# Film Boiling Analysis in Porous Media

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- | Film boiling | + | [[Image:boiling_g_(1).gif|thumb|600 px|alt= Film boiling in porous media|<center>'''Film boiling in porous media.'''</center>]] |

- | + | Film boiling of liquid saturated in a porous medium at an initial temperature of <math>{T_\infty } < {T_{sat}}</math> next to a vertical, impermeable heated wall at a temperature of <math>{T_w} > {T_{sat}}</math> is analyzed (see figure). Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. The liquid adjacent to the vapor layer is dragged upward by the vapor. The temperature at the liquid-vapor interface is at the saturation temperature. There are velocity and thermal boundary layers in the liquid phase adjacent to the vapor film. The solution of the film boiling problem requires solutions of vapor and liquid flow, as well as heat transfer in both the vapor and liquid phases. It is assumed that boundary layer approximations are applicable to the vapor film and to convection heat transfer in the liquid phase. It is further assumed that the vapor flow is laminar, two-dimensional; Darcy’s law is applicable in both the vapor and liquid phases. The continuity, momentum, and energy equations in the vapor film are | |

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- | <math>{T_w} > {T_{sat}}</math> is analyzed (see | + | |

<center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1) </math></center> | <center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1) </math></center> | ||

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<center><math>{u_v} = - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2) </math></center> | <center><math>{u_v} = - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2) </math></center> | ||

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<center><math>{u_v}\frac{{\partial {T_v}}}{{\partial x}} + {v_v}\frac{{\partial {T_v}}}{{\partial y}} = {\alpha _{mv}}\frac{{{\partial ^2}{T_v}}}{{\partial {y^2}}}\qquad \qquad(3) </math></center> | <center><math>{u_v}\frac{{\partial {T_v}}}{{\partial x}} + {v_v}\frac{{\partial {T_v}}}{{\partial y}} = {\alpha _{mv}}\frac{{{\partial ^2}{T_v}}}{{\partial {y^2}}}\qquad \qquad(3) </math></center> | ||

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where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor. | where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor. | ||

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<center><math>\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4) </math></center> | <center><math>\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4) </math></center> | ||

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<center><math>{u_\ell } = \frac{K}{{{\mu _\ell }}}{\rho _\infty }g{\beta _\ell }({T_\ell } - {T_\infty })\qquad \qquad(5) </math></center> | <center><math>{u_\ell } = \frac{K}{{{\mu _\ell }}}{\rho _\infty }g{\beta _\ell }({T_\ell } - {T_\infty })\qquad \qquad(5) </math></center> | ||

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<center><math>{u_\ell }\frac{{\partial {T_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {T_\ell }}}{{\partial y}} = {\alpha _{m\ell }}\frac{{{\partial ^2}{T_\ell }}}{{\partial {y^2}}}\qquad \qquad(6) </math></center> | <center><math>{u_\ell }\frac{{\partial {T_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {T_\ell }}}{{\partial y}} = {\alpha _{m\ell }}\frac{{{\partial ^2}{T_\ell }}}{{\partial {y^2}}}\qquad \qquad(6) </math></center> | ||

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where <math>{\alpha _{m\ell }}</math> is thermal diffusivity of the porous medium saturated with the liquid. | where <math>{\alpha _{m\ell }}</math> is thermal diffusivity of the porous medium saturated with the liquid. | ||

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, & {y = 0} \\ | , & {y = 0} \\ | ||

\end{array}\qquad \qquad(7) </math></center> | \end{array}\qquad \qquad(7) </math></center> | ||

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<center><math>T = {T_w}\begin{array}{*{20}{c}} | <center><math>T = {T_w}\begin{array}{*{20}{c}} | ||

, & {y = 0} \\ | , & {y = 0} \\ | ||

\end{array}\qquad \qquad(8) </math></center> | \end{array}\qquad \qquad(8) </math></center> | ||

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+ | It should be pointed out that <math>{u_v}</math> is not equal to zero at the heating surface under Darcy’s law, i.e., slip occurs at the surface. The boundary condition in the liquid that is far from the heated surface is | ||

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<center><math>{u_\ell } = 0\begin{array}{*{20}{c}} | <center><math>{u_\ell } = 0\begin{array}{*{20}{c}} | ||

, & {y \to \infty } \\ | , & {y \to \infty } \\ | ||

\end{array}\qquad \qquad(9) </math></center> | \end{array}\qquad \qquad(9) </math></center> | ||

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<center><math>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}} | <center><math>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}} | ||

, & {y \to \infty } \\ | , & {y \to \infty } \\ | ||

\end{array}\qquad \qquad(10) </math></center> | \end{array}\qquad \qquad(10) </math></center> | ||

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- | The mass balance at the liquid-vapor interface is | + | The mass balance at the liquid-vapor interface is (see [[Film Boiling Analysis]]): |

<center><math>{\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_v} = {\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_\ell }\begin{array}{*{20}{c}} | <center><math>{\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_v} = {\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_\ell }\begin{array}{*{20}{c}} | ||

, & {y = {\delta _v}} \\ | , & {y = {\delta _v}} \\ | ||

\end{array}\qquad \qquad(11) </math></center> | \end{array}\qquad \qquad(11) </math></center> | ||

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The temperature at the liquid-vapor interface is equal to the saturation temperature: | The temperature at the liquid-vapor interface is equal to the saturation temperature: | ||

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, & {y = \delta } \\ | , & {y = \delta } \\ | ||

\end{array}_v}\qquad \qquad(12) </math></center> | \end{array}_v}\qquad \qquad(12) </math></center> | ||

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- | The above film boiling problem can be solved using a similarity solution like that for film condensation in porous media | + | The above film boiling problem can be solved using a similarity solution like that for [[Film Condensation on an Inclined Wall|film condensation in porous media]]. |

- | + | ==References== | |

- | + | Cheng, P., and Verma, A.K., 1981, “The Effect of Subcooling Liquid on Film Boiling about a Vertical Heated Surface in a Porous Medium,” ''International Journal of Heat and Mass Transfer'', Vol. 24, pp. 1151-1160. | |

- | + | Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA | |

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- | + | Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced Heat and Mass Transfer'', Global Digital Press, Columbia, MO. | |

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Nield, D.A., and Bejan, A., 1999, ''Convection in Porous Media'', 2<sup>nd</sup> ed., Springer-Verlag, New York. | Nield, D.A., and Bejan, A., 1999, ''Convection in Porous Media'', 2<sup>nd</sup> ed., Springer-Verlag, New York. |

## Current revision as of 01:59, 9 July 2010

Film boiling of liquid saturated in a porous medium at an initial temperature of next to a vertical, impermeable heated wall at a temperature of *T*_{w} > *T*_{sat} is analyzed (see figure). Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. The liquid adjacent to the vapor layer is dragged upward by the vapor. The temperature at the liquid-vapor interface is at the saturation temperature. There are velocity and thermal boundary layers in the liquid phase adjacent to the vapor film. The solution of the film boiling problem requires solutions of vapor and liquid flow, as well as heat transfer in both the vapor and liquid phases. It is assumed that boundary layer approximations are applicable to the vapor film and to convection heat transfer in the liquid phase. It is further assumed that the vapor flow is laminar, two-dimensional; Darcy’s law is applicable in both the vapor and liquid phases. The continuity, momentum, and energy equations in the vapor film are

where α_{mv} is thermal diffusivity of the porous medium saturated with the vapor.
The governing equations for the liquid boundary layer are

where is thermal diffusivity of the porous medium saturated with the liquid.

The boundary conditions at the heated wall (*y* = 0) are

It should be pointed out that *u*_{v} is not equal to zero at the heating surface under Darcy’s law, i.e., slip occurs at the surface. The boundary condition in the liquid that is far from the heated surface is

The mass balance at the liquid-vapor interface is (see Film Boiling Analysis):

The temperature at the liquid-vapor interface is equal to the saturation temperature:

The above film boiling problem can be solved using a similarity solution like that for film condensation in porous media.

## References

Cheng, P., and Verma, A.K., 1981, “The Effect of Subcooling Liquid on Film Boiling about a Vertical Heated Surface in a Porous Medium,” *International Journal of Heat and Mass Transfer*, Vol. 24, pp. 1151-1160.

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.

Nield, D.A., and Bejan, A., 1999, *Convection in Porous Media*, 2^{nd} ed., Springer-Verlag, New York.