# Interaction between primary dendrite arm spacing and velocity of fluid flow during solidification of Al–Si binary alloys

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## Abstract

A new and more efficient numerical algorithm to simulate the solidification of binary metallic alloys, wherein for the first time, the undercooling of the liquidus temperature prior to solidification event and optimized thermo-physical properties was incorporated, has been recently developed and validated by various experiments. Subsequently, experiments were carried out to evaluate the validity of various theoretical models in the literature used to predict the dendrite arm spacing (DAS) and quantify the critical interaction between fluid flow and transient DAS during unsteady state solidification. Typically, models of solidification processes such as casting, welding and galvanizing assume a constant value of fluid flow to predict the DAS and in many cases unable to obtain validation. This practice is erroneous and the transient fluid flow developed during solidification has a significant effect on the transient DAS, thermal gradient (*G*), solidification velocity (*R*) and morphology of the mushy zone. The Bouchard–Kirkaldy model (DAS prediction) coupled with the Lehmann model to incorporate fluid flow velocity was the only valid theoretical model in binary alloy solidification.

## List of symbols

*c*_{ps}Specific heat of solid as a function of temperature (J kg

^{−1}K^{−1}) [1]*c*_{pl}Specific heat of liquid (J kg

^{−1}K^{−1}) [1]*C*_{L}Liquid concentration (wt%)

*C*_{o}Average alloy composition (wt%)

*C*_{S}Solid concentration (wt%)

*D*Solute diffusivity coefficient of liquid [6.25 × 10

^{−9}(m^{2}s^{−1})] [2]*G*Temperature gradient in liquid at the mushy zone/liquid interface (°C mm

^{−1})*k*Average partition coefficient (0.116) [3]

*K*_{s}Thermal conductivity of solid as a function of temperature (W m

^{−1}K^{−1}) [1]*K*_{l}Thermal conductivity of liquid (W m

^{−1}K^{−1}) [1]*L*Latent heat of fusion (J kg

^{−1}) [3]*m*The slope of liquidus line [− 6.675 (K wt%

^{−1})] [3]*p*Pressure (Pa)

*R*Velocity of mushy zone/liquid interface (mm s

^{−1})*t*Time (s)

*T*Temperature (°C)

*T*_{liq}Liquidus temperature (°C) [3]

*T*_{ini}Initial temperature of liquid (°C)

*T*_{m}Melting temperature of pure aluminum (660 °C) [3]

*T*_{eut}Eutectic temperature (578.6 °C) [3]

- \( \dot{T} \)
Instantaneous tip cooling rate =

*G*×*R*(°C s^{−1})- Δ
*T* Undercooling of

*T*_{liq}(°C)*U*_{r}Velocity in r direction (mm s

^{−1})*u*_{y}Velocity in y direction (mm s

^{−1})*U*Flow velocity in the liquid of mushy zone/liquid interface (mm s

^{−1})*β*Contraction ratio \( \left[ {\beta = \frac{{\rho_{s} - \rho_{l} }}{{\rho_{l} }}} \right] \) (volumetric shrinkage during solidification) [1]

*β*_{C}Solute expansion coefficient [− 4.26 × 10

^{−4}(K^{−1})] [1]*β*_{T}Thermal expansion coefficient [1.39 × 10

^{−4}(K^{−1})] [1]*Γ*Gibbs–Thomson coefficient [1.97 × 10

^{−7}(K m^{−1})] [4]*ϕ*Liquid fraction

*ρ*_{l}Liquid density (kg m

^{−3}) [1]*ρ*_{s}Solid density (kg m

^{−3}) [1]*µ*Dynamic viscosity 1.3 × 10

^{−3}(Pa s) [1]- \( \lambda_{1}^{0} \)
Primary arm spacing if no fluid flow effect is considered (µm)

*λ*_{1}Primary arm spacing (µm)

## Notes

### Acknowledgements

The authors wish to extend their gratitude to the Discovery Grant Program of the Natural Science and Engineering Research Council of Canada, for providing the funding for this research project.

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