For solidification and melting of a pure substance, phase change occurs at a distinct melting temperature, . 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 , denoted , 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,

(15–8)

(15–9)

where is the partition coefficient of solute , is the mass fraction of solute , and is the slope of the liquidus surface with respect to . If the value of the mass fraction exceeds the value of the eutectic mass fraction , then is clipped to when calculating the liquidus and solidus temperatures. It is assumed that the last species material of the mixture is the solvent and that the other species are the solutes.

Ansys Fluent expects that you will input a negative value for the liquidus slope of species (). If you input a positive slope for , Ansys Fluent will disregard your input and instead calculate it using the Eutectic temperature and the Eutectic mass fraction :

(15–10)

Updating the liquid fraction via Equation 15–3 can cause numerical errors and convergence difficulties in multicomponent mixtures. Instead, the liquid fraction is updated as,

(15–11)

where the superscript indicates the iteration number, is a relaxation factor with a default value of 0.9, is the cell matrix coefficient, is the time step, is the current density, is the cell volume, is the current cell temperature and is the interface temperature.

Ansys Fluent offers two models for species segregation at the micro-scale, namely the Lever rule and the Scheil rule. 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 , is calculated as:

(15–12)

where is the number of species.

The Scheil rule evaluates as:

(15–13)

For information about how back diffusion (that is, a finite amount of diffusion of the solute species) can be incorporated into the formulation, see the section that follows.

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

(15–14)

where is the reaction rate and is given by

(15–15)

is the velocity of the liquid and is the solid (pull) velocity. 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

(15–16)

The liquid () and solid () mass fractions are related to each other by the partition coefficient :

(15–17)

When the Scheil model is selected, Ansys Fluent solves for as the dependent variable  [677]:

(15–18)