Effects of spontaneous nanoparticle adsorption on the bubble-liquid and bubble-bubble interactions in multi-dispersed bubbly systems – A review

International Journal of Heat and Mass Transfer 120 (2018) 552–567

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

Review

Effects of spontaneous nanoparticle adsorption on the bubble-liquid and bubble-bubble interactions in multi-dispersed bubbly systems – A review Yang Yuan a, Xiangdong Li b, Jiyuan Tu a,b,⇑ a

School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia Key Laboratory of Ministry of Education for Advanced Reactor Engineering and Safety, Institute of Nuclear and New Energy Technology, Tsinghua University, PO Box 1021, Beijing 100086, China b

a r t i c l e

i n f o

Article history: Received 14 January 2017 Received in revised form 29 November 2017 Accepted 11 December 2017

Keywords: Bubble interface Nanoparticles Adsorption Modifications Modelling

a b s t r a c t Robust predictive models of dynamic bubbly systems of nanoparticle-liquid mixtures are vital to the design and assessment of relevant industrial systems. Previous attempts to model bubbly flows of dilute nanofluids using the classic two-phase flow models were unsuccessful although the apparent hydrodynamic properties of the dilute nanoparticle-liquid mixtures were only negligibly different to those of their pure base liquids. Emerging studies demonstrated that when bubbles exist in the mixture, nanoparticles tend to spontaneously aggregate at the bubble interface, forming a layer of ‘‘colloidal armour” and making the bubble interface partially rigid and less mobile. The colloidal armour also significantly modifies the characteristics of the bubble-liquid and bubble-bubble interactions. Therefore, it was proposed that the key job when developing a predictive model based on classic two-phase flow models is to reformulate the bubble-liquid and bubble-bubble interactions. However, the adsorption of nanoparticles in dynamic bubbly systems has rarely been studied. The lack of mechanistic understanding has severely hindered the model development. Therefore, this study reviews the common findings yielded from experimental and numerical investigations reported in literature, with the aim to clarify the critical points to address when modelling bubbly flows containing nanoparticles using the classic two-fluid and MUSIG models. Ó 2017 Published by Elsevier Ltd.

Contents 1. 2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanoparticle adsorption at phase interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The influences of nanoparticles on bubble-liquid interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The bubble-liquid interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The lift force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. The Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. The wake effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. The Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. The wake effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The influences of nanoparticles on bubble-bubble interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The bubble-bubble interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The thinning process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Surface mobility and rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author at: School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia. E-mail address: [email protected] (J. Tu). https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.053 0017-9310/Ó 2017 Published by Elsevier Ltd.

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4.3.

The rupture process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Fluctuation damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Energy barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conflict of interest statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Nomenclature AH E0 E1

DE1 CD CL CL,p CL,v CLS CLW CTD CW1, CW2 c ct db dcr dH Eo Eo0 F FD FK FL FTD FW g h h0 hf k kB kd ks L nw Reb

Hamaker constant, (10
12 ergs) Initial interfacial energy, (J m
2) Interfacial energy after the adsorption of particles, (J m
2) Adsorption or detachment energy, (J m
2) Drag coefficient, (dimensionless) Lift coefficient, (dimensionless) Lift coefficient due to pressure, (dimensionless) Lift coefficient due to viscous stress, (dimensionless) Shear-induced lift coefficient, (dimensionless) Wake-induced lift coefficient, (dimensionless) Turbulent dispersion coefficient, (dimensionless) Lubrication coefficient, (dimensionless) Solute concentration, (mol m
3) Critical solute concentration, (mol m
3) Bubble diameter, (m) Critical bubble diameter, (m) Maximum bubble horizontal dimension, (m) Eötvös number, (dimensionless) Modified Eötvös number, (dimensionless) Compressing force, (N) Drag force, (N m
3) Interfacial force, (N m
3) Lift force, (N m
3) Turbulent dispersion force, (N m
3) Wall lubrication force, (N m
3) Gravitational acceleration, (m s
2) Liquid film thickness, (m) Initial film thickness, (m) Critical film thickness, (m) Thermal conductivity, (W m
1 K
1) Boltzmann constant, (J K
1) Empirical constant in the film drainage time calculation, (dimensionless) Empirical constant in the drag coefficient calculation, (dimensionless) Thickness of the polymer layer, (m) Outward vector normal to the wall surface Bubble Reynolds number, (dimensionless)

Rex R0 rb rp rm S T tdr trp U us We yW

Vorticity Reynolds number, (dimensionless) Ideal gas constant with the value of, R0 = 8.31 (J K
1 mol
1) Bubble radius, (m) Particle radius, (m) The radius of the meniscus, (m) Mean distance between the attachment points, (m) Temperature, (K) Liquid film drainage time, (s) Liquid film rupture time, (s) Velocity, (m s
1) Liquid slip velocity, (m s
1) Weber number, (dimensionless) Adjacent point normal to the wall surface, (m)

Greek symbols a Void fraction, (dimensionless) C Surfactant surface concentration, (mol m
2) c Reduced surface potential, (V) d Separation between the surfaces, (m) h Contact angle, (°) hc Stagnant-cap angle, (°) hs An angle from the front stagnant point, (°) j Debye screening length, (m
1) l Viscosity, (Pa s) Pc Capillary pressure, (N) Pc Capillary pressure for bubbles with nanoparticles, (N) Pe Electrostatic double layer force, (N) Ps Steric repulsion force, (N) q Density, (kg m
3) q1 Number density of ion in the bulk solution, (sites m
3) r Interfacial tension, (N m
1) sij Bubble contact time, (s) Kc Critical wavelength of the thermal fluctuations, (m) Subscripts g Gas phase l Liquid phase

1. Introduction Bubbly flows, where discrete small bubbles are dispersed or suspended in liquid continuums, are widely encountered in various industries such as chemical, petroleum, mining and food processes that require large interfacial areas for efficient mixing of competing gas-liquid interactions. Maintaining the bubbly flow regime and enlarging the interfacial area have always been the interests of studies aiming to improve gas-liquid mixing during the past decades [1]. In pursuing of larger interface area concentrations (IACs), surfactants have been commonly added into the two-phase systems as they are efficient in increasing the gas-liquid interfacial area and stabilizing bubbles [2–4]. In recent years, thanks to the fast advances of nanotechnology, nanoparticles have been increas-

ingly utilized as substitute for surfactants [5] due to their unparalleled merits such as the excellent physical and chemical stabilities. It has long been aware that nanoparticles are capable of stabilizing bubbles in quiescent liquid, such as those in foams where the volume fraction of air could be as high as 99% [6,7]. In recent years, it was found that nanoparticles are also promising to stabilize dynamic multi-dispersed bubbly systems. Wang and Bao [8] found that the bubbly-to-slug flow regime transition in a vertical tube occurred at a higher gas superficial velocity when CuO nanoparticles (0.5 wt%) were added to nitrogen-water two-phase flows. This indicated that nanoparticles could help maintain a bubbly flow pattern with a higher void fraction than pure water. Park and Chang [9] experimentally investigated the two-phase flow

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dynamics of air-c-Al2O3/water nanofluid (0.1v%) and found bubbles generated through injecting air into the mixture were between 2 mm and 5 mm in diameter, which were much smaller than the bubbles in pure water (3 mm to 10 mm) injected under the same experimental conditions. The experiments also revealed that the radial void fraction distribution had a more flattened and uniform centre-peaked shape with the existence of nanoparticles in the water. The IAC of bubbles was as high as 300 m
3 in the nanoparticle-water mixture, which was almost twice of that in pure water. At the meantime, nanoparticles are mixed with the base liquid at a near-molecular level due to their small sizes. A nanoparticleliquid mixture behaves hydrodynamically like its pure base liquid when the nanoparticle load is low. A number of experimental measurements [10,11] particularly the benchmark study by Buongiorno et al. [12] have proven that when the nanoparticle concentration was less than 0.1% by volume, the properties of the mixture such as the density, viscosity, specific heat capacity and saturation temperature were only negligibly different from those of the base liquid. This has allowed numerically treating the nanoparticle-liquid mixture as a single-phase fluid in order to develop predictive models for the design and assessment of the industrial systems involving nanoparticle-liquid mixtures [13]. For the modelling of bubbly systems containing nanoparticles, the Eulerian-based two-fluid model [14,15] and Multi-SizeGroup (MUSIG) model [16] has been widely used where the mixture is approximated as a pseudo-liquid despite the fact that it is composed of pure liquid and solid particles. The approximation, however, although had been successful in single-phase systems [13], led to significant predictive discrepancies in two-phase systems and failed to revive the aforementioned novel phenomena [16,17]. Emerging studies proposed that some microscopic processes which are exclusive to two-phase systems of nanoparticle-liquid mixtures may have been neglected. Our recent studies [16,17], for the first time, attributed the modified two-phase flow structures to a thin layer of adsorbed nanoparticles at the bubble-liquid interface. It was demonstrated that the nanoparticles suspended in the liquid tend to spontaneously migrate towards and aggregate at the bubble interface, significantly modifying the rigidity and slip conditions of the interface, which consequently changes the characteristics of bubble-liquid and bubblebubble interactions. Parametric studies [18] indicated that introducing correction coefficients to the established correlations of interfacial forces can significantly improve the predictions. However, the coefficients are empirical and can be case-sensitive. In order that the two-phase flow dynamics of nanoparticle-liquid mixtures can be mechanistically predicted, fundamental knowledge about the nanoparticle adsorption and its effects on the bubble behaviours is of great importance. However, previous studies on nanoparticle aggregation and interfacial behaviours were mostly limited to static systems while analogous phenomena in dynamic systems have rarely been investigated. An enormous gap still exists between the current knowledge and a mechanistic predictive model. This paper aims to review the studies on nanoparticle adsorption on bubble interfaces and its effect on the bubble-liquid and bubble-bubble interactions, with an attempt to clarify the key jobs when developing predictive models for bubbly flows containing nanoparticles. However, this paper focuses only on bubbly flows of nanofluids without phase change, where the bubbles are injected into the liquid rather than being generated from nucleate boiling on heater surfaces. In the case of nucleate boiling of nanofluids, bubbles are usually larger than in their pure base liquids [15,19,20], which is contrary to the cases reviewed in this paper and believed to be a result of the larger bubble departure

sizes caused by the improved wettability of heater surfaces induced by nanoparticle deposition during the boiling process [15]. 2. Nanoparticle adsorption at phase interfaces The phenomenon of nanoparticle adsorption at gas-liquid interfaces has long been recognized and vastly utilized to stabilize bubbles in liquid foams [6]. Shown in Fig. 1 are microscopic images of nanoparticles adsorbed at the interfaces of bubbles and liquid drops submerged in another fluid. Fig. 1(a) illustrates that MAGSILICAÒ H8 nanoparticles (a single-domain iron oxide core with a fully closed silica shell with a diameter of 16 ± 10 nm) suspended in liquid aggregated at the surface of air bubbles submerged in an ethanol/water mixture (COH = 5 vol%) and formed a thin layer covering the bubble [21]. This thin layer of adsorbed nanoparticles was also clearly observed by Lin et al. [22] in a CdSe nanoparticletoluene/water mixture using Scanning Force Microscopy (SFM) (Fig. 1(b)). Using Transmission Electron Microscopy (TEM) method, Bӧker et al. [23] further demonstrated that the adsorption occurs in three steps. Firstly, free nanoparticles diffuse to the interface. Then the particles pack closer and form clusters which grow to form a closely packed particle array, lowing the interfacial tension. Finally, thermally activated exchange between adsorbed and incoming particles is observed, leading to a tightly packed layer (Fig. 2). According to Lin et al. [22], the adsorption of nanoparticles at the gas-liquid interface is driven by the spontaneous reduction of interfacial free energy. The placement of a single particle with an effective radius rp at the interface leads to the decrease of the initial interfacial energy from E0 to E1 yielding an energy difference of DE1 [24]:

Fig. 1. (a) Transmission Electron Microscopy (TEM) image of air bubbles with MAGSILICAÒ H8 nanoparticles in ethanol/water mixture [21]. (b) Fluorescence confocal microscope image of the adsorbed CdSe nanoparticles at toluene/water interface [22].

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or thermal agitation [21]. As a result, this closely-packed layer of nanoparticles at the interface generates a sort of ‘‘colloidal armour” [26]. This colloidal armour, on one hand, is found to create a steric barrier which is capable of stabilizing bubbles in liquid foams by inhibiting or overwhelmingly stopping bubble coalescence [5,27]. On the other hand, the properties and slip conditions of the bubble surfaces are speculated to be significantly changed due to the presence of the colloidal armour. Bubbles coated with a layer of nanoparticle would deform less and consequently be more like a rigid sphere [28]. In addition, the part of nanoparticles immersed in the gas phase can immobilize the bubble surface and change the slip condition from free-slip to no-slip, resulting in the partially or completely suppressed inner circulation flow [29]. Since bubbleliquid and bubble-bubble interactions which control the bubbles’ movement, size and distribution are significantly influenced by the bubble surface properties and slip conditions, it is crucial to clarify the mechanisms of the effects of nanoparticles on these two interactions. 3. The influences of nanoparticles on bubble-liquid interactions 3.1. The bubble-liquid interactions Hydrodynamic interactions between the gas and liquid phases are responsible for the complexity of gas-liquid flows. Interfacial forces are always the dominant components of these interactions and their formulations are critical to the effective prediction of gas-liquid flows. Forces exerted on a bubble moving in continuous liquid include drag force FD, lateral lift force FL, wall lubrication force FW and turbulent dispersion force FTD. The total interfacial force FK on the bubble is:

Fig. 2. Series of TEM images of 6 nm nanoparticle adsorption to the toluene/water interface in different adsorption steps [23]. (a) Step 1. (b) Step 2. (c) Step 3.

DE1 ¼ rr2p pð1  cos hÞ

ð1Þ

where the sign within the brackets is negative for particle removal into water (h < 90°) and positive for particle removal into air (h > 90°). r and h are the surface tension and contact angle, respectively. DE1 is the so-called adsorption energy or detachment energy. Following Eq. (1) the energy required for a nanoparticle with diameter of 50 nm and contact angle of 80° to be detached from the water-air interface is approximately DE1 = 65,000 kBT (kB is the Boltzmann constant and T is the absolute temperature), which is much higher than that of surfactants (generally several kBT [25]). Therefore, being different to surfactant molecules which can dynamically adsorb to and desorb from an interface, nanoparticles can irreversibly absorbed, which means it is almost impossible to force them out of the interface, either by shrinkage of the bubble

F K ¼ F D þ F L þ F W þ F TD

ð2Þ

F D ¼ 3CD ag ql jU g
U l jðU g
U l Þ=4db

ð3Þ

F L ¼
ag ql CL ðU g
U l Þ  ðr  U l Þ

ð4Þ

F W ¼
ag ql jU l
U g j maxð0; CW1 þ CW2 db =yW ÞnW

ð5Þ

F TD ¼
CTD ql kl ral

ð6Þ

where CD, CL, CW1 and CW2, and CTD denote drag coefficient, lift coefficient, wall lubrication coefficients and turbulent dispersion coefficient, respectively. The formulation of these coefficients has been strongly empirical due to the extreme complexity. Although dozens of correlations have been proposed for these coefficients, considerable uncertainties and discrepancies remain due to their empirical nature. It is worth noting that these interfacial forces are all closely related to liquid velocity filed surrounding the bubble [30]. An insight into the liquid flow around a nanoparticle-covered bubble is thus beneficial. When a spherical gas bubble with a clean interface moves at a constant velocity U through a continuous liquid phase, its streamlines are open, and in particular there are no wakes behind [31] because no shear exists on the bubble interface (Fig. 3(a)). With an increasing Weber number (We = qU2rb/r), the inertial distorts the bubble from spherical to oblate-ellipsoidal and spherical cap shapes. When the distortion is significant, flow separation and wake occur at the back end if the bubble Reynolds number Reb is larger than 125 [32] (Fig. 3(b)). However, for solid spheres, as long as the Reynolds number is larger than about 12, flow separation and wake formation can always occur [33,34]. This fact suggests that the reduction of a bubble’s interfacial mobility can cause wakes to form at a lower bubble Reynolds number [35–37]. When it comes to nanoparticle-containing systems, due to the colloidal

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3.2. The lift force

(a)

The lift force generally acts in the direction normal to the relative motion of fluid and bubbles, and largely controls the transverse motion of bubbles in a vertical flow. For small spherical bubbles in pure liquid shear flow, a lateral force is caused by the pressure difference due to a liquid velocity gradient (Fig. 4(a)). This lateral force is the so-called shear-induced lift force, which acts towards the descending liquid velocity gradient, or in another word, towards the pipe wall for a spherical bubble rising in an upward liquid flow (Ug > Ul). The lift coefficient CL is thus positive with a value ranging from 0.25 to 0.5 depending on the bubble Reynolds number and liquid viscosity [42–44]. For distorted bubbles in pure liquid, besides the shear-induced lift force, another lateral force arises due to the complex interactions between the bubble wake and the liquid shear filed [45]. According to Tomiyama et al. [46] this wake-induced lift force acts in an opposite direction of the shear-induced lift force and causes a direction reversal when the wake becomes strong enough (Fig. 4(b)). Tomiyama et al. [47] developed an empirical CL correlation which has allowed modelling the transverse migration of spherical and distorted bubbles in pure liquid.

(b)

8 0 Eo0 6 4 > < min½0:288 tanhðReb ; f ðEo ÞÞ CL ¼ f ðEo0 Þ ¼ 0:00105Eo03
0:0159Eo02
0:0204Eo0 0:474 4 < Eo0 6 10 > :
0:27 10 < Eo0

ð7Þ

Wake

(a)

Shear-induced li force

(c)

Liquid velocity distribuon

Wake

Fig. 3. Flow field surrounding the bubble. (a) Spherical bubbles in pure liquid. (b) Distorted bubbles in pure liquid. (c) Spherical bubbles in nanoparticle-containing system.

armour of nanoparticles, bubbles will be partially rigid and immobile, and become more resistant to deform [38]. A wake region could probably form behind the nanoparticle-covered bubble [35,39]. Meanwhile, under the action of liquid velocity gradient and fluid shear, the nanoparticle-covered bubbles tend to develop a rotating movement [40]. This rotating movement has been demonstrated to induce wake asymmetries, as illustrated in Fig. 3(c) [41].

(b)

Shear-induced li force

Wake-induced li force

Wake

Liquid velocity distribuon

Fig. 4. The lift force acing on (a) Spherical bubbles in pure liquid. (b) Distorted bubbles in pure liquid.

Y. Yuan et al. / International Journal of Heat and Mass Transfer 120 (2018) 552–567

where Eo0 is the modified Eötvös number based on the maximum bubble horizontal dimension dH. The Tomiyama lift coefficient is plotted against the bubble size in Fig. 5. For air bubbles rising in pure water, the negative-to-positive transition occurs at a critical bubble diameter of dcr = 5.8 mm [48–50]. When bubbles are coated with nanoparticles, they are more likely to behave like rigid spheres rather than deformable bubbles due to the increased rigidity and restricted mobility. As for the rigid sphere, both Kurose and Komori [51] and Bagchi and Balachandar [52] showed that the lift coefficient CL decreases with increasing bubble Reynolds number and takes near-zero value at Reb = 100. Beyond this value, CL keeps slightly decreasing and it takes the small negative value, indicating that the lift force on a rigid sphere acts in the opposite direction of that on a freeslipping bubble. In fact, similar findings have been obtained in studies of bubbles contaminated with surfactants [39,53]. The authors [16,17] also found that CL can be negative for small spherical nanoparticle-coated bubbles. In one of our recent studies [17], the Tomiyama correlation (Eq. (7)) [47] was incorporated in the two-fluid model employed to simulate air-water bubbly flows with nanoparticles. The numerical results were compared against the experimental data of Park and Chang [9]. The computations yielded a wall-peaked void fraction distribution despite a factual centre-peaked distribution was observed in the experiments (Fig. 6(a)). The reason that led to this discrepancy was found to be Eq.7 which generated a positive lift coefficient of 0.288 at an experimentally measured average bubble diameter of 3 mm. Parametric study revealed that when a negative value CL =
0.025 was used in the simulation, a good agreement with the experimental data was achieved (Fig. 6(b)). Since the void fraction distribution reflects the bubble distribution, this result indicated that the lift coefficient for small spherical bubbles in nanoparticle-containing system can be negative and under the action of which the bubbles tend to migrate towards the pipe centre. The widely accepted Tomiyama correlation (Eq. (7)) is thus not feasible to nanoparticle-covered bubbles. The positive-to-negative transition of the lift coefficient is expected to occur at a much smaller critical bubble diameter, as shown in Fig. 5. In order to develop a model appropriate for the lift force in nanoparticle-containing system, two plausible mechanisms that how nanoparticles reverse the direction of the lift force are analysed in this study. 3.2.1. The Marangoni effect The lift force acting on a surfactant-contaminated bubble in a linear shear flow was numerically studied by Fukuta et al. [39].

Fig. 5. The predicted lift coefficient as a function of bubble diameter.

557

Fig. 6. Comparison of predicted flow parameters against experimental data of bubbly flows containing nanoparticles with (a) Tomiyama model (Eq. (7)). (b) CL =
0.025.

They found the lift force decreased from a positive value to a negative value, when a clean bubble was gradually contaminated. For the first time, they related the reduction to a non-axisymmetric distribution of pressure on the bubble surface which was caused by the Marangoni effect. Fukuta et al. [39] explained that an uneven concentration distribution of surfactant exists along the bubble surface because the surfactant is swept off the front part and accumulates in the rear part of the bubble as it rises. Due to the surfactant accumulation in the rear part, a variation of surface tension along the surface is developed and causes a tangential shear stress on the bubble surface. This is known as the Marangoni effect and the tangential shear stress is the so-called Marangoni stress. Since nanoparticles act in many ways like surfactants [54] and tend to accumulate in the rear part of a rising bubble, it is reasonable to extrapolate that the Marangoni effect may affect the lift force acting on a nanoparticle-coated bubble in a similar way. Due to the Marangoni stress, both pressure and viscous stress on the bubble surface can become non-asymmetrically distributed. The lift coefficients due to pressure CL,p and due to viscous stress CL,v (CL = CL,p + CL,v) are thus inevitably changed. Fig. 7(a) illustrates that with more surfactants adsorbed on the bubble surface (corresponding to an increasing Langmuir number), the coefficient CL,p decreases dramatically to a negative value. When the bubble is fully coated (maximum Langmuir number), the viscous stress contribution CL,v becomes dominant, resulting in a negative value of the apparent lift coefficient CL. In addition, when a bubble is fully covered with nanoparticles and behaves like a rigid sphere, the lift force acting on this bubble is also influenced by the Reynolds number Reb [51]. Shown in Fig. 7 (b) are the contributions of pressure CL,p and viscous stress CL,v acting on a rigid sphere in a homogeneous linear shear flow with

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shear-induced lift coefficient CLS and wake-induced lift coefficient CLW:

CL ¼ CLS þ CLW

ð8Þ

Combining the experimental data with numerical data, the total lift coefficient in turbulent shear flows was correlated by Moraga et al. [55] in terms of the bubble Reynolds number Reb and vorticity Reynolds number Rex:



CL ¼ 0:17 exp

Reb Rex



ð9Þ

4:2  107

According to Eq. (9), no wake-induced lift force is expected when the Reynolds number is below 300 and consequently the shear effect becomes dominant. As the Reynolds number increases, the wake effect becomes increasingly important and eventually reverses the sign of the lift coefficient to negative (Fig. 8). Actually, both Eq. (7) or Eq. (9) are empirical. As pointed out by Moraga et al. [55], an accurate determination of the magnitude of the lift force induced by the wake effect is still very difficult, the main problems being the complexity of the wake structure and its elusiveness to an analytical treatment. The wide range of nanoparticle concentration on the bubble interface further intensifies the complexity. Fundamental and analytical studies are urgently needed to address the effects of aforementioned factors. 3.3. The drag force

Fig. 7. Contributions of pressure CL,p and viscous stress CL,v to the total lift coefficient acting on (a) A contaminated bubble [39]. (b) A rigid sphere [40].

a shear rate of a⁄ = 0.2. Both coefficients CL,p and CL,v change their signs from positive to negative in the range of 1  Reb  100. In Park and Chang’s experiment [9], the bubble Reynolds number was estimated to be 1000. Obviously, the lift coefficient could be negative even for small spherical nanoparticle-coated bubbles. 3.2.2. The wake effect As shown in Fig. 3(c), a slanted wake region induced by the immobile surface and rotating movement can be found behind the nanoparticle-coated bubble. Since the size of the wake could have the same order as that of the bubble itself, its effect on body forces cannot be neglected. In the wake region, when a vortex is shed, the space it originally occupied behind the bubble is replenished by liquid moving slower than the rotational velocity of the vortex [55]. A significant velocity reduction occurs here due to the sharp turn of the incoming fluid to occupy the volume immediately after the body. As a result, the pressure increases due to the decreasing fluid velocity. Therefore, when a vortex is shed, a transient lateral force on the bubble arises [55–57]. Sakamoto and Haniu [58] discovered that liquid vortices at the higher relative velocity side always grow faster and larger than those at the lower relative velocity side. Then the smaller vortices could be engulfed by the larger ones before they form a separate vortex and detach. In the absence of shedding, the lateral force thus always points toward the lower relative velocity side, which is opposite to the direction of the shear-induced lift for a bubble rising in an upward flow. The apparent lift force on a nanoparticle-coated bubble is thus expected to be a consequence of two competing factors: shear and wake effects. The total lift coefficient CL is given by the sum of

The drag force is one of the most important forces acting on bubbles as it dominantly controls the rise velocity of bubbles. The apparent drag force is a result of the shear and form drags, which are due to viscous surface shear stress and uneven pressure distribution around the bubble, respectively. According to Ishii and Zuber [59], four different regimes were assumed to model the drag coefficient CD: stokes, viscous, distorted and churn regimes. The stokes (0 < Reb < 0.2) and viscous (0.2 < Reb < 1000) regimes are characterized by the ‘‘undistorted particles” where the distortions of the bubbles are negligibly small and the drag coefficient CD mainly depends on the bubble velocity and liquid viscosity. As the bubble diameter increases, the shape of the bubble is gradually changed from spherical to oblate-ellipsoidal and then spherical cap. A vortex could develop behind the bubble, where the vortex departure creates a large wake region. This process happens in the distorted and churn regimes which are known as the ‘‘distorted particle” regime (1000 < Reb). In these two regimes, the distortion and irregular motions become pronounced and the drag coefficient

Shear-induced li force

Wake-induced li force

Wake Liquid velocity distribuon

Fig. 8. The lift force acting on spherical bubbles in nanoparticle-containing system.

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CD becomes proportional to the bubble radius and Reynolds number. Thus a mixture viscosity model was developed by Ishii and Zuber [59] to formulate the drag coefficient correlations for each individual flow regime. The drag coefficient as calculated is plotted in terms of the bubble Reynolds number in Fig. 9.

8 > < 24=Reb 0:75 CD ¼ 24ð1 þ 0:1Reb Þ=Reb > : 0:5 2Eo =3

0 < Reb 6 0:2 0:2 < Reb 6 1000

ð10Þ

1000 < Reb

Besides the influences of aforementioned bubble size and Reynolds number, the surface properties and slip condition also play important roles. It is well known that the drag coefficient of a solid particle can be as large as three times of that of a bubble with the same diameter and Reynolds number. When a clean bubble is contaminated with impurities, these impurities such as surfactants and nanoparticles can bridge the gap between the behaviours of a clean bubble and a solid particle by partially immobilizing the bubble surface [33,60,61]. As a result, the drag force on a contaminated bubble sits somewhere between a clean bubble and a rigid sphere [62]. Apart from that, Tomiyama et al. [28] believed the aggregation of impurities can also increase the shear drag by inducing the no-slip condition and hindering the internal circulation within the bubble (Fig. 10). Considering this effect, an empirical correlation was proposed to account for the drag enhancement (Eq. (11)).

CD ¼ max½24ð1 þ 0:15Re0:687 Þ=Reb ; 8Eo=3ðEo þ 4Þ b

ð11Þ

Recently, McClure et al. [63] improved the classic Grace model by multiplying an empirical constant (ks = 1.6–2.2) to take into account the contribution of adsorbed surfactants to the drag enhancement [61]:

CD ¼ ks CD ðgraceÞfðaÞ

ð12Þ

In the case of nanofluids, it is extrapolated that adsorbed nanoparticles might play a similar role in increasing the shear drag. Following the modified Grace model, our recent study [16] introduced the an empirical constant ks to the Ishii-Zuber model [59] (Eq. (10)) to account for the influence of nanoparticles and further expanded its range to ks = 1.6–3.0:

C D ¼ ks C D ðishiiÞ

ð13Þ

The numerical results (Fig. 11) showed that adding a simple empirical coefficient has very limited impact on the predicted bubble velocity. There must be some other factors needing consideration. Some plausible mechanisms are analysed as follows.

Fig. 10. Schematic overview of the effect of contaminants on the drag force [29]. (a) Ultra-pure liquid with free-slip boundary condition. (b) Slightly contaminated liquid with a limited circulation inside the bubble. (c) Fully contaminated bubble with no-slip boundary condition.

Fig. 9. The predicted drag coefficient as a function of bubble Reynolds number with Ishii-Zuber model.

3.3.1. The Marangoni effect Fukuta et al. [39] found that the Marangoni effect induced by the accumulation of surfactants on bubble surface not just influences the lift force, but increases the drag force. As mentioned before, when surfactants are adsorbed on the bubble surface, a

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tangential shear stress can develop. This implies that a shear-free boundary condition is no longer imposed in the liquid at the gasliquid interface, leading to an increased drag force. Duineveld [64] and Bel Fdhila and Duineveld [65] conducted experimental studies of air bubbles rising in water contaminated with surfactants. They found that below a critical bulk concentration, the equilibrium rise velocity is insensitive to the presence of surfactants, whereas the rise velocity decreases abruptly to a value corresponding to a solid sphere when the critical bulk concentration was reached. Numerous efforts have been devoted, trying to understand the novel phenomenon. The most widely acknowledged theory is the stagnant-cap model [66]. According the model, the bubble surface is divided into two distinct regions separated by a stagnant-cap angle hc. When the angle from the front stagnant point (hs) is smaller than hc, the surfactant surface concentration C is zero and the liquid remains free to slip along the interface; whereas in the rear part of the bubble (hs > hc), C is nonzero and the relative velocity of the fluid along the interface us vanishes. Sadhal and Johnson [67] proposed a correlation to estimate the drag coefficient for a surfactant-contaminated bubble:

  CD ðhc Þ
CD ðpÞ 1 1 2ðp
hc Þ þ sin hc þ sin 2hc
sin 3hc ¼ CD ðhc Þ
CD ðpÞ 2p 3 ð14Þ Similarly, the Marangoni effect also exists on nanoparticlecoated bubbles. The stagnant-cap theory is probably capable of depicting the distribution of nanoparticles at bubble surfaces. 3.3.2. The wake effect When the boundary condition around hs = hc abruptly changes from a shear-free to a no-slip condition, a marked peak in the interfacial vorticity is produced. There is more vorticity injected in the flow than in the case of a uniform no-slip condition, resulting in a wake region with larger length [62] and volume [37] behind surfactant-contaminated bubbles than solid spheres moving at the same Reynolds number. Moreover, with stronger wake effect and distortion and irregular motion of the bubbles when the bubble Reynolds number is high [55], the contaminated bubbles may consequently experience a drag enhancement similar to that in the distorted regime when the bubbly Reynolds number is in the range of 300–1000. As a result, it is expected that the transition point from the viscous regime to distorted regime may occur at a smaller Reynolds number in nanoparticle-containing system, as shown in Fig. 12.

Fig. 12. The predicted drag coefficient as a function of bubble Reynolds number with different drag models.

However, due to the lack of experimental data, taking the Marangoni and wake effects into account when modelling the drag force is still very challenging. For the stagnant-cap model, the major difficulty is the determination of the cap angle hc as a function of nanoparticle concentration along the interface. Since nanoparticles are irreversibly absorbed, which is different from surfactant molecules that can dynamically adsorb to and desorb from the surface, how to emphasize this difference and substitute a suitable cap angle still remains answered.

4. The influences of nanoparticles on bubble-bubble interactions 4.1. The bubble-bubble interactions In gas-liquid flows, the coalescence and break-up of bubbles have attracted considerable attention as they largely influence the temporal and spatial distribution of the two-phase flow parameters. Compared to break-up, coalescence was dominant in the case of bubbly flows in small-diameter vertical tubes [68]. In view of this, only bubble coalescence is considered in this paper. According to the film drainage model proposed by Shinnar and Church [69], bubble coalescence occurs within three steps (see Fig. 13): contact, thinning and rupture. Firstly, two bubbles come to contact with each other, flattening the bubble surfaces against each other and trapping a thin liquid film between them. The initial thickness h0 of this film was typically estimated to be 10
4 m [70]. The first step is controlled by the hydrodynamics of the bulk liquid flow. Secondly, the liquid film thins to a critical thickness hf (usually estimated as 10
8 m [71]) before it ruptures. If this thinning process takes longer than the time the bubbles maintain contacted, coalescence will not occur. The second step is controlled by the hydrodynamics of the liquid film. Finally, once the film is sufficiently thin it will rupture via an instability mechanism. This step is very rapid in comparison to the first two steps and it is usually not counted when calculating the coalescence time. According to the film drainage model [69], not all collisions can lead to coalescence. The concept of collision efficiency k is introduced to account for the probability of bubble coalescence:

  tdr k ¼ exp


s

Fig. 11. Comparison of predicted bubble velocity against experimental data of bubbly flows containing nanoparticles with different drag models.

ð15Þ

A larger collision efficiency leads to a larger mean bubble diameter and vice versa. In Park and Chang’s experiment [9], the measured bubble diameters were between 2 mm and 5 mm in the

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561

Fig. 13. Schematic overview of the coalescence process of two bubbles.

air-nanofluid bubbly flows, which were much smaller than those (3–10 mm) in the air-water system, despite the exactly same bubble injection condition and initial bubble size when injected. Therefore, it is reasonable to say that the smaller bubble size in the air-nanofluid system is a result of the reduced coalescence efficiency. According to Eq. (15), the coalescence efficiency is determined by the contact time s and the drainage time tdr [72]. The contact time s is controlled by the external liquid flow and turbulence in the bulk [73]. Considering the very low nanoparticle concentration (0.1v%) in the bulk liquid in the Park and Chang’s experiments [9], the nanoparticles would probably have not remarkable influence the contact time. On the other hand, the drainage time tdr, which is the time required for the thinning process, is determined by the flow in the liquid film between the bubbles. Du et al. [5] experimentally investigated the stability of quiescent bubbles coated with silica particles (primary diameter of 20 nm) and concluded that the adsorbed nanoparticles hindered the water flow at bubble surface and slowed down the film thinning process. Apparently, the drainage time of the liquid film in nanoparticle-containing systems can be elongated. However, this effect is currently unable to be mechanistically modelled due to the complexity. Alternatively, the numerical study by Yuan et al. [18] introduced a correction coefficient kd ranging from 1.0 to 2.0 into the classic PrinceBlanch [74] model in order to quantify the contribution of nanoparticle adsorptions to the prolonged film drainage time.

t0dr ¼ kd tdr ðPrince
BlanchÞ  tdr ¼

r3b ql 16r

1=2 ln

h0 hf

size and concentration on the bubble interfaces. Neither experimental nor theoretical studies are sufficiently conduced to develop a model that considers all these parameters. 4.2. The thinning process According to Oolman and Blanch [75], the thinning process of a clean liquid film is predominantly driven by the capillary pressure induced by the variations of the curvature of the gas-liquid interface. The interface is very close to flat at the centre of the film and the pressure at that point equals to the pressure inside the bubble. Outside the film a surface tension force towards the centre of the bubble is induced by the curvature. This surface tension force has to be balanced by a change in pressure across the interface. Thus the pressure in the bulk liquid outside the film is smaller than the pressure at the film’s centre. It is the pressure difference (Eq. (18)) that pushes the liquid in the film to flow outside:

Pc ¼

2r rb

ð18Þ

The thinning process could be governed using the conservation equations of mass and momentum (Eqs. (19) and (20)).

ð16Þ ð17Þ

It was found that when the correction coefficient took the value of kd = 1.02, the model achieved close predictions of the void fraction with the experiment data, as shown in Fig. 14. This indicates that the drainage time is indeed elongated by nanoparticles. For the purpose of further comparison, the predicted bubble size distributions are shown in Fig. 15. When the coefficient kd increased from 1.0 to 2.0, the bubble group with the highest fraction moved from Group 4 (4.5–6 mm) to Group 2 (1.5–3.0 mm), which demonstrated the inverse relationship between the drainage time and bubble diameters. Although a good agreement was achieved [18], it should be noticed that the correction coefficient kd is case sensitive and subject to a number of factors particularly the nanoparticle material,

Fig. 14. Comparison of predicted void fraction against experimental data of the bubbly flows containing nanoparticles [16].

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olate that restricting the mobility of the bubble surface through nanoparticle adsorption might be one of the important mechanisms responsible for the elongated drainage time. Considering this, it is reasonable to say that Eq. (17), which was proposed by Prince and Blanch under the assumption that the bubble surface was fully mobile and zero-stress (Fig. 17(a)), is no longer applicable to liquid film contaminated with nanoparticles. When nanoparticles gradually assemble at the interface and partially cover the bubble (Fig. 17. (b)), the liquid flow becomes quasi-steady creeping. Chesters [73] defined the drainage time for partially mobile interfaces.

tdr ¼

@u q s ¼ gDus
rP þ qg @t

ð19Þ ð20Þ

tdr ¼

Where h is the film thickness, us is the liquid velocity, and P is the pressure gradient (Fig. 16). These two equations must be closed using appropriate boundary conditions at the gas-liquid interface, which is crucial to properly determine the drainage dynamics. Mysels [76] investigated the drainage of a foam film and proposed two limiting cases of drainage, depending on the mobility of the interfaces: zero stress at a mobile interface and zero velocity at an immobile interface. Rio and Biance [77] compared the results obtained with the mobile [78] and immobile [79] boundary conditions and found that it took almost 80 ls for the immobile film to reach 10
8 m from 10
6 m, whereas it needs only 0.7 ls in the mobile case. This indicates that the immobility of the interface can significantly increase the film drainage time. A number of factors can affect the film thinning process, which are summarized as follows. 4.2.1. Surface mobility and rigidity A number of experimental and numerical studies have demonstrated the adsorbed nanoparticles can restrict the mobility of bubble interface. Lin and Slattery [80] developed a theoretical model of the thinning of the liquid film which is formed as a bubble approaches an interface. They found that very small surface tension gradients are sufficient to immobilize the interface. Worthen et al. [7] further suggested the addition of nanoparticles could increase the effective viscosity of the injected gas bubbles and hence reduce the bubble mobility. It is thus reasonable to extrap-

air liquid

y

h

x

air Fig. 16. Drainage of a liquid film under capillary pressure [77].

ð21Þ

When bubbles are fully covered with nanoparticles, their surfaces become fully immobile. According to Marrucci [81], the viscous effects instead of the inertial effects dominantly control the film thinning process. The liquid is expelled from between these immobile surfaces by a laminar flow. As illustrated in Fig. 17(c), the velocity profile in the film becomes parabolic with a zero slip at the bubble interface. Considering the fully restricted mobility, Chesters [73] derived the drainage time.

Fig. 15. Comparison of predicted bubble size fraction when kd takes the value of kd = 1.0–2.0 [16].

@h @hus þ ¼0 @t @x

plg F1=2  1 1 
2ð2pr=rb Þ3=2 hf h0

3l l F 2 1 1 r
16pr2 b h2f h20

!

ð22Þ

Since the drainage time of films with immobile interfaces is predicted to be tens or hundreds times longer than that of films with mobile interfaces, predicting the transition from very rapid to very slow coalescence becomes an important issue. Marrucci [81] correlated the transition to the nanoparticle concentration c and proposed a model for the critical concentration:

ct ¼ 0:084R0 TðrA2H =rb Þ

1=3

ð@ r=@cÞ
2

ð23Þ

where AH is the Hamaker constant and R’ is the ideal gas constant. It should be noted that the above analysis was based on the parallel model which assumes that the surfaces of coalescing bubbles deform into two parallel discs (Fig. 18(a)). In fact, when nanoparticles fully cover the bubbles, their surfaces can be slightly deformed and like rigid spherical particles (Fig. 18(b)). For two nondeformable spheres, the drainage time is defined by [73]

tdr ¼

3pll 2 h0 r ln 2F b hf

ð24Þ

4.2.2. Pressures Another factor influencing the thinning process is the pressing force that brings two bubbles to coalescence. This pressing force is usually described as the capillary pressure between the bubbles and the inter-film fluid. When nanoparticles have a very small contact angle, they can completely rest in the liquid film, lying between the two bubbles (Fig. 19). As drainage occurs, the bubbles form a meniscus around the particle. The curvature of the meniscus induces a net surface tension force towards the centre of the bubble which has to be balanced by a changed pressure. The pressure at the centre of the film is no longer equal to the pressure inside the bubble but becomes much smaller. As a result, the capillary pressure Pc of bubbles with nanoparticles is much smaller than the capillary pressure Pc of bubbles without nanoparticles.  Pic can be expressed as:

Pc ¼

  2r rb 1
rb rm

ð25Þ

where rm is the curvature radius of the meniscus. With a smaller capillary pressure, a slower thinning process happens, which leads

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Fig. 18. The geometry of the liquid film. (a) Deformable surfaces. (b) Nondeformable surfaces.

air h

liquid air

Fig. 19. Schematic overview of the liquid film with particles residing in [6].

Fig. 17. The velocity profile of the liquid in the film with (a) Fully mobile interfaces. (b) Partially mobile interface. (c) Fully immobile interfaces.

to a longer drainage time. As shown in Eq. (25), not only the curvature of the meniscus can affect the capillary pressure, but the surface tension r plays an important role. In a number of studies, nanoparticles have been demonstrated to be effective in lowering the surface tension of interfaces [23]. During the thinning process, the surface area increases whereas the concentration of adsorbed nanoparticles decreases. Since surface tension is an inverse function of nanoparticle concentration, a surface tension gradient can

develop along the bubble surface. According to Oolman and Blanch [75], the change of surface tension r due to the existence of impurities like nanoparticles can be expressed as:

Dr ¼

  2 1 2c @r h R0 T @c

ð26Þ

It is thus important to consider the change of surface tension when calculating the capillary pressure. For pure liquid, it is well known that the capillary pressure is the only pressure acting on the liquid film. But Oolman and Blanch [75] found that when a second component exists in the liquid, other pressures resisting the film thinning can develop. These

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process would become slower so that the rupture time even can be comparable to the drainage time.

pressures include the electrostatic double layer force and steric repulsion force. Through a mechanistic analysis, Langevin [82] pointed out that these disjoining pressures are mainly responsible for stabilizing foams. In fact, for nanoparticles with ionisable surface groups (e.g. latex or silica), the part of the particle immersed in the aqueous phase will become charged (Fig. 20). Thus an electrostatic double layer can be established. Sagert and Quinn [83] investigated the effect of electrostatic forces on thinning process and found proposed that this repulsive force can balance the capillary pressure and cause the film thinning to stop at an equilibrium film thickness. The equation of electrostatic double layer force is given by [84]:

Pe ¼ ð64pkB Trb q1 c2 =j2 Þ expð
jhÞ

4.3. The rupture process It is widely accepted that the growth of thermodynamic instability of the liquid film is the main contributor to the film rupture [86]. The instability is caused by the thermal fluctuations which can corrugate a deformable interface. Initial amplitude of surface wave at a single interface is very small, approximately 1  10
10 to 5  10
10 m [87]. During the thinning process, the amplitude of the surface waves keeps growing. Once the wave amplitude reaches to the critical film thickness hf, the film will rupture and the two bubbles start to coalesce. Thermal corrugations of the interfaces of liquid films were first observed through light-scattering experiments (Fig. 21(a)) [86]. These fluctuations are inhibited by surface tension but enhanced by Van der Waals attractive interactions between both sides of the film. Considering this effect, Vrij and Overbeek [88] formulated the critical wavelength Kc of the thermal fluctuations.

ð27Þ

where q1, c, and j represent the density of electric charge in the bulk solution, reduced surface potential and Debye screening length, respectively. In addition, Samanta and Ghosh [85] believed that the reduced bubble coalescence in contaminated systems is mainly due to the steric force imparted by the adsorption of amphiphilic contaminants at the gas-liquid interfaces. The adsorbed layer encounters a reduction in entropy when confined in a very small space as the bubbles approach to each other. Since the reduction in entropy is thermodynamically unfavourable, their approach is thus inhibited. According to Böker et al. [23], some nanoparticles such as Janus-particles like polymers have two surface regions: polar surface region and apolar surface region. These nanoparticles are surface-active and amphiphilic [23]. It is reasonable to extrapolate that when two bubbles approach to each other, similarly to the polymeric surfactant, the hydrated head groups of adsorbed nanoparticles will be overlapped, generating a steric repulsion force. This force could be calculated by [85].

Ps ¼

kB T s3

"   3=4 # 9=4 2L d
d 2L

sffiffiffiffiffiffiffiffiffiffiffiffi prh4 Kc ¼ 2 p AH

ð29Þ

The rupture time, which is the time required for a surface wave to develop to the critical film thickness hf, is calculated by

trp ¼ 10  96p2 ll rA
2 H hf

5

ð30Þ
8

where trp is estimated to be 330 ms for hf=10 m when the Hamaker constant AH is 10
20 J [88]. The rupture time increases with the film thickness h. When the equilibrium thickness is larger than 10
8 m due to the nanoparticle-induced disjoining pressures, the rupture time will be comparable to the drainage time and cannot be ignored. In addition, Rio and Biance [77] proposed that the presence of impurities can also limit the film rupture by the following two mechanisms: damping the fluctuations and providing an energy barrier.

ð28Þ

where d, L and s represent the distance between the interfaces, the thickness of the polymer layer, the mean distance between the attachment points. As drainage occurs, the above-mentioned disjoining pressures withstand the capillary pressure. The pressure gradient P in Eq. (20) thus decreases, slowing down the liquid flow in the film and elongating the drainage time. Langevin [82] further found that as the liquid film thins, the disjoining pressures can also affect the equilibrium thickness of the liquid film. In the absence of nanoparticles and the induced disjoining pressures, the capillary pressure drains the liquid film to the critical thickness hf and the film surface waves rupture the film rapidly. Therefore, in pure liquid the film rupture time is much smaller than the drainage time and usually not counted in the coalescence time. With the existence of nanoparticles, the disjoining pressure can equilibrate the capillary pressure at a thickness larger than the critical thickness hf. When this happens, the rupture

4.3.1. Fluctuation damping Bergeron [89] experimentally investigated the influences of surfactants on the liquid film stability using a rectangular ‘‘openframe” probe and the porous plate technique. He found the energy cost associated with thermal fluctuation is increased by the elasticity of the surfactant layer at the gas-liquid interface. This effect tends to decrease the probability of spatial fluctuations. Blute et al. [90] found silica nanoparticles (5–40 nm) and surfactants have similar effects on increasing the surface elasticity. When the interface gradually adsorbs nanoparticles and becomes rigid, the surface elasticity can exceed the surface tension and reduce the probability of expansion of a fluctuation (Fig. 21(b)). According to Rio and Biance [77], this effect, which is known as the GibbsMaragoni effect, is the common mechanism describing how the

air Electrostac double layer force air Fig. 20. Electrostatic double layer force between two nanoparticle-adsorbed bubble interfaces.

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(a)

h

(b)

h

Fig. 21. Corrugations of bubble interfaces [77]. (a) Without the adsorption of nanoparticles. (b) With the adsorption of nanoparticles.

presence of impurities can reduce the rupture and increase the stability of the liquid film. 4.3.2. Energy barrier Through a theoretical analysis of the nucleation of a hole in a thin liquid film, Wennerström et al. [91] found larger curvature energy (a part of surface free energy) of interfaces covered by surfactants also helps stabilizing the thin liquid films. The nucleation of a hole in a thin film is associated with a large curvature, which has an energetic cost that increases the energetic barrier to overcome for rupture [77]. This energy is larger when surfactants are attached to the interface. Similar explanations were also found in Timothy et al.’s study [6] where the role of particles in stabilizing foams was investigated. It was stated that in the rupture stage an energy barrier must be overcome to form a critically-sized hole in the liquid film. The stability of the film can be considered in line with the energy required for the hole formation. Because of the high free energies involved with strongly adsorbed particles, they are far more likely to be laterally moved along the contact interface, rather than expulsed into the open liquid. Thus the hole formation and expansion with the existence of nanoparticles can be much more difficult, which consequently elongates the film rupture time. When the film rupture time becomes long that it is comparable to the film drainage time, it should be incorporated in Eq. (15) for an improved calculation of the coalescence efficiency. However, the realistic situations are very diverse and complicated. As pointed out by Rio and Biance [77], the film rupture even can be stochastic if the drainage time is smaller than the time required to develop an instability. In view of the fact that all the abovementioned mechanisms are closely related to the nanoparticle layer at the bubble interface, more in-depth knowledge about the structure of the adsorbed nanoparticle layer is needed. In recent years, the development of Molecular Dynamics simulations may has provided a plausible solution to the formulation of the interfacial terms in order to address the effects of adsorbed nanoparticles.

armour”, which improves the rigidity and reduces the mobility of the bubble interface, and makes the bubble behaviours sit somewhere between a clean bubble and a solid particle. Consequently, flow separation occurs and a slanted wake region forms behind the nanoparticle-adsorbed bubble at a smaller Reynolds number. Both the pressure and viscous stress on bubble interface become asymmetrically distributed. As a result, the lift force acting on a nanoparticle-coated bubble reverses its direction at a smaller bubble diameter and the drag force increases much faster with the bubble Reynolds number when compared with those in pure liquids. In addition, the interactions between nanoparticles such as the electrostatic double layer force and steric repulsion force significantly resist the approach of two bubbles and hinder the rapture of liquid film, which lead to a remarkably reduced bubble coalescence rate in nanoparticle-liquid mixtures than in pure liquids. This was probably the reason that despite the same bubble injection conditions in Park and Chang’s experiments [9], the bubbles in the Al2O3-water nanofluid were significantly smaller than in pure water. However, theoretical modelling of bubbly systems of nanoparticle-liquid mixtures remains very challenging due to the difficulties in formulating the modified bubble behaviours induced by the adsorbed nanoparticles. Traditional two-phase flow theories seem to have encountered a bottleneck. Alternatively, particle-based methods [92,93] such as molecular dynamics, Brownian dynamics, dissipative particle dynamics and Mente Carlo simulations may be capable of achieving an insight into the embedded physics and generating promising closure models for the two-phase flow models. Acknowledgements The financial support provided by the Australian Research Council (Project ID DP130100819) is gratefully acknowledged. Conflict of interest statement The authors declared that there is no conflict of interest.

5. Summary Appendix A. Supplementary material The experimental and theoretical studies on the spontaneous nanoparticle adsorption on bubble interfaces and its effects on the bubble-liquid and bubble-bubble interactions were reviewed. It was addressed that the adsorbed nanoparticles form a ‘‘colloidal

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2017.12.053.

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