Effect of surface tension on condensation in porous media
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Yuwen Zhang (Talk  contribs) m (moved Effect of Surface Tension on Condensation in Porous Media to Effect of surface tension on condensation in porous media) 

(9 intermediate revisions not shown)  
Line 1:  Line 1:  
  The analysis in the  +  The analysis in the [[gravity dominated filmcondensation in a porous medium]] is valid for <math>\text{Bo}\gg 1</math>. When the condensation is gravitycapillary forces dominated (<math>\text{Bo}\sim 1</math>) or capillary force dominated (<math>\text{Bo}\ll 1</math>), there will be a twophase region that is saturated by a mixture of liquid and vapor. The fraction of liquid in the pore space is defined as saturation: 
  +  
  ) or capillary force dominated (<math>  +  { class="wikitable" border="0" 
  ), there will be a twophase region that is saturated by a mixture of liquid and vapor  +   
  +   width="100%"   
  \gamma _\ell  +  <Center><math>{{\gamma }_{\ell }}=\frac{{{\varepsilon }_{\ell }}}{\varepsilon }</math></center> 
  </math></center>  +  {{EquationRef(1)}} 
  +  }  
  where <math>\varepsilon _\ell  +  
  +  where <math>{{\varepsilon }_{\ell }}</math> and <math>\varepsilon </math> 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 twophase region is  The continuity equation for the twophase region is  
  +  
  \rho _\ell  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
  where  +  <center><math>{{\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</math></center> 
  +  {{EquationRef(2)}}  
  \dot m''_\ell  +  } 
  </math></center>  +  
  +  where ''y<sub>2</sub>'' is measured from the interface between liquid and the twophase region (see Fig. 8.29). The mass fluxes for liquid and vapor are governed by Darcy’s law, as follows:  
  +  
  \dot m''  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
  where <math>  +  <center><math>{{{\dot{m}}''}_{\ell }}=\frac{K{{K}_{r\ell }}}{{{\nu }_{\ell }}}\nabla {{p}_{\ell }}</math></center> 
  +  {{EquationRef(3)}}  
  +  }  
  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. (  +  
  +  { class="wikitable" border="0"  
  \frac  +   
  </math></center>  +   width="100%"  
  +  <center><math>{{{\dot{m}}''}_{v}}=\frac{K{{K}_{rv}}}{{{\nu }_{v}}}\nabla {{p}_{v}}</math></center>  
+  {{EquationRef(4)}}  
+  }  
+  
+  where <math>{{K}_{r\ell }}</math> and <math>{{K}_{rv}}</math> 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. (2) is reduced to  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>\frac{\partial {{u}_{\ell }}}{\partial x}+\frac{\partial {{v}_{\ell }}}{\partial {{y}_{2}}}=0</math></center>  
+  {{EquationRef(5)}}  
+  }  
The velocity components in the x and y directions are  The velocity components in the x and y directions are  
  +  
  +  { class="wikitable" border="0"  
  </math></center>  +   
  +   width="100%"   
  +  <center><math>{{u}_{\ell }}={{K}_{r\ell }}{{u}_{D}}=\frac{K{{K}_{r\ell }}({{\rho }_{\ell }}{{\rho }_{v}})g}{{{\mu }_{\ell }}}</math></center>  
  +  {{EquationRef(6)}}  
  </math></center>  +  } 
  +  
  where  +  { class="wikitable" border="0" 
  +    
  </math>  +   width="100%"  
  +  <center><math>{{v}_{\ell }}=\frac{K{{K}_{r\ell }}}{{{\mu }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial {{y}_{2}}}=\frac{K{{K}_{r\ell }}}{{{\mu }_{\ell }}}\frac{\partial {{p}_{cap}}}{\partial {{y}_{2}}}</math></center>  
  +  {{EquationRef(7)}}  
  +  }  
  </math></center>  +  
  +  where <math>{{u}_{D}}=K({{\rho }_{\ell }}{{\rho }_{v}})g/{{\mu }_{\ell }}</math> is Darcian velocity. The capillary pressure can be written as  
  where f(s) is a Leverett’s function  +  
  +  { class="wikitable" border="0"  
  f(s) = 1.417(1  s)  2.120(1  s)^2  +   
  </math></center>  +   width="100%"  
  +  <center><math>{{p}_{cap}}=\frac{\sigma }{\sqrt{K/\varepsilon }}f(s)</math></center>  
  and s is dimensionless saturation defined as  +  {{EquationRef(8)}} 
  +  }  
  s = \frac{{\gamma _\ell  +  
  </math></center>  +  where ''f''(s) is a Leverett’s function: 
  +  
  where <math>\gamma _{\ell i} </math>  +  { class="wikitable" border="0" 
  +    
  +   width="100%"   
  +  <center><big><big><math>f(s)=1.417(1s)2.120{{(1s)}^{2}}+1.263{{(1s)}^{3}}</math></big></big></center>  
  </math></center>  +  {{EquationRef(9)}} 
  +  }  
  Substituting eqs. (  +  
  +  and ''s'' is dimensionless saturation defined as  
  3\frac  +  
  </math></center>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>s=\frac{{{\gamma }_{\ell }}{{\gamma }_{\ell i}}}{1{{\gamma }_{\ell i}}}</math></center>  
+  {{EquationRef(10)}}  
+  }  
+  
+  where <math>{{\gamma }_{\ell i}}</math> is irreducible saturation, below which liquid flow will not occur. The relative permeability in eqs. (6) and (7) is obtained from  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>{{K}_{r\ell }}={{s}^{3}}</math></center>  
+  {{EquationRef(11)}}  
+  }  
+  
+  Substituting eqs. (6) – (11) into eq. (5), the following form of the dimensionless continuity equation is obtained:  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>3\frac{\partial s}{\partial x}+\left[ \frac{\sigma /\sqrt{K/\varepsilon }}{({{\rho }_{\ell }}{{\rho }_{v}})g} \right]\left[ (3{f}'+s{f}''){{\left( \frac{\partial s}{\partial x} \right)}^{2}}+s{f}'\frac{{{\partial }^{2}}s}{\partial y_{2}^{2}} \right]=0</math></center>  
+  {{EquationRef(12)}}  
+  }  
+  
which is subjected to the following boundary conditions:  which is subjected to the following boundary conditions:  
  +  
  s = 0\begin{  +  { class="wikitable" border="0" 
  , &  +   
  \end{  +   width="100%"  
  </math></center>  +  <center><math>s=0\begin{matrix} 
  +  , & x=0 \\  
  +  \end{matrix}</math></center>  
  s = 1\begin{  +  {{EquationRef(13)}} 
  , & {  +  } 
  \end{  +  
  </math></center>  +  { class="wikitable" border="0" 
  +    
  +   width="100%"   
  s = 0\begin{  +  <center><math>s=1\begin{matrix} 
  , & {  +  , & {{y}_{2}}=0 \\ 
  \end{  +  \end{matrix}</math></center> 
  </math></center>  +  {{EquationRef(14)}} 
  +  }  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>s=0\begin{matrix}  
+  , & {{y}_{2}}\to \infty \\  
+  \end{matrix}</math></center>  
+  {{EquationRef(15)}}  
+  }  
+  
Introducing the following similarity variable:  Introducing the following similarity variable:  
  +  
  \eta  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
  eqs. (  +  <center><math>\eta ={{y}_{2}}{{\left[ \frac{({{\rho }_{\ell }}{{\rho }_{v}})g}{(\sigma /\sqrt{K/\varepsilon })x} \right]}^{2}}</math></center> 
  +  {{EquationRef(16)}}  
  s'' = \frac  +  } 
  </math></center>  +  
  +  eqs. (12) – (15) are transformed to  
  +  
  s = 1\begin{  +  { class="wikitable" border="0" 
  , &  +   
  \end{  +   width="100%"  
  </math></center>  +  <center><math>{s}''=\frac{3\eta {s}'2(3{f}'+s{f}''){{{{s}'}}^{2}}}{2s{f}'}</math></center> 
  +  {{EquationRef(17)}}  
  +  }  
  s = 0\begin{  +  
  , &  +  { class="wikitable" border="0" 
  \end{  +   
  </math></center>  +   width="100%"  
  +  <center><math>s=1\begin{matrix}  
  The numerical solution of eq. (  +  , & \eta =0 \\ 
  +  \end{matrix}</math></center>  
+  {{EquationRef(18)}}  
+  }  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>s=0\begin{matrix}  
+  , & \eta \to \infty \\  
+  \end{matrix}</math></center>  
+  {{EquationRef(19)}}  
+  }  
+  
+  The numerical solution of eq. (17) was obtained using the RungeKutta method and it is concluded that <math>\eta =1.296</math> can be chosen to determine the thickness of the twophase 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  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  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
  +  <center><math>\frac{\partial u_{\ell }^{+}}{\partial {{x}^{+}}}+\frac{\partial v_{\ell }^{+}}{\partial {{y}^{+}}}=0</math></center>  
  \frac{{\partial ^2 u_\ell ^ +  +  {{EquationRef(20)}} 
  </math></center>  +  } 
  +  
  +  { class="wikitable" border="0"  
  \frac{{\partial ^2 \theta _\ell  +   
  </math></center>  +   width="100%"  
  +  <center><math>\frac{{{\partial }^{2}}u_{\ell }^{+}}{\partial {{y}^{+2}}}+1u_{\ell }^{+}=0</math></center>  
+  {{EquationRef(21)}}  
+  }  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>\frac{{{\partial }^{2}}{{\theta }_{\ell }}}{\partial {{y}^{+2}}}=0</math></center>  
+  {{EquationRef(22)}}  
+  }  
+  
where the dimensionless variables are defined as  where the dimensionless variables are defined as  
  
  
  
  
  
  
  
  
  
  
  
  
  Model 1. At the interface between the liquid and the twophase region, the shear stress is zero, i.e., <math>\partial u_\ell ^ +  +  { class="wikitable" border="0" 
  +    
  , which is the same as in classical Nusselt analysis. The velocity profile in the liquid layer is  +   width="100%"  
  +  <center><math>u_{\ell }^{+}=\frac{{{u}_{\ell }}}{{{u}_{D}}}\text{, }v_{\ell }^{+}=\frac{{{v}_{\ell }}}{{{u}_{D}}}\text{, }{{x}^{+}}=\frac{x}{\sqrt{K}}\text{, }{{y}^{+}}=\frac{y}{\sqrt{K}}\text{, }\delta _{\ell }^{+}=\frac{{{\delta }_{\ell }}}{\sqrt{K}}\text{, }\theta =\frac{T{{T}_{w}}}{{{T}_{sat}}{{T}_{w}}}</math></center>  
  u_\ell ^ +  +  {{EquationRef(23)}} 
  </math></center>  +  } 
  +  
+  Compared with velocity profile in the [[gravity dominated filmcondensation in a porous medium]], eq. (21) allows for nonslip conditions at the wall. The boundary conditions at the wall are  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>u_{\ell }^{+}=v_{\ell }^{+}=\theta =0\begin{matrix}  
+  , & {{y}^{+}}=0 \\  
+  \end{matrix}</math></center>  
+  {{EquationRef(24)}}  
+  }  
+  
+  The boundary conditions at the interface between the liquid film and the twophase region require that the velocity and shear stress in these two regions match, which makes the solution of the condensation problem very challenging. [[#ReferencesMajumdar and Tien (1990)]] proposed three models to handle the boundary condition at the interface between the liquid and the twophase regions; two of them are discussed below.  
+  
+  '''Model 1.''' At the interface between the liquid and the twophase region, the shear stress is zero, i.e., <math>\partial u_{\ell }^{+}/\partial {{y}^{+}}=0</math> at <math>\tilde{y}=\delta _{\ell }^{+}</math>, which is the same as in classical Nusselt analysis. The velocity profile in the liquid layer is  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>u_{\ell }^{+}=1\cosh {{y}^{+}}+\tanh \delta _{\ell }^{+}\sinh {{y}^{+}}</math></center>  
+  {{EquationRef(25)}}  
+  }  
+  
The liquid layer thickness can be obtained from an energy balance at the interface, and the result is  The liquid layer thickness can be obtained from an energy balance at the interface, and the result is  
  +  
  (1  {\  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
+  <center><math>(1\text{sec}{{\text{h}}^{2}}\delta _{\ell }^{+})\frac{d\delta _{\ell }^{+}}{d{{x}^{+}}}+(1\text{sech}\delta _{\ell }^{+})\frac{0.373}{\sqrt{\text{Bo}{{x}^{+}}}}=\frac{\text{J}{{\text{a}}_{\ell }}}{\delta _{\ell }^{+}\text{R}{{\text{a}}_{K}}}</math></center>  
+  {{EquationRef(26)}}  
+  }  
+  
where  where  
  +  
  {\  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
  is the Rayleigh number based on permeability, and <math>\alpha  +  <center><math>\text{R}{{\text{a}}_{K}}=\frac{{{K}^{3/2}}({{\rho }_{\ell }}{{\rho }_{v}})g}{{{\mu }_{\ell }}{{\alpha }_{e}}}</math></center> 
  +  {{EquationRef(27)}}  
  +  }  
  Model 2. This model also employs eq. (  +  
  +  is the Rayleigh number based on permeability, and <math>{{\alpha }_{e}}={{k}_{eff}}/({{\rho }_{\ell }}{{c}_{p\ell }})</math> is effective thermal diffusivity. Analytical solution of eq. (26) is not possible and it must be solved numerically.  
  . 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  +  
  +  '''Model 2.''' This model also employs eq. (21) to obtain the velocity in the liquid layer, except that the boundary condition at <math>{{y}^{+}}=\delta _{\ell }^{+}</math> is changed to <math>u_{\ell }^{+}=1</math>. Although it is not as rigorous as Model 1, it is an improvement over [[#ReferencesCheng (1981)]] because it uses a nonslip condition at the wall. The velocity profile in the liquid layer is  
  u_\ell ^ +  +  
  </math></center>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>u_{\ell }^{+}=1\cosh {{y}^{+}}+\coth \delta _{\ell }^{+}\sinh {{y}^{+}}</math></center>  
+  {{EquationRef(28)}}  
+  }  
and the overall energy balance at the interface is  and the overall energy balance at the interface is  
  +  
  \left(  +  { class="wikitable" border="0" 
  </math></center>  +   
  +   width="100%"   
+  <center><math>\left( 1\frac{1}{1+\cosh \delta _{\ell }^{+}} \right)\frac{d\delta _{\ell }^{+}}{d{{x}^{+}}}+\frac{0.373}{\sqrt{\text{Bo}{{x}^{+}}}}=\frac{\text{J}{{\text{a}}_{\ell }}}{\delta _{\ell }^{+}\text{R}{{\text{a}}_{K}}}</math></center>  
+  {{EquationRef(29)}}  
+  }  
+  
which also needs to be solved numerically.  which also needs to be solved numerically.  
  In the liquid region, eq. (  +  In the liquid region, eq. (22) will yield a linear temperature profile, and the local Nusselt number is 
  +  
  +  { class="wikitable" border="0"  
  </math></center>  +   
  +   width="100%"   
  +  <center><math>N{{u}_{x}}=\frac{{{x}^{+}}}{\delta _{\ell }^{+}}</math></center>  
  +  {{EquationRef(30)}}  
  +  }  
  +  
  +  ==References==  
  +  
+  Cheng, P., 1981, “Film Condensation Along an Inclined Surface in a Porous Medium,” International Journal of Heat and Mass Transfer, Vol. 24, pp. 983990.  
+  
+  Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA  
+  
+  Majumdar, A., Tien, C.L., 1990, “Effects of Surface Tension on Film Condensation in a Porous Medium,” ASME J. Heat Transfer, Vol. 112, pp. 751757.  
+  
+  White, S.M., and Tien, C.L., 1987, “An Experimental Investigation of Film Condensation in Porous Structures,” presented at the 6th International Heat Pipe Conference, Grenoble, France. 
Current revision as of 08:34, 26 July 2010
The analysis in the gravity dominated filmcondensation in a porous medium is valid for . When the condensation is gravitycapillary forces dominated (Bo˜1) or capillary force dominated (), there will be a twophase region that is saturated by a mixture of liquid and vapor. The fraction of liquid in the pore space is defined as saturation:

where and 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 twophase region is

where y_{2} is measured from the interface between liquid and the twophase region (see Fig. 8.29). The mass fluxes for liquid and vapor are governed by Darcy’s law, as follows:


where and K_{rv} 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. (2) is reduced to

The velocity components in the x and y directions are


where is Darcian velocity. The capillary pressure can be written as

where f(s) is a Leverett’s function:

and s is dimensionless saturation defined as

where is irreducible saturation, below which liquid flow will not occur. The relative permeability in eqs. (6) and (7) is obtained from

Substituting eqs. (6) – (11) into eq. (5), the following form of the dimensionless continuity equation is obtained:

which is subjected to the following boundary conditions:



Introducing the following similarity variable:

eqs. (12) – (15) are transformed to



The numerical solution of eq. (17) was obtained using the RungeKutta method and it is concluded that η = 1.296 can be chosen to determine the thickness of the twophase 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



where the dimensionless variables are defined as

Compared with velocity profile in the gravity dominated filmcondensation in a porous medium, eq. (21) allows for nonslip conditions at the wall. The boundary conditions at the wall are

The boundary conditions at the interface between the liquid film and the twophase 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 twophase regions; two of them are discussed below.
Model 1. At the interface between the liquid and the twophase region, the shear stress is zero, i.e., at , which is the same as in classical Nusselt analysis. The velocity profile in the liquid layer is

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

where

is the Rayleigh number based on permeability, and is effective thermal diffusivity. Analytical solution of eq. (26) is not possible and it must be solved numerically.
Model 2. This model also employs eq. (21) to obtain the velocity in the liquid layer, except that the boundary condition at is changed to . 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

and the overall energy balance at the interface is

which also needs to be solved numerically. In the liquid region, eq. (22) will yield a linear temperature profile, and the local Nusselt number is

References
Cheng, P., 1981, “Film Condensation Along an Inclined Surface in a Porous Medium,” International Journal of Heat and Mass Transfer, Vol. 24, pp. 983990.
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Majumdar, A., Tien, C.L., 1990, “Effects of Surface Tension on Film Condensation in a Porous Medium,” ASME J. Heat Transfer, Vol. 112, pp. 751757.
White, S.M., and Tien, C.L., 1987, “An Experimental Investigation of Film Condensation in Porous Structures,” presented at the 6th International Heat Pipe Conference, Grenoble, France.