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24.2.4 Species Equations

In the case of solidification/melting with species transport, the following species equation is solved:

 \frac{\partial}{\partial t} (\rho Y_i) + \nabla \cdot \left(... ... v}_p Y_{i,{\rm sol}}]\right) = - \nabla \cdot \vec{J_i} + R_i (24.2-10)

where $\vec{J_i}$ is given by

 \vec{J_i} = -\rho [\beta D_{i,m,{\rm liq}} \nabla Y_{i,{\rm liq}} + (1-\beta) D_{i,m,{\rm sol}} \nabla Y_{i,{\rm sol}}] (24.2-11)

Here $Y_i$ is the average species mass fraction in a cell:

 Y_i = \beta Y_{i,{\rm liq}} + (1-\beta) Y_{i,{\rm sol}} (24.2-12)

$Y_{i,{\rm liq}}$ and $Y_{i,{\rm sol}}$ are related by the partition coefficient $K_i$:

 Y_{i,{\rm sol}} = K_i Y_{i,{\rm liq}} (24.2-13)

${\vec v}_{\rm liq}$ 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

 {\vec v}_{\rm liq} = \frac{({\vec v} - {\vec v}_p(1-\beta))}{\beta} (24.2-14)

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