Finite Element Multi-scale Modeling of Chemical Segregation in Steel Solidification Taking into Account the Transport of Equiaxed Grains

Abstract

The transport of solid crystals in the liquid pool during solidification of large ingots is known to have a significant effect on their final grain structure and macrosegregation. Numerical modeling of the associated physics is challenging since complex and strong interactions between heat and mass transfer at the microscopic and macroscopic scales must be taken into account. The paper presents a finite element multi-scale solidification model coupling nucleation, growth, and solute diffusion at the microscopic scale, represented by a single unique grain, while also including transport of the liquid and solid phases at the macroscopic scale of the ingots. The numerical resolution is based on a splitting method which sequentially describes the evolution and interaction of quantities into a transport and a growth stage. This splitting method reduces the non-linear complexity of the set of equations and is, for the first time, implemented using the finite element method. This is possible due to the introduction of an artificial diffusion in all conservation equations solved by the finite element method. Simulations with and without grain transport are compared to demonstrate the impact of solid phase transport on the solidification process as well as the formation of macrosegregation in a binary alloy (Sn-5 wt pct Pb). The model is also applied to the solidification of the binary alloy Fe-0.36 wt pct C in a domain representative of a 3.3-ton steel ingot.

Appendices

Appendix A: Nomenclature and notations

b	Body force
C d	Drag coefficient
c p	Specific heat
D	Diffusion coefficient
d g	Grain diameter
D M	Artificial diffusion coefficient
g	Gravity vector
g	Phase fraction
$g_{c}^{\text{s}}$	Packing solid fraction
h	Enthalpy per unit mass
$h_{e}^{{{\mathbf{v}}^{\text{s}} }}$	Characteristic mesh size of an element e in direction of velocity vs
j	Solute flux vector
$J_{ }^{\varGamma }$	Interfacial solute transfer due to phase change
$J_{ }^{j}$	Interfacial solute transfer due to diffusion
$J_{ }^{\varPhi }$	Interfacial solute transfer due to nucleation
k p	Partition coefficient
l	Heat conduction length
L f	Latent heat of fusion
${\mathbf{M}}_{ }^{d}$	Interfacial momentum transfer due to interfacial stress
${\mathbf{M}}_{ }^{\varGamma }$	Interfacial momentum transfer due to phase change
${\mathbf{M}}_{ }^{\varPhi }$	Interfacial momentum transfer due to nucleation
n	Number of micro-time steps over a macro-time step
n	Unit vector normal to the liquid–solid interface
N	Grain density
$\dot{N}$	Generation rate of grain density
$p$	Pressure
q	Heat flux vector
$Q_{ }^{\varGamma }$	Interfacial heat transfer due to phase change
$Q_{ }^{j}$	Interfacial heat transfer due to diffusion
$Q_{ }^{\varPhi }$	Interfacial heat transfer due to nucleation
R	Resistance coefficient
R g	Grain radius
S v	Interfacial area concentration
$T$	Temperature
$t$	Time
δt	Micro time step
$\Delta t$	Macro time step
T ext	Exterior temperature
v	Growth velocity of grains
v	Velocity vector
v center	Velocity at the center of an element
w	Solute mass concentration
α	First constant parameter of the artificial diffusion coefficient
α t	Transition function
β	Second constant parameter of the artificial diffusion coefficient
β shr	Shrinkage coefficient
β T	Thermal expansion coefficient
β w	Solutal expansion coefficient
δ	Solute diffusion length
Γ	Rate of exchanged mass due to phase change
κ	Thermal conductivity
λ 2	Characteristic length for permeability
μ	Dynamic viscosity
φ i	Interpolation function associated with node i
ρ	Mass density
τ	Deviatoric stress tensor
Φ	Rate of transferred mass due to grain nucleation
$\nu$	Iteration
Subscripts
gr	Growth
$i, j$	Indexes of nodes
nucl	Nucleation
packed	Packed-bed regime
regime	Flux regime
ref	Reference
slurry	Slurry regime
tr	Transport
proj	Projection
modif	Modification
0	Initial state
Superscripts
*	Interface
B	Buoyancy
T	Transpose
l	Liquid phase
m	Mixture
s	Solid phase
α	Phase α
$\nu$	Iteration
Supplementary symbols
$\langle \rangle$	Volume average over all phases
$\left\langle {^{\alpha } } \right\rangle$	Volume average in phase α
$\left\langle {^{\alpha } } \right\rangle ^{\alpha }$	Intrinsic volume average in phase α
⊗	Tensor product
∇	Gradient operator
∇ ·	Divergence operator
$\overline{{}}$	Averaging operator
Nn	Number of nodes
Re	Reynolds number
tanh	hyperbolic tangent
$\| \|$	Magnitude of a vector

Appendix B: Solute Diffusion Lengths and Area Concentration

The solute diffusion lengths are taken from the work of Tveito et al.,[41] as the following formulations.

2.1 Solute Diffusion Length in the Liquid Phase

$$ \delta^{\text{l}} = \frac{{w^{\text{l*}} - \langle w^{\text{l}} \rangle^{\text{l}} }}{{ - \left. {\frac{{\partial w^{\text{l}} }}{{\partial {\mathbf{n}}}}} \right|^{*} }} $$

(B1)

$$ = \frac{d}{{\frac{d}{{R_{g} }} - \frac{{f\left( {R_{g} ,\Delta } \right) + g\left( {R_{f} ,R_{g} ,\Delta } \right)}}{{d\left[ {R_{g} + d - \left( {R_{g} + \Delta + d} \right)e^{ - \Delta /d} } \right] - f\left( {R_{g} ,\Delta } \right) + g\left( {R_{f} ,R_{g} ,\Delta } \right)\left( {e^{ - \Delta /d} - 1} \right)}}}}, $$

(B2)

where

$$ d = \frac{{D^{\text{l}} }}{v}\quad f\left( {R_{g} ,\Delta } \right) = \frac{{\left( {R_{g} + \Delta } \right)^{2} - R_{g}^{2} }}{2} $$

(B3)

$$ g\left( {R_{f} ,R_{g} ,\Delta } \right) = \frac{{R_{f}^{3} - \left( {R_{g} + \Delta } \right)^{3} }}{{3\left( {R_{g} + \Delta } \right)}}\quad \Delta = \hbox{min} \left( {R_{f} - R_{g} ; \frac{{2R_{g} }}{{{\text{Sh}}_{\text{conv}} }}} \right) $$

(B4)

$$ {\text{Sh}}_{\text{conv}} = \frac{2}{{3g^{\text{l}} }}{\text{Sc}}^{1/3} {\text{Re}}^{{n\left( {\text{Re} } \right)}} \quad {\text{Sc}} = \frac{{\mu^{\text{l}} }}{{\rho^{\text{l}} D^{\text{l}} }} $$

(B5)

$$ {\text{Re}} = \frac{{g^{\text{l}} 2R_{g} \left\| {\left\langle {{\mathbf{v}}^{\text{l}} } \right\rangle^{\text{l}} - \left\langle {{\mathbf{v}}^{\text{s}} } \right\rangle^{\text{s}} } \right\|}}{\nu }\quad n\left( {\text{Re} } \right) = \frac{{2\text{Re}^{0.28} + 4.65}}{{3\left( {\text{Re}^{0.28} + 4.65} \right)}}. $$

(B6)

2.2 Solute Diffusion Length in the Solid Phase

$$ \delta^{\text{s}} = \frac{{R_{g} }}{5} $$

(B7)

The area concentration is calculated as follows:

$$ S_{\text{v}} = 4\pi \left( {R_{g} } \right)^{2} N. $$

(B8)

Appendix C: Thermophysical Data

See Table C1.

Table C1 Thermophysical Data of Sn-5 Wt Pct Pb Alloy[8]

Appendix D: Analytical Solution for the 1D Test Case

The 1D Test Case consists in pure sedimentation of a column of preexisting globular grains with fixed size in a uniform temperature domain. Considering constant and equal densities of the solid and liquid phases, as well as no phase change and no nucleation, the average total mass conservation simplifies to $g^{\text{s}} \left\langle {\varvec{v}^{\text{s}} } \right\rangle^{\text{s}} + g^{\text{l}} \left\langle {\varvec{v}^{\text{l}} } \right\rangle^{\text{l}} = 0$. For the sake of simplicity, a constant settling value of the solid velocity is imposed, set to $\left\langle {\varvec{v}^{\text{s}} } \right\rangle_{0}^{\text{s}} = - 1\,{\text{mm}}\,{\text{s}}^{ - 1}$. The 1D domain height and the initial conditions are defined in Figure 4: a continuous and uniform 60-mm mushy zone region is initially present between heights 20 mm and 80 mm, with a uniform average grain density per unit volume, $N_{0} = 10^{9} \,{\text{grains}}\,{\text{m}}^{3}$, and volume fraction of solid, $g_{0}^{\text{s}} = 0.1$. One can easily derive the value for the liquid velocity in the mushy zone, $\left\langle {\varvec{v}^{\text{l}} } \right\rangle^{\text{l}} = - g_{0}^{\text{s}} \left\langle {\varvec{v}^{\text{s}} } \right\rangle_{0}^{\text{s}} /\left( {1 - g_{0}^{\text{s}} } \right) = 0.11\,{\text{mm}}\,{\text{s}}^{ - 1}$. Similarly, the radius of the grains, R_g,0, is simply given by using the definition of the fraction of solid, $g_{0}^{\text{s}} = N_{0} \left( {4/3} \right)\pi R_{g,0}^{3}$, leading to the value R_g,0 = 0.288 mm. Considering the fixed settling velocity and the packing limit at which the grain stop, g ^s _c  = 0.3, the time evolution of the distribution of the mushy zone is simply derived by considering that the total fraction of the solid phase is unchanged over the entire domain, while not exceeding g ^s _c in the packed bed. Values are reported in Table D1. The temperature is fixed to 498 K (224.856 °C), i.e., below the liquidus temperature of the Sn-5 wt pct Pb alloy, that is 498.72 K (225.57 °C) according to the thermophysical properties listed in Table C1 of Appendix C.[8] The average solute mass composition is defined by $w = g^{\text{s}} \left\langle {w^{\text{s}} } \right\rangle^{\text{s}} + g^{\text{l}} \left\langle {w^{\text{l}} } \right\rangle^{\text{l}}$. At any time, as the system is closed with respect to mass transfer, integration over the entire domain must retrieve the nominal composition of the alloy, $w_{0} = 5 {\text{wt}}\;{\text{pct}}\;{\text{Pb}}$. The initial composition profile assumes no macrosegregation. This means that the average composition is equal to w₀ at any position along the domain. However, assuming complete mixing in both liquid and solid phases, the lever rule holds and one can derive the equilibrium intrinsic composition of the liquid and solid phases, $\left\langle {w^{\text{l}} } \right\rangle^{\text{l}} = 5.556 {\text{wt}}\;{\text{pct}}\;{\text{Pb}}$ and $\left\langle { w^{\text{s}} } \right\rangle^{\text{s}} = 0.364\;{\text{wt}}\;{\text{pct}}\;{\text{Pb}}$, respectively. Knowing the distribution of solid and liquid and their initial and intrinsic compositions, one can directly compute the average compositions by tracking the change of phases due to sedimentation. Computed values are reported in Table D1.

Table D1 Time Evolution of the Distribution of the Solid Phase Along with 1D Simulation Domain (Dashed Lines in Figs. 5 and 6)