Mathematical Modeling of the Solidification Structure Evolution in the Presence of Ultrasonic Stirring

Abstract

Ultrasonic treatment (UST) was studied in this work to improve the quality of the cast ingots as well as to control the solidification structure evolution. Ultrasonically induced cavitation consists of the formation of small cavities (bubbles) in the molten metal followed by their growth, pulsation, and collapse. These cavities are created by the tensile stresses that are produced by acoustic waves in the rarefaction phase. The pressure for nucleation of the bubbles (e.g., cavitation threshold pressure) may decrease with increasing the amount of dissolved gases and especially with the amount of inclusions in the melt. Modeling and simulation of casting solidification of alloys with UST requires complex multiscale computations, from computational fluid dynamics (CFD) macroscopic modeling through mesoscopic to microscopic modeling, as well as strategies to link various length-scales emerged in modeling of microstructural evolution. The developed UST modeling approach is based on the numerical solution of the Lilley model (that is founded on Lighthills’s acoustic analogy), fluid flow, heat transfer equations, and mesoscopic modeling of the grain structure. The CFD analysis tool is capable of modeling acoustic streaming and ultrasonic cavitation. It is used in this work to study ingot solidification under the presence of ultrasound. The UST model was applied to low-temperature alloys including Al- and Mg-based alloys. Although the predicted ultrasonic cavitation region is relatively small, the acoustic streaming is strong and, thus, the created/survived bubbles/nuclei are transported into the bulk liquid quickly. The predicted grain size under UST condition is at least one order of magnitude lower than that without UST.

Appendix: Approach for Modeling of Ultrasonic Vibration in Fluids

For planar propagation in x-direction, the ultrasonic pressure, p _us (in Pa), is given by:

$$ p_{us} = a\omega \rho v\sin \left( {\omega t - kx} \right) $$

(A1)

The ultrasonic intensity, I _us (in W/m²), can be calculated with:

$$ I_{us} = 0.5a^{2} \omega^{2} \rho v\exp ( - \alpha x) = I_{us}^{o} \exp ( - \alpha x)\; {\text{with}}\;\alpha = C\frac{{f^{2} }}{{v^{3} \rho }}\eta $$

(A2)

In Eqs. [A1] and [A2], ρ is the material density, ω is the angular frequency, k is a coefficient of proportionality, η is the material dynamic viscosity, I ^o _us is the reference ultrasonic intensity at x = 0, f is the sound frequency, v is the velocity, a is the amplitude of the oscillations, α is the absorption coefficient, t is time, C is a material constant, and x is the distance. For example, α for metals at f = 20 kHz is of the order of 10⁻⁶ [m⁻¹].

The fluid flow with ultrasonic vibration can be described by the continuity (mass conservation) and Navier–Stokes momentum equations, where the additional source term is the volumetric force, F _us ,  that is calculated as:

$$ F_{us} = \frac{{\partial p_{us} }}{\partial x} = - a\omega \rho vk\cos \left( {\omega t - kx} \right)\; {\text{with}}\;F_{us} \;{\text{in}}\;{\text{N}}/{\text{m}}^{3} $$

(A3)

Heat transfer is described by the energy equation where the additional source term is the rate of ultrasonic heating, Q _us :

$$ Q_{us} \; = \;\frac{{\partial \;I_{us} }}{\partial \;x}\; = \; - \;\frac{{I_{us}^{o} }}{\alpha }\;\exp ( - \alpha \;x)\; {\text{with}}\;{Q}_{us} \;{\text{in}}\;{\text{W}}/{\text{m}}^{3} $$

(A4)