# Nucleate Boiling in a Wicked Surface

Figure 1: Modes of heat transfer in wicks

A wicked surface is formed by coating a surface with a porous layer. It can be applied in the wick structures of a heat pipe as well as form a modified surface for enhanced heat transfer. Boiling in porous media is more complicated than that from plain surfaces due to the presence of capillary structures. Also, heat transfer and vapor formation may change with different types of wick structures and working fluids. As a result, different heat transfer modes may exist in the wicks. For a thin layer of wick structure saturated with liquid as it is applied in heat pipes, four basic modes of heat transfer and vapor formation are shown schematically in Fig. 1 (Faghri, 1995).

Mode 1: Conduction-convection. The whole wick is filled with liquid; conduction occurs across the liquid layer and evaporation takes place from its surface. No boiling occurs within the wick. However, natural convection may take place within some thick wicks under a gravitational field. This is the situation for nonmetallic working fluids under a low heat flux and metallic working fluids under a low or moderately high heat flux. The heat transfer across the wick can be calculated by a conduction model with sufficient accuracy.

Mode 2: Receding liquid. As heat flux is increased, the evaporation at the liquid surface intensifies. The capillary or body forces available in the heat pipe may not be capable of driving enough liquid back to the evaporator zone. As a result, the liquid layer begins to recede into the wick structure. Until the liquid is completely depleted, the heat transfer across the liquid layer is still by conduction, and liquid vaporization takes place at the liquid-vapor interface. No boiling occurs within the wick structure. This is applicable to both nonmetallic and metallic working fluids.

Mode 3: Nucleate boiling. For some nonmetallic working fluids, a large temperature difference across the wick may cause nucleate boiling within the wick. Bubbles grow at the heated wall, escape to the liquid surface and burst rapidly. As noted before, nucleate boiling has significance for heat transfer improvements in many applications; except some cases such as wicked heat pipes which is one of the limits of operation. For wicked heat pipes that require capillary force to sustain liquid circulation, nucleate boiling in the wick indeed represents a heat transfer limit for the following reasons: (a) large bubbles bursting at the liquid surface may disrupt the menisci established at the liquid-vapor interface and eliminate the capillary force circulating the liquid condensate; and (b) vapor bubbles generated at the evaporator section may block the liquid return from the condenser section.

Mode 4: Film boiling. As the temperature difference across the wick increases, a large quantity of bubbles is generated at the heated wall. Before escaping to the surface, these bubbles coalesce and form a layer of vapor adjacent to the heated wall. This layer prevents the liquid from reaching the wall surface. As a result, the wall temperature will increase rapidly and the heat pipe may burn out. This heat transfer limit is similar to film boiling in pool boiling heat transfer, and is a common heat transfer limit for wicked heat pipes.

Heat transfer from wicked surfaces has been extensively studied in recent years. Since the wick provides additional sites for nucleation, heat transfer in the wick is more complicated than boiling heat transfer from plain surfaces. In addition, the superheat ΔT for boiling incipience in a wick should be lower than that for a smooth surface. Marto and Lepere (1982) experimentally studied pool boiling heat transfer from various porous metallic coatings and enhanced surfaces.

Ferrell and Alleavitch (1970) experimentally studied the heat transfer from a horizontal surface covered with beds of Monel® beads. The bed depth ranged from 3.2 mm to 25.4 mm, and water was the working fluid at atmospheric pressure. They concluded that the heat transfer mechanism at lower ΔT was conduction through the saturated wick-liquid matrix to the liquid-vapor interface, which corresponds to Mode 1 in Fig. 1. At higher superheats, nucleate boiling occurs and the experimental data deviates from the curve predicted by the conduction model. The experimental data intercepts the boiling curves from the plane surface and Rohsenow’s (1985) pool boiling correlation at a higher ΔT. The super heat at which nucleate boiling occurs is lower than that of pool boiling.

Figure 2: Bubble formation at the wall-wick interface

Analysis of boiling in the wick involves the formation of bubbles (nucleation) as well as their subsequent growth and motion. Some vapor nuclei or small bubbles always exist within the wick structure, but the superheat is essential for these bubbles to grow. A small, trapped, hemispherical vapor bubble with effective radius Rb in the vicinity of the wall-wick interface is shown in Fig. 2. The bubble as well as the liquid adjacent to the wall is at the wall temperature Tw. The vapor pressure inside the vapor bubble is pb,v, while the liquid pressure adjacent to the bubble is ${p_\ell }$. The temperature and pressure of the vapor adjacent to the meniscus are Tv and pv. At equilibrium,

${p_{b,v}} - {p_\ell } = \frac{{2\sigma }}{{{R_b}}}\qquad\qquad(1)$

The relation between the liquid pressure and the vapor-space pressure is

${p_v} - {p_\ell } = \frac{{2\sigma }}{{{R_{men}}}}\qquad\qquad(2)$

where Rmen is the radius of the liquid-vapor meniscus. Combining of eqs. (1) and (2) yields

${p_{b,v}} - {p_v} = 2\sigma \left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right)\qquad \qquad(3)$

The vapor pressure in the bubble, pv,b, is related to the saturation pressure corresponding to the wall temperature, psat(Tw) , by

${p_{b,v}} = {p_{sat}}({T_w})\left( {1 - \frac{{2\sigma {\rho _v}}}{{{p_{sat}}({T_w}){R_b}{\rho _\ell }}}} \right)\qquad \qquad(4)$

Combining eqs. (3) and (4), one obtains

${p_{sat}}({T_w}) - {p_v} = 2\sigma \left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right) + \frac{{2\sigma {\rho _v}}}{{{R_b}{\rho _\ell }}}\qquad \qquad(5)$

Assuming the vapor adjacent to the meniscus is at saturation state, and applying the Clausius-Clapeyron equation between (pv,Tv) and (psat(Tw),Tw), a critical temperature difference is obtained:

$\Delta {T_{crit}} = {T_w} - {T_v} = \frac{{{R_g}{T_v}{T_w}}}{{{h_{\ell v}}}}\ln \left[ {1 + \frac{{2\sigma }}{{{p_v}}}\left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right) + \frac{{2\sigma {\rho _v}}}{{{p_v}{R_b}{\rho _\ell }}}} \right]\qquad \qquad(6)$

If

$\frac{{2\sigma }}{{{p_v}}}\left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right) + \frac{{2\sigma {\rho _v}}}{{{p_v}{R_b}{\rho _\ell }}} < 1$

eq. (6) can be rewritten as

$\Delta {T_{crit}} = {T_w} - {T_v} = \frac{{{R_g}{T_v}{T_w}}}{{{h_{\ell v}}}}\left[ {\frac{{2\sigma }}{{{p_v}}}\left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right) + \frac{{2\sigma {\rho _v}}}{{{p_v}{R_b}{\rho _\ell }}}} \right]\qquad \qquad(7)$

considering

$\frac{{2\sigma {\rho _v}}}{{{p_v}{R_b}{\rho _\ell }}} \ll \frac{{2\sigma }}{{{p_v}}}\left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right)$

and assuming ${T_v}{T_w} \approx T_v^2$, a commonly-used critical superheat for bubble formation is obtained (Faghri, 1995):

$\Delta {T_{crit}} = \frac{{2\sigma {T_v}}}{{{h_{\ell v}}{\rho _v}}}\left( {\frac{1}{{{R_b}}} - \frac{1}{{{R_{men}}}}} \right)\qquad \qquad(8)$

which indicates that the required superheat in a wick is lower than that for a plain surface.

## References

Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Washington, DC.

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.

Ferrell, J.K., and Alleavitch, J., 1970, “Vaporization Heat Transfer in Capillary Wick Structures,” Chemical Engineering Progress Symposium Ser., 66, Vol. 2.

Marto, P.J., and Lepere, V.J., 1982, “Pool Boiling Heat Transfer from Enhanced Surfaces to Deelectric Fluids,” ASME Journal of Heat Transfer, Vol. 104, pp. 292-299.

Rohsenow, W.M., 1952, “A Method for Correlating Heat-Transfer Data for Surface Boiling of Liquids,” Transactions of ASME, Vol. 74, pp. 969-976.