IHTC15-KN02 MULTI-SCALE INTERFACIAL PHENOMENA AND HEAT TRANSFER ENHANCEMENT

Proceedings of the 15th International Heat Transfer Conference, IHTC-15 August 10-15, 2014, Kyoto, Japan

IHTC15-KN02

MULTI-SCALE INTERFACIAL PHENOMENA AND HEAT TRANSFER ENHANCEMENT A. L. N. Moreira Instituto Superior Técnico, Universidade de Lisboa, Department of Mechanical Engineering Center for Innovation, Technology and Policy Research, IN+ Avenida Rovisco Pais 1049-001 Lisbon, Portugal

ABSTRACT Recent inventions in micron- and submicron- scale systems driven by the rapidly expanding capability of micromachining technology have shown tremendous benefits in flow processes in many established and emerging fields of growing economic importance and great potential for innovation. The detailed understanding of the governing mechanism are at the heart of realizing many future technologies, but poses several new challenges relating multidisciplinary scientific areas: surface energy becomes increasingly important at the micron- and submicron- scales and processes turn out to be largely affected by the physical geometry of the domain and the molecular structure of the surface. Interfacial transport phenomena integrate information from microfluidics, surface chemistry, biological sciences, micro fabrication, to develop research dedicated to improve an in-depth understanding of the basic physics in the above fields, as well as to provide useful information for direct applications. It refers to mass, momentum, energy and entropy transfer across and along fluid/fluid and fluid/solid interfaces, including the interfacial kinetics in multiphase combustion systems. In this context, wettability becomes an important influential parameter which can be changed by chemically treatment, patterning, or only different material deposition onto the surface. New micro/nano structured surface fabrication techniques allow tailoring special lyophobic and lyophilic mixed surfaces with great potential for miniaturized heat transfer devices and are currently driving a renewable research interest for both, experimental and theoretical, studies in multi-scale transport phenomena. This paper intends to review the interfacial transport phenomena from the perspective of their potential to enhance both, the heat transfer and the critical heat flux, in multi-scale heat transfer applications, namely pool boiling, spray and droplet cooling.

KEY WORDS: Two-phase/Multiphase flow, Wettability, Nano/Micro scale measurement and simulation, Thermal management

1. INTRODUCTION Recent inventions in micron- and submicron- scale systems driven by the rapidly expanding capability of micromachining technology have shown tremendous benefits in flow processes in many established and emerging fields of growing economic importance and great potential for innovation. Examples include labon-a-chip technology, micro-propulsion, and micro-thermal technologies. The detailed understanding of the governing mechanism of these systems are at the heart of realizing many future technologies, but poses several new challenges: surface energy becomes increasingly important at the micron- and submicron- scales and processes turn out to be largely affected by the physical geometry of the domain and the molecular *Corresponding Author: [email protected]

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IHTC15-KN02 structure of the surface. But a more general understanding of interfacial transport at micron-sized scale is still required to progress in the design and operation of many modern devices which evolution involves the integration of multidisciplinary scientific areas. This is in the scope of the universal discipline of transport phenomena. In engineering, transport phenomena are irreversible processes resulting in the movement of physical entities such as mass, momentum, energy, or entropy within a physical system, which occurs in homogeneous systems with no interfaces, but also in heterogeneous systems consisting of homogeneous parts, or subsystems, separated by natural interfaces - such as a liquid and its vapour - or by semipermeable membranes. When a difference, or drop, in pressure, or temperature appears in a heterogeneous system, irreversible flows of mass and heat arises across the interface between subsystems. Interfacial transport phenomena thus include all effects associated with momentum, mass and energy transfer combined with reaction kinetics and entropy generation, at fluid/fluid or fluid/solid phase interfaces and may occur in various ways over a range of length scales including the molecular level. However, while at the macroscopic or larger length scales, self-similarity criteria often apply and transport processes can be described with classical laws of transport, surface forces dominate at the micron- and submicron- scales and processes turn out to be largely affected by the molecular structure of the interface and the physical geometry of the domain. Therefore, interfacial transport at micron-sized scale is a multidisciplinary field intersecting engineering, physics, chemistry, microtechnology and biotechnology. This paper considers fluid/solid interfaces occurring in liquid cooling systems, namely those where the coolant vaporizes. In these systems, the ability of the liquid to maintain and/or rupture the contact with the solid surface, on which depends the dynamics of the interfaces, is the effective factor determining two-phase heat transfer and, ultimately, the performance of the cooling system. The underlying phenomena are governed by the surface and interfacial interactions, acting usually at small (a few nanometers for van der Waals or electrostatic interactions) or very small (molecular) distances but in a large span of time scales. A classical hydrodynamic description breaks down at the moving contact line and microscopic features have to be invoked. As a result, both large-scale (hydrodynamic) features and short-scale (molecular) dynamics are inextricably linked into a truly multiscale problem. In addition, new concepts are being introduced to influence the dynamics of wetting, such as patterned surfaces, surfactants, and non-Newtonian flow. By altering the chemical properties and/or the geometry of the solid surface, one modifies the contact energy of the surface with the vaporizing liquid and hence the wetting properties. Again, the development of various heat transfer applications related with these special surfaces have been accelerated by new micro/nano structured fabrication techniques.

2. MULTISCALE DESCRIPTION OF WETTABILITY Despite wetting of a solid by a liquid is a crucial part of many processes, both natural and industrial, which have motivated much research over many years, the precise mechanism by which a liquid front advances across a solid remains only partially understood and our ability to predict wetting behaviour and to model processes that are dependent on wetting is significantly restricted (e.g., Blake and Ruschak, 1997, Popescu et al., 2012). 2.1 The equilibrium contact angle as a first approach to wettability Wettability refers to how a liquid deposited on a solid (or liquid) substrate spreads out or the ability of liquids to form boundary surfaces with solid states. The most commonly used means of evaluating wettability uses the sessile drop method and measures the boundary conditions of the tertiary system formed by the solid surface, the drop and the surrounding vapour when mechanical equilibrium (zero velocity), chemical equilibrium (chemical potential matching for each component present), and thermal equilibrium

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IHTC15-KN02 (temperature matching) - together referred to as thermodynamic equilibrium - is reached. Therefore, for a chemically homogeneous and stable substrate at ambient temperature, equilibrium corresponds to a state of minimum energy and can be deduced from a force balance at the contact line, expressed by the Young’s equation: 𝜎𝑠𝑣 = 𝜎𝑠𝑙 + 𝜎𝑙𝑣 cos 𝜃𝑒𝑞 (1) where the subscript sv refers to the surface-vapor interface and the subscript lv denotes the liquid-vapor surface tension and eq is the equilibrium contact angle that the liquid creates with the solid surface. A wetting liquid is a liquid that forms an equilibrium contact angle with the solid which is smaller than 90º, while a non-wetting liquid forms an angle larger than 90o. According to the theory of wetting statics which is behind equation (1), the equilibrium contact angle is unique for a given system of solid, liquid and vapour and, thus, quantifies the wettability of a solid surface by a liquid. Although the contact angle in equation (1) has been deduced from a macroscopic balance of forces, thermodynamic equilibrium of the three-phase system (vapour/liquid/solid) is determined by the cohesive forces of similar molecules and by the adhesive forces between dissimilar molecules. From this point of view, true thermodynamic equilibrium may be extremely difficult to achieve, especially in the vicinity of the contact line. In practice there is contact angle hysteresis, ranging from the advancing contact angle (when the contact surface Figure 1. Water droplet contact angle measurements increases) to the receding contact angle (when the contact on different surfaces surface decreases) and the static angle has a value between (from Sumner et al., 2004) both, depending on the nature of the surface. The dynamics of wetting studies the dependencies of the contact angle on the wetting rate that are due to the processes at molecular level (e g., Voinov, 2004) and provides an essential understanding of the complexity of contact angle phenomena. 2.2 The importance of length and time scales Macroscopic and microscopic approaches have been developed to describe and predict the wetting behavior. Hydrodynamic models only apply to regions where space and time variations are slow in the scales of molecular processes, collision mean-free path and inverse collision frequency, a restriction represented in the space of wavenumber, , and frequency, , by a range of low (). However, since the discrepancies appear in a subtle and gradual manner as one moves way from the region of low (), research in non-equilibrium statistical mechanics have been developed towards a unified theory that retains the basic structure of the equations of continuous fluid dynamics but replace the thermodynamic derivatives and transport coefficients by functions which can vary in space and time, e. g., de Gennes (1985), Cazabat et al. (1997), Blake (2010) and Boon and S. Yip (1980). Within this purpose, modelling of the behaviour of the contact line considers two spatial scales (e g., Ren et al 2010), an inner region near the contact line where the movement of the line is described by a microscopic contact angle and an outer region, away from the contact line described by an apparent contact angle (ap). In this context, eq in equation (1) is understood to be measured macroscopically, on a scale above that of longranged intermolecular forces according to which the uniquely defined macroscopic angle is the apparent contact angle (ap). In this approach, the challenges in modelling the moving contact line phenomena are i) to describe the flow close to the contact line, and to understand its consequences for the flow problem on a large scale (for example, to compute ap); ii) to match the hydrodynamic part of the problem to a microscopic neighbourhood of the contact line (of the size of a nanometer), where a molecular description needs to be adopted.

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IHTC15-KN02 The liquid film is then divided in three regions (e g., Bonn et al., 2009, Yuan and Zhao, 2013, Snoeijer and Bruno Andreotti, 2013), see Figure 1: i) first there is the bulk fluid region where the curvature of the interface is constant and the macroscopic flow in the drop is on the length scale of capillary radius, or the droplet radius, R. Here, the angle between the tangent to the liquid-gas interface and the liquid-solid interface is termed the apparent contact angle, or the apparent dynamic contact angle, if the contact line is not static; ii) radially towards the contact line is the transition region of continuous increasing adhesive forces where viscous forces are balanced by surface tension forces. In this region the thickness of the liquid film becomes measurable and the liquid-vapor interface strongly bent, with a point of inflection where it turns from concave to convex. iii) further closer to the contact line, there is the adsorbed film region with a size of the order of nanometers, dominated by adhesive forces and where it is not possible to uniquely determine the contact angle. The shape of the interface is obtained by solving the coupled interface-viscous flow problem in the transition region, where microscopic effects are negligible but the thickness of the liquid film is measurable, and, then, by matching the solution to the small-scale behavior in the film region near the contact line, on one side, and to the macroscopic bulk region on the opposite side. Microscopic effects, which are thought to dominate at length scales of the order of 0,1 m or less, are mostly confined to the neighborhood region of the contact line and, at some Figure 2 – Shape of liquid-vapor interface extent, to the transition region. However, from the (from Stephan and Fuchs, 2009) aforesaid is clear that microscopic effects alter the boundary conditions between the adsorbed and the transition regions and, by virtue of that, the behavior of the bulk region. A better understanding of those effects would, therefore, allow one to forecast the micro-scale effects determining momentum, heat and mass transfer at the solid-liquid interface, which are aimed to be addressed in the present paper. It is worth noting, at this point, that one is considering the case of inert, smooth and chemically homogenous surfaces, where microscale effects are solely due to purely chemical interfacial tension. 2.3. The effects of heat transfer Scale effects on the evaporation during phase-change heat transfer have been studied by Wayner (1999) and Stephan (2002) based on the influence of dispersion forces on phase equilibria between thin liquid films and their vapor. But heat transfer can be comprehensively analyzed based on the thermal resistances of the liquid film and the liquid-vapor interface, as in Khandekara et al (2010): the total resistance tends to infinity in the adhesion zone where attractive forces inhibit evaporation; is also high in the bulk fluid region due to the relatively liquid thick film; but is minimum in the intermediate transition region where the film thickness and the interfacial resistance decrease. This interpretation is corroborated by experiments reported by Höhmann and Stephan (2002) who observed that the temperature drastically drops in the micro transition region where the evaporation rate is about half of the total evaporation - see Figure 3 – and by the analytical analysis of (Morris, 2003) who showed that an important contribution to evaporation during phase-change heat transfer across menisci, comes from the liquid film region. Figure 3 - Temperature distribution near to the Therefore, the evaporation mechanism of the thin-film near the

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meniscus (from Höhmann and Stephan, 2002)

IHTC15-KN02 solid–liquid–vapor contact line region is of great interest to many heat transfer processes, such as the evaporating menisci in micro channels and grooves, micro-layers under growing bubbles and droplets spreading on a heater surface, and numerous studies have been undertaken over the past several decades, to investigate the character of this region both theoretically and experimentally (e.g, Ball et al., 2003, Du et al., 2001, Zhao et al., 2011). The variation of the chemical potential per unit volume at the liquid-vapor interface where evaporation occurs is related with the pressure jump between the pressure in the liquid side, plv and the pressure in vapor side, pv as: 𝑝

∆𝜇 = 𝜌𝐿 𝑅𝑔 𝑇𝐿𝑉 𝑙𝑛 ( 𝑃𝑙𝑣 )

(2)

𝑣

where the interfacial vapor pressure, plv, can be determined from the variation of the chemical potential, µ, with the pressure and the temperature jump across the liquid-vapor interface, (𝑇𝐿𝑉 − 𝑇𝑉 ), which, for a singlecomponent system can be expressed as in Gokhale et al (2003) by ∆𝜇 = −(𝑝𝑐 + Π − 𝑝ℎ ) +

𝜌𝐿 ℎ𝑓𝑔 (𝑇𝑙 𝑇̅

− 𝑇𝑣 )

(3)

where pc is the capillary pressure given by 𝛿 ′′

𝑝𝑐 = 𝜅𝜎 with the curvature given by 𝜅 = (1+𝛿 ′2 )3⁄2

(4)

Π is the disjoining pressure introduced by Derjaguin (1936) to describe the relative magnitude of adhesive and cohesive forces: 1

𝜕𝐺

𝐴

Π = − 𝐴𝑟𝑒𝑎 ( 𝜕𝑥 ) = − 𝛿 3

(5)

(with A being is the dispersion or modified Hamaker constant accounting for the van der Waals long-range forces), ph is the hydrostatic pressure 𝑝ℎ = 𝜌𝐿 𝑔𝑧, and (𝑇𝑙 − 𝑇𝑣 ) is the temperature jump across the liquidvapor interface, and hfg the latent heat of vaporization at the average phase change temperature, 𝑇̅ = (𝑇𝑙 + 𝑇𝑣 )⁄2. The temperature jump is related to the interfacial pressure difference 𝑝𝑣 − 𝑝𝑙 = 𝜅𝜎 + Π

(6)

by the Clausius–Clapeyron equation (Zhao et al., 2011): 𝑑𝑝

(𝑑𝑇)

𝑠𝑎𝑡

=

ℎ𝑓𝑔

(7)

1 1 𝑇̅ ( − ) 𝜌𝑣 𝜌𝑙

The interface liquid-vapor in the transition region is concave towards the vapor in the adsorbed thin film region so, both kL and Π, are positive and (pv – pl ) > 0 giving the interfacial free energy per unit volume or the pressure jump at the interface. It is also worth noting at this point that he disjoining pressure, Π, dominates over the adsorbed thin film region where the film thickness is small and the interface is flat, while the curvature effects in pc dominate in a large portion of the transition region and in the thicker meniscus (>0,1 mm) where  is large and the curvature non-zero. Equation (4) gives the shape of a general meniscus interface, (x), such as that of a liquid film confined by a growing vapor bubble in a microchannel (e g., Wee et al., 2005, Panchamgam et al., 2008, Ball et al., 2013), or of a liquid droplet spreading over a solid surface (e g., Bonn et al., 2013). The heat transfer and

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IHTC15-KN02 temperature distribution in the micro-meniscus region are then obtained from the thickness profile, (x) and the fluid flow calculated from the liquid pressure gradient, which in turn, is determined from the thickness dependent pressure jump across the liquid–vapor interface (e g., Wayner, 1991 and Stephan, 2002). Combining equations (4 - 6), considering surface tension a function of temperature, =[T(x)] and taking the derivative with respect to x, one obtains a third-order differential equation for the shape of the liquid film, (x): 1,5

𝛿 ′′′ =

(1+𝛿 ′2 ) 𝜎(𝑥)

3𝐴

3𝛿 ′ 𝛿′′2

𝜕

[ 𝛿 4 𝛿 ′ − 𝜕𝑥 (𝑝𝑣 − 𝑝𝑙 )] + (1+𝛿′2 ) −

𝛿 ′′ 𝜕𝜎 𝜎(𝑥) 𝜕𝑥

(8)

Equation (8) captures the effects of curvature and the Marangoni stress at the liquid-vapor interface. Several different boundary conditions have been used in solving the previous models, which may cause theoretical conflicts depending of the physical nature of the flow under analysis, e g., Zhao et al. (2011), Du and Zhao (2011). It is out of the scope of this presentation to address a sensitivity analysis of the physical significance of the diverse solutions, it is enough to close the discussion saying that the boundary conditions for equation (8) must specify the values of the film height, , and its slope, ’, near the contact line and match the curvature of the solution, ’’, to the curvature of the macroscopic solution in the bulk region, given by the apparent contact angle. The resulting profile, obtained for intermediate scales, thus relates to the microscopic physics at the contact line, where no continuum theory is able to capture molecular processes driving the motion of the contact line on one side, and serves as the effective boundary condition for the macroscopic hydrodynamic problem on the other side. The pressure gradient, given by the curvature and disjoning pressure, as established in equation (6), is the driving force for the liquid mass to flow from the bulk region towards the thinner transition region where evaporation occurs and is obtained from an hydrodynamic model of the flow, e g., DeGupta (1994), Snoeijer, 2006, Ball et al. (2013). As an example, using the one-dimensional lubrication approximation together with the Marangoni stresses at the liquid–vapor interface and a slip condition at the liquid-solid interface, Panchamgam et al. (2008) obtained the one dimensional mass flow rate: 𝑚̇(𝑥) =

1 𝜕𝜎 𝜕𝜅 + (𝜎𝛿 3 + 3𝜎𝛿 2 𝛽) ] [(𝜅𝛿 3 + 3𝛽𝜅𝛿 2 ) 3𝜈 𝜕𝑥 𝜕𝑥 𝛿′ 12𝛽𝐴 + 3𝜐𝛿 2 [(−4𝐴 − 𝛿 )] 1 𝜕𝜎 + [1,5𝛿 2 + 3𝛽𝛿] 3𝜐 𝜕𝑥

(9)

where  is the slip coefficient at the solid-liquid interface. The authors re-arranged the variables in equation (9) so to isolate the diverse contributions to the liquid flow: the first parcel in the right-hand-side represents the contribution of capillary forces; the second parcel that of the disjoining pressure and the last parcel that of the Marangoni stresses. Since the rate of liquid flowing from the bulk to the transition region evaporates in the meniscus, equation (9) shows the multiscale effects on the evaporation rate. The net heat flux in the evaporating meniscus is expressed as a function of the interfacial temperature and pressure jumps as given by the Kelvin-Clapeyron model obtained by combining equations (2) and (3) for non-equilibrium conditions, ∆𝜇 ≠ 0, together with the interfacial kinetic theory, e g., Wayner (1991), Ball et al (2013): 𝑐𝑙 (𝑇 𝑘𝑙 𝑞 ′′ = ℎ𝑙𝑣 (𝑇𝑙𝑣 − 𝑇𝑣 ) = ℎ𝑙𝑣 (10) 𝑙𝑣 − 𝑇𝑣 ) − ℎ𝑙𝑣 (Π + 𝑘𝜎𝑙 − 𝜌𝑙 𝑔𝑥 ) where the classical heat transfer coefficient, hlv, which is viewed as a function of only the temperature jump 𝑐𝑙 𝑘𝑙 is replaced by coefficients for the temperature (ℎ𝑙𝑣 ) and pressure (ℎ𝑙𝑣 ) jumps. 2.4. The effects of surface inhomogeneity Validity of Eq. (1) is restricted to ideal surfaces and plane interfaces, which are rarely found in engineering practice. The presence of chemical inhomogeneity, roughness or any microscale structure on the surface can

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IHTC15-KN02 significantly change the dynamics of wettability, the equilibrium contact angle and rate of evaporation, depending on the relative scales, (e g., Shibuichi, 1996, Lipowsky et al., 2000, Bico et al., 2002, Kubiak et al., 2011, Yuan and Zhao, 2013, Singh et al., 2013, Antonini et al., 2014) and is currently a strategy widely studied to control and enhance heat transfer efficiency (e g., You et al., 2003, Kim et al., 2006, Li et al., 2008, Chen et al., 2009). This remains the subject of intensive research of, both experimentalists and theorists, particularly for engineered surfaces. Approaches to the problem range from hydrodynamic simulations of spreading to quasi-static considerations but the equilibrium phenomena provides us a systematic understanding of how the morphology of the interface alters wettability, e.g., Seeman et al. (2005) or Tasinkevych et al. (2006 and 2007), though the dynamic aspects of wettability are those of interest to the applications to heat transfer devices. In this context, two limiting equilibrium states are often considered to define the macroscopic wetting behaviour of drops on a structured surface, e g. Berthier (2008): the Wenzel state, in which the liquid fully wets the grooves of the rough surface and the Cassie state, in which the drop sits on top of the rough peaks. One may have lower energy than the other, so it is argued that it is possible a droplet to transit from one state to the other if the droplet overcomes the energy barrier. However, neither the Cassie-Baxter nor the Wenzel theory apply in practice, since a droplet spreading towards equilibrium encounter several pinning states in which it can be trapped and never reach the theoretical result as stressed by Alava and Dubé (2012) who analysed the effects of surface heterogeneity on the equilibrium contact angle of liquid droplets looking at the pinning and depinning dynamics as the contact line spreads on the substrate. The authors suggested that the dynamics is initially controlled by surface tension imbalance but disorder effects start to dominate as the droplet further spreads and the contact line comes to a halt before an equilibrium state can be reached. In fact, the Wenzel and Cassie theories support the concept that the contact angle is affected by the area of contact between the liquid and solid, which is valid when the size ratio of the drop to the wavelength of roughness or chemical heterogeneity is sufficiently large, e g., Marmur and Bittoun (2009). When the underlying assumptions do not hold, the key to the kinetics of the contact line is the activation energy that must be overcome for the contact line to move from one metastable state to another, e g. Extrand (2003) and Gao and McCarthy (2004) and neither theory can be directly applied at microscopic scale. From the microscopic point of view, in a chemically homogeneous substrate, the disjoining pressure given by equation (1.5) is independent of lateral coordinates parallel to the surface and the equilibrium wetting film thickness  is such that it minimizes the interfacial interaction energy. But, in fact, the disjoining pressure is given by the gradient normal to the interface, y, of the effective interface potential which is the cost in free energy per unit area to maintain a wetting film of prescribed mean Figure 4 - Dependence of the effective interface potential 𝜔(ℎ)and thickness Π(y) = −𝜕𝜔(𝑦)⁄𝜕𝑦 . The 𝜕𝜔(ℎ) of the corresponding disjoining pressure Π = − from the minimum of Π(y) determines the wetting 𝜕ℎ film thickness h for a partially wetting substrate. film thickness when thermal equilibrium (from Rauscher et al., 2008) with the vapour is reached, e g., Raucher et al. (2008). Lateral variations of the substrate composition cause a lateral variation of the effective interface potential and thus of the wetting properties. Similarly, viewing a topographically structured substrate as an extreme case of a chemical structuring where parts of the substrate are replaced by air and some parts by substrate material, droplets in the vicinity of topographic surface features would behave similarly to droplets in the vicinity of chemical heterogeneities, e g., Rauscher et al. (2008). In many

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IHTC15-KN02 studies a step is modelled by an abrupt change of the disjoining pressure, as in Moosavi et al. (2009), Asgari and Moosavi (2013). But the complex liquid–solid interactions which occur during the dynamic wetting and determine the equilibrium contact angle can only be accurately described by a multiscale approach. As a general simple rule, Bonn et al. (2009) state that “if surface structures are on the nanoscale, the fine details of the spatial variation of intermolecular forces and the effects of thermal fluctuations such as interfacial capillary waves become important but if, on the other hand, surface structure is not too small (above 100 nm, say), macroscopic laws can still be applied locally and in the bulk, and surface and line contributions can be resolved as separate entities”. Calculations of the equilibrium menisci of wetting films on rough surfaces performed by Sweeney et al. (1993) suggested that the volume liquid film above the solid surface can be thought of as being due to the competitive effects of disjoining pressure and capillarity: while the first contributes to generate a uniform thin film conforming to the shape of the underlying solid, the second contributes to reduce the interfacial area. Analysis is then based on the definition of a critical film thickness, c, below which the liquid surface follows the roughness of the solid surface and surface tension plays a negligible role and above which the fluctuations due to the solid roughness fall off, e g., Plawsky et al. (2008). The physical explanation of Yuan and Zhao (2013) is instructive: when a droplet is deposited on a substrate patterned with an array of lyophilic pillars, the microscopic disjoining pressure drives the precursor film to propagate within the pillars and to form a liquid molecular layer on the pillar surface and the hydrodynamic pressure drives the spreading film to advance on the base of the precursor film and to fill the space among the pillars. Therefore, while the grooves between the pillars provide the excess driving force required by the precursor film and the spreading film; the pillars act as extra potential barriers to the precursor film.

3. TWO-PHASE HEAT TRANSFER PROCESSES 3.1 Pool Boiling In pool boiling heat transfer, the rate at which bubbles are generated and released from the surface are important parameters to delay the critical heat flux (CHF) and to increase the heat transfer coefficient (HTC), which depend on the thermo-physical properties of the liquid and surface material, on the dynamics of wettability and on the surface micro-geometry, e g., Nam et al. (2009) and Varanasi et al (2009). Both the advancing and the receding contact angles play an important role: highly wetting fluids produce less nucleation sites and facilitate the subsequent rewetting of the dry surface as bubbles lift off the boiling surface, while promoting evaporation during bubble growing, as the vapour pushes liquid away to create a larger dry patch. The dynamics and thermal transport along the triple contact line solid-liquid-vapor at the leading edge of the liquid film is, therefore, of particular importance as it drives the entire boiling process. In this context, the study of single vapour bubble dynamics is the key to understand the complex nucleate boiling phenomena. A bubble generated on a heated surface grows differently, as represented in the Figure below, depending on whether the surface is lyophilic or lyophobic. Heat transfer on a lyophilic surface follows the description given in section 2.2, based on the thermal resistances of the liquid film and the liquidvapor interface: zone I is the adhesion zone where attractive forces inhibit evaporation; zone II is the intermediate transition Figure 5 - Heat transfer zones on: a) hydrophilic and b) hydrophobic surfaces region where the liquid film thickness and the interfacial (from Phan et al., 2009a) resistance decreases down to a minimum and zone III is the bulk fluid region where heat transfer is similar to that of single phase flow. For a lyophobic surface, there is no liquid film underneath the bubbles and the heat flows through a dry patch zone which deteriorates heat transfer and the detachment of bubbles from the wall, e g., Demiray and Kim (2004).

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IHTC15-KN02 Phenomena in the micro region are thus of significant influence to the overall nucleate boiling heat transfer and the heat transfer rate depends on the bubble growth rate, departure diameter and frequency of bubble release. The bubble departure diameter can be estimated from a balance of forces acting on the growing bubble, namely buoyancy, adhesion forces and dynamic forces caused by the motion of the liquid–vapour interface, such as inertia, drag and lift forces. For wetting surfaces, considering a static equilibrium condition and balancing the buoyant and adhesive forces, the bubble departure diameter made dimensionless by the capillary length 𝐿𝑐 = Figure 6 - Schematics of the dynamics of the contact line [𝜎⁄(𝜌𝑙 − 𝜌𝑣 )]0,5 has been taken as proportional to the during bubble growth, where θμ is the microscopic equilibrium contact angle, 𝐷𝑏 ⁄𝐿𝑐 = 0,0208 𝜃 (usually contact angle: a) the bubble growth initiates, b) the contact line stops displacing at θμ=90°, c) the contact known as the correlation of Fritz, 1965) where  is the line starts moving toward the bubble axis, d) bubble static equilibrium contact angle. However, bubble detachment. growth is a dynamic process during which the (from Phan et al. 2010) curvature of the liquid–vapor interface changes, starting with the appearance of a microscopic contact ◦ angle greater than 90 and a dynamic contact angle, function of the interface velocity, should be considered instead. A static contact angle value may still be used but, in such case, the advancing contact angle is more appropriate, as it has more influence on the bubble growth than the receding contact angle, e g., Mukherjee and Kandlikar (2007). A multi-scale approach should then be considered as in Stephan and Fuchs (2009) and illustrated by Phan et al (2009a) as follows: the micro-scale contact angle in the meniscus of the interface (as defined in section 2.2) is the important parameter as the bubble starts growing; the macroscopic contact angle in the bulk region of the meniscus (see section 2.2) determines the size of the bubble - the bubble being bigger and its growing time longer for a smaller contact angle. Moreover, if the initial radius of the triple contact line is assumed to be equal to that of the nucleation site and taking geometric arguments, the energy required to activate the nucleation site, as well as the time to initiate the generation of a next bubble, are larger for a lower contact angle. With these assumptions, the correlation proposed by Phan et al (2009b) for the bubble departure diameter (made dimensionless with the capillary length Lc) provides a picture of the influence of wettability: 𝐷𝑏 𝐿𝑐

𝜌

= 1,94 ( 𝑙 )

−1⁄2

𝜌𝑣

(

𝜌𝑙 𝜌𝑣

− 1)

1⁄3

𝑡𝑔𝜃 −1⁄6

(for 0° < θ < 90°)

(11)

The correlation considers the micro-scale contact angle and shows that the bubble diameter decreases as the contact angle increases. Now, since the rate of heat removed by liquid vaporization is the product of the mass of each bubble by the number of bubbles released by unit time, 𝑄̇ = 𝑓𝑏 (𝜌𝑣 𝜋𝐷𝑏3 ⁄6)ℎ𝑓𝑔 , if the cooling rate is to be maximized, it is also necessary to know the influence of the wettability on the frequency of bubble release. 3

𝜌 𝜋𝐷 𝑄̇ = 𝑓𝑏 ( 𝑣 𝑏 ) ℎ𝑓𝑔

(12)

6

fb is typically found in experiments to be inversely proportional to the departure diameter. Multi-scale modelling of bubbles released from a single nucleation site showed that the bubble release frequency – including the growth time and waiting time – increases exponentially with the static contact angle, e g., Hazi and Markus (2009), which is attributed by the authors to the speed up of the rewetting process of the dry spot and to the increased residence time of the liquid layer adjacent to the wall. Both effects are consequences of the fact that an increase in the static contact angle is due to increasing molecular attractive forces. Combining

9

IHTC15-KN02 the trends of Db and fb, it is, therefore, expected the rate of heat removal to be maximum for a range of contact angles and to deteriorate for contact angles approaching either down to 0o or up to 90o. The picture of the pool boiling given above may be over simplified and the conclusions hindered by the complex interactions induced by other parameters not accounted in the analysis: surface defects, chemical inhomogeneity and roughness determine the size and shape of the nucleation cavities and alter the initial steps of nucleation and bubble growth, e g., Lee et al. (2010); horizontal coalescence caused by the presence of multiple nucleation sites and vertical coalescence at high rates of bubble release determine convection in the vicinity of the surface. Additionally, the dynamics of the three-phase contact line cause the heat removed from the surface to vary periodically as bubbles nucleate, grow and detach from the surface: numerical results reported by Stephan and Fuchs (2009) based on a multiscale approach of a single nucleation site, showed that the resultant transient heat induces fluctuations of the wall temperature as high as 20% suggesting that also the assumption of a constant and homogeneous wall surface temperature does not hold. Strategies based on the modification of the surface to enhance both, the heat transfer coefficient (HTC) and the critical heat flux (CHF), consider either a promotion of nucleation and HTC in the isolated bubble regime of low heat flux, or an improvement of liquid transport in the regime of slugs and columns of high heat flux. The former can be achieved by reducing wettability (Nam et al., 2008, Phan et al., 2009), by etching cavities, e g., Betz et al (2010) or adding microporous coatings, e g., Furberg et al. (2009); while the second can be achieved by increasing surface wettability (Takata, 2005, Nam et al., 2009, Jones et al., 2009). Also, very often, methods to increase wettability based on micro or nanostructuring the surface adds the benefit of easing nucleation, e.g., Betz et al. (2013). In this context, surface roughness has been thought to be an effective mechanism to enhance pool boiling heat transfer in the nucleate boiling regime, which has been widely reported to increase transfer coefficients, e. g., Jones et al (2009). The argument is that roughness activates nucleation sites, thus contributing to increase the latent heat removal, at the same time that it can promote the rate of bubble detachment and vapour convection, which increases bulk convection as reported by Chen and Chung (2002). However, a too high rate of bubble detachment can be detrimental, since the interaction mechanisms between successive and adjacent bubbles may inhibit the removal of heat, particularly for very high heat fluxes, which are the main target in the cooling of high energy density systems. Despite many researchers look for what surface parameters actually govern the heat flux, a consensus has not been achieved yet, partially because the several surface parameters act simultaneously in non-similar physical phenomena which cannot be scaled in a single dimensionless group, as suggested by Hanley et al. (2013). In an attempt to derive an universal correlation, modifications have been suggested to the correlation of Kandlikar (2001) (which considers the effects of the equilibrium contact angle alone) to account for the dynamic wetting, making use of the receding contact angle and for a roughness-surface amplified force, including a term to account, e g., Chu et al (2012): 𝑞̇ 𝐶𝐻𝐹 = [

1+𝑐𝑜𝑠𝜃 2(1+𝛼) 𝜋 ] [𝜋(+𝑐𝑜𝑠𝜃) + 4 (1 + 16

0,5

𝑐𝑜𝑠𝜃)𝑐𝑜𝑠∅]

× {𝜌𝑔0,5 ℎ𝑓𝑔 [𝜎𝑔(𝜌𝑓 − 𝜌𝑔 )]

0,25

}

(13)

where = 𝑟𝑐𝑜𝑠𝜃𝑟 , being r the receding contact angle. However, not many studies have addressed the interaction and coalescence of neighbouring bubbles and most consider simplified surface geometries with a limited number of micro-cavities or micro-pillars, but a deterioration of the heat transfer coefficient has been reported due to the horizontal coalescence of bubbles resulting in bulks of vapor, which isolate the surface from the liquid and reduce the induced liquid motion, e.g. McHale and Garimella (2010), Moita et al. (2011). The Critical Heat Flux is then limited, either by the hydrodynamics of the pool, or by the capillary pumping force bringing liquid back to the heated surface and thereby preventing the drying out of the heated area. In this context, results of research are clear that it is not only the size of nucleation sites, but also the intersite distance that matters, e g., Mukherjee and Dhir (2004), Teodori et al. (2013) and an optimal pattern of the structural elements can be devised, in particular for high heat fluxes, which besides augmenting the liquid/solid contact area, keeps the inhibitive interaction among

10

IHTC15-KN02 nucleation sites under acceptable intensity, thus ensuring an overall improved cooling performance, e. g., Nimkar et al. (2006). Significant enhancement of both critical heat flux (∼200 W/cm2) and heat transfer coefficient was reported by Chen et al. (2009) which was attributed to the properties of nanowires, such as high nucleate site density, super-hydrophilicity, and enhanced capillary pumping effect, suggesting that further enhancement may be possible by rational design and synthesis of nanowire arrays; the use a porous-layer coating with a periodic conical variation of thickness, e g., Liter and Kaviany (2001), provided a three-time increase of the Critical Heat Flux from that for a smooth surface and further enhancement was anticipated if the modulation wavelength could be reduced so to optimize regions of low resistance for the vapor to escape with high capillary-assisted liquid draw. Surface topography design is, therefore, known to play a major role in this process though, up to now, it has not been possible to easily correlate nucleate boiling data with surface roughness parameters. However, simplified geometries may lack similarity with the rough surfaces used in practical applications and extrapolation of the results is often of questionable accuracy. To overcome the difficulties of not having accurate design tools, empirical approaches have been suggested, consisting of maps for site activation and bubble coalescence, to allow predict the interaction mechanisms and the the relative weight of each, as in Bonjour et al. (2000) and Zhang and Shoji (2003).

3.2 Spray cooling Spray cooling allows a uniform temperature distribution and is less susceptible to disruption by vigorous boiling, mainly due to the combined effects of forced convection and nucleate boiling, which is why it has been accepted as one of the most promising cooling techniques to boost performance in microelectronic device, Mudwar (2001). However, the heat transfer from the wall results from complex interactions between diverse phenomena and it is difficult to find a dominant governing mechanism. Possibly the most important parameter is the thickness of the liquid film layer that forms onto the heated surface: for thick films, heat is primarily transferred as in traditional boiling but with mixing being enhanced by incoming droplets, besides gas entrainment and evaporation; at the opposite extreme of sparse sprays and/or at high surface temperature, droplet evaporation occurs onto a mostly dry surface at a rate strongly dependent on droplet impact conditions, such as their sizes and velocities. 3.2.1. Spray cooling with thick liquid films One proposed mechanism of spray cooling is by means of thin film evaporation, which occurs at small liquid throughputs where the spray is believed to form a thin liquid layer on the heated surface: heat is conducted through the liquid film from the solid surface to the liquid-vapor interface which is maintained at the saturation temperature and where liquid molecules outflow into the ambient, e g., Pais et al. (1992); as the liquid film becomes thinner, the thermal resistance decreases resulting in a higher cooling performance. Another mechanism is proposed when the spray creates a thick liquid film, where the basic boiling phenomena of boiling in the absence of an incoming spray are an essential condition to the more complex process of spray cooling. In this context, it is appropriate to extrapolate the results reported by Horacek et al (2005) that the heat flux depends more on the contact line length, rather than on the wetted area fraction of liquid on the surface, to argue that the three phase contact line developed at nucleation sites dominates over the conductive mechanism observed in thin liquid films, at least for moderate impact rates. As the impact rate and the impact momentum increase, in addition to bubbles generated at nucleation sites on the surface cavities, many more are generated within the liquid layer as liquid droplets from spray strike on the liquid surface and entrain vapor, which greatly enhance boiling heat transfer and increase turbulent mixing within the film promoting convection and free surface evaporation, e g., Rini et al. (2002). In both cases, of moderate and high liquid throughputs, the effect of increasing the droplet number flux is to improve the density of primary and secondary nuclei. In this context, spraying the surface intermittently has been

11

IHTC15-KN02 suggested to be an efficient method to optimize the rate of liquid supplied to the surface with the capability to remove the vapor from the surface, by adjusting the dead time interval between consecutive injections, e g., Panão and Moreira, (2009a,b). An intermittent spray cooling system has the joint capability of removing heat by promoting phase-change at lower duty cycles, and through a thin film boiling mechanism as the duty cycle approaches the working condition of a continuous spray. Laboratory experiments have showed about 30% increase of the heat flux (e g., Panão and Moreira, 2009a) or 35% decrease of the liquid flow rate for the same heat flux (e g., Zhang et al., 2013). In the limit of high flow rates of coolant heat transfer may be controlled by many other important parameters. Of these parameters, spray properties have received the most attention but apparent contradictions can be found in the literature, about which parameters actually govern heat transfer at spray impact. While some argue that heat transfer is governed by the velocity component of impacting droplets normal to the surface, e g., Chen et al. (2004), other authors argue that liquid flux is of much greater significance – either the volume flux, e.g. Rybicki and Mudawar (2006), the mass flux (e.g. Yao and Cox (2002) and Cabrera and Gonzalez 2003), or the number flux of droplets, e.g. Rini et al. (2002). Instead, for Pikkula et al. (2001), the heat transfer rate is dominated by the ratio between inertial and surface tension forces at impact (Weber number), and for Chen and Hsu (1995), by the wall superheating degree (T w–Tsat). Many apparent contradictions can be explained by the fact that the parameter(s) governing spray/wall heat transfer actually depend in a complex way on the boundary conditions of the problem, which in turn, are determined by the experimental or operating conditions. The practical result is that any derived empirical correlation lacks universality its applicability depends significantly on the liquid mass flux. From the mechanisms described above, it is admissible to conclude that the mechanisms behind the critical heat flux in spray cooling differ from those in typical boiling systems. For high heat fluxes for which the production of large bubble is intense, the vapor escaping from the surface and simultaneous ejection of liquid from bubbles bursting at the surface prevent fresh liquid droplets from hitting the surface. Based on the concept of a macro-layer of vapor formed onto the surface which is behind the concept of the thermal nonwetting regime, Chow et al. (1995) developed a correlation for the critical heat flux which accounts for the effects of surface tension: ′′ 𝑞̇ 𝐶𝐻𝐹 ̅ 𝜌𝑙 𝑄̇ " ℎ𝑓𝑔

1⁄2 1⁄2 𝜌𝑔 𝜎 𝑝𝑠 1⁄4 ) ( ) ( ) 2 ̅̅̅" 𝑑 𝜌𝑓 𝑝𝑎 𝜌𝑙 𝑄 32

= 0,38 (

(14)

In addition to the effect of the mean volumetric flow flux, 𝑄̅̇ " , this correlation accounts for the effect of surface tension and of the impacting velocity in the ratio of the stagnation to the ambient pressure, (𝑝𝑠 /𝑝𝑎 ). Another correlation has been proposed by Mudawar and Estes (1996), which spray angle additionally accounts for the effects of droplet diameter, d32, and surface subcooling, ∆𝑇𝑠𝑢𝑏 : " 𝑞̇ 𝐶𝐻𝐹 𝜃 𝜃 = 1,467 [(1 + 𝑐𝑜𝑠 ( )) 𝑐𝑜𝑠 ( )] ̅ " 2 2 𝜌𝑔 𝑄̇ ℎ𝑓𝑔

[

3.2.2.

𝜎

𝜌𝑓 ( ) 𝜌𝑔

0,35

2

𝜌𝑓 𝑄̅̇ " 𝑑32

0,3

]

× [1 + 0,0019

0,3

×

𝜌𝑓 𝐶𝑝,𝑓 ∆𝑇𝑠𝑢𝑏 𝜌𝑔 ℎ𝑓𝑔

]

(15)

The influence of surface finishing

Surface finish has been proven to significantly influence spray cooling, as it alters the flow field, altering the maximum liquid film thickness, the bubble diameter, vapor entrapment, bubble departure characteristics, and the ability to transfer heat. However, the benefits of micro-structuring the surface may be contradictory, depending on flow pattern and film thickness (thick film, thin film and dryout) as well as on spray operating conditions.

12

IHTC15-KN02 In the deep film regime, when spray cooling occurs with the formation of a liquid film thicker than mean roughness, the dominant effect of surface roughening is to increase the density of active nucleation sites density, similarly to boiling heat transfer. Enhancements of the order of 50% at the critical heat flux have been reported at microporous surfaces with modest flow rates due to capillary pumping of liquid on top and within the surface, e.g. Kim et al. (2004), though weaker effects may be expected for high spray rates, as the additional influence of spray parameters on mixing assumes increased relevance. However, careful must be taken since, at the same time it promotes nucleation, surface roughness also move the nucleate boiling regime to lower temperature with the disadvantage of also moving the critical heat flux to lower temperatures as reported, for example by Martínez-Galván (2013). In the opposite extreme, where heat transfer is primarily governed by a thin evaporative film, equal or smaller than surface roughness, the level of surface wetting, determined by the balance between the amount of liquid absorbed by capillary force over roughness and the amount of liquid evaporated, is the important factor in determining the performance of spray cooling: i) for too high rough peaks or too thin liquid film, increasing surface roughness creates channels with larger conduction resistance through the thick film to the free surface where evaporation occurs and peaks where a too thin liquid film can cause prematurely dry out, e g., Pais et al., 1992 - the result being that increasing the surface roughness may decreases the heat transfer; ii) in the complete evaporative wetting regime, a microporous surface performs better than a smooth surface, e. g., Kim et al. (2009) and the performance is affected by flow rate and coating thickness as these factors influence the level of liquid wetting. However, spray parameters are used as control variables to manage and optimize the amount of heat removed in a dynamic situation such as, for example, in electronic devices. The worth conclusion at this point is that the effects of spray parameters act differently on the performance of spray cooling depending on the ratio between surface roughness and the thickness of the liquid film. 3.2.3. Dilute sprays Dilute sprays and/or high superheats may be considered as a system of discrete droplets for which the dynamic wetting is determined by diverse individual drop impact phenomena - the droplet can deposit and spread onto the surface, splash with formation of smaller satellite droplets moving away from the surface, totally or partially rebound from the surface – the outcome depending on the impact conditions (velocity, Uo, direction relative to its surface, drop size, Do), properties of the liquid (dynamic viscosity  and surface tension ), the surface or interfacial tension, including the effect of roughness and wettability, e g., Moreira et al. (2010). Many studies are reported in the literature, both theoretical and experimental, addressing the transition from a wetting behaviour associated with droplet spreading onto the surface to a non-wetting behaviour associated with the disintegration of the droplet at impact, e.g., Cossali et al. (2005), Akhtar and Yule (2001), Akhtar et al. (2007a, b), Fujimoto et al. (2010). Though, in general, the energy of the droplet at impact gives a threshold for transition onto cold surfaces, e. g., Moreira and Moita (2011) - which can be altered due to surface inhomogeneity - heat transfer modifies momentum transfer and the characteristic time scales of droplet spreading and induces different disintegration mechanisms, e.g., Richter et al. (2005), Moreira et al. (2007), Cossali et al. (2008), and Müller et al. (2008). For the purpose of removing heat from the impacted surface, a low impact energy may be preferable as the consequent droplet deposition and spreading increases the area and lifetime of the liquid-solid interface; but at high impact energies, at which secondary atomization can cause surface dryout, strategies can be devised to better wet the surface with the secondary spray as in Heffington et al. (2002) and Vukasinovic et al. (2004). The first extreme is determined by the wettability effects in the space of wavenumber, , and frequency, , by a range of low (), as discussed in section 2.2, associated with the meniscus shape of the interface, which is encountered in a variety of practical situations, such as that of a droplet of a cooling liquid spreading over a heated solid surface; the growth and detachment of a bubble formed at the surface of a heated solid in

13

IHTC15-KN02 quiescent surroundings, or the evaporation of the liquid coolant in a microchannel heat dissipator, e g., Plawsky et al. (2008). As discussed in 2.4, wettability of a surface by an impacting droplet is altered by surface inhomogeneity. However, studies dealing with it often make use of the static and dynamic contact angles, which are quite difficult to implement, since a droplet spreading towards equilibrium encounter several pinning states in which it can be trapped, thereby never reaching the theoretical result. The effects of inhomogeneity may be considered in two groups, one for which the periodicity of the heterogeneity is much smaller than the size of the droplet, another for which droplets are smaller than a statistical wavelength of heterogeneities. It can be argued that droplets in the first group keep spreading until reaching ultra-thin surface layer: Tsekov et al. (2013) observed that a periodical surface roughness with the contour of lyophobic pillars on lyophilic surface makes the last one completely wettable; similarly, for chemical patterning, if the heterogeneity is completely contained with the contact area and does not intersect the contact line, wetting of heterogeneous surfaces is controlled by interactions at the contact line, not the contact area, e.g., Extrand (2003). On the contrary, for droplets smaller than the periodicity of the patterning, a mechanism can be figured as suggested by Yuan and Zhao (2013) for the wetting of droplets on lyophilic pillars where the smooth and initially circular contact line is compressed to propagates between the pillar pattern, the excess area of the rough surface imposing an excess driving force when the liquid reaches the pillars and also pinning the contact line until the slow part of the fringe catches up. However, as the scale of surface patterns goes down, the corresponding wetting structures cannot be understood using the same theoretical framework as for the macroscopic scale. Also, with decreasing droplet size to the meso and nanoscale, the line tension, described as the excess free energy arising from the three phase contact line, e.g., Fan (2006), becomes increasingly important as described, for example, by JódarReyes (2007). A plane line tension replaces the surface tension acting at the three-phase contact line but, in contrast with surface tension, the line tension changes sign and is positive if it contracts the wetting perimeter and negative if it expands the perimeter. In general, chemical patterning of substrates leads both to a wettability contrast and to a contrast of the line tension (e.g., Blecua, 2006), so sufficiently small structures may be governed by contributions from the contact lines as observed by Marmur (2006). But surface patterning also alters the transition threshold from deposition to disintegration, a mechanism which alters heat transfer. This transition is usually described by a “splashing parameter” first suggested by Mundo et al. (1995) for the impact onto cold surfaces, but which has been shown to assume the same functional form for structured (e.g., Pan et al. 2010; Palacios et al. 2010) and heated surfaces, i. e., K = WeaOhb, e. g., Moreira et al. (2010). Coefficients a and b depend on the physical mechanism of disintegration, which is strongly related to wettability (including topography) but also to surface superheating. In fact, the temperature of the surface is found to be an important parameter determining secondary atomization, depending on the heat transfer regime as defined by the well-known boiling curve of Nukyiama, e g., Moita et al., (2012): i) secondary atomization in the nucleate boiling regime is mainly thermally induced and surface topography plays a secondary role; ii) in the homogeneous wetting regime, the topographical micro-patterns with larger roughness ratio promote larger secondary droplets whose size is found to correlate as 𝐷𝑠 ⁄𝐷𝑜 = 𝐴𝑊𝑒 −𝑎 𝑅𝑒 −𝑏 𝐽𝑎−𝑐 𝑓(𝛿𝐿 ) 𝑓(𝑟𝑓 ), where δL is the thickness of the deposited film; iii) in the film boiling regime, the size of secondary droplets depends only on the dynamics of impact as given by 𝐷𝑠 ⁄𝐷𝑜 = 𝐴𝑊𝑒 −0,6 𝑅𝑒 −0,23 , e. g., Moita and Moreira (2009) and Müller et al. (2009). It is worth noting at this point that, at the same time that the surface can be artificially patterned, also nanofluids can be though to be tailored to meet the desired transport properties depending on the engineering application such as higher conductivity, enhancement of boiling heat transfer and critical heat flux, as well as the wetting properties of a surface can be altered with the deposition of nanoparticles. It has been shown that, not only the equilibrium contact angle of nanofluids but also the dynamic behavior during evaporation can be significantly affected by the size and concentration of nanoparticles, e. g., Leeladhar et al. (2009), Sefiane and Bennacer (2009) and Chen et al. (2010). This is, however, a complex issue that is not aimed here.

14

IHTC15-KN02 6. CLOSURE The present paper aims at reviewing current knowledge on fundamental interfacial phenomena endeavoring to enhance heat transfer. Specifically, the paper highlights multi-scale phenomena in interdisciplinary areas of physics of fluids such as the physics of the contact line, wetting and thermo-capillarity; two-phase systems behavior including drops, sprays and bubbles; phase change phenomena such as boiling; and multi-scaled heterogeneous surfaces. Theoretical and experimental studies have been cited that reveal the complex structured behaviour of a liquid in the vicinity of solid surface and clarify the importance for the overall wetting dynamics of the microscopic processes which take place near the contact line. Manipulation made at very small length scales, in particular of surface structure, either chemical or mechanical, can produce sensible changes of the contact angle and corresponding modifications of the length and time scales of the heat and mass transfer at the interface. As new experimental methods are being developed to construct patterns with even smaller domain sizes in the nanometer range, surface homogeneity plays an increased important role in devising interfaces with optimal transport properties. In this context, effective improvements of the heat transfer can be achieved with an appropriate design of micro-patterns with the ability to balance the positive effects to promote the activation of nucleation sites with the negative effect of an excessive interaction between them. However, accurate design tools are hindered by a lack of understanding of the strong interactions between the transport phenomena on very different length scales from the nanometer to the millimeter scale and depend on the developments in high resolution measurements techniques. Empirical approaches have been suggested, such as maps for site activation and bubble coalescence in pool boiling, to allow predict the interaction mechanisms and the the relative weight of each. A further understanding of the dynamics of wetting on very different length scales from the nanometer to the millimeter scale still depend on the developments in high resolution measurement techniques. Wettability studies are usually based on contact angle measurements, namely the sessile drop technique, which makes use of conventional visualization techniques with drawbacks when conventional optics are used, in particular for not uniform, not planar or rough base surfaces. 3D reconstruction techniques have recently been reported making use of X-ray micro-computed tomography or confocal microscopy and the results have proved promising for surface characterization and to get more accurate and detailed information about wettability characteristics.

ACKNOWLEDGMENTS Research has been funded by Fundação para a Ciência e a Tecnologia (FCT) in the framework of projects PTDC/EME-MFE/109933/2009 “The effects of surface wettability and roughness on microchannel evaporative heat transfer - application to cooling systems” and RECI/EMS-SIS/0147/2012 “Dynamics of interfacial transport phenomena in micro scale energy conversion systems”. Some results reported here were obtained within the research activities of the group on Multi-Scale Interfacial Phenomena. Among others, the contributions of Dr. Ana Moita, Dr. Miguel Panão, Mr. Emanuel Teodori and Mrs. Vânia Silvério are gratefully acknowledged, also for the many helpful discussions and criticisms.

NOMENCLATURE Ca CHF d32 Db fb

capillary number, Ca = U/ Critical Heat Flux Sauter mean diameter Bubble departure diameter Frequency of bubble release

T hfg p Lc Do

(-) (W/m2) (mm) (mm) (s-1)

15

temperature latent heat of vaporization pressure Capillary length, 𝐿𝑐 = [𝜎⁄(𝜌𝑙 − 𝜌𝑣 )]0,5 Droplet diameter at surface impact

(K) (kJ/kg) (N/m2) (-) (m)

IHTC15-KN02 Uo We Oh Re Ra 𝑄̅̇ " Kc rf

Droplet velocity at surface impact Weber number (We =Uo2Do/) Ohnesorge number (Oh = We/Re1/2) Reynolds number (Re = UoDo/) mean roughness Mean volumetric flow flux splashing parameter roughness ratio

(m/s) (-) (-) (-) (m) (m/s) (-) (-)

Greek symbols  chemical potential  surface tension  equilibrium contact angle θμ microscopic contact angle θR receding contact angle θA advanced contact angle  dynamic viscosity

Subcripts f g lv v l s

liquid saturated conditions vapour saturated conditions interface liquid-vapour bulk vapour bulk liquid secondary (reatomized) droplet

(J/mol) ( N/m )

(N s/m2)

𝛿 ′′ (1+𝛿 ′2 )3⁄2



curvature 𝜅 =

    

specific weight thichness of the liquid film effective interface potential Disjoining pressure Angular orientation

(kg/m3) (mm) (J/mol) (N/m2)

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