Effects of Force Fields on Interface Dynamics in view of Two-Phase Heat Transfer Enhancement and Phase Management in Space Applications

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University of Pisa

Department of Energy, Systems,

Territory and Construction Engineering

PhD Dissertation

Effects of Force Fields on Interface Dynamics in view

of Two-Phase Heat Transfer Enhancement and

Phase Management in Space Applications

Giacomo Saccone

May 2018

Supervisor: Professor Paolo Di Marco

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2

Giacomo Saccone

Effects of Force Fields on Interface Dynamics in view of Two-Phase

Heat Transfer Enhancement and Phase Management in Space

Applications

Giacomo Saccone, 2018

Supervisor:

Professor Paolo Di Marco University of Pisa, Dept. of Energy,

Systems, Territory and Construction

Engineering, Italy

Revised by:

Professor Catherine Colin Institut de Mécanique des Fluides de

Toulouse (IMFT), France

Professor Akio Tomiyama Graduate School of Engineering,

Kobe University, Japan

Department of Energy, Systems, Territory and Construction Engineering

University of Pisa

Largo Lucio Lazzarino, 1

56122 Pisa, Italy

Reference to this publication should be written as:

Saccone, G. (2018). Effects of Force Fields on Interface Dynamics in view of

Two-Phase Heat Transfer Enhancement and Phase Management in Space

Applications. PhD Dissertation, University of Pisa.

ISBN xxx-xxx-xxx. doi: nn/xxx

Pisa, Italy 2018

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4

Giacomo Saccone

Monsieur le Président de la République,

Me permettez-vous, dans ma gratitude pour le bienveillant accueil que vous m’avez fait un jour,

d’avoir le souci de votre juste gloire et de vous dire que votre étoile, si heureuse jusqu’ici, est menacée

de la plus honteuse, de la plus ineffaçable des taches? Vous êtes sorti sain et sauf des basses

calomnies, vous avez conquis les coeurs. Vous apparaissez rayonnant dans l’apothéose de cette fête

patriotique que l’alliance russe a été pour la France, et vous vous préparez à présider au solennel

triomphe de notre Exposition Universelle, qui couronnera notre grand siècle de travail, de vérité et

de liberté. Mais quelle tache de boue sur votre nom - j’allais dire sur votre règne - que cette

abominable affaire Dreyfus! Un conseil de guerre vient, par ordre, d’oser acquitter un Esterhazy,

soufflet suprême à toute vérité, à toute justice. Et c’est fini, la France a sur la joue cette souillure,

l’histoire écrira que c’est sous votre présidence qu’un tel crime social a pu être commis. Puisqu’ils

ont osé, j’oserai aussi, moi. La vérité, je la dirai, car j’ai promis de la dire, si la justice,

régulièrement saisie, ne la faisait pas, pleine et entière. Mon devoir est de parler, je ne veux pas être

complice. Mes nuits seraient hantées par le spectre de l’innocent qui expie là-bas, dans la plus

affreuse des tortures, un crime qu’il n’a pas commis. Et c’est à vous, monsieur le Président, que je

la crierai, cette vérité, de toute la force de ma révolte d’honnête homme. Pour votre honneur, je suis

convaincu que vous l’ignorez. Et à qui donc dénoncerai-je la tourbe malfaisante des vrais coupables,

si ce n’est à vous, le premier magistrat du pays?

[…] Mais cette lettre est longue, monsieur le Président, et il est temps de conclure. J’accuse le

lieutenant-colonel du Paty de Clam d’avoir été l’ouvrier diabolique de l’erreur judiciaire, en

inconscient, je veux le croire, et d’avoir ensuite défendu son oeuvre néfaste, depuis trois ans, par les

machinations les plus saugrenues et les plus coupables. J’accuse le général Mercier de s’être rendu

complice, tout au moins par faiblesse d’esprit, d’une des plus grandes iniquités du siècle. J’accuse le

général Billot d’avoir eu entre les mains les preuves certaines de l’innocence de Dreyfus et de les

avoir étouffées, de s’être rendu coupable de ce crime de lèse- humanité et de lèse-justice, dans un but

politique et pour sauver l’état-major compromis. J’accuse le général de Boisdeffre et le général Gonse

de s’être rendus complices du même crime, l’un sans doute par passion cléricale, l’autre peut-être

par cet esprit de corps qui fait des bureaux de la guerre l’arche sainte, inattaquable. J’accuse le

général de Pellieux et le commandant Ravary d’avoir fait une enquête scélérate, j’entends par là une

enquête de la plus monstrueuse partialité, dont nous avons, dans le rapport du second, un

impérissable monument de naïve audace. J’accuse les trois experts en écritures, les sieurs Belhomme,

Varinard et Couard, d’avoir fait des rapports mensongers et frauduleux, à moins qu’un examen

médical ne les déclare atteints d’une maladie de la vue et du jugement. J’accuse les bureaux de la

guerre d’avoir mené dans la presse, particulièrement dans L’Éclair et dans L’Écho de Paris, une

campagne abominable, pour égarer l’opinion et couvrir leur faute.

J’accuse enfin le premier conseil de guerre d’avoir violé le droit, en condamnant un accusé sur une

pièce restée secrète, et j’accuse le second conseil de guerre d’avoir couvert cette illégalité, par ordre, en

commettant à son tour le crime juridique d’acquitter sciemment un coupable. En portant ces

accusations, je n’ignore pas que je me mets sous le coup des articles 30 et 31 de la loi sur la presse

du 29 juillet 1881, qui punit les délits de diffamation. Et c’est volontairement que je m’expose.

Quant aux gens que j’accuse, je ne les connais pas, je ne les ai jamais vus, je n’ai contre eux ni

rancune ni haine. Ils ne sont pour moi que des entités, des esprits de malfaisance sociale. Et l’acte

que j’accomplis ici n’est qu’un moyen révolutionnaire pour hâter l’explosion de la vérité et de la

justice. Je n’ai qu’une passion, celle de la lumière, au nom de l’humanité qui a tant souffert et qui

a droit au bonheur. Ma protestation enflammée n’est que le cri de mon âme. Qu’on ose donc me

traduire en cour d’assises et que l’enquête ait lieu au grand jour! J’attends. Veuillez agréer,

monsieur le Président, l’assurance de mon profond respect.

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6

Giacomo Saccone

Nomenclature

𝑞

Heat Flux

𝑞

_"##

_{Critical Heat Flux}

ℎ

_%&

Latent Heat of Vaporization

𝜌

_&

Vapor density

𝜌

_%

, 𝜌

₎

Liquid density

𝜎, 𝛾

Surface Tension

𝑔

Gravity acceleration

𝑞

_-.##

_{Heat Flux for Nucleate Boiling}

𝜇

_%

Liquid viscosity

𝑃𝑟

_%

Liquid Prandtl Number =

23,454 64

𝑐

_8,%

Liquid Specific Heat

𝑘

_%

Liquid conductivity

𝑇

_;

Wall Temperature

𝑇

_<=>

Saturation Temperature

𝑝

Pressure

𝐾&

micro-structures permeability

𝑃2

Capillary Pressure

𝐸_;

Energy of wetting

𝑉

Volume

𝜃,

𝜗

Apparent Contact Angle

𝑣

Velocity

𝑡

_G,6

Element of Maxwell Stress Tensor

𝑇_H

𝐾

Interface Curvature

𝐾

_I

Interface Curvature at interface top

𝑝

_J

Gas Pressure

𝑝

₎

Fluid Pressure

𝑅

L

, 𝑅

M

Curvature Radii

𝑛

Unitary normal vector

∆𝑓

_H,-

Electric Stress

𝑇

_H,)

Maxwell Stress Tensor on the fluid side

𝑦

Coordinate of the interface from bubble top

𝑟

Radial coordinate of the interface

𝐾

_L

, 𝐾

_M

Local Curvatures

𝐸

_JH-

Heat Generated

𝐸

G-

Heat entering the control volume

𝐸

_ST>

Heat leaving the control volume

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10

Giacomo Saccone

𝐸

<>

Heat stored in the control volume

𝐼

_V

Heater Power

𝑉

_V

Heater Voltage

𝐴

_V

Heater Area

𝑉

Volume

𝐿

Interface Length

𝑧

𝑧 = 𝐻 − 𝑦

𝐻

Interface height

𝑚

Mass

𝑡

Time

𝑝

_H^>

Pressure outside the interface

𝑅

Interface radius (for droplets 𝑅 = 𝐷 2)

d_bc

Equivalent diameter

𝑚

Mass flow rate of imbibition

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Abstract

Phase change is generally the most effective method to transfer heat. When

phase change is present, an interface between the two phases exists. The

dynamics of interfaces is a very important topic for everything concerning

heat transfer.

Since the two phases are characterized by a strong density difference,

gravity is in most cases essential to separate them. When gravity lacks, the

two phases are mixed together and there is no preferential direction where

the less dense phase can go.

The purpose of this work is to prove the feasibility, reproducibility and

reliability in the use of the electric field as a replacement of gravity in

separating the two phases and enhancing heat transfer.

The dissertation will discuss several years of experimental investigations

dealing with droplets and bubbles, with and without transfer of heat,

momentum and mass across the interface, with and without the gravity field,

with and without the electric field.

Finally, the used tools, literature and new models implemented to derive all

forces acting on the interfaces will be discussed in order to complete the

analysis from an experimental and theoretical point of view.

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12

Giacomo Saccone

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Chapter I. Introduction

Heat transfer is a ubiquitous phenomenon. Many applications on earth and in space require a high efficiency in transferring heat: nuclear and thermal power plants for electric power production, boilers, heat exchangers, air conditioning systems, condensers; as well as satellites and space stations. The wide implementation of heat transfer, and the demand to reduce size and costs recalls an increase in efficiency. Heat can be transferred via conduction, convection and/or radiation. Convection and, in particular, forced convection is the best solution in terms of efficiency to transport the energy associated to heat. But we can do more.

Convective heat transfer requires a fluid and a closed hydraulic loop in case of forced convection. It is well known in literature that within convective heat transfer, the highest performances can be reached only with the use of phase change.

Phase change implies two aspects: the exploitation of the latent heat of vaporization/condensation, which enables very high heat transfer coefficients (see Table 1); and the introduction of an interface between the liquid phase and the vapor phase. Conditions of heat transfer Heat Transfer Coefficient _[W/m₂_K] Gases in free convection 5 – 40 Water in free convection 100 – 1500 Gas flow in tubes and between tubes 10 – 400 Water flowing in pipes 500 – 20000 Molten metals flowing in tubes 2000 – 45000 Water nucleate boiling 2000 – 100000 Film condensation of water vapor 4000 - 20000 Dropwise condensation of water vapor 30000 – 150000 Table 1: Approximate values of heat transfer coefficients Thus, heat transfer with phase change is one of the most interesting and efficient but also challenging methods. The challenge derives from the tricky behavior of the interface and, in particular, of the three-phase contact line (the line where vapor, liquid and solid meet), which strongly affects the heat transferred. For what concerns space applications, space management systems have the duty to maintain all components within acceptable temperature ranges. Nowadays, environmental cooling on the International Space Station is a two-fluid, single-phase system. Water is circulated through pipes in contact with the station interiors and brings heat to a heat exchanger provided with anhydrous ammonia

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Giacomo Saccone

circulating in pipes exposed to the external space environment [1]. Several problems derive from the present configuration: the low efficiency, which means a larger and heavier system; freezing problems of the fluid in contact with outer space and, last but not least, ammonia introduces health risks in case of leaks. A more efficient method of heat transfer makes use of flow boiling in a single liquid two-phase cooling system. In two-phase systems, the coolant has a low enough boiling temperature that when it absorbs ambient heat it begins to boil. The vapors are then cycled outside the station where they condense back into a liquid that can be cycled back [1]. One candidate fluid for a two-phase cooling system aboard the International Space Station is a member of the perfluorohexane family. The commercial product FC-72 (manufactured by 3M) has been chosen due to low boiling point, the fact that it is a liquid at standard temperature and pressure, its dielectric constant, low viscosity, as well as the absence of flammability and toxic effects in humans [1] (see Table 2). Water on the other side, would be perfect as heat transfer fluid, but space engineers already worry about ammonia freezing when it gets in contact with the outer space. And ammonia freezes at –78°C at 1 atm. Boiling Point (1 atm) 56°C Pour Point

-90°C

Estimated Critical Temperature

449 K Estimated Critical Pressure

1.83 x 106_Pa Vapor Pressure 30.9 x 103_Pa Latent Heat of Vaporization (1atm) 85 kJ/kg Liquid Density (25°C) 1690 kg/m3 Kinematic Viscosity 0.0038 cm2_/s Absolute Viscosity 640 µPa s Liquid Specific Heat 1100 J/kg K Liquid Thermal Conductivity 57 mW/m K Surface Tension 10 mN/m Dielectric Constant 1.75 Electrical Resistivity 1 x 105_Ω cm Table 2: FC-72 Properties from 3M [2] and REFPROP (perfluorohexan): Reference Fluid Thermodynamic and Transport Properties Database Once the importance of boiling, condensation and evaporation has been assessed, due to their incomparable efficiency in transferring heat, a new entity has to be introduced: the interface. Every time a system contains different phases, interfaces are present to separate them. The two phases exchange heat, mass and momentum across interfaces and thus, the behaviour, shape and size of the interface strongly affects these transfers. In the next section, we will discuss the relevant applications.

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I. A world using interfaces in everyday life

Boiling, condensation and evaporation are of relevance in various processes of engineering, physics, biology, life science and climatology. We can refer to the condensation cone forming around aircraft during the sonic boom, bubble formation in liquid fuels of rockets, vapor explosion in volcanoes, heterogeneous reaction on droplet and aerosol surfaces in the atmosphere.

Interfaces are important also for controlling crystallization and shape selective crystal growth; phenomena occurring at the cell membrane - water interface control the docking of proteins, the transmission of signals as well as transport of molecules in and out of the cell.

Clouds are natural assemblages of tiny water droplets which coalesce due to changes in the atmosphere and lead to rainfall or other forms of precipitation. Natural water systems such as lakes and oceans contain air in dissolved form as well as bubbles, and make up a component that is essential to marine life. On the other hand, for what concerns industrial systems, such as nuclear power plants, bubbles and droplets are present in boiling water reactors, drops in all spray cooling components; and, in general, they are of concern for all safety issues related to nuclear industry [3]. In chemical reactors, drops and bubbles commonly transport both reactants and products. Combustion engines utilize sprays of atomized liquid hydrocarbons as fuel. Industrial boilers for steam generation and condensers in steam power plants are the most trivial examples. Within almost all of these applications, the importance of interfaces is related to the transport of heat and/or mass. And in general, the continuous demands for smaller, portable, light and less expensive apparatus, opened up an entirely new area for research, and novel techniques were developed to augment the heat transfer performance.

Not only space, but also the narrow dimensional requirements associated with the cooling of recent electronic equipment produced new challenges at the micro-scale level and in boiling and condensation.

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Giacomo Saccone

Figure 1: Condensation Cone Figure 2: Boiling Water Reactor Scheme Figure 3: Vapor enters the pressure suppression

pool – Direct Contact Condensation [3] Figure 4: CPU demand up to now in commercial computers

II. The use of two-phase heat transfer in space applications

Buoyancy, and thus gravity, in most cases governs the dynamics of interfaces. The larger is the density difference between the phases, the stronger is gravity importance.

Bubble detachment generally happens when buoyancy prevails over the attaching force. It is clear that, lacking buoyancy, detachment will be impaired.

In space applications, the need to reduce size and weight of all thermal management systems (power generation, electronics cooling, cabin temperature control, waste management, and regenerative fuel cells) suggests to replace current single-phase cooling systems with two-phase ones. In fact, this technique can lead

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to order-of-magnitude enhancement in heat transfer coefficients, giving the possibility to reduce sensibly heat exchangers surface, size and weight.

It is particularly in space applications that using electric field in such systems shows its relevance [4].

It has been assessed also in terrestrial gravity environment that the application of electric field enhances heat transfer coefficient in boiling ([4][5][6][7][8][9]), condensation and evaporation [10][11].

Thought, the most interesting aspect is the possibility to act as a replacement of gravity where it lacks: experiments of pool boiling and evaporating droplets conducted in microgravity assessed that the electric field can stabilize two-phase

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Giacomo Saccone

Chapter II. State of the art

As previously said, boiling and evaporation are ubiquitous phenomena. In both phenomena, gravity has a relevant role: it is generally responsible of bubble detachment during boiling [14], [15] and “flattens” the sessile droplets [16] aiding the heat transfer in both processes.

The present study is aimed to extend and use these mechanisms, which are the most efficient heat transfer methods, to space applications. Phase separation and thermal management of space stations and satellites are becoming of growing interest and the reduction of size and weights calls the increase of systems performances and efficiencies [4].

The lack of gravity in such applications can be balanced by the application of a tunable electric field (EF), which creates body forces that substitute gravity. Many published works are available in literature proving the viability of this solution for microgravity applications [7][10][11][13][17][18][19].

I. Bubbles and boiling

We already discussed the importance of boiling heat transfer: it provides a large heat transfer coefficient and thus, a high efficiency. Certainly, it has a limit: Critical Heat Flux (CHF) is an important performance-limiting condition in almost all boiling heat transfer processes. It is generally associated with the formation of an irreversible dry spot, responsible of a drastic reduction of the local heat transfer coefficient, which can lead to burnout of the boiling surface. Thus, it is important not only to understand the mechanisms that lead to CHF, but also to explore and identify ways to enhance it. When we refer to boiling, we talk about the mechanisms of phase change which brings to the formation of bubbles. In the present work, we are discussing pool boiling: the liquid boils from a heater submerged in a stagnant pool. Fluid flows will be generated only due to bubbles motion, natural convection and electric field effects.

In1934 Nukiyama [20] performed an experiment consisting of a wire heater and a pool of water. He derived the so called “boiling curve”, schematized in

Figure 5

. The relation between the heat flux provided and wall superheat (temperature of the wall minus saturation temperature) is clearly represented. Looking at

Figure

5

from left to right: we find the natural convection zone where bubbles are not present yet (Region I), when we reach the onset of nucleation, we have the formation of isolated bubbles (point A). A is the Onset of Nucleate Boiling (ONB). Moving from this point, rising the heat flux brings smaller increase of the superheat: the gradient of the curve drastically changes and small increase in wall temperature coincides with much higher heat fluxes. Region II just described is the region where nucleate boiling is fully developed (FDNB). Now increasing wall temperature, we have more and more bubbles, growing in size and frequency of

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detachment, they cover the whole heater surface. The more we approach the point C, the more bubbles cannot be distinguished one from the following and structures similar to column jets begin. Point C is called Critical Heat Flux. It represents the maximum heat flux exchangeable in safe conditions: at this point, jets coalesce one with another preventing the liquid from rewetting the surface: a vapor layer now covers the heater, leading to a strong and sudden degradation of the heat transfer coefficient (ratio between heat flux and wall superheat). From this point, increasing the heat flux generates high and, generally, dramatic increase of surface temperature (C to E points).

Most of the applications work in imposing heat flux. And almost all of them want to avoid such temperature overshoot. Instead, if we work by imposing wall temperature, moving from point C and increasing wall superheat, we have a degradation of the exchangeable heat transfer. The vapor blanket is instable and we can get back to vapor jets. C is the minimum heat flux which allows a stable vapor blanket. From this point, the curve brings to very high wall superheat (temperature can reach the melting point) and a larger and larger amount of the heat is transmitted by radiation.

Figure 5: Nukiyama Boiling Curve [20] For what concerns the Critical Heat Flux (CHF), several studies have addressed and speculated on the mechanisms leading to it. Some of the most famous works were published ([14][15][21][22][23][24][25][26][27][28][32]), their intents, and thus models, developed during the years, to understand and predict CHF can be divided in two categories [21]: the first and less recent approach refers to hydrodynamic considerations; the second and more recent, considers energetic and thermal aspects. In particular, the latter consider the influence of the contact angle and attempts to describe how the liquid can rewet the surface.

The oldest model for CHF is the one presented by Kutateladze and Zuber [22][23][14] who considered CHF analogous to a flooding phenomenon. In

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Giacomo Saccone

particular, Kutateladze set up an expression on the basis of dimensional analysis, and Zuber proposed a hydrodynamic explanation of CHF, which attributes the boiling crisis to the breakup of the two-phase flow stability close to the boiling surface: when vapor generation is too intense that it creates hydrodynamic instabilities (Kelvin-Helmoltz instabilities) which destroy the columnar jet configuration, pushing the liquid away from the surface, and causing surface dryout. According to this model, CHF can be expressed as

𝑞

_"##

_{= 𝐾ℎ}

%&

𝜌

&I.e

𝜎𝑔 𝜌

%

− 𝜌

& I.Me

(1)

where K is a tuning parameter taken equal to 0.157 for Kutateladze [22][23] and to 0.131 for Zuber [14].

Zuber developed a study based on Taylor and Helmholtz instability. The model consists in analyzing the transition zone, where the heater is covered by a blanket of vapor, in an instable configuration. When the blanket collapses and liquid enters the vapor film, Taylor instability takes place. Taylor wave length is the length of the wave which dominates during collapse. This wave is the result of a balance between surface tension, inertia and gravity. During transition, vapor rises in jets which distributes on the surface in a regular square array. In turn, the collapse of vapor jets occurs when vapor velocity reaches a critical value. Helmholtz instability arises when jets surface tension can no longer balance pressure forces induced by the flow field. Heat flux has to be balanced with the latent heat of vaporization brought away by vapor jets.

Later, Katto and Yokoya [24], and Haramura and Katto [25], addressed to vapor columns instability, present in the so called “macrolayer”, and to dryout as a cause of boiling crisis. The model assumes that heat transfer is related to macrolayer evaporation: the heater surface is wet by tiny liquid film underneath the blanket of vapor, which feed the blanket itself. When the mass of the blanket is big enough (buoyancy equals inertial forces due to the motion of the surrounding liquid), the agglomerate detaches and a new big bubble replaces it. The balance between inertia and buoyancy allows the determination of a growth period for such big bubbles. Hydrodynamics takes place because of the relative rising speed of vapor form the macrolayer. This situation refers to Helmholtz instability. The height of the macrolayer has been set to a quarter of Helmholtz wave length, but this value has been debated later. Despite this, dryout takes place when the liquid film is completely evaporated; and thus, CHF is linked to liquid film refeeding mechanisms. In pool boiling, for an infinite surface as soon as bubbles detach, liquid penetrates. When all the liquid evaporates within a bubble period, CHF occurs. From these considerations and with the aid of an energy balance, they found an expression for CHF:

fghh ijk4j lm(iopiq) iqs t u

=

vu Mtt_ws t tx yq y z {

1 −

yq y z tx }o }q

+ 1

LL L• }o }q

+ 1

€ z z tx

(2)

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Unlikely, the ratio between vapor cross sectional area

𝐴

_• and heated area

𝐴

was found by comparing the RHS to 0.131 [25], and thus, the results of this correlation will be the same one that can be find by using Zuber correlation [14]. Despite this, on the basis of photographic evidence, consensus arose for the existence of the tiny liquid film underneath the vapor bubbles in FDNB. This layer is referred as “macrolayer” and has been experimentally observed [26]. Lienhard and Dhir [31] critically evaluated the assumptions made in the Zuber’s theory and modified the vapor velocity condition at which instability would set in. Consequently, they showed that Zuber’s equation would underpredict the CHF by 14 percent [32]. More recently, Kandlikar [21][32] proposed a theory which includes hydrodynamic as well as non-hydrodynamic effects, he investigated the dependence of CHF on receding contact angle 𝛽 and surface orientation ∅. He proposed a theoretical expression for CHF in which the receding contact angle 𝛽 appears explicitly, since the liquid would start to recede at the onset of the boiling crisis:

𝑞

_"##

_{= ℎ}

%&

𝜌

&I.e L„2S< …_L• _vM

+

v_†

1 + 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠 ∅

I.e

𝜎𝑔 𝜌

_%

− 𝜌

_& I.Me

(3)

Experiments confirm that a low contact angle (highly wetting liquid) will result in a higher value of CHF, while a high contact angle, such as a non-wetting surface, will result in drastic reduction in CHF, as also confirmed by Costello and Frea [33]. Unfortunately, in our case, the working fluid will have a static contact angle of few degrees, while the correlation is supposed to work for receding contact angles between 20° and 110°.

In any case, Kandlikar’s correlation can predict an increase of CHF with hydrophilic surfaces and a degradation of it for hydrophobic surfaces. See Figure 6 for a comparison between Kandlikar [32] for ∅ = 0° and ∅ = 90°, Zuber [14] and Kutateladze [22][23] models.

The value reported is the ratio present in almost all models:

𝑞

_"##

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22

Giacomo Saccone

Figure 6: Comparison of CHF models vs Contact Angle

Concerning nucleate boiling performance, Rohsenow and Griffith [27] suggested that the heat transfer enhancement under these conditions is the result of local liquid circulation in the region close to the heating surface, promoted by successive bubble detachments. The correlation reads:

Ž••hh 54 V4j ‘ J }4’}j I.e

=

_"L “” t •

𝑃𝑟

_%’ “ •

23,4 –—’–“˜™(8) V4j t •

(5)

where r and s for water are 0.33 and 1.0, respectively. Rohsenow and Griffith introduced an empirical constant 𝐶œ• to account for the material and the roughness of the boiling surface. Later, Jabardo et al. [28], Liaw and Dhir [29] and Vachon et al. [30] identified a clear dependence between 𝐶œ• and the contact angle during pool boiling of water on copper. Furthermore, by varying the surface roughness, they obtained different contact angles and showed that contact angle and 𝐶_œ• decrease as the surface is made smoother. They studied experimentally the variation of the wall heat flux with the contact angles in fully developed nucleate boiling, with heat fluxes close to CHF. Their results showed that as the surface wettability is improved, that is, as the contact angle decreases, a higher superheat is necessary to achieve the same heat flux. Many other models have been proposed for nucleate boiling in past half century, and illustrating all of them is beyond the scope of this work.

24 _{Paolo Di Marco – Pool Boiling} UIT Summer School 2017

24

Vapor recoil theory

Kandlikar (2001) considered the force balance of a bubble

in a direction parallel to the heater surface, assuming the

bubble as a truncated sphere and equating surface

tension and pressure forces to vapor recoil force (reaction

due to momentum change of evaporating fluid at the

interface); surface inclination

φ is also considered. When

the recoil force cannot be balanced, bubble starts

spreading laterally and CHF occurs. The resulting

expression is

(

)

0.5

(

)

0.25 0.5 ,

1 cos

2 "

1 cos cos

16

4 ( , ) "

CHF g fg f g CHF Z

q

h

g

K

q

+

β

π



 



_

_

=

_

_{ }

+

β

φ ρ

_

_

σ

ρ −ρ

_

π



 



= β φ

CHF decreases with surface inclination (

φ = 0

horizontal upward,

φ = 90° vertical) and with

increasing contact angle. Note that

β is the

receding contact angle.

Agreement with

experimental data in literature is found in the

range

β = 20° to 110°. Note that CHF is zero

for a superhydrophobic surface (

β= 180°).

( the bubble diameter is assumed one half of the

Taylor wavelength)

0 0.05 0.1 0.15 0.2 0 30 60 90 120 150 180 Ka p p a ( b e ta ) beta, [deg] vertical horizontal K=0.131 K=0.157

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II. Boiling with microstructured surfaces

While all these correlations apply relatively well to traditional boiling systems, nowadays several techniques can be adopted to enhance CHF beyond the limits predicted by these correlations. Specifically, CHF can be enhanced using passive or active techniques, such as micro-structured surfaces and electric field, respectively. It is well established that in CHF the dynamics of the contact line plays an important role, and surface conditions, e.g. roughness, wettability and porosity, or engineered features play a role too. Hydrophilic micro structures, e.g. micro pillars, create a preferential path for the liquid to penetrate underneath bubbles and delay the formation of dry spots: liquid imbibition is a function of fluid and surface properties, such as the intrinsic wettability (microscopic contact angle), as well as geometrical parameters (spacing, dimension and height of the structures). CHF models based on roughness fail in capturing trends of CHF on textured surfaces [34] since they predict a monotonic increase with the micro structures density, which has been experimentally disproved [35][36]. Instead, models based on the balance between capillary pressure (driving force) and viscous forces (opposing force) can explain the liquid entrainment underneath dry spots [34][37].

Dhillon et al. [34] speculated that CHF happens when the boiling surface temperature reaches a maximum value (𝑇2žG>), measured experimentally. Using a one dimensional thermal lumped capacitance model the heating timescale has been evaluated and compared to a rewetting time scale given by the sum of the imbibition time scale (due to micro-pillars) and the sloshing time scale (expressing the role played by gravity, which promotes sloshing of the surrounding liquid towards the center of the bubble footprint). CHF is expected to occur when the heating time scales gets smaller than the rewetting time scale. The result of the balance between these time scales gives:

𝑞

_"Ÿ##

_{= 𝜌}

<

𝑐

<

𝑡

<

𝑇

2žG>

− 𝑇

I †∆}J € ‘ L/†

1 − 𝜏

ž

+

M∆}J£_5‘j¤¥

(6)

where 𝜌_< is the heater density (silicon substrate), 𝑐_< is the heater specific heat and 𝑡< is the heater thickness. 𝑇I = 137°𝐶 is the boiling surface temperature after bubble departure and 𝑇_2žG> = 149°𝐶 is the critical surface temperature when CHF happens (both measured experimentally). 𝜏_ž = ∆𝑝_ž 𝑔∆𝜌𝐷 is a non-dimensional term accounting for the resistance of partially wetting surfaces to rewetting, ∆𝑝ž = 2𝜎 1 − 𝑐𝑜𝑠 𝜃 𝐻 is the surface wetting pressure reduction term, and 𝑃₂= − ∆𝐸; ∆𝑉 is the capillary pressure. Finally, 𝐾& is the micro-structures permeability and 𝜃 is the apparent contact angle at the micro-texture level.

This model predicts consistently pool boiling CHF values measured on textured heaters consisting of pillars with an average pillar side and height of 10µm and different spacing, as shown in

Figure 7

.

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24

Giacomo Saccone

Figure 7: Experimental results and analytic predictions of Eq. (6) from [34] However, while this model can qualitatively predict the effect of pillar spacing on CHF with water at atmospheric pressure, it cannot predict the effect of pillar size. In fact, to an infinite pillar size corresponds a permanent increase around 100% of CHF value (see Figure 8).

Figure 8: Inconsistency of Eq. (6) for large micropillar dimension In the next chapter we will analyze a new model to calculate liquid imbibition and we will discuss electric field effects at the microstructures level. using the capillary and viscous pressures given by equations (1) and (2) while noting that the viscous pressure drop in this case only acts along a length (1–f)l:

P_c _! mvið1 " fÞl

Kv ) vi !

K_vP_c

m 1 " fð Þl ð6Þ

As the imbibition liquid front travels ahead of the sloshing liquid front, it travels the entire length of the dry spot and so the rewetting timescale can be expressed as:

tw ¼ _2vl

i ¼ 1 " fð Þtw;i; tw;i ¼ ms

2DrgKvPc ð7Þ

where tw,i denotes the imbibition-induced component of the

rewetting timescale. By comparing the rewetting timescales

obtained from equations (5) and (7), we find that f ¼ tw,i/(tw,gþ

tw,i). The rewetting timescale can therefore be written as:

tw ¼ 1=t# w;gþ max 0; 1=t! w;i"$" 1 ð8Þ

where the max operator is used to preclude a negative contribution to the rewetting timescale in cases where the

capillary pressure Pc is negative (non-imbibing textures). In

Fig. 6e, we plot this calculated rewetting timescale versus micropillar spacing b for the micro-textured surfaces and see that it agrees quite well with an experimental timescale obtained by extrapolating the silicone oil imbibition results to water (Supplementary Note 5) corresponding to a measured silicon– water equilibrium contact angle of yB30! (see Methods and Supplementary Fig. 5). It is interesting to note that the dry spot

rewetting timescale tw exhibits exactly the same trend as the

heating timescale th, strengthening our hypothesis that the

phenomena of boiling crisis is dictated by a competition between the heating and rewetting of a dry spot on the boiling surface.

Now that we have obtained expressions for the dry spot heating and rewetting timescales, and verified them against experimental data, the value of CHF can be obtained by equating them using equations (4) and (8): q00_CHF ¼ rsCstsðTcrit" ToÞ' 4Drg3 s % & 1=4 1 " tr ð Þ þ max 0;2DrgK_msvPc % & " # ð9Þ To verify the applicability of the scaling model and check its

accuracy, we compare the calculated rewetting (tw) and heating

(th) timescales at the experimentally observed CHF values for all

the micro- and nano-textured boiling samples in Fig. 7a. The

ratio of tw and th is plotted against the micropillar spacing b

using experimentally measured values of micropillar width (a), height (h) and spacing for each sample (Table 1). Recall that the underlying hypothesis of the model was that CHF is encountered as soon as the rewetting timescale exceeds the heating timescale, which is borne out by the fact that most of the CHF data points in

Fig. 7a fall on the horizontal line tw/th¼ 1.

Figure 7b plots the CHF curves obtained from equation (9) for both the micro-textured and nano-textured surfaces versus the micropillar spacing b, and compares them with the experimental CHF data. The CHF model nicely captures the maxima observed in the experimental CHF data and most of the CHF data points fall on the model curves within the margin of error. At large micropillar spacings (bZ200 mm), the rewetting of the dry spot is purely gravity-induced because either the liquid does not imbibe

into the micropillars (Pco0 for micro-textured samples) or the

imbibition liquid-front velocity vi is smaller than the velocity of

the sloshing liquid front vg (nano-textured samples). In this

regime, the slightly higher CHF for the perfectly wetting (y0¼ 0)

nano-textured surface is explained by the non-zero

sloshing-liquid pressure reduction term (DPr) for the partially wetting

micro-textured surface (y0B30!). From bB200 mm to bB50 mm, the CHF increases for both the surfaces, albeit due to different reasons. The smaller increase for the micro-textured surface is

due to a reduction in DPr (decreasing y0), whereas the larger

increase for the nano-textured surface is because of

imbibition-induced rewetting of the dry spot becoming active (vi4vg). For

bo50 mm, imbibition-induced rewetting becomes active for both

the surfaces with the CHF enhancement larger for the nano-textured surface because of a smaller micro-texture level

liquid–solid contact angle y1. Below a micropillar spacing of

B10–20 mm, CHF for both surfaces starts to decrease with further reductions in b. This is because reducing b now increases

the viscous pressure drop Pv to a larger degree than the

imbibition capillary pressure Pc, resulting in a reduced imbibition

liquid-front velocity vi. For small micropillar spacings (booa

and booh), the capillary pressure scales as PcB1/b, whereas the

viscous pressure drop scales as PvB1/b2 (or KvBb2).

Discussion

We can summarize the role of the micropillars and the nanograss (or nanopillars) in CHF enhancement as follows: (i) For a

a

b

2.0 1.5 1.0 0.5 !w / !h 0.0 100 250 200 150 100 50 0 100 101 102 103 104 101 Micropillar spacing, b (µm) Micropillar spacing, b (µm) Heat flux, q ″ (W cm –2 ) 102 ₁₀3 Micro-texture Nano-texture CHF, micro-texture CHF, nano-texture Scaling model Scaling model 104

Figure 7 | CHF scaling model results. (a) Plot of the ratio of calculated rewetting (tw) and heating (th) timescales at the experimentally observed CHF values for both the micro- and nano-textured surfaces versus measured micropillar spacing b. The actual measured values of a, b and h have been used for all the samples (Table 1). The measured contact angle of water on silicon isyB30! (see Methods and Supplementary Fig. 5) and Tcrit–ToB12 !C. (b ) Plot of experimental CHF data and theoretical curves obtained using the CHF scaling model versus micropillar spacing b for the micro- and nano-textured surfaces. Average micropillar width of a ¼ 10 mm and height of h ¼ 12.75 mm were used for generating the theoretical CHF curves.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9247

ARTICLE

NATURE COMMUNICATIONS| 6:8247 | DOI: 10.1038/ncomms9247 | www.nature.com/naturecommunications 9

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III. Droplets, evaporation and condensation

The problem of droplet dynamics has been discussed widely in literature. The interest on this topic is supported by its importance in many industrial processes, such as: oil recovery [38], lubrication, liquid coating [39], spray quenching [40] and ink-jet printing [41]. The application of an external electric field finds its relevance in performances improvement of the processes mentioned above, but is also interesting in space applications for phase separations and management [42], in processes involving heat and mass transfer across an interface, thanks to the promising enhancement in those transfer processes [10], and in processes needing an active method of wettability control [43][44].

The first study on wetting phenomena was probably reported by Young [74], giving the name to the well-known Young’s equation; resulting from the well-known equation expressing equilibrium of contact line along the surface:

𝛾

_©ª

= 𝛾

_©«

+ 𝛾

_«ª

𝑐𝑜𝑠 𝜃

(7)

Figure 9: Sketch of a sessile droplet Where the meaning of the different terms is clarified by Figure 9. We distinguish three phases: the liquid L, the solid surface S and the surrounding atmosphere G. θ is the equilibrium contact angle. The contact angle can be between θ = 0 and θ = 180° (See Figure 10). The former case corresponds to perfect wettability; the latter corresponds to perfect non-wettability. More often, as for the surfaces used in this study and for most metallic surfaces, partial wetting occurs.

Figure 10: Behaviour of a sessile droplet on a surface, from hydrophilic (left) to hydrophobic (right) Furthermore, it has to be noticed that real surfaces are neither perfectly smooth nor totally homogeneous. This difference with the ideal case is at least partly responsible of contact angle hysteresis. In reality, the static contact angle can be within the receding contact angle, θr, and the advancing contact angle, θa, defined

(26)

26

Giacomo Saccone

as the ultimate contact angle for which the liquid droplet starts to advance or recede on the surface, respectively. Even more challenging is the characterization of contact angle when the triple line moves advancing or receding on the solid surface, as in the case of spreading liquid droplets or even boiling phenomena. The dynamics of liquid spreading and receding is not yet completely understood, different correlations and theories have been proposed. This section addresses to the most commonly used theories: the hydrodynamic theory and the molecular kinetic theory. More in details, Blake and Haynes [47] proposed the Molecular Kinetic Theory (MKT), Tanner [48] gave an important contribution founding a general tendency in contact angle behaviour during spreading, Cox [45] and Voinov [49] proposed a Hydro Dynamic Theory (HDT). Thanks to the works made by Bayer and Megaridis [50] and Blake [51], a short insight in the two most important theories is proposed hereafter. The former concentrates the treatment on a purely hydrodynamic point of view, the latter focuses only on the molecular aspects of liquid displacement in the neighborhood of the wetting line, neglecting the dynamics of the flow. When Navier-Stokes Equations are solved in the proximity of a moving contact line, there is a singularity in stress at the contact line, leading the drag force to non-finite values on the solid boundary [52], [53], [54]. The stress diverges as 1/r when r → 0 (r represents the distance along the surface from the contact line). In order to avoid this singularity, slip has been postulated to occur between the liquid and the solid surface at small distances, LS, from the contact line. Four types of slip conditions have been postulated by different researchers [55]:

• Zero tangential stress at the solid surface within a distance LS from the contact line and no slip for distances greater than LS.

• Difference in tangential velocity between liquid and solid (slip velocity) proportional to the local shear velocity gradient at the solid surface. • Slip velocity algebraically dependent upon distance from the contact line. • Slip velocity proportional to the power of the local shear velocity gradient. Cox [45] considered the triple line motion for a general geometry in which one fluid displaces another. The main assumptions of Cox’s viscous theory are that the triple line motion is steady, the flow is entirely viscous, the liquid is advancing and the surface is ideally smooth. Cox defined the macroscopic contact angle in terms of the asymptotic angle that the interface makes with the solid surface as the contact line is approached at the macroscopic (experimental) length scale (LH). Using the

interface shape close to the contact line, Cox found an expression for the macroscopic contact angle in terms of the Triple Line Velocity (TLV) 𝑉"« and the microscopic contact angle θm, which was defined as the angle the liquid interface

forms with the solid surface at distances of the order of the slip length LS from the

(27)

Figure 11: Dynamic and microscopic contact angles at the triple line (figure from [51]) In its simplest form, the resulting formula describing the change in the dynamic contact angle due to the viscous bending of the liquid–gas interface (Figure 11) may be written in terms of the capillary number as:

𝐶𝑎 =

Jj -®’Jj(-¯) °± o²_o³ (8)

𝐶𝑎 =

5 &_´ (9) where the function g_¶ θ is given by

𝑔

_&

𝜃 =

_I-¸’œ¹± ¸ º»œ ¸_{M œ¹± ¸}

𝑑𝜙

(10)

For static or dynamics contact angles smaller than 3π/4, as in most metallic surfaces, the integrand may be approximated by θw _{9 ; hence Eq. (8) becomes:}

𝜃

_¾w

_{− 𝜃}

¿w

= 9 𝐶𝑎 𝑙𝑛 𝐿 𝐿

¿

𝑤𝑖𝑡ℎ 𝜃

¿

= 𝜃 𝑎𝑛𝑑 𝜃

¾

< 3𝜋/4

(11) There is, as pointed out by Cox, some experimental evidence to suggest that for some systems at least, the microscopic contact angle is a constant, whose value depends only on the particular liquid and solid surface involved. However, for systems in which the microscopic contact angle could depend on the spreading velocity, owing perhaps to effects at the molecular scale, Cox’s theory is still valid but with 𝜃¿ = 𝑓(𝑉"«). This last assumption is supported by Voinov [49] and Ramé et al. [56]. Shikhmurzaev [57] also suggested that the contact angle is not only velocity dependent, but also sensitive to the entire flow field near the wetting line. On the other side, MKT uses the theory of absolute reaction rates and states that the essential triple line motion takes place by molecules “jumping” along the solid surface from the liquid to the vapour side of the contact line [47], the macroscopic behaviour of the triple line now depends on the general statistics of the molecular

(28)

28

Giacomo Saccone

displacements. These displacements occur in the three-phase zone, as sketched in Figure 12. The molecular-kinetic theory postulates that the entire energy dissipation occurs at the moving triple line. The wetting line moves with velocity 𝑉"«, and the liquid shows a dynamic advancing contact angle 𝜃₌ such that 𝜃₌ > 𝜃, where 𝜃 is the equilibrium (static) contact angle.

According to this theory, the velocity of the triple line is determined by the frequency κ and length λ of each molecular displacement. These displacements take place at the adsorption sites on the solid surface. The length of the molecular displacement λ is influenced by the size of the liquid molecules and depends strongly on the spacing of successive adsorption sites on the target surface.

Figure 12: Motion of molecules at the contact line (figure from [51]) The triple line velocity (𝑈 in the figure) is then given by 𝑉"«= 𝜅 𝜆, where 𝜅 is the net frequency of molecular displacement (jump frequency). For the contact line to move, work must be done to overcome the energy barriers that prevent molecular displacement. This work is done by the surface tension force, which is:

𝛾 cos 𝜃 − cos 𝜃

₌ (12) expressed per unit length of the contact line. The work done by this force is entirely developed in the contact point zone. Combining these ideas and using Frenkel–Eyring activated rate theory of transport in liquids, the following relationship between 𝜃 and 𝑉_"« was obtained by Blake & Haynes [47]:

𝑉

_"«

= 2 𝜅

_;

𝜆 sinh

´ Ís M 6 –

cos 𝜃 − cos 𝜃

¾

(13)

where k and T denote Boltzmann’s constant and the absolute temperature respectively. The number of absorption sites per unit area on the surface (𝑛) is

(29)

related to λ by λ ≈ n−1/2_{. In addition, the equilibrium jump frequency 𝜅}_;_{is related to}

the effective molar activation energy of wetting 𝛥𝐺_Ð by:

𝜅

_;

=

6 –_V

exp

’ ÔªÕ_Ö

× 6 – (14)

where 𝑁_y is Avogadro’s number and

ℎ

stands for Planck’s constant.

For viscous flow in simple liquids, 𝛥𝐺_Ð is about 10 kJ/mol, as reported by Blake [58]. High or low values of 𝛥𝐺_Ð imply, respectively, strong or weak dependence of the contact angle on triple line velocity.

As noted by Bayer and Megaridis [50], a limitation of κÚ expression is that it lacks of consideration for viscous losses at the triple line, although this property may have a strong influence on the dynamic contact angle. In fact, both solid-liquid interactions and viscous molecular interactions are expected to operate at the triple line while spreading over the solid surface.

A better approximation to the real mechanism of dissipation at the contact line is to combine viscous and liquid/solid interactions by writing 𝛥𝐺Ð= 𝛥𝐺<+ 𝛥𝐺&, where ΔG_œ is the contribution arising from the influence of the surface, and 𝛥𝐺_& is the contribution due to the influence of the liquid interactions at the molecular scale.

On the basis of those assumptions and with the help of the theory of absolute reaction rates [47], two new terms of interaction frequency can be identified: the first associated to fluid/molecules interaction and the other to solid/liquid interaction:

𝜇 =

V_&

exp

_Ö Ôªj × 6 – (15)

𝜅

_<

=

6 – V

exp

’Ôª“ Ö× 6 – (16)

𝜅

;

= 𝜅

<_5 &V (17)

Where, here, 𝑣 is the specific volume of the fluid. Consequently, Eq. (13) can be rewritten as:

𝑉

_"«

=

2 𝜅

<

ℎ 𝜆

𝜇 𝜈

sinh

𝛾

2 𝑛 𝑘 𝑇

cos 𝜃 − cos 𝜃

¾ (18) Molecular-kinetic parameters can be thus obtained by the use of this equation with nonlinear least-squares-fit analysis of the experimental data and subsequent use of Eqs.(15)(16)(17). According to the theory outlined above [50], if liquid molecular interactions are weak, i.e. Δ𝐺& is negligible, then 𝜅;≈ 𝜅<. If solid-liquid interactions are also weak, i.e. 𝜅_< is large, then Eq. 23 predicts 𝜃₌ to be weakly dependent on the triple line velocity. Small equilibrium contact angles θ mean strong solid/liquid interactions, large equilibrium contact angles mean weak solid/liquid interactions.

(30)

30

Giacomo Saccone

Furthermore, Δ𝐺< Δ𝐺&≫ 1 means that the liquid is likely to interact strongly with the solid surface, as for example in the case of aqueous glycerol on glass. If Δ𝐺< Δ𝐺&≪ 1, the solid-liquid interactions are instead relatively weak, as for example in the case of silicon oils on glass.

Since none of these approaches is complete, efforts were made to combine those two theories.

Blake [51] made an interesting review about those topics and pointed out the limitation of data fitting to correlate the relevant parameters in both theories. Due to this inconvenience, numerical simulation is possible only after fitting of experimental results.

IV. Electric field Effects

To explore CHF enhancement using active methods, an external electric field can be used. Electric-field-induced improvements in boiling heat transfer and CHF have been already observed also on ground [6][7][8][9][12][17]. Analyzing the force balances proposed and validated in literature for growing gas bubbles [66][67] [68][69][70], the importance of buoyancy manifests itself: bubble departure is achieved when surface tension can no longer withstand buoyancy (pressure force can be generally neglected in comparison [68]).

The present work is focused on such force and power balances because they are important tools in assessing the role of electric forces and the modification induced by the change in the relevant parameters. As already mentioned, many authors performed force balances in order to predict bubble detachment diameter as well as bubble departure frequency as well as boiling heat transfer. In general, bubble will not detach until the balance is satisfied. The introduction of the electric force alters the balance leading to a different equilibrium which results in an elongated bubble, a higher curvature radius at its top and a different contact angle. One important first step in deriving an expression for the volumetric electric force (𝑓H###) that acts on a fluid has been proposed by Landau and Lifsitz [71]:

𝑓

_H'''

_{= 𝜌}

)

𝐸 −

âãä s M

𝛻𝜀

ç

+

âã M

𝛻 𝜌

èâé è} –

𝐸

M

₍₁₉₎

where

𝜌

₎ is the free electric charge density, E the electric field intensity, e0 the

vacuum dielectric permittivity, eR the relative electric permittivity and the

subscript T the temperature. Only the first term (electrophoresis) depends on the sign of the electric field, but it is present only when free charge, 𝜌₎, buildup occurs. It doesn’t play a role in the present study, where pure dielectric fluids are involved. The second and third terms are termed dielectrophoresis and electrostriction. The former one depends on the gradient of the dielectric constant (𝜀_ç, which may occur

(31)

either from thermal gradients and/or from phase discontinuities) and the latter one is due to the gradient of 𝐸M_{. It is worth noting that, in a homogeneous and} isothermal, medium the dielectrophoretic force acts just at the interfaces, where a gradient of electric permittivity is encountered. The volumic electric force can be inserted into the Navier-Stokes equation (single phase), yielding

𝜌

_%ê&_ê>

= −𝛻𝑝 + 𝜇

_%

𝛻

M

_{𝑣 + 𝜌}

%

𝑔 −

âãä s M

𝛻𝜀

ç

+

âã M

𝛻 𝜌

èâé è} –

𝐸

M

₍₂₀₎

where p is the pressure and g is the gravity acceleration (note that since this approach is referred to adiabatic growing gas bubbles, constant viscosity is considered). In the following, the choice is made to lump the electrostrictive force with mechanical pressure p into an irrotational term, defining an electrostrictive pressure

𝑝

_H<,

𝑝

_H<

=

L_M

𝜀

_I

𝑏𝐸

M

₍₂₁₎

𝑏 = 𝜌

èâé è} –

(22)

where the sign has been changed according to the standard assumption that pressure is positive when the medium is compressed, and to consider the dielectrophoretic force as the divergence of an electric stress tensor, named after Maxwell [72]: 𝑇_H is the Maxwell stress tensor.

𝑓

_H###

_{= 𝛻𝑇}

H

, 𝑡

G,6

= 𝜀

I

𝜀

ç

𝐸

G

𝐸

6

−

âãâéä s M

𝛿

G,6

(23)

Substituting Eqs. (21)-(23), Eq. (20) becomes:

𝜌

ê& ê>

= −𝛻(𝑝 − 𝑝

H<

) + 𝜇𝛻

M

_{𝑣 − 𝜌𝛻 𝑔𝑦}

₍₂₄₎

Equation (24) can be used to determine the pressure in a still (𝑣 = 0) and homogeneous

𝛻𝜀

_ç

= 0

fluid in the presence of electric field, in the absence of free charge in the medium, where y is the Cartesian coordinate directing downwards.

𝛻 𝑝 − 𝑝

H<

− 𝜌𝑔𝑦 = 0

(25)

From which the mechanical pressure p is given as 𝑝 = 𝜌𝑔𝑦 +âã.äs

M + 𝑐𝑜𝑛𝑠𝑡

(26)

Therefore, unless the coefficient b is negative, the electrostriction increases the mechanical hydrostatic pressure in the medium. This contribution can give rise to the so called “electric field wind” and is responsible of an increase in the convective heat transfer part of boiling and evaporation overall heat transfer process.

(32)

32

Giacomo Saccone

For non-polar fluids, like the one considered here, the term

𝑏

is given by the Clausius-Mossotti law [72]:

𝑏 = 𝜌

èâé è} –

=

âé’L âé„M w

(27)

In a gas, no significant electrostrictive pressure is present since 𝑏 ≅ 0. But at the interface,

𝛻𝜀

ç

≠ 0.

We already said that as soon as we have two-phase we have an interface. It is straightforward to say that as soon as we have an interface we have curvature. Young-Laplace Equation, which comes out from a balance of forces at the interface:

𝑝

J

− 𝑝

)

= 2 𝜎 𝐾 = 𝜎

_çtL

+

_çsL

(28)

𝑔 refers to the gas/air phase (inside the interface), 𝑓 refers to the fluid phase

(outside the interface). Viscous effects are not present since we are referring

to a quasi-static gas bubble.

When electric field is applied:

𝑝

_J

− 𝑝

₎

+ ∆𝑓

_H,-

= 2 𝜎 𝐾 = 𝜎

_çL t

+

L çs

(29)

For the Maxwell Stress Tensor at the interface, just the dielectrophoretic part has to be considered since electrostriction was already considered in the pressure term.

∆𝑓

_H,-

= 𝑛 ∙ 𝑇

_H,)

− 𝑇

_H,J

∙ 𝑛 =

âã M 1 −

𝜀

𝑅,𝑓

𝜀

𝑅,𝑓

𝐸

𝑛,𝑓 2

_{+ 𝐸}

𝑡,𝑓 2

(30)

Where the subscripts

𝑛 and 𝑡 denote the components normal and tangential

to the interface, respectively. Considering a bubble over a surface, at its top

(subscript 0) we have:

𝑝

_J,I

− 𝑝

_),I

+ ∆𝑓

_H,-,I

= 2 𝜎 𝐾

_I

(31)

Subtracting Eq. (31) from Eq. (29), considering that

𝑦 = 0 at bubble top and

positive going downwards:

𝑝

_J

− 𝑝

_J,I

− 𝑝

₎

− 𝑝

_),I

+ ∆𝑓

_H,-

− ∆𝑓

_H,-,I

= 2 𝜎 𝐾 − 𝐾

_I

(32)

Applying Eq. (26):

𝜌_J

−

𝜌₎

𝑔𝑦 −

𝜀0𝑏 2

𝐸

) M

_{− 𝐸}

),IM

+ ∆𝑓

H,-

− ∆𝑓

H,-,I

=

2 𝜎 𝐾 − 𝐾

_I

(33)

The expression for the hydro-electro-static pressure along the interface turns out:

(33)

∆𝑓

_H,G->Hž)=2H

= ∆𝑓

_H,-

− ∆𝑓

_H,-,I

−

𝜀0𝑏 2

𝐸

) M

_{− 𝐸}

),IM

=

2 𝜎 𝐾 − 𝐾

_I

−

𝜌_J

−

𝜌₎

𝑔𝑦

(34)

In this way we can express curvature in function of curvature at top taking into account electric field overpressure. This formulation will be verified in Chapter IV [73][79]. In summary, the momentum equation for a single phase and homogeneous fluid reads:

𝜌

𝐷𝑣

𝐷𝑡

= −𝛻 𝑝 − 𝑝

H<

+ 𝜇𝛻

M

𝑣 − 𝜌𝛻 𝑔𝑦

(35)

While the momentum equation for two incompressible Newtonian fluids separated by an interface reads:

𝜌

𝐷𝑣

𝐷𝑡

= −𝛻 𝑝 − 𝑝

H<

+ 𝜇𝛻

M

𝑣 − 𝜌𝛻 𝑔𝑦 + 2𝜎𝐾𝑛𝛿

G->

−

𝜀

I

𝐸

M

2 𝛻𝜀

ç

(36)

And the resulting force acting on a surface 𝑆 (𝑛, its unit vector) will be expressed as:

𝐹

_H

=

_©

𝑇

_H

∙ 𝑛 𝑑𝑆

(37)

This force enters in the balance between the forces acting on a bubble. Problems arise from the evaluation of the local electric field, which is affected by bubble size and shape while, in turn, it influences bubble size and shape. Despite these issues, a numerical work has been conducted by Di Marco et al. [73]. The code agrees very well with the experimental data and gives important consideration about the relevance of the terms inside the electric force. In particular, the attention can be focused on the magnitude of the terms: the electric force, at a dielectric interface, mainly consists of the dielectrophoretic force. Electrostriction appears only in the liquid phase and alters the pressure distribution in it. From this result, it seems that in Eq. (19) we can neglect the first and third terms. Another evaluation of forces has been conducted via COMSOL Multiphysics, a FEM software, where the actual shape of the bubble, taken from video images, has been imported and the local value of the electric field calculated just solving Gauss law. Thanks to this tool, a complete evaluation of Maxwell stress tensor became possible [67].

Landau and Lifsitz [71] demonstrated that for a small spherical or elliptical gas bubble inside a non-conducting fluid, the resulting force due to dielectrophoresis is: 𝐹H = 𝜋 𝑑HŽw 6 𝜀J− 𝜀% 𝑛𝜀J+ 1 − 𝑛 𝜀%𝜀%𝛻𝐸 M ₍₃₈₎