Mass balance equation

The multi-field mass balance equation for the phase k is written as follows:

$\frac{\partial}{\partial t} (\alpha_k \rho_k) + \frac{\partial}{\partial x_i} (\alpha_k \rho_k U_{k,i}) = \Gamma_k$   (6)

where $\alpha_k, \rho_k, U_{k,i}$ represent the volumetric fraction, the density and the mean velocity of phase k, respectively and $\Gamma_k$ is the interface mass transfer rate on phase k, sum of all the other phases contribution.

$\Gamma_k = \sum_{p≠k} {{{\Gamma} ^ {c}}_{p→k} + {\Gamma^ {nuc}}_{w→k}}$   (7)

With ${\Gamma^c}_{p→k}$ the interface mass transfer from phase p to phase k (bulk transfer) and ${\Gamma^{nuc}}_{w→k}$ the mass transfer contribution to phase k induced by wall nucleate boiling.

In the model used for the study, these terms are calculated near the heated wall for the two phases, liquid and vapor (k = 1, 2), thus

${\Gamma^{nuc}}_{w→1}+{\Gamma^{nuc}}_{w→2}=0$   (8)

where $\Gamma^{nuc}_{w→2} \ge 0$ since vapour is produced by nucleation.

Finally, conservation relations for mass and volume lead to

$\sum \alpha_k=1$   (9)

$\sum \Gamma_k=0$, since ${\Gamma^c}_{p→k}+{\Gamma^c}_{k→p}=0$   (10)