3.5 Boiling and Condensation

Pool Boiling Regimes

Mechanisms and Characteristics

Pool boiling occurs when a heated surface is submerged in a large volume of stagnant (quiescent) liquid. The liquid near the surface absorbs enough energy to undergo a phase change to vapor. What determines the behavior of this process is the excess temperature $\Delta T_{excess} = T_s - T_{sat}$, the difference between the surface temperature and the saturation temperature of the liquid.

As you increase $\Delta T_{excess}$, the boiling passes through four distinct regimes. These are best understood by following the boiling curve, which plots heat flux $q''$ against excess temperature.

The Four Regimes

• Natural convection (low $\Delta T_{excess}$, typically < ~5°C for water at 1 atm): Heat transfer occurs through buoyancy-driven convection currents in the liquid. No significant bubble formation happens yet. Think of heating water in a pot before you see any bubbles.
• Nucleate boiling (moderate $\Delta T_{excess}$, roughly 5–30°C for water at 1 atm): Vapor bubbles form at nucleation sites on the heated surface, grow, and detach. The vigorous mixing caused by bubble motion produces very high heat transfer rates. This is the regime you want in most engineering applications. Visually, this is water at a rolling boil with bubbles streaming up from the bottom of the pot.
• Transition boiling (higher $\Delta T_{excess}$): Vapor bubbles begin to merge into unstable vapor patches that intermittently blanket portions of the surface. The liquid can only contact the surface sporadically, so the heat transfer coefficient decreases with increasing $\Delta T_{excess}$. This regime is unstable and generally avoided in design.
• Film boiling (high $\Delta T_{excess}$): A stable, continuous vapor film covers the entire heated surface. Because vapor is a poor thermal conductor compared to liquid, this film acts as an insulating layer and heat transfer rates drop significantly. The Leidenfrost effect is a familiar example: a water droplet placed on a very hot skillet levitates on a cushion of its own vapor instead of wetting the surface.

Critical Heat Flux in Boiling

Why Critical Heat Flux Matters

The critical heat flux (CHF) is the maximum heat flux achievable during nucleate boiling. Beyond this point, the surface transitions toward film boiling, the heat transfer coefficient drops sharply, and the surface temperature can spike to dangerous levels. In applications like nuclear reactor cooling, exceeding CHF can lead to catastrophic failure, so understanding and predicting it is essential.

Factors Influencing CHF

Several factors raise or lower the critical heat flux:

• Surface roughness: Rougher surfaces provide more nucleation sites, which helps sustain nucleate boiling to higher heat fluxes, increasing CHF.
• Surface wettability: Highly wettable (hydrophilic) surfaces promote liquid rewetting and delay vapor film formation, raising CHF.
• Liquid properties: Fluids with higher latent heat of vaporization ($h_{fg}$), higher thermal conductivity, and lower surface tension tend to have higher CHF values because they can absorb and transport more energy before film formation.
• System pressure: Increasing pressure generally increases CHF up to a point, because it changes the relative densities of liquid and vapor and affects bubble dynamics.

Estimating CHF with the Zuber Correlation

The most widely used correlation for predicting CHF on a large flat horizontal surface is the Zuber correlation:

$q''_{CHF} = C \, \rho_v \, h_{fg} \left[\frac{\sigma \, g \, (\rho_l - \rho_v)}{\rho_v^2}\right]^{1/4}$

where:

• $C \approx 0.131$ (Zuber's theoretical constant for a large horizontal plate; Kutateladze proposed $C \approx 0.149$)
• $\rho_v$ = vapor density
• $\rho_l$ = liquid density
• $h_{fg}$ = latent heat of vaporization
• $\sigma$ = surface tension
• $g$ = gravitational acceleration

Example setup for water at 1 atm: Using $\rho_l = 958 \, \text{kg/m}^3$, $\rho_v = 0.6 \, \text{kg/m}^3$, $h_{fg} = 2257 \, \text{kJ/kg}$, $\sigma = 0.0589 \, \text{N/m}$, and $C = 0.131$:

1. Calculate the bracketed term: $\frac{\sigma \, g \, (\rho_l - \rho_v)}{\rho_v^2} = \frac{0.0589 \times 9.81 \times (958 - 0.6)}{0.6^2}$

2. Take the fourth root of that result.

3. Multiply by $C \, \rho_v \, h_{fg}$ to get $q''_{CHF}$.

This yields a CHF on the order of $1 \times 10^6 \, \text{W/m}^2$ for water at atmospheric pressure, which is consistent with experimental data.

Film vs. Dropwise Condensation

Mechanisms and Characteristics

Condensation occurs when vapor contacts a surface whose temperature is below the vapor's saturation temperature. The vapor releases its latent heat and transitions to liquid. How that liquid behaves on the surface determines the type of condensation and, critically, the heat transfer performance.

• Film condensation: The condensate forms a continuous liquid film that flows down the surface under gravity. This film adds thermal resistance between the vapor and the cold surface, reducing the heat transfer rate. This is the more common mode in practice. A typical example is steam condensing on a cold vertical plate, where you can see a smooth sheet of water flowing downward.
• Dropwise condensation: The condensate forms as discrete droplets that nucleate, grow, merge, and eventually roll off the surface. Between droplets, the bare surface is directly exposed to vapor, which dramatically reduces thermal resistance. Heat transfer coefficients for dropwise condensation can be 5 to 20 times higher than for film condensation under the same conditions.

Nusselt Analysis for Film Condensation

For film condensation on a vertical plate, the average heat transfer coefficient can be estimated using the classical Nusselt analysis:

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

where:

• $\rho_l$, $\rho_v$ = liquid and vapor densities
• $g$ = gravitational acceleration
• $h'_{fg} = h_{fg} + 0.68 \, c_{p,l}(T_{sat} - T_s)$ = modified latent heat (accounts for subcooling of the condensate film)
• $k_l$ = liquid thermal conductivity
• $\mu_l$ = liquid dynamic viscosity
• $T_{sat}$ = saturation temperature, $T_s$ = surface temperature
• $L$ = vertical length of the plate (or tube)

A few things to note about this correlation:

• All liquid properties should be evaluated at the film temperature $T_f = (T_{sat} + T_s)/2$.
• The heat transfer coefficient decreases as $L$ increases, because the film thickens as it flows downward.
• The modified latent heat $h'_{fg}$ is often used in place of $h_{fg}$ to improve accuracy. Some textbooks use just $h_{fg}$, so check which version your course expects.
• For horizontal tubes, the same form applies but with the constant 0.725 and the tube diameter $D$ replacing $L$.

Boiling and Condensation Applications

Problem-Solving Approach

Boiling and condensation calculations appear throughout power generation (steam condensers, boilers), refrigeration (evaporators, condensers), and chemical processing (distillation, heat exchangers). A systematic approach keeps these problems manageable:

1. Identify the regime or mode. Is it nucleate boiling, film boiling, film condensation, or dropwise condensation? The excess temperature and surface conditions will tell you.
2. Select the appropriate correlation. Use the Rohsenow correlation for nucleate boiling, the Zuber correlation for CHF, the Nusselt analysis for film condensation, etc.
3. Gather fluid properties at the correct reference temperature (usually the saturation temperature or the film temperature, depending on the correlation).
4. Calculate the heat transfer coefficient from the correlation.
5. Determine the heat transfer rate or other unknowns using Newton's law of cooling or an energy balance.

Calculating Heat Transfer Rates

The heat transfer rate for both boiling and condensation follows Newton's law of cooling:

$q = \bar{h} \times A_s \times (T_{sat} - T_s)$

For condensation, you can also find the condensation rate $\dot{m}$ from an energy balance:

$\dot{m} = \frac{q}{h'_{fg}}$

This tells you how much vapor is condensing per unit time, which is often what you need for condenser sizing.

Example: Film condensation on a vertical tube. Steam at $T_{sat} = 100°\text{C}$ condenses on a vertical tube with $T_s = 80°\text{C}$, length $L = 1 \, \text{m}$, and outer diameter $D = 0.02 \, \text{m}$.

1. Evaluate water properties at the film temperature $T_f = (100 + 80)/2 = 90°\text{C}$.

2. Calculate $h'_{fg}$ using the modified latent heat expression.

3. Plug values into the Nusselt correlation to find $\bar{h}$.

4. Compute the surface area: $A_s = \pi D L = \pi (0.02)(1) \approx 0.0628 \, \text{m}^2$.

5. Calculate the heat transfer rate: $q = \bar{h} \times A_s \times (T_{sat} - T_s)$.

6. If needed, find the condensation rate: $\dot{m} = q / h'_{fg}$.