A Generalized Approach for Macroscopic Modelling of Solidification of Ternary Alloys with Dendritic Arm Coarsening

Abstract

A generalized macroscopic two-dimensional mathematical model has been developed to simulate the transport phenomena occurring during the solidification of ternary alloy systems. The model is based on a fixed-grid enthalpy based control volume approach and is capable of capturing the effects of dendritic arm coarsening on the transport characteristics of the solidification process. Microscopic features pertaining to complex thermo-solutal transport mechanisms and dendritic arm coarsening are numerically modelled through a novel formulation of latent enthalpy evolution, consistent with the phase change morphology of general multi-component alloy systems. Numerical simulations are performed for ternary steel alloy and the resulting convection and macrosegregation patterns are analyzed. It is observed that dendritic arm coarsening leads to an increased effective permeability of the mushy region resulting in an enhanced macrosegregation. The numerical results are also tested against experimental results reported in the literature, and very good agreement is found in this regard.

1. Introduction

In recent years, significant efforts have been devoted towards development of macroscopic models to study transport phenomena occurring during the phase change process pertaining to the solidification of multicomponent alloys.^1,2,3,4) There exists various techniques for macroscopic modeling of alloy solidification,^5,6,7) however the most popular method is ‘enthalpy-porosity’ approach.⁵⁾ This method eliminates the need of explicit tracking the solidification fronts, by modelling the entire macro-domain with a single set of volume-averaged continuum conservation equations. In this approach, latent enthalpy of a phase changing cell is related to the local porosity, which in turn, prescribes a frictional resistance on account of solidified phase distribution within the problem domain. This simplifies the numerical modelling requirements of a phase-change process considerably, by establishing a direct interlinkage between fluid flow, heat transfer and change of phase.

Due to the relatively simple phase-change characteristics of binary alloys, most of the enthalpy-based macroscopic models of alloy solidification have classically dealt with binary systems. Voller and co-workers⁸⁾ initially applied this formulation for modelling mushy zone phase change problems, followed by more advanced numerical studies for complicated cases of alloy solidification problems.^9,10,11,12) It can be noted here that prediction of solidification behaviour of such system would effectively require a more detailed assessment of the nature of solute redistribution and the associated phase equilibrium relationships. While the above mentioned studies contributed significantly towards understanding of physical phenomena involved in solidification of multicomponent alloys, a systematic study towards the development of enthalpy-based thermodynamically-consistent generalized simulation strategies for multicomponent alloy solidification has largely been ignored. In particular, macroscopic model for multicomponent systems, which can represent pertinent microscopic issues within the macroscopic framework in a metallurgically consistent manner is yet to be found in the literature. Typical microscopic issues include the morphology of the mushy region offering flow resistance, non-equilibrium solidification, macrosegregation, dendritic arm coarsening etc.

In the present work, a general mathematical procedure to formulate enthalpy-porosity based phase change modelling of a ternary alloy system is developed with effective representation of situations related to dendritic arm coarsening occurring during the solidification process. Towards this, suitable numerical schemes are devised to prescribe the permeability of the mushy zone as a function of the coarsening kinetics. In addition, a novel algorithm is developed that can effectively update the nodal latent heat on every iteration, consistent with the metallurgy of the dendritic arm coarsening. The model is capable of quantifying the effects of dendritic arm coarsening on macroscopic transport of momentum, heat and solute during solidification. As a demonstration of the numerical model developed in this study, solidification of three-element steel alloy is simulated and the resulting flow and macrosegregation patterns are analysed.

2. Mathematical Modelling

2.1. Governing Equations

The physical situation considered here is a two-dimensional rectangular mould containing a ternary metal alloy, initially in the molten state. The left boundary of the rectangular domain is subjected to a prescribed heat transfer coefficient (heat loss), whereas all other boundaries are kept insulated. Therefore the melt starts solidifying from the left vertical boundary and proceeds unidirectionally. The solid phase is assumed to be rigid and stationary, and fluid flow is two-dimensional and laminar. The problem domain is shown schematically in Fig. 1, describing the initial and boundary conditions.

Fig. 1.

Computational domain with boundary specifications.

Following a fixed-grid continuum formulation with a single-domain approach,⁹⁾ the equations governing conservation of mass, momentum, energy and species concentration appropriate to the present ternary system can be written as follows   

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \stackrel{\rightharpoonup }{u}\right)=0$$(1)

  

$$\frac{\partial (\rho \stackrel{\rightharpoonup }{u})}{\partial t}+\rho \stackrel{\rightharpoonup }{u}\cdot (\nabla \stackrel{\rightharpoonup }{u})=\nabla \cdot \left(\mu \nabla \frac{\rho }{{\rho }_l}u\right)-\nabla p-\frac{\mu \stackrel{\rightharpoonup }{u}}{K}+S_b$$(2)

Equations (1) and (2) are continuity equation and momentum equation respectively.

Here ρ, $\overrightarrow{u}$ , μ, p, K are the density, velocity, effective viscosity, pressure and permeability respectively, and the subscript ‘l’ denotes the liquid phase.

The buoyancy source term, S_b, appearing in Eq. (2), includes both thermal and solutal effects, and is given by the differences between local and initial values of temperature and liquid solute concentrations, respectively. Considering the effect of three components for the case of a ternary alloy, the buoyancy term in Eq. (2) can be written as:   

$${\text{S}}_{\text{b}}=\rho g[{\beta }_T(T-T_0)+{\beta }_{s1}(C_{l1}-C_{01})+{\beta }_{s2}(C_{l2}-C_{02})]$$(3)

Here, β_T and β_s are the thermal volumetric coefficient of expansion and solutal volumetric coefficient of expansion respectively, T₀ is the initial temperature in the domain and C₀ is the initial solute concentration. Subscript ‘1’ and ‘2’ refer to element 1 and element 2 respectively.

The thermal energy conservation equation is given as follows:   

$$\begin{array}{r}\frac{\partial }{\partial t}(f_s{c}_{ps}+f_l{c}_{pl})(\rho T)+\nabla \cdot (f_s{c}_{ps}+f_l{c}_{pl})(\rho \overrightarrow{u}T)\\ =\nabla \cdot \{(f_s{k}_s+f_l{k}_l)\nabla T\}+S_h\end{array}$$(4)

where,   

$$S_h=-\left[\frac{\partial }{\partial t}(\rho f_l\mathrm{\Delta }H)+\nabla \cdot (\rho \overrightarrow{u}\mathrm{\Delta }H)\right]$$(5)

here, f denotes mass fraction, and subscripts l and s refer to liquid and solid phases respectively. T is the temperature, c_p is the specific heat, and k is the thermal conductivity.

The source term, S_h, is a consequence of latent heat evolution during the solidification process, and is related to the liquid mass fraction of a computational cell as: f_l = ΔH/L, where L is the latent heat of freezing and ΔH is the latent enthalpy content of the computational cell under consideration.

The species conservation equation is given as follows:   

$$\frac{\partial }{\partial t}\left[\rho C_{mix}\right]+\nabla .(\rho \overrightarrow{u}{C}_l)=\nabla .({\rho }_s{g}_s{D}_s\nabla C_s+{\rho }_l{g}_l{D}_l\nabla C_l)$$(6)

Here, C_mix is the mixture composition, g is the volume fraction and D is the mass diffusion coefficient. C_l and C_s are species concentration in the liquid and solid respectively.

The term ρC_mix appearing in Eq. (6) is evaluated as   

$$\rho C_{mix}={\rho }_l{g}_l{C}_l+{\rho }_s{g}_s{C}_s$$(7)

It needs to be mentioned here that for the present case of a ternary alloy system, two independent species transport equations need to be essentially invoked; one equation for each constituent species. A major challenge in this regard would be to postulate a thermodynamically-consistent scheme that eventually relates the evolution of latent enthalpy with the local temperature, phase fraction distribution and composition of the constituent species that are obtained as solutions of the above equations. To achieve this purpose, a novel enthalpy updating strategy is proposed here (to be elaborated in the subsequent sub-section), which plays the pivotal role towards successful implementation of the present numerical scheme.

Furthermore, solute diffusion in the solid state is expressed as¹³⁾   

$$\frac{\partial \left(\rho f_s{C}_s\right)}{\partial t}=C_s^{*}\frac{\partial (\rho f_s)}{\partial t}+\frac{36{\rho }_s{D}_s}{{\lambda }_2^2}\left(C_s^{*}-C_s\right)$$(8)

where the superscript * denotes values at the solid/liquid interface, and λ₂ is the secondary dendritic arm spacing. Assuming local equilibrium at the interface, the following relationships can be written for the two phase region   

$$C_l=\frac{\left(T-T_{melt}\right)}{m_1}$$(9)

  

$$C_s^{*}=k_p{C}_l$$(10)

In Eq. (9), m_l is the slope of the liquidus line of the phase diagram.

The effective viscosity μ is calculated as¹⁴⁾   

$$\mu ={\mu }_l{\left[1-\left(\frac{F_{\mu }{g}_s}{0.3}\right)\right]}^{-2}$$(11)

Here, the function F_μ, is given by   

$$F_{\mu }=0.5-\frac{1}{\pi }{tan}^{-1}\left[100\left(g_s-g_{cr}\right)\right]$$(12)

In above equation, g_cr is a critical solid fraction as described in the literature.⁹⁾

Accordingly, the permeability K is assumed to vary as a function of liquid fraction¹⁴⁾   

$$K=\frac{{\lambda }_2^3{f}_l^3}{180{\left(1-f_l\right)}^2\left(1-F_{\mu }\right)}$$(13)

In the above equation (Eq. (13)), permeability K is a function of the porosity of the mushy zone, which is characterized by a coherent dendritic network in the molten metal matrix. In the enthalpy-porosity approach, porosity is modeled as the volume fraction of liquid in a computational cell, which, in turn, depends on the latent enthalpy content ΔH of the control volume under consideration during the phase change process. In a physical sense, K describes the friction between the solid and the liquid, as determined from the fraction of liquid and the microstructural morphology. This is in accordance with the physical situation prevailing during the columnar solidification of alloys. In a typical columnar structure, the interdendritic paths are long enough to support the backflow of melt through the solid-liquid mixed zone.¹⁵⁾ Due to dendritic arm coarsening, the solid-liquid interface moves in the microscopic length scale with progress in solidification. The coarsening occurs by remelting of some of the secondary dendritic arms during solidification.¹⁵⁾ This gives rise to a localized fluid flow that would effectively determine the final macrosegregation, apart from the effects of multicomponent thermosolutal convection. Hence, it is important to link the permeability of the mushy zone with the kinetics of dendritic arm coarsening. Eq. (13) specifically accomplished this task, in which the secondary dendritic arm spacing λ₂ is a variable during the evolution of the solidification process.

2.2. Boundary Conditions

The boundary conditions consistent with the above set of differential equations as well as the physical problem depicted in Fig. 1 are as follows:

(a) Left wall: u = 0, v = 0, $\begin{array}{l}-k\frac{\partial T}{\partial x}=h_c\left(T_{\infty }-T\right),\\ \mathrm{\hspace{0.17em} }\partial {(C_l)}_1/\partial x=0,\mathrm{\hspace{0.17em} }\partial {(C_l)}_2/\partial x=0\end{array}$

(b) Right wall: u = 0, v = 0, $\frac{\partial T}{\partial x}=0,\mathrm{\hspace{0.17em} }\partial {(C_l)}_1/\partial x=\partial {(C_l)}_2/\partial x=0$

(c) Top wall: u = 0, v = 0, $\begin{array}{r}\frac{\partial T}{\partial x}=0,\mathrm{\hspace{0.17em} }\partial {(C_l)}_1/\partial y=0,\\ \partial {(C_l)}_2/\partial y=0\end{array}$

(d) Bottom wall: u = 0, v = 0, $\begin{array}{r}\frac{\partial T}{\partial x}=0,\mathrm{\hspace{0.17em} }\partial {(C_l)}_1/\partial y=0,\\ \partial {(C_l)}_2/\partial y=0\end{array}$

Here, u and v are velocity components along x-axis and y-axis respectively, h_c is the convective heat transfer coefficient. Subscript ‘1’ and ‘2’ refer to element 1 and 2 respectively.

The initial conditions appropriate to the physical situation are as:

At t = 0, u = 0, T = T_i, (C_l)₁ = (C_i)₁, (C_l)₂ = (C_i)₂

3. Numerical Modelling

The coupled governing differential equations for fluid flow, heat transfer and mass transfer are discretized using a fully time-implicit finite volume methodology.¹⁶⁾ The fluid flow equations are solved using a pressure based primitive variable formulation according to the SIMPLER algorithm.¹⁶⁾ Convection-diffusion terms in the discretized governing equations are modelled by employing a power-law differencing scheme zone.¹⁶⁾ The above algorithm is appropriately extended for incorporating new features pertaining to dendritic arm coarsening, which distinguish the present model from the existing two-dimensional macroscopic models describing non-equilibrium solidification situations of multicomponent alloys. These new features are described below.

3.1. Modelling of Mushy Zone Permeability

It may be mentioned here that the mushy zone is assumed to be a saturated porous medium that offers frictional resistance toward fluid flow in that region. Permeability of the mushy zone is an important parameter which needs to be prescribed in accordance with the morphology of the phase-changing domain. In the present context, the theory developed by Mortenson¹⁷⁾ is adopted to calculate the secondary dendritic arm spacing at representative control point locations i,j at every time step,   

$$\begin{array}{l}{\left({\lambda }_2\right)}_{i,j}^3=\hfill \\ \frac{27\mathrm{\Gamma }{D}_l}{4}\underset{t}{\overset{t+\Delta t}{\int }}\frac{dt}{\left[1-{\left(k_p\right)}_{i,j}\right]\left[1-{\left(g_l\right)}_{i,j}\right]\left\{1-{\left[1-{\left(g_l\right)}_{i,j}\right]}^{1/2}\right\}\left(T_{melt}-T_{i,j}\right)}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left({\lambda }_{2,0}\right)}_{i,j}^3\end{array}$$(14)

where λ_2,0 is the secondary dendrite arm spacing corresponding to the previous time level and Γ is the Gibbs-Thomson coefficient. It can be noted that such temporal updating of λ₂ is carried out only within the range 0.07 < g_l < 0.9. For g_l > 0.9, the secondary dendrite arm spacing is assumed to be 3 μm¹³⁾ as an initial choice for the present situation. The partition coefficient value can be varied according to requirements in the present numerical scheme. The integral can be evaluated by following numerical integration method.

The enthalpy-porosity source term for the control volume i,j is given as   

$$S=S_c+S_p{u}_p,\hspace{1em}{S}_c=0,\hspace{1em}{S}_p=-\mu /K$$(15)

where μ is described by Eq. (11) and K varies according to (λ₂)_i,j as per Eq. (13).

3.2. Numerical Scheme for Enthalpy Update

The general form of the enthalpy updating expression given by Brent et al.⁵⁾ is first written as follows:   

$${[\mathrm{\Delta }{H}_P]}_{n+1}={[\mathrm{\Delta }{H}_P]}_n+\frac{a_P}{a^0{\hspace{0.17em}}_P}\lambda [{\left\{h_p\right\}}_n-c_p{F}^{-1}{\left\{\Delta H_P\right\}}_n]$$(16)

where ΔH_P is the latent heat content of the computational cell surrounding the grid point P, h is the sensible enthalpy, c_p is the specific heat, λ is a relaxation factor, n is the iteration level, a_p and a⁰_p are the coefficients of the finite volume discretization equation,¹⁶⁾ and F^–1 is the inverse of the latent heat function. It is to be noted here that in the above equation, F^–1 needs to be devised consistently with the microscopic physics pertaining to the dendritic arm coarsening phenomena in ternary metal alloy solidification, so that physically meaningful results can be obtained from the specific solidification model.

Considering the condition of local thermodynamic equilibrium, the liquidus surface in the phase diagram can be used to specify a relationship between the liquid concentrations and the temperature. In a functional form, this is expressed as   

$$T=T\left[{(C_l)}_1,{(C_l)}_2,\mathrm{..............},{(C_l)}_N\right]$$(17)

where N + 1 is the total number of species present in the alloy system. In a dilute ternary alloy system(i = 2), assuming linearized phase diagrams of the constituent binaries, a relationship between the temperature and liquid concentrations of the solutes along the liquidus surface can be written as¹⁸⁾   

$$T=T_m+m_l^1{(C_l)}_1+m_l^2{(C_l)}_2$$(18)

where $m_l^1$ and $m_l^2$ are the liquidus slopes of element 1 and element 2 respectively, and are given as   

$${(m_l)}^i={\left(\frac{\partial T_l}{\partial C_l}\right)}^i$$(19)

In Eq. (18), T_m represents the melting temperature of the pure solvent, which happens to be an important input to the thermodynamic calculation database. Further, the equations describing the relationship between the concentrations and liquid fraction in a ternary system can be given as¹⁹⁾   

$${\left(\frac{C_l}{C_0}\right)}_i={\left(1-y_i{f}_s\right)}^{-\frac{x_i}{y_i}},\mathrm{\hspace{0.17em} }\text{for component}\mathrm{\hspace{0.17em} }i$$(20)

Here, f_s is the mass fraction of the solid, y_i = 1 – 2γk*_p,i and x_i = 1 – k*_p,i, where k*_p,i is the equilibrium partition coefficient for each solute ‘i’ and is given as (C_s)_i = k*_p,i(C_l)_i. γ is the back diffusion parameter²⁰⁾ and represents the degree of back diffusion in the dendritic solid. In case of zero back diffusion (i.e., γ = 0), the Scheil model becomes applicable, and in case of complete back diffusion (i.e., γ = 1), the lever rule holds. A microscopically-consistent estimate of the parameter γ, for more general situations, can be obtained as¹⁹⁾   

$$\gamma =2\alpha [1-e^{-1/\alpha }]-e^{-1/2\alpha }$$(21)

where $\alpha =\frac{D_s{t}_f}{l_s^2}$ is the Fourier diffusion number. D_s is the diffusion coefficient in the solid, l_s is the solidification length scale (typically, of the order of secondary dendritic arm spacing), and t_f is the local solidification time scale.

Now, from Eq. (20) in Eq. (18), one can arrive at an expression that relates T with f_l , as   

$$T=T_m+\sum _{i=1}^N\left[m_l^i{(C_o)}_i(1-y_i)+m_l^i{(C_o)}_i{y}_i{f}_l^{-\frac{x_i}{y_i}}\right]$$(22)

It needs to be noted here that Eq. (14) is valid for a general N + 1 component alloy system, with N = 2 representing the special case of a ternary alloy system.

In order to obtain the latent heat function (F), the latent enthalpy (ΔH) is related with the phase-changing temperature, through the sensible enthalpy (h), as   

$$h=F^{-1}(\mathrm{\Delta }H)$$(23)

where h can be written as (c_pT). From Eqs. (22) and (23), an expression for the inverse of the latent heat function, corresponding to the case of multi-component alloy solidification systems, can be obtained as   

$$F^{-1}(\mathrm{\Delta }H)=T_m+\varphi$$(24)

where ϕ is given by   

$$\mathrm{\varphi \phi }=\sum _{i=1}^N\left[m_l^i{(C_o)}_i(1-y_i)+m^i{\hspace{0.17em}}_l{(C_o)}_i{y}_i{f}_l{\hspace{0.17em}}^{-\frac{x_i}{y_i}}\right]$$(25)

In order to ensure that the resultant liquid fraction conforms to the microscopic features of dendritic arm coarsening, additional microscopic issues must be considered. At this stage, condition of locally constant cooling rate is assumed, which is valid for a variety of practical applications such as directional solidification and crystal growth.²¹⁾ For such cases, the liquid mass fraction (f_l) can be expressed as²¹⁾   

$$f_l=1-\frac{1-\theta }{1-k_p}+\frac{k_p}{{\left(1-k_p\right)}^2}{\theta }^{-1/\left(1-k_p\right)}{\left(\frac{1}{\theta }-1\right)}^{-n}\times F\left(\theta :k_p,n\right)$$(26)

where   

$$F\left(\theta :k_p,n\right)=\underset{1}{\overset{1/\theta }{\int }}{\xi }^{\left(2k_p-1\right)/\left(1-k_p\right)}{\left(\xi -1\right)}^{n+1}d\xi$$(27)

Here n represents an experimentally determined coarsening exponent, often specified in the range of 0.29–0.35.²¹⁾ It may be noted here that the above case reduces to the specific case of Scheil’s equation when n = 0. However, irrespective of the value of n, the updated latent heat from Eq. (23) must yield a liquid fraction consistent with expression (26). Therefore, in order to obtain a metallurgically consistent solution for the above set of equations, the following steps are sequentially performed:

Step 1: Determine C_l(i,j) from the latest iteration, using the macroscopic species conservation Eq. (6)

Step 2: Evaluate the integral in expression (27)

Step 3: Evaluate f_l(i,j) using Eq. (26), indicated as f_l1(i,j)

Step 4: Obtain f_l(i,j) from Eq. (23), indicated as f_l2(i,j)

Step 5: Update f_l as   

$$\begin{array}{r}{f}_l(i,j)=f_{l2}(i,j)+{\alpha }_{1}sign\left\{1,\left[f_{l2}(i,j)-f_{l1}(i,j)\right]\right\}\\ \times \left[f_{l1}(i,j)-f_{l2}(i,j)\right]\end{array}$$(28)

where α₁ is a relaxation factor.

The algorithm for correction of the liquid fraction basically attempts to neutralize the difference between values predicted by Eqs. (23) and (26), respectively. In order to ensure smooth convergence for solution, value of λ₁ is kept low (~0.05–0.1).

Step 6: Solve for u(i,j), v(i,j), and T(i,j), using the updated values of liquid fraction during next iteration within the same time-step. The fluid flow equations are solved using a pressure based primitive variable formulation according to the SIMPLER algorithm. The process described above is repeated until convergence is achieved. Once the convergence check is satisfied, simulation is advanced to the next time step and continued until the domain solidifies completely.

4. Results and Discussion

For the purpose of illustration of the present numerical model, simulations are performed for the case of a three element (Fe-0.08wt%C-0.1wt%Mn) steel alloy with iron, carbon and manganese as the constituent elements. The liquidus temperature of the ternary steel alloy system for the present case is 1805 K, while the melt initially is at a uniform temperature of 1830 K inside the domain. The heat transfer coefficient (h_c) at the left wall is taken as 125 W/m².K. Table 1 summarizes the species-dependent thermodynamic characteristics, as adopted in the present work. The relevant thermophysical properties for steel are listed in Table 2. Different grid sizes (40 × 40, 60 × 60 and 100 × 100) are used for the resolution of the computational domain of size (0.1 m × 0.1 m) to check for grid independence. It is found that a further refinement in grid spacing does not alter the numerical predictions appreciably.

Table 1.Thermodynamic parameters of elements for ternary alloy.

Element	Cin (Wt%)	βs (1/Wt%)	Dl (m2/s)	Ds (m2/s)	∂Tl/∂Cl (K/Wt%)	kp (Wt%/Wt%)
Carbon (C)	0.08	1.10 × 10–2	2 × 10–9	5.6 × 10–10	–78	0.34
Manganese (Mn)	0.1	1.92 × 10–3	2 × 10–9	1.2 × 10–13	–3.32	0.75

Table 2.Thermophysical properties of steel.

Parameter	Value
Specific heat (c )	787 J/Kg. K
Thermal conductivity of solid (ks)	30 W/m.K
Thermal conductivity of liquid (kl)	27 W/m.K
Thermal expansion coefficient (βT)	2.0 × 10–4 K–1
Density (ρ)	7300 kg/m3
Viscosity (μ)	6.0 × 10–4 kg/m.s
Latent heat of fusion (L)	270 × 103 J/kg

Figure 2 shows the convection pattern for the ternary (Fe-0.08wt% C-0.1wt%Mn) steel alloy system, at time, t = 80 s after commencement of solidification. Once the left vertical wall of the rectangular mould is subjected to cooling, thermal gradients establish flow in the melt, with thermal buoyancy forces governing the transport processes at the initial stages. Consequently, the cold fluid descends along the interface and takes a counterclockwise turn near the bottom of the cavity, as can be observed from the streamlines in Fig. 2. As solidification progresses, the rejected solute being lighter (for both carbon and manganese) than the bulk fluid, tends to reduce the density of fluid in the mushy zone and drive the fluid in the upward direction. However, in the present situation, the relative strength of solutal buoyancy is smaller than the thermal buoyancy, and therefore the flow field is dominated by the thermal buoyancy effect. Presence of a minor vortex near the bottom corner of the cavity is due to the local solutal gradient build-up around the region.

Fig. 2.

Streamfunction plots at time = 80 s for Fe–C–Mn steel alloy showing the convection pattern for the ternary (Fe-0.08wt% C-0.1wt%Mn) system, at time, t = 80 s after commencement of solidification.

Figure 3 shows the flow pattern at a later stage (t = 160 s) of flow development. Due to the effect of directional heat transfer in the present case, flow is observed to progress in a longitudinal direction. Effect of localized solutal convection is negligible in the present case.

Fig. 3.

Streamfunction plots at time = 160 s for Fe–C–Mn steel alloy showing the convection pattern for the ternary (Fe-0.08wt% C-0.1wt%Mn) system, at time, t = 160 s after commencement of solidification.

Figure 4 depicts the composition variation of carbon along the longitudinal (x-axis) direction at different time intervals during the solidification of the present alloy system. For the purpose of comparison, results from the present model are plotted along with those without coarsening considerations. Figures 4(a)–4(b) shows the composition variation at t = 60 s and t = 180 s, and at a vertical location y = 0.05 m. A sharp increase in solute concentration is noticed at the vicinity of the interface between mushy region and liquid region, because of the rejection of solute during solidification. Due to the thermal buoyancy effect, the rejected solute is carried downward along the interface leading to a composition variation in the vertical direction. The effect of dendrite arm coarsening is to decrease the frictional resistance towards fluid flow in the mushy region. This leads to an enhanced convective flow which transports the rejected solute in the bulk flow resulting in composition variation in the bulk portion of the domain. Since models considering fixed dendritic arm spacing are unable to capture the enhanced advective field, results from such model predicts almost homogeneous composition in the bulk portion of the domain with a flatter curve. Similar behavior is also observed for the composition variation for manganese.

Fig. 4.

Variation of liquid concentration in terms of mass fraction of solute (carbon) along the longitudinal direction at different time intervals and at a vertical location y = 0.05 m, (a) at t = 60 sec (b) at t = 180 sec.

The macrosegregation level of the constituent elements (carbon and manganese) during solidification is shown in Fig. 5. The figure plots the average redistribution of the components integrated over the whole volume of the casting as a function of time. In Fig. 5, macrosegregation effects are quantified as follows:   

$$M(i)={\left[\frac{1}{V}{\underset{V}{\int }\left(\frac{C_{mix,i}}{C_{o,i}}-1\right)}^{2}dV\right]}^{1/2}$$(29)

where the subscript ‘i’ denotes solute element under concern.

Fig. 5.

Predicted carbon and manganese macrosegregation levels during solidification.

It can be seen that the development of macrosegregation, for both carbon and manganese, are similar in nature. Deviation from the present model results is also observed for the case when no dendrite arm coarsening is considered. For the particular case of fixed dendrite spacing, the solute redistribution level rises initially for both the elements, till the mushy zone engulfs almost the entire solidification domain, causing the fluid flow to weaken progressively, leading to an asymptotic saturation of the solutal concentration distribution. On the other hand, owing to a reduced frictional resistance with coarsening of the dendrites, convective flow intensifies, which transports the rejected solute species in the bulk domain more effectively, leading to an enhanced value of segregation. The solute concentration profile shows a rising trend as compared to the case where the coarsening of dendrites are not considered. It may be noted that severity of segregation is dependent on the partition coefficient of the element. In the present case, the partition coefficient value of carbon being smaller than manganese, the M_c value (i.e carbon macrosegregation) is higher than M_Mn (i.e macrosegregation for manganese), as seen from the Fig. 5.

4.1. Comparison with Experimental Results

In order to assess the quantitative predictive capabilities of the present model, a comparative study between experimental results reported in the literature and the corresponding outcomes from the present study is performed. Towards this, simulation results from the present model are compared with the experimental data corresponding to unidirectional solidification of an Al8.1wt%-Cu3wt%-Si ternary alloy system in a vertical water-cooled mold growth.²²⁾ The casting conditions and thermophysical properties of the alloy has been detailed in literature.^22,23) Figure 6 shows the numerically predicted temperature values and the corresponding experimental data,²³⁾ at two different vertical locations within the mold. Model predictions considering dendrite arm coarsening and without coarsening are shown in the figure. Results of the present model are found to match well with the experimental results. Further, the simulated macrosegregation level of copper has been compared with experimental observations in Fig. 7, and good agreement is observed. In both the cases (Figs. 6, 7), the experimental results are more accurately predicted by the present model than by existing ones. This is due to the fact that models considering fixed dendritic arm spacing are unable to capture the enhanced advective field which has a very important bearing on the overall solidification transport and macrosegregation behavior.

Fig. 6.

Comparison of numerically predicted temperature values (considering dendrite arm coarsening and without coarsening) and corresponding experimental data²³⁾ at two different heights of the mold (h = 9 mm and h = 70 mm).

Fig. 7.

Comparison between experimental results and numerically predicted concentration values (considering dendrite arm coarsening and without coarsening) of Cu for Al–Cu–Si ternary alloy.

5. Conclusions

In the present work, a macroscopic numerical model is developed which is capable of capturing the effects of secondary dendritic arm coarsening on the multicomponent solidification process. The macroscopic transport of mass, momentum, heat and solute is analysed using a fixed-grid enthalpy-based mathematical model, and the mushy zone is modeled according to the coarsening kinetics of dendrite arms. Solidification for ternary steel alloy is numerically simulated and the resulting flow and macrosegregation patterns are studied. From the simulation results, it is observed that the macrosegregation patterns for the constituent elements (carbon, manganese) are qualitatively similar, with the severity of segregation dependent on the partition coefficient of the respective constituents. Numerical predictions also indicate that the dendritic coarsening leads to an increased permeability of the mushy zone, which in turn results in a more pronounced macrosegregation. Results from the present model are also compared with the experimental measurements reported in the literature and very good agreements are observed.

Acknowledgement

The authors would like to thank the management of Tata Steel, India, for giving permission to publish this work.

Nomenclature

c: specific heat

C: species concentration

D: mass diffusion coefficient of species

f: mass fraction

g: volume fraction

h: specific enthalpy

h_c: convective heat transfer coefficient

ΔH: latent enthalpy

K: permeability

L: latent heat

T: temperature

$\overrightarrow{u}$ : velocity vector

u: axial component of velocity

v: radial component of velocity

μ: dynamic viscosity

ρ: density

Γ: Gibbs-Thomson coefficient

 

Subscripts

i: initial

l: liquid

s: solid

mix: mixture

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