Introduction
Macrosegregation is a term that denotes a concentration inhomogeneity at the scale of a casting. It poses a severe quality issue for big ingots or continuously cast slabs. It is generally believed that the smaller the dimension of a casting the less distinctive are macrosegregations. This opinion is motivated by the fact that in most cases. the relative motion between solute-enriched liquid and solute-depleted crystals necessary for the formation of macrosegregation is caused by (i) natural or forced convection, (ii) sedimentation of crystals, and/or (iii) solidification-induced feeding flow.[
1‐
4] However, if deformation during solidification is the main reason for the relative motion between solid and liquid, then one can conclude that macrosegregation can also form in thin products such as in sheets produced by twin-roll casting.
Twin-roll casting is a technology that has been used over the recent decades for commercial production of thin metal strips. During this process, the molten metal is injected into the gap between two water-cooled counter-rotating rolls. As a result, it solidifies and the corresponding solid shell that forms on each of the moving roll surfaces is subjected to a considerable amount of hot rolling before reaching the roll nip.
This technology has many advantages compared with conventional casting techniques. The major advantage has to do with the reduced number of operational steps in the production line. This leads to a reduction in investment and processing costs, as well as less-rigorous logistics and labor requirements.[
5,
6] According to Barekar and Dhindaw,[
7] it also yields fewer greenhouse gas emissions. Furthermore, due to the very high cooling rates, the final strand is expected to have a refined microstructure with improved mechanical properties.[
8,
9]
On the other hand, combining metal solidification with hot rolling into one single step makes it more sensitive to process conditions,[
10] and eventually susceptible to a number of casting defects.[
11‐
14] In fact, over the years, as the emphasis has been growing for improving productivity, increasing attention has been placed on the understanding of the corresponding complex melt flow and solidification patterns.
The behavior of metallic alloys in the semisolid state is complex as it has been found that its behavior changes dramatically as the solid fraction increases. For flows with very dilute solid particles, the viscosity of the mixture is usually dependent on the solid fraction and the liquid viscosity. As the solid fraction increases, at specifically above a certain level (hereafter referred as transition solid fraction), the material develops a considerable resistance to deformation,[
15] usually reflected by an abrupt increase in the viscosity field.
In view of the above reason, many research groups in this field simply assume that, at this stage, the mixture behaves as a fixed, rigid solid body. For instance, Ni and Beckermann[
16] suggested a volume-averaged two-phase model which described the transport phenomena during solidification. Based on this pioneering numerical model, other studies ensued by applying the concept to equiaxed,[
17] columnar,[
18‐
20] and the most general columnar/equiaxed[
21] solidification.
However, although valuable progress has been achieved in the simulation of transport of crystals, the assumption of fixed rigid solid body for large solid fractions is still an oversimplification of the general solidification process (as pointed out above). Furthermore, another drawback of such concept is that it is only valid when the solid particles are entirely surrounded by liquid. This assumption allows one to consider that the pressure at the solid–liquid interface can be approximated by the average liquid pressure, and finally results in one single pressure field for both phases. This is a key point that greatly simplifies the solution procedure, and it is an important reason as to why it is so popular in the metallurgical community. However, it should be fairly reasonable to say that such conditions are not usually fulfilled in most industrial applications.
On the other hand, a more complex description of the semisolid slurry is to treat it as a viscoplastic continuous solid skeleton saturated with interstitial liquid.[
22] Such a state is characterized by a continuous coherent structure that is able to sustain significantly higher stresses. In this case, the macroscopic velocity gradients in the solid can no longer be neglected since the deformation and motion of the skeleton is now strictly connected to the hydrodynamic properties of the liquid flow. In fact, under these circumstances, solid and liquid phases become inherently coupled: if, on the one hand, pressing the solid skeleton drives the fluid flow behavior, on the other, the resulting pressure distribution in the interstitial liquid affects the equivalent stress experienced by the solid phase.
Suéry and his team had a substantial contribution in this field by analyzing the rheological and mechanical behavior of alloys,[
22‐
24] and by proposing constitutive equations for viscoplastic porous metallic materials saturated with liquid.[
25,
26] Based on the investigation of the mechanical behavior of an aluminum alloy in the semisolid state, Nguyen
et al.[
22] suggested that the onset of the viscoplastic behavior occurs at a solid fraction of 57 pct. Similarly, Fachinotti
et al.[
27] developed a thermomechanical and macrosegregation model for solidification of binary alloys. Contrary to the previous models, the one detailed in Fachinotti
et al.[
27] is not limited to isothermal conditions as the authors were able to consider the mass transfer between phases. This feature makes it applicable to actual solidification processes.
Regarding specifically scenarios replicating twin-roll casting, one can find that it is usually done by using single phase finite element codes, originally designed for pure rolling, and by treating the liquid as a solid with low viscosity.[
28,
29] On the other hand, if a liquid core still exists at the roll nip, an approximate solution can be achieved by neglecting the mechanical framework in the model.[
30] However, the viscoplasticity of the semisolid slurry must be considered if solidification has already reached the strand center before reaching the roll nip. In fact, this scenario is a great example where the deformation of the semisolid metallic alloy plays an important role in the outcome of the process, and therefore, in this case the viscoplastic regime should not be ignored.
The current study is an attempt to fill this gap that still prevails in the research community regarding the solidification process, and particularly when the material is then subjected to imposed deformation. A stand-alone numerical model has been developed to account for the transport and growth of equiaxed crystals during solidification. Furthermore, in order to account for mush deformation, a solution algorithm has been developed to include a viscoplastic regime when the concentration of solid is above the transition solid fraction. The model is implemented in OpenFOAM and is applied to a test case replicating a twin-roll casting process.
Model Description
In the current study, the simulation of the twin-roll casting relies on a two-phase Eulerian–Eulerian volume-averaged model. This approach solves mass, momentum, species, and enthalpy conservation equations for both phases. The two phases considered here are the liquid melt (
\( \ell \)) and the solid equiaxed crystals (
\( s \)), with the sum of their volume fractions being always equal to unity. Furthermore, the model also takes into account a transport equation for the grain number density, with a predetermined value being assumed at the beginning of the simulation. Conservation equations, sources terms, and auxiliary equations are summarized in Table
I with the subscript
\( i \) referring to one of the two phases. Note that when
\( i = \ell \), the minus sign from “
\( \mp \)” must be taken, whereas when
\( i = s \) the parameter in question is positive. In any case, all the parameters are properly identified in the nomenclature section. Further assumptions include neglecting the effect of gravity, and considering constant density fields in both phases.
Table I
Volume-Averaged Conservation Equations
Mass Cons.: |
\( \frac{{\partial g_{i} \rho_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} ) = \mp M_{ls} \)
| (11) |
Momentum Cons.: |
\( \frac{{\partial g_{i} \rho_{i} {\mathbf{v}}_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} {\mathbf{v}}_{i} ) = - g_{i} \nabla p + \nabla \cdot g_{i} {\varvec{\uptau}}_{i}^{eff} \mp {\mathbf{U}}_{ls} \)
| (12) |
Species Cons.: |
\( \frac{{\partial g_{i} \rho_{i} c_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} c_{i} ) = \nabla \cdot (g_{i} \rho_{i} D_{i} \,\nabla c_{i} ) \mp C_{ls} \)
| (13) |
Enthalpy Cons.: |
\( \frac{{\partial g_{i} \rho_{i} h_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} h_{i} ) = - \nabla \cdot \left( {\frac{{\lambda_{i} }}{{c_{{p_{i} }} }}\nabla h_{i} } \right) \mp H_{ls} \)
| (14) |
Grain Transport: |
\( \frac{\partial n}{\partial t} + \nabla \cdot ({\mathbf{v}}_{s} n) = 0 \)
| (15) |
All quantities are volume averaged in the present formulation, although they are not distinguished as such in Table
I. This compromise on the detail of the equations is expected to facilitate the reading process.
From the set of conservation equations, one can realize that the solid phase is treated in a very similar manner as the liquid. The major distinction has to do with how the deviatoric stress term presented in the momentum equations should be expressed. For the liquid phase, it is assumed that it behaves as an incompressible Newtonian fluid and so the viscous stress is proportional to the deviatoric part of the strain rate:
$$ {\varvec{\uptau}}_{l}^{\text{eff}} = {\varvec{\uptau}}_{\ell } = 2\mu_{\ell } {\text{dev}}({\dot{\mathbf{\varepsilon }}}_{\ell } ), $$
(1)
where the strain rate tensor is given by
\( {\dot{\mathbf{\varepsilon }}}_{\ell } = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\left( {\nabla {\mathbf{v}}_{\ell } + (\nabla {\mathbf{v}}_{\ell } )^{T} } \right) \) and the liquid viscosity is assumed to be constant (
\( \mu_{\ell } = 0.013\,{\text{Pa}} \cdot {\text{s}} \)). Note that the flow is assumed to be laminar in the present model. Considering that the characteristic size
\( L \) (casting size) in the domain is 0.01 m, and the corresponding liquid velocity is 0.04 m/s, the system Reynolds number
\( \left( {\text{Re} = L\,{\mathbf{v}}_{l} \,\rho_{l} /\mu_{l} } \right) \) is about 80, which is less than the critical number 2100 for the onset of turbulence flow.
As for the solid phase, the deviatoric stress term will depend on the volume fraction in the region under consideration. In fact, whether the semisolid lies within the viscoplastic regime or not will influence the solution procedure. Below the viscoplastic threshold (
\( g_{s}^{t} \)), the approach is similar to the liquid phase, even though the viscosity is no longer kept constant throughout the regime (as shown below). On the other hand, as the solid fraction increases above
\( g_{s}^{t} \), the mechanical properties of the solid phase change dramatically, and the behavior of the solid phase has been modeled according to a compressible viscoplastic model.[
22] Combining both approaches into a general mathematical statement yields:
$$ {\varvec{\uptau}}_{s}^{\text{eff}} = \left\{ {\begin{array}{ll} {2\mu_{s} {\text{dev}}\left( {{\dot{\mathbf{\varepsilon}}}_{s} } \right)} & {{\text{for }}g_{s} \le g_{s}^{t} } \\ {2{\frac{{\mu_{s}^{\text{app}} }}{A}{\text{dev}}}\left( {{\dot{\mathbf{\varepsilon }}}_{s} } \right) + \mu_{s}^{\text{app}} \left( {\frac{1}{9B}} \right){\text{tr}}({\dot{\mathbf{\varepsilon }}}_{s} ){\mathbf{I}}} & {{\text{for }}g_{s} > g_{s}^{t} } \\ \end{array} } \right.. $$
(2)
where
\( {\dot{\mathbf{\varepsilon }}}_{s} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\left( {\nabla {\mathbf{v}}_{s} + (\nabla {\mathbf{v}}_{s} )^{T} } \right) \).
Comparison between the equations defined for each regime in Eq. [
2] suggests that the first term in the viscoplastic regime
\( (g_{s} > g_{s}^{t} ) \) is somewhat analogous to the viscous stress defined in the lower solid fraction range
\( (g_{s} \le g_{s}^{t} ) \), with the corresponding viscosity parameter being defined by the apparent solid viscosity and the rheological parameter
A. In addition, a second term appears in the viscoplastic regime, which can be referred as the compression term with the bulk viscosity coefficient being equal to
\( {{\mu_{s}^{\text{app}} } \mathord{\left/ {\vphantom {{\mu_{s}^{\text{app}} } {9B}}} \right. \kern-0pt} {9B}} \). Besides the previous quantities already highlighted, it takes into account a second rheological parameter
B which can be interpreted as a compressibility factor. According to Nguyen
et al.,[
22] the rheological parameters
A and
B can be empirically modeled as follows:
$$ A = \frac{3}{{g_{s}^{6.47} }}\,{\text{and}}\,B = \frac{1}{{g_{s}^{6.94} }} - 1. $$
(3)
It can be seen that both coefficients are maximal (but finite) when the solid fraction is equal to
\( g_{s}^{t} \) (which corresponds to the start of the viscoplastic regime) and then they exhibit an exponential decay as
\( g_{s} \) increases.
The terms
\( \mu_{s}^{{}} \) and
\( \mu_{s}^{\text{app}} \) represent the solid viscosity terms in each of the regimes. For low solid fractions, Ishii[
31] postulated that the mixture viscosity of a set of particles can be expressed according to the Power-Law viscosity model:
$$ \mu_{mix} = \mu_{l} \left( {1 - \frac{{g_{s} }}{{g_{s}^{p} }}} \right)^{{ - 2.5g_{s}^{p} }} $$
(4)
with
\( g_{s}^{p} \) being defined as the random packing limit, which establishes the threshold from which the equiaxed grains create a rigid structure. It is usually expressed in the literature as loose or close packing limit depending on the compaction protocol. In the current study, the loose random packing of spheres of 58.5 pct proposed by Olmedilla
et al.[
32] has been adopted. It is worth mentioning that the original physical interpretation of this parameter is not particularly complying with the viscoplastic approach proposed in the current study, as the solid phase is considered to become a rigid structure above the packing limit. Nevertheless,
\( g_{s}^{p} \) is required in Eq. [
4] to calculate the solid viscosity in the nonviscoplastic regime, and therefore, should be regarded as a fitting constant that affects the evolution of the viscosity in this regime.
Assuming the validity of the mixture rule for the entire simulation (
\( \mu_{mix} = g_{s} \mu_{s} + g_{l} \mu_{l} \)), one can determine the corresponding solid viscosity as follows:
$$ \mu_{s} = \frac{{\mu_{\ell } }}{{g_{s} }}\left( {\left( {1 - \frac{{g_{s} }}{{g_{s}^{p} }}} \right)^{{ - 2.5g_{s}^{p} }} - (1 - g_{s} )} \right). $$
(5)
It is worth mentioning at this point that the transition solid fraction assumed above corresponds to the volume fraction at which the crystals start to become able to sustain tensile loads. This threshold is usually referred to as rigidity point in the literature.[
33] On the other hand, the coherency point is defined as the moment at which the solid fraction is high enough such that bridges between crystals start to form, but no tensile loads can be sustained.[
33] Such a threshold is not established explicitly but can be identified in the nonviscoplastic regime by the exponential increase in the solid viscosity, given by Eq. [
5].
Regarding the viscoplastic regime, once the solid fraction exceeds
\( g_{s}^{t} \), the solid phase is assumed to exhibit viscoplastic behavior. The apparent solid viscosity generally accepted in the literature takes the form given in the Norton–Hoff model:
$$ \mu_{s}^{\text{app}} = 3K_{v} \left( {\sqrt 3 \dot{\varepsilon }_{s}^{\text{eq}} } \right)^{m - 1} , $$
(6)
where
\( K_{v} \) and
m are the viscoplastic consistency and the strain-rate sensitivity, respectively. The latter parameter is usually assumed as a fixed value which depends on the material properties. Following the study of Nguyen
et al.,[
22] it is set as 0.213. On the other hand, the viscoplastic consistency can be calculated by combining the strain-rate tensor proposed by Nguyen
et al.[
22] and the Norton–Hoff stress–strain-rate relation, yielding
$$ K_{v} = \sqrt 3^{( - 1 - m)} \left( {\beta \exp \left( { - \frac{Q}{RT}} \right)\alpha^{1/m} } \right)^{ - m} . $$
(7)
Besides the strain-rate sensitivity defined above and the molar gas constant (
\( R = 8.31446\, \) J/mol/K), this equation involves additional three material-dependent parameters
\( \beta \),
\( Q \) , and
\( \alpha \), which have been set with the following values based on the experimental trials carried out by Nguyen
et al.[
22] as
\( 4.98 \times 10^{17} \), 257 kJ/mol, and 0.03 MPa, respectively. The authors refer that these constants characterize the deformation behavior of an aluminum alloy in the solid state that is close to solidus and are usually temperature dependent. However, in the current study, they have been assumed as constant throughout the process range. Introducing all assumed quantities into Eq. [
7] results in a viscoplastic consistency value of 6.31 × 10
6 Pa s.
As for the equivalent strain rate presented in Eq. [
6], it is given by the following expression
$$ \dot{\varepsilon }_{s}^{\text{eq}} = \sqrt {\frac{2}{A}({\dot{\mathbf{\varepsilon }}}_{s} :{\dot{\mathbf{\varepsilon }}}_{s} ) - \left( {\frac{2}{3A} - \frac{1}{9B}} \right){\text{tr}}({\dot{\mathbf{\varepsilon }}}_{s} )^{2} } , $$
(8)
which again depends on the rheological parameters defined in Eq. [
3].
The microscopic phenomena resulting from the solidification process are taken into account by considering exchange terms in the conservation equations.
\( M_{\ell s} ,{\mathbf{U}}_{\ell s} ,C_{\ell s} \), and
\( H_{\ell s} \) are the mass, momentum, species, and energy exchange rates between solid and liquid and are outlined in Table
II.
Table II
Exchange Terms Used in the Conservation Equations
Mass Transfer: |
\( M_{\ell s} = v_{r} S_{\ell s} \rho_{s} \varPhi_{\text{imp}} \)
| (16) |
Momentum Transfer: |
\( {\mathbf{U}}_{\ell s} = {\mathbf{U}}_{{_{\ell s} }}^{d} + {\mathbf{U}}_{{_{\ell s} }}^{p} = K_{\ell s} ({\mathbf{v}}_{\ell } - {\mathbf{v}}_{s} ) + u^{*} M_{\ell s} \)
| (17) |
Species Transfer: |
\( C_{\ell s} = c_{s}^{*} M_{\ell s} \)
| (18) |
Enthalpy Transfer: |
\( H_{ls}^{{}} = h_{c} (T_{\ell } - T_{s} ) \)
| (19) |
The mass-transfer rate (for solidification or melting) is calculated as a function of the specific surface area (
\( S_{\ell s} = n \cdot 4\pi r^{2} \)), impingement factor (
\( \varPhi_{\text{imp}} = \hbox{min} \left[ {g_{l} /\left( {1 - \pi \sqrt 3 /8} \right),1} \right] \)), and crystal growth velocity (
\( v_{r} \)). The latter quantity determines how fast the solid interface grows (or shrinks) with solidification (or melting). It is calculated according to the following equation:
$$ v_{r} = \frac{{D_{l} }}{{r_{f} \left( {1 - {r \mathord{\left/ {\vphantom {r {r_{f} }}} \right. \kern-0pt} {r_{f} }}} \right)}}\frac{{c_{l}^{*} - c_{l} }}{{c_{l}^{*} \left( {1 - k} \right)}} $$
(9)
It can be seen that the difference between the average species mass fraction in the liquid at the interface (
\( c_{l}^{*} = (T - T_{f} )/m_{ls} \)) and the volume-averaged liquid mass fraction predicted by the species conservation equation (
\( c_{l} \)) is the major driving force for crystal growth.
The momentum exchange term can be caused by mechanical interactions (superscript
d) or by phase change (superscript
p).
\( {\mathbf{U}}_{{_{\ell s} }}^{d} \) can be interpreted as the drag force between the two phases, with
\( K_{\ell s} \) being the drag coefficient. In the current study,
\( K_{\ell s} \)follows the submerged object approach in the lower solid fraction regime, whereas in the viscoplastic regime the porous medium approach with Kozeny–Carman-type permeability is adopted instead. It can then be written as follows:
$$ K_{ls} = \left\{ {\begin{array}{*{20}c} {18g_{l}^{2} \frac{{\mu_{l} g_{s} C_{\varepsilon } }}{{d^{2} }}\,\,\,{\text{for}}\,\,\,g_{s} < g_{s}^{t} } \\ {g_{l}^{2} \frac{{\mu_{l} }}{K}\,\,\,\,\,\;\;\;\;\;\;\;\;{\text{for}}\,\,\,g_{s} \ge g_{s}^{t} } \\ \end{array} } \right. $$
(10)
where the terms
\( C_{\varepsilon } = 10{{g_{s} } \mathord{\left/ {\vphantom {{g_{s} } {g_{l}^{3} }}} \right. \kern-0pt} {g_{l}^{3} }} \) and
\( K = K_{0} {{g_{l}^{3} } \mathord{\left/ {\vphantom {{g_{l}^{3} } {g_{s}^{2} }}} \right. \kern-0pt} {g_{s}^{2} }} \) represent the settling ratio and the overall flow permeability, respectively.
\( K_{0} \) is an empirical parameter that has been set to
\( {{d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {180}}} \right. \kern-0pt} {180}} \), so
\( K_{ls} \) maintains a uniform trend during the entire spectrum of volume fractions.
On the other hand, the contribution to the phase change in the momentum exchange term (
\( {\mathbf{U}}_{{_{\ell s} }}^{p} \)) depends on the average velocity value, given by the parameter
\( u^{*} \) (with
\( u^{*} = {\mathbf{v}}_{l} \) during solidification and
\( u^{*} = {\mathbf{v}}_{s} \) during melting), and the mass-transfer rate. The product of the two terms corresponds to a momentum force (per unit volume and time) that is added to or subtracted from the corresponding phase momentum during a phase change. A similar mathematical expression is also set for the species source term, as it can be seen in Eq. [18]. The solute exchange caused by the phase change is proportional to the mass-transfer rate, with the constant of proportionality being the equilibrium solid mass fraction,
\( c_{s}^{*} \). Note that further details about the contribution of the phase change in the momentum exchange term have been given somewhere else,[
17] so only a short description is presented here.
Lastly, the enthalpy exchange term is employed to enforce thermal equilibrium between the phases. Even though two energy conservation equations are considered—and so different temperatures are expected in each phase—\( H_{ls} \) is set with a very large volume heat-transfer coefficient (\( h_{c} \) = 109 W/m3/K) between the phases. Such a procedure is a trade off between trying to accomplish the precondition of thermal equilibrium, and at the same time keeping the numerical calculation results as stable as possible.
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