17.7 Species Equations

For solidification and melting of a pure substance, phase change occurs at a distinct melting temperature, $T_{\rm melt}$. For a multicomponent mixture, however, a mushy freeze/melt zone exists between a lower solidus and an upper liquidus temperature. When a multicomponent liquid solidifies, solutes diffuse from the solid phase into the liquid phase. This effect is quantified by the partition coefficient of solute $i$, denoted $K_i$, which is the ratio of the mass fraction in the solid to that in the liquid at the interface.

ANSYS FLUENT computes the solidus and liquidus temperatures in a species mixture as,

$$T_{\rm solidus} = T_{\rm melt} + \sum_{\rm solutes}{ m_i Y_i / K_i }$$ (17.7-1)

$$T_{\rm liquidus} = T_{\rm melt} + \sum_{\rm solutes}{ m_i Y_i }$$ (17.7-2)

where $K_i$ is the partition coefficient of solute $i$, $Y_i$ is the mass fraction of solute $i$, and $m_i$ is the slope of the liquidus surface with respect to $Y_i$. It is assumed that the last species material of the mixture is the solvent and that the other species are the solutes.

The liquidus slope of species $i$, $m_i$, is calculated from the Eutectic temperature and the Eutectic mass fraction as,

Updating the liquid fraction via Equation  17.4-3 can cause numerical errors and convergence difficulties in multicomponent mixtures. Instead, the liquid fraction is updated as,

where the superscript $n$ indicates the iteration number, $\lambda$ is a relaxation factor with a default value if 0.9, $a_p$ is the cell matrix co-efficient, ${\Delta}t$ is the time-step, $\rho$ is the current density, $V$ is the cell volume, $T$ is the current cell temperature and $T^{*}$ is the interface temperature.

ANSYS FLUENT offers two models for species segregation at the micro-scale, namely the Lever and Scheil rules. The former assumes infinite diffusion of the solute species in the solid, and the latter assumes zero diffusion. For the Lever rule, the interface temperature, $T^{*}$, is calculated for a binary mixture as,

where

The Scheil rule evaluates $T^{*}$ as,

For the Lever rule, species transport equations are solved for the total mass fraction of species $i$, $Y_i$:

where $R_i$ is the reaction rate and $\vec{J_i}$ is given by

${\vec v}_{\rm liq}$ is the velocity of the liquid and ${\vec v}_p$ is the solid (pull) velocity. ${\vec v}_p$ is set to zero if pull velocities are not included in the solution. The liquid velocity can be found from the average velocity (as determined by the flow equation) as

The liquid ( $Y_{i,{\rm liq}}$) and solid ( $Y_{i,{\rm sol}}$) mass fractions are related to each other by the partition coefficient $K_i$:

When the Scheil model is selected, ANSYS FLUENT solves for $Y_{i,{\rm liq}}$ as the dependent variable [ 359]:

$$\frac{\partial}{\partial t} (\rho Y_{i,{\rm liq}}) + \nabla \cdot... ...\rm liq} Y_{i,{\rm liq}} + (1-\beta) {\vec v}_p Y_{i,{\rm sol}}]\right) = R_i +$$

$$\nabla \cdot (\rho \beta D_{i,m,{\rm liq}} \nabla Y_{i,{\rm liq}... ...\rho (1-\beta)) + \frac{\partial}{\partial t} (\rho (1-\beta) Y_{i,{\rm liq}} )$$ (17.7-12)