# Formation of and heat transfer through thin liquid films

Figure 1 Thin evaporating film on a fragment of the rough solid surface

The next two sections describe the heat and mass transfer analysis of an axially-grooved heat pipe (AGHP) to illustrate an application of the ideas presented thus far. The operating principles of the heat pipe are outlined in Section 1.6.4. A partial cross-section of the particular axially-grooved heat pipe configuration considered here is shown in Figures 2 (a) and (b) and 1. Slots with walls of angle γ relative to the radial direction, and of mouth width W, are cut along the inner surface length (z-axis) of a circular pipe; this forms a series of fins through which heat is transferred between the working fluid and the heat pipe structure. For this analysis, the heat transfer processes in the heat pipe container and the working fluid are considered to be one-dimensional in the radial direction; as a result, axial heat conduction (into or out of the page in Fig. 2) is neglected. The radial direction is represented by the x-axis in Fig 2. The working fluid forms a meniscus of angle θmen with the slot wall as shown in Fig. 2(a), and the fluid may form a thin film of thickness δ over the fin top surface, particularly during the condensing portion of the heat pipe cycle.

Since the thermal resistance of a low-temperature AGHP depends primarily on the film thickness in the condenser and evaporator sections, evaluation of the thin liquid film heat transfer will be the focus of this analysis. Moreover, since the heat transfer and fluid dynamic processes in a thin film are similar in both the evaporator and condenser sections of the heat pipe, the formation of the films may be described by the same equations as long as one takes into account the different directions of the temperature potential. During the condensation process, liquid in the subcooled thin film flows toward the meniscus region along the s-coordinate. During evaporation, liquid in the superheated thin film flows from the meniscus region in the opposite direction. The model presented here assumes a given meniscus contact angle and accounts for the effects of interfacial thermal resistance, disjoining pressure, and surface roughness.

In the evaporating section of the heat pipe, the thin liquid film flows over the crown of the fin and undergoes evaporation caused by heat transfer from the heat-loaded fin surface, which has the curvature Kw. The local heat flux through the liquid film due to heat conduction is (see Fig. 1)

Figure 2 Cross-section of the characteristic element of (a) an axially-grooved condenser, and (b) an evaporator (Khrustalev and Faghri, 1995; A and B are shown in Fig. 1).
Figure 2 Cross-section of the characteristic element of (a) an axially-grooved condenser, and (b) an evaporator (Khrustalev and Faghri, 1995; A and B are shown in Fig. 1).
 ${q}''={{k}_{\ell }}\frac{{{T}_{w}}-{{T}_{\delta }}}{\delta }$ (1)

where the local thickness of the liquid layer δ and the temperature of the free liquid film surface Tδ are functions of the s-coordinate. For small Reynolds numbers (less than 10), it is valid to assume a fully-developed laminar liquid flow velocity profile (Khrustalev and Faghri, 1995):

 ${{u}_{\ell }}=-\frac{1}{2{{\mu }_{\ell }}}\frac{d{{p}_{\ell }}}{ds}\left( 2\eta \delta -{{\eta }^{2}} \right)$ (2)

where η is the coordinate normal to the solid-liquid interface. In arriving at eq. (2), it is assumed that the shear stresses at the interface are negligible. It is further assumed that the vapor pressure is constant along the s-coordinate, and that the liquid flow is driven mainly by the surface tension and the adhesion forces.

 $\frac{d{{p}_{\ell }}}{ds}=-\sigma \frac{dK}{ds}+\frac{d{{p}_{d}}}{ds}-K\frac{d\sigma }{d{{T}_{\delta }}}\frac{d{{T}_{\delta }}}{ds}+\frac{d}{ds}\left( \rho _{v}^{2}v_{v,\delta }^{2} \right)\left( \frac{1}{{{\rho }_{v}}}-\frac{1}{{{\rho }_{\ell }}} \right)$ (3)

where K is the local interface curvature and pd is the disjoining pressure. The impact of the last two terms on the results was found to be negligible in the present analysis, and therefore they are omitted in the following equations.

The continuity and energy equations for the evaporating liquid layer lead to

 $\frac{d}{ds}\int_{0}^{\delta }{{{u}_{\ell }}d\eta }=\frac{{{q}''}}{{{h}_{\ell v}}{{\rho }_{\ell }}}$ (4)

Substituting eqs. (1) – (3) into eq. (4) gives the following relation for the thickness of the evaporating film, $\delta \left( s \right)$:

 $\frac{1}{3{{\mu }_{\ell }}}\frac{d}{ds}\left[ {{\delta }^{3}}\frac{d}{ds}\left( {{p}_{d}}-\sigma K \right) \right]=\frac{{{k}_{\ell }}\left( {{T}_{w}}-{{T}_{\delta }} \right)}{{{h}_{\ell v}}{{\rho }_{\ell }}\delta }$ (5)

The film surface curvature K expressed in terms of the solid surface curvature Kw and film thickness is

 $K={{K}_{w}}+\frac{{{d}^{2}}\delta }{d{{s}^{2}}}{{\left[ 1+{{\left( \frac{d\delta }{ds} \right)}^{2}} \right]}^{-3/2}}$ (6)

It is assumed that the absolute value of the vapor core pressure at any z-location along the groove is related to vapor temperature by the saturation conditions

 pv = psat(Tv) (7)

and therefore can be defined for a given Tv using the saturation tables.

The interfacial temperature Tδ, which is affected by the disjoining and capillary pressure, also depends on the value of the interfacial resistance. For the case of a comparatively small heat flux, interfacial resistance is defined by the following relation [see eq. (17)]:

 ${{{q}''}_{\delta }}=-\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{\left( {{p}_{\text{sat}}} \right)}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]$ (8)

where pv and ${{({{p}_{sat}})}_{\delta }}$ are the saturation pressures corresponding to Tv (assumed to be a constant along s) in the bulk vapor and at the thin liquid film interface (varies with s), respectively.

The relation between the vapor pressure over the thin evaporating film, ${{({{p}_{sat}})}_{\delta }}$ –which is affected by the disjoining pressure – and the saturation pressure corresponding to Tδ, psat(Tδ), is given by eq. (18):

 ${{({{p}_{sat}})}_{\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left[ \frac{{{({{p}_{sat}})}_{\delta }}-{{p}_{sat}}({{T}_{\delta }})+{{p}_{d}}-\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right]$ (9)

This reflects the fact that under the influence of disjoining and capillary pressures, the liquid free surface saturation pressure ${{({{p}_{sat}})}_{\delta }}$ differs from the normal saturation pressure psat(Tδ) and varies along the thin film (i.e., s-coordinate); pv and Tv are, however the same for any value of s at a given z-location. ${{({{p}_{sat}})}_{\delta }}$ also varies due to the fact that Tδ changes along s. Under steady-state conditions, the heat flux obtained by eqs. (1) and (8) are the same. Combining these two expressions yields

 ${{T}_{\delta }}={{T}_{w}}+\frac{\delta }{{{k}_{\ell }}}\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{({{p}_{sat}})}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]$ (10)

Equations (9) and (10) determine the interfacial temperature Tδ and pressure psat(Tδ). The wall temperature Tw, which results from solving heat conduction between grooves (i.e., heat conduction by the fins), must be provided as an input to the solution procedure.

As the liquid film thins, the disjoining pressure pd and the interfacial temperature Tδ increase. A nonevaporating film thickness is present under specific conditions, as demonstrated in Section 5.4.1, that gives the equality of the liquid-vapor interface and solid surface temperatures:

 Tδ = Tw (11)

Substituting eq. (11) into eq. (10), one obtains

 ${{({{p}_{sat}})}_{\delta }}={{p}_{v}}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}}$ (12)

Substituting eq. (12) into eq. (9), one obtains the disjoining pressure for the nonevaporating film thickness as

 ${{p}_{d}}=-{{p}_{v}}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}}+{{p}_{sat}}\left( {{T}_{w}} \right)+{{\rho }_{\ell }}{{R}_{g}}{{T}_{w}}\ln \left( \frac{{{p}_{v}}}{{{p}_{sat}}\left( {{T}_{w}} \right)}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}} \right)+\sigma K$ (13)

At the nonevaporating film thickness, the disjoining pressure can also be obtained by eq. (19), i.e.,

 ${{p}_{d}}=-{A}'\delta _{0}^{-B}$ (14)

Combining eqs. (13) and (14) yields the nonevaporating film thickness:

 ${{\delta }_{0}}={{\left\{ \frac{1}{A'}\left[ {{p}_{v}}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}}-{{p}_{sat}}\left( {{T}_{w}} \right)-{{\rho }_{\ell }}{{R}_{g}}{{T}_{w}}\ln \left( \frac{{{p}_{v}}}{{{p}_{sat}}\left( {{T}_{w}} \right)}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}} \right)-\sigma K \right] \right\}}^{-1/B}}$ (15)

which is applicable to nonpolar liquids. For water, however, the following logarithmic dependence of disjoining pressure on the liquid film thickness is preferable (Holm and Goplen, 1979):

 ${{p}_{d}}={{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}\ln \left[ a{{\left( \frac{\delta }{3.3} \right)}^{b}} \right]$ (16)

where a = 1.5336 and b = 0.0243.

## Contents

 ${{{\dot{m}}''}_{\delta }}=\frac{{{{{q}''}}_{\delta }}}{{{h}_{\ell v}}}=\left( \frac{2\alpha }{2-\alpha } \right)\sqrt{\frac{{{M}_{v}}}{2\pi {{R}_{u}}}}\left( \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{p}_{\ell }}}{\sqrt{{{T}_{\ell }}}} \right)$ (17)
 ${{p}_{v,\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left\{ \frac{[{{p}_{v,\delta }}-{{p}_{sat}}({{T}_{\delta }})-{{p}_{cap}}+{{p}_{d}}]}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right\}$ (18)
 pd = − A'δ − B (19)

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Khrustalev, D., and Faghri, A., 1994, “Thermal Analysis of a Micro Heat Pipe,” ASME Journal of Heat Transfer, Vol. 116, No. 1, pp. 189-198.

Khrustalev, D. K., and Faghri, A., 1995, “Heat Transfer during Evaporation and Condensation on Capillary-Grooved Structures of Heat Pipes,” ASME Journal of Heat Transfer Vol. 117, pp. 740-747.

Holm, F.W., and Goplen, S.P., 1979, “Heat Transfer in the Meniscus Thin-Film Transition Region,” ASME Journal of Heat Transfer, Vol. 101, No. 3, pp. 543-547.