Mollaret_experimental and Numerical Investigation of the Evaporation Into Air of a Drop on a Heated Surface

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0263–8762/04/$30.00+0.00# 2004 Institution of Chemical Engineers

www.ingentaselect.com=titles=02638762.htm Trans IChemE, Part A, April 2004Chemical Engineering Research and Design, 82(A4): 471–480

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF

THE EVAPORATION INTO AIR OF A DROP ON A

HEATED SURFACE

R. MOLLARET1

, K. SEFIANE1,*, J. R. E. CHRISTY

1, and D. VEYRET

2

1School of Engineering and Electronics, University of Edinburgh, Edinburgh, UK 2 Laboratoire IUSTI, UMR CNRS 6595, Technopo le de Cha teau-Gombert-5, Marseille, France

Understanding the wetting and evaporation behaviour of volatile droplets on heated surfaces is very important for many industrial applications. In this paper the behaviour of a sessile drop evaporating on a heated surface is investigated both experimentally and

numerically. Results are reported for the evaporation of water drops on two different substrates at various temperatures. A numerical model, based on a finite element method, has been developed

to describe the hydrodynamics inside the evaporating drop and the effect of the humidity on theevaporation process, assuming the droplet to be a spherical cap. The energy and Navier–Stokesequations are solved within the droplet and the vapour concentration is computed using thediffusion equation. The drop volume and flow and temperature fields within the drop are obtained and the evolution of the volume in time is compared with the experimental results.

Keywords: wetting; evaporation; drop; modelling.

INTRODUCTION

The evaporation of sessile droplets has been studied

extensively over the last few decades, due to the widerange of interest and of applications. Both experimentaland numerical studies have been performed to elucidate theinteractions between wetting characteristics and the evapora-tion process. Birdi and Winter (1989) measured the changeof weight of evaporating water droplets on a glass surfaceand concluded that the evaporation rate remains constant for most of the period of evaporation. Following this investiga-tion (Birdi, 1993), they reported that the rates of evaporationof sessile droplets both of water on glass and of n-octane onTeflon surfaces are constant whilst the contact line is pinned.

Bourges-Monnier and Shanahan (1995) investigated theinfluence of evaporationon thecontact angle. Theevaporation

of sessile droplets of water and n-decane was studied onvarious substrates using a projection method for determina-tion of contact angle. Drop dimensions (height, base width)and contact angle were measured as a function of time. Thevariability of the contact angle originated from roughness and chemical heterogeneity. A decrease in evaporation rate wasobserved with increasing initial contact angle. During theevaporation process distinct stages were observed. In the first stage the droplet base remained constant as the contact angleand droplet height decreased. In the second stage, the contact angle remained roughly constant as thedroplet base decreased

until eventually the droplet disappeared. Bernardin et al.(1997) presented an experimental investigation of thetemperature-dependence of the quasi-static advancing contact

angle of water on an aluminiumsurface. Twodistincttempera-ture dependent regimes were observed. In the lower tempera-ture regime, below 120C, a relatively constant contact angleof 90 was observed. In the high temperature regime, above120C, the contact angle decreased in a fairly linear manner.

More recently Hu and Larson (2002) researched theinternal fluid flow in a volatile droplet. The fluid flow inan evaporating droplet was investigated experimentally and the results were compared to the theoretical model of Deegan (1998). The volume of the droplet was found todecrease linearly with time when left to evaporate in acontrolled environment. Fluid flow within the droplet wasmeasured by suspending small tracer particles within the

bulk. The particles were found to circulate, movingoutwards along the liquid–gas interface towards the edgeof the droplet and returning along the substrate, as evapora-tion proceeded. Both the radial and vertical velocities of the particles were calculated. In the first stage observed, thecontact line was found to be pinned and the contact angledecreased. In the second stage, the contact line receded while the contact angle remained very small. The first stageoccupied 90–95% of the droplet lifetime, thus the authorsfocussed on the evaporation of a sessile droplet with a pinned contact line (stage one). They investigated the phenomena experimentally and by computation using thefinite element method (FEM) for water on a glass substrate.The experimental observation that the net evaporation rate

from the droplet remains almost constant with time for asmall initial contact angle (y< 40) was consistent with the

471

*Correspondence to: Dr K. Sefiane, School of Engineering and Electronics,University of Edinburgh, Kings Buildings, Mayfield Road, EdinburghEH9 3JL, UK.E-mail: [email protected]



prediction of the model. They also determined a criticalcontact angle at which the contact line starts to recede and found that it is around 4.

Hegseth et al. (1996) demonstrated experimentally that when a drop evaporates it exhibits a vigorous interior flow.They postulate that this flow is driven by surface tensiongradients that arise because of the non-uniform temperature

on the free surface resulting from variations in local evapora-tion rates in proximity to solid surfaces. Kavehpour et al.(2002) reported the effect of thermocapillary stresses on thesurface of an evaporating drop. They demonstrated that thiseffect is more important for more volatile liquids and for more thermally conductive substrates. The authors showed how these thermocapillary effects can play a role in thespreading of a silicon oil drop on various substrates. Ruizand Black (2002) recently proposed a model to describe thehydrodynamics inside an evaporating drop on a heated surface. They used the model to investigate the convectionrolls induced by thermocapillary effects. The simulationconfirms that surface tension gradients on the free surface

of the evaporating drop could account for the observed internal flows. The model, however, did not fully account for the effect of diffusion and concentration in the vapour phase above the evaporating drop. This is a significant limitation of the model as the state of saturation of the gas phase is known to be of paramount importance.

Another important aspect of the wetting of volatiledroplets is the anchoring of the triple line. Blake et al.(1997) demonstrated that molecular dynamics allows acomplete study of the dynamics of droplet spreading at the microscopic scale. Particular focus was given to thestudy of the net fluxes inside the droplet during spreading.The authors concluded that, at the molecular level, themechanism of spreading could be viewed as a competition

between a surface tension driving force and friction betweenthe liquid and solid atoms. This result supports theDeGennes and Cazabat (1990) model of spreading.

It is clear that, despite the amount of work devoted to thesubject, many issues are still not well understood. Elucida-tion of the controlling parameters for wetting hysteresis isone of the problems still under investigation (Nakae et al.,1998). The mechanism behind the depinning phenomenonand the role of the evaporation rate have not been fullydetermined. The modelling of the process, however, islimited either to ideal cases (Anderson and Davis, 1995)or to a single stage of the process (Shanahan, 1995). In thestages described above, the depinning point is crucial in

understanding the wetting behaviour of the drop. It is alsocritical for many applications: hence it is essential to under-stand the factors behind this phenomenon.

The objectives of this work are firstly to investigateexperimentally the role of the substrate temperature on thewetting behaviour of sessile volatile droplets, secondly todevelop a numerical model describing the hydrodynamicswithin the droplet as well as the vapour diffusion in the gas phase and finally to compare the output of the developed model with experimental results.

EXPERIMENTAL SET-UP AND PROCEDURE

The apparatus consists of a substrate (aluminium or

PTFE) heated from below using a cartridge heater, athermocouple inserted into the substrate to monitor and

allow for control of the temperature and a syringe (operated by a stepper motor) enabling the injection of a 10ml liquid droplet at a rate of 2 ml s1. The substrate was placed in asemi-open cell, so that it was protected from the convectionthat occured in the laboratory, whilst allowing the vapour generated by evaporation to diffuse into the atmosphere.The drop profile was videotaped as it evaporated and the

contact angle, base width, height and volume were measured using FTA drop analysis software (First Ten AngstromsCompany). The substrate was cleaned with a cloth soaked with acetone before each run. The experiment wasconducted on a low-vibration optical table to avoid anyexternal disturbances.

The results are presented in both dimensional form and anormalized form, where all the quantities except contact angle (time, volume and base width) were normalized usingthe maximum values. Two sets of results are presented, thefirst set is for water droplets evaporating on an aluminiumsubstrate and the second one is for water on a PTFEsubstrate. The two substrates are selected because of the

difference in thermal conductivity and surface energy, which plays a major role in wettability.

WATER ON ALUMINIUM

Figures 1–3 show the evolution of the droplet volume intime for various temperatures. There is a linear trend throughout the lifetime of the droplet at higher tempera-tures, whereas at lower temperatures the behaviour tendsto deviate from the linear trend towards the end of thedroplet lifetime. The trend in the contact angle (Figure 4)also indicates that at lower temperatures (20, 40, or 60C)the behaviour differs from that at higher temperatures: at lower temperatures there is a period when the contact angle remains constant (following depinning of the contact line).

Figures 5 and 6 show that there is a depinning of the contact line at lower temperatures (20, 40, or 60C),whereas at higher temperatures (80, 100C) the contact lineremains pinned. There is a clear shift in the time at whichdepinning occurs as the temperature is increased. The

Figure 1. Variation of the droplet volume in time for water on aluminium

at 40C. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Trans IChemE, Part A, Chemical Engineering Research and Design, 2004, 82(A4): 471–480

472 R. MOLLARET et al.



Figure 2. Variation of the droplet volume in time for water on aluminiumat 80C. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 3. Variation of the droplet volume in time for water on aluminiumat various temperatures. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 4. Variation of wetting contact angle in time for water on aluminium

at various temperatures. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 5. Variation of the droplet base width in time for water on aluminiumat various temperatures. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 6. Variation of the droplet normalized base width in time for water onaluminium. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 7. Variation of wetting contact angle for water on aluminium at

various temperatures. This figure is available in colour via www.ingentas-elect.com=titles=02638762.htm.


EVAPORATION OF A DROP ON A SURFACE 473



influence of the depinning process on wetting behaviour isseen in Figures 4, 7 and 8, where the contact angle behaviour clearly changes when the triple line depins, with the contact angle remaining almost constant for a period after depinning before dropping off as the droplet nears the end of its lifetime.

WATER ON PTFE

Figures 9 and 10 present the evaporation rate and Figures 11–14 the wetting behaviour of a water droplet ona PTFE substrate. The results indicate that there is very littleeffect of temperature on the behaviour of the evaporation process and wetting dynamics. The droplet remains pinned for a period of time before depinning, at which stage the base width decreases. At the same time the contact anglechanges from a decreasing regime towards a rather constant Figure 8. Variation of normalized wetting contact angle as a function

of the drop volume. This figure is available in colour via www.ngentaselect.com=titles=02638762.htm.

Figure 9. Variation of the droplet volume in time of the drop volumefor water on PTFE. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 10. Variation of normalized volume as a function for water on PTFE


Figure 11. Variation of the droplet base width in time for water on PTFEat various temperatures. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 12. Variation of normalized base width in time for water on PTFE






Figure 13. Variation of the droplet contact angle in time for water on PTFEat various temperatures. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 14. Variation of wetting contact angle with normalized timeon PTFE. This figure is available in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 15. Comparison of wetting behaviour of water droplet on aluminiumand PTFE at 20C. This figure is available in colour via www.

ingentaselect.com=titles=02638762.htm.

Figure 16. Configuration parameters.

Figure 17. Structure of solution algorithm.





regime. It is worth noting that the receding contact angle isapproximately the same at all investigated temperatures.

Figure 15 shows a comparison of the wetting behaviour of a water droplet evaporating on aluminium and PTFE

substrates at 20C. The initial contact angle is greater onPTFE than on aluminium; furthermore the wetting beha-viour is clearly different. On aluminium the contact angledecreases linearly, before suddenly increasing slightly and

Figure 18 . Time sequence of evaporation predicted by the model (water, T H ¼ 80C; humidity ¼ 50%; ambient temperature ¼ 20C).







model will thus be applied to the case of water dropletsevaporating on the aluminium substrate. The droplet shape isconsidered as a spherical cap (Figure 16) throughout theevaporation process. Two existing models were combined:inside the droplet the energy and Navier–Stokes equationsare solved and in the gas phase the vapour concentrationabove the droplet is calculated using the diffusion equation.

Cylindrical coordinates are used with the z -axis vertical. Thesaturated vapour concentration on the droplet surface dependson the local temperature field (obtained from the previoustime step). The temperature at the liquid–gas interface is not constant, which initiates Marangoni convection.

The general conservation equations are written in theliquid phase:

H ~V V ¼ 0 (1)

r@ ~V V

@t þ r( ~V V H) ~V V ¼ H P þ rnH2 ~V V (2)

@T

@t

þ ~V V HT ¼ aH2T (3)

where ~V V is the velocity vector with radial and axialcomponents, respectively u and v. In the vapour, the diffu-sion equation is written:

@C

@t ¼ DH2C (4)

The concentration as r ! 1, z ! 1 is taken as theambient humidity.

Surface tension is written as a decreasing linear functionof temperature.

s ¼ s0 d s

d T (T T 0) (5)

The dimensionless equations in the liquid phase and the boundary condition are written following Ruiz and Black (2002). The scaling factors are the initial drop radius ( Ro), avelocity scale [U sc ¼ (d s=d T )DT =m] and a temperaturescale, (DT ¼ T H T 1). The pressure was made dimension-less using rU 2sc.

1

~ r r

@ ~ r r ~ uuð Þ

@~ r r þ

@~ vv

@~ z z ¼ 0 (6)

@~ uu

@~ t t þ

1

~ r r

@ ~ r r ~ uu2 @~ r r

þ@ ~ uu~ vvð Þ

@~ z z ¼

@ ~ P P

@~ r r þ

1

~ r rRe

@

@~ r r ~ r r

@~ uu

@~ r r

~ uu

~ r r 2 Reþ

1

Re

@

@~ z z

@~ uu

@~ z z

(7)

@~ vv

@~ t t þ

1

~ r r

@ ~ r r ~ uu~ vvð Þ

@~ r r þ

@ ~ vv2 @~ z z

¼ @ ~ P P

@~ z z þ

1

~ r rRe

@

@~ r r ~ r r

@~ vv

@~ r r

þ 1

Re

@

@~ z z

@~ vv

@~ z z

þ

Ra

Re2 Pr ~ T T

(8)

@ ~ T T

@~ t t þ

1

~ r r

@ ~ r r ~ uu ~ T T @~ r r

þ@ ~ vv ~ T T @~ z z

¼ 1

~ r rRe Pr

@

@~ r r ~ r r

@ ~ T T

@~ r r

!

þ

1

Re Pr

@

@~ z z

@ ~ T T

@~ z z !

(9)

The dimensionless initial conditions are:

at ~ t t ¼ 0, ~ uu ¼ ~ vv ¼ 0, ~ T T ¼T T 1DT

¼ 0 (10)

The dimensionless boundary conditions are, for contact at the heated surface (i.e. ~ z z ¼ 0):

~ uu ¼ ~ vv ¼ 0, ~

T T ¼ 1 (11)for the axis of symmetry (i.e. ~ r r ¼ 0):

~ uu ¼ 0, @~ vv

@r ¼ 0,

@ ~ T T

@r ¼ 0 (12)

For the free surface, boundary conditions are written expres-sing respectively, mass conservation at the interface anormal stress balance, a tangential stress balance (includingthermocapillary term) and energy conservation.

~ V V n ~ V V 1,n ¼ ~ mm00 (13)

~ P P ¼ ~ kk

Re

1

Ca

~ T T þ 2

Re

@ ~ V V n

@n

~V V

@~nn

@n0@ 1A (14)

@ ~ V V t @n

þ@ ~ V V n@ s

~V V

@~ s s

@nþ

@~nn

@ s

¼

@ ~ T T

@ s (15)

@ ~ T T

@n ¼ Bi ~ T T þ

Ma

Ja~ mm00 (16)

The local mean curvature at the interface is given byk ¼ H ~nn.

In Ruiz and Black (2002), the evaporation rate, ~ mm00, inequation (13) was determined from the mass transfer coeffi-cient and the saturation vapour density. The environment surrounding the droplet was assumed to be dry air, but the

surrounding environment vapour concentration could be set as desired. The mass transfer coefficient was determined byanalogy with natural convection heat transfer.

As the droplet evaporates with a pinned triple line thecontact angle decreases and the liquid–vapour interfacedescends towards the substrate. The algorithm in Figure 17is solved at every time step until convergence is reached.

The numerical technique used to solve the system of equations is based on a finite-element method (FEM). Becauseit is a moving boundary problem, the algorithm is applied for every time step, with a new grid generated for each time step.

Ruiz and Black (2002) assumed a mass transfer coefficient for diffusion of vapour in the gas phase, based on natural

convection heat transfer. We have adapted this model bysolving the diffusion equation for vapour in the gas phase.The model allows us to compute the isoconcentrations and couple the concentration field to the local evaporation rate. At the liquid–vapour interface of the drop, the local evaporationflux is expressed as (following Hu and Larson, 2002):

~ J J (r ,t ) ¼ DHC (17)

Table 1. Simulation parameters and dimensionless numbers.

Case 2 Ro (mm) yo T H (C) Ca Pr

1 3 94 80 0.1628 2.2

2 3.18 95.43

70 0.1356 2.23 3.18 97.96 60 0.1085 2.2





The model has been compared with the experimentalresults performed for water droplets evaporating on analuminium substrate. The experimental initial conditions(initial radius, initial contact angle) have been used in themodel to allow comparison (Table 1); the model shows avery good agreement with the experimental results prior todepinning (Figure 19).

Figure 18 shows results for three time stages during thecourse of evaporation from the droplet. Within the droplet, therecirculation is shown on the left and the isotherms on the right and in the vapour phase lines of constant concentration aredepicted.Figure 19 provides a comparison of thevolumevs time

for the output of the model with results from our experiments.

DISCUSSION AND ANALYSIS

The evaporation process of a volatile sessile droplet must be analysed in conjunction with the wetting behaviour.These two processes are closely related. As shown by previous authors the evaporation of a sessile droplet takes place in various stages depending on the environmentalconditions and the substrate material (Figure 20). The first regime is the constant base regime (A), where, as the dropevaporates the contact angle decreases, whilst the base widthremains constant. The second regime is the constant angle

regime (B); the base line recedes and the contact angleremains constant. As the droplet evaporates it tends to keepthe equilibrium contact angle; if the triple line is allowed tomove freely this regime is observed. It worth noting that thiswould be valid for ideal surfaces where no roughness or

chemical heterogeneity of the substrate is present to prevent the contact line from receding. On real surfaces (rough),however, the first regime can be observed, and sometimesthe combination of both regimes, where the contact lineremains pinned for a while before it depins. In this latter case, as the evaporation progresses, the contact angledecreases and the Young unbalanced force pulling on thetriple line increases. When it overcomes the adhesion forcesinduced by roughness, the triple line depins (C). The Youngunbalanced force is expressed in many theories (Blake et al.,1997; DeGennes, 1985) as follows:

F ¼ slg(cos y cos ye) (18)

where ye is the equilibrium contact line as described by theYoung equation:

cos ye ¼ss ssl

slg

(19)

The results for water on aluminium show that, as temperature is increased, the wetting behaviour shifts from adepinning regime to a constant base regime (fully pinned).Considering the force balance at the triple line (Figure 21),if the Young disequilibrium force, which tends to depin thetriple line, is weakened by reduction of the surface tensionat increased substrate temperature, the roughness adhesionforces will be able to overcome the Young disequilibrium

force and the contact line remains pinned throughout thelifetime of the drop.In the case of PTFE, the surface energy is low, so that the

Young disequilibrium force can always overcome the fric-tional force once the contact angle drops to a low enoughangle. This value does not appear to depend significantly ontemperature.

In the model introduced by Ruiz and Black (2002), theenvironment surrounding the droplet was assumed to be dryair, therefore the concentration of the vapour in the far field was assumed to be zero. However, Bourges-Monnier and Shanahan (1995) have demonstrated that saturation of thevapour phase can dictate evaporation and wetting behaviour.Therefore a model that includes diffusion in the vapour is

required if such behaviour is to be predicted. The model of Ruiz is based on a single mass transfer coefficient predicted

Figure 19. Comparison of experimental and predicted evaporation rate of water on aluminium substrate, relative humidity ¼ 40%. This figure isavailable in colour via www.ingentaselect.com=titles=02638762.htm.

Figure 20. Schematic diagram showing evaporating drop behaviour.

Figure 2 1. Force balance exerted on the contact line.





for the whole surface, using a correlation for naturalconvection heat transfer and the analogy between heat and mass transfer. Whilst the concentration gradient can bealtered by allowing for a level of saturation of the bulk air, the effect of local variations in saturation conditions onthe mass transfer coefficient is not accounted for.

In our revision of this model, evaporation is calculated

from the diffusion equation applied to a spatial mesh in thegas phase. The bulk concentration of vapour is set as a boundary condition far from the surface and the gas inimmediate contact with the liquid is assumed to be fullysaturated at the liquid temperature. The results (Figure 19)show the very close agreement achieved with this revised model to the experimental data for the change in volume of the drop with time. The deviation in the case of water at 60C is due to depinning of the drop during the experiment.

CONCLUSION

The evaporation and wetting of a water sessile droplet is

investigated on a heated substrate. Two substrates were used,namely aluminium and PTFE. The substrates were heated in a temperature range 20–100C. The two substrates had different thermal conductivities and surface energies whichled to very different wetting behaviour. On aluminium thedroplet exhibited a depinning of the triple line at tempera-tures up to 60C. From 80 to 100C the triple line remained pinned. There was a clear shift in the depinning time astemperature was increased. This behaviour is explained bylooking at the force balance at the triple line. Increasingtemperature reduces the surface tension, therefore reducingthe Young unbalanced force pulling on the triple line. Thus,above a critical temperature, the Young unbalanced force isnot strong enough to overcome the adhesion forces due to

roughness.Furthermore a numerical model was developed to

describe the evaporation process and investigate the hydro-dynamics inside the drop. The model was compared with theexperimental results on the aluminium substrate and good agreement was achieved.

NOMENCLATURE

Bi Biot number, Bi ¼ hnc Ro=k Ca capillary number, Ca ¼ jd s=d T jDT =s0

C concentration D diffusion coefficient hfg latent heat of vaporisation

hnc natural convection heat transfer coefficient ~ J J mass flux Ja Jacob number, Ja ¼ hfg=c pDT k liquid thermal conductivityk air air thermal conductivitym droplet massm00 evaporative flux

Ma Marangoni number, Ma ¼ jd s=d T jDTRo=ma~nn unit normal vector to droplet surface Nu Nusselt number, Nu ¼ hnc D=k air

P pressure Pr Prandtl number, Pr ¼ n=ar radial coordinate

Ra Rayleigh number, Ra ¼ g bDTR30=va

Re Reynolds number, Re ¼ U sc R0=n Ro

initial contact radius of drop~ s s unit vector tangential to surface

t timeT temperature

T H substrate temperatureDT temperature difference between the wall and ambient U sc velocity scaleu radial component of the velocityv axial component of the velocity~V V velocity vector

z axial coordinate

Greek symbolsa liquid thermal diffusivityb coefficient of expansionk local mean curvature of interfacey contact anglem liquid dynamic viscosityn liquid kinematic viscosityr densitys surface tension

Subscripts

ls liquid–solid lg liquid–gassg solid–gasI interfaceo reference staten normal component

t tangential component

Superscripts

dimensionless form

REFERENCES

Anderson, D. and Davis, S., 1995, The spreading of volatile liquid dropletson heated surfaces, Phys Fluids, 7: 248.

Bernardin, J.D., Walsh, C.B. and Franses, E.I., 1997, Contact angletemperature dependence for water droplets on practical aluminiumsurfaces, Int J Heat Mass Transfer , 40: 1017–1033.

Birdi, K.S., 1993, Wettability and the evaporation rates of fluids from solid surfaces, J Adhes Sci Technol , 7: 485.

Birdi, K.S., and Winter, A., 1989, A study of the evaporation rates of smallwater drops placed on a solid surface, J Phys Chem, 93: 3702.

Blake, T., Clarke, A., De Connik, J. and De Ruijter, M.J., 1997, Contact angle relaxation during droplet spreading: comparison between molecular kinetic theory and molecular dynamics, Langmuir , 13: 2164–2166.

Bourges-Monnier, C. and Shanahan, M., 1995, Influence of evaporation oncontact angle, Langmuir , 11: 2820–2829.

Deegan, R., 1998, Patternformation in drying drops, Phy Rev E , 61(1):475–485.DeGennes, P., 1985, Wetting: statistics and dynamics, Rev Mod Phys, 57:

827–863.DeGennes, P. and Cazabat, A., 1990, Spreading of a stratified incompres-

sible droplet, C R Acad Sci, 310: 1601.Hegseth, J.J., Rashidnia, N. and Chai, A., 1996, Natural convection in

droplet evaporation, Phys Rev E , 54(2): 1640–1644.Hu, H. and Larson, G., 2002, Evaporation of a sessile droplet on a substrate,

J Phys Chem, 106: 1334–1344.Kavehpour, P., Ovryn, B. and McKinley, G.H., 2002, Evaporatively-driven

marangoni instabilities of volatile liquid films spreading on thermally

conductive substrates, Colloids Surf , 206(1–3): 409–423. Nakae, H., Inui, R., Hirata, Y. and Saito, H., 1998, Effects of surfaceroughness on wettability, Acta Mater , 46(7): 2313–2318.

Ruiz, O.E. and Black, W.Z., 2002, Evaporation of water droplets placed on aheated horizontal surface, J Heat Transfer , 24: 854–863.

Shanahan, M.E.R., 1995, Simple theory of ‘‘stick–slip’’ wetting hysteresis, Langmuir , 11: 1041–1043.

ACKNOWLEDGEMENTS

This work is the result of a fruitful collaboration with Ecole Polytech.,Marseille, France. The collaboration was supported by the Royal Society of Engineering under Grant =R36668.

This paper was presented at the 8th UK National Heat Transfer Conference held in Oxford, UK, 9–10 September 2003. The paper was

received 22 August 2003 and accepted for publication after revision4 February 2004.



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Mollaret_experimental and Numerical Investigation of the Evaporation Into Air of a Drop on a Heated Surface