# Jump and boundary conditions at interfaces

(Difference between revisions)
 Revision as of 09:59, 28 June 2010 (view source)← Older edit Current revision as of 13:42, 5 August 2010 (view source) (6 intermediate revisions not shown) Line 3: Line 3: ==Conservation of mass at interface== ==Conservation of mass at interface== +
${\dot m''_\delta } = {\rho _\ell }\left( {{V_{\ell ,{\mathbf{n}}}} - {V_{I,{\mathbf{n}}}}} \right) = {\rho _v}\left( {{V_{v,{\mathbf{n}}}} - {V_{I,{\mathbf{n}}}}} \right)$
''See Main Article'' [[Conservation of mass at interface]]. ''See Main Article'' [[Conservation of mass at interface]]. ==Conservation of momentum at interface== ==Conservation of momentum at interface== +
${p_v} - {p_\ell } = \sigma (T)\left( {\frac{1}{{{R_I}}} + \frac{1}{{{R_{II}}}}} \right) - {p_d}$
''See Main Article'' [[Conservation of momentum at interface]]. ''See Main Article'' [[Conservation of momentum at interface]]. - ==Energy== + ==Conservation of energy at interface== - ''See Main Article'' [[Conservation of energy at interface|Energy]]. +
$\left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} = {\dot m''_\delta }{h_{\ell v}}$
+ ''See Main Article'' [[Conservation of energy at interface]]. - ==Mass Species== + ==Conservation of mass species at interface== - ''See Main Article'' [[Conservation of mass species at interface|Mass Species]]. + For a general interface between phases $k$ and $j$ in a multi-component system, a local balance in mass flux of species $i$ must be upheld. The total species mass flux, ${\dot m''_i}$, at an interface is: + +
${\dot m''_i} = {\rho _{k,i}}\left( {{{\mathbf{V}}_{k,i}} - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}} = {\rho _{j,i}}\left( {{{\mathbf{V}}_{j,i}} - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}}$
- ==Supplementary Conditions== + ''See Main Article'' [[Conservation of mass species at interface]]. - ''See Main Article'' [[Supplementary conditions at interfaces|Supplementary Conditions]]. + + ==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. + + ==Further Reading== + + ==External Links==

## Current revision as of 13:42, 5 August 2010

Shape of the liquid-vapor interface near a vertical wall.

The conservation equations introduced above can be applied within each phase and up to an interface. However, they are not valid across the interface, where sharp changes in various properties occur. Appropriate boundary conditions at the interface must be specified in order to solve the governing equations for heat, mass, and momentum transfer in the two adjoining phases. The interface conditions will serve as boundary conditions for the transport equations in the adjacent phases. Jump conditions at the interface can be obtained by applying the basic laws (conservation of mass, momentum, energy, and the second law of thermodynamics) at the interface. It is the objective of this subsection to specify mass, momentum, and energy balance at a non-flat liquid-vapor interface (see figure), as well as species balance in solid-liquid-vapor interfaces. For solid-liquid or solid-vapor interfaces, these jump conditions can be significantly simplified.

## Conservation of mass at interface

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

See Main Article Conservation of mass at interface.

## Conservation of momentum at interface

${p_v} - {p_\ell } = \sigma (T)\left( {\frac{1}{{{R_I}}} + \frac{1}{{{R_{II}}}}} \right) - {p_d}$

See Main Article Conservation of momentum at interface.

## Conservation of energy at interface

$\left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} = {\dot m''_\delta }{h_{\ell v}}$

See Main Article Conservation of energy at interface.

## Conservation of mass species at interface

For a general interface between phases k and j in a multi-component system, a local balance in mass flux of species i must be upheld. The total species mass flux, ${\dot m''_i}$, at an interface is:

${\dot m''_i} = {\rho _{k,i}}\left( {{{\mathbf{V}}_{k,i}} - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}} = {\rho _{j,i}}\left( {{{\mathbf{V}}_{j,i}} - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}}$

See Main Article Conservation of mass species at interface.

## 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.