Natural Convection, Boiling, and Condensation

Introduction and Context

Heat transfer involving motion caused by density differences and phase change is central to many mechanical-engineering systems. Natural convection cools electronic equipment, buildings, transformers, and nuclear-reactor components without mechanical fans or pumps. Boiling provides very high heat-transfer rates in steam generators, evaporators, refrigeration systems, and thermal-management devices. Condensation is used in power-plant condensers, distillation equipment, refrigeration systems, and process heaters.

These processes are highly effective because fluid motion and latent heat transport can greatly exceed the heat-transfer capacity of ordinary single-phase conduction or convection. They also involve important stability and safety limits. Examples include critical heat flux, dryout, tube vibration, pressure excursions, and the loss of cooling-film stability.

Learning Objectives

After studying this chapter, the student should be able to:

1. Explain buoyancy-driven flow and distinguish natural from forced convection.
2. Calculate natural-convection heat-transfer coefficients using appropriate dimensionless correlations.
3. Describe pool-boiling regimes and interpret the boiling curve.
4. Estimate nucleate-boiling heat transfer and identify critical heat flux.
5. Distinguish pool boiling from flow boiling and explain pressure-drop effects.
6. Analyze filmwise and dropwise condensation.
7. Select and evaluate practical phase-change equipment while recognizing operating limits.

1. Buoyancy-Driven Flow

Natural convection occurs when a temperature difference produces density differences in a gravitational field. A warmer fluid generally becomes less dense and rises, while cooler, denser fluid descends. The resulting circulation transports energy without an externally imposed flow.

For a small temperature range, fluid density may be approximated by

where ρ is density, ρ_ref is density at reference temperature T_ref, β is the volumetric thermal-expansion coefficient, and T is temperature. For an ideal gas, β = 1/T when absolute temperature is used. For liquids, β is obtained from property data.

The buoyancy force per unit volume is approximately ρgβ(T − T_ref), where g is gravitational acceleration. The direction of buoyant motion depends on the geometry and temperature distribution. A hot vertical wall produces an upward boundary layer; a hot horizontal upward-facing surface usually produces rising plumes.

Dimensionless groups

Natural-convection correlations are commonly expressed using the Grashof, Prandtl, and Rayleigh numbers:

Here L is the characteristic length, T_s is surface temperature, T_∞ is the far-fluid temperature, ν is kinematic viscosity, α is thermal diffusivity, and Pr is the Prandtl number. The Rayleigh number measures the relative importance of buoyancy-driven motion compared with viscous and thermal diffusion effects.

The heat-transfer coefficient is defined through Newton’s law of cooling:

where q̇'' is heat flux and h is the convective heat-transfer coefficient. The average Nusselt number is

where k is fluid thermal conductivity. A general correlation has the form Nu_L = f(Ra_L, Pr, geometry, boundary conditions).

Typical natural-convection correlations

For a vertical plate, a widely used engineering correlation is

This Churchill–Chu form is applicable over a broad range of laminar and turbulent conditions, approximately 10⁻¹ < Ra_L < 10¹², although property evaluation and geometry limits must still be respected.

For a vertical plate in a predominantly laminar boundary layer, a simpler relation often used for moderate Rayleigh numbers is

For a horizontal surface, the correlation depends strongly on whether the hot surface faces upward or downward. A hot surface facing upward promotes instability and strong convection; a hot surface facing downward suppresses motion and usually gives a lower heat-transfer coefficient. Therefore, a correlation must match the orientation and characteristic length defined by its source.

Property evaluation

For external natural convection, fluid properties are usually evaluated at the film temperature:

For gases over moderate temperature differences, this is often adequate. Near phase-change conditions, near the critical point, or for large temperature differences, properties should be evaluated more carefully and variable-property correlations may be needed.

Example 1: Natural convection from a vertical plate

A vertical plate 0.50 m high and 1.0 m wide is maintained at 80 °C in still air at 20 °C. At the film temperature of 50 °C, take k = 0.028 W/(m·K), ν = 1.80 × 10⁻⁵ m²/s, α = 2.58 × 10⁻⁵ m²/s, β = 1/323 K⁻¹, and g = 9.81 m/s². Estimate the average heat-transfer coefficient using the laminar relation.

The Rayleigh number is

This is below approximately 10⁹, so a laminar estimate is reasonable for an introductory calculation. Thus,

Therefore,

The heat loss from both sides, neglecting edge effects, is approximately

The result is an estimate: radiation, ambient drafts, property variation, and transition to turbulence may alter the actual heat loss.

2. Pool Boiling

Pool boiling occurs when a heated surface is immersed in a liquid that is otherwise substantially stationary. The surface temperature exceeds the saturation temperature corresponding to the local pressure. Define the wall superheat as

where T_sat is the saturation temperature and ΔT_e is the excess temperature.

Boiling can be classified as subcooled or saturated. In subcooled boiling, the bulk liquid temperature is below T_sat, although vapor bubbles may form near the heated wall. In saturated boiling, the bulk liquid is approximately at saturation conditions.

Nucleation and bubble dynamics

Boiling generally begins at microscopic cavities or surface imperfections containing vapor embryos. A bubble grows when evaporation at the liquid–vapor interface exceeds condensation caused by surrounding liquid. Bubble departure is influenced by surface tension, buoyancy, inertia, wettability, pressure, and the local thermal boundary layer.

Surface tension is important at small scales. The pressure difference across a curved interface is described by the Young–Laplace relation:

where σ is surface tension and R₁ and R₂ are principal radii of curvature. For a spherical bubble, Δp = 2σ/R.

The pool-boiling curve

The boiling curve plots heat flux q̇'' against excess temperature ΔT_e. Its principal regions are:

1. Natural convection region: The liquid is heated without sustained bubble formation.
2. Onset of nucleate boiling: Isolated bubbles begin forming at active nucleation sites.
3. Fully developed nucleate boiling: Bubble formation and departure produce vigorous mixing and high heat transfer.
4. Critical heat flux: The maximum heat flux before the surface becomes partially blanketed by vapor.
5. Transition boiling: Vapor coverage is unstable; heat flux may decrease as surface temperature rises.
6. Film boiling: A relatively continuous vapor layer separates the surface from the liquid.

The critical heat flux is also called the burnout or departure-from-nucleate-boiling limit. Exceeding it can cause a rapid increase in wall temperature because vapor has much lower thermal conductivity and density than liquid.

For a large horizontal heater in a quiescent pool, the Zuber critical-heat-flux estimate is

where h_fg is latent heat of vaporization, ρ_l and ρ_v are saturated-liquid and saturated-vapor densities, and σ is surface tension. The coefficient is an engineering approximation for an idealized, large, horizontal surface. Geometry, liquid motion, confinement, pressure, dissolved gases, and surface condition can significantly change the result.

For nucleate boiling, empirical correlations are commonly used. The Rohsenow correlation is one example:

where μ_l is saturated-liquid viscosity, c_p,l is liquid specific heat, Pr_l is liquid Prandtl number, C_sf is a surface–fluid combination constant, and n is an empirical exponent. Values of C_sf and n depend on the fluid and surface; they must not be assumed universal.

Safety significance of critical heat flux

A design should not operate near the predicted critical heat flux without an appropriate margin. A simple engineering margin may be expressed as

A value greater than unity is necessary but not sufficient for safe operation. Transients, fouling, local hot spots, flow maldistribution, and uncertainty in correlations must also be considered. In high-energy equipment, protection may include trips, pressure relief, redundant temperature measurements, controlled ramp rates, and emergency cooling.

3. Flow Boiling

Flow boiling occurs when a liquid moves through a heated channel and vapor is generated. It is the operating mode of evaporators, boilers, steam generators, and many cooling channels. Unlike pool boiling, the flow rate strongly affects wall temperature, vapor distribution, pressure drop, and dryout.

Typical flow regimes include bubbly flow, slug or plug flow, churn flow, annular flow, and mist or dispersed flow. The regime depends on mass flux, quality, channel size, pressure, fluid properties, and orientation.

Vapor quality is defined as the mass fraction of vapor in a two-phase mixture:

where ṁ_v and ṁ_l are vapor and liquid mass flow rates. Quality is meaningful in the two-phase region and is not the same as void fraction, which is the volume fraction occupied by vapor.

A useful energy balance for an evaporator is

where ṁ is total mass flow rate and h is specific enthalpy. If saturated liquid enters and saturated vapor leaves at nearly constant pressure,

Pressure drop in a boiling channel consists of frictional, acceleration, and gravitational components. Since saturation temperature changes with pressure, pressure drop also affects local boiling conditions. Excessive pressure drop may reduce system capacity, promote instability, or cause flow starvation.

Flow-boiling stability

Parallel channels can experience unstable flow redistribution. A channel receiving less flow may heat more, generate more vapor, and experience a greater pressure-drop change. Depending on the system characteristic, this feedback can either stabilize or destabilize the flow. Engineering controls include restrictors, careful manifold design, feedback control, and operation away from dryout.

4. Condensation

Condensation occurs when vapor changes to liquid after rejecting latent heat. The condensate may form a continuous liquid film or discrete droplets.

Filmwise condensation

In filmwise condensation, the condensate wets the surface and forms a continuous film. Heat passes through the condensate by conduction, often making the film the dominant thermal resistance. As the film thickens under gravity, its thermal resistance increases and local heat transfer decreases.

For laminar film condensation on a vertical plate, the Nusselt average correlation is

Thus,

Here L is plate height, k_l is condensate thermal conductivity, μ_l is condensate dynamic viscosity, ΔT = T_sat − T_w, and properties are usually evaluated at a suitable film temperature. The model assumes a smooth laminar film, negligible vapor shear, negligible radiation, constant properties, and a vertical isothermal surface.

The local film thickness increases downward. Consequently, the local heat-transfer coefficient is highest near the top and decreases toward the bottom. Subcooling of the condensate and vapor shear may require more advanced models.

Condensation inside tubes

In condensers, vapor may condense inside or outside tubes. Important design variables include tube diameter, orientation, vapor velocity, condensate drainage, noncondensable gases, fouling, pressure drop, and allowable approach temperature. Noncondensable gases are particularly harmful because they accumulate near the condensing interface and create an additional diffusion resistance.

Dropwise condensation

In dropwise condensation, liquid forms discrete droplets that repeatedly grow, coalesce, and drain. Much of the surface remains exposed to vapor, so heat-transfer coefficients can be several times higher than in filmwise condensation. The challenge is maintaining a water-repellent or otherwise nonwetting surface under temperature cycling, chemical exposure, erosion, and contamination.

Filmwise condensation is usually the conservative design assumption because it is stable and easier to maintain. Dropwise condensation can be valuable in specialized equipment but requires surface engineering and reliable long-term performance.

Condensation heat-transfer resistance

For a heat exchanger, the overall heat-transfer coefficient may be represented by

where U is based on a consistent area basis, h_i and h_o are inside and outside convection coefficients, R_f are fouling resistances per unit area, δ_w is wall thickness, and k_w is wall conductivity. If inside and outside areas differ significantly, area ratios must be included. The heat-transfer rate is

for a heat exchanger with approximately constant overall coefficient, where ΔT_lm is the log-mean temperature difference.

5. Phase-Change Equipment

Boilers and evaporators

A boiler adds heat to produce vapor, while an evaporator removes solvent or concentrates a solution. Design must address heat flux, circulation, pressure control, liquid level, fouling, carryover, and relief protection. Natural-circulation boilers use density differences between heated and unheated legs; forced-circulation units use pumps.

Condensers

Surface condensers keep the condensate separate from the cooling fluid and are common in power plants. Direct-contact condensers mix the streams and can achieve high effectiveness, but the condensate and coolant must be compatible. Air-cooled condensers avoid water consumption but require larger heat-transfer surfaces and are sensitive to ambient temperature.

Refrigeration evaporators and condensers

In refrigeration, the evaporator absorbs heat at low pressure and the condenser rejects heat at high pressure. Proper refrigerant distribution, superheat control, oil return, and avoidance of liquid carryover into the compressor are essential. Environmental regulations influence refrigerant selection, particularly regarding ozone-depletion potential and global-warming potential.

6. Illustrative Case: Cooling a High-Heat-Flux Electronics Module

Background. An electronics manufacturer must cool a compact power module dissipating 2 kW. Air cooling would require a large fan and heat sink, while the module has limited surface area.

Problem. A boiling-based cold plate is considered. The designer must prevent critical heat flux, maintain uniform cooling, and avoid pressure excursions.

Analysis. The cold plate uses a dielectric working fluid circulating through parallel channels. The heat load is converted into a required enthalpy rise:

The channel geometry is evaluated for local heat flux, pressure drop, vapor quality, and flow distribution. The predicted critical heat flux is compared with the maximum local operating heat flux using a design margin. Sensors monitor inlet pressure, outlet pressure, wall temperature, and flow rate.

Approach and solution. The design uses inlet flow restrictors to improve parallel-channel distribution, a vapor separator, a condenser, and an expansion-control device. The control system reduces electrical power if wall temperature rises rapidly or if flow falls below its permitted value. The heat-transfer surface is manufactured with controlled roughness to promote stable nucleation but is inspected for defects that could create local hot spots.

Outcome. The system meets the thermal requirement at normal load and maintains a wall-temperature margin below the critical condition during expected transients. The main trade-off is that increased flow improves cooling but raises pumping power and pressure-drop requirements.

Learning points. Phase-change cooling is not selected solely because its heat-transfer coefficient is high. A successful design also requires flow stability, pressure control, material compatibility, sensor redundancy, fault response, and a demonstrated margin from critical heat flux.

7. Limitations, Best Practices, and Contemporary Developments

Correlations are empirical or semi-empirical and should not be extrapolated beyond their stated ranges. Common errors include using bulk rather than saturation properties, confusing quality with void fraction, selecting the wrong characteristic length, neglecting radiation in natural convection, and applying a pool-boiling correlation to flow boiling.

Best practice includes:

• Define geometry, boundary conditions, pressure, and operating regime before selecting a correlation.
• Use consistent SI units and property data at appropriate temperatures and pressures.
• Check whether the flow is single phase, two phase, transitional, or near dryout.
• Include fouling, contact resistance, manufacturing tolerances, and measurement uncertainty.
• Validate critical predictions with experiments, validated computational models, or accepted design codes.
• Provide relief devices and shutdown logic for credible overpressure and loss-of-cooling events.
• Treat noncondensable gases, dissolved gases, and surface contamination as performance variables.

Contemporary research includes microchannel boiling, structured and porous surfaces, nanostructured coatings, dielectric immersion cooling, additive-manufactured heat exchangers, and advanced refrigerants. Artificial intelligence can assist with fault detection, soft sensors, reduced-order prediction, and control optimization, but AI models must be validated against conservation laws and operating data. They should not replace independent safety limits or certified protection systems.

Chapter Summary

Natural convection is driven by buoyancy generated by density differences in a gravitational field. Its behavior is organized using Grashof, Prandtl, Rayleigh, and Nusselt numbers. Correct geometry, orientation, property evaluation, and correlation range are essential.

Boiling transfers heat through sensible heating and latent vaporization. Pool boiling progresses from natural convection through nucleate boiling, critical heat flux, transition boiling, and film boiling. Nucleate boiling gives high heat-transfer rates, but exceeding critical heat flux can cause dangerous wall-temperature escalation. Flow boiling adds effects of mass flux, vapor quality, pressure drop, flow regime, and channel stability.

Condensation may be filmwise or dropwise. Filmwise condensation is stable and commonly used for design; dropwise condensation can provide higher heat-transfer coefficients but requires durable surface control. Phase-change equipment must be designed from coupled thermal, hydraulic, mechanical, control, environmental, and safety perspectives.

Key Terms

Buoyancy: Body force caused by density differences in a gravitational field.

Natural convection: Heat transfer accompanied by fluid motion generated internally by buoyancy.

Rayleigh number: Dimensionless measure of buoyancy relative to viscous and thermal diffusion effects.

Nusselt number: Dimensionless ratio of convective to conductive heat transfer across a characteristic length.

Film temperature: Approximate average of surface and free-stream temperatures used for property evaluation.

Pool boiling: Boiling from a heated surface immersed in a substantially stationary liquid.

Flow boiling: Boiling in a moving liquid flowing through a heated channel.

Saturation temperature: Temperature at which liquid and vapor coexist at a specified pressure.

Excess temperature: Difference between wall temperature and saturation temperature.

Nucleate boiling: Boiling regime characterized by active bubble formation at surface nucleation sites.

Critical heat flux: Maximum heat flux attainable before significant vapor blanketing begins.

Film boiling: Regime in which a continuous vapor layer separates the surface from the liquid.

Vapor quality: Mass fraction of vapor in a two-phase mixture.

Void fraction: Volume fraction occupied by vapor in a two-phase flow.

Condensation: Phase change from vapor to liquid accompanied by latent-heat rejection.

Filmwise condensation: Condensation that produces a continuous liquid film on the surface.

Dropwise condensation: Condensation occurring as discrete droplets on a nonwetting surface.

Latent heat: Energy transferred during phase change without a change in temperature at equilibrium.

Dryout: Loss of liquid wetting at a heated surface, often causing a sharp temperature increase.

Fouling resistance: Thermal resistance caused by unwanted deposits on heat-transfer surfaces.

Review Questions

5 Short Answer Questions

1. What physical mechanism produces natural convection?
2. Define the Rayleigh number and identify the three dimensionless groups from which it is formed.
3. What is meant by excess temperature in pool boiling?
4. Distinguish vapor quality from void fraction.
5. Why can noncondensable gases reduce condensation heat transfer?

5 Long Answer Questions

1. Explain the natural-convection boundary layer along a heated vertical plate and discuss the roles of buoyancy, viscosity, and thermal diffusion.
2. Describe the complete pool-boiling curve and explain why the critical heat flux is a safety limit.
3. Compare pool boiling and flow boiling with respect to flow rate, pressure drop, vapor distribution, and dryout.
4. Derive or explain the main assumptions behind laminar film condensation on a vertical plate.
5. Compare filmwise and dropwise condensation and discuss why the latter is difficult to maintain in industrial equipment.

3 Analytical / Application Questions

1. Air at 25 °C flows naturally around a vertical plate maintained at 75 °C. Identify the properties and characteristic length needed to estimate the heat-transfer coefficient, and explain how you would judge whether a laminar correlation is suitable.
2. A boiling channel experiences a local heat-flux increase while mass flow remains constant. Analyze the likely changes in vapor generation, pressure drop, wall temperature, and risk of dryout.
3. A condenser performs well during commissioning but loses capacity after several months. Develop a diagnostic sequence considering fouling, noncondensable gases, condensate drainage, cooling-fluid flow, and instrumentation error.

Multiple Choice Questions with Answers

Question 1

Which dimensionless number most directly compares buoyancy effects with viscous and thermal diffusion effects in natural convection?

A. Reynolds number

B. Rayleigh number

C. Biot number

D. Fourier number

Correct Answer: B. Rayleigh number

Question 2

For a vertical plate in natural convection, fluid properties are commonly evaluated at which temperature for a first engineering estimate?

A. The wall temperature only

B. The free-stream temperature only

C. The film temperature, the average of wall and free-stream temperatures

D. The saturation temperature in every case

Correct Answer: C. The film temperature, the average of wall and free-stream temperatures

Question 3

What is the principal danger of exceeding critical heat flux in pool boiling?

A. The liquid becomes incompressible

B. The wall temperature may rise rapidly because of vapor blanketing

C. The surface temperature becomes equal to ambient temperature

D. The latent heat becomes infinite

Correct Answer: B. The wall temperature may rise rapidly because of vapor blanketing

Question 4

Vapor quality in a two-phase mixture is defined as the:

A. Ratio of vapor volume to total volume

B. Ratio of vapor mass to total mixture mass

C. Ratio of liquid mass to vapor mass

D. Ratio of vapor density to liquid density

Correct Answer: B. Ratio of vapor mass to total mixture mass

Question 5

Which condensation mode generally provides the higher heat-transfer coefficient when it can be maintained?

A. Dropwise condensation

B. Filmwise condensation

C. Sublimation

D. Natural convection without phase change

Correct Answer: A. Dropwise condensation

Question 6

If a heated vertical plate has L = 0.50 m, k = 0.028 W/(m·K), and Nu_L = 87.5, what is the average heat-transfer coefficient?

A. 0.49 W/(m²·K)

B. 4.90 W/(m²·K)

C. 49.0 W/(m²·K)

D. 1,563 W/(m²·K)

Correct Answer: B. 4.90 W/(m²·K)

Question 7

Which change generally increases natural-convection heat transfer from a heated surface, all else being comparable?

A. Reducing the temperature difference to zero

B. Eliminating gravity

C. Increasing the buoyancy-producing temperature difference

D. Replacing the fluid with a vacuum

Correct Answer: C. Increasing the buoyancy-producing temperature difference

Question 8

Which condition is especially harmful to condensation heat transfer?

A. Removal of noncondensable gases

B. A clean condensing surface

C. Accumulation of noncondensable gas near the interface

D. Adequate condensate drainage

Correct Answer: C. Accumulation of noncondensable gas near the interface

Question 9

For an evaporator with negligible pressure change and saturated liquid entering and saturated vapor leaving, the approximate heat duty is:

A. Q̇ ≈ ṁh_fg

B. Q̇ ≈ ṁg

C. Q̇ ≈ h_fg/ṁ

D. Q̇ ≈ ṁ/k

Correct Answer: A. Q̇ ≈ ṁh_fg

Question 10

Which statement about empirical boiling correlations is correct?

A. They are valid for all fluids and geometries without adjustment

B. They require appropriate surface-fluid constants and applicable operating ranges

C. They eliminate the need for safety margins

D. They apply only when no phase change occurs

Correct Answer: B. They require appropriate surface-fluid constants and applicable operating ranges