whole Process of Electroslag Remelting and the Process Simulation. Acta Metall Sin, 2017, 53(4): 494504.
Fig.1 Schematic of geometric domain and boundaries for electromagnetic field calculation
No.
A˙x
A˙r
?˙
①
?A˙x?x=0
0
?˙0
②


Medium boundary condition
③


Medium boundary condition
④


Medium boundary condition
⑤


Medium boundary condition
⑥


Medium boundary condition
⑦


Medium boundary condition
⑧
0
0

⑨
?A˙x?x=0
0
0
⑩
?A˙x?r=0
?A˙r?r=0
??˙?r=0
Table 1 Boundary conditions of equations for electromagnetic field calculation shown in Fig.1
Fig.2 Schematic of geometric domain and boundaries for fluid velocity and temperature distribution calculations (u and v are axial and radial velocities, respectively)
No.
u
v
k
ε
T
①
u=0
v=0
?k?x=0
Boundary node value calculated with k
Tm*
②
u=0
v=0
?k?r=0
Boundary node value calculated with k
Tm*
③
u=0
?v?x=0
?k?x=0
?ε?x=0
Radiation boundary
④
u=0
v=0
?k?r=0
Boundary node value calculated with k
Ts*
⑤
u=0




⑥
u=0
v=0
?k?x=0
Boundary node value calculated with k

⑦




Tb,1
⑧




Tb,2
⑨
?u?r=0
v=0
?k?r=0
?ε?r=0
?T?r=0
Table 2 Boundary conditions of transport equation calculations shown in Fig.2
Fig.3 Simulated distributions of magnetic intensity amplitude (H) and current density (J)
Fig.4 Simulated distributions of heat (P) and electromagnetic force (F)
Fig.5 Simulated distributions of fluid velocity (v), temperature (T) and liquid volume fraction (f_{l}) under different melt rates (w_{0}—the critical melt rate below which no melt pool formed. Each picture is composed of two parts, the right half represents the calculation domain and the left half symmetric with the right, for the axisymmetrical formulation of the model. The left part is used for temperature distribution and the right for fluid velocity and liquid volume fraction) (a) w_{0} (b) 1.3w_{0} (c) 1.7w_{0} (d) 2.8w_{0} (e) 3.5w_{0} (f) 4w_{0} (g) 5.3w_{0} (h) 7w_{0} (i) 8.6w_{0}
Fig.6 Variations of simulated melt pool depth, mushy zone distance, and maximum temperature in slag pool corresponding to different melting speeds (w′—the ratio of melt rate to the critical value of w_{0}, h_{p}—melt pool depth, h_{m}—mushy zone distance, T_{M}—maximum temperature in slag pool )
Fig.7 Model results of changed distributions of v (the upper right of each picture), T (the left part) and f_{l} (the right part) during the whole process in practical test situations
Fig.8 Variations of simulated h_{p}, h_{m} and T_{M} in slag pool in slag pool with the increased ingot height (H_{ing}) (The arrow points to the specific height of ingot where melt pool depth was measured)
Fig.9 Analysis photo of melt pool shape on the ingot axial section
Fig.10 Model results of distributions of secondary dendrite arm spacing (SDAS) in ingot (The arrow points to the position where dentrite morphologies were analyzed)
Fig.11 Morphologies of dendrite structure analysis on the ingot axial section
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