Condensation Heat Transfer Analysis
Theory: the condensate film resists heat
Overview
Condensation is just vapour cooling into liquid. What makes the heat transfer coefficient hard to calculate?
Vapour turns to liquid at the wall very quickly, but the liquid forms a film that blocks the heat path. Heat must cross the film to reach the wall, so the coefficient is set mostly by film thickness. The film thickens as it drains downward, so taller walls have lower averages. Nusselt put this into equations in 1916. Let's compare vertical walls and horizontal tubes, and see how film waves and tube rows matter.
Vertical wall (Nusselt)
$L$ is wall height; $h'_{fg}$ adds film subcooling to the latent heat. The coefficient is inversely proportional to the quarter power of $L$ and temperature difference, so a 16× taller wall only halves it.
Horizontal tubes and banks
On a horizontal tube liquid only flows half the circumference, so the film is thin. In an N-row bank, condensate from upper tubes drips onto lower ones and thickens their films, so the average scales by $N^{-1/4}$ (Nusselt's estimate; in practice splashing softens the drop, and some designs use $N^{-1/6}$).
Nusselt's 1916 paper
In 1916 Germany's Nusselt, in a short paper on steam condensation, derived the heat transfer coefficient from film flow and conduction alone — an early example of analytically combining fluid mechanics and heat transfer, still found at the start of textbooks. His name survives in the Nusselt number, the dimensionless heat transfer coefficient. That a century-old formula remains usable in design, once corrected for waviness, shows its assumptions captured the essential physics.
Worked example: steam at 1 atm
Wall height and film
Saturated steam (100 °C) condensing on a 90 °C wall (10 K difference):
| Wall height | Coefficient (Nusselt) | Film at bottom | Film Reynolds number | With waviness |
|---|---|---|---|---|
| 0.1 m | 11,440 W/m²K | 0.079 mm | 67 | 11,860 (+4%) |
| 0.5 m | 7,650 W/m²K | 0.118 mm | 225 | 8,710 (+14%) |
| 1.0 m | 6,430 W/m²K | 0.140 mm | 379 | 7,670 (+19%) |
The film is only about 0.1 mm, but water's low conductivity makes it most of the thermal resistance. Above a film Reynolds number of about 30 the surface becomes wavy, thinning the film locally and raising the coefficient.
Effect of temperature difference (0.5 m)
| Temperature difference | Coefficient | Heat flux |
|---|---|---|
| 5 K | 9,080 W/m²K | 45 kW/m² |
| 10 K | 7,650 W/m²K | 76 kW/m² |
| 20 K | 6,450 W/m²K | 129 kW/m² |
| 40 K | 5,460 W/m²K | 218 kW/m² |
A larger difference condenses more, thickening the film, so the coefficient falls; heat flux grows only as the 3/4 power of the difference.
Vertical tube, horizontal tube, banks
| Arrangement | Mean coefficient |
|---|---|
| Vertical tube (1 m long) | 6,430 W/m²K |
| Horizontal tube (25 mm diameter) | 12,500 W/m²K |
| Horizontal, 5 rows | 8,360 W/m²K |
| Horizontal, 10 rows | 7,030 W/m²K |
| Horizontal, 20 rows | 5,910 W/m²K |
One horizontal tube nearly doubles the vertical value, yet 20 rows fall below vertical.
Horizontal tubes win because the liquid travels a short distance. Comparing the coefficients and lengths, a 25 mm horizontal tube matches a vertical wall only 7 cm tall. But stacking rows accumulates falling condensate, effectively the same as a long flow path. So large condensers limit row count or drain condensate partway. In reality droplets splash, so the fall-off is gentler than Nusselt's estimate — some designs take about 0.6 rather than 0.47 for 20 rows.
Analysis and design tips
- Compute the film Reynolds number to choose laminar, wavy-laminar or turbulent correlations.
- Vent non-condensables such as air; even small amounts accumulate near the wall and sharply cut the coefficient.
- In banks, consider row effects and vapour shear.
- Dropwise condensation is several times higher but hard to sustain; design normally assumes filmwise.
- When modelling condensation in CFD, check the phase-change model and film resolution (about 0.1 mm).
“The condenser that lost performance to an air leak”
At a power plant, condenser vacuum worsened and turbine output fell though cooling-water temperature and flow were unchanged. The cause was a small air leak at a shaft seal. As steam condenses, air is left behind near the tube surface as a thin layer the steam must diffuse through, badly hindering condensation. Fixing the leak and checking the air ejector restored vacuum. Condenser performance depends not just on the film but on an invisible air layer on the steam side.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ignoring non-condensables | Performance overestimated | Venting and diffusion |
| Single-tube formula for banks | Coefficient too high | Row correction |
| Ignoring waviness (high Re) | Coefficient too low | Wavy-laminar correlation |
| Properties at vapour temperature | Error | Use film temperature |
| Designing for dropwise | Shortfall over time | Design for filmwise |
I'd like to learn related topics.
Try the vertical-wall film condensation tool and steam tables. Related pages include boiling heat transfer, two-phase heat exchangers, heat pipes and thermosiphons.
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