Effects Of Force Fields on
Droplets Evaporation
Tesi di Laurea Magistrale
Ingegneria Meccanica
Università di Pisa
Scuola di Ingegneria
Dipartimento di Ingegneria Civile e Industriale Corso di Laurea Magistrale in Ingegneria Meccanica
Effects Of Force Fields on Droplets
Evaporation
Thesis by
Alekos I. Garivalis
Supervisors
Prof. Paolo Di Marco
. . . .Prof. Sauro Filippeschi
. . . .Candidate
Alekos I. Garivalis
. . . .Sessione di Laurea 10 Ottobre 2018 Anno Accademico 2017/2018
Purpose of this thesis is contributing to understand the phenomena that determine and rule the evaporation of droplets of fluid deposed on a heated surface. Despite the decennary research and due to some theoretical and practical problems we are still not able to provide a full description of all the aspects of the process, which anyhow is very important for natural phenomena and for daily, industrial and space applications. The results of several experimental tests have been analyzed, part of which performed on board an ESA aircraft that, through the execution of special parabolic trajectories, cancels for about 22 seconds effects of gravity. Run the experiments in weightless conditions has a dual purpose: the first is observing directly the droplets behavior in these conditions for space heat and mass transfer applications, the second is getting a greater understanding of the phenomena to be extended to cases with gravity. In addition to gravity field effects on evaporation those of a constant electric field have been studied. In weightlessness indeed convection, one of the most important heat transfer mode based on fluids density gradient, is strongly limited. Using electric fields is a way of compensating the lack. Summing up, this work analyzes droplets evaporation and the effects gravitational and electric fields have on it.
Lo scopo di questa tesi è contribuire alla comprensione dei fenomeni che governano l’evapo-razione di gocce di un fluido depositate su una superficie riscaldata. Nonostante la ricerca decennale e a causa di alcune difficoltà di ordine teorico e pratico non siamo tutt’ora in grado di fornire una descrizione completa di tutti gli aspetti del processo, che risulta tuttavia di grande importanza nei fenomeni naturali, in applicazioni quotidiane, industriali e spaziali. Sono stati analizzati i risultati di alcune prove sperimentali, parte delle quali eseguita a bordo di un aereo messo a disposizione da ESA che, attraverso l’esecuzione di particola-ri traiettoparticola-rie paraboliche, annulla per circa 22 secondi l’effetto della gravità. Eseguire gli esperimenti di evaporazione in assenza di peso ha un duplice scopo: il primo è osservare direttamente il comportamento delle gocce in tali condizioni per applicazioni collegate allo scambio di massa e calore in ambito spaziale, il secondo è avere una maggiore comprensio-ne dei fenomeni da estendere anche ai casi in cui la gravità è presente. Oltre agli effetti del campo gravitazionale sull’evaporazione sono stati studiati quelli di un campo elettrico costante. In assenza di peso infatti la convezione, una delle modalità di scambio termico più importanti che si basa sul gradiente di densità dei fluidi, è fortemente limitata. Un modo per compensare la mancanza è l’utilizzo di campi elettici. Rissumendo, questo lavoro analizza l’evaporazione di gocce e l’effetto che hanno su di essa il campo gravitazionale ed elettrico.
1 Introduction 1
2 Droplets wetting and evaporation: state of art 5
2.1 Static conditions . . . 5
2.1.1 Surface tension . . . 5
2.1.2 Mechanical equilibrium on a curved interface . . . 7
2.1.3 Wettability and equilibrium contact angle . . . 10
2.1.4 Sessile droplet and capillary equation . . . 12
2.1.5 Advancing/receding angles and hysteresis . . . 15
2.2 Dynamic conditions . . . 16
2.2.1 Contact angles on a moving interface . . . 16
2.2.2 Triple line velocity . . . 16
2.2.3 Droplets evaporation . . . 19
2.3 Effects of electric field . . . 21
2.4 Droplets internal convection . . . 25
3 Measurement methods 27 3.1 Initial image processing . . . 27
3.2 Calibration . . . 29
3.3 Center of gravity, height, base diameter and radius of curvature calculation 29 3.4 Volume calculation . . . 31
3.5 Contact angle calculation . . . 31
3.6 Force balances . . . 33
3.6.1 Vertical balance . . . 34
3.6.2 Radial balance . . . 35 i
3.7 Triple Line Velocity . . . 36
3.8 Inputs and outputs statement of the Matlab routine . . . 37
3.9 Uncertainty analysis . . . 38 4 Evaporating droplets 41 4.1 Experimental activities . . . 41 4.1.1 Parabolic flights . . . 42 4.1.2 Experimental apparatus . . . 45 4.1.3 Data conditioning . . . 52 4.1.4 Environment conditions . . . 52 4.1.5 Experiments execution . . . 53
4.2 Analysis of the experimental results . . . 55
4.2.1 On ground results . . . 55
4.2.2 Parabolic flights results . . . 67
4.2.3 Comparison graphs . . . 75
4.2.4 Heat transfer analysis . . . 78
4.3 Discussion . . . 82
5 Conclusions 85 5.1 Developments . . . 86
1.1 Photo of a sessile water droplet during experimental set-up. . . 2
2.1 Molecular density variation at interface (Figure by [1]). . . 6
2.2 Generic curved surface. . . 7
2.3 Tangential stress balance. . . 9
2.4 Definition of interfaces and equilibrium contact angle. . . 10
2.5 Different behaviours of contact angles. . . 12
2.6 Definition of reference system and coordinates for capillary equation. . . 12
2.7 Effects of hysteresis for a drop on a tilt surface. . . 15
2.8 Dynamic contact angle and hysteresis (Figure by [9]). . . 16
2.9 Viscous bending on the mesoscale for an advancing meniscus (Figure by [9]). 17 2.10 Dynamic wetting according to the molecular-kinetic theory (Figure by [9]). . 18
2.11 Droplet evaportion at the contact line. (a) hydrophilic droplet (b) hydropho-bic droplet (Figure by [32]). . . 20
2.12 Qualitative effect of force fields on droplets shape. . . 22
2.13 Thermal waves in evaporating droplets (a) water (no thermal patterns), (b) methanol (Figure by [20]). . . 26
2.14 Qualitative convection motions from a three dimensional simulation of a droplet on a heated substrate (Figure by [29]). . . 26
3.1 Frame obtained from the camera. . . 27
3.2 Difference between frames before and after contrast increase. . . 28
3.3 Black and white image after using im2bw command. . . 28
3.4 Final image and edge. . . 29
3.5 The screw used to do the calibration. . . 30 iii
3.6 Example of the top of a droplet: experimental measured points in red, fitted
parabola in blue. . . 30
3.7 Right edge of a droplet: experimental measured points in red, fitted parabola in blue. . . 32
3.8 Common trend to stabilization for a certain number of points (Figures by [30]) . . . 32
3.9 Control volume and droplet. . . 33
3.10 Radial force balance: pressure of the liquid in red, pressure of the air in orange, surface tension in blue. . . 35
3.11 Radial force balance, sectional view. . . 35
3.12 Edge definition. . . 38
4.1 Microgravity facilities (Figure from [7]). . . 42
4.2 The parabolic trajectory and the Novespace plane nose down. . . 43
4.3 Accelerations during a parabola. . . 44
4.4 Experiment rack. . . 45
4.5 Detail of heater section. . . 46
4.6 Schematic of droplet deposition system. . . 46
4.7 Schematic of droplet suction system. . . 47
4.8 Overall schematic. . . 49
4.9 The apparatus on the aircraft ready for parabolic flights. . . 50
4.10 The apparatus in the laboratory. . . 50
4.11 Detail of the foil and the support (containment celc removed). . . 51
4.12 Detail of the apparatus with a droplet in evaporation with electric field on. 51 4.13 Detail of IR camera placed under the foil. . . 52
4.14 Psicrometric diagram. . . 53
4.15 Evaporation trend, experiment on ground: GF = 1 g, q = 1015W/m2, EF = 0kV. 56 4.16 Experiments on ground: GF = 1 g, q = 695W/m2, EF = 0kV. . . . . 57
4.17 Experiments on ground: GF = 1 g, q = 1015W/m2, EF = 0kV. . . . . . 58
4.18 Experiments on ground: GF = 1 g, q = 1385W/m2, EF = 3, 5kV.. . . . 59
4.20 Comparison between theoretical and measured values (q = 695W/m2, EF = 0kV).
. . . 61
4.21 Comparison between theoretical and measured values (q = 1385W/m2, EF = 0kV). . . . 62
4.22 Comparison between theoretical and measured shape of the droplet. . . 62
4.23 Vertical force balance; q = 1015W/m2, EF = 0kV.. . . . 64
4.24 Radial force balance; q = 1015W/m2, EF = 0kV. . . . . 64
4.25 Vertical force balance in the presence of electric field (electrode distance 6 mm); q = 1385W/m2, EF = 5kV. . . 65
4.26 Radial force balance in the presence of electric field (electrode distance 6 mm); q = 1385W/m2, EF = 5kV. . . 65
4.27 Vertical force balance in the presence of electric field in volume (electrode distance 6 mm); q = 1385W/m2, EF = 5kV. . . . . 66
4.28 Radial force balance in the presence of electric field in volume (electrode distance 6 mm); q = 1385W/m2, EF = 5kV. . . . . 66
4.29 Day 3 parabolas number 5 and 6. . . 68
4.30 Day 3 parabolas number 9 and 10. . . 69
4.31 Day 3 parabolas number 24 and 25. . . 70
4.32 Vertical force balance in volume, in microgravity; q = 667W/m2, EF = 0kV, day 3 parabolas number 3, 4, 5 and 6. . . 72
4.33 Radial force balance in volume, in microgravity; q = 667W/m2, EF = 0kV, day 3 parabolas number 3, 4, 5 and 6. . . 72
4.34 Vertical force balance in volume, in microgravity; q = 1342W/m2, EF = 3, 5kV, day 3 parabolas number 23, 24 and 25, electrode distance 6 mm. . . 73
4.35 Radial force balance in volume, in microgravity; q = 1342W/m2, EF = 3, 5kV, day 3 parabolas number 23, 24 and 25, electrode distance 6 mm. . . 73
4.36 Vertical force balance in volume, in microgravity; q = 1342W/m2, EF = 5kV, day 3 parabolas number 26, 27 and 28, electrode distance 6 mm. . . 74
4.37 Radial force balance in volume, in microgravity; q = 1342W/m2, EF = 5kV, day 3 parabolas number 26, 27 and 28, electrode distance 6 mm. . . 74
4.39 Evaporation rate function of base diameter, in normal gravity and micrograv-ity, q = 1342W/m2, EF = 0kV . . . . 75
4.40 Evaporation rate function of base diameter, in normal gravity and in the presence of electric field, q = 1342W/m2, GF = 1g. . . . 76
4.41 Evaporation rate function of base diameter, in microgravity and in the pres-ence of electric field, q = 1342W/m2, EF = 0kV , comparison with
micro-gravity. . . 76 4.42 Heat flow at a single pixel element. . . 78 4.43 Frames by IR camera acquisitions. . . 80 4.44 Heat fluxes for a water droplet in normal gravity, Db = 4 mm, EF = 11 kV.
Figure from [32]. . . 81 A.1 Specification of the coating made by ENBIO company. . . 88
3.1 Input values . . . 37
3.2 Output values . . . 37
3.3 Experimental uncertainty table. . . 39
4.1 Data aquired by sensors. . . 49
4.2 On ground test matrix: there are scheduled the current and the electric field values for experiments performed in laboratory. . . 54
4.3 69th PFC test matrix: for each day and parabola are scheduled the current and the electric field values. . . 55
A.1 Foil data. . . 87
Nomenclature
b electrostriction coefficient Ca capillary number Db base diameter (m) E electric field (V/m) F force (N) g acceleration of gravity (N/m2) H droplet height (m) L interface lenght (m) k surface curvature (m−1) p pressure (Pa) Rb base radius (m)R0 apex of droplet radius of curvature (m)
t time (s)
T temperature (K)
U vetting line velocity (m/s)
V volume (m3)
ε0 vacuum dielectric permittivity (F/m)
εR relative electric permittivity
λc capillary lenght (m)
λe electrical lenght (m)
θ contact angle (deg)
θ contact angle (deg)
θa advancing contact angle (deg)
θD dynamic contact angle (deg)
θm microscopic contact angle (deg)
θr receding contact angle (deg)
θS static contact angle (deg)
ρF free electric charge density (C/m3)
ρ density (kg/m3)
σ surface tension (N/m)
Suffixes
d deviatoric e electric f liquid g gas n normal R radial s solid t tangentialAcronyms
CCA constant contact angle
CCR constant contact radius
CL contact line
EF electric field
ESA European Space Agency
GF gravitational field HDT hydrodynamic theory HF E hydrofluoroether HY D electrohydrodynamics IR infrared M KT molecular-kinetic theory N I National Instruments P EEK polyrthereketone
P F C Parabolic Flight Campaign
Introduction
Despite two centuries of research on wettability, the precise phenomena involved in the in-teraction between fluids and solid surfaces are still not completely understood. If the final state of a system of interest is controlled by the statics of wetting, which is the most inves-tigated and understood part, the process by which the final state is achieved is governed by dynamics of wetting, harder to study and control. Among the various aspects of wetting dy-namics the evaporation of droplets, whereby a single droplet evaporates in the surrounding air, is of particular importance because is primary for both natural and industrial processes as coating, painting, inkjet printing, DNA mapping and nanotechnologies [20]. Moreover in heat transfer phase change cooling techniques are very important. For example evaporative spray cooling consists of cooling a surface by latent heat absorbition that transforms liquid droplets in vapour phase; tipically phase change heat transfer rate is greater than air cool-ing techniques, and is used in many areas of engineercool-ing such as metallurgical applications, nuclear and electronic industries and fire suppression systems. Phase change heat transfer is very important for space applications too, where weightless limits convective phenomena. Many factors contribute to droplet evaporation dynamics. The first part of this work is a review of the current knowledge about the topic. Starting from statics we need to in-troduce capillarity, i.e. the interfaces phenomena. Describing the surface tension we will get the Young’s equation, that binds surface tension, pressure and geometrical shape. A very important parameter contained in the equation is the equilibrium contact angle, that, if measured, can be used to calculate tensions or vice versa can be calculated starting by the Young’s equation and hydrostatic equations; an important application is represented by capillary equation. The contact angle is the way to quantify wettability, the affinity of liquid
for solids. Some of the factors involved are surface inhomogeneity, surface roughness and impurities, temperature and species concentration. All these concepts are valid in static conditions, and things get more complicated in dynamic conditions. We will show that the contact angle is not unique anymore when the interface moves (introducing aspects such tiple line velocity and contact angle hysteresis) and the most common theories available in the literature will be discuss (Hydrodynamic theory and Molecular-kinetic theory). Even evaporation (mass transfer) makes change the contact angle during the process and the trend depends on heating conditions.
One of the main purpose of the thesis is to find out how external force fields (in particular gravitational and electric field) affect the dynamic phenomena like shape, heat and mass transfer of a sessile droplet in order to control such aspects in industrial and space heat transfer applications and get insight about the physical mechanism ruling the phenomenon. For that reason the main aspects of electric fields effects and electric forces are described. After a view of the main aspects of the problem and the ways to measure and predict (when possible) the significant parameters involved, we are going to discuss the results of several experiments on droplets evaporation conducted under different conditions, with or without electric or gravitational fields and with different heat flux magnitude.
Figure 1.1: Photo of a sessile water droplet dur-ing experimental set-up.
The experimental campaign has been con-ducted at the laboratory of DESTEC and on ESA experimental parabolic flights in the past months, with the collaboration of the Trinity College of Dublin.
Is important to underline that the dynamic of droplets evaporation is a hard problem to study and still now not completly under-stood. As often happens in thermofluidody-namic, it is very easy to get to manage many parameters all togheter, risking to make a great mess and fail to glimpse the physical sense of the phenomenon lost in a large amount of data and physical aspects involved. For that reasons the experiment was designed with care, it is the simplest apparatus that performs all the measurements (and, despite this, there are a lot of variables to take into
ac-count). Ultimately simplify the experiments and attempt to identify and isolate the effects of the several actors taking part in the heat and mass transfer of a sessile droplet is useful to open the way to the study of more complex phenomena and practical application.
Droplets wetting and evaporation:
state of art
In Chapter 2 the state of art knoweledge necessary to understand the thesis topic is exposed. The physics behind the studied phenomena is described and the definitions of the quantities and properties used are given. The purpose of the work concerns drops in dynamic con-ditions, which includes heat and mass exchange, but a description of the system in static conditions is essential. The effect of the electric and gravitational field is also described.
2.1
Static conditions
For the purpose of the current work the static condition is defined as follows: a thermody-namic system at equilibrium with no mass, momentum or energy variations, i.e. mass and heat transfer and inertia forces null or negligible. A system can also be quasi-static when the variations take place in very long times. A good characterization of systems composed by fluids, solids and their interfaces has been given and it is the theme of the following paragraphs.
2.1.1 Surface tension
The physical origin of surface tension can be understood considering a liquid-vapour inter-face: a molecule in the bulk liquid is subject to forces of repulsion from its close neighbor and forces of attraction by all others. Both types of forces act symmetrically so the resul-tant on each molecule is zero. A molecule in the surface loses half of its interactions. For
that reason density is lower in the interface region then in the bulk liquid phase, and the mean spacing of the molecules in the liquid near the interface is greater that un the bulk liquid. Therefore the energy per molecule is greater in the interfacial region than in the bulk liquid; that means that a molecule at the surface is in an unfavorable energy state. From
Figure 2.1: Molecular density variation at interface (Figure by [1]).
a thermodynamic point of view the system has an additional free energy per unit area of the interface due of the presence of interface [1]. This is the reason that liquids adjust their shape in order to expose the smallest surface area. Said the above we define the surface tension σ as a direct measure of the energy deficit between the molecule in the bulk and the molecule in the interfacial region per unit surface area, its unit is energy on area, so N m−1.
Another way to explain surface tension is considering the molecules at interface subject to Van Der Waals forces of bulk liquid and vapour: the forces originating from the vapour are weaker than those originating from the liquid, so the molecules at interface are attrected to the bulk liquid.
It is important to point out that the surface tension is a function of temperature. The value of the surface tension generally decreases with temperature (in effect at the critical temperature the properties of the two phases are identical so the surface tension is zero); for some mixtures this is not true. Many curve-fit equations have been proposed, but often the linear approximation is sufficient [1]. Even species concentration in the liquid makes the value of surface tension change. In the presence of surface tension gradient motion of the liquid can be established in order to balance the gradient by shear stress. That is called Marangoni effect.
2.1.2 Mechanical equilibrium on a curved interface
(a) Generic curved surface with surrounding volume. (b) Unit vectors definition. Figure 2.2: Generic curved surface.
Surface tension has an important role in the determination of the shape of the liquid-vapour interface. As a matter of fact the interface shape takes a configuration in order to minimize the free energy of the system. But let’s take a step back for a while and let’s consider a generic interface moving in motion with velocity w (see figure 2.2); referring to the approach followed in [4], the integral mass (equation 2.1) and momentum (equation 2.2) balances in a volume of vanishing thickness surrounding the interface read
Z Z S ρAnA· (vA− w) − ρBnB· (vB− w) dA = 0 (2.1) Z Z S ρAnA· (vA− w)vA− ρBnB· (vB− w)vB dA = Z Z S
(nA· TA,d+ nA· TA,e− nApA) − (nB· TB,d+ nB· TB,e− nBpB)
dA + Z BA mσ ds (2.2)
Considering transformation of line integral to a surface integral Z BA mσ ds = Z Z S (∇σ − kσn) dA (2.3)
As the integration domain S is arbitrary, it is possible to get the local equilibrium equations (2.4, 2.5)
ρA· (vA− w) − ρB· (vB− w) = 0 (2.4)
˙
m0(vA− vB) =
(n · TA,d+ n · TA,e− npA) − (n · TB,d+ n · TB,e− npB) − ∇σ + σkn
(2.5)
where
• pA is the pressure in the A side (Pa).
• pB is the pressur in the B side (Pa).
• σ is the surface tension (N/m).
• k is the surface curvature, defined as k = 1 R1 +
1
R2, where R1 is the first radius and
R2 is the second radius (m−1).
• ˙m0 is the mass flow trough surface S (kg/m2s).
• TA,d and TB,d the deviatoric stress tensors acting on the two sides (Pa).
• TA,e and TB,e the Maxwell stress tensors acting on the two sides (Pa).
• w is the interface velocity (m/s). • vAand vB the fluids velocities (m/s).
In detail, the first term of equation 2.5 is the recoil force due to moving mass flow; the second term represents shear and pressure stresses; the third term is due to surface tension and surface tension gradient.
Starting from equation 2.5, and considering the tangential components of momentum balance along l and q and the normal component along n (see figure 2.3) in the absence of
Figure 2.3: Tangential stress balance.
mass flow and electric stress the following expression are obtained 0 = (τnl,A− τnl,B) − dσ dl (2.6) 0 = (τnq,A− τnq,B) − dσ dq (2.7) 0 = (τnn,A− τnn,B) − (pA− pB) + σk (2.8)
Placed in this form the equations are very interesting and easier to understand: con-cerning the first two, in the presence of variations of surface tension along the surface, due e.g. to temperature), a jump in shear stress is present, which is the origin of Marangoni flow. From the third one, the noted Laplace equation can be obtained.
Laplace equation
The Laplace equation (sometimes Young-Laplace equation) is the principal mathematical tool to predict the shape of the interface.
It can be derived from radial equilibrium exposed above, in the absence of normal viscous stress. Considering the interface of a bubble or droplet, for which an internal and an external sides can be clearly defined, the equation reads
pin− pout = σ 1 R1 + 1 R2 = σk (2.9)
The pressures are the internal and external to the considered system, the radii character-ize the local surface curvature and are measured in two perpendicular planes containing the normal to the surface, at last there is the local value of the surface tension. Until the surface
tension is different from zero the pressure inside the bubble or droplet must be greater thn outside, furthermore the smaller the object the greater is internal overpressure.
The Laplace equation relates interfacial tension, interface geometry (quantified by the radii of curvature) and pressure differences between the fluid at each point of interface. It can be used with equations of hydrostatics to compute the shape of static interface or if the shape of interface can be determined experimentally the interfacial tension can be calculated. Solve the equation can be very hard; in many cases geometric symmetry is properly used. A typ-ical simple example is a certain volume of liquid only subject to different pressures accoring with Laplace equation that takes nearly a spherical shape (as for example is noticeable in the rain).
2.1.3 Wettability and equilibrium contact angle
After discussing surface tension another important property has to be introduced: wettabil-ity. That is defined as the affinity of liquid for solids. Different behaviours can be observed in the interaction between a liquid in contact with a solid surface depending on the surface and the type of liquid. A weak affinity of a liquid with the solid surface produces beads of the liquid on the surface, high affinity makes the liquid spread on the surface forming a film to maximize the liquid-solid contact area. This is very important for that research because vaporization or condensation process ultimately takes place at the liquid-vapor interface and the heat and mass transfer depends very strongly on the way the two phases contact the solid, as pointed out in [6].
Figure 2.4: Definition of interfaces and equilibrium contact angle.
flat and perfectly smooth surface. The contact angle θ quantifies the wettability and it is the angle between the liquid-vapor interface and the solid surface. The system in figure 2.4 has three interfaces: one between the vapour and liquid, to which the σlg surface tension
is associated, other two between solid-liquid and solid-vapour interfaces, to which also are associated surface tensions σsl and σsg of appropriate values. The contact line or triple line
is where the three phases meet.
Figure 2.4 reports also the force balance (per unit lenght) of the contact line at point O (the vector quantities are written in bold).
σsg = σsl+ σlg (2.10)
In the horizontal direction requires that:
σlgcos θ = σsg− σsl (2.11)
Rearranged as equation 2.11 it is called Young’s equation or Neumann’s formula ([1], [2]). Regarding the vertical direction the force is balanced by a reaction of the solid, but it is usually so small and the modulus of elasticity of the surface is so high that there is not significative deformation on the surface (as a counterexample if the material is soft like rubber or paint, or it is a liquid, it does distort: that is the reason why a water drop on a fresh coat of paint leaves behind circular mark).
From a practical point of view is pretty hard to measure the surface tensions between solid and fluids, contrary to the tension of the liquid-vapour interface. The balance is usually used calculating the difference σsg− σsl note σlg and the contact angle.
Young’s equation is derived under some conditions: ideal surface, no adsorbed liquid on the surface and constant values of surface tension along the interfaces. Furthermore because surface tensions are equilibrium properties even the contact angle defined by the equation is an equilibrium property. For that reason is called equilibrium contact angle. It is important to point up that such angle is defined in static conditions and macroscopic scale; afterwards in next paragraphs the contact angle definition will be multiplied.
2.1.4 Sessile droplet and capillary equation
Figure 2.5: Different behaviours of contact angles.
It is interesting to apply the concepts previously seen to sessile droplets, as the object of this study. A sessile droplet is an amount of liquid deposited on a solid surface that assumes axial symmetrical shape. The contact angle and consequently the droplet shape is function of the liquid-solid couple. Typically the behavior of the droplet is distinguished according to the value of the contact angle: if it is greater than ninety degrees the droplet is hydrophobic, if smaller is hydrophilic. A liquid for which θ = 0◦ is said to completely wet
the surface, while for θ = 180◦ is completely nonwetting.
Effect of gravity can be considered in order to predict droplet shape in static conditions. For the droplet case consider a reference system like in figure 2.6. The relationship between internal and external pressure at any point of the surface is given by Laplace-Young equation
Figure 2.6: Definition of reference system and coordinates for capillary equation.
pf = pg+ σ 1 R1 + 1 R2 (2.12)
In static conditions pressures are related to the vertical coordinate by the Stevin equa-tion1
pf = p0,f+ ρfg y, pg= p0,g+ ρgg y (2.13)
where p0,f and p0,g are the pressures of the two phases at the apex of the liquid and the
gas respectively. Due to the axial symmetry the radii at the apex are equal (R0) so the link
between the two pressures is given by Laplace equation
p0,f = p0,g+
2σ
R0 (2.14)
Combining 2.12 with 2.13 and 2.14 eliminating pressures, the capillary equation is obtained σ 1 R1 + 1 R2 = 2σ R0 + (ρf− ρg) g y (2.15)
This interesting equation is very useful in problems concerning sessile droplets and it can be used for bubbles spreading in a liquid too (see [7], [8], [19]) simply changing the sign of the gravity term (as a matter of fact a spreading bubble is a gas in a liquid, a sessile droplet is a liquid in a gas). A very interesting fact is that a droplet upside down (gravity changes sign) has the same shape of a spreading bubble, so as a bubble upside down has the same shape of a sessile droplet.
The last term of 2.15, the gravity one, can be considered as an added overpressure caused by the gravity field
∆pgravity = (ρf − ρg) g y (2.16)
A similar term can be introduced for an overpressure added by an electric field (∆pelectric).
In this case the capillary equation becomes
σ 1 R1 + 1 R2 = 2σ R0 + ∆pgravity+ ∆pelectric (2.17)
1The famous equation of Simon Stevin (Bruges, 1548 - L’Aia, 1620) that asserts that the pressure in a
It can be used to evaluate ∆pelectric once the curvature is measured. However capillary
equation can be applied in static conditions in the presence of gravity field to calculate the interface shape. If both overpressures are equal to zero (no presence of gravitational either electric field) is clear from the equation that the droplet take the form of a spherical cap. There is a particular lenght, called capillary lenght λc, that gives information about the
im-portance of gravity force respect to surface tension force. It can be estimated by comparing the Laplace pressure (due to surface tension) to the hydrostatic pression (Stevin equation), in a liquid submitted to earth’s gravity [2]. The capillary length is defined as
λc=
p σ/ρg
The capillary length is generally of the order of few mm (2.7 mm for water in normal gravity); if one wants to increase λcin a liquid it is necessary to work in a microgravity environment
or replace air with a non-miscible liquid whose density is similar to that of the original liquid.
2.1.5 Advancing/receding angles and hysteresis
In static conditions values of the angle different from the equilibrium one can be obtained: the receding contact angle θr and the advancing contact angle θa are defined as the
ulti-mate contact angle for which the liquid droplet starts to advance or recede on the surface, respectively. Compared to the equilibrium contact angle the measured values are θr ≤ θS
and θa ≥ θS. When there are different values of contact angle in receding and advancing
conditions it is called hysteresis. The unperfect smothness or homogenity of the surface are at least partly responsible of that contact angle hysteresis; that explains why rain drops can stay fixed on a window or a windscreen for example. In figure 2.7 a drop can stay fixed on a tilt surface due to the different value of the angles (the difference of the components parallel to the surface balances the force of gravity).
2.2
Dynamic conditions
What previously described is only a partial aspect of the studied phenomena. In dynamic conditions contact angle is not unique anymore, volume and shape changes are involved, the contact line moves. And all these factors are bound to evaporation.
2.2.1 Contact angles on a moving interface
The main parameters used to quantify the dynamics of wetting are the relative velocity at which the liquid moves across the solid, i.e. the wetting-line velocity U, and the dynamic contact angle θD. Regarding the contact angle, its variations can be easily observed if the
liquid is moving on the solid surface (or more precisely if relative speed exist between solid and liquid). As a matter of fact in dynamic conditions occurs a dynamic contact angle θD,
that can assume different values from equilibrium, receding and advancing ones. Farther a dependence on the contact line velocity exist: in figure 2.8 is schematically shown the contact angle behaviour, with and without the effect of hysteresis.
(a) Dynamic contact angle θDfunction of triple line
velocity..
(b) Contact angle hysteresis.
Figure 2.8: Dynamic contact angle and hysteresis (Figure by [9]).
2.2.2 Triple line velocity
The triple line as mentioned before is the region where three phases meet, and is involved quite strongly in dynamics. Even at equilibrium the molecules of the triple line zone are subjected to constant thermal activity, so that the wetting line fluctates about its mean po-sition. When there is no equilibrium the triple line moves with its overall direction imposed by external forces or potential gradient. Because dynamic wetting occurs at a finite rate, with changes in the wetted area, the processes involved must be thermodynamically
irre-versible and therefore dissipative. Indeed, the fact that the observed dynamic contact angle is velocity-dependent, and differs from its equilibrium value, is clear evidence of this [10]. The correlation between dynamic contact angle and triple line velocity is a tricky problem still not completely understood. Several correlation and models have been proposed. The most used are described in next paragraphs.
Hydrodynamic theory (HDT)
Figure 2.9: Viscous bending on the mesoscale for an advancing meniscus (Figure by [9]).
Most common liquids, Newtonian ones, show zero relative velocity at solid surfaces, and this is the normally used boundary condition to solve Navier-Stokes equation (no-slip con-dition). But this assumption produces a singularity in stress at the contact line, leading the drag force to non-finite values [3]. To avoid singularity several slip conditions in the vicinity of the triple line have been postulated [11].
The Hydrodynamic Theory (HDT), proposed by Cox and Voinov [12], assumes that there is a viscous bending of the interface within a mesoscopic region below the scale of observation and this bending is due to the dynamic contact angle θD at macroscopic scale. The actual
contact angle is a microscopic angle θm assumed to be governed by short-range
intermolec-ular forces. For this reason its equilibrium value is usually taken (θm = θS). Figure 2.9
shows schematically the assumptions, and is clear that there are therefore three relevant length scales.
formula describing the change in the dynamic contact angle due to bending of the liq-uid–vapour interface may be written in terms of the capillary number (Ca = µv
σ , where µ is
dynamic viscosity, v is characteristic velocity and σ is surface tension) and approximating the result for θD < 3π/4as explained in [9] it becomes
θD3− θm3 = 9Ca ln
L Lm
, θm = θS, θD < 3π/4 (2.18)
where L and Lm are appropriately chosen macroscopic and microscopic length scales,
respectively.
For what concernes the value of θm Cox, supported by Voinov, asserts the dependency
on the liquid and solid surface only in the light of some experimental evidences [12]; the dependency on flow velocity U is conserved.
Molecular-kinetic theory (MKT)
Figure 2.10: Dynamic wetting according to the molecular-kinetic theory (Figure by [9]).
This approach, proposed by Blake and Haynes [13], focuses on the process of attachment or detachment of fluid molecules to or from the solid surface (figure 2.10). Unlike HDT the microscopic contact angle is assumed equal to the measured contact angle θm = θD. There
are two lenght scales that are the molecular scale (where the dissipation occurs) and the macroscopic scale.
displacements, and λ, the average distance of each displacement. The idea at the base of the theory is that the out-of-balance surface tension force that arises when equilibrium is disturbed produces the force that allows the triple line to move: Fw = σ(cos θS− cos θD).
That force does the work necessary to overcome the energy barriers that prevent molecular displacement.
The final equation for the triple line velocity is [9]:
U = 2k0λ sinhσ(cos θS− cos θD)λ2/2kBT
(2.19) where kB is the Boltzmann constant and T the absolute temperature. For the k0 expression
subsequent developments of the theory proposed proportionality inverse of viscosity [10]. If the argument of the sinh function is small, i.e. the cosine of the dynamic contact angle is close to che equilibrium one, equation 2.19 reduces to its linear form (because sinh x ≈ x).
U = σ(cos θS− cos θD)k0λ3/kBT (2.20)
2.2.3 Droplets evaporation
Droplets evaporation exist in daily life and industrial applications such as phase change cooling, inkjet printing, spraying, combustion and so on. A vast amount of literature is avaiable, and only a short synthesis will be made here. Picknett and Bexon performed some investigations on the evaporation of methyl droplets [18] describing two evaporation modes: the constant contact radius (CCR) and the constant contact angle (CCA) modes. In CCR mode, the contact angle reduces while the contact radius is pinned, and in CCA mode the contact radius contracts while the contact angle is kept constant. However the evaporation sometimes is based on a combination of modes, called mixed mode or stick-slip mode. Typically on hydrophilic surfaces CCR mode dominates, while on hydrophobic surfaces CCA dominates [5]. When contact angle becomes too small, a transition on mixed mode occurs.
Many researchers focused on droplets evaporation rate, that seems related to the evaporation mode. It is typical to use dimensionless parameters as V/V0 (where V is volume and V0 the
initial volume) and t/tT (where t is the time variable and tT the total evaporation time). In
β = 1(the so called 1/1 power law) as highlighted in [23] and [24]. Other researchers ([25], [26]) found that in CCA mode β = 2/3. Seems that when the contact angles are larger than 150◦, β can be smaller than 2/3, and further dependencies on the volume and initial
contact angle can not be excluded a priori [27].
The experiments of the various studies are based on droplets laid on a heated surface. The surfaces can be heated at constant temperature or constant heat flux.
Figure 2.11: Droplet evaportion at the contact line. (a) hydrophilic droplet (b) hydrophobic droplet (Figure by [32]).
It seems that the evaporation mechanisms are quite different for hydrophilic and hy-drophobic droplets [32]. For hydrophilic droplets, the adsorbed layer forms as a result of the strong adhesion forces between the liquid and the solid (see figure 2.11.a). The adsorbed film region is characterised by long range intermolecular forces. This disjoining pressure re-sult in a flat liquid-vapour interface of a few nanometers thick and prevents evaporation occurring in this region [33]. The transition region is defined by growing film thickness which results in a reduction in the long range intermolecular forces. This region experience the highest heat fuxes across the droplet as a result of the low thermal resistance from the small film thickness [33]. As the film thickness increases from the transition region into the intrinsic meniscus and micro-convection regions, so too does the thermal resistance resulting in a decrease in the local heat flux. It is well established that the low thermal resistance of the liquid in the transition region results in proportionately high evaporative heat and mass flux. From a heat transfer point of view, this has been observed as a cofee-ring shaped heat flux distribution beneath a hydrophilic droplet on heated substrates [33].
For hydrophobic droplets, the droplet mechanics are significantly different since the adhesion forces between the solid and liquid are weak. It is predicted that there is a greatly reduced adsorbed film region [32] and this mitigates the formation of a micrometer-scale transition region, as shown diagrammatically in figure 2.11.b. A a temperature gradient establishes along the droplet interface, with the highest temperature occurring near the contact line due to the proximity with the hotter surface. This established the highest evaporation flux due to the increase in the local saturation pressure which induces a high vapour fraction gradient, which drives vapour diffusion and overcomes the geometric confinement in this re-gion which tends to impede diffusion. Evaporation is also highest at the contact line though due to quite different mechanisms compared with hydrophilic droplets [32].
2.3
Effects of electric field
The effect of electric force depends on the configuration of imposed electric field. In the following, reference is made to the EF created by an electrode laid parallel to the surface in where the drop resides. The introduction of electric force alters the force balance leading to a different equilibrium condition and influencing dynamics of heat and mass transfer, driving the vapour towards the zone of lower electric field, separationg phases and increasing boiling heat transfer in normal and microgravity conditions [14]. For this electric fields can be used successfully to replace gravitational field in space applications [14], [15]; farther the electric force is conservative and shares several characteristics with buoyancy. Electric forces produce visible modifications in the shape as an increase in curvature radius at top, an increase in the height of the interface, a lower radial size and it seems like if the interface is pulled away from the surface. Figure 2.12 shows the electric field effects with images aquired during experimental activities exposed afterwards.
Electrohydrodynamics (EHD) describes the interations between an electric field with electric charges (free and polarized) in a fluid. A term of volumetric force due to electric field interaction is added to the momentum balance in Navier-Stokes equations. According to Landau and Lifšitz [16] the volumic electric force that acts on a fluid is
fe000= ρFE − ε0E2 2 ∇εR+ 1 2ε0∇(bE 2) (2.21)
di-(a) Droplet in 1 g and 0 kV . (b) Droplet in µg and 0 kV .
(c) Droplet in 1 g and 3 kV . (d) Droplet in µg and 3 kV . Figure 2.12: Qualitative effect of force fields on droplets shape.
electric permittivity, εRthe relative electric permittivity and b the electrostriction coefficient
defined as
b = ρ∂εR ∂ρ
T (2.22)
where ρ is density and T temperature. This coefficient is close to zero for gases and there are several expressions for polar and non-polar fluids ([14], [15]). The three terms of vo-lumic electric force are called electrophoretic, dielectrophoretic and electrostrictive forces, respectively. The first term, also Coulomb’s force, is the only one that depends on the sign of electric field. The other two depend on the gradient of the dielectric constant (related to thermal gradients or phase discontinuities), the gradient of the electric field, and on the magnitude of E2, thus being independent of the field polarity. In details the second term is
a body form due to non-homogeneiteis of electric permittivity and the third term is due to non-uniformities of the electric field; the last term is also irrotational and consequently can be interpreted as an added electrically induced pressure [14].
According to continuum mechanics, volumic forces can be expressed as the divergence of a stress tensor, that for the specific case is the Maxwell stress tensor Te. Exploiting the
Gauss-Ostrogradskij divergence theorem the resulting force on a surface S with normal outward n is
Fe=
Z Z
S
The components of Te according to Panofsky and Phillips
tik= ε0εREiEk−
ε0E2
2 (εR− b)δik (2.24)
Because of the change of geometry due to the movement of the phases alters the electric field distribution, electric field and hydrodynamics equations are coupled: when an electric field acts in an incompressible fluid, a term for electric force is added to Navier-Stokes equations. Farther the Maxwell equations have to be solved, but for poorly conducting fluids magnetic induction effects can be neglected and so only the first two Maxwell equations need to be considered [17]. At last all the equations needed are
∇ · v = 0 (2.25) ρDv Dt = −∇p + µ∇ 2v + ρg + f000 e (2.26) ∇ · (ε0εRE) = ∇ · D = ρF (2.27) ∇ × E = 0 (2.28)
where v is the velocity, p the pressure, µ dynamic viscosity and f000
e is given by
equa-tion 2.21. Starting from the Navier-Stokes equaequa-tion in the presence of EF and in the absence of free charge in the medium
ρDv Dt = −∇p + µ∇ 2v + ρg −ε0E2 2 ∇εR+ 1 2ε0∇(bE 2) (2.29)
Equation 2.29 can be rearranged considering electrostriction part of the pressure term, and y the Cartesian coordinate directing downwards
ρDv
Dt = −∇(p − pes) + µ∇
2v + ρ∇(g y) −ε0E2
2 ∇εR (2.30)
where pes = 12ε0∇(bE2). Assuming steady conditions (v = 0) and ∇εR = 0, equation
2.30 yields
∇(p − pes− ρ g y) = 0 (2.31)
p = ρ g y + 1
2ε0∇(bE
2) + const (2.32)
In the gases, no electrostrictive pressure term is present, as b is almost zero.
In the presence of an electrical normal stress at the interface ∆fe,n the Young-Laplace
equation can be generalized in its extended version:
(pf − pg) + ∆fe,n = σk (2.33)
It can be shown that either for a perfectly dielectric or conducting fluid the tangential component of electric stress vanishes [15]. As exposed i section 2.1.4, the capillary equation for a droplet (or a bubble) is obtained by subtracting from 2.33 its particular expression at the droplet apex (where with 0 properties at the apex are indicated):
(pf − pf,0) − (pg− pg,0) + (∆fe,n− ∆fe,n,0) = σ(k − k0) (2.34)
For the Maxwell stress at the interface, Te, just the dielectrophoretic part has to be
considered, as electrostriction was already considered in the pressure term, and only the component normal to the interface, that reads
∆fe,n= nf · (Te,f− Te,g) · nf =
ε0 2 (1 − εR,f) εR,fEn,f2+ Et,f2 (2.35) Substituting 2.35 in 2.34 the following equation is obtained
(ρf− ρg) g y −
1
2ε0∇bf(Ef
2− E
f,02) + (∆fe,n− ∆fe,n,0) = σ(k − k0) (2.36)
which finally yields the overpressure due to electric field effect
∆fint= (∆fe,n− ∆fe,n,0) −
1 2ε0∇bf(Ef 2− E f,02) = ε0 2 (1 − εR,f) εR,f(En,f2− En,f,02) + (Et,f2− Et,f,02) −1 2ε0∇bf(Ef 2− E f,02) (2.37)
eval-uation of the electric overpressure value. Starting from these eqeval-uations Di Marco conducted a wide experimental and numerical work [15], that validated the assumptions and evaluated the Maxwell stress tensor completely. Other works concern the transient and steady [7], [14] behavior of sessile droplets under the action of electric forces. When a sessile droplet is ex-posed to an electric field, its shape is defined by the balance of surface tension, electrostatic and gravitational forces. The surface tension component tends to make the droplet spher-ical (see section 2.1.1), the electrostatic force elongates the droplet towards the electrode and the gravitational one flattens the droplet. Since the ∆fint≈ ε0εRE2 [34], it is possible
to define the electrical lenght λe, that gives information about the importance of electrical
force respect to surface tension force. It can be estimated by comparing the Laplace pressure (due to surface tension) to the electric overpressure due to a costant electric field. It reads:
λe= σ/ε0εRE2
For the conditions studied in this thesis, water and electric fields ranging from 5 to 8.3 MV/m, λe is about 1 mm. The droplet shape and its evaporation rate are driven by triple
line movement too and force fields do have a role on triple line dynamics and heat transfer. However results are difficult to compare due to the great number of parameters involved: nature of the fluid and the substrate, mode of application of electric field and its geometry, experiments configurations [7]. Literature is not so wide but this is a very fertile branch of research for a better understanding of nature and for the possibility to replace buoyancy in microgravity for space applications.
2.4
Droplets internal convection
Inside the droplet convective motions are present, due to two flows: capillary flow driven by continuity and Marangoni or thermocapillary flow driven by surface tension gradients. For the first one is assumed that evaporation occurs at the triple line and the fluid flows radially outwards to replace the evaporated fluid; the droplet usually maintains a constant radius and the contact angle decreases. Depending on the temperature gradient, Marangoni flow could have opposite behavior [28]. It is however found that the temperature in the bulk of the droplet is not uniform. This sometimes revealed thermal patterns which were identified as hydrothermal waves (even in reduced gravity environment). Most of the studies concluded
that the hydrothermal waves could not be seen in evaporating water droplets, as reported in figure 2.13.
Figure 2.13: Thermal waves in evaporating droplets (a) water (no thermal patterns), (b) methanol (Figure by [20]).
Many authors investigated the flow patterns inside the evaporating sessile drops, by numerical and experimental ways [21], [20]: still the exact role of the thermal waves and the contributions of the flows on evaporation kinetics remains to be fully quantified. Cer-tainly,the competition of flow induced by the privileged evaporation near the contact line, the thermo-capillarity and the buoyancy in the liquid is strong during evaporation, and the prevalence of one or two effects rules the flow direction inside the drop and consequently evaporation dynamics. A three dimensional simulation of a droplet on a heated substrate performed by Sáenz and Sefiane and exposed in [29] get the idea of the flows behaviour.
Figure 2.14: Qualitative convection motions from a three dimensional simulation of a droplet on a heated substrate (Figure by [29]).
Measurement methods
All properties related to the droplet shape from which informations about force balance can be obtained are determined by image processing. The evolution of the drop in evaporation is filmed by a high resolution camera (Ximea MQQ022MG-CM), and each frame is analyzed by a Matlab routine. Chapter 3 describes in detail the operations performed by this routine. These methods where used successfully with bubbles by Saccone and Di Marco ([7], [8]) and the further considerations in the following are applicable to both bubbles and droplets.
3.1
Initial image processing
Matlab was used because it offers many useful functions for processing images in the Image Processing Toolbox. Videos are imported in the Matlab workspace as a four-dimensional matrix which associates a grayscale value to each pixel in the frame for each frame: pixels are identified by the coordinates x and y with origin on the upper left corner of the frame and the gray scale goes from 0 to 255 (0 is total black and 255 total white). An example of the treated images is shown in figure 3.1.
Figure 3.1: Frame obtained from the camera. 27
The image is properly cut and contrast is increased in order to better distinguish the outlines, important for the next steps (figure 3.2).
(a) Before contrast adjust. (b) After contrast adjust. Figure 3.2: Difference between frames before and after contrast increase.
Then the image is converted from grayscale to black and white with the command im2bw; the definition of a level between 0 and 1 is required to separate the pixels that will be black from those that will be white. This is a very delicate and sensitive parameter for the the goodness of the analysis. If everything has functioned correctly, an image of the type shown in the figure 3.3 is presented:
Figure 3.3: Black and white image after using im2bw command.
The image has been cut according to parameters set manually (in figure 3.3 for example is evident that it has been cut too low, as a part of the reflected droplet image is also included) . It is not necessary to be excessively precise as the routine automatically adjusts the image size afterwards. To do that with the command regionprops many geometric properties associated with the droplet shape are calculated a first time through internal algorithms based on numerical analysis of pixels. Some of these are:
• Area (number of pixels)
• Centroid (center of mass of the region)
• Length in pixels of the semi-major and semi-minor axes of the ellipse of inertia • Extreme points of the region
• Angle between the horizontal and the semi-major axes of the ellipse of inertia
Exploiting the extreme points and the orientation the image is cut precisely at the droplet base. During this operation the color of the pixels is also inverted (figure 3.4 (a)). This method is very effective in most cases, but manual cutting is not excluded for certain images.
(a) Final image. (b) Edge found with the Canny method. Figure 3.4: Final image and edge.
At last the command regionprops is used again to calculate the same propertis described above on the final image and the edge of the droplet is obtained with the command edge. To find the edges recurs to the Canny method ([8], [7]): “this method finds edges by looking for local maxima of the gradient of an image. The gradient is calculated using the derivative of a Gaussian filter” (from the Matlab documentation of the edge function).
The final image as that in figure 3.4 (a) is the starting point for the following steps.
3.2
Calibration
Up to this point the measurements of lengths and areas are in pixels, because extrapolated directly from the image elaboration. Before each experimental session the calibration is performed, which consists in acquiring the image of an object of known size (or easily measurable) in order to calculate the scale factor to convert pixels lenghts into mm lenghts. An example of calibration is shown in the figure 3.5, in which a screw is placed where the droplet is placed.
3.3
Center of gravity, height, base diameter and radius of
cur-vature calculation
The position of the center of gravity is calculated during image processing and is given in pixels. It is sufficient to convert it into mm thanks to the scale factor calculated with the
Figure 3.5: The screw used to do the calibration.
calibration. A 2-D array containing the coordinates in mm (using the scale factor again) of the points on the edge is assembled for next steps. Using the extrema points the base diameter is calculated. The height of the drop is calculated finding the maximum and minimum in vertical direction.
The direct measurement of the radius of curvature at the top of the droplet is performed selecting a certain number of pixels starting from the one on the top to do a parabolic regression (polyfit function). In this way the radius can be calculated as the radius of the osculating circle of the parabola, which in its origin results 1
2a (describing the parabola as
y = ax2+ bx + c). 2.5 3 3.5 4 4.5 1.8 2 2.2 2.4 Measured points Fitted parabola
Figure 3.6: Example of the top of a droplet: experimental measured points in red, fitted parabola in blue.
3.4
Volume calculation
To calculate the volume the droplet is supposed to be axisymmetric, so the following ex-pression can be used:
V = Ls3 N
X
k=1
2πrp,k
where V is the volume, Ls the scale factor, N the number of pixels of the half of the figure
area, rp,k the distance of each pixel from the axis of symmetry. The above summation
considers each pixel as a unitary area figure that rotates around the vertical axis passing through the barycenter of the droplet. The many volumes obtained by the rotation of each pixel are added together to have the final volume; naturally the pixels of the left and right part are used separately with an angle of 360◦. The final volume is an average of the two in
order to compensate for the slight differences that may exist between one side of the droplet and the other; if the difference between the two volumes is too large a warning message is issued. Lastly the value thus obtained is multiplied by the cube scale factor with the aim of having the result in mm3.
This is a numerical version of Pappo-Guldino theorem for rotation solids obtained by rotat-ing a surface:
Theorem (of Pappus and Guldinus). The volume of a rotation solid obtained by rotating a plane figure of an angle α ∈ [0, 2π] around an axis complanar to the area is given by V = αdA, where V is the volume, A the area, d the distance of the barycenter of the figure from the axis and α the angle.
3.5
Contact angle calculation
Regarding the contact angle its determination is among the most delicate that subject to greater uncertainty. The method consists of achieve a least square regression on a number of points of the edge (figure 3.7) and evaluate the gradient at the extremity (triple line). Points are weighted in function of the proximity to the extremity. The interface should be as sharp as possible in order to reduce errors in finding the profile; reflections on droplet shape and unwanted vibrations (inevitable during parabolic flight for example) compromise sharpness increasing error.
3 3.05 3.1 3.15 3.2 3.25 3.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Measured points
Least square regression
Figure 3.7: Right edge of a droplet: experimental measured points in red, fitted parabola in blue.
sensitive parameter for the analysis. Previously a work on the dependence of the calculation on the number of points has been done, with droplets of HFE 7100 [30]. A strong effect on the contact angle measurement due to the number of pixels and the tendency to stabilize the value from a certain point onwards was observed.
20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 Npix Con tact Angle θc [degrees]
(a) Dependency on pixel number of contact angle calculation.
(b) Common trend to stabilization for a many im-ages.
Figure 3.8: Common trend to stabilization for a certain number of points (Figures by [30]) .
For water used for this work a good indication of the number of points was provided by capillary length (see section 2.1.4). Good results in terms of contact angle have been achieved setting a number of points corresponding to one tenth of the capillarity length, converting lengths into pixels with the scale factor; using fewer points leads to bigger errors and more points to increasing in calculation times. All this is valid in the range that goes from 20◦ to 90◦. This setting turned out appropriate for the cases with no gravity (which
3.6
Force balances
The routine is also able to compute force balances using physical properties such liquid density and surface tension, measures from the experimental apparatus such gravity accel-eration and electric field value and the geometrical properties computed as exposed in the previous paragraphs.
Considering the momentum balance in a control volume surrounding the droplet on the liquid-vapour interface and on the base. Following the approach used by Cattide [8] the momentum balance is espressed in equation 3.1.
Figure 3.9: Control volume and droplet.
d dt Z Z Z V ρfv dV = Z Z Z V ρfg dV + Z Z S Tv,g− pgI + ρgvg(vg− vs) + Te,g ·n dS + Z Z A Tv,f − psfI + ρfvf(vf − vs) + Te,f ·n dA + I CL σf gtf gdL (3.1)
The different terms are: variation of momentum, gravitational force acting in the volume, hydrostatic pressure, viscous stress, momentum inflow/outflow (evaporation or condensa-tion), electric stress (Maxwell tensor), integrated all over the gas side of drop interfaces and all over the liquid side contact area, and surface tension forces acting on the contact line. For a quasi-static interface with no mass transfer the equation is simplified in the 3.2
I CL σf gtf gdL + Z Z Z V ρfg dV − Z Z S pgI· n dS − Z Z A pfI · n dA + Z Z S Te,g· n dS + Z Z A Te,f · n dA = 0 (3.2)
In the routine methods to evaluate numerically the integrals above are implemented, based on previous image elaboration, in vertical and radial direction.
3.6.1 Vertical balance
Projecting 3.2 in vertical direction it becomes:
Fp+ Fσ+ Fb+ Fe= 0 (3.3)
The terms are explained as follows, applied to the axisymmetric droplet. They are pressure force, surface tension force, buoyancy force, electric field force:
Fp = πD2b 4 2σ R0 + (ρf − ρg)gH (3.4) Fσ = −πDbσ sin θ (3.5) Fb = V g(ρf − ρg) (3.6) Fe= Z Z S Te,g· n · j dS + Z Z A Te,f· n · j dA (3.7)
The first three forces (Fp, Fσ, Fb) can be evaluated once the geometrical parameters of
the droplet (Db, R0, H, θ, V) are obtained from the images, and must add to zero in absence
of electric field. This force balance has been verified experimentally for bubbles and droplets in the absence of electric field in several references [6], [8], [7]. When the electric field is applied its force can be obtained as the unbalanced component as shown in equation 3.9.
Fp+ Fσ+ Fb = Fres= 0 (3.8)
3.6.2 Radial balance
For radial balance consider half of the droplet. In figure 3.10 forces acting on the slice are shown: pressure of the liquid in red, pressure of the air in orange, surface tension in blue.
Figure 3.10: Radial force balance: pressure of the liquid in red, pressure of the air in orange, surface tension in blue.
Figure 3.11: Radial force balance, sectional view.
Expressing the forces due to pressures and surface tension the momentum balance in radial direction is written as follows (pg is projected on surface A):
Z Z A (pf − pg) dA − Z L σ dL + Z CL σ cos θ sin ϕ dCL = 0 (3.10)
Following the same approach as what exposed in section 2.1.4 and replacing the pressures with equations 2.12, 2.13 and 2.14 the previous expression becomes (assuming constant contact angle and surface tension)
Z Z A 2σ R0 + (ρf − ρg) g y dA = σ Z L dL − cos θ Z CL sin ϕ dCL (3.11) and finally, after some steps
2 Z H 0 Z r(y) 0 2σ R0 + (ρf − ρg) g y dr dy = σ L − 2Rbcos θ (3.12) The first term of equation 3.12 is the radial pressure force Fp,R, whilst the second is the
radial surface tension force Fσ,R. The equilibrium of the drop reads Fp,R = Fσ,R; also in
this case, when electric field is applied, the resulting electric force is derived from the sum of the others terms (Fe,R = Fσ,R− Fp,R) and they have to be experimentally verified in the
absence of electric field.
3.7
Triple Line Velocity
Starting from the base diameter calculation for each frame of the video and the acquisition frequency of the camera the triple line velocity is calculated using forward Euler method.
3.8
Inputs and outputs statement of the Matlab routine
In the tables 3.1 and 3.2 the significant parameters managed by the routine are reported.Table 3.1: Input values
Name Description Units
σ Surface tension of the liquid-vapour interface N/m
ρf Liquid density kg/m3
ρg Vapour density kg/m3
g Array of measured acceleration in vertical direction m/s2 HV Array of measured electric field between the two electrodes kV
Table 3.2: Output values
Name Description Units
θ Contact angle deg
V Droplet volume mm3
R0 Radius of curvture at the top mm
H Height mm
Db Base diameter mm
L Interface lenght mm
Fp Pressure force N
Fσ Surface tension force N
Fb Buoyancy force N
Fe Electrical force N
Fp,R Radial pressure force N
Fσ,R Radial surface tension force N
3.9
Uncertainty analysis
Figure 3.12: Edge definition. The experimental uncertainty has been
evaluated. Apart from the measurement er-rors of the instrumentation another source of uncertainty is the image elaboration phase, during which lengths, volume, cur-vature radii and contact angles are mea-sured. One pixel of uncertainty has been considered for lengths. For volume has been calculated the defect and excess vol-ume adding one pixel along the edge and applying Pappo-Guldino. More difficult is calculate the contact angle error, because it
is determined by a parabolic regression that depends on the number of points used and the image quality. To estimate that a comparison between the measured angle and the one calculated with Laplace equation in quasi-static conditions with no electric field (see section 4.2.1 for more details) has been made and assumed equal in the other conditions. Concerning the quantities which derive from direct calculations, the error propagation meth-ods have been used. Called F a function that depends on the unrelated variables contained in the vector x, the error is given by:
∆F (x, ∆x) = v u u t Xn k=1 ∂F ∂xi ∆xi 2
There are less quantifiable error sources. The droplet is backlit by a LED and this can produce reflections on the edge and in the bulk that sometimes make measure accuracy worse. Thereby some pixels, brighter than than they should be, are excluded from the cal-culations. Another source always present during parabolic flights is represented by residual vertical acceleration (g-jitter) and lateral accelaration (due to aircraft maneuvers), which make the droplet mova and this can produce blurred images, very annoynig for volume and edge measures. Also in this case a good feedback on measures correctness derives from the static balance of the forces: if the resultant of the forces (calculated using physical
proper-ties and geometrical measured values) is zero in static conditions with no electric field that is an indication of the good success of the measures.
Table 3.3: Experimental uncertainty table.
Parameter PU [%] θ ± 28.6 V ± 0.005 R0 ± 8.1 H ± 1.4 Db ± 0.2 L ± 0.18 Fp ± 6.1 Fσ ± 5.2 Fb ± 0.001 Fe ± 11.3 Fp,R ± 6.9 Fσ,R ± 4.5 Fe,R ± 11.4
Evaporating droplets
Chapter 4 contains the main part of the work. First of all the experimental set-up is de-scribed in all its parts and an explanation of the experiments execution is provided. Where-upon the results of the analysis are shown and commented in order to reach conclusions in the next chapter.
4.1
Experimental activities
The most important part of the present work are the performed experiments. Substantially the experiment consist of a metal foil on which a droplet is deposited. The foil is heated by Joule effect as it is clamped between two copper bus bars that are connected to a DC power supply, in order to produce droplet evaporation with constant heat flux. Above the heater a whaser-shaped electrode is placed in order to generate an electrostatic field between itself and the heater, which is grounded. The collected data derive from three subsystems. First there is the optical camera, located in front of the droplet that aquires images that are used to derive the droplet profile; the used tecnique is called shadowgraphy as the images are taken with the aid of a backlight illumination by a white led lamp. Then there is a IR camera located below the foil to monitor the local temperature of the heater, in order to derive a map of heat flux. Lastly there is a data acquisition system that receives the signals from temperature sensors, pressure transducers, accelerometers, current and voltage sensors, in order to measure all values necessary for the analysis or monitor the experiment conditions in real time. All the systems are inserted inside a very compact rack in order to perform experiments during parabolic flights too.
