Phase Change Heat Transfer

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 $T_{sat}$.

Pool boiling is boiling on a heated surface submerged in a stagnant liquid. The behaviour depends on the excess temperature $\Delta T_e = T_w - T_{sat}$.

Regimes

 q"  ^           . Critical (burnout)
 log |          /  \
     |        /      \       _.-- film
     |      /          \  _.-'    boiling
     |    /  nucleate    '.  Leidenfrost
     |  / A    B     C   D   E
     +-------------------------> log dT_e

The minimum of the curve is the Leidenfrost point.

Critical heat flux

In heat flux-controlled equipment (electric heaters, nuclear fuel rods), exceeding $q_{max}$ makes the surface temperature jump to the film boiling value, often above the melting point of the material (burnout). So the design heat flux is kept well below $q_{max}$.

Correlations

• Rohsenow (nucleate boiling): $q'' = \mu_lh_{fg}\left[\dfrac{g(\rho_l-\rho_v)}{\sigma}\right]^{1/2}\left[\dfrac{c_{pl}\Delta T_e}{C_{sf}h_{fg}Pr_l^n}\right]^3$, $C_{sf}$ depends on the surface-liquid pair.
• Zuber (critical flux): $q''_{max} = 0.131\,\rho_v^{1/2}h_{fg}\left[\sigma g(\rho_l-\rho_v)\right]^{1/4}$.

(a) Average coefficient (Nusselt)

$\Delta T = 100 - 80 = 20$ K, $L = 0.5$ m, $g = 9.81$ m/s$^2$.

$$\begin{aligned} \bar h &= 0.943\left[\frac{\rho_l(\rho_l-\rho_v)\,g\,h_{fg}\,k_l^3}{\mu_l\,L\,\Delta T}\right]^{1/4}\\ &= 0.943\left[\frac{6.34\times10^{12}}{3.148\times10^{-3}}\right]^{1/4}\\ &= 0.943\,[2.014\times10^{15}]^{1/4} = 0.943\times6699\\ &= 6317\ \text{W/m}^2\text{K} \end{aligned}$$

(b) Heat transfer rate

$$Q = \bar h A\,\Delta T = 6317\times(0.5\times1)\times20 = 63\,175\ \text{W} = 63.2\ \text{kW}$$

(c) Condensate rate

$$\dot m = \frac{Q}{h_{fg}} = \frac{63\,175}{2.257\times10^6} = 0.0280\ \text{kg/s} = 100.8\ \text{kg/h}$$

(d) Film Reynolds number

$$Re_f = \frac{4\dot m}{b\,\mu_l} = \frac{4\times0.0280}{1\times314.8\times10^{-6}} = 356$$

$Re_f$ lies between 30 and 1800, so the film is wavy laminar. Nusselt's equation is then conservative: ripples usually raise $\bar h$ by about 20 percent (Kutateladze correction), so the true value may be near 7500 W/m$^2$K.

Answer: $\bar h \approx 6320\ \text{W/m}^2$K; $Q \approx 63.2$ kW; $\dot m \approx 101$ kg/h; $Re_f \approx 356$ (wavy).