When boiling occurs on a solid surface at low superheat, bubbles can be seen to form repeatedly at preferred positions called nucleation sites. Nucleate boiling can occur in Pool Boiling and in Forced-Convective Boiling. The heat transfer coefficients are very high but, despite many years of research, empirical correlations for the coefficients have large error bands. Much of the difficulty arises from the sensitivity of nucleate boiling to the microgeometry of the surface on a micron length scale and to its wettability; it is difficult to find appropriate ways of quantifying these characteristics. There is still disagreement about the physical mechanisms by which the heat is transferred so phenomenological models for nucleate boiling at present do no better, and often worse, than the empirical correlations. An empirical correlation of wide application has been given by Gorenflo (1991), based on the general scaling of fluid thermal and transport properties with reduced pressure p/pc and reduced temperature T/Tc. (see Reduced Properties.) Recent reviews of the voluminous research literature on mechanisms in boiling include those by Dhir (1990) and Fujita (1992). This article describes the features of nucleate boiling on which there is broad agreement and indicates the areas of disagreement and further development.
The approach to modeling of nucleate boiling at low wall superheats has been to try to understand separately how many nucleation sites are active at a specified superheat, how bubbles grow and depart and how they influence heat transfer. We shall see that the processes are in fact linked, that wall superheat cannot be specified by a single value and that the flow conditions of the bulk liquid in pool boiling or convective boiling have some influence when nucleation sites are widely spaced in the so-called 'isolated-bubble' regime. First, however, we consider an idealized situation, the conditions for equilibrium of a small spherical vapor bubble of radius re in pure, uniformly superheated liquid and the consequences of departures from equilibrium. 'Superheat' and 'subcooling', which occur so frequently in the descriptions of boiling, are defined relative to the saturation temperature Tsat(p0) corresponding to the system pressure po, being the condition for equilibrium between liquid and vapor at an interface of zero curvature, Figure 1. A spherical bubble of finite radius r has an interface of curvature 2/r and this has two effects: (1) for mechanical equilibrium of the bubble interface there must be an excess internal pressure of 2σ/re to resist the collapsing membrane stress caused by the surface tension σ; (2) the vapor pressure for a given interfacial temperature is decreased (Kelvin equation)
where σ is surface tension, v specific volume of the liquid, M molecular weight and the universal gas constant.
There is a similar effect with exponent of opposite sign for the vapor pressure in equilibrium with a droplet of liquid. The effect is negligible for radii greater than about 10 nm.
From (1) and (2) the vapor pressure must be greater than p0 by 2σ/re and the interface must be superheated, Figures 1 and 2(a). In a uniformly superheated liquid maintained at constant pressure the equilibrium of the bubble is unstable against any disturbance. A decrease in radius leads to a requirement for a higher vapor pressure for equilibrium; this cannot be provided so collapse continues. An increase in radius leads to a requirement for a lower vapor pressure and therefore a lower interfacial superheat. The resulting temperature gradient from the bulk liquid to the interface drives the heat flow that provides the latent heat for continued growth, Figure 2(b). A radial pressure difference is required to drive the motion of the liquid but this declines as growth proceeds, also the interfacial temperature approaches the saturation temperature as 2σ/re becomes negligible. Then the rate of growth of the bubble is controlled by the rate of heat transfer, which can be modeled approximately by transient conduction in the liquid:
where λ1 is thermal conductivity of the liquid, κ1 thermal diffusivity of the liquid, h1g latent heat of evaporation, ρg density of the vapor, ρ1 density of the liquid and c1 specific heat capacity of the liquid.
In homogeneous nucleation the unstable nuclei from which growth commences are supposed to be formed by the random fluctuations in local energy in the superheated liquid. The number distribution of clusters of high-energy molecules (i.e., bubbles) depends on their work of formation, including a contribution by surface free energy (surface tension). Some of the clusters will be above the critical size for unstable equilibrium, some below. By combining the expression for cluster size distribution with a model for rate of growth or collapse, the net rate of bubble nucleation can be predicted, Skripov (1974), Blander and Katz (1975). The rate is extremely sensitive to temperature, increasing by many orders of magnitude over a very small range of temperature so that an effective homogeneous nucleation temperature Tn can be calculated. Lienhard (1976) obtained an approximate generalization of the analyses in the form
For system pressures well below the critical pressure, the homogeneous nucleation temperature is approximately 0.91 Tc, whatever the value of p0, corresponding to very high superheats Tn – Tsat. These superheats can be achieved in very carefully-controlled experiments but generally bubbles nucleate at superheats far smaller than those predicted by homogeneous nucleation models, particularly at solid walls.
When water boils at atmospheric pressure on a heated metal wall, bubbles appear at wall superheats of around 10K, compared to the superheat of 216K required for homogeneous nucleation. Similarly low superheats are required for the boiling of most other liquids on heated solid walls. Superheats approaching the homogeneous nucleation values can be achieved only by subjecting the liquid-wall system to prolonged periods of high pressure at low temperature (subcooling) or sometimes in the initiation of boiling of extremely well-wetting liquids such as fluorocarbons [Bar-Cohen, (1992)]. Wetting can be quantified by measurement of the Contact Angle θ between the liquid-vapor interface and the surface of the solid but the contact angle can exhibit hysteresis, its value depending on the direction and rate of motion of the contact line between liquid, vapor and solid, Figure 3. Microscopic examination, combined with the sensitivity of the boiling superheats to the previous temperature-pressure history, wettability and surface finish of the heated wall, provides strong evidence that the preferred sites for bubble formation are cavities with dimensions of a few microns or smaller, that trap and stabilize liquid-vapor interfaces at far larger radii of curvature than those of the nuclei in homogeneous nucleation. Thus 'nucleate boiling' is a misnomer, since new clusters of vapor phase do not have to be created. Instead bubbles form repeatedly from tiny reservoirs of continuously-maintained vapor. The processes inside such small cavities cannot be observed directly.
The supposed mechanisms by which liquid-vapor interfaces are stabilized are summarized in Figure 4 Most systems in which boiling occurs are initially filled with cold liquid so the liquid-vapor interface must be stabilized when subcooled liquid first enters the cavity, i.e., when the vapor pressure pg is less than the system pressure p0. This requires reversal of the curvature of the interface. This could occur at a region in the cavity that is so poorly wetted that the local contact angle greatly exceeds 90° (Figure 4a). However, contact angles measured on large plane surfaces generally range from nearly zero for cryogenic and fluorocarbon liquids on clean metals to around 70° for water on poorly cleaned stainless steel. Reversal of interfacial curvature when θ is much less than 90° requires a re-entrant geometry (Figure 4b). The presence of trapped or dissolved noncondensible gas can have a large effect on the stability of the interface under subcooled conditions by increasing the total pressure in the reservoir of gas plus vapor so that it exceeds the system pressure; the effect may be time-dependent as soluble gas diffuses between the interface and the interior of the liquid. When the temperature of the wall surrounding the cavity is increased, the vapor pressure increases until the curvature of the liquid-vapor interface reverses again and the trapped vapor becomes unstable to growth by evaporation (Figure 4c). The excess pressure and the corresponding superheat for equilibrium may go through several local maxima before the vapor finally emerges from the cavity and 'nucleates' the growth of a visible bubble, (Figure 4d). The highest of these superheats determines the wall superheat for the inception of bubble production. Maintaining production may then be possible at a lower superheat that only has to overcome the local maximum at the mouth of the cavity, position 6 in Figure 4d. This model explains the sensitivity of the onset of boiling of well-wetting fluids to pre-boiling conditions and the hysteresis between boiling curves for increasing and decreasing heat flux (Figure 5).
As an embryo bubble emerges from a cavity it encounters a large negative temperature gradient in the liquid surrounding it, resulting from the efficient heat transfer driven by the motion of previous bubbles produced by the cavity itself or by adjacent nucleation sites. This gradient has been modeled by transient or steady conduction into the liquid. It reduces the effective superheat at the interface of a spherical bubble at the mouth of a cavity (Figure 6) and limits the size range of cavities that can be active. When combined with information about the size distribution of cavities actually present on a surface (which may be difficult to obtain) and the further assumption that the wall superheat is uniform, this model should define the number of nucleation sites active at any superheat. Increasing the superheat should activate progressively smaller cavities, causing the steep gradient of the nucleate boiling curve. However, the model is oversimplified and does not take into account the inherent patchiness of nucleate boiling heat transfer which, in some circumstances, can lead to large local variations from the mean value of the wall superheat [Kenning (1992)].
Cavities which are stable traps for subcooled vapor prior to boiling may not be the only nucleation sites for bubbles once boiling has been established. Rather shallow cavities which are poor vapor traps may be 'seeded' with vapor from bubbles growing at more stable sites [Judd and Chopra (1993)]. Small bubbles bursting through the liquid layers under larger bubbles may produce clouds of tiny bubbles that act as secondary nucleation sites that are not associated with surface cavities [Mesler (1992)], by a process for which there is as yet no quantifiable model. Because of the various mechanisms by which nucleation sites can be created and interact, it is not possible to specify the number of active nucleation sites without also considering bubble motion and the localized processes of heat transfer.
The mechanism of growth of a bubble in uniformly superheated liquid, described previously, is modified when nucleation occurs at a solid wall. Growth as a perfect hemisphere (Figure 7a) is prevented by the difficulty of displacing liquid from the solid boundary so a microlayer of liquid is left under the base of the bubble (Figure 7b). The curvature at the periphery of the bubble depends on the local viscous and inertial stresses. It is sometimes sharp enough to give the appearance of a contact angle between the bubble and the wall but there is no triple contact line so the properties of the wall can exert no influence. The thickness of the microlayer at the bubble boundary can be estimated from viscous boundary layer theory without detailed consideration of the bubble shape. As it grows, the bubble displaces liquid so by the time it reaches a point at distance R from the nucleation site in time t the liquid at R has been in motion for time t and the boundary layer of slow-moving liquid that is overtaken by the bubble is of thickness δRo, where
where ν1 is the kinematic viscosity of the liquid.
The bubble grows by transient conduction of heat to its interface, as in Equations (2) and (3) but modified by the temperature gradient in the liquid near the wall, and by additional conduction through the microlayer so approximately
From Equations (5) and (6) the initial thickness of the microlayer under a growing bubble increases approximately linearly with radius to a thickness ranging from a few microns for small, fast-growing bubbles to tens of microns under slow-growing bubbles in pool boiling at low wall superheats. Once formed, the microlayer decreases in thickness by evaporating into the bubble as heat is conducted from the superheated wall across the thin microlayer. As the bubble sticks further out from the wall into liquid that is less superheated, or even subcooled, its rate of growth decreases and it starts to move away from the wall under the combined influence of hydrodynamic and hydrostatic forces. In saturated pool boiling on a horizontal wall the bubble lifts off vertically and the periphery of the base of the bubble moves back towards the nucleation site (Figure 7c). Initially it moves over wall that is still covered by the microlayer but at small radii it may encounter a region where the microlayer has evaporated to dryness and then it would be appropriate to refer to a dynamic advancing contact angle at the base of the bubble. Cooper and Chandratilleke (1981) have presented nondimensional functions to describe the evolution from near-hemispherical to near-spherical shape during the growth of bubbles under various idealized conditions but analytical models for bubble growth often assume inaccurately that the bubble is a truncated sphere. Correlation of the departure size by a balance between buoyancy and surface tension forces with a static contact angle θ (Figure 7d) gives no more than the right order of magnitude for the radius:
Improvements in the understanding of bubble departure are to be expected from numerical modeling that takes account of changes in bubble shape and the associated liquid inertia that can drive bubbles away from the wall, even against a hydrostatic buoyancy force. In subcooled boiling the bubbles recondense, either after moving away from the superheated wall at low subcooling, or in close proximity to the wall at large subcooling of the bulk liquid. (See also Bubble Growth.)
The overall mechanism of heat transfer must involve heat removal from the wall, followed by transport into the interior of the bulk liquid. In nucleate boiling, the bubbles somehow greatly reduce the thermal resistance that occurs close to the wall in heat transfer to a single-phase liquid. The mechanisms of heat removal from the wall, summarized in Figure 8, are generally supposed to be:
conduction across the very thin microlayers under growing bubbles;
quenching by relatively cold bulk liquid moving towards the wall as bubbles round off and detach, modeled by transient conduction into the liquid from 'areas of influence' on the wall about four times the maximum contact area of the bubbles;
further localized convective cooling by the motion of bulk liquid in the wakes of departing bubbles;
a general increase in turbulence in the liquid close to the wall.
Heat is transferred into the bulk liquid by the motion of bubbles away from the wall (latent heat transport), which may also carry some superheated liquid round each bubble, or by turbulent transport in the liquid. In subcooled boiling there may be a 'heat-pipe' effect of vapor evaporating at the base of bubbles and recondensing where the bubbles are in contact with the subcooled bulk liquid (Figure 8v).
Mechanisms (1), (2) and (3) are concentrated round the nucleation sites and fluctuate as bubbles grow and depart so there must be some unsteady lateral conduction of heat in the wall, (Figure 9). Only a wall made of a material with infinite thermal diffusivity can have a uniform, steady superheat. In experiments using very thin, electrically heated walls the local variations in temperature are accentuated and can be measured by observing a layer of thermochromic liquid crystal on the back of the wall, Kenning (1992) and Kenning and Yan (1995). In pool boiling of water at low heat fluxes such measurements confirm that there is strong cooling by microlayer evaporation (1); they show that mechanisms (2) and (3) are less effective than the transient conduction 'quenching' model suggests and that they operate on a wall area of influence that is no bigger than the maximum projected areas of the bubbles (Figure 10); the general level of convective cooling (4) is several times the level expected for single phase convection. The localized cooling round the nucleation sites interacts with the processes of bubble nucleation and growth (Figure 11). The waiting between bubbles depends on the rate of recovery of the local wall superheat after the departure of a bubble. This recovery may be interrupted by cooling by bubbles growing at adjacent sites or by fluctuations in the general convective cooling so that sites produce bubbles intermittently. The spatial variations in wall superheat affect the rate of microlayer evaporation that helps to drive bubble growth. The model for nucleation site activity based only on the mean wall superheat, summarized in Figure 6, cannot represent the intricacies of the real nucleation processes on thin walls. On thicker walls the variations in superheat should be smaller but they can only be measured at a few locations by microthermometers. However, the variations can be modeled numerically on a supercomputer and preliminary studies suggest that they influence nucleate boiling, even on a wall of high thermal conductivity such as copper [Sadasivan et al (1994)]. This sort of study should improve our understanding of nucleate boiling but the fundamental difficulties of specifying the microgeometry and internal wetting characteristics of the nucleation sites will remain.
Figure 10. Wall cooling during bubble growth 50-60: growth to maximum radius; 60–70: detachment; 70–80: rising bubble.
As the heat flux and the mean wall superheat are increased in saturated pool boiling, the active nucleation sites become so numerous that their bubbles start to coalesce a short distance from the wall. There is a transition to 'fully developed' nucleate boiling, in which the wall is covered by a liquid-rich 'macrolayer' less than 1 mm thick through which thin stems of vapor are connected to an overlying cloud of large 'mushroom' bubbles (Figure 12). Heat transfer is assumed to occur by conduction across the unsteady macrolayer causing evaporation at the bases of the mushroom bubbles, and by evaporation at the wall into the vapor stems feeding the bubbles. The fluctuating solid-liquid-vapor contact lines at the bases of the stems may be zones of efficient heat transfer. Wayner (1992) has described the processes of flow and heat transfer in liquid films so thin that they are influenced by van der Waals forces. There is still debate about the mechanisms of heat transfer in fully-developed nucleate boiling. The boiling curve (heat flux vs. mean wall superheat) loses the sensitivity to the orientation of the wall that is evident at lower heat fluxes [Nishikawa et al (1984)] but it is still sensitive to the surface condition of the wall, so there is no discontinuity in the curve. It is unclear what role is played by the individual nucleation sites as the heat flux is increased and the macrolayer gets thinner. Nucleate boiling breaks down when the macrolayer can no longer be replenished with liquid at a sufficient rate, or when local dry spots are stabilized by the resulting local increase in wall superheat (see Burnout in Pool Boiling).
In forced-convective boiling the heated walls form confining channels through which liquid is forced by an externally-applied pressure gradient. The conditions that have received most experimental attention are flow inside vertical and horizontal tubes and flow outside bundles of horizontal tubes. Most experiments involve uniform electrical heating, which does not always represent well the boundary conditions for boiling in heat exchangers, where the source of heat is a hot fluid. The liquid is usually subcooled when it enters the heated region. Vapor is first generated by nucleate boiling; the wall must be superheated to a value that depends on its microgeometry and wettability in order to activate nucleation sites, as in pool boiling. This superheat may be generated by increasing the heat flux, by decreasing the liquid flow rate or by decreasing the system pressure (or perhaps by a combination of all three in industrial systems). In uniformly heated systems boiling is initiated near the downstream end of the heated channel, where the wall temperature is highest and the pressure is lowest, giving the highest wall superheat. With further increases in heat flux, for instance, the initiation point moves upstream and flow boiling develops on its downstream side through regions of subcooled boiling in which the vapor bubbles condense at or close to the wall, bubbly flow in which bubbles move into the bulk flow (even if it is still slightly subcooled, i.e., the averaged thermodynamic quality is still negative), then the flow regimes corresponding to higher vapor fractions described in the articles on Forced Convection Boiling and Two-phase Flow. A typical wall temperature profile along a uniformly heated channel is shown in Figure 13. The wall superheat is approximately constant in the subcooled nucleate boiling region; as the quantity of flowing vapor increases, the wall superheat decreases as other mechanisms of heat transfer come into effect. (Wall superheats generally refer to time-space average values, since little work has been done in flow boiling on the local variations in superheat that have been shown to be important in pool boiling.) Like pool boiling, flow nucleate boiling of well-wetting liquids can exhibit hysteresis, which can modify the axial distributions of wall superheat [Wadekar (1993)].
Design correlations usually treat flow boiling as a combination of nucleate boiling and convection. Care is required because nucleate boiling is expected to be a nonlinear function of the wall superheat and convection may be driven either by the wall to bulk temperature difference in subcooled flows or by the wall superheat in saturated flows; it is safer to combine heat fluxes at a given wall superheat, rather than heat transfer coefficients. The nucleate boiling contribution is often based on information obtained from pool boiling experiments so it is subject to the usual difficulties of specifying surface conditions; this makes it difficult to obtain accurate correlations or even to choose between correlation schemes. One such scheme is the simple addition of the nucleate boiling and liquid convective heat fluxes, which seems to work reasonably well for large nucleate boiling fluxes at large liquid subcooling [e.g., del Valle and Kenning (1985)], although the nucleate boiling flux depends on the subcooling. This is not surprising because subcooling has a large effect on bubble behavior, reducing size but increasing frequency. This simple scheme does not work when the mass fraction x of vapor becomes significant in saturated flow boiling. The presence of the vapor increases the convective heat flux by mechanisms that depend on the flow regime. The velocity and turbulence of the liquid near the wall may be increased, sliding bubbles may continuously create thin liquid microlayers analogous to the transient microlayers in pool boiling [Cornwell (1990)], the flow may oscillate in the slug-churn flow regime or liquid may flow on the wall as a thin but highly disturbed film in the annular flow regime. These effects may be represented approximately by multiplying the single-phase liquid heat transfer coefficient by an enhancement factor F that is a function of the local quality x and the fluid properties to obtain the convective heat flux . The changes in flow conditions at the wall should have an effect on the nucleate boiling heat flux at a given wall superheat. The nuclei are exposed to larger temperature gradients, which from Figure 6 may suppress the activity of some sites, and hydrodynamic forces cause bubbles to detach at smaller sizes than in pool boiling. On the other hand, nucleation may be aided by the seeding of unstable sites as bubbles slide along the wall or by the entrainment of microbubbles created at liquid-vapor interfaces in the interior of the two-phase flow. The details of these processes are not understood but it is assumed that they can be combined in a suppression factor S that is a function only of the local flow conditions and which reduces the basic nucleate boiling heat flux. Chen (1966) introduced a correlation scheme for heat , of the form
of which there have been many subsequent developments (see article on Boiling). However, some experimental data are better represented by setting equal to whichever is larger of or , [Kenning and Cooper (1989)].
This article has so far dealt with nucleate boiling on surfaces that are nominally smooth. We have seen that nucleation depends on microscopic cavities that are accidental consequences of the method of manufacture of the surface. Prolonged service may modify the nucleation characteristics by corrosion or by deposition of corrosion products or dissolved solids. For non-fouling service, tubing is commercially available with 'enhanced' surfaces designed to provide large but stable nucleation sites, sometimes combined with short fins to extend the surface area. Thome (1990) provides a detailed description of boiling on enhanced surfaces (see Augmentation of Heat Transfer, Two Phase).
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