Effect of surface tension on condensation in porous media

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The analysis in the preceding subsection is valid for gravity-dominated condensation in porous media ({\rm{Bo}} \gg 1 ). When the condensation is gravity-capillary forces dominated (Bo˜1 ) or capillary force dominated ({\rm{Bo}} \ll 1 ), there will be a two-phase region that is saturated by a mixture of liquid and vapor, as shown in Fig. 8.29. The fraction of liquid in the pore space is defined as saturation:


\gamma _\ell   = \frac{{\varepsilon _\ell  }}{\varepsilon }

(8.386) where \varepsilon _\ell

and \varepsilon 
are volume fraction of the liquid and porosity in the porous media. The effect of surface tension on condensation in porous media was studied by Majumdar and Tien (1990) and their work will be briefly described below.

The continuity equation for the two-phase region is


\rho _\ell  \left( {\frac{{\partial u_\ell  }}{{\partial x}} + \frac{{\partial v_\ell  }}{{\partial y_2 }}} \right) + \rho _v \left( {\frac{{\partial u_v }}{{\partial x}} + \frac{{\partial v_v }}{{\partial y_2 }}} \right) = 0

(8.387) where y2 is measured from the interface between liquid and the two-phase region (see Fig. 8.29). The mass fluxes for liquid and vapor are governed by Darcy’s law, as follows:


\dot m''_\ell   =  - \frac{{KK_{r\ell } }}{{\nu _\ell  }}\nabla p_\ell

(8.388)


\dot m''_v  =  - \frac{{KK_{rv} }}{{\nu _v }}\nabla p_v

(8.389) where K_{r\ell }

and Krv
are relative permeabilities (dimensionless) for liquid and vapor phases, respectively. Their products with the permeability K represent the permeability for liquid and vapor flow in the porous media.  

Since the density of the vapor is much lower than that of the liquid, the pressure variation in the liquid phase is very insignificant. Also, the change of capillary pressure is due mainly to change of liquid pressure. Consequently, the vapor flow is negligible compared to the liquid flow, so that eq. (8.387) is reduced to


\frac{{\partial u_\ell  }}{{\partial x}} + \frac{{\partial v_\ell  }}{{\partial y_2 }} = 0

(8.390)

The velocity components in the x- and y- directions are


u_\ell   = K_{r\ell } u_D  = \frac{{KK_{r\ell } (\rho _\ell   - \rho _v )g}}{{\mu _\ell  }}

(8.391)


v_\ell   =  - \frac{{KK_{r\ell } }}{{\mu _\ell  }}\frac{{\partial p_\ell  }}{{\partial y_2 }} = \frac{{KK_{r\ell } }}{{\mu _\ell  }}\frac{{\partial p_{cap} }}{{\partial y_2 }}

(8.392)

where

u_D  = K(\rho _\ell   - \rho _v )g/\mu _\ell
is Darcian velocity. The capillary pressure can be written as 

p_{cap}  = \frac{\sigma }{{\sqrt {K/\varepsilon } }}f(s)

(8.393) where f(s) is a Leverett’s function (see Section 4.6.6):

f(s) = 1.417(1 − s) − 2.120(1 − s)2 + 1.263(1 − s)3

(8.394) and s is dimensionless saturation defined as


s = \frac{{\gamma _\ell   - \gamma _{\ell i} }}{{1 - \gamma _{\ell i} }}

(8.395) where \gamma _{\ell i}

is irreducible saturation, below which liquid flow will not occur. The relative permeability in eqs. (8.391) and (8.392) is obtained from

K_{r\ell }  = s^3

(8.396) Substituting eqs. (8.391) – (8.396) into eq. (8.390), the following form of the dimensionless continuity equation is obtained:

Failed to parse (syntax error): 3\frac{{\partial s}}{{\partial x}} + \left<center><math> {\frac{{\sigma /\sqrt {K/\varepsilon } }}{{(\rho _\ell - \rho _v )g}}} \right
\left
Failed to parse (syntax error): {(3f' + sf'')\left( {\frac{{\partial s}}{{\partial x}}} \right)^2 + sf'\frac{{\partial ^2 s}}{{\partial y_2^2 }}} \right
= 0

</math></center> (8.397) which is subjected to the following boundary conditions:


s = 0\begin{array}{*{20}c}
   , & {x = 0}  \\
\end{array}

(8.398)


s = 1\begin{array}{*{20}c}
   , & {y_2  = 0}  \\
\end{array}

(8.399)


s = 0\begin{array}{*{20}c}
   , & {y_2  \to \infty }  \\
\end{array}

(8.400) Introducing the following similarity variable:

Failed to parse (syntax error): \eta = y_2 \left<center><math> {\frac{{(\rho _\ell - \rho _v )g}}{{(\sigma /\sqrt {K/\varepsilon } )x}}} \right
^2

</math></center> (8.401) eqs. (8.397) – (8.400) are transformed to


s'' = \frac{{3\eta s' - 2(3f' + sf'')s'^2 }}{{2sf'}}

(8.402)


s = 1\begin{array}{*{20}c}
   , & {\eta  = 0}  \\
\end{array}

(8.403)


s = 0\begin{array}{*{20}c}
   , & {\eta  \to \infty }  \\
\end{array}

(8.404) The numerical solution of eq. (8.402) was obtained using the Runge-Kutta method and it is concluded that η = 1.296

can be chosen to determine the thickness of the two-phase region.

For the liquid region, the model in the preceding subsection overpredicts the heat transfer coefficient because of slip in the velocity at the wall. They presented a new model based on the following assumptions: (a) the boundary layer assumption applies in the liquid film, (2) convection terms in the energy equation are negligible, (3) subcooling in the liquid is negligible, and (4) the fluid properties are constants. The governing equations for the liquid film in dimensionless form are


\frac{{\partial u_\ell ^ +  }}{{\partial x^ +  }} + \frac{{\partial v_\ell ^ +  }}{{\partial y^ +  }} = 0

(8.405)


\frac{{\partial ^2 u_\ell ^ +  }}{{\partial y^{ + 2} }} + 1 - u_\ell ^ +   = 0

(8.406)


\frac{{\partial ^2 \theta _\ell  }}{{\partial y^{ + 2} }} = 0

(8.407) where the dimensionless variables are defined as


u_\ell ^ +   = \frac{{u_\ell  }}{{u_D }}{\rm{,   }}v_\ell ^ +   = \frac{{v_\ell  }}{{u_D }}{\rm{,   }}x^ +   = \frac{x}{{\sqrt K }}{\rm{,   }}y^ +   = \frac{y}{{\sqrt K }}{\rm{,   }}\delta _\ell ^ +   = \frac{{\delta _\ell  }}{{\sqrt K }}{\rm{,   }}\theta  = \frac{{T - T_w }}{{T_{sat}  - T_w }}

(8.408) Compared with eq. (8.356) in Cheng’s (1981) model, eq. (8.406) allows for nonslip conditions at the wall. The boundary conditions at the wall are


u_\ell ^ +   = v_\ell ^ +   = \theta  = 0\begin{array}{*{20}c}
   , & {y^ +   = 0}  \\
\end{array}

(8.409) The boundary conditions at the interface between the liquid film and the two-phase region require that the velocity and shear stress in these two regions match, which makes the solution of the condensation problem very challenging. Majumdar and Tien (1990) proposed three models to handle the boundary condition at the interface between the liquid and the two-phase regions; two of them are discussed below.

Model 1. At the interface between the liquid and the two-phase region, the shear stress is zero, i.e., \partial u_\ell ^ +  /\partial y^ +   = 0

at \tilde y = \delta _\ell ^ +  

, which is the same as in classical Nusselt analysis. The velocity profile in the liquid layer is


u_\ell ^ +   = 1 - \cosh y^ +   + \tanh \delta _\ell ^ +  \sinh y^ +

(8.410) The liquid layer thickness can be obtained from an energy balance at the interface, and the result is


(1 - {\rm{sech}}^2 \delta _\ell ^ +  )\frac{{d\delta _\ell ^ +  }}{{dx^ +  }} + (1 - {\rm{sech}}\delta _\ell ^ +  )\frac{{0.373}}{{\sqrt {{\rm{Bo}}x^ +  } }} = \frac{{{\rm{Ja}}_\ell  }}{{\delta _\ell ^ +  {\rm{Ra}}_K }}

(8.411) where


{\rm{Ra}}_K  = \frac{{K^{3/2} (\rho _\ell   - \rho _v )g}}{{\mu _\ell  \alpha _e }}

(8.412) is the Rayleigh number based on permeability, and \alpha _e  = k_{eff} /(\rho _\ell  c_{p\ell } )

is effective thermal diffusivity. Analytical solution of eq. (8.411) is not possible and it must be solved numerically.

Model 2. This model also employs eq. (8.406) to obtain the velocity in the liquid layer, except that the boundary condition at y^ +   = \delta _\ell ^ +

is changed to u_\ell ^ +   = 1

. Although it is not as rigorous as Model 1, it is an improvement over Cheng (1981) because it uses a nonslip condition at the wall. The velocity profile in the liquid layer is


u_\ell ^ +   = 1 - \cosh y^ +   + \coth \delta _\ell ^ +  \sinh y^ +

(8.413)

and the overall energy balance at the interface is


\left( {1 - \frac{1}{{1 + \cosh \delta _\ell ^ +  }}} \right)\frac{{d\delta _\ell ^ +  }}{{dx^ +  }} + \frac{{0.373}}{{\sqrt {{\rm{Bo}}x^ +  } }} = \frac{{{\rm{Ja}}_\ell  }}{{\delta _\ell ^ +  {\rm{Ra}}_K }}

(8.414) which also needs to be solved numerically. In the liquid region, eq. (8.407) will yield a linear temperature profile, and the local Nusselt number is


Nu_x  = \frac{{x^ +  }}{{\delta _\ell ^ +  }}

(8.415) Figure 8.31 shows comparison between the results predicted from Models 1 and 2, together with the experimental results by White and Tien (1987) for constant porosity media. The parameter R in the figure is defined as


R = \frac{{{\rm{Ra}}_K }}{{{\rm{Bo}}}} = \frac{{\sigma \sqrt {K\varepsilon } }}{{\mu _\ell  \alpha _e }}

(8.416) which reflects the ratio of surface tension force and viscous force. It can be seen from Fig. 8.31(a) that for aluminum foam, the agreement between Model 1 and experimental results is very good, but Model 2 significantly overpredicts the heat transfer rates. This is expected because Model 2 is not as rigorous as Model 1. The slight overprediction of Model 1 may be attributed to the fact that shear stress at the interface is neglected. For polyurethane foam, on the other hand, Model 1 also significantly overpredict heat transfer rate. This overprediction is due to the fact that surface tension drag at the liquid region interface is neglected in Model 1. Since R in Fig. 8.31 (b) is one order of magnitude higher than that in Fig. 8.31 (a), it is expected that surface tension plays a more significant role in Fig. 8.31 (b).