# Film condensation on cylinders and spheres

## External condensation on a vertical cylinder

If the thickness of the liquid film is smaller than the diameter of the cylinder by at least an order of magnitude, the heat transfer and Nusselt number expressions for a vertical cylinder are the same as those for a vertical plate.

## Horizontal cylinders and spheres

When the cold surface is curved, the tangential direction of the gravity varies along the condensate film. The empirical correlations for other configurations also have almost the same forms as that of a vertical plate, except the leading coefficient and the characteristic length differ. For example, for laminar film condensation on a horizontal cylinder, the leading coefficient changes from 0.943 to 0.729 and the characteristic length changes from L to D, the diameter of the cylinder, i.e.,

 $\overline{Nu}=\frac{{{\overline{h}}_{D}}D}{{{k}_{\ell }}}=0.729{{\left[ \frac{{{D}^{3}}{{h}_{\ell v}}g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}{{{k}_{\ell }}{{\nu }_{\ell }}({{T}_{sat}}-{{T}_{w}})} \right]}^{1/4}}$ (1)

which can be obtained by applying Nusselt analysis.

For laminar film condensation on a sphere, the average heat transfer coefficient can also be obtained by Nusselt analysis.

 $\overline{Nu}=\frac{{{\overline{h}}_{D}}D}{{{k}_{\ell }}}=0.815{{\left[ \frac{{{D}^{3}}{{{{h}'}}_{\ell v}}g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}{{{k}_{\ell }}{{\nu }_{\ell }}({{T}_{sat}}-{{T}_{w}})} \right]}^{1/4}}$ (2)

## Vertical column of n horizontal tubes

Another expression of great importance in heat exchanger design is that of a vertical column of n horizontal tubes. The average heat transfer coefficient for all n cylinders can be obtained by:

 $\overline{Nu}=\frac{{{\overline{h}}_{D,n}}D}{{{k}_{\ell }}}=0.729{{\left[ \frac{{{D}^{3}}{{{{h}'}}_{\ell v}}g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}{n{{k}_{\ell }}{{\nu }_{\ell }}({{T}_{sat}}-{{T}_{w}})} \right]}^{1/4}}$ (3)

which indicates that the average heat transfer coefficient for all n cylinder, ${{\bar{h}}_{D,n}}$, is related to the average heat transfer coefficient for the first tube of this array, ${{\bar{h}}_{D}}$, by

 ${{\overline{h}}_{D,n}}=\frac{{{\overline{h}}_{D}}}{{{n}^{1/4}}}$ (4)

This calculation would lead to a lower average heat transfer coefficient for an array than that found for a single tube. In practice, however, it has been found that these calculations significantly underestimate the heat transfer capabilities of this system, due to splashing effects that occur as the condensate (in the form of sheet or droplets) falls from one tube to a tube underneath.

## Condensation on the outside of a horizontal tube in crossflow

The correlations that we discussed thus far have been limited to cases in which the flow of the condensate is driven by gravity. When the vapor is forced to flow over the cooler surface, the vapor interacts with the condensate and drags liquid in the vapor flow direction. For condensation on the outside of a horizontal tube in crossflow, the heat transfer coefficient is affected by both free-steam velocity of vapor, ${{u}_{\infty }}$, and gravitational force (Shekriladze and Gomelauri, 1966), so that

 $\overline{Nu}=\frac{\overline{h}D}{{{k}_{\ell }}}=0.64\operatorname{Re}_{D}^{1/2}{{\left[ 1+{{\left( 1+1.69\frac{g{{{{h}'}}_{\ell v}}{{\mu }_{\ell }}D}{u_{\infty }^{2}{{k}_{\ell }}\left( {{T}_{sat}}-{{T}_{w}} \right)} \right)}^{1/2}} \right]}^{1/2}}$ (5)

where ${{\operatorname{Re}}_{D}}={{u}_{\infty }}D/{{\nu }_{\ell }}$ is based on the viscosity of liquid. Equation (5) is valid for a Reynolds number up to 106.

## References

Bejan, A., 1991, “Film Condensation on a Upward Facing Plate with Free Edges,” International Journal of Heat and Mass Transfer, Vol. 34, pp. 578-582.

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.

Shekriladze, I.G., and Gomelauri, V.I., 1966, “Theoretical Study of Laminar Film Condensation of Flow Vapour,” International Journal of Heat and Mass Transfer, Vol. 9, pp. 581-591.