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Effect of Diffusion Length in Modeling of Equiaxed Dendritic Solidification under Buoyancy Flow in a Configuration of Hebditch–Hunt Experiment

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Abstract

Modeling of equiaxed solidification is vital for understanding the solidification process of metallic alloys. In this work, an extended literature review is given for the models currently used for equiaxed solidification simulations. Based on this analysis, we present a three-phase multiscale equiaxed solidification model in which some approximations regarding solute transport at microscopic scale are put together in a new way and incorporated into macroscopic transport equations. For the latter, a term relating to the momentum exchange between the two phases is revised, and a modification for the grain packing algorithm is proposed. A modernized model for equiaxed dendrite growth is tested using a case of solidification of Sn-5 wt pct Pb alloy in a parallelepiped cavity that mimics the Hebditch–Hunt experiment. The results obtained using two approaches to calculate diffusion length are presented and compared both with each other and with numerical results from elsewhere. It is demonstrated that diffusion length has a crucial effect on the final segregation pattern.

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Abbreviations

\( c_{i} \) :

Solute concentration in i-phase (wt pct)

\( S_{\rm e}^{J} \) :

Diffusion surface concentration of the grain phase (\( {\text{m}}^{ - 1} \))

\( c_{\rm mix} \) :

Mix solute concentration (wt pct)

t :

Time (s)

\( c_{\rm eut} \) :

Eutectic concentration (wt pct)

\( T \) :

Temperature (K)

\( c_{\rm l}^{\ast} , c_{\rm s}^{\ast} \) :

Equilibrium concentration at f–s or d–s interface (wt pct)

\( T_{\rm eut} \) :

Eutectic temperature (K)

\( c_{\rm env} \) :

Concentration at l–d interface (wt pct)

\( T_{0} \) :

Initial temperature (K)

\( c_{\rm p}^{i} \) :

Specific heat for i-phase (\( {\text{J kg}}^{ - 1} {\text{K}}^{ - 1} \))

\( {\Updelta}T \) :

Constitutional undercooling (K)

\( c_{0} \) :

Initial concentration (wt pct)

\( {{\Updelta }}T_{\rm nucl} \) :

Nucleation temperature (K)

\( d_{\rm e} , d_{\rm s} \) :

Grain diameter (m)

\( \vec{u}_{i} \) :

Velocity vector of i-phase (\( {\text{m s}}^{ - 1} \))

\( D_{\rm l} \) :

Diffusion coefficient in liquid (\( {\text{m}}^{2} {\text{s}}^{ - 1} \))

\( \vec{U}_{ij}^{D} \) :

Momentum transfer rate from i-phase to j-phase (\( {\text{kg m}}^{ - 2} {\text{s}}^{ - 2} \))

\( f_{i} \) :

Volume fraction of i-phase (1)

\( v_{\rm env} \) :

Envelope growth velocity (\( {\text{m s}}^{ - 1} \))

\( f_{\rm p}^{\rm e} ,f_{\rm p}^{s} \) :

Packing limit fraction (1)

\( v_{\rm ds} \) :

d–s interface growth velocity (\( {\text{m\,s}}^{ - 1} \))

\( f_{\rm s}^{\rm e} \) :

Solid fraction within grains (1)

\( \vec{F}_{\rm Bi} \) :

Buoyancy force of i-phase (\( {\text{kg\,m}}^{ - 2} {\text{s}}^{ - 2} \))

\( h_{i} \) :

Enthalpy of i-phase (\( {\text{J\,kg}}^{ - 1} \))

\( H^{\ast} \) :

Interfacial heat transfer coefficient (\( {\text{W\,m}}^{ - 3} {\text{K}}^{ - 1} \))

J ij :

Species transfer rate from i-phase to j-phase (\( {\text{m\,s}}^{ - 1} \))

k i :

Thermal conductivity for i-phase (\( {\text{W\,m}}^{ - 1} {\text{K}}^{ - 1} \))

k :

Solute partition coefficient (1)

\( K_{ij} \) :

Liquid-equiaxed drag coefficient (\( {\text{kg\,m}}^{ - 3} {\text{s}}^{ - 1} \))

\( l_{i} \) :

Diffusion length of i-phase (m)

L :

Latent heat (\( {\text{J kg}}^{ - 1} \))

m :

Slope of liquidus in phase diagram (K)

\( M_{{{\Upphi }}} \) :

Mass transfer rate from nucleation (\( {\text{kg m}}^{ - 3} {\text{s}}^{ - 1} \))

\( M_{ij} \) :

Mass transfer rate from i-phase to j-phase (\( {\text{kg m}}^{ - 3} {\text{s}}^{ - 1} \))

\( n_{\rm max} \) :

Maximum equiaxed grain density (\( {\text{m}}^{ - 3} \))

n :

Grain number density (\( {\text{m}}^{ - 3} \))

\( N_{{{\Upphi }}} \) :

Nuclei production rate (\( {\text{m}}^{ - 3} {\text{s}}^{ - 1} \))

Re :

Reynolds number (1)

Sh :

Sherwood number (1)

\( S_{\rm e}^{\rm M} \) :

Surface concentration of the equivalent sphere (\( {\text{m}}^{ - 1} \))

\( \vec{u}_{i} \) :

Velocity vector of i-phase (\( {\text{m\,s}}^{ - 1} \))

\( \vec{U}_{ij}^{\rm D} \) :

Momentum transfer rate from i-phase to j-phase (\( {\text{kg\,m}}^{ - 2} \,\text{s}^{-2}\))

\( v_{\rm env} \) :

Envelope growth velocity (\( {\text{m\,s}}^{ - 1} \))

\( v_{\rm ds} \) :

ds interface growth velocity (\( {\text{m\,s}}^{ - 1} \))

\( \beta_{\rm T} \) :

Thermal expansion coefficient (\( {\text{K}}^{ - 1} \))

\( \beta_{\rm c} \) :

Solutal expansion coefficient (1)

\( {\Upphi}_{\text{M}} \) :

Shape factor of dendritic envelope growth (1)

\( {\Upphi}_{\text{J}} \) :

sphericity of dendritic envelope (1)

\( {\Upgamma} \) :

Gibbs Thomson coefficient (m K)

\( \lambda_{2} \) :

Secondary arm spacing (\( {\text{m}} \))

\( \mu_{i} \) :

Viscosity of i-phase (\( {\text{kg m}}^{ - 1} {\text{s}}^{ - 1} \))

\( \rho_{i} \) :

Density of i-phase (\( {\text{kg m}}^{ - 3} \))

\( \rho_{\rm ref} \) :

Reference density (\( {\text{kg m}}^{ - 3} \))

\( \rho_{\rm s}^{b} \) :

Density of solid phase for buoyancy

\( {\Uppsi}_{i} \) :

A intensive medium property (–)

d:

Interdendritic liquid phase

e: d + s:

Exquiaxed grain phase

f: l + d:

Liquid phase

l:

Extradendritic liquid phase

s:

Interdendritic solid phase

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Acknowledgments

This work is a joint cooperation between the SIMAP laboratory of Grenoble INP (France) and the Key Laboratory of EPM of Northeastern University (P. R. China). The authors gratefully acknowledge financial support from the National Nature Science Foundation of China (Grant No. U1760206), the National Key R&D Program of China (Grant No. 2017YFE0107900), the Project of Introducing Talents of Discipline Innovation to Universities 2.0 (the 111 Project of China 2.0, No. BP0719037), and China Scholarship Council (No. 201706080074). The SIMAP laboratory acknowledges the financial support provided by the ESA-MAP MICAST Project Contract 14347/01/NL/SH.

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Correspondence to Engang Wang.

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Manuscript submitted April 30, 2019.

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Wang, T., Semenov, S., Wang, E. et al. Effect of Diffusion Length in Modeling of Equiaxed Dendritic Solidification under Buoyancy Flow in a Configuration of Hebditch–Hunt Experiment. Metall and Materi Trans B 50, 3039–3054 (2019). https://doi.org/10.1007/s11663-019-01703-z

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