Conservation of energy at interface

Neglecting the work done by surface tension and disjoining pressure, the energy balance at the interface is $\left( {{{{\mathbf{q''}}}_\ell } - {{{\mathbf{q''}}}_v}} \right) \cdot {\mathbf{n}} - ({\mathbf{n}} \cdot {{\mathbf{\tau '}}_\ell }) \cdot {{\mathbf{V}}_{\ell ,rel}} + ({\mathbf{n}} \cdot {{\mathbf{\tau '}}_v}) \cdot {{\mathbf{V}}_{v,rel}} = {\dot m''_\delta }\left[ {\left( {{e_v} + \frac{{{\mathbf{V}}_{v,rel}^2}}{2}} \right) - \left( {{e_\ell } + \frac{{{\mathbf{V}}_{\ell ,rel}^2}}{2}} \right)} \right] \qquad \qquad(1)$

If the velocity of the reference frame is taken as the interfacial velocity VI, eq. (1) can be rewritten as $\begin{array}{l} \left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} - ({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_\ell }) \cdot ({{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}) + ({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_v}) \cdot ({{\mathbf{V}}_v} - {{\mathbf{V}}_I}) \\ = {{\dot m''}_\delta }\left\{ {\left[ {{e_v} + \frac{{{{({{\mathbf{V}}_v} - {{\mathbf{V}}_I})}^2}}}{2}} \right] - \left[ {{e_\ell } + \frac{{{{({{\mathbf{V}}_\ell } - {{\mathbf{V}}_I})}^2}}}{2}} \right]} \right\} \\ \end{array} \qquad \qquad(2)$

where the heat fluxes in the liquid and vapor phases have been determined by Fourier’s law of conduction.

Equation (2) can be rewritten in terms of enthalpy (h = e + p / ρ), i.e., $\begin{array}{l} \left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} - ({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_\ell }) \cdot ({{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}) + ({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_v}) \cdot ({{\mathbf{V}}_v} - {{\mathbf{V}}_I}) \\ = {{\dot m''}_\delta }\left[ {{h_{\ell v}} - \left( {\frac{{{p_v}}}{{{\rho _v}}} - \frac{{{p_\ell }}}{{{\rho _\ell }}}} \right) + \frac{1}{2}{\mathbf{V}}_v^2 - {{\mathbf{V}}_v} \cdot {{\mathbf{V}}_I} - \frac{1}{2}{\mathbf{V}}_\ell ^2 + {{\mathbf{V}}_\ell } \cdot {{\mathbf{V}}_I}} \right] \\ \end{array} \qquad \qquad(3)$

where ${h_{\ell v}}$ is the difference between the enthalpy of vapor and liquid at saturation, i.e., the latent heat of vaporization. The stress tensor in eq. (3) can be expressed as $\tau ' = - p{\mathbf{I}} + 2\mu {\mathbf{D}} - \frac{2}{3}\mu \left( {\nabla \cdot {\mathbf{V}}} \right){\mathbf{I}} = - p{\mathbf{I}} + \tau \qquad \qquad(4)$

Since the relative velocities at the interface satisfy $({V_{\ell}}-{V_I})n = {\dot m''_{\delta}}/{{\rho}_{\ell}}$ and $({V_v}-{V_I})n = {\dot m''_{\delta}}/{{\rho}_v}$, we have ${p_\ell }\left( {{{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}} - {p_v}\left( {{{\mathbf{V}}_v} - {{\mathbf{V}}_I}} \right) \cdot {\mathbf{n}} = {\dot m''_\delta }\left[ { - \left( {\frac{{{p_v}}}{{{\rho _v}}} - \frac{{{p_\ell }}}{{{\rho _\ell }}}} \right)} \right] \qquad \qquad(5)$

Substituting eqs. (4) and (5) into eq. (3), the energy balance can be written as $\begin{array}{l} \left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} - \left( {{\mathbf{n}} \cdot {\tau _\ell }} \right)\left( {{{\mathbf{V}}_\ell } - {{\mathbf{V}}_I}} \right) + \left( {{\mathbf{n}} \cdot {\tau _v}} \right)\left( {{{\mathbf{V}}_v} - {{\mathbf{V}}_I}} \right) \\ = {{\dot m''}_\delta }\left[ {{h_{\ell v}} + \frac{1}{2}{\mathbf{V}}_v^2 - {{\mathbf{V}}_v} \cdot {{\mathbf{V}}_I} - \frac{1}{2}{\mathbf{V}}_\ell ^2 + {{\mathbf{V}}_\ell } \cdot {{\mathbf{V}}_I}} \right] \\ \end{array} \qquad \qquad(6)$

To simplify the energy equation, the kinetic energy terms are considered negligible and no-slip conditions are assumed at the interface, ${V_{\ell ,{\mathbf{t}}}} = {V_{v,{\mathbf{t}}}} = {V_{I,{\mathbf{t}}}} \qquad \qquad(7)$

Therefore, the energy equation can be rewritten as $\begin{array}{l} \left( {{k_v}\frac{{\partial {T_v}}}{{\partial {x_{\mathbf{n}}}}} - {k_\ell }\frac{{\partial {T_\ell }}}{{\partial {x_{\mathbf{n}}}}}} \right) \\ = {{\dot m''}_\delta }\left[ {{h_{\ell v}} + \frac{4}{3}{\nu _\ell }\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{\mathbf{n}}}}} - \frac{2}{3}{\nu _\ell }\left( {\frac{{\partial {V_{\ell ,{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right)} \right. \\ \left. {{\rm{ }} - \frac{4}{3}{\nu _v}\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{\mathbf{n}}}}} + \frac{2}{3}{\nu _v}\left( {\frac{{\partial {V_{v,{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right)} \right] \\ \end{array} \qquad \qquad(8)$

where ${v_{\ell}}$ and vv are kinematic viscosities of liquid and vapor phases, respectively.

The energy balance at the interface can be simplified by assuming that the change in the kinetic energy across the interface is negligible, i.e., $\left( {{k_v}\nabla {T_v} - {k_\ell }\nabla {T_\ell }} \right) \cdot {\mathbf{n}} = {\dot m''_\delta }{h_{\ell v}} \qquad \qquad(9)$

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.