# Gravity dominated film-condensation in a porous medium

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- | When condensation is dominated by gravity, the effect of surface tension is negligible, and consequently, no two-phase region exists. Condensation along an inclined wall in a porous medium (See Fig. 8.30) will be discussed in this subsection. A porous medium saturated with dry vapor at its saturation temperature, Tsat, is bounded by an inclined impermeable wall with a temperature Tw (<math> | + | When condensation is dominated by gravity, the effect of surface tension is negligible, and consequently, no two-phase region exists. Condensation along an inclined wall in a porous medium (See Fig. 8.30) will be discussed in this subsection. A porous medium saturated with dry vapor at its saturation temperature, Tsat, is bounded by an inclined impermeable wall with a temperature Tw (<math>{{T}_{w}}<{{T}_{sat}}</math>). Since the wall temperature is below saturation temperature, film condensation occurs on the inclined wall and the condensate flows downward due to gravity. It is assumed that the condensation is gravity-dominated and, therefore, that the liquid and vapor are separated by a sharp interface, not a two-phase region. In addition, the following assumptions are made by [[#References|Cheng (1981)]] in order to obtain an analytical solution: |

- | ). Since the wall temperature is below saturation temperature, film condensation occurs on the inclined wall and the condensate flows downward due to gravity. It is assumed that the condensation is gravity-dominated and, therefore, that the liquid and vapor are separated by a sharp interface, not a two-phase region. In addition, the following assumptions are made by Cheng (1981) in order to obtain an analytical solution: | + | |

- | 1. The condensate film is very thin compared to the length of the inclined wall (<math>\delta _\ell | + | 1. The condensate film is very thin compared to the length of the inclined wall (<math>{{\delta }_{\ell }}\ll L</math>) so that boundary layer |

- | + | assumption is valid. | |

- | + | ||

- | + | 2. The properties for the porous medium, liquid, and vapor are independent from temperature. | |

- | + | ||

- | + | 3. The inclination angle, <math>\phi </math>, is small enough for the gravity component in the normal direction of the surface to be negligible. | |

- | + | ||

+ | 4. Darcy’s law is valid for both liquid and vapor phases. | ||

+ | |||

+ | 5. The saturation temperature, Tsat, is constant. | ||

Under these assumptions, the continuity, momentum, and energy equations for the liquid layer are | Under these assumptions, the continuity, momentum, and energy equations for the liquid layer are | ||

- | <center><math> | + | |

- | \frac | + | <center><math>\frac{\partial {{u}_{\ell }}}{\partial x}+\frac{\partial {{v}_{\ell }}}{\partial y}=0</math> |

- | </math | + | (8.355)</center> |

- | (8.355) | + | |

- | <center><math> | + | <center><math>{{u}_{\ell }}=\frac{K}{{{\mu }_{\ell }}}({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi </math> |

- | + | (8.356)</center> | |

- | </math | + | |

- | (8.356) | + | <center><math>{{u}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial x}+{{v}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial y}={{\alpha }_{\ell }}\frac{{{\partial }^{2}}{{T}_{\ell }}}{\partial {{y}^{2}}}</math> |

- | <center><math> | + | (8.357)</center> |

- | + | ||

- | </math | + | |

- | (8.357) | + | |

The boundary conditions at the wall are | The boundary conditions at the wall are | ||

- | <center><math> | + | |

- | + | <center><math>{{v}_{\ell }}=0\begin{matrix} | |

- | , & | + | , & y=0 \\ |

- | \end{ | + | \end{matrix}</math> |

- | </math | + | (8.358)</center> |

- | (8.358) | + | |

- | <center><math> | + | <center><math>{{T}_{\ell }}={{T}_{w}}\begin{matrix} |

- | + | , & y=0 \\ | |

- | , & | + | \end{matrix}</math> |

- | \end{ | + | (8.359)</center> |

- | </math | + | |

- | (8.359) | + | |

At the interface, the boundary conditions are | At the interface, the boundary conditions are | ||

- | <center><math> | + | |

- | + | <center><math>{{T}_{\ell }}={{T}_{sat}}\begin{matrix} | |

- | , & | + | , & y={{\delta }_{\ell }} \\ |

- | \end{ | + | \end{matrix}</math> |

- | </math | + | (8.360)</center> |

- | (8.360) | + | |

- | <center><math> | + | <center><math>\dot{{m}''}={{\rho }_{\ell }}\left( {{u}_{\ell }}\frac{d{{\delta }_{\ell }}}{dx}-{{v}_{\ell }} \right)\begin{matrix} |

- | \dot m'' = \rho _\ell | + | , & y={{\delta }_{\ell }} \\ |

- | , & | + | \end{matrix}</math> |

- | \end{ | + | (8.361)</center> |

- | </math | + | |

- | (8.361) | + | <center><math>\dot{{m}''}{{h}_{\ell v}}=-{{k}_{m\ell }}\frac{\partial T}{\partial y}\begin{matrix} |

- | <center><math> | + | , & y={{\delta }_{\ell }} \\ |

- | \dot m'' | + | \end{matrix}</math> |

- | , & | + | (8.362)</center> |

- | \end{ | + | |

- | </math | + | where <math>\dot{{m}''}</math> is mass flux of condensate across the interface, and <math>{{k}_{m\ell }}</math> is thermal conductivity of the porous medium saturated with liquid. Combining of eqs. (8.361) and (8.362) yields |

- | (8.362) | + | |

- | where <math>\dot m''</math> | + | <center><math>{{\rho }_{\ell }}{{h}_{\ell v}}\left( {{u}_{\ell }}\frac{d{{\delta }_{\ell }}}{dx}-{{v}_{\ell }} \right)=-{{k}_{m\ell }}\frac{\partial T}{\partial y}\begin{matrix} |

- | + | , & y={{\delta }_{\ell }} \\ | |

- | + | \end{matrix}</math> | |

- | <center><math> | + | (8.363)</center> |

- | \rho _\ell | + | |

- | , & | + | |

- | \end{ | + | |

- | </math | + | |

- | (8.363) | + | |

Introducing stream function | Introducing stream function | ||

- | <center><math> | + | |

- | + | <center><math>{{u}_{\ell }}=\frac{\partial \psi }{\partial y}\begin{matrix} | |

- | , & { | + | , & {{v}_{\ell }}=-\frac{\partial \psi }{\partial x} \\ |

- | \end{ | + | \end{matrix}</math> |

- | </math | + | (8.364)</center> |

- | (8.364) | + | |

and the following similarity variables: | and the following similarity variables: | ||

- | <center><math> | + | |

- | \eta | + | <center><math>\eta =\sqrt{R{{a}_{\ell x}}}\frac{y}{x}</math> |

- | </math | + | (8.365)</center> |

- | (8.365) | + | |

- | <center><math> | + | <center><math>f(\eta )=\frac{\psi }{{{\alpha }_{\ell }}\sqrt{R{{a}_{\ell x}}}}</math> |

- | f(\eta ) = \frac{\psi }{{\alpha _\ell | + | (8.366)</center> |

- | </math | + | |

- | (8.366) | + | <center><math>\theta (\eta )=\frac{{{T}_{\ell }}-{{T}_{sat}}}{{{T}_{w}}-{{T}_{sat}}}</math> |

- | <center><math> | + | (8.367)</center> |

- | \theta (\eta ) = \frac{{ | + | |

- | </math | + | |

- | (8.367) | + | |

where | where | ||

- | <center><math> | + | |

- | + | <center><math>R{{a}_{\ell x}}=\frac{({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi Kx}{{{\mu }_{\ell }}{{\alpha }_{\ell }}}</math> | |

- | </math | + | (8.368)</center> |

- | (8.368) | + | |

the governing equations and the corresponding boundary conditions become | the governing equations and the corresponding boundary conditions become | ||

- | <center><math> | + | |

- | f' = 1 | + | <center><math>{f}'=1</math> |

- | </math | + | (8.369)</center> |

- | (8.369) | + | |

- | <center><math> | + | <center><math>2{\theta }''+f{\theta }'=0</math> |

- | 2\theta '' + f\theta ' = 0 | + | (8.370)</center> |

- | </math | + | |

- | (8.370) | + | <center><math>f(0)=0</math> |

- | <center><math> | + | (8.371)</center> |

- | f(0) = 0 | + | |

- | </math | + | <center><math>\theta (0)=1</math> |

- | (8.371) | + | (8.372)</center> |

- | <center><math> | + | |

- | \theta (0) = 1 | + | <center><math>\theta ({{\eta }_{\delta }})=0</math> |

- | </math | + | (8.373)</center> |

- | (8.372) | + | |

- | <center><math> | + | <center><math>\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}f({{\eta }_{\delta }})</math> |

- | \theta (\eta _\delta | + | (8.374)</center> |

- | </math | + | |

- | (8.373) | + | |

- | <center><math> | + | |

- | {\ | + | |

- | </math | + | |

- | (8.374) | + | |

where | where | ||

- | <center><math> | + | |

- | \eta _\delta | + | <center><math>{{\eta }_{\delta }}=\sqrt{R{{a}_{\ell x}}}\frac{{{\delta }_{\ell }}}{x}</math> |

- | </math | + | (8.375)</center> |

- | (8.375) | + | |

is the dimensionless liquid film thickness and | is the dimensionless liquid film thickness and | ||

- | <center><math> | + | |

- | {\ | + | <center><math>\text{J}{{\text{a}}_{\ell }}=\frac{{{c}_{p\ell }}({{T}_{sat}}-{{T}_{w}})}{{{h}_{\ell v}}}</math> |

- | </math | + | (8.376)</center> |

- | (8.376) | + | |

is Jakob number that measures the degree of subcooling at the wall. | is Jakob number that measures the degree of subcooling at the wall. | ||

+ | |||

Integrating eq. (8.369) and considering eq. (8.371), one obtains | Integrating eq. (8.369) and considering eq. (8.371), one obtains | ||

- | <center><math> | + | |

- | f = \eta | + | <center><math>f=\eta </math> |

- | </math | + | (8.377)</center> |

- | (8.377) | + | |

which can be substituted into eqs. (8.370) and (8.374) to get | which can be substituted into eqs. (8.370) and (8.374) to get | ||

- | <center><math> | + | |

- | 2\theta '' + \eta \theta ' = 0 | + | <center><math>2{\theta }''+\eta {\theta }'=0</math> |

- | </math | + | (8.378)</center> |

- | (8.378) | + | |

- | <center><math> | + | <center><math>\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}{{\eta }_{\delta }}</math> |

- | {\ | + | (8.379)</center> |

- | </math | + | |

- | (8.379) | + | |

The solution of eq. (8.378) with eqs. (8.372) and (8.373) as boundary conditions is | The solution of eq. (8.378) with eqs. (8.372) and (8.373) as boundary conditions is | ||

- | <center><math> | + | |

- | \theta (\eta ) = 1 - \frac | + | <center><math>\theta (\eta )=1-\frac{\text{erf}(\eta /2)}{\text{erf(}{{\eta }_{\delta }}/2)}</math> |

- | </math | + | (8.380)</center> |

- | (8.380) | + | |

where the dimensionless film thickness can be obtained by substituting eq. (8.380) into eq. (8.379): | where the dimensionless film thickness can be obtained by substituting eq. (8.380) into eq. (8.379): | ||

- | <center><math> | + | |

- | {\ | + | <center><math>\text{J}{{\text{a}}_{\ell }}=\sqrt{\frac{\pi {{\eta }_{\delta }}}{2}}\exp \left( \frac{\eta _{\delta }^{2}}{4} \right)\text{erf}\left( \frac{{{\eta }_{\delta }}}{2} \right)</math> |

- | </math | + | (8.381)</center> |

- | (8.381) | + | |

The heat flux at the wall is | The heat flux at the wall is | ||

- | <center><math> | + | |

- | q'' | + | <center><math>{{{q}''}_{w}}=-{{k}_{m\ell }}{{\left( \frac{\partial {{T}_{\ell }}}{\partial y} \right)}_{y=0}}=\frac{{{k}_{m\ell }}({{T}_{sat}}-{{T}_{w}})\sqrt{\text{R}{{\text{a}}_{\ell x}}}}{x}{{{\theta }'}_{\ell }}(0)</math> |

- | </math | + | (8.382)</center> |

- | (8.382) | + | |

and the local Nusselt number is | and the local Nusselt number is | ||

- | <center><math> | + | |

- | + | <center><math>N{{u}_{x}}=\frac{{{{{q}''}}_{w}}x}{{{k}_{m\ell }}({{T}_{sat}}-{{T}_{w}})}=\frac{Ra_{\ell x}^{1/2}}{\sqrt{\pi }\text{erf(}{{\eta }_{\delta }}/2)}</math> | |

- | </math | + | (8.383)</center> |

- | (8.383) | + | |

- | where <math>\eta _\delta | + | where <math>{{\eta }_{\delta }}</math> is function of Jakob number, <math>\text{J}{{\text{a}}_{\ell }}</math>, as indicated by eq. (8.381). [[#References|Cheng (1981)]] recommended that eq. (8.383) can be approximated using |

- | + | ||

- | , as indicated by eq. (8.381). Cheng (1981) recommended that eq. (8.383) can be approximated using | + | <center><math>N{{u}_{x}}={{\left( \frac{1}{2J{{a}_{\ell }}}+\frac{1}{\pi } \right)}^{1/2}}Ra_{\ell x}^{1/2}</math> |

- | <center><math> | + | (8.384)</center> |

- | + | ||

- | </math | + | |

- | (8.384) | + | |

In practical application, the average Nusselt number is often of the interest. It can be obtained by integrating eq. (8.384): | In practical application, the average Nusselt number is often of the interest. It can be obtained by integrating eq. (8.384): | ||

- | <center><math> | + | |

- | \overline {Nu} | + | <center><math>\overline{Nu}=\frac{\bar{h}L}{{{k}_{m\ell }}}={{\left( \frac{1}{J{{a}_{\ell }}}+\frac{2}{\pi } \right)}^{1/2}}Ra_{\ell L}^{1/2}</math> |

- | </math></center> | + | (8.385)</center> |

+ | |||

+ | ==References== |

## Revision as of 17:54, 31 May 2010

When condensation is dominated by gravity, the effect of surface tension is negligible, and consequently, no two-phase region exists. Condensation along an inclined wall in a porous medium (See Fig. 8.30) will be discussed in this subsection. A porous medium saturated with dry vapor at its saturation temperature, Tsat, is bounded by an inclined impermeable wall with a temperature Tw (*T*_{w} < *T*_{sat}). Since the wall temperature is below saturation temperature, film condensation occurs on the inclined wall and the condensate flows downward due to gravity. It is assumed that the condensation is gravity-dominated and, therefore, that the liquid and vapor are separated by a sharp interface, not a two-phase region. In addition, the following assumptions are made by Cheng (1981) in order to obtain an analytical solution:

1. The condensate film is very thin compared to the length of the inclined wall () so that boundary layer assumption is valid.

2. The properties for the porous medium, liquid, and vapor are independent from temperature.

3. The inclination angle, φ, is small enough for the gravity component in the normal direction of the surface to be negligible.

4. Darcy’s law is valid for both liquid and vapor phases.

5. The saturation temperature, Tsat, is constant.

Under these assumptions, the continuity, momentum, and energy equations for the liquid layer are

The boundary conditions at the wall are

At the interface, the boundary conditions are

where is mass flux of condensate across the interface, and is thermal conductivity of the porous medium saturated with liquid. Combining of eqs. (8.361) and (8.362) yields

Introducing stream function

and the following similarity variables:

where

the governing equations and the corresponding boundary conditions become

*f*' = 1 (8.369)

*f*θ' = 0 (8.370)

*f*(0) = 0 (8.371)

_{δ}) = 0 (8.373)

where

is the dimensionless liquid film thickness and

is Jakob number that measures the degree of subcooling at the wall.

Integrating eq. (8.369) and considering eq. (8.371), one obtains

*f*= η (8.377)

which can be substituted into eqs. (8.370) and (8.374) to get

The solution of eq. (8.378) with eqs. (8.372) and (8.373) as boundary conditions is

where the dimensionless film thickness can be obtained by substituting eq. (8.380) into eq. (8.379):

The heat flux at the wall is

and the local Nusselt number is

where η_{δ} is function of Jakob number, , as indicated by eq. (8.381). Cheng (1981) recommended that eq. (8.383) can be approximated using

In practical application, the average Nusselt number is often of the interest. It can be obtained by integrating eq. (8.384):