Film Boiling Analysis in Porous Media

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<center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1) </math></center>
<center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad(1) </math></center>
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(10.255)
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<center><math>{u_v} =  - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2) </math></center>
<center><math>{u_v} =  - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad(2) </math></center>
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(10.256)
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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>
<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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(10.257)
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where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor.  
where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor.  
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<center><math>\frac{{\partial {u_\ell }}}{{\partial x}} + \frac{{\partial {v_\ell }}}{{\partial y}} = 0\qquad \qquad(4) </math></center>
<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>
<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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(10.259)
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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>
<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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(10.260)
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where <math>{\alpha _{m\ell }}</math> is thermal diffusivity of the porous medium saturated with the liquid.  
where <math>{\alpha _{m\ell }}</math> is thermal diffusivity of the porous medium saturated with the liquid.  
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   , & {y = 0}  \\
   , & {y = 0}  \\
\end{array}\qquad \qquad(7) </math></center>
\end{array}\qquad \qquad(7) </math></center>
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<center><math>T = {T_w}\begin{array}{*{20}{c}}
<center><math>T = {T_w}\begin{array}{*{20}{c}}
   , & {y = 0}  \\
   , & {y = 0}  \\
\end{array}\qquad \qquad(8) </math></center>
\end{array}\qquad \qquad(8) </math></center>
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(10.262)
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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
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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   , & {y \to \infty }  \\
   , & {y \to \infty }  \\
\end{array}\qquad \qquad(9) </math></center>
\end{array}\qquad \qquad(9) </math></center>
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<center><math>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}}
<center><math>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}}
   , & {y \to \infty }  \\
   , & {y \to \infty }  \\
\end{array}\qquad \qquad(10) </math></center>
\end{array}\qquad \qquad(10) </math></center>
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(10.264)
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The mass balance at the liquid-vapor interface is [see eq. (10.152)]:
The mass balance at the liquid-vapor interface is [see eq. (10.152)]:
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   , & {y = {\delta _v}}  \\
   , & {y = {\delta _v}}  \\
\end{array}\qquad \qquad(11) </math></center>
\end{array}\qquad \qquad(11) </math></center>
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(10.265)
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The temperature at the liquid-vapor interface is equal to the saturation temperature:  
The temperature at the liquid-vapor interface is equal to the saturation temperature:  
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   , & {y = \delta }  \\
   , & {y = \delta }  \\
\end{array}_v}\qquad \qquad(12) </math></center>
\end{array}_v}\qquad \qquad(12) </math></center>
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(10.266)
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The above film boiling problem can be solved using a similarity solution like that for film condensation in porous media discussed in Section 8.5.2. The results   
The above film boiling problem can be solved using a similarity solution like that for film condensation in porous media discussed in Section 8.5.2. The results   
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  {\rm{J}}{{\rm{a}}_\ell } = \frac{{{c_{pv}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}} \\  
  {\rm{J}}{{\rm{a}}_\ell } = \frac{{{c_{pv}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}} \\  
  \end{array}\qquad \qquad(13) </math></center>
  \end{array}\qquad \qquad(13) </math></center>
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(10.267)
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where Jakob numbers <math>{\rm{J}}{{\rm{a}}_v}</math> and <math>{\rm{J}}{{\rm{a}}_\ell }</math>, measure the degrees of superheat in the vapor and subcooling in the liquid. For all cases shown in Fig. 2, the effect of liquid subcooling on the heat transfer is insignificant. The effect of vapor superheat on heat transfer is significant when <math>{\rm{J}}{{\rm{a}}_v}</math> is less than 2. The following asymptotic result can be obtained from Fig. 2:
where Jakob numbers <math>{\rm{J}}{{\rm{a}}_v}</math> and <math>{\rm{J}}{{\rm{a}}_\ell }</math>, measure the degrees of superheat in the vapor and subcooling in the liquid. For all cases shown in Fig. 2, the effect of liquid subcooling on the heat transfer is insignificant. The effect of vapor superheat on heat transfer is significant when <math>{\rm{J}}{{\rm{a}}_v}</math> is less than 2. The following asymptotic result can be obtained from Fig. 2:
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   , & {{\rm{Ja}}}  \\
   , & {{\rm{Ja}}}  \\
\end{array}_v} \to \infty \qquad \qquad(14) </math></center>
\end{array}_v} \to \infty \qquad \qquad(14) </math></center>
-
(10.268)
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==References==
==References==

Revision as of 14:57, 1 June 2010

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

 Film boiling in porous media
Figure 1 Film boiling in porous media.

Tw > Tsat is analyzed (see Fig. 1; Cheng and Verma, 1981; Nield and Bejan, 1999). 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 eq. (10.152)]:

{\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 discussed in Section 8.5.2. The results

 Heat transfer for film boiling on a vertical wall in porous media
Figure 2 Heat transfer for film boiling on a vertical wall in porous media (Cheng and Verma, 1981; Reprinted with permission from Elsevier).

obtained by Cheng and Verma (1981) are shown in Fig. 2. The dimensionless parameters used in Fig. 2 are defined as

\begin{array}{l}
 {\rm{N}}{{\rm{u}}_x} = \frac{{{h_x}x}}{{{k_{mv}}}} = \frac{{{{q''}_w}x}}{{{k_{mv}}({T_w} - {T_{sat}})}}\begin{array}{*{20}{c}}
   , & {{\rm{R}}{{\rm{a}}_{xv}} = \frac{{({\rho _\ell } - {\rho _v})gKx}}{{{\mu _v}{\alpha _{mv}}}}}  \\
\end{array} \\ 
 \begin{array}{*{20}{c}}
   {R = \frac{{{\rho _v}}}{{{\rho _\ell }}}\left[ {\frac{{{\mu _\ell }{\alpha _{mv}}({\rho _\ell } - {\rho _v}){c_{p\ell }}}}{{{\mu _v}{\alpha _{m\ell }}{\rho _\ell }{\beta _\ell }{h_{\ell v}}}}} \right],} & {{\rm{J}}{{\rm{a}}_v} = \frac{{{c_{p\ell }}({T_{sat}} - {T_\infty })}}{{{h_{\ell v}}}},} & {}  \\
\end{array} \\ 
 {\rm{J}}{{\rm{a}}_\ell } = \frac{{{c_{pv}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}} \\ 
 \end{array}\qquad \qquad(13)


where Jakob numbers Jav and {\rm{J}}{{\rm{a}}_\ell }, measure the degrees of superheat in the vapor and subcooling in the liquid. For all cases shown in Fig. 2, the effect of liquid subcooling on the heat transfer is insignificant. The effect of vapor superheat on heat transfer is significant when Jav is less than 2. The following asymptotic result can be obtained from Fig. 2:

N{u_x} = 0.5642{\rm{Ra}}_{xv}^{1/2}{\begin{array}{*{20}{c}}
   , & {{\rm{Ja}}}  \\
\end{array}_v} \to \infty \qquad \qquad(14)


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.

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

Further Reading

External Links