Film Boiling Analysis in Porous Media

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===10.3.4 Nucleate Site Density===
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[[Image:boiling_g_(1).gif|thumb|600 px|alt= Film boiling in porous media|<center>'''Film boiling in porous media.'''</center>]]
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The knowledge of distribution of nucleation sites is an important factor in determining the boiling characteristics of a surface under specific operating conditions. The number density of sites, or total number of active sites per unit area, is a function of contact angle, cavity half angle, and heat flux (or superheat) (Fig. 10.6), i.e.,
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<center><math>{N''_a} = f(\theta ,\phi ,\Delta T,{\rm{fluid properties}}) \qquad \qquad( ) </math></center>
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Film boiling of liquid saturated in a porous medium at an initial temperature of <math>{T_\infty } < {T_{sat}}</math> next to a vertical, impermeable heated wall at a temperature of <math>{T_w} > {T_{sat}}</math> is analyzed (see figure). Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. The liquid adjacent to the vapor layer is dragged upward by the vapor. The temperature at the liquid-vapor interface is at the saturation temperature. There are velocity and thermal boundary layers in the liquid phase adjacent to the vapor film. The solution of the film boiling problem requires solutions of vapor and liquid flow, as well as heat transfer in both the vapor and liquid phases. It is assumed that boundary layer approximations are applicable to the vapor film and to convection heat transfer in the liquid phase. It is further assumed that the vapor flow is laminar, two-dimensional; Darcy’s law is applicable in both the vapor and liquid phases. The continuity, momentum, and energy equations in the vapor film are
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(10.78)
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<center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1) </math></center>
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Equation (10.11) indicated that for a given local heat flux or superheat, a cavity will be active if <math>{R_{\min }}</math> is greater than <math>{R_b}</math>,
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<center><math>{u_v} =  - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2) </math></center>
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<center><math>{R_{\min }} \ge \frac{{2\sigma {T_{sat}}}}{{{h_{\ell v}}{\rho _v}\Delta T}} \qquad \qquad( ) </math></center>
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<center><math>{u_v}\frac{{\partial {T_v}}}{{\partial x}} + {v_v}\frac{{\partial {T_v}}}{{\partial y}} = {\alpha _{mv}}\frac{{{\partial ^2}{T_v}}}{{\partial {y^2}}}\qquad \qquad(3) </math></center>
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(10.79)
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Obviously each cavity on a real surface has a specific <math>{R_{min}}</math> that is a function of geometry and the contact angle. Considering eqs. (10.78) and (10.79), one expects that as the wall superheat increases, <math>{R_{min}}</math> decreases and the number of active sites having cavity radii greater than <math>{R_{min}}</math> increases.
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where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor.  
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[[#References|Lorenz et al. (1974)]] counted total active sites/cm<sup>2</sup> on a #240 (sand paper) finished copper surface for different working fluids as a function of <math>{R_{min}}</math>, which is shown in Fig. 10.14. 
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The governing equations for the liquid boundary layer are
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<center><math>\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4) </math></center>
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<center><math>{u_\ell } = \frac{K}{{{\mu _\ell }}}{\rho _\infty }g{\beta _\ell }({T_\ell } - {T_\infty })\qquad \qquad(5) </math></center>
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<center><math>{u_\ell }\frac{{\partial {T_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {T_\ell }}}{{\partial y}} = {\alpha _{m\ell }}\frac{{{\partial ^2}{T_\ell }}}{{\partial {y^2}}}\qquad \qquad(6) </math></center>
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[[Image:boiling_j_(9).jpg|thumb|400 px|alt=Number density of active sites for boiling on a copper surface|Figure 10.14 Number density of active sites for boiling on a copper surface [[#References|(Lorenz et al., 1974)]]. ]]
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where <math>{\alpha _{m\ell }}</math> is thermal diffusivity of the porous medium saturated with the liquid.  
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The boundary conditions at the heated wall (''y'' = 0) are
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[[#References|Kocamustafaogullari and Ishii (1983)]] have correlated various existing experimental data of <math>{N''_a}</math> for water on a variety of surfaces and pressure ranges from 1 to 198 atm by
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<center><math>{v_v} = 0\begin{array}{*{20}{c}}
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  , & {y = 0} \\
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<center><math>{N''_a} = D_d^2{\left[ {{{\left( {\frac{{{D_c}}}{{{D_d}}}} \right)}^{ - 0.44}}F} \right]^{1/4.4}} \qquad \qquad( ) </math></center>
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\end{array}\qquad \qquad(7) </math></center>
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(10.80)
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Where
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<center><math>F = 2.157 \times {10^{ - 7}}{\left( {\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{ - 3.2}}{\left[ {1 + 0.0049\left( {\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)} \right]^{4.13}} \qquad \qquad( ) </math></center>
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<center><math>T = {T_w}\begin{array}{*{20}{c}}
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(10.81)
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  , & {y = 0} \\
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\end{array}\qquad \qquad(8) </math></center>
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<center><math>{D_c} = 4\sigma \left[ {1 + ({\rho _\ell }/{\rho _v})} \right]/{p_\ell } \cdot \left\{ {\exp \left[ {{h_{\ell v}}({T_v} - {T_{sat}})/({R_g}{T_v}{T_{sat}})} \right] - 1} \right\} \qquad \qquad( ) </math></center>
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It should be pointed out that <math>{u_v}</math> is not equal to zero at the heating surface under Darcy’s law, i.e., slip occurs at the surface. The boundary condition in the liquid that is far from the heated surface is
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(10.82)
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<center><math>{D_d} = 0.0208\theta \sqrt {\frac{\sigma }{{g({\rho _\ell } - {\rho _v})}}}  \cdot 0.0012{\left( {\frac{{{\rho _\ell } - {\rho _v}}}{{{\rho _v}}}} \right)^{0.9}} \qquad \qquad( ) </math></center>
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<center><math>{u_\ell } = 0\begin{array}{*{20}{c}}
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(10.83)
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  , & {y \to \infty } \\
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\end{array}\qquad \qquad(9) </math></center>
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where <math>{R_g}</math> is the gas constant for the vapor.
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<center><math>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}}
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[[#References|Wang and Dhir (1993a, 1993b)]] have studied number density for boiling of water at 1 atm on a mirror-finished copper surface, and they provided a mechanistic approach for relating the cavities that are present on the surface to the cavities that actually nucleate.  
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  , & {y \to \infty }  \\
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\end{array}\qquad \qquad(10) </math></center>
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The mass balance at the liquid-vapor interface is (see [[Film Boiling Analysis]]):
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<center><math>{\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_v} = {\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_\ell }\begin{array}{*{20}{c}}
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  , & {y = {\delta _v}}  \\
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\end{array}\qquad \qquad(11) </math></center>
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The temperature at the liquid-vapor interface is equal to the saturation temperature:
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<center><math>{T_v} = {T_\ell } = {T_{sat}}{\begin{array}{*{20}{c}}
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  , & {y = \delta }  \\
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\end{array}_v}\qquad \qquad(12) </math></center>
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The above film boiling problem can be solved using a similarity solution like that for [[Film Condensation on an Inclined Wall|film condensation in porous media]].  
==References==
==References==
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Cheng, P., and Verma, A.K., 1981, “The Effect of Subcooling Liquid on Film Boiling about a Vertical Heated Surface in a Porous Medium,” ''International Journal of Heat and Mass Transfer'', Vol. 24, pp. 1151-1160.
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Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA
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Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO.
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Nield, D.A., and Bejan, A., 1999, ''Convection in Porous Media'', 2<sup>nd</sup> ed., Springer-Verlag, New York.
==Further Reading==
==Further Reading==
==External Links==
==External Links==

Current revision as of 01:59, 9 July 2010

 Film boiling in porous media
Film boiling in porous media.

Film boiling of liquid saturated in a porous medium at an initial temperature of {T_\infty } < {T_{sat}} next to a vertical, impermeable heated wall at a temperature of Tw > Tsat is analyzed (see figure). Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. The liquid adjacent to the vapor layer is dragged upward by the vapor. The temperature at the liquid-vapor interface is at the saturation temperature. There are velocity and thermal boundary layers in the liquid phase adjacent to the vapor film. The solution of the film boiling problem requires solutions of vapor and liquid flow, as well as heat transfer in both the vapor and liquid phases. It is assumed that boundary layer approximations are applicable to the vapor film and to convection heat transfer in the liquid phase. It is further assumed that the vapor flow is laminar, two-dimensional; Darcy’s law is applicable in both the vapor and liquid phases. The continuity, momentum, and energy equations in the vapor film are

\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1)


{u_v} =  - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2)


{u_v}\frac{{\partial {T_v}}}{{\partial x}} + {v_v}\frac{{\partial {T_v}}}{{\partial y}} = {\alpha _{mv}}\frac{{{\partial ^2}{T_v}}}{{\partial {y^2}}}\qquad \qquad(3)


where αmv is thermal diffusivity of the porous medium saturated with the vapor. The governing equations for the liquid boundary layer are

\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4)


{u_\ell } = \frac{K}{{{\mu _\ell }}}{\rho _\infty }g{\beta _\ell }({T_\ell } - {T_\infty })\qquad \qquad(5)


{u_\ell }\frac{{\partial {T_\ell }}}{{\partial x}} + {v_\ell }\frac{{\partial {T_\ell }}}{{\partial y}} = {\alpha _{m\ell }}\frac{{{\partial ^2}{T_\ell }}}{{\partial {y^2}}}\qquad \qquad(6)


where {\alpha _{m\ell }} is thermal diffusivity of the porous medium saturated with the liquid.

The boundary conditions at the heated wall (y = 0) are

{v_v} = 0\begin{array}{*{20}{c}}
   , & {y = 0}  \\
\end{array}\qquad \qquad(7)


T = {T_w}\begin{array}{*{20}{c}}
   , & {y = 0}  \\
\end{array}\qquad \qquad(8)


It should be pointed out that uv is not equal to zero at the heating surface under Darcy’s law, i.e., slip occurs at the surface. The boundary condition in the liquid that is far from the heated surface is

{u_\ell } = 0\begin{array}{*{20}{c}}
   , & {y \to \infty }  \\
\end{array}\qquad \qquad(9)


{T_\ell } = {T_\infty }\begin{array}{*{20}{c}}
   , & {y \to \infty }  \\
\end{array}\qquad \qquad(10)


The mass balance at the liquid-vapor interface is (see Film Boiling Analysis):

{\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_v} = {\left( {\rho u\frac{{d\delta }}{{dx}} - \rho v} \right)_\ell }\begin{array}{*{20}{c}}
   , & {y = {\delta _v}}  \\
\end{array}\qquad \qquad(11)


The temperature at the liquid-vapor interface is equal to the saturation temperature:

{T_v} = {T_\ell } = {T_{sat}}{\begin{array}{*{20}{c}}
   , & {y = \delta }  \\
\end{array}_v}\qquad \qquad(12)


The above film boiling problem can be solved using a similarity solution like that for film condensation in porous media.

References

Cheng, P., and Verma, A.K., 1981, “The Effect of Subcooling Liquid on Film Boiling about a Vertical Heated Surface in a Porous Medium,” International Journal of Heat and Mass Transfer, Vol. 24, pp. 1151-1160.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Nield, D.A., and Bejan, A., 1999, Convection in Porous Media, 2nd ed., Springer-Verlag, New York.

Further Reading

External Links