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>T_w  < T_{sat} </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:
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). 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:
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1. The condensate film is very thin compared to the length of the inclined wall (<math>\delta _\ell   \ll L</math>
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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  
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2. ) so that boundary layer assumption is valid.
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assumption is valid.
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3. The properties for the porous medium, liquid, and vapor are independent from temperature.
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-
4. The inclination angle, <math>\phi </math>
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2. The properties for the porous medium, liquid, and vapor are independent from temperature.
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5. , is small enough for the gravity component in the normal direction of the surface to be negligible.
+
 
-
6. Darcy’s law is valid for both liquid and vapor phases.
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3. The inclination angle, <math>\phi </math>, is small enough for the gravity component in the normal direction of the surface to be negligible.
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7. The saturation temperature, Tsat, is constant.
+
 
 +
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
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<center><math>
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-
\frac{{\partial u_\ell }}{{\partial x}} + \frac{{\partial v_\ell }}{{\partial y}} = 0
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<center><math>\frac{\partial {{u}_{\ell }}}{\partial x}+\frac{\partial {{v}_{\ell }}}{\partial y}=0</math>
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</math></center>
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(8.355)</center>
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(8.355)
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<center><math>
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<center><math>{{u}_{\ell }}=\frac{K}{{{\mu }_{\ell }}}({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi </math>
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u_\ell   = \frac{K}{{\mu _\ell }}(\rho _\ell   - \rho _v )g\cos \phi  
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(8.356)</center>
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</math></center>
+
-
(8.356)
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<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>
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<center><math>
+
(8.357)</center>
-
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>
+
-
(8.357)
+
The boundary conditions at the wall are
The boundary conditions at the wall are
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<center><math>
+
-
v_\ell   = 0\begin{array}{*{20}c}
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<center><math>{{v}_{\ell }}=0\begin{matrix}
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   , & {y = 0} \\
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   , & y=0  \\
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\end{array}
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\end{matrix}</math>
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</math></center>
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(8.358)</center>
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(8.358)
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-
<center><math>
+
<center><math>{{T}_{\ell }}={{T}_{w}}\begin{matrix}
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T_\ell   = T_w \begin{array}{*{20}c}
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   , & y=0  \\
-
   , & {y = 0} \\
+
\end{matrix}</math>
-
\end{array}
+
(8.359)</center>
-
</math></center>
+
 
-
(8.359)
+
At the interface, the boundary conditions are
At the interface, the boundary conditions are
-
<center><math>
+
-
T_\ell   = T_{sat} \begin{array}{*{20}c}
+
<center><math>{{T}_{\ell }}={{T}_{sat}}\begin{matrix}
-
   , & {y = \delta _\ell }  \\
+
   , & y={{\delta }_{\ell }}  \\
-
\end{array}
+
\end{matrix}</math>
-
</math></center>
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(8.360)</center>
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(8.360)
+
-
<center><math>
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<center><math>\dot{{m}''}={{\rho }_{\ell }}\left( {{u}_{\ell }}\frac{d{{\delta }_{\ell }}}{dx}-{{v}_{\ell }} \right)\begin{matrix}
-
\dot m'' = \rho _\ell \left( {u_\ell \frac{{d\delta _\ell }}{{dx}} - v_\ell } \right)\begin{array}{*{20}c}
+
   , & y={{\delta }_{\ell }}  \\
-
   , & {y = \delta _\ell }  \\
+
\end{matrix}</math>
-
\end{array}
+
(8.361)</center>
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</math></center>
+
-
(8.361)
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<center><math>\dot{{m}''}{{h}_{\ell v}}=-{{k}_{m\ell }}\frac{\partial T}{\partial y}\begin{matrix}
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<center><math>
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   , & y={{\delta }_{\ell }}  \\
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\dot m''h_{\ell v} = - k_{m\ell } \frac{{\partial T}}{{\partial y}}\begin{array}{*{20}c}
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\end{matrix}</math>
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   , & {y = \delta _\ell }  \\
+
(8.362)</center>
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\end{array}
+
 
-
</math></center>
+
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>
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<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}
-
is mass flux of condensate across the interface, and <math>k_{m\ell } </math>
+
   , & y={{\delta }_{\ell }}  \\
-
is thermal conductivity of the porous medium saturated with liquid. Combining of eqs. (8.361) and (8.362) yields
+
\end{matrix}</math>
-
<center><math>
+
(8.363)</center>
-
\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{array}{*{20}c}
+
 
-
   , & {y = \delta _\ell }  \\
+
-
\end{array}
+
-
</math></center>
+
-
(8.363)
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Introducing stream function  
Introducing stream function  
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<center><math>
+
-
u_\ell   = \frac{{\partial \psi }}{{\partial y}}\begin{array}{*{20}c}
+
<center><math>{{u}_{\ell }}=\frac{\partial \psi }{\partial y}\begin{matrix}
-
   , & {v_\ell   = - \frac{{\partial \psi }}{{\partial x}}}  \\
+
   , & {{v}_{\ell }}=-\frac{\partial \psi }{\partial x}  \\
-
\end{array}
+
\end{matrix}</math>
-
</math></center>
+
(8.364)</center>
-
(8.364)
+
 
and the following similarity variables:
and the following similarity variables:
-
<center><math>
+
-
\eta = \sqrt {Ra_{\ell x} } \frac{y}{x}
+
<center><math>\eta =\sqrt{R{{a}_{\ell x}}}\frac{y}{x}</math>
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</math></center>
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(8.365)</center>
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(8.365)
+
-
<center><math>
+
<center><math>f(\eta )=\frac{\psi }{{{\alpha }_{\ell }}\sqrt{R{{a}_{\ell x}}}}</math>
-
f(\eta ) = \frac{\psi }{{\alpha _\ell \sqrt {Ra_{\ell x} } }}
+
(8.366)</center>
-
</math></center>
+
-
(8.366)
+
<center><math>\theta (\eta )=\frac{{{T}_{\ell }}-{{T}_{sat}}}{{{T}_{w}}-{{T}_{sat}}}</math>
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<center><math>
+
(8.367)</center>
-
\theta (\eta ) = \frac{{T_\ell   - T_{sat} }}{{T_w  - T_{sat} }}
+
 
-
</math></center>
+
-
(8.367)
+
where  
where  
-
<center><math>
+
-
Ra_{\ell x} = \frac{{(\rho _\ell   - \rho _v )g\cos \phi Kx}}{{\mu _\ell \alpha _\ell }}
+
<center><math>R{{a}_{\ell x}}=\frac{({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi Kx}{{{\mu }_{\ell }}{{\alpha }_{\ell }}}</math>
-
</math></center>
+
(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></center>
+
(8.369)</center>
-
(8.369)
+
-
<center><math>
+
<center><math>2{\theta }''+f{\theta }'=0</math>
-
2\theta '' + f\theta ' = 0
+
(8.370)</center>
-
</math></center>
+
-
(8.370)
+
<center><math>f(0)=0</math>
-
<center><math>
+
(8.371)</center>
-
f(0) = 0
+
-
</math></center>
+
<center><math>\theta (0)=1</math>
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(8.371)
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(8.372)</center>
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<center><math>
+
-
\theta (0) = 1
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<center><math>\theta ({{\eta }_{\delta }})=0</math>
-
</math></center>
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(8.373)</center>
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(8.372)
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-
<center><math>
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<center><math>\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}f({{\eta }_{\delta }})</math>
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\theta (\eta _\delta ) = 0
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(8.374)</center>
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</math></center>
+
 
-
(8.373)
+
-
<center><math>
+
-
{\rm{Ja}}_\ell \theta '(\eta _\delta ) = - \frac{1}{2}f(\eta _\delta )
+
-
</math></center>
+
-
(8.374)
+
where  
where  
-
<center><math>
+
-
\eta _\delta   = \sqrt {Ra_{\ell x} } \frac{{\delta _\ell }}{x}
+
<center><math>{{\eta }_{\delta }}=\sqrt{R{{a}_{\ell x}}}\frac{{{\delta }_{\ell }}}{x}</math>
-
</math></center>
+
(8.375)</center>
-
(8.375)
+
 
is the dimensionless liquid film thickness and  
is the dimensionless liquid film thickness and  
-
<center><math>
+
-
{\rm{Ja}}_\ell   = \frac{{c_{p\ell } (T_{sat} - T_w )}}{{h_{\ell v} }}
+
<center><math>\text{J}{{\text{a}}_{\ell }}=\frac{{{c}_{p\ell }}({{T}_{sat}}-{{T}_{w}})}{{{h}_{\ell v}}}</math>
-
</math></center>
+
(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></center>
+
(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></center>
+
(8.378)</center>
-
(8.378)
+
-
<center><math>
+
<center><math>\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}{{\eta }_{\delta }}</math>
-
{\rm{Ja}}_\ell \theta '(\eta _\delta ) = - \frac{1}{2}\eta _\delta
+
(8.379)</center>
-
</math></center>
+
 
-
(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{{{\rm{erf}}(\eta /2)}}{{{\rm{erf(}}\eta _\delta /2)}}
+
<center><math>\theta (\eta )=1-\frac{\text{erf}(\eta /2)}{\text{erf(}{{\eta }_{\delta }}/2)}</math>
-
</math></center>
+
(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>
+
-
{\rm{Ja}}_\ell   = \sqrt {\frac{{\pi \eta _\delta }}{2}} \exp \left( {\frac{{\eta _\delta ^2 }}{4}} \right){\rm{erf}}\left( {\frac{{\eta _\delta }}{2}} \right)
+
<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></center>
+
(8.381)</center>
-
(8.381)
+
 
The heat flux at the wall is
The heat flux at the wall is
-
<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 {{\rm{Ra}}_{\ell x} } }}{x}\theta '_\ell (0)
+
<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></center>
+
(8.382)</center>
-
(8.382)
+
 
and the local Nusselt number is
and the local Nusselt number is
-
<center><math>
+
-
Nu_x  = \frac{{q''_w x}}{{k_{m\ell } (T_{sat} - T_w )}} = \frac{{Ra_{\ell x}^{1/2} }}{{\sqrt \pi {\rm{erf(}}\eta _\delta /2)}}
+
<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></center>
+
(8.383)</center>
-
(8.383)
+
 
-
where <math>\eta _\delta </math>
+
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  
-
is function of Jakob number, <math>{\rm{Ja}}_\ell </math>
+
-
, 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>
-
Nu_x  = \left( {\frac{1}{{2Ja_\ell }} + \frac{1}{\pi }} \right)^{1/2} Ra_{\ell x}^{1/2}  
+
 
-
</math></center>
+
-
(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} = \frac{{\bar hL}}{{k_{m\ell } }} = \left( {\frac{1}{{Ja_\ell }} + \frac{2}{\pi }} \right)^{1/2} Ra_{\ell L}^{1/2}  
+
<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 (Tw < Tsat). 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 ({{\delta }_{\ell }}\ll L) 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

\frac{\partial {{u}_{\ell }}}{\partial x}+\frac{\partial {{v}_{\ell }}}{\partial y}=0 (8.355)
{{u}_{\ell }}=\frac{K}{{{\mu }_{\ell }}}({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi (8.356)
{{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}}} (8.357)

The boundary conditions at the wall are

{{v}_{\ell }}=0\begin{matrix}
   , & y=0  \\
\end{matrix} (8.358)
{{T}_{\ell }}={{T}_{w}}\begin{matrix}
   , & y=0  \\
\end{matrix} (8.359)

At the interface, the boundary conditions are

{{T}_{\ell }}={{T}_{sat}}\begin{matrix}
   , & y={{\delta }_{\ell }}  \\
\end{matrix} (8.360)
\dot{{m}''}={{\rho }_{\ell }}\left( {{u}_{\ell }}\frac{d{{\delta }_{\ell }}}{dx}-{{v}_{\ell }} \right)\begin{matrix}
   , & y={{\delta }_{\ell }}  \\
\end{matrix} (8.361)
\dot{{m}''}{{h}_{\ell v}}=-{{k}_{m\ell }}\frac{\partial T}{\partial y}\begin{matrix}
   , & y={{\delta }_{\ell }}  \\
\end{matrix} (8.362)

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

{{\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} (8.363)

Introducing stream function

{{u}_{\ell }}=\frac{\partial \psi }{\partial y}\begin{matrix}
   , & {{v}_{\ell }}=-\frac{\partial \psi }{\partial x}  \\
\end{matrix} (8.364)

and the following similarity variables:

\eta =\sqrt{R{{a}_{\ell x}}}\frac{y}{x} (8.365)
f(\eta )=\frac{\psi }{{{\alpha }_{\ell }}\sqrt{R{{a}_{\ell x}}}} (8.366)
\theta (\eta )=\frac{{{T}_{\ell }}-{{T}_{sat}}}{{{T}_{w}}-{{T}_{sat}}} (8.367)

where

R{{a}_{\ell x}}=\frac{({{\rho }_{\ell }}-{{\rho }_{v}})g\cos \phi Kx}{{{\mu }_{\ell }}{{\alpha }_{\ell }}} (8.368)

the governing equations and the corresponding boundary conditions become

f' = 1 (8.369)
2θ'' + fθ' = 0 (8.370)
f(0) = 0 (8.371)
θ(0) = 1 (8.372)
θ(ηδ) = 0 (8.373)
\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}f({{\eta }_{\delta }}) (8.374)

where

{{\eta }_{\delta }}=\sqrt{R{{a}_{\ell x}}}\frac{{{\delta }_{\ell }}}{x} (8.375)

is the dimensionless liquid film thickness and

\text{J}{{\text{a}}_{\ell }}=\frac{{{c}_{p\ell }}({{T}_{sat}}-{{T}_{w}})}{{{h}_{\ell v}}} (8.376)

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

2θ'' + ηθ' = 0 (8.378)
\text{J}{{\text{a}}_{\ell }}{\theta }'({{\eta }_{\delta }})=-\frac{1}{2}{{\eta }_{\delta }} (8.379)

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

\theta (\eta )=1-\frac{\text{erf}(\eta /2)}{\text{erf(}{{\eta }_{\delta }}/2)} (8.380)

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

\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) (8.381)

The heat flux at the wall is

{{{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) (8.382)

and the local Nusselt number is

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)} (8.383)

where ηδ is function of Jakob number, \text{J}{{\text{a}}_{\ell }}, as indicated by eq. (8.381). Cheng (1981) recommended that eq. (8.383) can be approximated using

N{{u}_{x}}={{\left( \frac{1}{2J{{a}_{\ell }}}+\frac{1}{\pi } \right)}^{1/2}}Ra_{\ell x}^{1/2} (8.384)

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

\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} (8.385)

References