A mechanistic model for embryo size prediction at boiling incipience: ‘Work of formation’ based approach

International Journal of Heat and Mass Transfer 110 (2017) 921–939

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A mechanistic model for embryo size prediction at boiling incipience: ‘Work of formation’ based approach Janani Srree Murallidharan a, B.V.S.S.S. Prasad a, B.S.V. Patnaik b,⇑ a b

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, 600036, India

a r t i c l e

i n f o

Article history: Received 28 October 2016 Received in revised form 6 February 2017 Accepted 3 March 2017 Available online 3 April 2017 Keywords: Boiling Nucleation Gibb’s free energy Embryo Phase-change

a b s t r a c t The initial size of the embryo, which is formed at the inception of boiling, plays a vital role in the accurate prediction of component scale wall boiling phenomenon. Embryo size predictions are typically calculated using the classical theory of nucleation. However, in recent times, the predictive capability of this theory was found to have limitations. Hence, there is need for a more fundamental and mechanistic model to overcome some of the drawbacks. In this paper, we propose a ‘work of formation’ based model for the embryo formation. This model is mechanistic and includes a Van der Waals based real gas treatment for the vapour. It also incorporates Lewins surface tension model that is a function of the boilingnucleus size. The present model also accounts for the boiling occurrence in the presence of undissolved nanobubbles on the surface. The embryo formation model has been extensively tested for both low and high pressures, horizontal and vertical test section orientation, and for different surfaces and fluids. The energy required for the embryo formation was found to be higher, when the initial gas bubble is intact compared to when the gas bubble diffuses into the embryo. Some of the contradictory claims on the suitability of classical theory of nucleation to nanosurfaces have been tested. From the present embryo formation model, the physics of nucleation such as, the effect of pressure fluctuations and energy dissipation mechanisms involved in the formation is explained. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Boiling and condensation influences a wide spectrum of industrial systems. The design and safety of such systems is highly dependent on the precise understanding associated with the phase change process. To be able to better design the safety standards, several plausible scenarios need to be addressed. To this end, computational techniques can play a prominent role in building models with phase-change that can accurately mirror the physics. The Eulerian-Eulerian multiphase model (EEMF), is one of the popular choices for the computational framework in such componentscale systems. The heterogeneous nature of boiling occurring at the wall is modeled through the ‘wall heat flux partitioning’ model (WHFP). A detailed study of this framework for high pressure subcooled flow boiling conditions was recently discussed in Murallidharan et al. [1]. In that study, the wall bubble departure diameter (D) was identified as the most important parameter that needs to be modeled mechanistically and accurately. For a better prediction of D, it is essential to accurately obtain the initial size of the bubble ⇑ Corresponding author. E-mail address: [email protected] (B.S.V. Patnaik). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.011 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

nucleus that forms on the wall. Correlations such as Benjamin and Balakrishnan [2]’s nucleation site density model, Kirichenko [3]’s models, Wang and Dhir [4]’s model were analyzed in Ref. [1], and it was found that, the ‘radius of cavity’ term played a crucial role. In general, the classical theory of nucleation is used to determine the cavity size (or) embryo size [5]. The formulation of the classical theory, as shown in Eq. (1), is obtained by combining the Clausius - Clayperon equation and the Young’s Laplace equation as follows:

T l
T sat ðPl Þ >

2rlv T sat Dv lv hlv rmin

ð1Þ

where Tl and Tsat is the superheated liquid and saturation temperature respectively, rlv is the surface tension, Dmlv is the difference in specific volume between vapour and liquid, hlv is the latent heat of vaporization and rmin corresponds to the radius of the bubble embryo. However, the classical theory is only elementary and does not take into account key parameters such as, the wall contact angle and wall roughness [6]. Based on the current state of literature (presented in Section 2) it can be seen that an accurate and comprehensive framework to model nucleation is required. In this work, we propose a embryo formation model which is more mechanistic

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Nomenclature A D G h K n P Q” r S T U V v W

interface area (m2) bubble departure diameter (m) Gibb’s free energy (J) latent heat of vaporization (J/kg) Boltzmann constant number of moles of species (moles) pressure (bar) heat flux (W/m2) radius (m) entropy (J/K) temperature (K) internal energy (J) volume (m3) specific volume (m3/mole) expansion work (J)

Greek symbols D difference operator g diffusivity (m2/s) h contact angle (°) l chemical potential (J/mole) q density (kg/m3) m specific volume (m3/mole) r surface tension (N/m) X incipience of surface tension 1 planar surface value Subscripts and superscripts b base

and is based on ‘work of formation’. Additionally, the proposed model is applicable for component scale boiling models, interface tracking, DNS based studies etc. Currently in these studies, the initial embryo size is arbitrarily chosen and is based on mesh resolution [7]. It will be shown that, the proposed embryo formation model provides a mechanistically accurate alternative. 2. Background literature 2.1. Nucleation theories All existing nucleation theories can predominantly be classified as – phenomenological, kinetic and molecular. The phenomenological models compute the energy involved in the formation of a nucleus, primarily using the Gibb’s free energy function [8,9]. Computing the formation energy of the nucleus is based on the assumption that, both vapour and liquid are a continuum, and can be represented through the macroscopic properties such as, temperature, pressure and surface tension. Kinetic methods focus on computing rate of nucleation using models such as, the Boltzmann statistics of equilibrium cluster distribution and by using empirical coefficients such as the monomer evaporation coefficient [10]. The molecular based nucleation deals with very small length scales and accounts for atomic behavior (e.g.: molecular interaction potential) as well as interface movement (e.g.: density functional approach [11]). Models based on kinetic and molecular nucleation theories use small length scales (molecular) and considerable statistical empiricism (kinetic methods) and hence have only limited experimental support. In fact among the various nucleation theories there is a seven orders of magnitude difference between micro and macro length scales [12]. Moreover, all of them cannot be used to solve the problem at the same level due to length

c cr g gl i j l ls lv min O sat v vg vs wall ⁄ /

cavity critical point gas gas-liquid chemical species interface liquid liquid property at saturation temperature liquid-vapour minimum before boiling saturated vapour vapour-gas vapour property at saturation temperature value at the wall after boiling non-dimensionalised parameter w.r.t. critical point

Abbreviations CHF critical heat flux D-RGC diffused – real gas core EEMF Eulerian-Eulerian Multiphase model I-RGC intact-real gas core LSR liquid saturation ratio ONB onset of nucleate boiling OSV onset of significant void WHFP wall heat flux partitioning

and time scale issues and other computational constraints. From the categories stated above, the phenomenological approach of free-energy based nucleation can span various length scales without significant computational demand and mesh resolution. Furthermore, it can easily be implemented alongside existing boiling modules of general purpose computational fluid dynamics (CFD) solvers. More importantly, the Gibb’s free energy based approach is based on the fundamentals of continuum thermodynamics. Hence it is most suited for easy application to the component scale modeling of boiling. 2.2. Free-energy based nucleation The study of nucleation using Gibb’s free energy was first proposed by Volmer and Weber (see [13]). Lu and Peng [14] modeled the dynamic evolution of nucleation for different types of cavities such as inside, outside, and twice-nucleation. They have concluded that, the structure of the cavity (cavity size) and surface characteristics strongly influence the type of nucleation. However, their study was not validated against experiments and was not tested for different operating conditions. They have treated the newly formed nucleus as a cluster of atoms and used the kinetic equations of nucleation for modeling its distribution and rate of growth. Their study did not account for the presence of pre-existing gas nuclei at the wall, although they are usually observed in reality. Wu et al. [15] improved upon this study and modeled nucleation near a wall by introducing a temperature gradient in the vicinity of the wall. Validations were performed by comparing the predictions of heat flux variation vs. wall superheat with that of the experiments. They have studied the variation in the free-energy of the system for various temperature gradients and contact angles. However, no direct validation of predicted embryo size

J.S. Murallidharan et al. / International Journal of Heat and Mass Transfer 110 (2017) 921–939

was attempted. The model was very elementary and did not explicitly account for the chemical potential and surface energy variation during phase change. Though Wu et al. [15] did not treat the nucleus as a cluster of atoms or molecules like Lu and Peng [14], both have not attempted the real gas model. Quan et al. [8] developed a model based on Gibbs free energy to study the nucleation for surfaces with different wettability (different contact angles). Although they have studied the variation of nucleation with contact angle and temperature gradient, their study adopted a different temperature profile at the wall when compared to Wu et al. [15]. They have formulated the problem in terms of the availability function [8,9]. In Ref. [9] it was concluded that, the criterion formulated from Gibbs free energy is the necessary condition for the onset of bubble nucleation, while the criterion based on availability function is not a necessary (but sufficient) condition for the onset of nucleation. They have found that, the critical radius predicted by the availability criterion is larger than the Gibb’s criterion. Unlike Wu et al. [15], Quan et al. [8] explicitly modeled the change in chemical potential and surface energy during boiling. However, the study treated the gas as an ideal gas and did not account for any pre-existing gas bubbles. Chen et al. [16] have accounted for the transient effects by studying pulse heating conditions, by extending the model of Quan et al. [8]. Comparisons of wall heat flux predictions for different superheats were done against experiments in which water, alcohol and R113 were used as test fluids. It was found that among the three fluids, the embryo size and the work of formation required for water was the largest [8,9]. Yarom and Marmur [17] performed nucleation studies, in the presence of a pre-existing insoluble gas nucleus. Though this study was an improvement, the model formulation was still highly idealized (ideal gas model with constant surface tension). Yuan et al. [18] developed a method that took into account the presence of an initial insoluble gas, temperature gradient, and the line tension. Their study performed a comprehensive analysis on the variation of the formation energy with contact angle, temperature, etc. However, the method was not validated against any experiments. Moreover, insoluble gas was modeled as an ideal gas and only considered surface tension variation with temperature and not with radius. Abyzov and Schmelzer [19] modeled heterogeneous nucleation for both condensation and boiling on planar solid surfaces. They have accounted for the changes in the state parameters such as density in relation to the supersaturation of the system and have chosen Van der Waals fluid model. Although this is more realistic than previous models, their model still has some idealism and does not account for the presence of gas nuclei in the system. 2.3. Experimental data The formation of bubble nucleus at boiling incipience is hard to observe in most experimental flow visualization studies. However, studies of bubbles in the later stages of growth are not difficult to capture and hence, are commonly found. Hence, understanding the incipience, and development of a nucleation model are even more difficult. In this section, different bubble nucleation and growth experiments found in the literature have been reviewed [20–27]. Most studies in heterogeneous nucleation, visualize photographically, the bubble growth process from its formation to its departure. These images are used in determining bubble size with time. Most studies cover low pressure conditions, with several combinations of temperatures, surfaces and fluids. They encompass pool boiling conditions [20], flow boiling [21], bubbles sliding along the wall [22], etc. However, in all these studies, the embryo size and its formation have not been captured, due to the difficulty in experimentation. For example, the first image captured in the visualization experiments of Situ et al. [21], the bubble size is less than a millimeter in size [21], the smallest being 50–100 lm [20].

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With surface roughness parameters being of sub-micrometer values (e.g.: 0.25 lm in [20]), the sizes of embryos are expected to be even smaller. The complexity only increases with system pressure. There are a few ‘high pressure’ studies using refrigerants [23]. However, these are still relatively very low pressures in the context of fluids such as water. Sakashita [24] captured the growth of bubble in water at 44.7 bar. However, the first observable size of the bubble is 10 lm for 45 bar [24]. Hence, these studies could not present the embryo size. A few studies, in addition to visualizing the bubble growth, also monitor the temperature of the surface. They estimate the likely footprint of the bubble based on the radius of the temperature contours [25]. Recently, Sedmak et al. [26] have attempted ‘submicron’ thermal imaging of pool boiling. The smallest base size they could capture was 10 lm. Jung and Kim [27] also used high-speed video and infra-red camera to monitor the variation in the thickness of the microlayer and the ‘dryspot’ radius as the bubble grows. They measured the microlayer thickness to be of the order of a few micrometers. Although these studies are highly promising, most embryos are expected to be of the order of a micron or lesser which is smaller than that which could be measured. Hence, it should be pointed out that even advanced techniques are unable to capture the embryo size and the solid’s temperature response at the time of its formation. The effect of surface roughness and its composition on bubble formation and growth has been extensively studied. The surface types include smooth [28], hydrophilic [29], superhydrophilic [30], superhydrophobic [31], composite surfaces [32], textured [31,33], etc. Though some insights on bubble growth, coalescence [32] and the effect of different fluids (dielectric liquid [33], refrigerants [32], water [25,30–32], butane [28]) is gained from these studies, no data is available on embryo inception. Some of the studies also attempted to recreate experimental observations through interface tracking/DNS based simulations [7,30,34]. Though most of these studies were largely successful in capturing the bubble growth at later stages, they have to assume an initial seed size for the bubble at the beginning of the simulation [7]. This is an arbitrary value based on mesh considerations and is not the actual embryo size. Hence, no significant insight could be gained from these computational studies with regard to the nucleus formation. Experiments studying smaller length scales such as nanosurfaces and nanobubbles were reviewed to see if they were successful in observing embryo formation in boiling. Nanobubbles are known to be very stable, have long lifetimes and to form generally on hydrophobic surfaces [35–37]. Zhang et al. [38] studied boiling in the presence of nanobubbles and observed that the nanobubbles aided boiling but did not loose their stability in the vicinity of the boiling nucleus. The most widely used technique for observing surface nanobubbles is atomic force microscopy (AFM) [36]. From these studies it was found that nanobubbles have very large liquid side contact angles. Consequently they have a height of only a few nanometers, but a base radius of several microns [35,36]. Nam and Ju [39] and Witharana et al. [40] conducted boiling experiments using nanosurfaces (natural [39] and with artificial cavities [40]). Though they were unable to observe the embryo formation, they have estimated the likely size using the classical theory of nucleation and the superheat required for boiling. Table 1 summarizes the experimental details of some of the important studies mentioned above. On the whole, it can be noticed that there is a lack of experiments to aid clear validation of a model of embryo size that is formed. There are a few experimental studies that visualize the very process of phase change. Though most of these studies do not have any information with regard to the resulting bubble embryo size, they provide significant insight into the underlying physics of the phenomena. For example, Frost and Sturtevant [41], in their

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Table 1 Summary of key experimental studies. Authors

Working fluid

Pool/flow boiling

Details of basic technique

Bubble size

1.

Luke and Cheng [20]

Propane

Pool

0.1–0.5 mm

2.

Situ et al. [21]

Water

Flow

3

Li et al. [22]

Water

Flow

4

Miglani et al. [23]

R-134a

Pool

5

Sakashita [24]

Water

Pool

6.

Gerardi et al. [25]

Water

Pool

7.

Sedmak et al. [26]

Water

Pool

8.

Jung and Kim [27]

Water

Pool

9.

Bon et al. [28]

Pentane Butane

Pool

10.

Sur et al. [29]

Water

Pool

11.

Nam et al. [30]

Water

Pool

12.

Wang et al. [31]

Water

Pool

13

Suszko and El-Genk [33]

PF-5060 dielectric liquid

Pool

14

Borkent et al. [37]

Water

Pool

15

Nam and Ju [39]

Water

Pool

16

Witharana et al. [40]

Water

Pool

High-speed video, semi-automated evaluation methods; observed activation, growth and departure of bubble Flow through heated annulus of 1.73 m length; A CCD camera to visualize the flow; video camera focused on active nucleation site, 2.6 s of images taken for each operating condition Sliding bubbles in rectangular channel visualized; high-speed digital camera with image processing algorithm; bubble numbers, velocities and equivalent diameters monitored Test section is aluminum block with quartz windows; non-invasive TLC based thermography and high resolution canon SLR cameras; temperature field below nucleation site and TLC response monitored, bubbles visualized Test section is cylindrical high-pressure cell with sapphire windows, heating surface is a 8 lm nickel foil; high speed video camera, images manually analyzed using a PC; observed bubble formation, growth and its coalescence; conducted on both horizontal and vertical heater surface Indium Tin Oxide is electrically heated; infrared high-speed camera with simultaneous high speed imaging system used Transient thermal imaging of temperature field beneath a growing bubble performed; test section is erbium-doped, heavy-metal glass, Er:ZBLALiP; high-resolution, fluorescence microscopy used for thermal imaging of nucleation site; variation in intensity of fluorescence emission measured, fluorescence images processed using the custom-developed MATLAB script Simultaneously monitors surface temperature, the liquid–vapour phase distribution, and the microlayer geometry; heater surface comprises of 700-nm-thick ITO film heater on CaF2 plate; high-speed video camera and HeNe laser used, two block prisms placed below CaF2 plate; dynamics of boiling bubble visualized, temperature distribution on boiling surface measured. Performed pool boiling experiments on brass, electro-polished stainless steel, and unpolished stainless steel surfaces; 2 sets of three E-type thermocouples imbedded at incremental distances from the boiling surface; data recorded for 200 iterations using Labview for every instance of measurement Boiling on a silicon wafer studied; high-speed optical imaging and infrared (IR) thermography used; bubble nucleation, growth and departure on a hydrophilic surface monitored, IR temperature measurements used to study the existence of the microlayer near the three-phase contact line Dynamics of single bubbles on a superhydrophilic surface with well-defined nucleation sites studied; polycarbonate test chamber, a quartz window, a base plate and two cartridge heaters forms the setup; high-speed camera captured video images, image analysis software used; bubble ebullition cycle visualized Bubble formation over superhydrophobic-micropatterned copper surfaces investigated; copper block is heat source; a K-type thermocouple was attached to bottom of source and connected to a multi-channel data logger, high-speed CCD camera was mounted on a microscope; bubble formation visualized, surface temperature monitored Transient growth on heated Cu surfaces with circular dimples investigated; high-speed video camera; sequential images of the growing bubbles obtained, the growth rate, departure diameter and detachment frequency estimated Cavitation experiments performed for studying nanobubbles; shock wave generator generated a pressure signal in fluid such that it is focused onto substrate; fiber optic probe hydrophone recorded the pressure signal, CCD camera with long-distance microscope captured images, AFM data acquired using a VEECO/Digital Instruments (DI) multimode AFM; four different kinds of probes used depending on substrate type and procedure of its preparation Photolithography used to prepare islands of smooth Teflon AFÒ of widths of 10–100 m on silicon substrate, setup was a polycarbonate test chamber of 200  93 mm dimension and quartz windows 50.8 mm in diameter, 2 cartridge heaters and K-type thermocouples maintained bulk liquid at saturation temperature, planar thin film heaters and K-type thermocouple wires at back side of the silicon substrate raised (and measured) surface temperature; high speed camera used; observed bubble nucleation and growth on islands Polyetherimide cylindrical boiling chamber; top sealed by thick quartz window and bottom had a silicon wafer, halogen spot lamp placed above the quartz window used to heat silicon wafer; IR camera and gold mirror measured wafer surface, images post-processed with custom MATLAB script and ImageJ; measured surface temperature required for bubble nucleation

50 lm to 1 mm 0.5–2 mm 0.3–1 mm

0.5–2 mm 20 lm

0.5–5 mm

–

2–6 mm

0.1–1 mm

–

0.3–0.9 mm

3–40 nm height 60–300 nm in diameter

–

–

J.S. Murallidharan et al. / International Journal of Heat and Mass Transfer 110 (2017) 921–939

0.01–0.1 mm

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experiment, immersed a droplet at the bottom of a vertical tube that is filled with a non-volatile host liquid. A positive temperature gradient is maintained in the host liquid right from the bottom to the top of the tube. As the droplet rose through the channel, the authors observed that, there was an explosive vaporization at the droplet interface at very high superheats. They found that, during nucleation, pressure waves were generated near the interface and were propagated into the host fluid. In the context of the present study this is indeed a useful insight. 2.4. Summary From the review of work of formation literature discussed in Section 2.2, it is clear that most of the current models have focused on capturing the external factors such as, wall temperature gradient, and line tension. However, the formation of an embryo is an instantaneous process governed mainly by the properties of the gas and liquid involved, and the newly formed interface. It should be pointed out that, among the different models, only Abyzov and Schmelzer [19] have attempted to include the real gas assumption. However they have not done so in the presence of dissolved gases. Most of the studies also do not model surface tension variation with radius. The necessity of the aforementioned modeling choices is explained in Section 3.1. From the discussion on experimental data presented in Section 2.3, it is observed that, a comprehensive comparison and testing at various operating conditions is missing from the literature. This is mainly due to the absence of suitable techniques in visualizing the nucleation phenomenon. However, it can be seen that, although direct measurements are not available for comparison, derived parameters such as temperature and static contact angle can be used to validate the proposed model. Given the above limitations in the open literature, the proposed embryo formation model focuses on aspects such as, surface tension, real gas, and chemical potential rather than modeling the less important external parameters such as, wall temperature variation and line tension. The width and range of applicability of the model is enhanced by covering all sizes, both diffuse and intact gas core bubble (for stable surface nanobubbles) etc. Suitable experiments have been identified for extensive comparison at various operating conditions. The present model has been tested for low and high pressures and for vertical and horizontal boiling conditions as well. Most importantly, through the proposed model, we have offered an explanation for contradictory experimental observations in boiling studies involving surface nanobubble. In Section 3, the mathematical framework of the embryo formation model is discussed. Section 4 presents a comparison of model predictions with experiments. In Section 5, a brief summary and conclusions are presented. 3. Mathematical model In this section, the mathematical framework for the determination of the size of a vapour embryo, formed during heterogeneous boiling, is detailed. This method is based on ‘Gibbs free energy’ i.e. ‘work of formation’ approach. Studies on boiling nuclei, like that of Yarom and Marmur’s [17,42] follow this approach. Their basic model assumes that, initially, an insoluble gas nucleus exists in a pool of liquid. Once boiling starts, vapour is assumed to form at the gas nucleus interface. This vapour is assumed to uniformly mix with the gas inside the bubble as shown in Fig 1. Using this system definition, the stability of boiling nucleus in the presence of an insoluble gas is analyzed. In their study, Yarom and Marmur [17] assume the gas and vapour to be following ideal gas laws and the interface that is newly formed, to have constant surface tension. This is a highly idealized system definition and hence is

Fig. 1. Description of the different stages of the system configuration before and after boiling.

not an accurate representation of reality. Hence, in the present study we propose improvements for realistically modeling the formation of the boiling-nucleus at inception. 3.1. Basis for the modeling assumptions The modeling choices for the present formulation are as follows:  The shape and size of a nucleus influences its formation energy [17,42]. The correct volume of the bubble can only be obtained by treating the bubble as a real gas, as opposed to an ideal gas. Moreover, it has been found that, the flattening of nanobubbles is due to Van der Waals attraction [43]. Since Van der Waals equation of state is well known as a good model for capturing real gas behavior, we have adopted the same in our embryo formation model.  The surface tension of the fluid influences the peak location of the formation energy curve [44]. Recent work on nucleation has accounted for the variation of temperature with surface tension. Infact, surface tension is a function of parameters such as temperature, radius, and density. Since the micrometer – sized embryo is expected to be well submerged within superheated liquid, it is reasonable to ignore the influence of temperature. Hence, no temperature gradient is imposed around the embryo. Furthermore, the liquid temperature around the bubble is assumed to be uniform and it is at the wall superheat. In the experiments of Attard [45], a 42% reduction in surface tension in nano-sized bubbles was observed. Infact, the influence of radius on surface tension in such nano-sized bubbles, was studied by Lewins [46,47]. Furthermore, super saturation around nano-sized bubbles may significantly reduce surface tension [48]. Hence, in this study we incorporate the surface tension model proposed by Lewins [47].  Heterogeneous nucleation occurs predominantly at the interface of a pre-existing gas nuclei trapped in the surface cracks, which is accounted in the present model. Sometimes these pre-existing nuclei are surface nanobubbles, which are found to be highly stable and retain their shape even when they are

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exposed to a boiling-nuclei [38]. More importantly they do not diffuse into the boiling embryo. Hence, our proposed model also accounts for intact gas bubble nucleation. 3.2. Fundamental formulation of the gas-liquid system Consider an isolated system, filled with liquid, at a constant temperature Tl and pressure Pl. A gas bubble is assumed to be in equilibrium with the liquid and is present within this system, at a pressure Pg (Fig. 1). Following the method of Yarom and Marmur [17], the internal energy of the system can be written as follows

U ¼ T l ðSl þ Sg Þ
V l Pl
V g Pg þ þ

X X ðll nl Þi þ ðlg ng Þi i

X ðrAÞj

i

ð2Þ

j

ð8Þ

j

In Eq. (8), the first two terms on the R.H.S denote the work done due to bubble expansion, while the third and fourth terms represent the chemical potential change due to mass transfer. The last term represents the change in surface energy due to bubble growth. For a quasi-static reversible process: @ DG=@r ¼ 0. Hence, change in Gibbs free energy (DG) against radius (‘r’) would have a maximum and/or minimum value [17]. The radial size of the bubble at which the change in Gibbs free energy is maximum corresponds to the unstable critical size of the bubble, while the minimum corresponds to the stable one. 3.4. Properties of gas and vapour

where U – Total internal energy of the system; Sl, Sg – Entropy of liquid and vapour; Vl, Vg – Volume occupied by liquid and vapour; ll, lg – Chemical potential of liquid and vapour; r – Surface energy per unit area; A – Interface Area; i - Chemical species; j - interface (liquid-solid, solid-vapour, liquid-vapour); n- number of moles of the chemical component. The Gibb’s Free Energy of a system is defined as,

G ¼ U
TS þ PV

ð3Þ

where S = Sl + Sg is the total entropy of the system and V = Vl + Vg is the total volume of the system. Substituting (2) in (3):

G ¼ ðPl
Pg ÞV g þ

X X X ðll nl Þi þ ðlg ng Þi þ ðrAÞj i

i

ð4Þ

j

where G represents the Gibbs free energy for a system filled with liquid, which is in equilibrium with a gas bubble. In the following sections, the application of this approach to boiling scenarios will be detailed. 3.3. Formulation of gas-vapour boiling system The system currently comprises of a gas bubble in equilibrium with the liquid (stage 1 in Fig. 1). The Gibbs free energy for stage 1 is same as Eq. (4).

GO ¼ ðPOl
POg ÞV Og þ lOl nOl þ lOg nOg þ

X ðrAÞjO

ð5Þ

j

where O denotes conditions before boiling. At this point boiling commences at the gas-liquid interface. This implies that, a portion of the surrounding liquid is converted into vapour. The bubble is now assumed to comprise of a uniform mixture of gas and vapour (stage 2 in Fig. 1). The Gibbs free energy for stage 2 is as follows,

G ¼ ðPl
Pv ÞV v þ ll nl þ lg ng þ lv nv þ

X

ðrAÞj

POl ¼ P l ¼ Pl ;

lOl ¼ ll

3.4.1. Lewins model of surface tension variation It has been shown through the Schroedinger wave equation that surface tension comes into existence only after 8–9 molecules of water gather together [47]. Hence, it is clear that surface tension varies with the size of the bubble. This is especially true for bubbles of very small sizes such as, vapour embryos or surface nanobubbles, which initiate heterogeneous boiling. The variation of surface tension with radius and temperature is given as [47],



rðr; TÞ ¼ r1 ðTÞ 1
exp

  r rX

ð9Þ

In this model, the surface tension tends to zero as r ? 0, and at large bubble sizes, the value reaches the plane surface value. In the formulation, ‘rX’, the incipient radius, is the only source of empiricism. The value of this term, known as Lewins parameter, is obtained by following a mechanistic approach. This parameter essentially sets the lower limit for the radius at which surface tension comes into effect and prevents the model from predicting a zero surface tension. Lewins calculates the gas spinoidal point for a given set of operating conditions for arriving at this value. For calculating the gas spinoidal location, one needs the equation of state of the gas. For this purpose, the Van der Waal’s equation is chosen. This procedure is further elaborated in the following section. 3.4.2. Van der Waals equation The Van der Waals equation is the simplest analytical real gas model. It can be formulated in non-dimensional form as follows,

8 T cr P 3 ¼
 2 Pcr 3 ttcr
1 t

ð10Þ

tcr

where ⁄ denotes after boiling. We use ‘g’ as the subscript for before boiling and use ‘v’ for after boiling. This would account for the fact that, before boiling the bubble is only filled with gas, while after boiling a mixture of vapour and gas exists within the bubble. It is to be noted that, the entire transition from stage 1 to stage 2 is assumed to be a quasi-static reversible process. For the transition process stated above, the following relations are valid,

nOg ¼ ng ;

In the embryo formation model, both the gas and vapour are assumed to be real gases. Here, the gas and vapour molecules are assumed to adhere to the Van der Waals equation of state. Additionally, the surface tension of the bubble interface is assumed to vary with the bubble size and Lewin’s model [46,47] is further adopted.

T

ð6Þ

j

nOl ¼ nl þ nv ;

DG ¼ ðPl
Pv ÞV v
ðPl
POg ÞV Og þ ðlg
lOg Þng þ ðlv
ll Þnv X þ ððrAÞj
ðrAÞOj Þ

ð7Þ

From Eqs. (5) and (6), the change in Gibbs free energy (DG) due to boiling can be written as follows:

where subscript ‘cr’ denotes the values at the critical point. A qualitative pressure versus specific volume plot of the Eq. (10) is shown in Fig. 2. The equation when plotted passes through two points of inflection. These points of maxima and minima are known as gas and liquid spinoides respectively. The spinodes are given by the following cubic equation,

@P= @ t=

!

¼ T

6 ðt

= Þ3




24T = ð3t=
1Þ

2

¼0

ð11Þ

Here P = ¼ PPcr , T = ¼ TTcr , v = ¼ vvcr refer to the non-dimensional pressure, temperature and specific volume respectively.

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Vg ¼

Fig. 2. Qualitative representation of pressure – volume variation for a real fluid and location of the spinode points.

The region between the two spinodes is considered unstable, which is extensively discussed in theories addressing nucleation [49]. Lewins [47] suggests that at a given temperature, the gas spinode represents a location beyond which, only liquid can exist. This implies that the gas spinoide is the maximum pressure at which the vapour can exist. This consequently means that the gas spinoidal pressure is the upper limit of the overpressure that can exist in the bubble. Hence, the hypothesis adopted in this study is that, the vapour embryo is first formed at this limited overpressure. From this upper limit of the pressure value, the ‘rX’ value can be calculated using the Laplace equation (Eq. (12)).

where h is the contact angle of the bubble with the surface. Consequently, Eq. (13) and Van der Waals equation (Eq. (10)) are simultaneously solved to obtain the initial gas bubble radius (rO). We notice that, while solving the Van der Waal’s equation, three roots will be encountered, of which the root corresponding to the gas spinoide location is chosen. From the specific volume at this location, the pressure inside the initial gas bubble and its radius (rO) can be determined. Eqs. (10) and (13) are solved iteratively until convergence is obtained for the initial gas bubble embryo size, which is constrained by the initially assumed number of moles of gas.  Chemical potential variation term: The terms, which denote the chemical potential variation in Eq. (8), is ðlg
lOg Þng þ ðlv
ll Þnv . The former term denotes the change in potential of the gas due to boiling, while latter refers to the change in potential in the vapour due to its phase change. The change in chemical potential at constant temperature conditions is given by dl ¼ v dP. The integral for the first term is re-written in terms of specific volume as follows

ðl
l  g

Z O gÞ

¼

dl ¼

Z

P v

PO g

mdP ¼ Pcr v cr

Z



ðP= Þg O ðP = Þg

m

v cr

 d

P Pcr



ð14Þ Eq. (14) can be rewritten using Eq. (11) as follows:

ðl
l  g

v cr

Z ðv = Þg

"

ðv = Þg

O

v

6

24T =

#

dv 2 ð3t=
1Þ ! " #ðv = Þg !
8T = lnj3v =
1j 8T = 6 þ ¼ Pcr v cr
= 3 9v =
3 v =O

O g Þ ¼ P cr

=

ðt= Þ

3




=

ð15Þ

In order to solve Eq. (8), several unknown terms need to be closed. The procedures followed for the same are detailed as follows.  Determination of incipient size of surface tension: This is the bubble size at which surface tension begins to act on the interface. This is calculated for both gas and vapour by computing the gas spinoide using Eq. (11). The roots of the cubic equation (Eq. (11)), i.e. the specific volume, which are positive and whose value is greater than 1/3 is only meaningful [50]. These roots are identified and the largest amongst the roots is defined as the gas spinoide specific volume. The Van der Waals equation (Eq. (10)) is then used to compute the vapour pressure for the gas spinoidal specific volume. The incipient radius is now calculated using the Young-Laplace equation,

2r 1 rX

ð13Þ

ðv Þg

3.5. Closure relations

Pincp
Pl ¼

1 3 pðrO Þ ð1 þ cos hÞ2 ð2
cos hÞ 3

ð12Þ

Once rX is known, we can calculate the surface tension at any radial size of the gas or vapour bubble using Eq. (9). It is to be noted here that, after boiling commences, it is assumed that the gas is completely diffused inside the vapour bubble. Hence, once boiling commences, the surface tension value for the mixed gas-vapour bubble is assumed to be same as that of a pure vapour bubble.  Determination of initial gas bubble radius: The number of moles of gas (nO g ) that is initially present in the system, is considered to be a known value. Given nO g , and assuming the bubble to be spherical (or truncated sphere if it is on the wall), the volume of the gas bubble can be formulated as

For term 2, which gives change in potential due to vaporization, a similar procedure is adopted. When there is phase change, the chemical potential of both the liquid (lls ) and vapour (lv s ), at the saturation temperature, is considered to be equal. This property enables us to rewrite the second term as follows:

 ðlv
ll Þnv ¼ ðlv
lv s Þ
ðll
lls Þ nv

ð16Þ

However, we already know that,

ðll
lls Þ ¼ v l ðPl
Psat Þ

ð17Þ

ðlv
ll Þ can be computed similar to Eq. (14). Using the above discussion, the number of moles of vapour (nv ) can be obtained as follows. When boiling commences, the gas – vapour bubble (rv) is assumed to grow from its initial gas-bubble size (rO). This is subsequently incremented by 0.1% of bubble radius as, rv = rO + dr. Using current vapour bubble radius (rv) both pressure and specific volume of gas – vapour mixture are computed. This in turn enables us to obtain the total number of moles of gas and vapour (nv þ ng ). Since number of gas moles (ng ) is already known, the difference between the two gives nv .  Surface area based energy term: Surface energy is the product of surface tension (r) and the surface area (A) denoted as (rA). Generally, studies use the value of surface tension (r) computed from a plane surface extension test (r1 ), as 4pr 2 r. However, if the surface tension were to vary with the radius of the curve, using 4pr2 r would be erroneous [47]. Hence, Lewins [47] proposed using D(rA). For a bubble growing at the wall, with h being the contact angle, the area is defined as follows,

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A¼

p 4

r2 ð8 þ 9 cos h
cos 3hÞ

ð18Þ

The change in surface energy for an initial gas bubble of size rO to grow to a gas-vapour bubble of size rv is as follows: The surface energy of the gas-vapour bubble at radius rv,

p ðrv l Av Þ ¼ r1 v l ð8 þ 9 cos h
cos 3hÞ 4   !# "  v 
rv r r 2 e X vl  ðr v Þ
2r 2X v l 1
1 þ rX v l ð19Þ O

The surface energy at radius r of the gas bubble is given as,

ðrgl AÞO ¼ r1

gl

p ð8 þ 9 cos h 4 " O 2


cos 3hÞ ðr Þ
2r X



2

gl

1
1þ

rO rX

 
rO  !# r e X gl

gl

ð20Þ Note: rv l denotes surface tension between gas-vapour mixture and the liquid and rgl denotes surface tension between gas and liquid. Hence, the change in surface energy due to heterogeneous boiling is given by

DðrAÞ ¼

X ððrAÞj
ðrAÞOj Þ

ð21Þ

j

 Expansion work term: The expansion work term is computed as follows:

DW exp ansion ¼ ðPl
P v ÞV v
ðPl
P Og ÞV Og

ð22Þ

The values of the pressure and volume can be calculated from the bubble radius, the bubble volume expression, the number of moles of gas and vapour (ng ; nv ) and the Young-Laplace equation (Eq. (12)). 3.6. Solution procedure 1. Initialize the conditions of the system such as, pressure (Pl), temperature (Tl), the initial number of moles of gas (nO g ) and the contact angle (h) of the bubble with the wall. 2. Using Pl and Tl, calculate the radius (rX) at which the concept of surface tension becomes meaningful. 3. Calculate the initial gas bubble size rO using Eqs. (10) and (13). 4. Increment the bubble size by small amounts i.e. r v ¼ rO þ dr. Here rv denotes the size of the gas-vapour bubble as it grows. 5. For every rv, calculate the corresponding vapour pressure Pv⁄ and volume Vv⁄ using Young’s Laplace Equation and Eq. (10). 6. Calculate the number of moles of vapour (nv ) present in the newly formed gas-vapour bubble. 7. Calculate the chemical potential (Eqs. (14)–(17)) and the surface energy (Eqs. (19)–(21)). 8. Calculate the expansion work using Eq. (22). 9. Calculate change in Gibbs free energy ðDGÞ due to boiling. 10. Repeat steps (5)–(9) for the next value of r v until convergence. Free energy denotes the thermodynamic potential that is required for a gas/liquid to move from one state to another. Fig. 3 is a qualitative depiction of change in free energy change required for the formation of the bubble radius. On plotting DG vs. r, the stable and unstable radius of the gas-vapour bubble can be identified. As mentioned in Yarom and Marmur [17] this curve

Fig. 3. Qualitative representation of the 3 types of variation of Gibb’s free energy with bubble radius based on the liquid saturation ratio (LSR).

has a minimum and a maximum point of inflection depending on the ratio of

Pl . Psat

If

Pl P sat

> 1, then a stable radius is possible and the

bubble is said to be in stable equilibrium. If

Pl P sat

< 1 (which refers

to boiling), it is likely that an unstable radius is formed. The bubble radius corresponding to the maximum change in Gibbs energy, if it exists, denotes the unstable radius of the bubble. Due to some perturbations in the system, if the bubble becomes slightly bigger at this point, then it continues to grow. On the other hand, if it becomes slightly smaller, then the bubble shrinks. Hence, for a vapour embryo formed at the wall, to be able to grow and depart from the wall, its radius should be equal to the unstable critical radius as indicated in Fig. 3. In the present study, we propose using this unstable critical radius instead of the ‘radius of cavity’. Note that, the latter is popularly used in macroscale boiling studies to determine the boiling nuclei. Since in this formulation, the initial gas bubble is assumed to diffuse into the boiling embryo, it is named as ‘Diffused-Real Gas Core’ (D-RGC) model. However, if the initial gas bubble is intact, the ‘Intact-Real Gas Core’ (I-RGC) model comes into picture. 3.7. I-RGC model If the initial gas bubble remains intact even after boiling begins, the vapour covers the gas bubble core and the interface between vapour and gas also remains intact (see Fig. 4). Gibbs free energy of such a system can be obtained as follows,

Fig. 4. Representation of formation of vapour phase over the intact gas bubble in the I-RGC model.

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The total internal energy of the system is given by,

U ¼ T l ðSv þ Sl þ Sg Þ
P l V l
Pv V v
Pg V g þ ll nl þ lv nv þ lg ng þ rv l Av l þ rv g Av g

ð23Þ

ðrv g
rOgl ÞAv g þ ðrv l Av l Þ

The total Gibbs free energy of the overall system is given as

G ¼ U þ Pl ðV l þ V v þ V g Þ
T l ðSl þ Sv þ Sg Þ

ð24Þ

Substituting Eq. (23) in Eq. (24), will give Gibb’s free energy of the system after heterogeneous boiling begins

G ¼ ðPl
Pv ÞV v þ ðPl
Pg ÞV g þ ll nl þ lv nv þ lg ng þ ðrv l Av l þ rv g Av g Þ

ð25Þ

Gibbs free energy of the system prior to boiling is same as Eq. (5) O

G ¼

ðPOl




POg ÞV Og

þl

O O l nl

þl

O O g ng

þ ðrgl Agl Þ

O

lOg ¼ lg ; AOgl ¼ Agv ; V Og ¼ V g

ð27Þ

Thus the change in Gibb’s free energy due to boiling can be written as follows:

DG ¼ ðPl
Pv ÞV v þ ðlv
ll ÞnV þ ðrv g
rOgl ÞAv g þ ðrv l Av l Þ

ð28Þ

3.7.1. Closure relations Since, the properties of the fluid and the interfaces are essentially same as that described in Section 3.4, the procedure for the determination of the incipient size of surface tension and the initial gas bubble radius is also similar to that described in Section 3.5. However, the following additional differences must be accounted.  Chemical potential variation term: Since the gas bubble core remains intact the ‘change in potential’ due to the diffusion of gas is not present. The term that needs to be calculated is only ðlv
ll Þnv and this is computed in Section 3.5. In calculating this term, the number of moles of vapour required, is determined as follows. The Young-Laplace equation is first applied to the vapour–liquid interface that is formed after boiling, with a radius of curvature rv. Using this, pressure P⁄v on the vapour side is calculated. The Van der Waals equation is now solved for this vapour pressure to obtain the specific volume of the vapour. This specific volume is essentially the volume occupied by one mole of vapour. Hence the volume of the annular shape occupied by the vapour is divided by the specific molar volume, to give the number of moles of vapour present.

h 

nv ¼

ðr v Þ3
ðrO Þ

3

i

p ð1 þ cos hÞ2 ð2
cos hÞ 3

v v

ð30Þ

Here, the only additional term when compared to the D-RGC case is the surface tension between the vapour and the gas. This surface tension suddenly comes into effect in the system at the time of the start of heterogeneous boiling, i.e. when the vapour starts to form on the gas bubble. This surface tension variation is assumed to follow Lewins model, which is computed as follows.

ðrv g ÞAv g ¼ r1 v g

p 4

ð8 þ 9 cos h "

  
rO  !# rO 2 r e X vg
cos3hÞ ðr O Þ
2r2X v g 1
1 þ rX v g

ð26Þ

Additionally, the relations in Eq. (7) is valid. Since the gas core does not change in size or shape, the following relations are also assumed to be valid

POg ¼ Pg ;

sion between gas and liquid will be different from that between gas and vapour. Consequently, for the I-RGC case, there is a different interface energy term in Eq. (28), when compared to the D-RGC model (Eq. (8)):

ð31Þ The two unknowns in the above equation are r1 v g and r X v g . It is assumed that the initial gas bubble is unchanged before and after boiling, i.e. the pressure inside the gas bubble and its size does not change before and after boiling, hence we can write Before boiling

P Og
Pl ¼

2rgl rO rO

ð32Þ

and after boiling

P g
Pv ¼

2rv g rO

rO

ð33Þ

At the liquid-vapour interface,

P v
Pl ¼

2rv l rv

rv

ð34Þ

In combining the three equations, we get the following relation

ð rv g

 rO Þ

¼ ðrgl

rO Þ

O


ðrv l

rv Þ



 O r rv

ð35Þ

This indicates the surface tension at the initial gas bubble radius rO. Using the assumption that r X gl ¼ r X v g ; r1 v g can be obtained. It is also possible to calculate r1 v g from the Lewins surface tension model.  Expansion work term: The expansion work term only comprises of ðPl
Pv ÞV v and can be easily calculated as mentioned in Section 3.5. This modified formulation of the proposed model is applied when the initial gas bubble remains intact and does not diffuse into the boiling embryo. This will be referred to as ‘Intact - Real Gas Core’ (I-RGC) in the present formulation.

ð29Þ

 Surface area based energy term: In the I-RGC model, the nucleation occurs, on the interface between the intact gas core, and the liquid bulk. This nucleation around the gas bubble is possible only when a specific number of molecules of liquid get converted into vapour. Below which, the nucleation event is not energetically favorable. Hence, as soon as nucleation begins, the system can be visualized as shown in Fig. 4. Here, it can be noticed that, the original gas-liquid interface prior to boiling is replaced by the newly formed gas-vapour interface. It is assumed in this study that this replacement is instantaneous. It is also assumed that, this instantaneous change will be reflected by a change in surface tension value from rgl to rv g . This is indeed physically accurate as well, since the surface ten-

3.8. Notes on adherence of model to physics  In the ‘work of formation’ method as well as in the classical nucleation theory (Eq. (1)), the system is assumed to be at constant temperature and pressure. However, in experiments such as Frost and Sturtevant [41], it was found that, at the time of nucleation, pressure waves propagate outwards from the interface towards the liquid domain. According to their study, the rise in pressure was around five times the initial pressure. Hence, embryo formation is not a constant pressure phenomenon. Consequently, in this study, though the temperature is maintained constant, the liquid pressure near the interface is not kept constant. The properties of the fluid are obtained corresponding to this constant temperature and its saturation pressure. During our computations it was found that, the value

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of the liquid pressure near the interface varied between this saturation pressure and the system pressure. It was also found from our study that, with increase in pressure, the range of values that the liquid pressure could take is reduced. This observation is corroborated by the experiments, where the pressure wave reduces in magnitude at higher pressure [41].  The ideal gas assumption simplifies the mathematical complexity involved in the calculation of chemical potential term. In fact, in some elementary studies, real gases are treated as ideal gases by using the fugacity of the real gas [51]. Fugacity of a real gas is essentially the pressure of the ideal gas whose chemical potential is equal to that of the real gas. Since an ideal gas treatment can be retained, it greatly simplifies the mathematical formulation of the problem being studied. However, the problems for which this treatment has been applied are very elementary. Moreover, the application of fugacity brings in additional terms of empiricism pertaining to the specific species/type of gas present in the system. This is perhaps possible and acceptable for a very simple study, but not in the context of boiling incipience involving pressure waves. The complexity of the problem only increases when one takes into account the presence of the wall and the possibility of different species of dissolved gases being present with the vapour. However, we believe the use of fugacity makes the formulation unnecessarily complicated without the assurance of better and physically accurate prediction. Moreover, there is a danger of too much empiricism, which would make it less accurate. Hence, in the present model, we have chosen to retain the real gas treatment. Hence, we propose to better capture (a) the slope of the l vs. P curve of vapour (b) value of Pv and the relative positions of Pv, Pl and Psat in the P vs. V plot, etc.  At the time of boiling incipience, pressure waves are found to deform the interface [41]. As a consequence, the surface tension is expected to fluctuate at this point. This is however too complex to capture using a simple model. Almost all studies in the literature use the surface tension value corresponding to a flat interface. We believe, this is too simplistic and is inaccurate. Hence, in the present study, the Lewins model [47], which has all the necessary mechanistic features, is implemented. 4. Results and discussion 4.1. Model validation The work of formation comprises of three components viz. work of expansion, surface energy and chemical potential terms. Their

(a) Fig. 5. (a) Components of total ‘work of formation’ for D-RGC with Marmur [17].

variation as a function of radius is plotted in Fig. 5a for the test condition ng = 10
16, T l = 373 K and

Pl P sat ðT l Þ

of 0.65. In this plot, the

surface energy increases exponentially, while the expansion work term decreases exponentially. However, it can be noticed that, the chemical potential energy term has a dominant influence on the total work of formation. The main physical criteria that validates the present study is the location of change in chemical potential of the liquid. Infact, this state indicates the occurrence of phase change. This corresponds to the maximum work of formation, which in turn represents the formation size of the embryo. This is also corroborated by the observations of Yarom and Marmur [17]. The mathematical formulation discussed in Section 3, has its basis in the approach adopted by Yarom and Marmur [17]. Hence we choose similar test conditions of their study for a comparison exercise. To this end, ng = 10
16, T l = 373 K and

Pl P sat ðT l Þ

of 0.49, etc.

are chosen. As described earlier, it is valid to assume that the initial embryo is completely submerged in superheated liquid for real life conditions. Hence, present computations are conducted assuming the temperature of liquid where embryo forms to be uniformly superheated at T l . The fluid properties and P sat are determined for this liquid temperature. The variation of the energy required for the formation of the boiling-nucleus against radius is presented in Fig. 5b. It can be seen that, the formation energy (DG) has two extrema, one maximum and one minimum. These two locations correspond to dG ¼ 0, which dr essentially represent the equilibrium positions of interest, that represent the embryo size. The minimum extrema refers to stable equilibrium, while the maximum refers to the unstable equilibrium. Hence, a bubble with radius corresponding to this maximum location is said to be in unstable equilibrium. Even with a very small disturbance in the system, the bubble will either continue to grow unrestricted or will collapse depending on, whether the bubble radius marginally increases or decreases due to this disturbance. This unstable equilibrium radius is of interest in bubble inception and growth studies. This is because, the initial boiling embryo is the one that starts to grow unrestricted into a macroscale bubble at the wall. This unstable equilibrium radius/size of the bubble is equivalent to the bubble cavity radius of bubble computed by classical nucleation theory in macroscopic boiling studies. From Fig. 5b, we can see that the bubble embryo radius predicted is smaller in comparison to the predictions given in Yarom and Marmur’s [17]. We believe that, our real gas bubble assumption should indeed predict a smaller bubble size. It should be pointed out that, ideal gas assumption of Yarom and Marmur [17] does not take into account

(b) Pl P sat ðT l Þ

= 0.65. (b) Comparison of predicted variation of ‘work of formation’ with radius with Yarom and

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the attraction forces between the molecules of gas and vapour. This results in the gas bubble size predicted by them to be bigger. Additionally, the energy required for forming a real gas bubble of equivalent size as that of ideal gas bubble, is higher, which is predicted in our computations. It also follows that, a larger real gas bubble will have more pressure and will require more energy for its formation l as depicted in Fig. 5b for PsatPðT = 0.59. Þ l

For the condition where the initial gas bubble remains intact and does not diffuse into the boiling embryo, the I-RGC formulation is used. The current test condition predicts that the energy required for the embryo to form is higher when the gas bubble is intact, compared to when the initial gas bubble diffuses into the embryo. However, the boiling will occur, only if this excess energy is available in the system. Also, though the formation energy required is different, the embryo radius is nearly the same for DRGC and I-RGC models. 4.2. Nucleation in pool boiling It is well known that nucleation in pool boiling commences with the formation of embryos in the surface crevices. The classical nucleation theory predicts the embryo size, by making the assumption that it is of the same size as that of the cavity. To this end, the predictions of the present model are tested under low and high pressure conditions. 4.2.1. Low pressure conditions In the pool boiling literature, most of the experimental data is confined to either water or refrigerants. Under low pressure conditions, Miglani et al. [23] have measured temperature fluctuations for an isolated nucleation site using the refrigerant R134a. The details of the chosen test conditions against the predictions of the present model are given in Table 2. From their visualization studies, the first radial data point at t = 0 ms, is the smallest bubble base radius that could be measured and was found to be 105 lm in size. This value is several times larger than the minimum embryo radius of 64 nm as predicted by our model in Table 2. This value is very close to the embryo radius predicted by the classical nucleation theory, which is 100 nm. Similar experiments were conducted by Duan et al. [34]’s using water, to study bubble nucleation, growth and its detachment. The system parameters that have been chosen for the test case is as detailed in Table 3. The first data point measurement of the radius, for this test case, was found to be approximately 0.25 mm. However, using the classical nucleation theory the embryo radius is obtained as 3.63 lm.

This compares well with our model, which predicts a base radius of 3.11 lm. Hence our model is able to predict nucleation at low pressures for both water as well as refrigerants. 4.2.2. High pressure condition The embryo formation model at high pressures, is assessed against the experiments of Sakashita [24] and the test conditions are detailed in Table 4. However, for this evaluation, a direct comparison between the experiments and computations is not possible, as the first measurable value is several orders higher than the embryo size. Moreover, the applicability of classical nucleation theory for these pressures is also not well established. Hence, in order to assess our model, the predictions are first analyzed, if with increase in pressure, the bubble growth rate is slower. It can be noticed from Table 4, that the predictions of the proposed model generally satisfy the formation of smaller bubble size, with increase in pressure. Trend wise this is comparable to the classical nucleation theory. However, a careful examination of Table 4 reveals that, our present model predicts same bubble size for 2 pressures 22.3 bar as well as 31.7 bar, while the classical theory predicts a smaller bubble size for the higher pressure (31.7 bar). To be able to find detailed answers, we perform computations over the entire pressure range of 1–180 bar. It was observed that, the present computations predict discrete jumps in embryo size unlike the classical nucleation theory. To analyze the detailed reasons, consider a bubble in an infinite medium subjected to two different wall superheats (2 K and 5 K). Table 5 and 6 present the work of formation and embryo size predictions for these two values of constant superheat. It can be noticed that, as a whole with increase in pressure Psat (Tl), the embryo size as well as the formation energy decreases. It can be seen that at higher superheats, smaller boiling-nuclei are formed (e.g.: 1.01325 bar and 15.179 bar of Table 5 and 6), correspondingly formation energy is also lesser. From Table 6 it can be noticed that, the entire pressure range of 70–120 bar, predicts the same size of boiling-nucleus. The question arises as to how this is possible. Whilst performing these computations, it became apparent that the initial number of moles of gas that is involved in the formation of the minimum-sized nucleus plays an important role in this phenomenon. It is found that for a given set of conditions, a specific embryo size is energy-wise favorable. To migrate to the next combination of higher ng with a bigger nucleus size, the energy required is higher. The system might not possess this required energy, till it reaches a specific pressure. Hence, till that pressure is reached, the system repeatedly forms the same nucleus

Table 2 Prediction of the boiling-nucleus size for the test condition of Miglani et al. [23] using D-RGC model. D-RGC model - h = 35° Range of radius of vapour nuclei Surface temperature (K)

Global superheat (K)

Initial quantity of gas (moles)

Pl P sat

Initial gas bubble radius r° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

311

6.24

10
20

0.85

12.120

0.112

2.185E
4

0.064

Table 3 Prediction of the boiling-nucleus size for the test condition of Duan et al. [34] using D-RGC model. D-RGC model - h = 85° Range of radius of vapour nuclei Surface temperature (K)

Global superheat (K)

Initial quantity of gas (mole)

Pl P sat

Initial gas bubble radius r° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

382

9

10
19

0.73

27.87

3.127

2.505

3.11

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Table 4 Prediction of boiling-nucleus size for Sakashita’s [24] study. Minimum radius of boiling-nucleus for h = 60° (nickel surface: 35° < h < 80°)

incipience

Local superheat (K)

Initial gas bubble radius r° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8 D-RGC model

Work of formation (DG/KT) ⁄ 10
8 I-RGC model

Bubble radius in experiment r (lm)

Classical nucleation theory (lm)

0.94 0.97 0.97

0.94 0.97 0.97

2.69 2.31 1.82

9.17 9.00 8.75

0.509 0.509 0.408

4.37e
2 3.77e
2 5.42e
2

4.42e
2 3.82e
2 5.46e
2

0.509 0.509 0.408

0.545 0.432 0.349

0.91 0.93

0.91 0.93

4.70 4.01

9.26 8.99

0.309 0.209

2.10e
2 6.58e
3

2.14e
2 6.95e
3

0.309 0.209

0.348 0.212

Surface temperature (K) (Pl)

Global superheat (K)

Initial quantity of gas (moles)

Global

Horizontal 494 (22.3 bar) 512 (31.7 bar) 531 (44.7 bar)

2.69 2.31 1.82

10
20 10
20 10
20

Vertical 488 (19.1 bar) 518 (34.0 bar)

4.70 4.01

10
20 10
20

LSR

Pl P sat

Local LSR Pl Psat

at

Table 5 Prediction of minimum boiling-nucleus size for 2 K superheat and at high pressures. Minimum radius of boiling-nucleus for h = 0° and 2K superheat (D-RGC model) Surface temperature (K)

Global superheat (K)

Pl (bar)

375 474 533 561 579 600 618 632

2 2 2 2 2 2 2 2

1.01 15.18 45.31 70.02 90.64 120.27 151.31 179.69

Global LSR

Pl Psat

0.94 0.96 0.97 0.97 0.97 0.97 0.98 0.98

Pl Psat

Initial quantity of gas (moles)

Local LSR

incipience

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

10
18 10
19 10
20 10
20 10
20 10
20 10
21 10
21

0.940 0.960 0.970 0.970 0.970 0.983 0.990 0.996

18.06 1.12 0.31 0.11 0.11 0.11 0.10 0.10

1.54e+2 3.37e
1 1.56e
2 1.91e
3 4.11e
4 3.06e
5 3.83e
5 1.29e
5

at

Table 6 Prediction of minimum boiling-nucleus size for 5 K superheat and at high pressures. Minimum radius of boiling-nucleus for h = 0° &5K superheat (D-RGC model) Surface temperature (K)

Global superheat (K)

Pl (bar)

378 477 536 564 582 603 621 635

5 5 5 5 5 5 5 5

1.01 15.18 45.31 70.02 90.64 120.27 151.31 179.69

Global LSR

Pl Psat

0.84 0.90 0.92 0.93 0.93 0.98 0.94 0.94

Pl Psat

Initial quantity of gas (moles)

Local LSR

incipience

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

10
18 10
19 10
20 10
20 10
20 10
20 10
21 10
21

0.840 0.900 0.920 0.941 0.966 0.985 0.991 0.997

6.02 0.42 0.11 0.11 0.11 0.11 0.10 0.10

1.69e+1 4.21e
2 1.56e
3 9.24e
6 6.28e
6 2.01e
5 3.00e
6 5.16e
6

size with the same ng. However, it should be noted that for the same nucleus size at higher pressure a lower formation energy is sufficient. Another point to be noted here is that, Tables 4–6, use two terms for Pl/Psat i.e. Global LSR (Liquid saturation ratio) and Local LSR. This refers to the liquid pressure maintained globally in the system and the liquid pressure that is required locally near the bubble interface. The significance of this will be discussed in detail in Section 4.3. 4.3. Nucleation on nanosurfaces It is observed from literature that, classical theory of nucleation is widely used in macroscale boiling problems. With recent advancements in nanotechnology, boiling on nanoscopically polished surfaces has gained prominence. However, the use of the classical nucleation theory to such surfaces has generated contrasting results. (For example, see the studies of Nam and Ju [39] and Witharana et al. [40]). In this section, the present model is evaluated for the width of applicability of classical nucleation theory. According

at

to Nam and Ju [39], the classical nucleation theory predicts to be an order of magnitude higher than what was observed on their experiments. Hence, they have proposed a modification by accounting for the contact angle effect. However, Witharana et al. [40] have disagreed with this claim and pointed out that, their results are likely to be erroneous due to deposits/dust on the surface and are not an actual representation of the physics. They have stated that, the classical theory accurately predicts nucleation on nanosurfaces. To this end, we investigate both the claims found in the literature. To start with, the similarities and differences between the two experiments are as follows. i. Both are experiments conducted in water on nanosurfaces with atmospheric test conditions. ii. The heating surface in Nam and Ju [39] has hydrophobic islands of Teflon AF on a silicon substrate, while Witharana et al. [40] have used a pure silicon substrate. The static contact angle of water with the former is >120° ± 2°, whilst for the latter it is <10°.

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was found in their experiments that the bubbles nucleate, when the liquid temperature near the surface is at 220 °C. In the present computation, Psat (Tl) is set as 23.12 bar, while the liquid pressure is initially set at the chamber pressure of 1 bar i.e. LSR = Pl/Psat = 0.04. By setting the initial number of moles of gas that is present in the system as ng = 10
20 moles, the D-RGC model predicts an initial gas bubble radius greater than rX. Fig. 6 depicts the work of formation required against the bubble size. Though the gas bubble as well as the boiling embryo size is bigger than the lower limit, the profile of the Gibb’s free energy variation with radius is not satisfactory for LSR = 0.04. The curve does not have a positive maxima, implying that an unstable radius is not possible for this LSR. Hence, Pl/Psat ratio is increased to verify if nucleation occurs at higher values of Pl/Psat. It can be noticed that, Pl/Psat = 0.61 (Pl = 14.12 bar) is the lowest pressure at which a boiling-nucleus is formed. From Table 8, it can be observed that, the D-RGC model predicts an embryo radius of 0.11 lm (with initial gas bubble radius of 8.97 nm). The work of formation required for the D-RGC and IRGC models is presented in Table 9. Here, the free energy required for the formation in the I-RGC case was found to be higher. For the present test condition i.e. a constant surface temperature of 220 °C and global pressure Pl = 1 bar, the experiments [40] have observed bubbles forming on the heater surface. However, as indicated by our computations (for LSR = 0.04), no nucleation is possible if the local fluid pressure near the heated surface is at 1 bar. For nucleation to occur, we have predicted that, the local pressure of the fluid needs to be at least 0.61 ⁄ Psat (Tl) (i.e. 14 bar). We surmise that, this pressure jump occurs through the heterophase fluctuations. Such pressure waves have indeed been observed in the experiments of Frost and Sturtevant [41], which further substantiate the present LSR value. The variation of pressure against volume for a pure substance such as water is indicated by the P-V diagram in Fig. 7. The local pressure Pl = 14.12 bar would lie in the metastable region of the curve (AB) of an isotherm. In this region, the liquid has a tendency to easily transform into vapour. Consequently, even a minor disturbance results in the transformation of the liquid into an embryo (from C to D). By drawing a horizontal at Pl = 14 bar, we find that

iii. Nam and Ju [39] have used a natural surface with the maximum peak-to-peak height of 5.20 nm, on the other hand, Witharana et al. [40] have employed artificial cavities of varying sizes. To this end, we test if these differences in experimental conditions were responsible for the contrasting observations. 4.3.1. Assumptions for computations Witharana et al. [40] conducted pool-boiling experiments in a cylindrical test chamber. Its top was sealed with a quartz window, with a heater at the bottom surface and covered by a silicon wafer. These wafers (with and without cavities) were mounted and temperature at which the bubbles nucleate was measured with the help of an IR camera. However, the authors have pointed out that, temperature measured is not of the surface, but refers to the average temperature of a thin layer of fluid (200 lm thickness), just adjacent to the silicon wafer. Based on the description of the experiment as well as the conclusions of Witharana et al. [40], the following settings have been adopted in our computations.  Witharana et al. [40] observed random surface nucleation, even at locations other than the cavity and independent of surface temperature. This can certainly be attributed to some residual gas trapped in the surface crevices. Though the set-up has been degassed, it is almost impossible to completely degas the system. Hence, our computations are performed with the assumption of minimal amount of gas entrapment, to be able to predict the formation of the smallest boiling-nucleus.  In our computations, the temperature of the liquid (Tl) is set at the superheat temperature. Consequently, the properties of the fluid are also determined at this temperature. However, the entire liquid within the chamber is not at the superheat temperature. Due to differential temperature natural convection is bound to occur. The heat input, the temperature variation and other inhomogeneities due to liquid motion is bound to induce local heterophase fluctuations [52]. These fluctuations are indeed essential in order to permit phase change from liquid to vapour. In fact experiments of Frost and Sturtevant [41] have observed fluctuations in the form of pressure waves propagating outward from the interface during boiling incipience. Hence, the local fluid pressure adjacent to the interface is varied, though the global chamber pressure is kept constant at 1 atm.  Witharana et al. [40] measured the static contact angle of water with that of silicon wafer as 7° and this is retained for our computation.  The initial size of gas bubble as well as the boiling-nucleus (i.e. embryo), are dictated by the concept of surface tension coming into effect, which is a function of the minimum number of moles of gas. In our computations, this is determined using Lewins surface tension model (see Eq. (9)). 4.3.2. Nucleation on surfaces: without cavities The following Witharana et al. [40] test condition of a silicon wafer with no cavities is first considered (given in Table 7). It

Fig. 6. Variation of ‘work of formation’ of boiling-nucleus with radius predicted by D-RGC model for Witharana et al.’s [40] test case of wafer with no cavity.

Table 7 Witharana et al.’s [40] data of cavity size on wafers, the respective surface temperatures and the corresponding fluid properties. Cavity radius (lm)

Surface temperature (°C)

Temperature (K)

Pressure (bar)

Specific volume of liquid (m3/mole)

Surface tension (N/m)

– 0.045 0.299 1.058 2.175

220.00 197.90 155.43 126.30 117.91

493.00 470.90 428.43 399.30 390.91

23.13 14.86 5.41 2.38 1.86

2.14E
05 2.08E
05 1.97E
05 1.92E
05 1.91E
05

0.0331 0.0382 0.0477 0.0538 0.0554

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Table 8 Prediction of minimum boiling embryo using D-RGC model (h = 7°) and comparison against Witharana et al.[40] D-RGC model - h = 7° Minimum radius of vapour nuclei at boiling incipience Surface temperature (K)

Global superheat (K)

Initial quantity of gas (moles)

Global

493.0 470.9 428.4 399.3 390.9

120.0 97.9 55.4 26.3 17.9

10
20 10
20 10
19 10
19 10
19

0.04 0.07 0.19 0.43 0.55

LSR

Pl P sat

incipience

Initial gas bubble radius r ° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

Cavity radius in Witharana et al. [40] rc (lm)

% Deviation (rb
rc)/rc

0.61 0.29 0.19 0.43 0.55

8.95 9.19 23.40 22.35 22.16

0.11 0.11 0.22 0.82 1.32

6.87E
6 3.79E
5 1.28E
2 2.46E
1 7.41E
1

0.013 0.013 0.026 0.099 0.161

– 0.045 0.300 1.058 2.175

–
71.20
91.33
90.64
92.59

Local LSR Pl Psat

at

Table 9 Comparison of the work of formation predicted by D-RGC and I-RGC models for a radius of boiling-nucleus comparable to the cavity size (h = 7°). Radius of vapour nuclei comparable to experimental set-up h = 7° Surface temperature (K)

Global superheat (K)

Local

493.0 470.9 428.4 399.3 390.9

120.0 97.9 55.4 26.3 17.9

0.61 0.82 0.93 0.95 0.97

Pl P sat

at incipience

the Tsat (Pl) = 196 °C. Hence, when the local pressure fluctuation near the surface equals Pl = 14 bar, the local superheat for the liquid is only 24 °C (220 °C
196 °C = 24 °C). Thus, the ‘phase change’ is actually due to the local superheat of 24 °C. The global superheat of 120 °C has indeed facilitated the required energy for the heterophase fluctuations for the local pressure to jump from 1 bar to 14 bar. Hence, the present nucleation model (both D-RGC and IRGC) explain the need for high value of global superheat apart from predicting the ‘boiling-nucleus’ size and the ‘work of formation’. To the best of authors knowledge, such an insight is being introduced for the first time, as to how high level of global superheat is transformed into pressure energy. 4.3.3. Nucleation on surfaces: with cavities In this section, test conditions that use a wafer with cavity are investigated. Computations are performed to determine the minimum boiling-nucleus as shown in Table 8. It should be pointed out that the predicted embryo size is smaller than the cavity size. Hence, a second set of computations are performed to calculate the required liquid saturation ratio (LSR), to obtain a nuclei size, comparable to the cavity size (Table 10). From

Fig. 7. Qualitative representation of P-v diagram for phase change.

D-RGC work of formation (DG/KT) ⁄ 10
8

I-RGC work of formation (DG/KT) ⁄ 10
8

6.87E
6 1.63E
2 2.16E0 33.40E0 1.71E+2

3.04E
4 1.95E
2 2.16E0 33.41E0 1.71E+2

the results presented in Table 8 and Table 10, the following observations can be drawn.  Boiling-nuclei much smaller than the cavity size can be formed for the given test conditions. Consider the case of a wafer with a 0.045 lm cavity, and liquid pressure 0.29 ⁄ Psat (Tl), where a nucleus of base radius 0.013 lm is formed. This value is much lower than the cavity size itself and clearly shows that boiling can form at surface incongruities much smaller than 0.045 lm. This reinforces the experimental observation of nucleation occurring randomly on a nano surface.  Unlike the aforementioned minimum embryo size predictions, for an embryo to be of a size comparable to the cavity size, the local Pl should be comparable to Psat (Tl), as shown in Table 9.  Furthermore, as the cavity size increases, the Pl is even higher and closer to Psat (Tl). This implies that the required local superheat is smaller. However, since Pl needs to be closer to Psat (Tl), the energy required for the embryo formation is also higher.  Both D-RGC and I-RGC models predict similar nuclei sizes. However, higher work of formation is required for an I-RGC embryo compared to D-RGC. Also, the energy required in both cases, is same at higher LSRs.

4.3.4. Nucleation on surfaces: effect of contact angle As mentioned earlier, contrasting observations were made by Witharana et al. [40] and Nam and Ju [39] on the validity of classical nucleation theory to boiling in nanosurfaces. The contact angle appears to be the main parameter, which is different in both studies. Hence, we investigate the role of contact angle in the embryo formation. The computations of Witharana et al. [40] are at 7° contact angle, which was investigated, in the earlier sections. However, when Witharana et al. [40] coated the interior of the cavity with florinated residue, the static contact angle was found to be 75°. Hence, a set of computations with contact angle of 75° is performed to investigate, if the nature of surface coating will have an influence. When the surface contact angle is 75°, the required LSR is found to be lower compared to 7° case. From the present computations we observe that, surfaces with an intermediate contact angle (75°) enable a similar embryo size to be reached at a lower

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J.S. Murallidharan et al. / International Journal of Heat and Mass Transfer 110 (2017) 921–939 Table 10 Prediction of radius of embryo comparable to Witharana et al. [40] using D-RGC model (h = 7°). D-RGC model - h = 7° Radius of vapour nuclei comparable to experiment’s cavity size Surface temperature (K)

Global superheat (K)

Initial quantity of gas (moles)

Global

493.0 470.9 428.4 399.3 390.9

120.0 97.9 55.4 26.3 17.9

10
20 10
19 10
19 10
19 10
19

0.04 0.07 0.19 0.43 0.55

LSR

Pl Psat

incipience

Local superheat (K)

Initial gas bubble radius r ° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

Cavity radius in Witharana et al. [40] rc (lm)

% Deviation (rb
rc)/rc

0.61 0.82 0.93 0.95 0.97

24.00 8.90 3.43 2.30 0.91

8.95 22.54 22.32 22.05 21.97

0.11 0.32 2.52 9.02 19.92

6.87E
6 1.63E
2 2.16E0 33.40E0 1.71E+2

0.013 0.039 0.306 1.098 2.426

– 0.045 0.300 1.058 2.174

–
13.33 +2.00 +3.78 +11.59

Local LSR Pl Psat

at

Table 11 Prediction of radius of embryo comparable to Witharana et al. [40] using D-RGC model (h = 75°). D-RGC model - h = 75° Radius of vapour nuclei comparable to experimental set-up Surface temperature (K)

Global superheat (K)

Initial quantity of gas (moles)

Global

493.0 470.9 428.4 399.3 390.9

120.0 97.9 55.4 26.3 17.9

10
20 10
20 10
19 10
19 10
19

0.04 0.07 0.19 0.43 0.55

LSR

Pl Psat

incipience

Local superheat (K)

Initial gas bubble radius r ° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

Cavity radius in Witharana et al. [40] rc (lm)

% Deviation (rb
rc)/rc

0.61 0.31 0.38 0.59 0.73

24.00 48.90 34.43 16.30 9.91

8.95 10.67 27.20 26.03 25.79

0.11 0.11 0.33 1.13 2.23

6.87E
6 2.22E
5 1.47E
2 3.37E
1 1.45E0

0.013 0.106 0.316 1.087 2.153

– 0.045 0.300 1.058 2.174

– +135.55 +5.33 +2.74
0.96

Local LSR Pl Psat

at

Table 12 Comparison of the work of formation predicted by D-RGC and I-RGC models for boiling-nucleus comparable to the cavity (h = 75°). Radius of vapour nuclei comparable to experimental set-up h = 75° Surface temperature (K)

Global superheat (K)

D-RGC Local incipience

493.0 470.9 428.4 399.3 390.9

120.0 97.9 55.4 26.3 17.9

0.61 0.31 0.38 0.59 0.73

LSR

Pl P sat

at

incipience

D-RGC initial gas bubble radius r° ⁄ 10
3 (lm)

I-RGC initial gas bubble radius r° ⁄ 10
3 (lm)

D-RGC work of formation (DG/KT) ⁄ 10
8

I-RGC work of formation (DG/KT) ⁄ 10
8

0.61 0.27 0.37 0.58 0.73

8.95 10.67 27.20 26.03 25.79

8.95 10.72 27.22 26.06 25.79

6.87E
6 2.22E
5 1.47E
2 3.37E
1 1.45E0

3.04E
4 4.44E
5 1.69E
2 3.24E
1 1.45E0

I-RGC Local LSR

Pl P sat

at

LSR (comparing Tables 9 and 11). The work of formation for the IRGC model is given in Table 12. This lowering of LSR is observed to cause an increase in local superheat. It should be pointed out that the static contact angles for surfaces with hydrophobic coatings are even higher (h > 110°). Hence, it is only reasonable to extend the present study to surfaces with very high contact angles. The test conditions given in Nam and Ju [39] are: Liquid Pressure = 1 atm; Surface Temperature = 382 °C (with an observed superheat of 9 °C. The Teflon AR surface is measured to have a water contact angle of 120 ± 2°. Hence, to start with, a static contact angle of 120° was used in our computations. Table 13 presents the predicted embryo size for the test conditions of Nam and Ju [39]. Our computations show that, nucleation occur at the global LSR itself. This implies no additional local fluctuations are required to aid in the nucleation. The embryo size predicted with h = 120° is slightly bigger than the surface incongruities of 5 nm height. Nam and Ju [39] state that, AFM studies generally measure surface nanobubbles to have a height of 5–100 nm and a base radius of 0.5 lm. From our computations, it was found that 167° contact angle rather than 120°, satisfies AFM measurements. Corresponding base radius is 0.632 lm with a base height of 72 nm (Table 13). Hence, a boiling-nucleus can form over a nano-sized gas bubble and grows with just 9° superheat, if the

contact angle is very large (a hydrophobic surface). Hence, the observations of low superheat of Nam and Ju [39] are indeed validated. So far, our computations have independently verified the experimental observations of Witharana et al. [40] and Nam and Ju [39]. Now, we compare a similar test condition to address the existing contrast, in both findings. Witharana et al.’s [40] test condition with surface temperature of 390.9 K is in the same range as Nam and Ju’s [39] condition of 382 K. Prediction of the embryo size by our model is presented in Fig. 8, for different surface contact angles and test conditions. Fig. 8a–c correspond to Witharana et al.’s [40] surface conditions and Fig. 8d and e correspond to Nam and Ju’s [39] surface conditions. Witharana et al.’s [40] main claim was that low superheat is only possible in micrometer-sized cavities and not for nanoscale surface incongruities. According to Nam and Ju [39] low superheats are possible for nanosurfaces. From the bubble size predictions in Fig. 8a–c, the only possibility of a nano-sized nucleus, is in Fig. 8a. Even this is not a viable option as the required superheat is very high (at 17.9). Until contact angle is 75°, low superheat nucleation on nanosurfaces is not possible. Fig. 8d has a low superheat but, nucleus is of micrometer size when the contact angle is h = 120°. For such low superheats, nucleation would be of nano-size, only when h = 167° (as seen in schematic 8e). Hence,

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Table 13 Prediction of the boiling-nucleus size for the test condition of Nam and Ju [39] using D-RGC model for h = 120°, 167°. D-RGC model - h = 120°, 167° Minimum radius of vapour nuclei Angle of contact

Surface temperature (K)

Global superheat (K)

Initial quantity of gas (moles)

Global

120° 167°

382 382

9 9

10
19 10
19

0.73 0.73

LSR

Pl P sat

incipience

Local superheat (K)

Initial gas bubble radius r° ⁄ 10
3 (lm)

Vapour nuclei radius rv (lm)

Work of formation (DG/KT) ⁄ 10
8

Predicted base radius of vapour nuclei rb (lm)

0.73 0.73

9 9

49.193 710.35

3.149 2.811

6.89 E
1 1.129 E
4

2.720 0.632

Local LSR Pl Psat

at

T l = 390.9 K ;

(a)

LSR Superheat Bubble Radius Base radius rb

Global

Local

0.55 17.9

0.55 17.9 1.32 µm 0.161 µm

T l = 390.9 K ;

(b)

LSR Superheat Bubble Radius Base radius rb

LSR Superheat Bubble Radius Base radius rb

Local

0.55 17.9

0.97 0.91 19.92 µm 2.426 µm

LSR Superheat Bubble Radius Base radius rb

Local

0.55 17.9

0.73 9.91 2.23 µm 2.153 µm

LSR Superheat Bubble Radius Base radius rb

= 120 o

Global

Local

0.73 9.00

0.73 9.00 3.15 µm 2.72 µm

T l = 382 K ;

(e)

= 7 5o

Global

T l = 382 K ;

(d)

= 7o

Global

T l = 390.9 K ;

(c)

= 7o

= 167 o

Global

Local

0.73 9.00

0.73 9.00 2.81 µm 0.63 µm

(f)

Fig. 8. Qualitative representation of the embryo size for similar test conditions of Witharana et al. [40] ((a)–(c)) and Nam and Ju [39] ((d) and (e)); (f) pictorial representation of different shapes of bubble. Note: not drawn to scale.

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Witharana et al.’s [40] highly generalized conclusion that, nucleation at low superheats is possible only on microscale cavities needs further qualification. From the present computations, we find that, it is mostly true, except for highly hydrophobic surfaces. The classical theory of nucleation predicts the required global superheat accurately for predominantly hydrophilic nanosurfaces, as observed in Witharana et al. [40]. However, it fails to predict the required global superheat for hydrophobic nanosurfaces. For such cases, the modified version of Nam and Ju [39] that accounts for the contact angle effect needs to be used. Thus, it should be pointed out that, both the classical theory and it’s modified version are not universally applicable. Hence, we believe, the present embryo formation model, is most comprehensive, and applies to all types of surfaces, as it accommodates for the essential local physics of nucleation.

4.4. Global and local superheat As discussed in Sections 3.3 and 3.6, for nucleation to occcur, the energy required for the formation of the embryo must be favourable and available within the system. The change in Gibb’s free energy due to nucleation provides an estimate for this energy. As pointed out earlier, this change in Gibb’s free energy is also termed as the ‘work of formation’ of the bubble and is in turn a summation of three components (Eq. (8)) i.e. the surface energy, the work of expansion and the change in chemical potential energy. The values of the latter two terms are mainly determined by the pressure and superheat near the bubble interface and volume of the bubble. In the computation of the volume of the bubble as well as the surface energy term, the contact angle of the bubble with the heater wall also plays a significant role. Hence, whether a bubble embryo forms or not mainly depends on the pressure and superheat near the interface as well as the contact angle conditions. In the present study, this superheat which is available near the interface and which aids phase change is termed as local superheat. On the other hand, the system’s superheat is termed as global superheat. For the conditions given in Table 2–4, the work of formation was computed by using the respective conditions of natural contact angles, system pressure and global superheats. By using the system pressure and global superheats for the nucleation calculation, it is implied that the pressure and local superheat near the bubble interface is the same as the system pressure and global superheat. From the resulting free energy calculations, it was observed that, embryo formation is possible for these combinations. This implies, the system has sufficient energy for phase change/nucleation to occur. Hence, no additional energy is required to nucleate the embryo. Consequently, the local phase

(a)

change superheat is same as the global superheat. On the other hand, in Witharana et al.’s [40] experiments, this was found to be different (see Table 8). The ‘work of formation’ was calculated using the natural contact angle of 7° and system pressure of P = 1 atm. It was found that, this combination could not support nucleation even for a high global superheat of 120 °C. The computations showed that the first event of nucleation occurred only when the local pressure near the interface is much higher than the system pressure (Pl/ Psat = 0.61 as opposed to Pl/Psat = 0.04). It was also observed that, the local superheat required for phase change was only 24 °C. This was much smaller than the global superheat of 120 °C. These values indicate that, at the instant of nucleation, though the local superheat required for the embryo to form is very small, the local pressure needs to be higher than the system pressure. The need for increased local pressure is possible only through heterophase fluctuations, which can be triggered only by providing additional energy. This is essentially achieved by ensuring a large heat energy input via the heater surface (global superheat). Therefore, the global superheat is higher than the local superheat. The experiments of Nam and Ju [39] observed bubbles nucleating at the wall (see Table 13) at a much lower global superheat (9 °C) even though they maintained the same system pressure as Witharana et al. [40]. The only difference was that, the contact angle was much higher for the Nam and Ju [39] case (120°). Hence, for the ‘work of formation’ calculations, the combination of 120°, local p = 1 atm and local superheat = 9 °C was used and it was found that the system was indeed capable of supporting nucleation. This can be explained as follows. As mentioned earlier, the ‘work of formation’ is a summation of three energy components. In Witharana et al. [40], the contact angle was small and hence a higher global superheat was required to ensure that the required energy levels are attained for bubble nucleation. Whereas, in Nam and Ju [39], the high contact angle was sufficient to ensure favourable energy and a high global superheat was not necessary. 4.5. Parametric sensitivities From the present computations, it was observed that, the liquid-surface contact angle (h) and the number of moles of gas (ng ) of the initial gas bubble are the most important parameters. The variation in the ‘work of formation’ due to changes in these two parameters is explained in this section, for an intact real gas l = 0.65). core model with test condition (ng = 10
16, T l = 373 K, PsatPðT Þ l

4.5.1. Effect of contact angle The variation of required formation energy against the radius of boiling nucleus is shown in Fig. 9a for different contact angles. The

(b)

Fig. 9. Variation in DG vs. r for different (a) contact angles and (b) number of moles of initial gas.

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computations show that, for small angles of up to 30°, there is no significant variation in the formation energy and the corresponding embryo size. However, at larger angles, the radius remains more or less the same but the formation energy decreases. Additionally, for the present test case, the system becomes unstable without the possibility of a boiling-nucleus at very high angles of 120° and above. 4.5.2. Effect of number of moles of gas The effect of the initial number of moles of gas on the embryo size is presented in Fig. 9b. It can be noticed that, higher the moles of gas, smaller the required formation energy and hence, bigger the embryo size. This also shows that a continuous increase in the moles of gas does not result in a continuous increase in bubble radius. The bubble remains locked at one size, until and unless it has sufficient energy in the system to form a bigger nucleation bubble. 5. Summary and conclusion In this study, a ‘work of formation’ based model for the prediction of embryo size at boiling incipience is proposed. This model is mechanistic and accounts for the physics of nucleation more accurately than the existing models. In this model, we have implemented real gas treatment for vapour and gas and incorporated the variation of surface tension with size. The embryo formation model has been extensively tested for both low and high pressures, horizontal and vertical conditions, and for different surfaces and fluids. The formulation also accounts for cases where, the initial gas bubble remains intact, thus making it applicable to studies dealing with stable surface nanobubbles. Both the ‘Diffuse- Real Gas Core’ (D-RGC) and ‘Intact-Real Gas Core’ (I-RGC) models were found to give good predictions. The energy required for the formation of an embryo when the initial gas bubble is intact (I-RGC) was found to be higher than, when the gas bubble (D-RGC) diffuses into the embryo. The extent of validity of the classical nucleation theory was tested by comparing two different studies [39,40] that had contradictory conclusions, with predictions from the present model. Main conclusions of the present study can be briefly summarized as follows:  The liquid pressure near the interface of the embryo can be higher than the chamber pressure in the range of 1–180 bar. We have shown that this pressure variation exemplifies through the heterophase pressure fluctuations.  From the present model, it was found that, the actual local superheat near the interface is lower than the global superheat in some cases. From the detailed computations, we postulate that, although such a high global superheat was measured experimentally, the required local superheat is indeed small. A significant portion of this energy is used to trigger the pressure fluctuations in the system, and hence, the actual superheat required for phase change is much smaller.  Although embryos’ formation is fairly random on a surface, we find that, cavities help in reducing the formation energy, through the contact angle effect.  Unlike the classical nucleation theory and its variants, the present work of formation model is able to predict. i. Nucleation in both hydrophobic and hydrophilic surfaces. ii. Different sizes that the bubble can take for the same set of operating conditions. iii. Discrete step jumps in sizes for changing operating conditions. This is attributed to proper accounting of contact angle effects as well as local pressure fluctuations.

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