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Nucleate boiling

Nucleate boiling is a phase-change heat transfer process in which vapor bubbles nucleate, grow, and detach from discrete sites on a heated surface in contact with a subcooled or saturated liquid pool, enabling efficient dissipation of high heat fluxes through the latent heat of vaporization and enhanced liquid agitation.^[1] This regime represents the most effective portion of the pool boiling curve, typically occurring between wall superheats of approximately 5–30°C, where heat transfer coefficients can reach values orders of magnitude higher than those in single-phase convection.^[2] It is distinguished from other boiling modes, such as film boiling, by the absence of a continuous vapor layer blanketing the surface, which would otherwise insulate the wall and reduce heat transfer efficiency.^[3] The mechanism of nucleate boiling begins with the formation of vapor embryos at favorable nucleation sites, such as microscopic cavities or surface imperfections on the heating wall, where trapped gas or vapor provides the initial superheat required for phase change.^[4] Once initiated, bubbles grow primarily through evaporation at the liquid-vapor interface, including contributions from microlayer evaporation beneath the bubble base and transient conduction from the superheated wall layer.^[3] Bubble departure occurs when buoyant forces exceed surface tension and drag, typically at diameters predicted by models like Fritz's correlation, leading to the release of vapor into the bulk liquid and the entrainment of cooler liquid toward the surface, which sustains the cycle.^[5] Key parameters influencing this process include surface wettability, roughness, fluid properties (e.g., surface tension and latent heat), and operating conditions like pressure and subcooling, all of which affect nucleation site density, bubble frequency, and overall heat flux.^[1] Nucleate boiling is subdivided into isolated bubble and churn-turbulent sub-regimes, transitioning as heat flux increases and bubbles begin to interact, coalescing into slugs or columns that further intensify mixing but approach the critical heat flux (CHF) limit, beyond which a vapor blanket forms and heat transfer deteriorates.^[6] Its importance stems from its application in diverse engineering systems requiring compact, high-performance cooling, including nuclear reactors, steam generators, electronics thermal management, and refrigeration cycles, where accurate prediction of heat transfer rates—often via correlations like Rohsenow's or mechanistic models—is essential for design and safety.^[7] Ongoing research focuses on enhancement techniques, such as surface modifications (e.g., nanostructures or porous coatings), to extend the nucleate boiling regime and elevate CHF values, addressing challenges in high-heat-flux environments.^[5]

Fundamentals of Boiling

Definition and Regimes

Nucleate boiling is a heat transfer regime in which discrete vapor bubbles form at specific nucleation sites on a heated surface submerged in a stagnant liquid, grow by evaporation at the bubble-liquid interface, and detach due to buoyancy, carrying latent heat away from the surface. This process typically occurs at wall superheats (the difference between the surface temperature and the saturation temperature of the liquid) ranging from 5 to 30°C for common fluids like water at atmospheric pressure. The foundational understanding of nucleate boiling emerged in the 1930s through experimental observations. Max Jakob and Wilhelm Linke conducted early measurements of heat transfer from a horizontal plate to boiling water, documenting the relationship between heat flux and temperature difference in the nucleate regime without identifying a critical limit.^[8] Shortly thereafter, Shiro Nukiyama's 1934 experiments using a thin platinum wire as both heater and thermometer established the classic boiling curve, plotting heat flux against wall superheat and revealing the sequence of boiling regimes, including the nucleate phase.^[9] The boiling curve provides a graphical representation of heat transfer behavior during pool boiling, with wall superheat on the x-axis and heat flux on the y-axis; nucleate boiling occupies the steep, high-efficiency portion between the onset of nucleate boiling (ONB) and the critical heat flux (CHF), where heat transfer coefficients can reach 10,000 to 100,000 W/m²K due to vigorous bubble activity. In pool boiling, this regime follows natural convection boiling (characterized by minimal bubble formation at superheats below 5°C) and precedes the unstable transition boiling (where partial vapor blanketing occurs) and stable film boiling (fully vapor-covered surface at high superheats).

Role in Heat Transfer

Nucleate boiling serves as a highly efficient heat transfer mechanism in engineering applications, achieving heat transfer coefficients that are typically 10 to 100 times higher than those in single-phase convection due to the latent heat absorbed during bubble formation and departure. This enhancement arises from the agitation and mixing induced by bubbles, which effectively transport heat away from the surface.^[10] The fundamental energy balance in this regime is expressed as the heat flux q = h \Delta T, where h is the boiling heat transfer coefficient specific to nucleate boiling and \Delta T is the wall superheat, yielding far superior performance compared to convective alternatives.^[10] In practical systems, nucleate boiling is integral to nuclear reactors, including boiling water reactors (BWRs) that rely on this regime for core cooling and pressurized water reactors (PWRs) that utilize subcooled nucleate boiling to manage high heat loads.^[11] It also plays a key role in electronics cooling for dissipating heat from high-power components, in refrigeration evaporators for efficient phase change, and in steam generators for power production.^[12] These applications benefit from the regime's ability to handle substantial heat removal rates, up to the critical heat flux (CHF) of 1-10 MW/m² for water under typical conditions, allowing for compact designs with high thermal performance.^[13] Despite its advantages, nucleate boiling is sensitive to surface conditions, including roughness and wettability, which can alter nucleation site density and overall efficiency.^[10] Enhancement techniques, such as surface roughening, promote more active nucleation sites to boost heat transfer rates without exceeding CHF limits.^[14] This sensitivity underscores the need for precise surface preparation in design to maintain reliable operation up to the CHF threshold.^[13]

Nucleation and Bubble Dynamics

Nucleation Process

Nucleate boiling begins with the nucleation process, where vapor bubbles initiate at specific sites on the heating surface, driven by local superheating that overcomes the energy barrier for phase change. This process is predominantly heterogeneous, occurring at surface imperfections such as cavities, pits, or roughness elements that trap vapor or gas pockets, serving as preferential nucleation sites. These sites lower the required superheat compared to bulk fluid nucleation, enabling bubble formation at practical temperatures. Surface preparation techniques, including oxidation or fouling, can create or modify these active sites by altering the surface morphology and chemistry, thereby influencing the density and activation of nucleation points.^[15]^[10]^[16] The wettability of the surface, characterized by the contact angle θ between the liquid-vapor interface and the solid, plays a critical role in site activation and embryo stability. Hydrophilic surfaces (low θ) promote wetting and may suppress nucleation by flooding cavities, while hydrophobic surfaces (higher θ) facilitate vapor trapping and easier bubble inception. This relationship is described by Young's equation:

\cos \theta = \frac{\sigma_{sg} - \sigma_{sl}}{\sigma_{lg}}

where σ_sg, σ_sl, and σ_lg are the solid-gas, solid-liquid, and liquid-gas interfacial tensions, respectively. The contact angle hysteresis further affects dynamic behavior during embryo formation.^[17]^[18] In contrast to heterogeneous nucleation, which dominates in practical boiling scenarios at superheats of approximately 10 K for water at atmospheric pressure, homogeneous nucleation occurs rarely in the bulk fluid and requires extreme superheats exceeding 100 K, often approaching 200 K or more. The activation energy barrier for forming a stable vapor embryo in homogeneous nucleation is given by

\Delta G = \frac{16\pi \sigma^3}{3 (\Delta P)^2}

where σ is the surface tension and ΔP is the pressure difference across the interface (related to superheat via the Clausius-Clapeyron equation). For heterogeneous nucleation on surfaces, this barrier is reduced by a geometric factor depending on the contact angle and cavity shape, such as a spherical cap configuration, making surface-initiated bubbles far more probable. Dissolved gases in the liquid can further assist by providing initial vapor embryos, lowering the effective superheat needed, while surface tension σ directly influences the critical embryo size and stability.^[15]^[10]^[19] Experimental studies employing high-speed imaging have visualized the onset of nucleation, revealing that bubble embryos emerge abruptly at active sites once the local superheat surpasses a minimum threshold, typically influenced by the interplay of surface tension, wettability, and gas content. These observations confirm the stochastic nature of site activation and the role of microscopic surface features in determining the minimum superheat for inception.^[20]^[21]

Bubble Growth and Departure

Once a vapor embryo forms at a nucleation site during nucleate boiling, the bubble undergoes radial growth governed primarily by the Rayleigh-Plesset equation, which describes the dynamics of the bubble radius R(t) under the influences of inertial, thermal, viscous, and pressure forces in the surrounding liquid.^[22] The equation balances the liquid's inertial effects with pressure differences across the interface, surface tension, and viscous drag, yielding:

R \ddot{R} + \frac{3}{2} \dot{R}^2 = \frac{1}{\rho_l} \left( p_g - p_\infty - \frac{2\sigma}{R} - \frac{4\mu \dot{R}}{R} \right),

where \rho_l is the liquid density, p_g is the gas pressure inside the bubble, p_\infty is the far-field pressure, \sigma is surface tension, and \mu is liquid viscosity; thermal effects enter through p_g, which depends on heat transfer driving evaporation.^[22] In the early inertial-controlled phase, growth is rapid due to superheat-induced pressure gradients, transitioning to viscous dominance at higher viscosities or smaller scales, while thermal control prevails in superheated liquids where heat diffusion limits mass transfer.^[23] For heat diffusion-limited growth in superheated liquids, the bubble radius evolves asymptotically as R(t) \approx \sqrt{\mathrm{Ja} \, \alpha \, t}, where Ja is the Jakob number (\mathrm{Ja} = \frac{c_p \Delta T \rho_l}{h_{fg} \rho_v}, with c_p as specific heat, \Delta T as superheat, h_{fg} as latent heat, and \rho_v as vapor density) and \alpha as thermal diffusivity; this square-root dependence arises from solving the heat conduction equation around the bubble interface, assuming thin thermal boundary layers.^[22] This model, derived for spherical bubbles in uniform superheat, captures the thermally controlled regime typical in pool boiling, where evaporation at the interface sustains growth until buoyancy or other forces intervene.^[24] Bubble departure occurs when the growing vapor bubble detaches from the heated surface, driven by a force balance where buoyancy overcomes surface tension and drag. The primary forces include upward buoyancy (F_b = \frac{\pi}{6} D_d^3 g (\rho_l - \rho_v)), attachment via surface tension (F_s = \pi D_d \sigma \sin \theta), and quasi-steady drag (F_d \approx C_d \frac{\pi}{4} D_d^2 \frac{1}{2} \rho_l v^2, with drag coefficient C_d and bubble rise velocity v); departure happens when F_b \approx F_s + F_d.^[25] A seminal correlation for the departure diameter D_d in pool boiling, based on static buoyancy-surface tension balance neglecting drag for low velocities, is the Fritz equation:

D_d = 0.0208 \theta \sqrt{\frac{\sigma}{g (\rho_l - \rho_v)}},

with \theta as the contact angle in degrees; this predicts diameters on the order of 1-5 mm for water at atmospheric pressure, scaling with fluid properties and wettability. In flow boiling, liquid velocity enhances drag, reducing D_d and promoting earlier detachment compared to pool conditions.^[25] The departure frequency, or inverse of the bubble cycle time (growth plus waiting periods), typically ranges from 10 to 1000 Hz, corresponding to cycle times of 1-100 ms, depending on superheat, fluid properties, and surface conditions; higher frequencies occur at moderate heat fluxes where rapid growth shortens residence time.^[25] In flow boiling, crossflow velocity increases frequency by accelerating departure, often modeled as f_d \propto \sqrt{g (\rho_l - \rho_v)/\rho_l} / D_d, linking it directly to departure size.^[26] High-speed photography reveals that as heat flux approaches critical heat flux (CHF), bubble density increases, leading to frequent coalescence where adjacent bubbles merge into larger vapor patches, blanketing the surface and impairing liquid rewetting; this coalescence-dominated regime, observed at frame rates exceeding 1000 fps, marks the transition from isolated bubble ebullition to vapor film formation.^[27]

Heat Transfer Analysis

Heat Flux Correlations

Heat flux correlations provide essential predictive tools for estimating the heat transfer rates during nucleate boiling, enabling engineers to design systems like boilers and electronic coolers by relating wall superheat to heat flux. These models, primarily empirical, are derived from extensive experimental data and incorporate fluid properties, surface characteristics, and gravitational effects to capture the enhanced convection due to bubble agitation.^[28] The Rohsenow correlation stands as the seminal empirical model for pool nucleate boiling heat flux, widely adopted for its simplicity and broad applicability across fluids and surfaces. It expresses the wall heat flux q'' as:

q'' = \mu_l h_{fg} \left[ \frac{g (\rho_l - \rho_v)}{\sigma} \right]^{1/2} \left( \frac{c_{p,l} \Delta T}{C_{sf} h_{fg} \Pr_l^n} \right)^3

where \mu_l is the liquid viscosity, h_{fg} the latent heat of vaporization, g gravity, \rho_l and \rho_v the liquid and vapor densities, \sigma the surface tension, c_{p,l} the liquid specific heat, \Delta T the wall superheat, \Pr_l the liquid Prandtl number, C_{sf} a fluid-surface interaction constant (typically 0.013 for water-copper), and n an exponent (1.0 for water, 1.7 for other fluids). This formulation accounts for the latent heat transport by departing bubbles and the microlayer evaporation beneath them, with the constants calibrated from experiments to reflect surface wettability effects.^[28]^[29] Rohsenow's model originates from an energy balance at the bubble-liquid interface, integrating the latent heat removed by bubble growth and departure with convective contributions from induced liquid motion; this leads to dimensionless groups such as the Jakob number (Ja = c_{p,l} \Delta T / h_{fg}) for buoyancy-driven flow and the boiling number (Bo = q'' / (G h_{fg}), though G is replaced by a characteristic velocity in pool boiling). The cubic dependence on superheat reflects the exponential increase in nucleation sites and bubble frequency with temperature excess, validated through dimensional analysis of experimental datasets. Bubble departure diameter, as predicted by models like Fritz's correlation, informs the characteristic length scale implicitly in the buoyancy term.^[28]^[29] Other notable correlations complement Rohsenow by focusing on specific mechanisms or conditions. The Forster-Zuber correlation emphasizes the bubble growth contribution to heat transfer, modeling the transient conduction across the vapor-liquid interface during expansion, yielding a heat flux proportional to the superheat and fluid thermal properties via the Jakob number; it is often integrated into composite models for pool boiling. For flow boiling scenarios, the Lazarek and Black correlation predicts the heat transfer coefficient as h = 30 \frac{k_l}{D_h} \mathrm{Re}^{0.857} \mathrm{Bo}^{0.714}, where D_h is the hydraulic diameter, Re the Reynolds number based on mass flux G, and Bo the boiling number, highlighting nucleate boiling dominance in small-diameter tubes. These models show comparable accuracy, with Rohsenow predicting water data at atmospheric pressure within ±20% over a range of superheats from 5–30°C.^[30]^[31] Validation of these correlations relies on experimental datasets from horizontal flat plates and vertical tubes, where measured heat flux versus superheat curves align closely with predictions; for instance, plots of predicted q'' against experimental values for water and refrigerants on copper surfaces demonstrate mean absolute errors under 15% in the fully developed nucleate regime, confirming their utility for engineering applications despite scatter from surface roughness variations.^[29]^[32]

Influencing Factors

Surface effects play a significant role in modulating nucleate boiling performance by altering nucleation site density and bubble dynamics. Increased surface roughness enhances the number of active nucleation sites, leading to improved heat transfer coefficients; for instance, in pool boiling of water and FC-77, roughness values up to Ra = 10.0 µm can double the heat transfer compared to polished surfaces (Ra ≈ 0.03 µm).^[33] Porous surfaces further augment this effect by providing additional cavities for bubble entrapment, with micro-porous copper layers (95–220 µm thick) achieving up to 17-fold increases in the maximum nucleate boiling heat transfer coefficient (from ~0.8 W/cm²K to 13.5 W/cm²K) and 40–70% higher critical heat flux in saturation boiling of PF-5060.^[34] Nanoparticle-enhanced coatings, such as silica thin-films on nickel surfaces, boost nucleate boiling heat transfer coefficients by over 100% and critical heat flux by up to 100% through modified wettability and increased site density, without substantial changes in roughness.^[35] Fluid properties influence bubble formation and stability in nucleate boiling. Elevated system pressure reduces bubble departure diameter due to higher vapor density and surface tension effects, while significantly increasing critical heat flux; for water, raising pressure from 1 atm to 20 atm can elevate the maximum heat flux by approximately fivefold, as vapor bubbles become smaller and more frequent.^[36] Subcooling, the temperature difference between saturation and bulk liquid, suppresses the onset of nucleate boiling by requiring higher wall superheat to initiate nucleation, as the subcooled liquid absorbs latent heat from growing bubbles, delaying their departure and increasing the required superheat for stable boiling. Orientation and flow conditions affect bubble departure and convective contributions in nucleate boiling. In pool boiling, gravity aids bubble removal, with horizontal upward-facing surfaces providing optimal performance due to efficient buoyancy-driven detachment, whereas downward-facing or vertical orientations can lead to vapor accumulation and reduced heat transfer.^[37] In flow boiling, forced convection enhances heat transfer by thinning the thermal boundary layer and promoting bubble stripping from the surface, resulting in higher heat fluxes compared to pool boiling under similar conditions.^[38] Contaminants in the working fluid can alter interfacial properties and nucleation behavior. Dissolved non-condensable gases, such as air or carbon dioxide, promote nucleation by providing pre-existing vapor embryos that lower the energy barrier for bubble inception, enabling boiling at reduced wall superheats. Surfactants reduce surface tension, facilitating smaller bubble formation and enhanced nucleation rates, though excessive concentrations may hinder bubble departure and degrade overall heat transfer. Recent studies on nanofluids (post-2020) demonstrate augmentation through nanoparticle deposition on surfaces during boiling, forming porous layers that increase active sites; for example, Al₂O₃-water nanofluids at low concentrations (0.001–0.01 vol.%) have shown up to 50% enhancement in nucleate pool boiling heat transfer coefficients on modified copper surfaces.^[39]

Critical Transitions

Onset of Nucleate Boiling

The onset of nucleate boiling (ONB) marks the critical transition from single-phase convection-dominated heat transfer to the nucleate boiling regime, where vapor bubbles begin to form and detach from nucleation sites on the heated surface. This shift occurs when the wall superheat overcomes the energy barrier for heterogeneous nucleation, allowing bubble embryos in surface cavities to grow and destabilize the convective boundary layer, thereby enhancing heat transfer rates. The process is influenced by fluid subcooling, flow velocity, and surface conditions, with ONB typically initiating at low superheats (around 5–15 K for water at atmospheric pressure).^[10] A characteristic feature of ONB is the presence of hysteresis in the boiling curve, where the path traced during increasing heat flux differs from that during decreasing flux. During heating, a higher superheat is often required to initiate boiling due to the need to fill cavities with vapor and overcome wetting forces, whereas during cooling, boiling persists at lower superheats because trapped vapor sustains nucleation sites. This hysteresis, first systematically observed in flow boiling experiments, can lead to delayed activation or premature quenching, affecting the stability of heat transfer in applications like nuclear reactors and electronics cooling.^[40] Prediction of ONB conditions relies on models that balance convective and nucleative contributions. The seminal Bergles-Rohsenow correlation, developed for forced-convection boiling, estimates the onset superheat \Delta T_{\text{onb}} by extending earlier graphical methods to account for subcooling and velocity effects; this model links ONB to flow parameters, predicting lower superheats under higher velocities and subcooling, and has been validated across a range of pressures (1–140 bar) for water and other fluids.^[41] Experimentally, ONB is identified by indicators such as a abrupt increase in heat flux (often by factors of 5–10 for constant wall temperature) or the onset of wall temperature fluctuations (with amplitudes up to several Kelvin and frequencies in the Hz range), signaling the activation of nucleation sites after an initial waiting period for vapor entrapment. These signatures arise from the intermittent bubble growth and departure, which disrupt the thermal boundary layer. In constant heat flux setups, the transition manifests as a sudden drop in superheat, confirming the enhanced evaporative cooling. Modern diagnostics have improved ONB detection precision, particularly through high-speed infrared thermography, which captures spatially resolved temperature fields to identify pre-nucleation hotspots and early bubble-induced perturbations since the 2010s. This non-intrusive technique reveals sub-millimeter thermal gradients and waiting times (typically 0.1–1 s) at incipient sites, enabling real-time monitoring in opaque flows and bridging gaps in traditional thermocouple-based methods.^[42]

Departure from Nucleate Boiling

Departure from nucleate boiling (DNB) marks the upper limit of the nucleate boiling regime, where the heat flux reaches its maximum value known as the critical heat flux (CHF), beyond which the heat transfer efficiency deteriorates sharply. This transition occurs when the vapor production overwhelms the liquid's ability to maintain contact with the heated surface, leading to the collapse of nucleate boiling dynamics. In pool boiling, the CHF represents the peak heat flux q_{\max} before the onset of transition boiling, typically predicted by the Zuber correlation derived from hydrodynamic considerations:

q_{\max} = \frac{\pi}{24} \rho_v h_{fg} \left[ \frac{\sigma g (\rho_l - \rho_v)}{\rho_v^2} \right]^{1/4},

where \rho_v and \rho_l are the vapor and liquid densities, h_{fg} is the latent heat of vaporization, \sigma is the surface tension, and g is gravitational acceleration.^[43] This correlation, developed through analysis of vapor column stability, provides a foundational estimate for CHF in saturated pool boiling on horizontal surfaces and has been validated across various fluids under atmospheric conditions.^[13] The primary mechanism driving DNB in pool boiling is hydrodynamic instability, specifically the Helmholtz instability at the vapor-liquid interface. As bubbles coalesce into vapor jets or columns emanating from the surface, the interface becomes susceptible to wave-like perturbations that grow when their wavelength exceeds the Helmholtz critical wavelength, approximately $2\pi \sigma / (\rho_l - \rho_v) u_v^2, where u_v is the vapor velocity. This instability prevents fresh liquid from reaching the surface, causing localized dryout and a vapor blanket formation that blankets portions of the heater.^[13] In flow boiling, DNB mechanisms differ, often involving subcooled or low-quality conditions where bubble crowding in the boundary layer leads to dry patch formation; for instance, in vertical upward flow, vapor slugs or annular flow patterns exacerbate liquid film thinning, triggering DNB at lower superheats compared to pool boiling.^[44] Post-DNB, the system enters transition boiling, culminating at the Leidenfrost point where a stable vapor film fully insulates the surface, drastically reducing heat transfer coefficients to those of single-phase vapor convection.^[45] The immediate consequences of DNB are severe, characterized by a rapid temperature excursion of the heated surface due to the insulating vapor layer, often termed burnout. Wall superheats can surge from tens of degrees Celsius in nucleate boiling to 100–1000°C in the ensuing film boiling regime, risking material failure in applications like nuclear reactors or electronics cooling.^[46] This thermal crisis underscores the need for CHF prediction and enhancement strategies; for example, porous surfaces, such as microporous coatings, can enhance CHF through mechanisms like capillary wicking that sustain liquid supply to evaporation sites and delay instability onset.^[47] Recent advancements in the 2020s have leveraged machine learning models, including physics-informed neural networks, to predict CHF more accurately by integrating experimental databases with hydrodynamic features for diverse conditions including flow boiling DNB.^[48]

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Aug 28, 2017 · Experimental studies have shown that microporous surfaces induce one of the highest enhancements in critical heat flux (CHF) during pool ...
[47]
Machine learning in critical heat flux studies in nuclear systems
This paper reviews recent advancements in the application of ML techniques to CHF predicting, categorizing the research into CHF value prediction and CHF event ...