Film Boiling Analysis in Porous Media

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(Created page with '===10.3.4 Nucleate Site Density=== The knowledge of distribution of nucleation sites is an important factor in determining the boiling characteristics of a surface under specific…')
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===10.3.4 Nucleate Site Density===
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===10.7.3 Film Boiling Analysis in Porous Media===
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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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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 
 +
[[Image:boiling_g_(1).gif|thumb|400 px|alt= Film boiling in porous media. | Figure 10.41 Film boiling in porous media.
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  ]]
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<math>{T_w} > {T_{sat}}</math> is analyzed (see Fig. 10.41; [[#References|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
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<center><math>\frac{{\partial {u_v}}}{{\partial x}} + \frac{{\partial {v_v}}}{{\partial y}} = 0\qquad \qquad( ) </math></center>
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(10.255)
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<center><math>{N''_a} = f(\theta ,\phi ,\Delta T,{\rm{fluid properties}}) \qquad \qquad( ) </math></center>
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<center><math>{u_v} = - \frac{K}{{{\mu _v}}}({\rho _\ell } - {\rho _v})g\qquad \qquad( ) </math></center>
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(10.78)
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(10.256)
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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{{\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( ) </math></center>
 +
(10.257)
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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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where <math>{\alpha _{mv}}</math> is thermal diffusivity of the porous medium saturated with the vapor.
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(10.79)
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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( ) </math></center>
 +
(10.258)
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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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<center><math>{u_\ell } = \frac{K}{{{\mu _\ell }}}{\rho _\infty }g{\beta _\ell }({T_\ell } - {T_\infty })\qquad \qquad( ) </math></center>
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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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(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( ) </math></center>
 +
(10.260)
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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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<center><math>{v_v} = 0\begin{array}{*{20}{c}}
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  , & {y = 0} \\
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\end{array}\qquad \qquad( ) </math></center>
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(10.261)
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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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<center><math>T = {T_w}\begin{array}{*{20}{c}}
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  , & {y = 0}  \\
 +
\end{array}\qquad \qquad( ) </math></center>
 +
(10.262)
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It should be pointed out that <math>{u_v} 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
 +
<center><math>{u_\ell } = 0\begin{array}{*{20}{c}}
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  , & {y \to \infty }  \\
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\end{array}\qquad \qquad( ) </math></center>
 +
(10.263)
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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>{T_\ell } = {T_\infty }\begin{array}{*{20}{c}}
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  , & {y \to \infty } \\
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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( ) </math></center>
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(10.80)
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(10.264)
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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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The mass balance at the liquid-vapor interface is [see eq. (10.152)]:
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(10.81)
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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}} \\
 +
\end{array}\qquad \qquad( ) </math></center>
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(10.265)
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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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The temperature at the liquid-vapor interface is equal to the saturation temperature:
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(10.82)
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<center><math>{T_v} = {T_\ell } = {T_{sat}}{\begin{array}{*{20}{c}}
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  , & {y = \delta } \\
 +
\end{array}_v}\qquad \qquad( ) </math></center>
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(10.266)
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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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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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(10.83)
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[[Image:boiling_j_(1).jpg|thumb|400 px|alt= Heat transfer for film boiling on a vertical wall in porous media | Figure 10.42 Heat transfer for film boiling on a vertical wall in porous media [[#References|(Cheng and Verma, 1981]]; Reprinted with permission from Elsevier).
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  ]]
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obtained by [[#References|Cheng and Verma (1981)]] are shown in Fig. 10.42. The dimensionless parameters used in Fig. 10.42 are defined as
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<center><math>\begin{array}{l}
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{\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}}
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  , & {{\rm{R}}{{\rm{a}}_{xv}} = \frac{{({\rho _\ell } - {\rho _v})gKx}}{{{\mu _v}{\alpha _{mv}}}}}  \\
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\end{array} \\
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\begin{array}{*{20}{c}}
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  {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}}}},} & {}  \\
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\end{array} \\
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{\rm{J}}{{\rm{a}}_\ell } = \frac{{{c_{pv}}({T_w} - {T_{sat}})}}{{{h_{\ell v}}}} \\
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\end{array}\qquad \qquad( ) </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. 10.42, 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. 10.42:
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<center><math>N{u_x} = 0.5642{\rm{Ra}}_{xv}^{1/2}{\begin{array}{*{20}{c}}
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  , & {{\rm{Ja}}}  \\
 +
\end{array}_v} \to \infty \qquad \qquad( ) </math></center>
 +
(10.268)
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where <math>{R_g}</math> is the gas constant for the vapor.
 
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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.
 
==References==
==References==

Revision as of 16:21, 29 May 2010

Contents

10.7.3 Film Boiling Analysis 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

 Film boiling in porous media.
Figure 10.41 Film boiling in porous media.

Tw > Tsat is analyzed (see Fig. 10.41; 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( )

(10.255)

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

(10.256)

{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( )

(10.257)

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( )

(10.258)

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

(10.259)

{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( )

(10.260)

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( )

(10.261)

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

(10.262)

It should be pointed out that {u_v} 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
	<center><math>{u_\ell } = 0\begin{array}{*{20}{c}}
   , & {y \to \infty }  \\
\end{array}\qquad \qquad( ) </center> (10.263)

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

(10.264)

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( )

(10.265)

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( )

(10.266)

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 10.42 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. 10.42. The dimensionless parameters used in Fig. 10.42 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( )

(10.267)

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. 10.42, 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. 10.42:

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

(10.268)


References

Further Reading

External Links