Phase-Change and Multiphase Heat Transfer

Introduction and Context

Phase-change heat transfer involves energy transfer associated with the transformation of a substance from one phase to another, such as liquid–vapour, solid–liquid, or solid–vapour change. Multiphase heat transfer is broader: it concerns heat transfer in systems containing two or more phases, whether or not phase change occurs. Examples include boiling water, condensing steam, melting metals, ice formation, spray cooling, gas–liquid reactors, fluidized beds, and cryogenic systems.

These processes are central to power generation, refrigeration, heat pumps, chemical processing, electronics cooling, thermal energy storage, metallurgy, and aerospace systems. Their analysis is more difficult than single-phase convection because phase distribution, interfacial motion, latent heat, surface condition, pressure, and flow regime interact strongly.

At postgraduate level, phase-change heat transfer should be treated as a coupled thermodynamic, fluid-mechanical, and transport phenomenon rather than as a collection of isolated correlations.

Learning Objectives

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

1. Explain the thermodynamic and transport foundations of phase-change heat transfer.
2. Distinguish nucleate boiling, transition boiling, film boiling, condensation modes, and solidification regimes.
3. Apply energy balances, latent-heat relations, dimensionless groups, and selected engineering correlations.
4. Interpret boiling curves, flow regimes, critical heat flux, minimum film boiling, and condensation resistance.
5. Analyse a practical evaporator or condenser and identify design limitations.
6. Evaluate modelling assumptions, experimental uncertainty, safety issues, and contemporary research directions.

1. Thermodynamic Foundations

1.1 Saturation and phase equilibrium

At a specified pressure, a pure substance changes phase at its saturation temperature, T_sat. Conversely, at a specified temperature, equilibrium occurs at the saturation pressure, p_sat. During equilibrium phase change of a pure substance at constant pressure, temperature remains approximately constant while latent heat is absorbed or released.

The latent heat of vaporization is

where h_fg is latent heat of vaporization in J/kg, and h_g and h_f are saturated-vapour and saturated-liquid enthalpies. For condensation, the same quantity is released. Latent heat varies significantly with pressure and approaches zero at the critical point.

For a saturated liquid–vapour mixture, the vapour quality is

and, for any specific mixture property approximated by equilibrium weighting,

Here m_g and m_f are vapour and liquid masses, respectively. Quality is meaningful only in the two-phase saturated region; it is not the same as void fraction, which is the fraction of volume occupied by vapour.

1.2 Energy balance

For a steady device with negligible kinetic and potential-energy changes, one-dimensional heat transfer associated with evaporation or condensation is

where Q_dot is heat-transfer rate in W and m_dot is mass flow rate in kg/s. If sensible heating or subcooling is also present, the complete enthalpy balance is

The latent-only form is applicable when inlet and outlet states differ primarily by phase change at nearly constant saturation conditions.

1.3 Interfacial energy balance

At a moving interface, heat may be consumed by phase change. A simplified local balance is

where q_double_prime is heat flux in W/m² and m_double_prime is interfacial mass flux in kg/(m²·s). More generally, heat conducted from both sides of an interface and interfacial energy storage must be included. This relation demonstrates why a phase boundary can remain near its equilibrium temperature while substantial heat is transferred.

2. Governing Dimensionless Groups

Dimensionless parameters help identify dominant mechanisms and transfer correlations. Important groups include:

Here rho, mu, k, c_p, u, L, sigma, g, alpha, and t denote density, dynamic viscosity, thermal conductivity, specific heat, velocity, characteristic length, surface tension, gravitational acceleration, thermal diffusivity, and time, respectively.

The Bond number is especially important for bubbles and droplets. At small Bond number, surface tension controls shape; at large Bond number, gravity strongly deforms the interface. The Jakob number indicates whether sensible heating is negligible compared with latent heat. Correlations must be used only within their documented ranges of pressure, geometry, fluid properties, and flow regime.

3. Pool Boiling

Pool boiling occurs when a heated surface is immersed in a relatively quiescent liquid. The surface superheat is

where T_s is surface temperature. The boiling curve relates heat flux to surface superheat.

3.1 Regimes of the boiling curve

1. Natural convection region: The liquid is below the nucleation threshold. Heat transfer is dominated by buoyancy-driven convection.
2. Onset of nucleate boiling: Vapour embryos become active at surface cavities.
3. Nucleate boiling: Bubbles repeatedly form, grow, detach, and induce vigorous liquid mixing. Heat-transfer coefficients are usually high.
4. Critical heat flux (CHF): The maximum heat flux sustainable before a vapour-rich condition develops over the surface.
5. Transition boiling: Surface coverage by vapour fluctuates between wetting and drying. Heat flux decreases as surface temperature increases.
6. Film boiling: A comparatively continuous vapour layer separates the surface from the liquid. Heat transfer is generally lower because vapour has low thermal conductivity, although radiation can become important at high temperature.

The critical heat flux is a safety limit in many systems. Exceeding it can cause a sharp surface-temperature excursion, material damage, burnout, or loss of thermal control.

3.2 Nucleation and surface effects

Boiling begins at microscopic cavities that can retain vapour embryos. Nucleation depends on surface roughness, wettability, dissolved gas, contamination, pressure, and thermal history. A highly polished surface may delay nucleation but may also produce fewer, larger active sites. Engineered coatings and microstructures can increase nucleation-site density and liquid replenishment, but they may deteriorate, foul, or lose mechanical integrity.

For engineering design, boiling heat flux is often written as

where h_b is the boiling heat-transfer coefficient. Unlike a single-phase coefficient, h_b can vary sharply with heat flux and surface condition.

3.3 Pool-boiling correlations

A widely used form for nucleate pool boiling is the Rohsenow correlation:

The symbols mu_l, rho_l, rho_v, c_p,l, and Pr_l refer to saturated-liquid viscosity, liquid and vapour densities, liquid specific heat, and liquid Prandtl number. C_sf is an empirical surface–fluid constant and n is an empirical exponent. The correlation assumes nucleate pool boiling and requires experimentally appropriate values of C_sf and n. It should not be applied indiscriminately to forced-flow boiling, very high pressures, microgravity, or unfamiliar surfaces.

4. Critical Heat Flux and Film Boiling

CHF is often interpreted as a hydrodynamic limit arising from competition between liquid replenishment and vapour escape. A classical pool-boiling estimate for a large horizontal surface is the Zuber relation:

where C_Z is an empirical constant, commonly near 0.13 for the idealized configuration. The relation is an estimate, not a universal law. Surface orientation, size, wettability, pressure, liquid subcooling, dissolved gas, and system geometry can alter CHF substantially.

Film boiling may be analysed using a vapour-film model. At high surface temperatures, thermal radiation between surface and liquid can contribute significantly. A practical radiation expression is

where epsilon is emissivity, sigma_SB is the Stefan–Boltzmann constant, and T_sur is the effective surrounding temperature. Absolute temperature must be used. Radiation should not be added to convection without considering whether surfaces are mutually visible and whether the vapour layer is participating or absorbing.

5. Forced-Flow Boiling

In forced-flow boiling, liquid moves through a channel while evaporation occurs at the wall or within the bulk. Important regimes include bubbly flow, slug or plug flow, churn flow, annular flow, and mist or dispersed flow. The regime depends on mass flux, quality, pressure, channel geometry, fluid properties, and gravity.

A useful overall energy balance is

where P_w is heated perimeter and L_h is heated length. If pressure drop is significant, saturation temperature varies along the channel and the enthalpy balance must be solved together with momentum and property relations.

The local boiling coefficient is commonly represented by a convective contribution modified by a nucleate-boiling contribution. For example, Chen-type approaches combine mechanisms through a form such as

where h_tp is the two-phase coefficient, h_sp is a single-phase liquid or vapour reference coefficient, h_nb is a nucleate-boiling coefficient, and F and S are empirical enhancement and suppression factors. Such models are engineering approximations and require validation for the fluid and geometry.

5.1 Pressure drop and flow instability

Two-phase pressure drop includes frictional, accelerational, and gravitational components:

The accelerational term arises because density changes as evaporation proceeds. In parallel channels, small differences in flow rate can cause unequal boiling, pressure-drop redistribution, and oscillations. Design therefore requires attention to inlet restrictions, flow distribution, compressibility of vapour, control valves, and transient operation.

6. Condensation

Condensation occurs when vapour is cooled below its saturation condition. The main modes are filmwise and dropwise condensation.

6.1 Filmwise condensation

In filmwise condensation, liquid condensate forms a continuous layer on the surface. Heat must conduct through this film, making its thickness a major resistance. For laminar condensation on a vertical plate, the classical Nusselt result for the average coefficient is

Here L is plate height and Delta T = T_sat − T_w is the temperature difference, with wall temperature T_w. The derivation assumes a smooth vertical plate, steady laminar liquid film, negligible vapour shear, constant properties, negligible superheating or subcooling, and heat transfer mainly by conduction across the condensate film. The local coefficient decreases with distance because the film thickens.

For a horizontal tube, a representative relation is

where D is tube diameter. Tube banks require correction for condensate drainage from upper to lower tubes.

6.2 Dropwise condensation

In dropwise condensation, droplets form discrete islands, repeatedly grow, coalesce, and depart. Much of the surface remains relatively clean and thinly covered, so heat-transfer coefficients can be far higher than in filmwise condensation. Stable dropwise behaviour is difficult to maintain because coatings degrade, surfaces become contaminated, and condensate chemistry changes. Consequently, conservative equipment design often assumes filmwise condensation unless long-term dropwise performance is demonstrated.

6.3 Condensation resistance network

For a wall separating hot vapour from a coolant, the approximate thermal resistance is

where h_c and h_cool are condensation- and coolant-side coefficients, delta and k_w are wall thickness and conductivity, and R_f represents fouling and contact resistances. The heat rate is

This network is valid when resistances can be approximated as linear and the relevant temperatures are adequately defined. In phase-change systems, property variation and condensate drainage may make the coefficients position-dependent.

7. Melting, Solidification, and the Stefan Problem

Solidification and melting involve a moving interface. For one-dimensional conduction-controlled solidification, the interface position s(t) is determined by the Stefan condition:

Here L is latent heat per unit mass, k_s and k_l are solid and liquid conductivities, and the temperature gradients are evaluated at the interface on the solid and liquid sides. Sign conventions depend on the coordinate direction; the equation expresses latent-energy consumption or release as the difference between conductive heat fluxes.

The classical Stefan problem assumes a sharp interface, local equilibrium at the melting temperature, known phase properties, and negligible convection in the liquid. Real materials may exhibit mushy zones, dendritic growth, constitutional supercooling, shrinkage, natural convection, and anisotropic microstructure. Enthalpy methods are often used in numerical models because they represent latent heat over a temperature interval and avoid explicit interface tracking.

8. Boiling and Condensation in Heat Exchanger Design

A phase-change heat exchanger requires simultaneous consideration of thermal, hydraulic, mechanical, and control requirements. The design workflow commonly includes:

1. Establishing inlet and outlet thermodynamic states.
2. Calculating required duty from enthalpy differences.
3. Selecting geometry and allowable pressure drop.
4. Estimating local heat-transfer coefficients by regime.
5. Including wall, fouling, and contact resistances.
6. Checking CHF, dryout, flooding, entrainment, vibration, and flow distribution.
7. Assessing startup, shutdown, turndown, non-condensable gases, and control response.
8. Verifying materials, pressure containment, relief protection, and applicable codes.

Non-condensable gases are particularly damaging in condensers. Even a small concentration can accumulate near the condensing surface, creating a diffusion resistance and lowering the local vapour partial pressure. Proper venting, deaeration, and layout are therefore as important as the nominal condensation coefficient.

9. Worked Example: Condensing Steam on a Vertical Plate

Saturated steam at 100 °C condenses on a smooth vertical plate maintained at 90 °C. The plate is 1 m high and 1 m wide. Use representative liquid-water properties at approximately 95 °C:

rho_l = 962 kg/m³, rho_v = 0.60 kg/m³, mu_l = 2.97 × 10^-4 Pa·s, k_l = 0.68 W/(m·K), and h_fg = 2.26 × 10⁶ J/kg. Estimate the average filmwise condensation coefficient and heat-transfer rate, neglecting wall resistance and subcooling of condensate.

The temperature difference is

Using the vertical-plate relation,

The quantity inside the fourth root is approximately 3.55 × 10¹6, giving

The plate area is A = 1 m², so

Thus the estimated heat-transfer rate is approximately 23 kW. The corresponding condensation rate is

This result is an idealized estimate. Vapour shear, condensate subcooling, non-condensable gases, surface waviness, and wall resistance could reduce the actual duty.

10. Applied Case: Refrigeration Evaporator with Uneven Flow Distribution

Background

Consider an illustrative refrigerant evaporator composed of many parallel microchannels. Liquid–vapour refrigerant enters a manifold, absorbs heat from an air stream, and exits as superheated vapour.

Problem

Initial tests show that some channels experience early dryout and high wall temperature, while others remain liquid-rich. The total measured duty is below the design value, and pressure oscillations occur during compressor-speed changes.

Analysis

The symptoms indicate maldistribution, coupled channel instability, and local deterioration of wetting. A channel receiving excessive vapour may have reduced liquid replenishment and a high wall temperature. A liquid-rich channel may have a larger pressure drop and divert flow elsewhere. The average heat-transfer coefficient alone cannot reveal this local hazard.

Approach

The engineering team should:

• measure inlet quality, mass flux, pressure drop, and wall temperature by channel group;
• redesign the inlet manifold using balanced flow resistances;
• install suitable inlet distributors or restrictors;
• evaluate a two-phase pressure-drop and dryout model over the operating envelope;
• test startup, low-load, and transient compressor conditions;
• provide temperature and pressure protection against local overheating;
• check compatibility of lubricant, refrigerant, seals, and materials;
• validate the design experimentally rather than extrapolating a correlation outside its range.

Outcome

A balanced manifold and controlled minimum mass flux reduce channel-to-channel variation. The evaporator reaches the required duty with lower peak wall temperature, although the added restrictions increase pumping or compressor-side pressure losses. The design decision therefore involves a trade-off between flow uniformity, thermal performance, parasitic power, manufacturability, and reliability.

Learning Points

Phase-change equipment must be designed using local as well as average quantities. Flow distribution, transient behaviour, and safety margins can dominate performance. A correlation is not a substitute for regime identification and validation.

11. Modelling, Measurement, and Research Issues

Two-phase systems are challenging to measure. Void fraction, interfacial area, local wall temperature, droplet size, and transient film thickness may vary rapidly and may be inaccessible. Thermocouples can disturb the flow; optical methods may require transparent sections; electrical impedance methods depend on calibration and phase distribution.

Computational approaches include volume-of-fluid methods, level-set methods, Eulerian multiphase models, Lagrangian particle or droplet tracking, phase-field methods, and enthalpy formulations. Direct numerical simulation can resolve fundamental mechanisms at small scales but is expensive at engineering scales. Reynolds-averaged and large-eddy approaches require closure models for turbulence, interfacial momentum, nucleation, wall boiling, coalescence, breakup, and phase change.

Machine learning is increasingly used to construct surrogate models, classify flow regimes, infer unmeasured variables, and optimize surfaces. Its use must be constrained by conservation laws, thermodynamic consistency, uncertainty quantification, and training-data validity. A model that performs well within one pressure and geometry range may fail under extrapolation.

12. Safety, Sustainability, and Best Practices

Phase-change equipment frequently operates at high pressure, low temperature, or with flammable, toxic, or environmentally sensitive fluids. Design should include pressure relief, containment, leak detection, safe venting, material compatibility, fatigue assessment, and compliance with applicable pressure-vessel and refrigeration standards.

Best practices include:

• use consistent property data and document the source;
• identify flow regime before selecting a correlation;
• apply correlations only within their validated range;
• include fouling, non-condensables, and manufacturing tolerances;
• distinguish local peak temperature from area-average temperature;
• perform uncertainty and sensitivity analyses;
• validate against representative experiments;
• design for safe failure and controlled transients;
• consider refrigerant global-warming potential, toxicity, flammability, and lifecycle impact;
• maintain traceable assumptions in simulation and design records.

Contemporary Developments

Current research includes micro- and nanoengineered boiling surfaces, enhanced condensers, dielectric immersion cooling for electronics, additive-manufactured porous structures, phase-change thermal batteries, refrigerant alternatives, two-phase cooling in data centres, cryogenic propellant management, and reduced-gravity boiling. Key research challenges remain the prediction of CHF and dryout in complex geometries, durable surface enhancement, scale-up from laboratory measurements, and reliable multiphysics models under transient conditions.

Chapter Summary

Phase-change heat transfer is governed by the interaction of latent heat, interfacial transport, fluid motion, surface condition, and thermodynamic equilibrium. Boiling progresses from convection through nucleate boiling, critical heat flux, transition boiling, and film boiling. Condensation may be filmwise or dropwise, with condensate-film resistance often controlling performance. Forced-flow boiling introduces flow-regime transitions, pressure drop, dryout, and instability. Melting and solidification involve moving interfaces and are described fundamentally by the Stefan condition.

Dimensionless groups such as Reynolds, Jakob, Bond, Weber, and Capillary numbers help identify dominant mechanisms, but empirical correlations must be applied within their assumptions and validation ranges. Practical design requires enthalpy balances, resistance networks, flow-distribution analysis, safety assessment, and experimental validation. Modern numerical and machine-learning methods are valuable only when constrained by conservation, uncertainty analysis, and physical validation.

Key Terms

Phase change: Transformation of matter between solid, liquid, and vapour states.

Latent heat: Energy absorbed or released during phase change without an ideal-equilibrium temperature change.

Saturation temperature: Phase-equilibrium temperature at a specified pressure.

Vapour quality: Mass fraction of vapour in a saturated liquid–vapour mixture.

Void fraction: Volume fraction occupied by a phase, usually vapour in a channel.

Pool boiling: Boiling from a heated surface in a relatively quiescent liquid.

Nucleate boiling: Boiling characterized by repeated formation and departure of bubbles.

Critical heat flux: Maximum heat flux before a major deterioration of liquid access to the surface.

Film boiling: Boiling with a substantial vapour layer separating liquid and surface.

Forced-flow boiling: Boiling in a moving stream through a channel or passage.

Dryout: Loss of liquid wetting from a heated surface or channel region.

Filmwise condensation: Condensation forming a continuous liquid film.

Dropwise condensation: Condensation occurring as discrete droplets.

Non-condensable gas: Gas mixed with vapour that does not undergo condensation under operating conditions.

Stefan problem: Moving-boundary phase-change problem involving latent heat.

Jakob number: Ratio of sensible energy to latent energy.

Bond number: Ratio of gravitational to surface-tension effects.

Void fraction: Volumetric, rather than mass-based, phase fraction.

Two-phase pressure drop: Pressure loss arising from frictional, gravitational, and accelerational effects in multiphase flow.

Review Questions

Short Answer Questions

1. Define latent heat of vaporization and explain why it depends on pressure.
2. Distinguish vapour quality from void fraction.
3. List the principal regions of the pool-boiling curve.
4. Why can non-condensable gases substantially reduce condenser performance?
5. State the main assumptions of the classical Nusselt vertical-plate condensation model.

Long Answer Questions

1. Explain the mechanisms of nucleate, transition, and film boiling, including the engineering significance of critical heat flux.
2. Derive the energy-balance framework for a forced-flow evaporator and discuss the effects of pressure drop and changing saturation temperature.
3. Compare filmwise and dropwise condensation in terms of mechanism, performance, durability, and design practice.
4. Explain the Stefan condition for solidification and discuss how mushy zones and convection complicate the classical model.
5. Discuss the role of dimensionless groups in selecting and validating phase-change heat-transfer correlations.

Analytical / Application Questions

1. A condenser receives 0.025 kg/s of saturated steam with latent heat 2.25 MJ/kg. Estimate the required heat-removal rate and identify two reasons why an actual condenser might require a larger area than an ideal calculation suggests.
2. A boiling channel exhibits oscillating pressure drop and unequal outlet qualities among parallel passages. Diagnose the likely mechanisms and propose a measurement and redesign plan.
3. Select a suitable analysis strategy for a transient thermal-energy-storage module using a phase-change material. Compare an enthalpy method with explicit interface tracking and identify the required experimental validation.

Multiple Choice Questions with Answers

Question 1

Which quantity represents the mass fraction of vapour in a saturated liquid–vapour mixture?

A. Void fraction

B. Vapour quality

C. Bond number

D. Jakob number

Correct Answer: B. Vapour quality

Question 2

In pool boiling, the critical heat flux marks the approximate point at which:

A. The liquid reaches its triple point

B. The wall becomes isothermal

C. Liquid access to the heated surface deteriorates sharply

D. All bubbles immediately collapse

Correct Answer: C. Liquid access to the heated surface deteriorates sharply

Question 3

For saturated-steam condensation, the latent-only heat balance is:

A. Q_dot = m_dot h_fg

B. Q_dot = h_fg/m_dot

C. Q_dot = m_dot/h_fg

D. Q_dot = m_dot c_p/T

Correct Answer: A. Q_dot = m_dot h_fg

Question 4

Which dimensionless group compares sensible energy with latent energy?

A. Reynolds number

B. Weber number

C. Jakob number

D. Bond number

Correct Answer: C. Jakob number

Question 5

The classical Nusselt vertical-plate condensation model primarily assumes that the condensate film is:

A. Turbulent and radiation-dominated

B. Laminar and conduction-dominated

C. A dispersed vapour mist

D. Absent from the wall

Correct Answer: B. Laminar and conduction-dominated

Question 6

A condenser receives m_dot = 0.010 kg/s of steam with h_fg = 2.0 × 10⁶ J/kg. Neglecting sensible effects, what is the heat-transfer rate?

A. 2 kW

B. 20 kW

C. 200 kW

D. 0.005 kW

Correct Answer: B. 20 kW

Question 7

Which statement correctly distinguishes quality from void fraction?

A. Both are always identical

B. Quality is volume-based and void fraction is mass-based

C. Quality is mass-based and void fraction is volume-based

D. Neither applies to two-phase flow

Correct Answer: C. Quality is mass-based and void fraction is volume-based

Question 8

A low Bond number for a bubble indicates that:

A. Gravity dominates surface tension

B. Surface tension strongly influences bubble shape

C. Viscosity is necessarily zero

D. The flow must be single phase

Correct Answer: B. Surface tension strongly influences bubble shape

Question 9

Why are non-condensable gases harmful in a condenser?

A. They increase the latent heat of the vapour

B. They create an additional diffusion resistance near the condensing surface

C. They eliminate pressure drop

D. They guarantee dropwise condensation

Correct Answer: B. They create an additional diffusion resistance near the condensing surface

Question 10

The Stefan condition is fundamentally used to analyse:

A. A moving phase-change interface

B. Electrical resistance in a wall

C. Single-phase pump efficiency

D. Radiation through vacuum only

Correct Answer: A. A moving phase-change interface