Boiling in Porous Media Heated from Below

Boiling in a porous medium.

When a thick porous medium saturated with liquid sits on a horizontal impermeable surface heated from below, boiling may occur when the horizontal surface temperature reaches the saturation temperature of the liquid. The experiments conducted by Sondergeld and Turcotte (1977) revealed that no significant superheat is required to initiate boiling in porous media. The heat transfer mode is similar to boiling in a wicked surfaces – discussed in the preceding subsection – except that evaporation on the top of the porous medium does not have much effect on the formation and growth of the vapor bubbles. In addition, the vapor bubbles cannot escape from the top of the porous medium as they can in the case of boiling on a wicked surface. Consequently, the entire porous medium can be divided into two regions: a two-phase region and a liquid region (see figure). The two-phase region is isothermal at the saturation temperature, and heat transfer in the two-phase region is achieved by counter-percolation of liquid and vapor. If, on the other hand, the temperature in the liquid region is below saturation temperature, convection may take place before and after boiling is initiated. Since the density of the liquid overlaying the two-phase region is higher than that of the two-phase region itself, the interface between liquid and the two-phase region may become unstable due to buoyancy and gravitational instabilities. The gravitational instability differs from the classical Rayleigh-Taylor instability because heat and mass transfer across the interface.

When the superheat on the surface is not very high, the vapor bubbles generated at the horizontal heating surface move upward while the liquid flows downward; this motion is similar to nucleate boiling in bulk liquid. The horizontal heating surface can be kept wet and the vaporization process can be sustained. As the superheat increases and the critical heat flux is exceeded, the vapor bubbles join together to form a vapor film that covers the horizontal surface and prevents it from coming into direct contact with the liquid. In this case, a vapor region is present and the entire porous medium is divided into three layers.

The characteristics of boiling in porous media is often described using the Bond number

${\rm{Bo}} = \frac{{g({\rho _\ell } - {\rho _v})K/\varepsilon }}{\sigma }\qquad \qquad(1)$

where K and ε are respectively permeability and porosity. Fukusako et al. (1986) conducted an experimental study of the boiling of water, R-11 and R-113 saturated in packed beds of spherical beads with diameters ranging from 1.0 to 16.5 mm. The height of the packed bed ranged from 10 to 300 mm. When there are no glass beads present (${\rm{Bo}} \to \infty$), Fukusako et al.’s (1986) experimental results are consistent with the conventional pool boiling curve (see Pool Boiling Regimes). The variation of the heat flux follows the order of nucleate, transient, and film boiling. The existences of maximum heat flux (transition from nucleate to transient boiling) and minimum heat flux (transition from transient to film boiling) are apparent. For boiling in a packed bed with large glass beads (Dp = 16.3 mm, Bo=0.61), the variation of heat flux follows a similar trend except that the maximum heat flux is lower. As the diameters of the glass beads decrease, the maximum heat flux continuously decreases, while the minimum heat flux is almost unaffected. As a result, the heat flux becomes a monotonic function of ΔTe when the diameter of the glass beads is less than 2.0 mm or Bo < 0.01. In this case, the transition from nucleate to film boiling occurred without going through a maximum heat flux.

For transition boiling in porous media, Fukusako et al. (1986) recommended the following empirical correlation based on the experimental results from their own and other results in the literature:

$\begin{array}{l} N{u_\ell } = \frac{{q''{D_p}}}{{\Delta {T_e}{k_{eff,\ell }}}} \\ = 0.075{\left( {\frac{{{D_p}}}{{\sqrt {\sigma /[g({\rho _\ell } - {\rho _v})]} }}} \right)^{0.9}}{\left( {\frac{{{h_{\ell v}}}}{{{c_{eff,\ell }}\Delta {T_e}}}} \right)^m}\Pr _\ell ^{2.37}{\left( {\frac{{{k_{eff,\ell }}}}{{{k_\ell }}}} \right)^n} \\ \end{array}\qquad \qquad(2)$

where ${k_{eff,\ell }} = \varepsilon {k_\ell } + (1 - \varepsilon ){k_{sm}}$ and ${c_{eff,\ell }} = \varepsilon {c_\ell } + (1 - \varepsilon ){c_{sm}}$ are thermal conductivity and specific heat of the porous media saturated with liquid. The exponents in eq. (2) are

$m = 1.3{\left( {\frac{{{D_p}}}{{\sqrt {\sigma /[g({\rho _\ell } - {\rho _v})]} }}} \right)^{0.6}}\Pr _\ell ^{ - 0.8}\qquad \qquad(3)$

and

$n = - 0.59\Pr _\ell ^{0.3}\qquad \qquad(4)$

Fukusako et al. (1986) showed that eq. (2) correlates with most experimental results obtained by them and other researchers within +-40%. For film boiling in porous media, Fukusako et al. (1986) recommended the following correlation:

$N{u_v} = \frac{{q''{D_p}}}{{\Delta {T_e}{k_{eff,v}}}} = 4.10{({\rm{Gr}}{\Pr _v})^{0.25}}{\left( {\frac{{{h_{\ell v}}}}{{{c_{eff,v}}\Delta {T_e}}}} \right)^{0.04}}{\left( {\frac{{{k_{eff,v}}}}{{{k_v}}}} \right)^{ - 0.95}}{\left( {\frac{{{D_p}}}{{{H_p}}}} \right)^{0.15}}\qquad \qquad(5)$

where ${k_{eff,v}} = \varepsilon {k_v} + (1 - \varepsilon ){k_{sm}}$ and ${c_{eff,v}} = \varepsilon {c_v} + (1 - \varepsilon ){c_{sm}}$ are thermal conductivity and specific heat of the porous media saturated with vapor. Equation (5) was obtained for the packed bed with height, Hp, ranging from 10 to 300 mm. The Grashof number is defined as:

${\rm{Gr}} = \frac{{({\rho _\ell } - {\rho _v}){\rho _v}gD_p^3}}{{\mu _v^2}}\qquad \qquad(6)$

They also obtained an empirical correlation for minimum heat flux as follows

$\frac{{{\rm{q}}_{{\rm{min}}}^{\rm{2}}{D_p}}}{{{\rho _v}\sigma h_{\ell v}^2}}{\rm{ = 9}}{\rm{.21}} \times {\rm{1}}{{\rm{0}}^{\rm{3}}}{\left( {\frac{{{D_p}}}{{\sqrt {\sigma /[g({\rho _\ell } - {\rho _v})]} }}} \right)^{0.65}}{\left( {\frac{{{\rho _v}}}{{{\rho _\ell }}}} \right)^{0.5}}{\left( {\frac{{{k_{eff,v}}}}{{{k_v}}}} \right)^{0.1}}\qquad \qquad(7)$

References

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.

Fukusako, S., Komoriya, T., Seki, N., 1986, “An Experimental Study of Transition and Film Boiling Heat Transfer in Liquid-Saturated Porous Bed,” ASME Journal of Heat Transfer, Vol., 108, pp. 117-124

Kaviany, M., 1995, Principles of Heat Transfer in Porous Media, Springer-Verlag, New York, NY.

Ramesh, P.S., Torrance, K.E., 1990, “Stability of Boiling in Porous Media,” International Journal of Heat and Mass Transfer, Vol. 33, pp. 1895-1990.

Sondergeld, C.H. and Turcotte, D.L., 1977, “An Experimental Study of Two- Phase Convection in a Porous Medium with Applications to Geological Problems,” Journal of Geophysical Research, Vol. 82, pp. 2045-2053.