---
title: "Nusselt theory of film condensation"
subject: "Heat Transfer"
chapter: 5
key: "semester_5/heat_transfer:nusselt-film-condensation-theory"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/nusselt-film-condensation-theory
site: BCT Engineering
---

# Nusselt theory of film condensation

Heat Transfer · Chapter 5: Phase Change Heat Transfer

- Asked in: year not given, 8 marks
- Video: [Heat Transfer L29 p1 - Condensation - Nusselt's Model](https://www.youtube.com/watch?v=K48RirCbu74) — Ron Hugo
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/nusselt-film-condensation-theory
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-5

## Question

Distinguish between film and dropwise condensation. State the assumptions of Nusselt's theory of film condensation on a vertical plate and derive the expression for the average heat transfer coefficient.

## Short answer

Film condensation forms a continuous liquid layer with higher thermal resistance and lower h; dropwise condensation forms droplets that roll off, giving 5–10 times higher h but is difficult to sustain. Nusselt's theory assumes laminar film, pure saturated quiescent vapor, uniform wall temperature, conduction across film, and constant properties. The average heat transfer coefficient is: h_bar = 0.943 [ρ_l(ρ_l−ρ_v) g h'_fg k_l^3 / (μ_l L ΔT)]^{1/4}.

## Answer

#### Film and dropwise condensation
| Point | Film condensation | Dropwise condensation |
|---|---|---|
| Surface | Wetted; liquid forms a continuous film | Not wetted (oily or coated); drops form and roll off |
| Resistance | Film acts as thermal resistance | Little film, bare surface exposed |
| $h$ | Lower | 5 to 10 times higher |
| Occurrence | Usual in industry (clean surfaces) | Hard to maintain; needs promoters |

Industrial condensers are designed for film condensation because it is the dependable condition.

#### Assumptions of Nusselt's theory
1. Laminar film flow with constant fluid properties.
2. Pure saturated vapour at rest, no shear at the liquid-vapour interface.
3. Wall at uniform temperature $T_w < T_{sat}$.
4. Heat moves across the film by conduction only (linear temperature profile); no convection or inertia in the film.
5. Subcooling of the liquid is included through $h'_{fg} = h_{fg}+0.68c_{pl}(T_{sat}-T_w)$; vapour density is small compared with liquid.

#### Derivation
Let the film thickness at height $x$ be $\delta$. Force balance on an element of liquid (gravity minus buoyancy balanced by viscous shear):

$$
\mu_l\frac{d^2u}{dy^2} = -g(\rho_l-\rho_v)
$$

With $u = 0$ at $y = 0$ (wall) and $du/dy = 0$ at $y = \delta$:

$$
u = \frac{g(\rho_l-\rho_v)}{\mu_l}\left(\delta y-\frac{y^2}{2}\right)
$$

Mass flow per unit width: $\dot m = \int_0^\delta\rho_l u\,dy = \dfrac{\rho_l g(\rho_l-\rho_v)\delta^3}{3\mu_l}$.

Heat transferred by conduction across the film equals latent heat released by the extra condensate $d\dot m$ in length $dx$:

$$
\frac{k_l(T_{sat}-T_w)}{\delta}dx = h'_{fg}\,d\dot m = h'_{fg}\frac{\rho_l g(\rho_l-\rho_v)\delta^2}{\mu_l}d\delta
$$

Integrating from $\delta = 0$ at $x = 0$:

$$
\delta = \left[\frac{4\mu_lk_l(T_{sat}-T_w)\,x}{g\rho_l(\rho_l-\rho_v)h'_{fg}}\right]^{1/4}
$$

The local coefficient is $h_x = k_l/\delta$. The average over height $L$ is $\bar h = \frac{4}{3}h_{x=L}$:

$$
\bar h = 0.943\left[\frac{\rho_l(\rho_l-\rho_v)\,g\,h'_{fg}\,k_l^3}{\mu_l\,L\,(T_{sat}-T_w)}\right]^{1/4}
$$

Properties of liquid are taken at film temperature $(T_{sat}+T_w)/2$ and $h_{fg}$ at $T_{sat}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement correctly compares film and dropwise condensation?
   A. Dropwise has a lower h because drops insulate the surface
   B. Film condensation occurs on non-wetted surfaces
   C. Both have the same h
   D. Dropwise gives 5 to 10 times higher h but is hard to maintain, so condensers are designed for film
   Answer: D. In dropwise condensation the bare surface is exposed, so h is much higher, but it is hard to sustain, so design uses film condensation. (Key point: Film vs dropwise)

2. Which is one of the Nusselt assumptions?
   A. Turbulent film flow
   B. Heat crosses the film by conduction only, with a linear temperature profile
   C. Vapour moves at high velocity, giving interfacial shear
   D. Wall temperature varies along the height
   Answer: B. Nusselt assumes laminar flow, still vapour with no shear, uniform wall temperature and conduction-only heat transfer in the film. (Key point: Nusselt assumptions)

3. What boundary conditions are used to get the film velocity profile?
   A. u = 0 at both wall and interface
   B. du/dy = 0 at the wall and u = 0 at the interface
   C. u = 0 at the wall and du/dy = 0 at the interface (no shear)
   D. u maximum at the wall
   Answer: C. No slip at the wall (y = 0) and zero shear at the liquid-vapour interface (y = delta) give u = g(rho_l - rho_v)/mu (delta y - y^2/2). (Key point: Velocity profile)

4. How is the average coefficient obtained from the local one?
   A. h_avg = h_x at x = L
   B. h_avg = (4/3) h_x at x = L
   C. h_avg = (3/4) h_x at x = L
   D. h_avg = 2 h_x at x = L
   Answer: B. Since h_x = k/delta varies as x^(-1/4), averaging over L gives 4/3 of the end value, leading to the 0.943 constant. (Key point: Average coefficient)

## Related in Heat Transfer

- [Condensation of steam on a vertical plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afilm-condensation-vertical-plate-numerical) — Heat Transfer, chapter 5
- [Natural convection from a vertical plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Anatural-convection-vertical-plate-heat-loss) — Heat Transfer, chapter 3
- [Laminar forced convection over a heated flat plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Alaminar-flow-over-heated-plate-air) — Heat Transfer, chapter 3

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