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 (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 2. ) so that boundary layer assumption is valid. 3. The properties for the porous medium, liquid, and vapor are independent from temperature. 4. The inclination angle, φ 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. 7. 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{array}{*{20}c}
   , & {y = 0}  \\
\end{array}

(8.358)


T_\ell   = T_w \begin{array}{*{20}c}
   , & {y = 0}  \\
\end{array}

(8.359) At the interface, the boundary conditions are


T_\ell   = T_{sat} \begin{array}{*{20}c}
   , & {y = \delta _\ell  }  \\
\end{array}

(8.360)


\dot m'' = \rho _\ell  \left( {u_\ell  \frac{{d\delta _\ell  }}{{dx}} - v_\ell  } \right)\begin{array}{*{20}c}
   , & {y = \delta _\ell  }  \\
\end{array}

(8.361)


\dot m''h_{\ell v}  =  - k_{m\ell } \frac{{\partial T}}{{\partial y}}\begin{array}{*{20}c}
   , & {y = \delta _\ell  }  \\
\end{array}

(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{array}{*{20}c}
   , & {y = \delta _\ell  }  \\
\end{array}

(8.363) Introducing stream function


u_\ell   = \frac{{\partial \psi }}{{\partial y}}\begin{array}{*{20}c}
   , & {v_\ell   =  - \frac{{\partial \psi }}{{\partial x}}}  \\
\end{array}

(8.364) and the following similarity variables:


\eta  = \sqrt {Ra_{\ell x} } \frac{y}{x}

(8.365)


f(\eta ) = \frac{\psi }{{\alpha _\ell  \sqrt {Ra_{\ell x} } }}

(8.366)


\theta (\eta ) = \frac{{T_\ell   - T_{sat} }}{{T_w  - T_{sat} }}

(8.367) where


Ra_{\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)


{\rm{Ja}}_\ell  \theta '(\eta _\delta  ) =  - \frac{1}{2}f(\eta _\delta  )

(8.374) where


\eta _\delta   = \sqrt {Ra_{\ell x} } \frac{{\delta _\ell  }}{x}

(8.375) is the dimensionless liquid film thickness and


{\rm{Ja}}_\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)


{\rm{Ja}}_\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{{{\rm{erf}}(\eta /2)}}{{{\rm{erf(}}\eta _\delta  /2)}}

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


{\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)

(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 {{\rm{Ra}}_{\ell x} } }}{x}\theta '_\ell  (0)

(8.382) and the local Nusselt number is


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)}}

(8.383) where ηδ

is function of Jakob number, {\rm{Ja}}_\ell  

, as indicated by eq. (8.381). Cheng (1981) recommended that eq. (8.383) can be approximated using


Nu_x  = \left( {\frac{1}{{2Ja_\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 hL}}{{k_{m\ell } }} = \left( {\frac{1}{{Ja_\ell  }} + \frac{2}{\pi }} \right)^{1/2} Ra_{\ell L}^{1/2}