Prediction of Macrosegregation of Fe-C Peritectic Alloy Ingot Through Coupling with Thermodynamic Phase Transformation Path
FENG Miaomiao1,2, ZHANG Hongwei1,2(), SHAO Jingxia1,2, LI Tie1,2, LEI Hong1,2, WANG Qiang1,2
1.Key Laboratory of Electromagnetic Processing of Materials, Ministry of Education, Northeastern University, Shenyang 110819, China 2.School of Metallurgy, Northeastern University, Shenyang 110819, China
Cite this article:
FENG Miaomiao, ZHANG Hongwei, SHAO Jingxia, LI Tie, LEI Hong, WANG Qiang. Prediction of Macrosegregation of Fe-C Peritectic Alloy Ingot Through Coupling with Thermodynamic Phase Transformation Path. Acta Metall Sin, 2021, 57(8): 1057-1072.
The phase transformation path is vital to enclosing the macroscopic transport equations for predicting alloy macrosegregation. However, the analytical approximations for micro-segregation, such as the lever rule (LR), are invalid because an actual alloy is a multi-component system with several coexisting solids. The LR only expresses the phase transformation between a single solid phase and a liquid phase and adopts a constant solute partition coefficient, which is insufficient for micro-segregation. In this study, a model combining the thermodynamic phase transformation path calculation with the macroscopic transport was adopted to predict the macrosegregation formation in an Fe-0.1%C (mass fraction) peritectic alloy, which considers the coexistence of multi-solids and the variance of the local partition coefficient at the solid/liquid and solid/solid phase interface with a solidification process. The phase transformation path from the liquid state cooling to room temperature within a certain range of the solute concentrations was obtained using the LR approximation combined with the thermodynamic equilibrium calculation (LR-TEC). By tabulating the phase transformation path and interpolating the local concentration and enthalpy, the corresponding temperature, phase fraction, phase concentration, and phase enthalpy required in the continuum macroscopic transport model were achieved. The latent heat released and the specific heat corresponding to the amounts of the two solid phases at the peritectic or eutectic phase transformation zone were updated along with their dependence on the local concentration and temperature. This method was validated through the benchmark macrosegregation test of the binary Sn-5%Pb alloy. Regarding the Fe-0.1%C alloy, the varied local partition coefficients and the other thermodynamic parameters with multi-solids precipitating during solidification resulted in a more severe macrosegregation profile in the ingot. At the end of the solidification calculation, the predicted minimum relative solute concentration for the Fe-0.1%C alloy was -2.22% at y = 45 mm from the bottom and x = 16 mm from the left wall of the ingot by LR-TEC. In contrast, it was -1.78% using the LR Analytical model near y = 55 mm at the left-side wall. The predicted maximum macrosegregation ratio at the right wall of the ingot by LR-TEC was 1.13% larger than that achieved using the LR Analytical model. Several solids, such as α and γ, α and cementite (CEM), or α, γ, and CEM at the left part (x < 0.0342 m), and δ and γ at the right (x > 0.0858 m), still coexisted in the region at the end of solidification calculation.
Fund: National Natural Science Foundation of China(51574074);National Natural Science Foundation of China and Shanghai Baosteel(U1460108);Natural Science Foundation of Liaoning Province(L20150183)
Fig.1 Phase transformation paths of Fe-0.1%C alloy predicted by LR-TEC (a) and LR Analytical (b) models (LR—lever rule, TEC—thermodynamic equilibrium calculation, CEM—cementite, L— liquid, S—solid, T—temperature)
Parameter description
Symbol
Unit
Sn-5%Pb
Fe-0.1%C
Initial concentration
c0
%
5.0
0.1
Initial temperature
T0
oC
226
1550
Pure solvent melting temperature
Tm
oC
232
1538
Partition coefficient
kp
-
0.0656[5]
0.2[8]
Liquidus temperature
Tliq
oC
224.86
1530.12
Liquidus slope
m
oC·%-1
-1.286[5]
-80.579[8]
Eutectic/peritectic temperature
Te
oC
181.41
1494.63
Liquid phase concentration at eutectic/peritectic point
ce
%
38.1
0.53
Solid phase concentration at eutectic/peritectic point
ces
%
2.2
0.09
Thermal expansion coefficient
βT
oC-1
6 × 10-5[5]
1 × 10-4[8]
Solute expansion coefficient
βs
%-1
-5.3 × 10-3[5]
4 × 10-5[8]
Reference concentration
cref
%
5.0
0.1
Reference temperature
Tref
oC
226
1550
Ambient temperature
Text
oC
25
25
Liquid density
ρ
kg·m-3
7000[5]
7020[8]
Dynamic viscosity
μl
Pa·s
1.0 × 10-3[5]
6.2 × 10-3[8]
Latent heat of solidification
L
J·kg-1
61000[5]
270000[8]
Specific heat capacity at constant pressure
cp
J·kg-1·K-1
260[5]
680[8]
Thermal conductivity of alloy
λ
W·m-1·K-1
55[5]
34[8]
Liquid diffusion coefficient
Dl
m2·s-1
1 × 10-8[5]
1 × 10-8[8]
Secondary dendrite arm spacing
λ2
μm
65
50
Heat transfer coefficient
α
W·m-2·K-1
300
800
Time step
Δt
s
0.01
0.05
Table1 The alloy physical properties used in the calculation
Fig.2 Variances of enthalpy (h), release of latent heat (ΔL) and cp of Sn-5%Pb alloy with T
Fig.3 Illustrated linear interpolations for temperature via h and concentration (c) in tabulated phase transformation path
Condition
T
fl
h > hliq (Liquid zone)
T = (h - L) / cp
fl = 1
he < h ≤ hliq (Mushy zone)
T = (h - Lfl) / cp
hsol < h ≤ he (Mushy zone)
T = Te
fl = (h - cpTe) / L
h ≤ hsol (Solid zone)
T = h / cp
fl = 0
Table 2 Relations between h, T and mass fraction of liquid (fl)
Fig.4 Dependences of kp on temperature in Fe-0.1%C alloy
Fig.5 Phase transformation paths of Sn-5%Pb alloy predicted by LR-TEC and LR Analytical models
Fig.6 Dependences of kp on temperature in Sn-5%Pb alloy predicted by LR-TEC and LR Analytical models
Fig.7 Mass fraction of liquid and velocity field solidified at 400 s for Sn-5%Pb alloy predicted by LR-TEC (|V|max = 0.79 mm/s) (a) and LR Analytical (|V|max = 0.67 mm/s) (b) models (|V|max—maximum absolute value of velocity in the flow field)
Fig.8 Distributions of relative solute concentration [(c - c0) / c0] × 100 for solute Pb solidified at 400 s for Sn-5%Pb alloy predicted by LR-TEC (a) and LR Analytical (b) models
Fig.9 Distributions of [(c - c0) / c0] × 100 for solute Pb at the end of solidification calculation for Sn-5%Pb alloy predicted by LR-TEC (t = 991.82 s) (a) and LR Analytical (t = 1202.37 s) (b) models
Fig.10 Distributions of [(c - c0) / c0] × 100 for solute Pb on the plane at heights of y = 5 mm (a), y = 25 mm (b), y = 35 mm (c), and y = 55 mm (d) from bottom of the ingot at end of solidification for Sn-5%Pb alloy
Fig.11 Phase distributions in 2D ingot section during cooling process of Fe-0.1%C alloy predicted by LR-TEC model (Calculation finishing at 570.2 s)
Fig.12 Distributions of predicted [(c - c0) / c0] × 100 for solute C during solidification of Fe-0.1%C alloy ingot
Fig.13 Influences of storage mode of temperature in tabulation on distributions of [(c - c0) / c0] × 100 for solute C at cross section of the Fe-0.1%C alloy ingot (Mode 1—intitial temperature Tini = Tliq (int type) + overheat 30oC, temperature step ΔT = -1oC; Mode 2—Tini = Tliq (real type) + overheat 30oC, ΔT = -1oC; Mode 3—Tini = Tliq (int) + overheat 30oC, ΔT = -1oC, with L/δ, L + δ/γ, γ/α, γ + α/CEM phase transformation points in tabulation; Mode 4—Tini = Tliq (real) + overheat 30oC, with temperature reset at each phase transformation point (real type) while keeping ΔT = -1oC in tabulation)
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