# Heat Transfer Coefficient for Dropwise Condensation

By substituting the expressions for temperature drops through the interface, capillary depression, and liquid droplets into the following equation

ΔTtotal = TvaporTw = ΔTvapor + ΔTδ + ΔTcap + ΔTdroplet

and neglecting the temperature drop in the vapor phase, the temperature drop is obtained:

 $\Delta T_{total}=T_{sat}-T_{w}=\frac{2q_{d}}{h_{\delta }\pi D^{2}}+\left ( T_{v}-T_{w} \right )\frac{D_{min}}{D}+\frac{q_{d}}{2k_{l}\pi D}$ (1)

The heat flux through a single droplet can then be given by rearranging eq. (1), i.e.,

 $q_{d}=\left ( \frac{\pi d^{2}}{2} \right )\Delta T_{total}\frac{\left ( 1-D_{min}/D \right )}{1/h_{\delta }+D/4k_{l}}$ (2)

To obtain an expression for the total heat transfer through all of the droplets, one must integrate over the total number of droplets and the whole size distribution of the droplets. To do this, we must know the droplet size distribution equal to the number of droplets with diameters between D and D + dD per unit area of surface. Therefore, to obtain the expression for total heat flux we multiply qd times the number density $n_{D}^{''}dD$ of droplets of size D and integrate over the whole range of droplet size (Carey, 1992):

 $q^{''}=\frac{\pi \Delta T_{total}}{2}\int_{D_{min}}^{D_{max}}n_{D}^{''}D^{2}\frac{\left ( 1-D_{min}/D \right )}{1/h_{\delta }+D/4k_{l}}dD$ (3)

The total heat transfer coefficient for the area covered by the liquid droplets can be found for the dropwise condensation process:

 $h_{drops}=\frac{\pi }{2}\int_{D_{min}}^{D_{max}}n_{D}^{''}D^{2}\frac{\left ( 1-D_{min}/D \right )}{1/h_{\delta }+D/4k_{l}}dD$ (4)

In dropwise condensation, the type of conduction that occurs at the wall is a direct result of the constricted heat flow around and between the large droplets on the surface of the wall. In filmwise condensation, conduction from the liquid-solid interface would be obtained by applying Fourier’s Law through the liquid film. However, in dropwise condensation, the resistance to thermal transport found in the liquid droplets is much higher than the resistance found in the liquid-free area between the droplets. Conduction through the walls would prefer to initiate at the bare areas. Therefore, the droplets effectively “squeeze” or “constrict” the heat flow toward the small bare areas. Mikic (1969) developed an expression for the resistance encountered in this constriction:

 $R_{cons}=\left ( \frac{1}{3\pi k_{l}} \right )\int_{D_{min}}^{D_{max}}\frac{n_{D}^{''}D^{2}dD}{[1-f(d)]}$ (5)

where f(D) is the fraction of the surface area covered by droplets with diameters greater than D. As can be seen above, this expression is a total expression for resistance, because it takes into consideration the full range and number of droplets.

Finally, if this resistance is combined and assumed to be in series with all the other resistances described in previous sections, the total heat transfer coefficient, including conduction through the wall (in this development, the constriction conduction mode was used due to the dropwise condensation discussion), is found to be

 $h_{total}=\left ( \frac{1}{h_{drops}}+R_{cons} \right )^{-1}$ (6)

This theoretical model cannot be used to predict the heat transfer coefficient for dropwise condensation unless we know the droplet size distribution $n_{D}^{''}$. Determination of droplet size is very difficult, especially for droplets smaller than 10 μm, which are major contributors to the heat transfer in dropwise condensation.

Among the possible combinations of fluids and surfaces, steam and well-promoted copper surfaces have been investigated extensively. Griffith (1983) recommended the following correlation for prediction of the heat transfer coefficient for dropwise condensation of steam:

 $h=\begin{Bmatrix} 51104+2044T_{sat} & 22^{\circ}C100^{\circ}C \end{Bmatrix}$ (7)

## References

Carey, V. P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D. C.

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.

Griffith, P., 1983, “Dropwise Condensation,” in Heat Exchange Design Handbook, edited by E.U. Schlunder, Vol. 2, Chapter 2.6.5, Hemisphere Publishing, New York, NY.

Mikic, B.B., 1969, “On Mechanism of Dropwise Condensation,” International Journal of Heat and Mass Transfer, Vol. 12, pp. 1311-1323.