liquid drops on vertical and inclined surfaces: i. an experimental study of drop geometry

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Journal of Colloid and Interface Science 273 (2004) 556–565www.elsevier.com/locate/jcis

Liquid drops on vertical and inclined surfacesI. An experimental study of drop geometry

A.I. ElSherbini and A.M. Jacobi∗

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana–Champaign, 1206 W. Green St., Urbana, IL 61801, USA

Received 17 July 2003; accepted 8 December 2003

Abstract

Experiments have been conducted to investigate the geometric parameters necessary to describe the shapes of liquid dropand inclined plane surfaces. Two liquids and eight surfaces have been used to study contact angles, contact lines, profiles, anddrops of different sizes for a range of surface conditions. The results show the contact-angle variation along the circumferenceto be best fit by a third-degree polynomial in the azimuthal angle. This contact-angle function is expressed in terms of the maximinimum contact angles of the drop, which are determined for various conditions. The maximum contact angle,θmax, is approximately equato the advancing contact angle,θA, of the liquid on the surface. As the Bond number,Bo, increases from 0 to a maximum, the minimucontact angle,θmin, decreases almost linearly from the advancing to the receding angle. A general relation is found betweenθmin/θA andBofor different liquid–surface combinations. The drop contour can be described by an ellipse, with the aspect ratio increasing withBo. Theseexperimental results are valuable in modeling drop shape, as presented in Part II of this work. 2004 Elsevier Inc. All rights reserved.

Keywords: Drop shape; Contact angle; Drop profile; Contact line

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1. Introduction

Determining the three-dimensional shape of a liquid don a plane surface is essential in many engineering acations. When liquid drops condense on or evaporate frosurface, the rate of heat transfer through a drop and the foacting on it depend on the drop geometry. Understandingshape of a drop is a prerequisite to obtaining its volumemass. In order to find the phase-change heat transferficient or the amount of liquid retained on a surface, inecessary to know the shapes of drops of different sizevarious surface conditions.

While the shapes of small liquid drops on horizontal sfaces can be defined by simple spherical caps, drops ofing sizes on vertical and inclined surfaces are more compbecause contact angles vary from the advancing to the reing contact angle along the drop circumference (see [1]this work, the advancing and receding contact angles refthe maximum and minimum contact angles, respectivela drop in the critical condition of incipient motion. A bri

* Corresponding author.E-mail address: [email protected] (A.M. Jacobi).

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2003.12.067

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discussion of earlier works on the shapes of drops onclined surfaces is presented in acompanion paper [2], wherwe consider the drop shape in a general way. In this p(Part I), we focus on key parameters affecting drop shasuch as contact angles and three-phase contact lines.

Several researchers [3–5] have predicted advancingtact angles significantly higher than the contact anglesserved on horizontal surfaces. These results are in conwith the observations of MacDougall and Ockrent [6], wreported that the advancing angle was almost equal tohorizontal-surface contact angle. Moreover, the advanand receding contact angles of drops were predictedto vary with surface inclination, although measurementsshowed them to be independent of inclination angles. Scalculations for advancing and receding angles under criticconditions cast doubt on the predictions for the maximand minimum contact angles at other conditions. To unstand drop shapes, there is a need to find the maximumminimum contact angles of drops under various conditionand relate them to the characteristic contact angles osurface.

Drop-shape studies have also disagreed on the conangle variation along the circumference of a drop. The c

http://www.elsevier.com/locate/jcis

A.I. ElSherbini, A. M. Jacobi / Journal of Colloid and Interface Science 273 (2004) 556–565 557

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putations of Dimitrakopoulos and Higdon [7] resulted in tcontact angle,θ , varying as a step function of the azimuthangle,φ. On the other hand, Brown et al. [4] and Milinazand Shinbrot [5] predicted smoothθ(φ) functions, but theirpredictions differed significantly from each other. Rotenbeet al. [8] imposed a contact angle function,θ(φ), as a boundary condition of the drop shape problem. They assumeddrop to be moving slowly and prescribed arbitrary relatiobetween dynamic contact angles and the normal veloof the contact line. Other researchers have imposed dient contact angle functions to calculate the surface-tenforces acting on drops. For example, Korte and Jacobassumedθ(φ) to be linear, while Extrand and Kumagai [1considered cosθ(φ) as linear. Other combinations of costant and linear relations have been assumed [11–13]. Iabsence of experimental measurements of the contact-variation with the azimuthal angle, it is not clear whichthe predicted or assumedθ(φ), if any, should be adopted ianalyzing drops on inclined surfaces.

The contact line (or contour) at the base of a dropanother parameter describing its shape. Several investions of drop shapes [4,5,14] have assumed the droptour to be circular. Experiments and predictions by Rotberg et al. [8] showed drop contours to be elongatedthe direction of gravity, with a front-to-back asymmetry.contrast, Dimitrakopoulos and Higdon [7] predicted drcontours that were elongated in the direction perpendlar to gravity. They added a side-expansion constraintheir numerical model and the resulting contours were cloto earlier experimental findings. Extrand and Kumagai [presented measurements of critical contour-elongationpect ratio, and front-to-back asymmetry for several liqusurface combinations. In their analysis of drops, ExtrandKumagai assumed the contour shape to be an ellipse.san V and Chow [12] assumed the drop contour to havearcs connected by parallel sides in the direction of gravbased on observations of moving drops by Bikerman [and Furmidge [16]. For drops of noncritical sizes reston inclined surfaces, the contour shapes have not beeequately addressed by prediction or observation. Therethe shapes of drop contoursunder various conditions havnot been determined. Furthermore, quantitative descriptof drop contours require relating contour parameters, suchelongation and aspect ratio, to drop and surface conditio

The discrepancies in the literature regarding contangle functions and contour shapes are reflected in the caculation of surface-tension forces,Fs , acting on drops. Thedimensionless retentive force can be related to the conangle hysteresis by

(1)Fs

γR= k(cosθR − cosθA),

wherek is a constant,γ is the liquid–vapor surface tensioandR is a length scale taken as the equivalent radius ofcontour, as recommended by ElSherbini [1]. Calculatedues of the retentive-force factor,k, based on the assumptio

e

--

-

-

-,

Fig. 1. Predictions of retentive-force factors based on the assumptionscalculations of earlier researchers. The large differences emphasize thportance of reliably determining drop-shape parameters.

or numerical predictions of earlier researchers range f1.0 to 3.14, as shown in Fig. 1. The deviation ink, which ex-ceeds 300%, emphasizes the significance of determinindrop-shape parameters in question.

In a related paper, we have provided a relatively simplemathematical model for the shapes of drops on surfacesThat model, like prior numerical work and modeling, relon contact-angle and drop-contour data; such data arefocus of this article. Herein we report experimental findinon drop shape, contact-angle variation, and base contoudrops over a wide range of conditions, and we present aeralized description of the results.

2. Setup and procedures

Experiments were conducted to investigate the thdimensional shapes of drops on plane surfaces. In particultests were aimed at studying contact angles and conof drops at various conditions. Moreover, drop profiles avolumes were examined to validate the proposed approxtions and verify the resulting predictions in Part II [2]. Tdrop-shape tests involved two liquids, several hydrophand hydrophobic surfaces, different drop sizes, and varsurface-inclination angles, in order to cover a wide rangconditions. The liquids used were water and ethylene glyThe tested surfaces had eight commercially available cings, designated by letters A through H, which were chaterized by their contact angles. The chosen liquids prodifferent ratios of density to surface tension (ρ/γ ). How-ever, since only two polar liquids were used, care shouldexercised when the findings are applied to liquids with otchemical characteristics.

Advancing and receding contact angles were measby feeding and withdrawing liquid, respectively, throughmicrosyringe onto a horizontal surface. A contact-angle goniometer (Rame-Hart 100-00) was used to take the mea

558 A.I. ElSherbini, A. M. Jacobi / Journal of Colloid and Interface Science 273 (2004) 556–565

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(a)

(b)

Fig. 2. (a) Front view and (b) cross-section of the apparatus used to stuthe shapes of drops on inclined surfaces.

ments. Three specimens of each surface were cut, wonce with alcohol for cleaning, and dried. The advancand receding angles were measured on three differentof each specimen, totaling 18 measurements for each liqsurface combination.

An apparatus was designed and constructed for the dshape investigation. The setup is shown schematicallFig. 2. The drop to be examined rests on the test surwhich is attached to a fixed plate. A camera, microscoand light source are connected to an arm that rotates ar

s

-

,

d

the drop. The fixed plate and rotating arm are attachedlarger plate that can be tilted to different angles of inclition. A single lens reflex (SLR) camera with a 1:1 macrolenwas initially used without mounting the microscope. Printof drop images with enlargements of up to 20 times weretained. The prints were computer-scanned to prepare imfor software processing. Also, film negatives were scanusing a high-resolution scanner with an optical resoluof 5000 (15,000 total) dots per inch (dpi). Analysis of tscanned negatives produced results that were close toptically enlarged images. Another method for recordinimages was connecting the microscope, which had a×magnification, to a digitalcamera with macromode capabity. This imaging method gave reasonable results, compto the other methods. The digital camera was connectethe microscope with a special adapter that could holdcamera against gravity when needed. A xenon light wasat an angle of 180◦ from the camera, illuminating arounthe drop profile. The light source was covered by blackvet to improve contrast. When water drops were testedfixed plate was replaced by another plate inside a vapor-tightransparent box that was saturated with water vapor to rethe drop evaporation rate. Tests were conducted to verifyrefraction through the box did not affect the recordedages. Using the box, no significant change was observthe profiles or contour of a drop over the time of the expment.

Images of drop contours were recorded from a locaperpendicular to the base of the drop. A removable fixwas constructed to mount the camera in a way ensuringit was positioned properly. Test images of grid marks wrecorded, and an image aspect ratio of 1.0 was obtaineduncovered krypton light source was used for contour imaA measuring grid was attached to the fixed plate to determthe dimensions of drop contours from recorded images.

Tests were conducted to examine the variation of conangle with azimuthal angle in drops. A rectangular piecethe test surface was cut and attached to the fixed plawas then cleaned by wiping it once using alcohol. Theface was tilted to the angle of inclination,α, to be testedA drop was injected onto the test surface from a microringe. Microsyringes with capacities of 1, 5, 10, and 25(1 µl= 1 mm3) were used to cover a range of drop volumwith good accuracy. The maximum uncertainty in the vume measured from syringes was 2.5%. For water dropswater was added to the box enclosing the fixed plate anlowed to saturate the air inside the box before injectingdrop. However, ethylene glycol did not show any significevaporation problems and a similar box was not needefact, it was selected for these experiments because of itsvapor pressure.

Profile images of drops wererecorded with the camerpositioned at various azimuthal angles. The camera awas changed by rotating the arm carrying the cameralight source fromφ = 0◦ to 165◦ or from 180◦ to 345◦,whereφ is measured as in Fig. 2a. At least two rounds


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pictures were recorded so that average values of contacgles were obtained. An image of a drop profile showsangles. For example, an image with a camera angle of◦shows contact angles atφ = 0◦ andφ = 180◦. Notice thata camera angle of 270◦ shows these same angles from tother side. The profile images were used to characteriznature of the contact angle functionθ(φ). Once the func-tion was determined, other drop tests involved measuonly the maximum and minimum angles under various cditions. Images of drop contours were also recorded anddimensions of the contact line were calculated. The dimsions of the drop contour and the contact angles wereused to predict the volume of a drop and compared to theume injected from the microsyringe. The tested drops ranin volume from 0.7 to 25.0 mm3, with diameters from 1.18 to5.80 mm. Measurements of contour dimensions resultemaximum uncertainties of 2.3% in equivalent diameter,D,and 4.6% in aspect ratio,β . The propagated uncertainties fthe Bond number and calculated volume reached 5.8%8.7%, respectively. The tested surfaces had advancinggles from 49◦ to 112◦ with receding angles from 12◦ to 78◦.Surface-inclination angles were varied from 0◦(horizontal)to 90◦(vertical).

Images of drops became available for software proceseither through scanning of prints or negatives, or by copyfrom the digital camera, as explained earlier. Three softwpackages were used to analyze profile images and ocontact angles. The profile image was rotated, when needeso that the base of the drop coincided with the horizonDigital-image processing was then used to adjust thetrast and find edges of the portions of the profile close to eend. Another software package was utilized to obtainx–y

coordinates of the edges. Each data file with edge coonates was analyzed by a third package to set an origin whethe drop base met the profile, fit a curve to the edge,find the slope of that curve at the origin. The contact angthe arctangent of the slope. A fourth-degree polynomialfound to best fit the edge coordinates. The goodness owas measured by the coefficient of determination,r2, whichcan be calculated from

(2)r2 = 1−∑

(yi − ycal)2/n∑

(yi − yav)2/n= 1−

∑(yi − ycal)

2∑y2i − (∑

yi

)2/n

,

whereyi is a coordinate value obtained from image analyycal is a calculated value from the polynomial,yav is the av-erage ofyi coordinates, and the summations are done ovi.The edge coordinates were fit by polynomials having coecients of determination,r2, that always exceeded 0.98.

Although this procedure for measuring contact angis lengthy, it is highly repeatable. Different operators cducting the experiments obtained very close angle measments. A test was performed to check uncertainty due tothe experimental procedure and the data analysis. Awas injected onto a test surface and the procedure of moing the camera and recording an image was repeatetimes. Analysis of the images showed the uncertainty

-

-

-

-

Fig. 3. Drop contour showing correction needed for azimuthal angles dthe contour being noncircular. The camera records contact angles atc andd , instead ofa andb, respectively.

be 3◦. Since the tested surfaces were real engineeringfaces, results showed higher scatter due to nonhomogeA quicker and easier method of finding contact angles frimages is fitting two circles to the profile and calculatingangles. The method, which gives comparable results, isscribed briefly in [2].

There is an error associatedwith measuring contact angles of drops with noncircular contours. A given profiletersects the contour at two points, say ‘a’ and ‘b’, as shownin Fig. 3. A camera views the profile from a perpendiclar direction, but because the contour is not circular it viepoints ‘c’ and ‘d ’ as the endpoints. In other words, the caera records the contact angles at the azimuthal angle oφc

instead ofφ. In order to calculateφc, the contour is assumeto be an ellipse. Contact-line images confirm this assution. Let the major and minor axes of the ellipse beL andw,respectively. The equation of the ellipse can be written a

(3)y =√

w2 − x2

β2

for the half withy � 0, whereβ is the aspect ratio. Lineab

has a slope of tanφ. The slope of the tangent to the ellipseany point is

(4)S = − x

β√

L2 − x2.

The perpendicular to lineab that is tangent to the ellipsepointc = (xc, yc), has a slope,Sc , of

(5)Sc = 1

tanφ.

Using Eq. (4), thex-coordinate of pointc can be expresseas

(6)xc = − ScβL√1+ S2

c β2.

The y-coordinate corresponding toxc can be found fromEq. (3). Substituting forSc from Eq. (5) and simplifying,


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Fig. 4. Drop on an inclined hydrophobic surface showing the apparent deter greater than the base diameter. The measured diameter was coso that the drop is identified by the base diameter.

tanφc can be calculated fromyc/xc to be

(7)tanφc = tanφ

β2.

Thus, for a given azimuthal angle,φ, and ellipse aspect ratio, β , the corrected azimuthal angle,φc , can be calculatefrom Eq. (7). Notice that the angle correction dependsboth the angle and aspect ratio. Also,φc = φ at 0◦, 90◦,180◦, and 270◦ for all aspect ratios. In studying the varition of θ with φ, azimuthal angles were corrected. Typicorrections were on the order of 4◦, but some azimuthal angles were corrected by up to 10◦.

For images of drop contours, ellipses were found to bfit the contact lines. The grid scale was used to determthe dimensions of the ellipse. The equivalent diameterdrops with contact angles greater than 90◦ needed to be corrected, since the “apparent” diameters were not the samthe base diameters. Fig. 4 shows an example of a dropbothθmax andθmin greater than 90◦. The deviation betweethe apparent and base diameter was within 5% for tedrops; however, this deviation triples in volume calculatand should not be ignored.

3. Results and discussion

3.1. Horizontal contact angles

Static contact-angle measurements for the tested liqand surfaces are presented in Table 1. The advancing aranged from 49◦ to 112◦, and the receding angles rangfrom 12◦ to 78◦. The contact-angle hysteresis spannerange of 23◦ to 51◦. The contact-angles measured on hizontal surfaces were compared to angles obtainedimages of drops on inclined surfaces. The results shothe advancing contact angles of some tilted surfaces tslightly higher than those forhorizontal surfaces. This finding is consistent with the measurements of MacDougallOckrent [6]. The advancing angles reported in the table hbeen adjusted whenever a slight increase was observed

d

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Table 1Contact angle measurements for liquids and surfaces used

Surface Ethylene glycol Water

θA θR θA θR

A 50 15 72 29B 50 15 49 12C 59 25 75 52D 59 27 83 56E 69 19 97 46F 65 30 88 58G 69 28 80 55H 89 59 112 78

3.2. Contact angle function

Measurements of the contact angle as it varied withazimuthal angle were recorded for drops on several surfat various conditions. The azimuthal angle of zero is taat the plane of symmetry pointing downward, as in Fig.Thus, (φ = 0) corresponds to the maximum contact anglethe drop,θmax, and (φ = 180◦) corresponds to the minimumcontact angle,θmin. Fig. 5a shows the resulting variationcontact angle with azimuthal angle for a drop on a vecal surface. The contact angle,θ , in Fig. 5a can be fit bya third-degree polynomial of the azimuthal angle,φ, witha coefficient of determination,r2, of 0.98. Notice that thisdrop is not necessarily of critical size, and the maximand minimum angles are designated byθmax and θmin, re-spectively. However, the examined contact angles spaalmost the whole range between the advancing and rececontact angles characteristic ofthe liquid–surface combination. The contact angle functionθ(φ) may be written as

(8)θ(φ) = a1φ3 + a2φ

2 + a3φ + a4,

whereφ andθ are in radians, anda1, a2, a3, anda4 are con-stants. The conditions used to determine the four consin Eq. (8) are

(9a)θ(0) = θmax,

(9b)θ(π) = θmin,

and from symmetry, the slope of theθ function should vanish atφ = 0 and 180◦,

(9c)dθ

dφ

∣∣∣∣∣φ=0

= 0

and

(9d)dθ

dφ

∣∣∣∣∣φ=π

= 0.

Thus, the equation fitting the contact angle functioncomes

(10)θ = 2θmax− θmin

π3 φ3 − 3θmax− θmin

π2 φ2 + φmax.

Another way of representing the data is by plotting coθ

againstφ. The cosine of the contact angle can also be


lree

(a) (b)

Fig. 5. (a) Variation of contact angle with azimuthal angle for a 0.75-mm3 droplet on surface C at an inclination angle of 90◦. The third-degree polynomiafrom Eq. (10) fits the data withr2 = 0.98. (b) Variation of the cosine of the contact angle with theazimuthal angle for the same droplet. The third-degpolynomial from Eq. (13) fits the data withr2 = 0.97.

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by a third-degree polynomial of the azimuthal angle, wa very similar value ofr2, as shown in Fig. 5b. ElSherbin[1] presented similar figures for drops ranging from 0.7520 mm3 in volume, with surface inclination angles betwe10◦ and 90◦ from the horizontal, andr2-values between 0.8and 0.98. The function cosθ can be expressed in a form similar to Eq. (8) withθ replaced by cosθ , and conditions (9aand (9b) changed accordingly. The slope boundary condtions remain the same. But since

(11)d(cosθ)

dφ= −sinθ

dθ

dφ,

the conditions can be written as

(12a)cosθ(φ = 0) = cosθmax,

(12b)cosθ(φ = π) = cosθmin,

(12c)d(cosθ)

dφ

∣∣∣∣φ=0

= 0,

and

(12d)d(cosθ)

dφ

∣∣φ=π

= 0.

Applying conditions (12) and solving, the equation descing the contact angles in Fig. 5b becomes

cosθ = 2cosθmax− cosθmin

π3 φ3 − 3cosθmax− cosθmin

π2 φ2

(13)+ cosθmax.

Equation (13) presents a reasonable fit for the relationtween the contact angle and the azimuthal angle in a droan inclined surface. Notice that knowledge ofθmax andθminfor a drop determines the function. For a horizontal drθmax = θmin, and Equation (13) reduces to cosθ = cosθmax.It is also worth noting that Figs. 5a and 5b present datathe same drop with almost equal values ofr2, indicating thatEqs. (8) and (13) fit the data equally well. Equation (13more useful when analyzing the forces acting on the dropshown in [1].

Experimental determination of the functionθ(φ) resolvesthe discrepancies in the literature concerning the naturthe curve. Fig. 6 compares earlier predictions of the con

Fig. 6. Comparison between predictions by earlier researchers for the cotact angle function and the third-degree function found in this work.horizontal contact angle, used as input, is 70◦ for the curve by Brown et al[4] and 90◦ for all other curves.

angle function to the results obtained in this work. Theure shows that previous numerical predictions ofθ(φ) do notresemble the experimentally based curve. In addition todifference in the shape of the curve, the maximum and mimum angles are also different. Results for these anglesdiscussed next.

3.3. Maximum and minimum contact angles

The contact angle function,θ(φ), describes the contacangle in a drop at any azimuthal angle,φ, in terms ofθmaxandθmin. It is therefore desired to obtain the maximum aminimum contact angles of drops for various conditions.mensionless groups are used to generalize the presenof results. The conditions of drop size, surface inclinatiand liquid properties are described by variables includethe Bond number,Bo. It is a ratio of gravitational to surfacetension forces,

(14)Bo = ρgD2 sinα

γ,

whereρ is the liquid density (neglecting air density),g is theacceleration of gravity,D is the equivalent drop diameteα is the surface inclination angle, andγ is the surface tension of the liquid–vapor interface. For a given liquid drop aconstant temperature, an increase inBo indicates an increas


eed by

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heres

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ex-tionoran-

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Fig. 7. Maximum contact angle, divided by the advancing angle for all thliquids and surfaces tested. The maximum angle can be approximatthe advancing angle for all tested conditions.σ = 0.06.

in drop diameter and/or surface inclination. The characteristic contact angles of the liquid–surface combination,θA andθR, are included as dimensionless parameters representithe measured contact angles.

The maximum contact angle,θmax, is normalized by theadvancing angle of the surface,θA. The dimensionless maximum angle (θmax/θA) is plotted against the Bond numberFig. 7 for all tested surfaces and conditions. Theθmax/θA

ratio is almost constant and equal to one. It is concluthat θmax ≈ θA for all drop sizes, surface inclinations, aliquid–surface combinations tested. For a drop depositea horizontal surface, spreading leads to a contact angleto θA. When the surface is tilted, the value ofθmax cannotincrease beyond the characteristicθA of the liquid–surfacecombination. Theθmax finding is in contrast with some ealier numerical studies [4,5], but agrees with the measments of [6,14].

The minimum contact angles of drops show a dramcally different behavior. Let the minimum contact angle brepresented by the dimensionless parameterΘ,

(15)Θ = θmin − θR

θA − θR

,

whereθA andθR are the advancing and receding contactgles of the surface, respectively. Fig. 8 showsΘ as a functionof the Bond number for one of the tested surfaces. TheΘ pa-rameter starts with a value of 1 atBo = 0, and goes to 0 atmaximum Bond number,Bomax, which translates toθmin de-creasing fromθA to θR in thatBo range. A line fits the datwith a coefficient of determination,r2, of 0.98.

Earlier works predicted [4,5] and measured [14]creases in minimum contact angle as the Bond numbecreased, with either the drop size or the surface inclinaangle being fixed. However, the rate of decrease inθmin var-ied for different drop sizes. The current findings generathe behavior of the minimum angle by showing that thecrease inθmin follows the same curve for different drop sizand various surface-inclination angles. In addition, thesults in Fig. 8 show that for the various conditions tested,Θ parameter approaches 0 at high Bond numbers and, tfore, θmin goes toθR. Thus, contrary to earlier prediction[4,5] that the criticalθmin varies with drop conditions, th

l

-

Fig. 8. Variation of the dimensionless minimum angle parameterΘ withthe Bond number for ethylene glycol drops of various sizes on surfacedifferent inclination angles.r2 = 0.98.

Fig. 9. Minimum contact angle, normalized by the maximum angle, avaries with the Bond number for all liquids and surfaces tested. A sicurve fits the data withr2 = 0.90.

present results suggest that the criticalθmin is almost con-stant and equal to the receding contact angle,θR , that ischaracteristic of the liquid–surface combination. This conclusion is consistent with the experiments of [6].

Theθmin results from the various surfaces and liquidsamined are combined in Fig. 9, which shows the variaof θmin/θmax with the Bond number for all tests. Data fall tested surfaces, liquids, drop sizes, and inclinationgles are well fitted by a single curve, withr2 = 0.90. It isan interesting result that the behavior of minimum conangles of drops is the same for different liquids on surfawith a wide range of contact angles. While Fig. 8 geneizes the relation betweenθmin and conditions such as drodiameter and surface inclination, the slope of theΘ(Bo) linevaries from one surface to another. On the other hand, Fgeneralizes the minimum contact angle results a step fuby showing that the rate of decrease inθmin relative toθmaxis the same for different liquid–surface combinations. Csequently, the decrease inθmin is independent ofθR. Thereceding contact angle acts only as the lower limit toθmin atwhich the drop cannot stay on the surface. The equationfits theθmin data is

(16)θmin

θA

= 0.01Bo2 − 0.155Bo + 0.97,

where θA ≈ θmax as found above. It is important to nothat different surfaces will have different values for tmaximum Bond number,Bomax. In other words, the minimum contact angle will follow the curve of Fig. 9 until th


cle.

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nesectesondowsd aFortwo11.forropsedse

ses,

nd

as-ot o

spec

,y at

s ofal-

ers.nd

aresd

the

s a

mea-s

h a

con-thcur-the

(a) (b)

Fig. 10. Typical images of drop contours fit by (a) an ellipse and (b) a cir

drop reaches the critical condition and slides. The maximumBond number,Bomax, associated with maximum drop sizand receding contact angle can be determined from a fbalance on the drop.

3.4. Drop contour

Frontal images of drops on vertical and inclined plashow that contact lines can be fit by ellipses, with aspratioβ = L/w. The dimensions of the major and minor axof the ellipse were recorded for tested drops. When the Bnumber is zero, the ellipse reduces to a circle. Fig. 10 shtwo typical images of drop contours fit with an ellipse ancircle. More images of drop contours can be found in [1].drops near critical sizes, the contour can be better fit byellipses sharing the same minor axis, as shown in Fig.However, using a single ellipse with an equivalent areasuch a large drop leads to excellent approximations of dvolume and retention forces. Therefore, one ellipse is uto fit each contour. The major axis of the equivalent ellipis equal to the average of the major axes of the two ellip

(17)L = LA + LR

2,

whereLA andLR are the major axes of the advancing areceding ellipses, respectively.

In order to model drop contours, a relation betweenpect ratio and drop size is needed. Fig. 12 presents a plthe aspect ratio of the contour,β , againstBo for all liquid–surface combinations tested. The figure shows that the aratio increases slightly withBo. All the data can be fit by asingle line having a small slope, with a standard deviationσ ,of 0.05. The aspect ratio increases from a value of unitBo = 0 to a value of 1.32 atBo = 3. Theβ equation fittingthe data is

(18)β = 1+ 0.096Bo.

Extrand and Kumagai [10] measured the aspect ratiodrop contours at the special case of incipient motion,though they did not report drop sizes or Bond numbUsing analysis of critical drops presented in [1], the Bonumbers were calculated from their data. Fig. 13 compEq. (18) to the aspect-ratio measurements of Extrand an

f

t

Fig. 11. Contour of a drop near critical size, fit by two ellipses sharingsame minor axis.

Fig. 12. Aspect ratio of the elliptical contour at the base of a drop afunction of the Bond number for all liquids and surfaces tested.σ = 0.05.

Fig. 13. Predicted critical aspect ratios of drop contours, compared tosurements by Extrand and Kumagai [10]. The empirical equation of thiwork fits their data withσ = 0.06.

Kumagai. The equation fits their data reasonably, witstandard deviation of 0.06.

Several researchers, such as Merte and Yamali [17],sidered the drop diameter,D, to be equal to the base lengof the cross section at the symmetry plane. However, therent findings indicate that the contour is an ellipse, andmentioned length is 2L, which overestimatesD by a factor


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. In

.

inedions.con-

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ts inles

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Fig. 14. An ellipse representing a drop contour with radiusζ at a point(x, y).

of√

β. An error of about 15% in diameter can result frotheir assumption, and such an error would triple in preding volume.

It can be shown that the equation describing the conin terms of azimuthal angle,φ, is

(19)ζ(φ) = L√cos2 φ + β2 sin2 φ

,

whereζ is the ellipse radius,β is the aspect ratio, andL isthe length of the major axis, as shown in Fig. 14. In thvaluable work on drops, Extrand and Kumagai [10] mistenly used another equation to describe the ellipse, buerror did not have significant effects on their results [1].

The drop-contour results, summarized by Eq. (18),plain why both circular and elliptical contours have bereported in the literature. For small drops, with small Bonumbers, the aspect ratio is close to unity and the contare essentially circular. Large drops have contours thaelliptical with higher values ofβ . Static drops did not exhibparallel-sided contours as was reported for sliding dr[15,16].

The present results describe the contact angles andtours of drops under general conditions. For a liquid dropon an inclined surface with advancing contact angleθA, themaximum contact angle is approximately equal toθA. TheBond number for the drop can be found from the liqproperties, drop equivalent diameter, and surface-inclinaangle. Equation (16) can be used to obtainθmin. With bothθmax and θmin known, the contact angle function,θ(φ), isfound from Eq. (10) or Eq. (13). The aspect ratio,β , of thedrop contour can be calculated from Eq. (18). The asratio and equivalent diameter lead to the value ofL. Thus,the contour function,ζ(φ), in Eq. (19) is determined for thdrop. The predictions of drop profiles and volumes arecussed in Part II.

The results of this investigation are of considerable sigicance to the study of drops on plane surfaces. The coangle and contour functions obtained in this work can lto reliable calculations of the forces acting on a dropwe show in another work [1]. Also, a sensitivity analypresented in [2] shows that some of the studied paramcan significantly influence volume prediction. Therefore,is important to use variables that are appropriate forconditions when describing a drop on a vertical or inclinsurface. Moreover, these experimental results can be

-

t

s

d

as boundary conditions for numerical models of dropsPart II, we use the current results as inputs to our model

4. Conclusions

Key parameters affecting the shapes of drops on inclsurfaces have been investigated under various conditA setup has been developed and used to study profiles,tact angles, contours, and volumes for liquid drops of difent sizes on surfaces at various inclination angles. Wateethylene glycol drops were tested on eight engineeringfaces covering a range of advancing contact angles from◦to 112◦ and receding angles from 12◦ to 78◦. The contactangle variation along the circumference of the drop,θ(φ),was determined experimentally to resolve disagreementhe literature. The maximum and minimum contact angcan be used to accurately express theθ and cosθ functionsas third-degree polynomials of the azimuthal angle,φ. Themaximum contact angle,θmax, is found to be approximatelequal to the advancing angle,θA, of the liquid–surface combination for all tested conditions. For various drop siand surface-inclination angles, the increase in Bond numfrom 0 to Bomax causes the minimum contact angle,θmin,to decrease almost linearly fromθA to θR. The minimum-contact-angle data from all tested liquids and surfaces cabe fit by a single curve. Therefore, it is concluded that a general relation exists between the minimum contact angledrop, the advancing angle of the surface, and the Bond nber, which is applicable for different liquids and surfacesis also concluded that the rate of decrease inθmin with re-spect toθA is independent of the receding contact angle.

The general shape of a drop contour is characterizedellipse, with the aspect ratio increasing slightly as the Bnumber increases. Aspect-ratio data from all tests werby a single line, which agreed with measurements repoin the literature. The current results are useful for modedrop shapes and for calculating their volumes and the foacting on them.

Acknowledgments

The authors acknowledge the support of the Air Contioning and Refrigeration Center (ACRC). The authorsalso grateful for access to facilities at the Imaging Technol-ogy Group of the Beckman Institute for Advanced Scieand Technology, and access to the goniometer of ProfeVinay Gupta.

References

[1] A.I. ElSherbini, Modeling Condensate Drops Retained on the ASide of Heat Exchangers, Ph.D. thesis, Univ. of Illinois at UrbaChampaign, Urbana, 2003.

[2] A.I. ElSherbini, A.M. Jacobi, J. Colloid Interface Sci., in press.


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[3] B.K. Larkin, J. Colloid Interface Sci. 23 (1967) 305.[4] R.A. Brown, F.M. Orr Jr., L.E. Scriven, J. Colloid Interface Sci.

(1980) 76.[5] F. Milinazzo, M. Shinbrot, J. Colloid Interface Sci. 121 (1988) 254[6] G. MacDougall, C. Ockrent, Proc. R. Soc. London Ser. A 180 (19

151.[7] P. Dimitrakopoulos, J.J.L. Higdon, J. Fluid Mech. 395 (1999) 181.[8] Y. Rotenberg, L. Boruvka, A.W. Neumann, J. Colloid Interfa

Sci. 102 (1984) 424.[9] C.M. Korte, A.M. Jacobi, J. Heat Transfer 123 (2001) 926.

[10] C.W. Extrand, Y. Kumagai, J. Colloid Interface Sci. 170 (1995) 51[11] C.W. Extrand, A.N. Gent, J. Colloid Interface Sci. 138 (1990) 431.[12] E.B. Dussan, R.T.-P. Chow, J. Fluid Mech. 137 (1983) 1.[13] E. Wolfram, R. Faust, in: J.F. Padday (Ed.), Wetting, Spreading,

Adhesion, Academic Press, New York, 1978, p. 213.[14] T. Tsukada, M. Hozawa, N. Imaishi, K. Fujinawa, J. Chem. E

Jpn. 15 (1982) 421.[15] J.J. Bikerman, J. Colloid Sci. 5 (1950) 349.[16] C.G.L. Furmidge, J. Colloid Sci. 17 (1962) 309.[17] H. Merte, C. Yamali, Warme Stoffubertrag. 17 (1983) 171.

liquid drops on vertical and inclined surfaces: i. an experimental study of drop geometry

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