# Modeling of marangoni-induced droplet motion and melt convection during solidification of hypermonotectic alloys

- Received:

DOI: 10.1007/s11661-003-0200-3

- Cite this article as:
- Wu, M., Ludwig, A. & Ratke, L. Metall and Mat Trans A (2003) 34: 3009. doi:10.1007/s11661-003-0200-3

## Abstract

A two-phase volume averaging approach to model Marangoni-induced droplet motion of the minority liquid phase and the convection in the parent melt during solidification of the hypermonotectic alloys is presented. The minority liquid phase decomposed from the parent melt as droplets in the miscibility gap was treated as the second-phase *L*_{2}. The parent melt including the solidified monotectic matrix was treated as the first phase *L*_{1}. Both phases were considered as different and spatially interpenetrating continua. The conservation equations of mass, momentum, solute, and enthalpy for both phases, and an additional transport equation for the droplet density, were solved. Nucleation of the *L*_{2} droplets, diffusion-controlled growth, interphase interactions such as Marangoni force at the *L*_{1}-*L*_{2} interface, Stokes force, solute partitioning, and heat release of decomposition were taken into account by corresponding source and exchange terms in the conservation equations. The monotectic reaction was modeled by adding the latent heat on the *L*_{1} phase during monotectic reaction, and applying an enlarged viscosity to the solidified monotectic matrix. A two-dimensional (2-D) square casting with hypermonotectic composition (Al-10 wt pct Bi) was simulated. This paper focused on Marangoni motion, hence gravity was not included. Results with nucleation, droplet evolution, Marangoni-induced droplet motion, solute transport, and macrosegregation formation were obtained and discussed.

### Nomenclature

*c*_{0}alloy concentration

*c*_{c}critical concentration

*c*_{1},*c*_{2}volume-averaged species concentration

*c**_{1},*c**_{2}interface concentration under thermal equilibrium

- Δ
*c*_{d} *c**_{2}−*c**_{1}*c*_{L2}*L*_{2}monotectic concentration*c*_{m}monotectic concentration

*c**interface species

- Δ
*c* *c*_{1}−*c**_{1}*C*_{12}(= −*C*_{21})species exchange rate

*C*_{12}^{d}(= −*C*_{21}^{d})species transfer at

*L*_{1}-*L*_{2}interface*C*_{12}^{p}(= −*C*_{21}^{p})solute partitioning due to phase change

*c*_{mix}mix concentration

*c*_{p(1)},*c*_{p(2)}specific heat

*D*_{1},*D*_{2}diffusion coefficient

*d*_{2}droplet diameter

*f*_{1},*f*_{2}volume fraction

**f**_{M}Marangoni force on single droplet

**f**_{st}Stokes force on a single droplet

**F**_{M}volume-averaged Marangoni force

**g**gravity

*H*heat-transfer coefficient at casting-mold interface

*H**volume heat-transfer coefficient between two liquid phases

*h*_{1},*h*_{2}enthalpy

*h*_{1}^{ref},*h*_{2}^{ref}enthalpy at

*T*_{ref}*h**interface enthalpy

- Δ
*h*_{d} heat of decomposition

- Δ
*h*_{M} latent heat of monotectic reaction

*K*_{21}(=*K*_{12})momentum exchange coefficient

*k*solute partitioning coefficient

*k*_{1},*k*_{2}thermal conductivity

*L*_{1},*L*_{2}two liquid phases

*M*_{12}(= −*M*_{21})mass-transfer rate per volume

*m*slope of liquidus in phase diagram at

*c*_{0}*m*_{12}mass-transfer rate for a single droplet

*N*droplet nucleation rate

*n*droplet density

*n*_{max}maximum droplet density

*p*pressure

*Q*_{12}(= −*Q*_{21})energy exchange rate

*Q*_{12}^{d}(= −*Q*_{21}^{d})energy exchange by heat transfer

*Q*_{12}^{p}(= −*Q*_{21}^{p})energy exchange due to phase change

*R*droplet radio

*S*_{A}solid-phase Al

*S*_{B}solid-phase Bi

*T*_{c}critical temperature

*T*,*T*_{1},*T*_{2}temperature

*T*_{f}^{A}melting point of pure metal (Al)

*T*_{f}^{B}melting point of pure metal (Bi)

*T*_{m}^{f}monotectic temperature

*T*_{ref}reference temperature for enthalpy definition

- ▽
*T* temperature gradient

- Δ
*T* undercooling

- Δ
*T*_{N} Gaussian distribution width of droplet nucleation law

- Δ
*T*_{σ} undercooling for maximum droplet nucleation rate

*t*time

**U**_{12}(= −**U**_{21})momentum exchange rate

**U**_{12}^{d}(= −**U**_{21}^{d})momentum exchange due to Stokes force

**U**_{12}^{p}(= −**U**_{21}^{p})momentum exchange due to phase change

*u*_{1},*u*_{2}velocity component in

*x*direction**u**_{1},**u**_{2}velocity vector

**u**_{12},**u**_{21}interphase velocity

**u***interface velocity

*v*_{1},*v*_{2}velocity component in

*y*direction*ρ*_{1},*ρ*_{2}density

*σ*surface tension at liquid-liquid interface

*σ*_{0}experimental parameter in Eq. [19]

*μ*_{1},*μ*_{2}viscosity

*τ*_{1},*τ*_{2}stress-strain tensors

- Subscripts 1, and 2
indicate first and second liquid phases.