Conservation of mass at interface

At a liquid-vapor interface, the mass balance is

${\dot m''_\delta } = {\rho _\ell }({{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}) \cdot {\mathbf{n}} = {\rho _v}({{\mathbf{V}}_v} - {{\mathbf{V}}_I}) \cdot {\mathbf{n}} \qquad \qquad(1)$

where ${\dot m''_\delta }$ is mass flux at the interface due to phase change and VI is the velocity of the interface. For a three-dimensional interface, there are three components of velocity: the normal direction, and two tangential directions, denoted by n, t1, and t2, respectively. Therefore, the velocity components should be defined according to these directions, as follows:

${\mathbf{V}} \cdot {\mathbf{n}} = {V_{\mathbf{n}}} \qquad \qquad(2)$
${\mathbf{V}} \cdot {{\mathbf{t}}_1} = {V_{{{\mathbf{t}}_1}}} \qquad \qquad(3)$
${\mathbf{V}} \cdot {{\mathbf{t}}_2} = {V_{{{\mathbf{t}}_2}}} \qquad \qquad(4)$

The interfacial mass balance can be rewritten in these terms by:

${\dot m''_\delta } = {\rho _\ell }\left( {{V_{\ell ,{\mathbf{n}}}} - {V_{I,{\mathbf{n}}}}} \right) = {\rho _v}\left( {{V_{v,{\mathbf{n}}}} - {V_{I,{\mathbf{n}}}}} \right) \qquad \qquad(5)$

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.