Nusselt theory of film condensation

Film and dropwise condensation

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 $T_{sat}$.