# Evaporation from Cylindrical Pore under Low to Moderate Heat Flux

Evaporation in a microchannel under low/moderate heat flux was studied by Khrustalev and Faghri (1995a). In the scenario under consideration, liquid evaporates from the surface of the menisci situated at the liquid-vapor interface. Figure 9.19 shows the schematic of the cylindrical pore (microchannel) and liquid meniscus. A description of the heat transfer during evaporation from a pore is given here with the two following main assumptions.

1. The temperature of the solid-liquid interface Ts can be considered constant along the s-coordinate for small s.

2. The curvature of the axisymmetrical liquid-vapor surface of the meniscus is defined by the main radius of curvature K = 2 / Rmen and is independent of s.

The validity of the second assumption has been proven numerically by Khrustalev and Faghri (1995b); it was shown that this assumption could give an error of less than 5% when calculating the overall heat transfer coefficient during evaporation from a capillary groove. Since heat transfer during evaporation from thin films in a pore is similar to that in a capillary groove, this assumption can be justified for the present analysis.

Figure 9.19 Schematic of evaporation from a cylindrical pore.

The local heat flux through the liquid film due to heat conduction is

 ${{{q}''}_{\ell }}={{k}_{\ell }}\frac{{{T}_{s}}-{{T}_{\delta }}}{{{\delta }_{\ell }}}$ (1)

where the local thickness of the liquid layer ${{\delta }_{\ell }}$ and the temperature of the free liquid film surface Tδ are functions of the s-coordinate. Tδ is affected by the disjoining and capillary pressures. It also depends on the value of the interfacial resistance, which is defined for the case of comparatively small heat flux at the interface, q''δ, by the following relation given by the kinetic theory:

 ${{{q}''}_{\delta }}=-\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v\delta }}}{\sqrt{{{T}_{v}}}}-\frac{{{\left( {{p}_{\text{sat}}} \right)}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]$ (2)

where pvδ and ${{({{p}_{sat}})}_{\delta }}$ are the saturation pressures corresponding to Tv and at the thin liquid film interface, respectively.

The relation between the saturation vapor pressure over the thin evaporating film, ${{({{p}_{sat}})}_{\delta }},$ which is affected by the disjoining pressure, and the normal saturation pressure corresponding to Tδ, psat(Tδ), is given by the following relation (see Chapter 5):

 ${{({{p}_{sat}})}_{\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left[ \frac{{{({{p}_{sat}})}_{\delta }}-{{p}_{sat}}({{T}_{\delta }})+{{p}_{d}}-\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right]$ (3)

which reflects the fact that under the influence of the disjoining and capillary pressures, the liquid free surface saturation pressure ${{({{p}_{sat}})}_{\delta }}$ is different from normal saturation pressure psat(Tδ). It varies along the thin film (or s-coordinate) while pvδ and Tv are the same for any value of s. This variation is also due to the fact that Tδ changes along s. While the evaporating film thins as it approaches the point s = 0, the difference between ${{({{p}_{sat}})}_{\delta }},$ given by eq. (3), and the pressure obtained for a given Tδ using the saturation table becomes larger. This difference is the reason for the existence of the thin non-evaporating superheated film, which is in equilibrium state in spite of the fact that Tδ > Tsat.

Under steady-state conditions, ${{{q}''}_{\ell }}={{{q}''}_{\delta }}$, and it follows from eqs. (1) and (2) that

 ${{T}_{\delta }}={{T}_{s}}+\frac{\delta }{{{k}_{\ell }}}\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v\delta }}}{\sqrt{{{T}_{v}}}}-\frac{{{({{p}_{sat}})}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]$
(4)

Equations (3) and (4) determine the interfacial temperature Tδ and pressure ${{({{p}_{sat}})}_{\delta }}$ for a given vapor pressure, pvδ, the solid-liquid interface temperature Ts, and the liquid film thickness ${{\delta }_{\ell }}(s)$.

As the liquid film thins, the disjoining pressure, pd, and the interfacial temperature, Tδ, increase. Under specific conditions, a non-evaporating film thickness is present which gives the equality of the interfacial and solid surface temperatures, Tδ = Ts. This is the thickness of the equilibrium nonevaporating film δ0, For the disjoining pressure in water, the following equation for was used in the present analysis (Holm and Goplen, 1979):


${{p}_{d}}={{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}\ln \left[ a{{\left( \frac{\delta }{3.3} \right)}^{b}} \right]$ (5)

where a = 1.5336 and b = 0.0243. From equations (3) – (5), the following expression for the thickness of the equilibrium film is given:

 ${{\delta }_{0}}=3.3{{\left\{ \frac{1}{a}\exp \left[ \frac{{{p}_{sat}}({{T}_{w}})-{{p}_{v}}\sqrt{{{T}_{w}}/{{T}_{v}}}+\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{w}}}+\ln \left( \frac{{{p}_{v}}}{{{p}_{sat}}({{T}_{w}})}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}} \right) \right] \right\}}^{1/b}}$ (6)

The total heat flow through a single pore is defined as

 ${{q}_{p}}=\int_{0}^{{{R}_{p}}}{\frac{{{T}_{s}}-{{T}_{\delta }}}{{{\delta }_{\ell }}/{{k}_{\ell }}}}2\pi rds=\int_{0}^{{{R}_{p}}}{\frac{{{T}_{s}}-{{T}_{\delta }}}{{{\delta }_{\ell }}/{{k}_{\ell }}}}2\pi {{R}_{\text{men}}}\sin \left[ \arctan \frac{{{R}_{p}}}{s+\sqrt{R_{\text{men}}^{\text{2}}-R_{p}^{2}}} \right]ds$ (7)

The surface of the pore wall is totally covered with microroughnesses, from which the characteristic size varies, for example, Rr = 10-8 to 10-6 m. Apparently, the thin liquid film formation can be affected by some of these microroughnesses. The following approximation for the liquid film thickness was given by Khrustalev and Faghri (1995b): for ${{\delta }_{0}}\le {{\delta }_{\ell }}\le {{\delta }_{0}}+{{R}_{r}}$ and ${{R}_{r}}\gg {{\delta }_{0}},$

 ${{\delta }_{\ell }}={{\delta }_{0}}+{{R}_{r}}-\sqrt{R_{r}^{2}-{{s}^{2}}}-{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{s}^{2}}+2{{R}_{\text{men}}}s\sin {{\theta }_{f}} \right)}^{1/2}}$ (8)

where for a surface with microroughness θf = 0, and the liquid film thickness in the interval, ${{\delta }_{\ell }}\ge {{\delta }_{0}}+{{R}_{r}},$ is

 $\delta ={{\delta }_{0}}+{{R}_{r}}-{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{s}^{2}}+2{{R}_{\text{men}}}s\sin {{\theta }_{\text{men}}} \right)}^{1/2}}$ (9)

For the smooth-surface model, θf is the angle between the solid-liquid and liquid-vapor interfaces at the point on s where the disjoining pressure and dK/ds become zero. Note that for small values of the accommodation coefficient (for example α= 0.05), the value of Rr has not affected the total heat flow rate through the liquid film (Khrustalev and Faghri, 1995b).

The interfacial radius of curvature is related to the pressure difference between the liquid and the vapor by the extended Laplace-Young equation:

 ${{p}_{v\delta }}-{{p}_{\ell \delta }}=\frac{2\sigma }{{{R}_{\text{men}}}}+\frac{\rho _{v}^{2}v_{v\delta }^{2}}{{{\varepsilon }^{2}}}\left( \frac{1}{{{\rho }_{\ell }}}-\frac{1}{{{\rho }_{v}}} \right)$ (10)

where vvδ is the vapor mean blowing velocity specified for a given meniscus, and $\varepsilon$ is the porosity. The latter is needed in this equation because the evaporation takes place into the dry region of the porous structure. The temperature of the saturated vapor near the interface, Tv, is related to its pressure by the saturation conditions:

 Tv = Tsat(pvδ) (11)

Then the heat transfer coefficient during evaporation from the porous surface is defined as

 ${{h}_{ep}}=\frac{{{\varepsilon }_{s}}{{q}_{p}}}{\pi R_{p}^{2}({{T}_{s}}-{{T}_{v}})}$ (12)

where ${{\varepsilon }_{s}}={{A}_{p}}/{{A}_{t}}$ is the surface porosity defined as the ratio of the surface of the pores to the total surface of the porous structure for a given cross-section (it is assumed that ${{\varepsilon }_{s}}=\varepsilon$; Khrustalev and Faghri, 1995a). Figure 9.20 shows that the heat transfer coefficient during evaporation from the porous surface, he, p, depends significantly on the curvature of the liquid meniscus. For smaller pore sizes, the values of he, p are larger because a larger relative surface is occupied by the thin films.

## References

Khrustalev, D. and Faghri, A., 1995a, “Heat Transfer in the Inverted Meniscus Type Evaporator at High Heat Fluxes,” International Journal of Heat and Mass Transfer, Vol. 38, pp. 3091-3101.

Khrustalev, D.K. and Faghri, A., 1995b, “Heat Transfer during Evaporation and Condensation on Capillary-Grooved Structures of Heat Pipes,” ASME Journal of Heat Transfer, Vol. 117, August, No. 3, pp. 740-747.