Thermal Resistances in the Dropwise Condensation Processes

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The condensation process must overcome a series of thermal resistances for the heat and mass transfer to occur. These resistances include the thermal resistance found in the vapor, thermal resistance encountered during the phase change from vapor to liquid, resistance caused by capillary depression of the equilibrium saturation temperature at the interface, thermal resistance found in the liquid phase, and thermal resistance found at the wall where heat is conducted from the surface into the wall. The mode of conduction into the wall depends on whether dropwise condensation or filmwise condensation occurs, as will be discussed below. In short, when filmwise condensation occurs uniformly along the surface, the heat flux can simply be found from a straight application of Fourier’s Law of conduction into a solid. However, if dropwise condensation occurs, conduction into the wall is constricted around the individual droplets and cannot occur uniformly over the solid wall. However, some resistances can be neglected in relation to others, except in special cases. These individual resistances and their importance to the overall resistance will be discussed in this section.

Figure 7.13 shows the resistance to heat flow associated with both filmwise and dropwise condensations, in which Rw is resistance resulting from conduction of heat through the cold wall, Rliquid is resistance resulting from heat conduction through liquid film or a droplet, Rcap is the resistance resulting from capillary depression of the equilibrium saturation temperature, Rδ is interfacial thermal resistance, Rv is resistance resulting from heat transfer in the vapor phase, and Rconv is the convective thermal resistance for the area not covered by the droplet in the dropwise conduction. Overall, the thermal resistance associated with vapor will be the same for both dropwise and filmwise condensation, as is also the case for the interfacial resistance and capillary depression resistance. The conduction resistances found in dropwise and filmwise condensation are different, i.e., heat is conducted differently through liquid droplets as compared to a liquid film. Conduction through an individual liquid droplet is a function of the size of the droplet (an expression that takes into account all droplets found on the wall surface will also be presented), while the conduction through a thin liquid film is a function of the film thickness and thus a function of position on the wall. The thin liquid film will be discussed in detail in the next section. It is assumed in this discussion that the wall temperature is held at a constant temperature, Tw, and therefore the overall temperature drop for the area covered by the droplet is as follows:

ΔTtotal = TvaporTw = ΔTvapor + ΔTδ + ΔTcap + ΔTdroplet

where the temperature differences are for the vapor, interface δ, capillary depression of the equilibrium saturation temperature cap, and conduction through the droplet, respectively.

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Resistance in the Vapor

The thermal resistance found in vapor can usually be ignored except in special cases, because it is usually an order of magnitude less than the other resistances found in the condensation process. This low contribution to the overall resistance is the result of its ability to mix extremely well if either free or forced convection is present; this in turn allows the heat and mass transfer towards the cooler surface to easily occur. However, if the vapor is superheated, this mixing process is severely limited, and the resistance of the vapor phase would have to be considered in the form of conduction through vapor. In that case, the controlling temperature difference would be (TvTsat), i.e., the temperature difference between the bulk superheated vapor and the saturated temperature at the liquid-vapor interface. Other cases in which the thermal resistance of the vapor would have to be taken into account include the condensation of vapor mixtures (described above in binary mixtures) and vapor mixtures that include an inert gas (noncondensable gas). The inert gas effectively insulates the conduction of heat through a vapor. However, in this discussion the thermal resistance of the vapor will be ignored because it is usually negligible except for the above cases.

Interfacial Resistance

The next resistance encountered in the thermal path from vapor to wall is that found at the vapor-liquid interface. The high heat transfer coefficients associated with the condensation process make it possible to achieve a large heat transfer rate with a small temperature drop. This is necessary because the temperature drop at the vapor-liquid interface in a condensation process is very small. This resistance is found in both filmwise and dropwise condensation and the expressions are identical. The heat flux at the interface can be obtained by (Faghri, 1995)

q_{\delta }^{''}=\left ( \frac{2\alpha }{2-\alpha } \right )\left ( \frac{h_{lv}^{2}}{T_{v}v_{lv}} \right )\sqrt{\frac{M_{v}}{2\pi R_{u}T_{v}}}\left ( 1-\frac{p_{v}v_{lv}}{2h_{lv}} \right )\left ( T_{v}-T_{l} \right )

where is the accommodation coefficient. The corresponding heat transfer coefficient across the interface is obtained by

h_{\delta }=\frac{q_{\delta }^{''}}{T_{v}-T_{l}}=\left ( \frac{2\alpha }{2-\alpha } \right )\left ( \frac{h_{lv}^{2}}{T_{v}v_{lv}} \right )\sqrt{\frac{M_{v}}{2\pi R_{u}T_{v}}}\left ( 1-\frac{p_{v}v_{lv}}{2h_{lv}} \right )

For most systems the second term in the last parentheses is very small compared to unity and, therefore, can be neglected to obtain the following:

h_{\delta }=\frac{q_{\delta }^{''}}{T_{v}-T_{l}}=\left ( \frac{2\alpha }{2-\alpha } \right )\left ( \frac{h_{lv}^{2}}{T_{v}v_{lv}} \right )\sqrt{\frac{M_{v}}{2\pi R_{u}T_{v}}}

The temperature drop across the interface of a single liquid droplet – assuming it is hemispherical – is

\Delta T_{\delta }=\frac{q_{d}}{h_{\delta }\left ( \pi D^{2}/2 \right )}

where πD2 / 2 = Alv is the surface area of the hemispherical liquid droplet.

Resistance Due to the Capillary Depression of the Equilibrium Saturation Temperature

Upon formation in their nucleation sites, droplets grow only if they form with a radius that exceeds the equilibrium radius. The minimum equilibrium radius of a newly formed droplet that will not spontaneously disappear is (Faghri and Zhang (2006):

r_{min}=\frac{2\sigma v_{l}}{R_{g}Tln[p_{v}/p_{sat}(T)]}

The relationship between saturation pressure and corresponding temperature can be obtained by Clapeyron-Clausius equation (Faghri, 1995):

\frac{dp}{dT}=\frac{h_{lv}p}{R_{g}T^{2}}

which can be integrated to obtain:

ln\frac{p_{v}}{p_{sat}}=-\frac{h_{lv}}{R_{g}}\left ( \frac{1}{T_{w}}-\frac{1}{T_{v}} \right )

Substituting eq. into eq. and assuming T = Tw and T_{v}T_{w}\approx T_{w}^{2}, the expression for minimum equilibrium radius size becomes

r_{min}=\frac{D_{min}}{2}=\frac{2v_{l}\sigma T_{v}}{h_{lv}\left ( T_{v}-T_{w} \right )}

According to Graham and Griffith (1973), a resistance exists due to the slight depression of the equilibrium interface temperature below that of the normal saturation temperature for a droplet of diameter D. Assuming that a droplet forms with the minimum equilibrium diameter and grows spontaneously to any nonstable diameter D, we can replace the temperature difference TsatTw with ΔTcap and the minimum droplet diameter Dmin with the actual size of the droplet D in eq. to get the following expression for the temperature drop across the capillary depression:

\Delta T_{cap}=\frac{4v_{l}\sigma T_{sat}}{h_{lv}D}

Combining eq. with the minimum equilibrium droplet size expression, eq. , one obtains

\Delta T_{cap}=\frac{\left ( T_{sat}-T_{w} \right )D_{min}}{D}

Resistance Due to Conduction through the Droplet

The conduction of heat through either a liquid droplet or liquid film is usually the controlling factor in resistance to heat flow. This is due directly to the fact that the largest temperature drop in the condensation process occurs in the liquid film or droplet, even though the conduction path is relatively short in comparison to the other heat flow lengths in the condensation process. This leads to high resistances and low heat transfer coefficients. Graham and Griffith (1973) also developed an expression for the conduction of heat through a single droplet of diameter D from the liquid-vapor interface to the wall. As the heat flux travels through the droplet from interface to wall, the planar area normal to the heat flux varies due to the droplet’s spherical shape. Also, the distance that the heat flux has to travel through the droplet depends on where it entered the droplet at the liquid-vapor interface. Therefore, Graham and Griffith (1973) took into account these variations in the development of the heat flux through a single droplet. This heat flux can be written in terms of the temperature drop through the droplet due to conduction as follows: