STATIC AND DYNAMIC DROP FORMATION ON DOWNWARD-FACING NOZZLES

Atomization and Sprays, 23 (7): 597–617 (2013)

STATIC AND DYNAMIC DROP FORMATION

ON DOWNWARD-FACING NOZZLES

M. Etzold, F. Durst, U. Acikel, R. Gautam, & M. Zeilmann∗

FMP TECHNOLOGY GMBH, Am Weichselgarten 34, 91058 Erlangen,Germany

∗Address all correspondence to M. Zeilmann

E-mail: [email protected]

Original Manuscript Submitted: 10/16/2012; Final Draft Received: 08/13/2013

Static drop formation on downward-facing nozzles was investigated, numerically and experimentally.A representation of the variation of the contact angle with respect to nozzle diameter and increasingdrop volume, during quasistatic drop formation, was determined and experimentally verified. Thecondition for the formation of single drops is given. It is pointed out that this limit exists for dynamicdrop formation on downward-facing nozzles. This formation is treated in an integral manner and theresults are presented in an Ohnesorge–Reynolds diagram. The latter diagram is explained in detail,since it allows a generalization of the authors’ experimental results. Experiments and correspondingtheoretical results are described and show very good agreement.

KEY WORDS: drop formation

1. INTRODUCTION AND AIM OF WORK

The formation of drops through downward-facing nozzles is aprocess that is well knownand is widely observed in nature, engineering, science, andmedicine. This drop forma-tion is dependent on the fluid properties such as the densityρ, the dynamic viscosityµ, and the surface tensionσ, and also the contact angleθD between the fluid and thenozzle material. Furthermore, the nozzle diameterD plays a role and also the flow rateV at which the fluid is supplied through the nozzle, or the corresponding mean ve-locity U = 4V /πD2. With these parameters, the drop formation can be described. Adimensional analysis was carried out which shows that the drop formation is generallydependent on the Reynolds and Ohnesorge numbers:

Re=ρUD

µ(Reynolds number) (1)

Oh =µ

√ρσD

(Ohnesorge number) (2)

1044–5110/13/$35.00 c© 2013 by Begell House, Inc. 597

598 Etzold et al.

NOMENCLATURE

a Laplace constant (m)D nozzle diameter (m)d drop diameter (m)F Harkins–Brown correction factorfc characteristic frequency (Hz)fT drop frequency (Hz)g acceleration due to gravity (m/s2)L length of cannula (m)Le entrance length (m)m mass flow rate through nozzle (kg/s)Oh Ohnesorge numberpa ambient pressure (N/m2)pL pressure in the liquid of the drop

(N/m2)pσ pressure due to surface tension

(N/m2)r coordinate in radial direction (m)r∗ dimensionless coordinate in radial

directionR0 vertex radius (at the bottom of the

drop,y = 0) (m)R∗

0 dimensionless vertex radiusR1,2 local radius of curvature of the drop

surface (m)rD nozzle radius (m)r∗D dimensionless nozzle radius (m)Re Reynolds number

tform time of drop formation (s)tresp characteristic fluid response

time (s)U mean velocity (m/s)Uc characteristic velocity (m/s)Ut actual velocity (m/s)V ∗ dimensionless volumeV flow rate (m3/s)Vc characteristic flow rate (m3/s)VD drop volume (m3)V ∗

max dimensionless maximum dropvolume

Vt actual flow rate (m3/s)y coordinate in axial direction (m)y∗ dimensionless coordinate in

axial directionyS drop height (m)yS

∗ dimensionless drop heightWe Weber number

Greek SymbolsθD contact angle between the fluid

and the nozzle material (◦)µ dynamic viscosity (Pa s)ρG density in the gas phase (kg/m3)ρL density in the liquid (kg/m3

σ surface tension (N/m)

for a given contact angleθD between the fluid and the nozzle material. For the presen-tation of the results, the Weber number We= Re2Oh2 can also be employed.

It is intuitively clear that the formation of individual drops on downward-facing noz-zles is limited, on one end, by the static drop formation atV = 0 and at the otherend by the jet formation. In Section 2 the theoretical treatment the static drop forma-tion is summarized and some numerical results and their experimental verification re-garding the behavior of the contact angle with respect to (increasing) drop volume aregiven.

Atomization and Sprays

Static and Dynamic Drop Formation on Downward-Facing Nozzles 599

As mentioned above, the formation of individual drops has a limit when the condi-tion for jet formation is reached, i.e., when the following force balance holds:

π

4D2

(

ρU2)

> πDσ (3)

This condition for the upper limit of individual drop formation, the jet formation, is elab-orated in Section 3.1 and it is shown in Section 3.2 that the drop formation on downward-facing nozzles can be described by the relation

Oh · Re=fT

fc(4)

wherefT is the measured drop frequency andfc a characteristic frequency resultingfrom Eq. (3). Using the quantities Oh, Re, andfc, the dynamic drop formation fromdownward-facing nozzles can be represented in an Oh-Re diagram. In this diagram,the physics of drop formation can be captured in a way well suited to understand thedrop formation in most practical applications. This is shown also in Section 3, whereconsiderations are summarized that permit insight into a wide range of drop formationprocesses.

To verify the theoretically derived relations, experiments were carried out using cap-illaries as downward-facing nozzles for the drop formation. By counting the number ofdrops and measuring the corresponding time, the actual dropfrequency could be de-termined, and by calculating the characteristic frequencyfrom the measured flow andfluid properties, the ratio of the drop frequency and the characteristic frequency couldbe evaluated. Through corresponding experiments, Eq. (4) could be experimentally veri-fied. This is summarized in Section 3 and conclusions, final remarks, and an outlook arepresented in Section 4.

Static drop formation was already studied by Bashfort and Adams (1883), followedby an extended calculation of the drop shape by Fordham (1948). Siemes (1954) andDurst and Beer (1969) also investigated the bubble shape during static bubble formation.From the computed shape of the bubble, Durst and Beer (1969) concluded that the bub-ble detachment takes place when, with increasing bubble volume, a bubble shape withmaximum volume is reached. Hence the bubble detachment was found not to take placewhen a gravity/surface tension force balance enforces bubble breakup. In their study theauthors found the same for the static drop formation. A detailed summary regarding thesolutions of the differential equations of static drop formation is given by Hartland andHartley (1976).

The drop formation on downward-facing nozzles has been extensively investigatedby various researchers, but no attempts are known to the authors to combine the staticand dynamic drop formation in a single diagram. This representation is claimed by theauthors to be new.

Lohnstein (1906) derived a relation for the weight and the size of a falling drop af-ter breakup. This relation was based on a force balance of thesurface tension and the

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600 Etzold et al.

gravity force. A correction factor was needed to capture theexperimentally measureddrop volume. This was further investigated by Harkins and Brown (1919). They devel-oped experimentally the Harkins–Brown correction factor (Harkins and Brown, 1919)to reach agreement with their experimental data for the detached drop size. The cor-responding results were given by Harkins and Brown (1919) interms of diagrams andtabulated values. Lando and Oakley (1967) used a curve-fitting procedure to derive ana-lytical relations for the Harkins–Brown correction factor.

Scheele and Meister (1968) worked on the prediction of the drop size for dynamicdrop formation at low velocities by a modification of the force balance that involvedterms for the Stokes’ drag, inertia forces, and the volume flow during the necking processof the drop. Their investigations embraced the treatment ofliquid drop formations intoa liquid environment. In their treatment a liquid drop forming in a gas phase turned outto be a special case. A good overview of the aforementioned analytical and empiricalresults is given by Clift et al. (1978), and the reader is referred to their work for furtherinformation on the existing literature on drop formation ondownward-facing nozzles.

In connection with the existing literature, the following papers should be mentioned:Zhang and Basaran (1995) studied dynamic drop formation experimentally with regardto the appearance of secondary drops. They compared their achieved drop sizes withthe model of Scheele and Meister (1968) and reported good agreement at low volumeflow rates. Computations of the full Navier–Stokes system for dynamic drop formationwere performed, for example, by Wilkes et al. (1999), leading to the shape, velocity, andpressure field inside the drop.

2. STATIC DROP FORMATION

2.1 Theoretical Considerations

The theoretical treatment of static drop formation on downward-facing nozzles is wellknown. The derivation of the differential equation for the drop shape can be treated ina way presented by, for instance, Siemes (1954), based on thepressure considerationsindicated below and with the essential features of the resultant drop illustrated in Fig. 1.

The actual drop shape is well represented by static pressureconsiderations, providedthat the surface tension forces of the drop are much higher than the forces due to flowmomentum of the fluid entering the drop through the nozzle. Itis assumed that the inertiaforces are permitted to have a magnitude of at most 1% of the surface tension forcesbefore dynamic effects need to be taken into account:

0.01 πDσ >π

4D2ρU2 (5)

Reformulating this relation leads to the following generalcondition for the validation ofthe derivation presented below for the static drop formation:



+

+2

− [ − ]

θ

Surrounding air Fluid in the drop

Equalization of air and fluid pressure

FIG. 1: Drop formation and associated coordinate axes and pressures.

Re· Oh < 0.02 (6)

This relation is independent of viscosity. By comparing thecharacteristic fluid responsetime tresp = µ3/

(

σ2ρL

)

with the time of drop formationtform = VD/V , viscosityeffects can be taken into account. Provided that the flow ratethrough the nozzle fulfillsthe following relation, the theory of static drop formationcan be applied:

tform > tresp (7)

VD

V>

µ3

σ2ρL(8)

whereV is the fluid supply flow rate through the nozzle opening. The right-hand sideof Eq. (8) represents the characteristic fluid response timethat can be derived by dimen-sional analysis.

Equation (8) specifies that the time needed for the formationof one drop [left-handside of Eq. (8)] has to be lower than the fluid response time to apply changes of the dropshape. Consequently, the assumption for the static drop formation in Eq. (8) impliesthat the fluid inside the drop has sufficient time to react to changes of the drop. Bothconditions in Eqs. (6) and (8) have to be fulfilled for the pressure relation of the theoryof static drop formation.

From Fig. 1, one can see that the following pressure relationship holds for the staticdrop formation at each location of the drop surface:

pL = pa + pσ (9)


602 Etzold et al.

wherepL is the pressure in the liquid,pa the atmospheric pressure, andpσ the pressuredue to the surface tension. For the surface tension caused bypressurepσ, one can expressthrough the Young-Laplace equation

pσ = σ

[

1

R1

+1

R2

]

(10)

whereR1 andR2 are the local radii of curvature of the drop surface. Taking their depen-dence ony andr into account, Eq. (9) can be rewritten by inserting Eq. (10) in Eq. (9)and replacingR1 by r andy, using the well-known description of the radius of curvatureby y

′

(r) andy′′

(r). R2 is also replaced by the mathematical expressiony′

(r):

pL (r, y) = pa + σ

y′′

[

1+y′2]3/2

+y′

r[

1 + y′2]1/2

(11)

As mentioned above, this equation holds for every point at the surface of the drop. Atthe very bottom of the drop, respectively aty = 0 (see coordinate system in Fig. 1),R1

andR2 are equal (R1 = R2 = R0); hence the pressure at the vertex is:

p0 = pL (r, y = 0) = pA +2σ

R0

(12)

At any arbitrary point on the surface of the drop, the pressure changes due to the hydro-static forces acting on the drop surface. By taking the pressure of the drop-surroundingair into account, the following pressure relation arises:

pL (r, y) = p0 − [ρL − ρG] gy = pA +2σ

R0

− [ρL−ρG] gy (13)

From Eqs. (11) and (13), the following differential equation for the shape of the bubblesurface can be derived:

σ

y′′

[

1 + y′2

]3/2+

y′

r[

1 + y′2

]1/2

=

2σ

R0

− [ρL − ρG] gy (14)

By introducing the Laplace constanta =√

2σ/[[ρL − ρG] g], the above differentialequation can be normalized by dividingr, y andR0 by a:

y∗′′

[

1 + y∗′2]3/2

+y∗

′

r∗[

1 + y∗′2]1/2

=2

R∗0

− 2y∗ (15)



This differential equation needs to be solved, starting aty∗ = 0, r∗ = 0, where foreach solution of the drop shapeR∗

0 is assumed, so that the bubble volume increasesfor each new computing. This assumption is similar to Durst and Beer (1969) taking acontinuous flow rateV through the nozzle into account. For everyR∗

0, the integrationtakes place up to a plane at which a certain radius, the nozzleradiusr∗D is reached orup to a plane where a certain angle of inclineθD is formed. As mentioned above, thisapproach to compute the static drop formation was taken up first by Durst and Beer(1969) to compute upward-facing static bubble formation onthe inside of nozzles. Itwas adapted by the authors to do their corresponding computations, and it was extendedto the computation of the drops forming at constantθD.

In the case of the computations of drop formation presented in this paper, the staticdrop formation was studied with respect to a constant contact angle between the drop andthe lower nozzle surfaceθD, and with respect to a constant nozzle radiusr∗D (see Fig. 1).The integration, to yield the drop shape, was carried out very similar to the integrationperformed by Durst and Beer (1969).

In Fig. 1, computed drop shapes are shown. They were computedas a sequence ofsolutions of the differential equation (15) choosing different values ofR∗

0. As mentionedabove, theR∗

0 selections are made in such a way that the volumeV ∗ (R∗0) of the drop is

increasing from one drop shape to the next, corresponding toa continuous flow rateVthrough the nozzle. In Fig. 2(a), the nozzle diameter has a constant value ofr∗D = 0.4(for water in air:rD = 1.56 mm with a = 3.9 mm), which implies that the contactangleθD changes during drop formation since the integration of Eq. (12) was stoppedwhenr∗D = 0.4 was reached. The contact angle, in this case, first increasesand thendecreases with increasing drop volume. Drop formations forconstant radiir∗D can formin two distinct ways. In one way the drop shows a necking, in the other case not. This isquantitatively shown in Fig. 2(a).

The other drop formation case can be seen in Fig. 2(b), where the drop forms at thelower surface of the nozzle, i.e., not inside of it. The contact angle (θD = 70◦) withthe downward-facing nozzle surface is kept constant. During the entire computed dropformation, fluid is conveyed through the nozzle opening. In this way a drop radius formson the lower surface of the nozzle. During drop formation this drop radius first increasesand then decreases when constantθD values are applied as boundary conditions. For thepractical usage of the results of the theory of static drop formation, the results presentedin Fig. 2 are not very suitable because the vertex radiusR∗

0, which is necessary as astarting value for the computation of the drop shape, is normally not known.

In Fig. 3, the relationship between the vertex radius and thenormalized drop vol-ume is shown for two different nozzle radii. This form of computational results wasobtained by numerical integration of the differential equation (13) for two different,but pregiven nozzle radiir∗D and changing vertex radiusR∗

0. A similar diagram wasgiven by Durst and Beer (1969). At the beginning of the drop formation, when the dropvolume is assumed to be zero, the vertex radius has to be chosen to be infinitely high


604 Etzold et al.

* = 0.02; = 23° ; * = 1.00 = 0.09; = 72° ; = 0.40 = 0.57; = 109° ; = 0.45

= 0.96; = 75° ; = 0.47

= 0.31; = 0.60; = 0.45

= 0.82; = 0.80; = 0.57

= 1.93; = 0.95; = 0.70 = 2.22; = 0.90; = 0.66

a)

b )

*−

*

*−

*

*

*

Drop shape with necking

Drop shapes without necking

* *

* *

* *

* **

* **

* **

* **

FIG. 2: Computed drop shapes (a) for a nozzle diameterr∗D = 0.4 and (b) for a contactangle with the downward-facing surface of the fluid–providing nozzleθD = 70◦.

(R∗0 → ∞). Furthermore, it can be seen that for the so-called “small”nozzle (r∗D = 0.5)

below r∗D = 0.648 [according to the computations of Siemes (1954) and Durst andBeer (1969) for example] the vertex radius first decreases, then increases, and finally(V ∗/V ∗

max > 0.9) decreases again before the drop reaches a maximum volume for thechosen “small radius.” At both turning points the contact angle is 90◦. For the largernozzle (r∗D = 2 > 0.648) the vertex radius decreases until the maximum drop volume isreached. Here, the contact angle is always below90◦. These nozzles are known as “big”nozzles.

To derive a relationship between the nozzle radius, the chosen contact angle, andthe drop volume without the vertex radiusR∗

0 being used for presentation, a specialalgorithm was implemented into the authors’ computationalprogram. With the help of



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.0 0.2 0.4 0.6 0.8 1.0

max⁄

= 0.5

= 2

*

**

*

*

FIG. 3: Relationship between relative drop volume and vertex radius for a small and abig nozzle.

this algorithm, the evolution of the contact angleθD with respect to the drop volumeV ∗

could be computed for one particular nozzle diameter. Therefore, with the knowledge oftheR∗

0 − V ∗ dependency (see Fig. 3), the vertex radius was changed in such a way thatuniform logarithmic volume steps were achieved.

The algorithm started at a certain nozzle radius with a very high value ofR∗0, being

close to an infinite radius with zero volume. ThenR∗0 was decreased at every step to

obtain the next (increasing) target value ofV ∗. For big nozzles this was carried out untilthe volume reached a maximum value, and then the computationwas stopped.

For small nozzles the vertex radius was decreased until it reached the minimum value(see Fig. 3). The contact angle at this point was always90◦, thenR∗

0 was increased untilit reached a maximum value and the contact angle again reached 90◦. Finally, R∗

0 wasdecreased until the drop volume reached the maximum value.

This was done for nozzle radii in the range ofr∗D = 0.01 to r∗D = 2, however theθD value at every drop volume step and the combined data are shown together in onediagram (see Fig. 4).

In Fig. 4, the drop evolution with constantr∗D represents drop formation with a con-stant nozzle radius from initially zero volume up to the maximum volume causing thedrop breakup [the case in Fig. 2(a)]. If the drop runs out at a certain material- and liquid-dependent contact angle, the formation process leaves the horizontal line when this con-tact angle is reached for the first time. Subsequently, it continues on an isoline of constantcontact angle in the direction of increasing volume. The formation after running out canalso be characterized, because then the nozzle radius becomes the drop radius at thenozzle surface [the case in Fig. 2(b)]. Its evolution can be read from the ordinate.


606 Etzold et al.

[ ° ]

*

*

Dθ

FIG. 4: Evolution of contact angleθD during drop formation with respect to drop vol-umeV ∗ and nozzle radiusr∗D. (The thin white and black lines are isolines of constantcontact angle, the thick black line is the maximum theoretical volume, the red line rep-resents the drop breakup point due to gravity, and the white arrows show the evolutionof one drop from a nozzle ofr∗D = 0.1 as an example.)

The white arrows in Fig. 4 illustrate an example of drop evolution at a nozzle radiusof r∗D = 0.1. It is supposed that the drop leaves the nozzle opening at a contact angle ofθD = 100◦. It can be seen that at first, with increasing drop volume,r∗D stays constantwhile θD increases. WhenθD = 100◦ is reached,r∗D increases and the drop evolutioncontinues on the line of the constant contact angleθD = 100◦, which implies that thedrop forms then outside of the nozzle.

With the diagram, developed and applied by the authors, it ispossible—withoutknowledge of the vertex radius behavior—to characterize the drop formation in terms ofcontact angle and nozzle radius at a certain volume.

Hartland and Hartley (1969) obtained similar results to those presented in Fig. 4.They performed numerical computations and also created a diagram characterizing thedrop-formation process by nozzle radius, drop volume, and contact angle.

The thick black line in Fig. 4 represents the maximum volume of a forming drop ona downward-facing nozzle. Higher volumes are not possible for surface-attached dropsbecause the differential equation provides no solution forlarger volumes at the corre-sponding nozzle diameter.

The red line shows the drop breakup due to gravity. It is computed by checkingthe following condition at every point of the drop shape during the integration of thedifferential equation:



[ρL − ρG] gV ≤ 2π σ r sinθ (16)

or, in dimensionless notation:V ∗ ≤ π r∗sinθ (17)

Figure 4 also shows that drops leaving nozzles with radii below 0.8 never break up dueto gravity. In the experiments described in the next section, one could observe that thedrops break up at the maximum volume (black line). The reasonmight be that dropsat the maximum volume tend to run out by further increasing the volume. If this is notpossible due to the use of a cannula with a very small wall thickness, probably theydetach. Further investigations are necessary regarding this aspect.

The red line exists up to a nozzle radius of approximatelyr∗D = 1.62. For nozzleradii above this value, the drop detaches right after the start of the drop formation. Thisbehavior can be explained by some analytical consideration.

Considering the drop at the start of the formation process tohave a spherical shape,its volume and contact angle can be expressed as follows:

V ∗ =π

3

[

R∗0 −

√

R∗02 − r∗D

2

]2 [

2R∗0 +

√

R∗02 − r∗D

2

]

(18)

θD = arcsinr∗DR∗

0

(19)

By rearranging Eq. (17) the following condition arises:

V ∗

π r∗DsinθD≤ 1 (20)

Inserting Eqs. (18) and (19) and taking the limiting value for R∗0 → ∞ yields an expres-

sion for the drop detachment at nearly zero volume which onlydepends on the nozzlediameter:

limR∗

0→∞

V ∗

π r∗DsinθD=

r∗D2

4≤ 1 (21)

From this relation it is obvious that drops forming from nozzles with a radius abover∗D = 2 detach at an infinitesimal volume. This clarifies the detachment of the drop atthe first step of numerical integration with the smallest volume for radii abover∗D =1.62. The discrepancy can be explained by the different drop shapes of the numericalintegration (real drop shape) and of the analytical solution (sphere shape).

A practical application of Fig. 4 is the contamination of nozzles. In order to avoidsuch contamination, the drop may not form outside of the nozzle. This means that thecontact angle, which depends on the material of the nozzle and the liquid, should notbe exceeded. Figure 4 can be used to find an optimal nozzle radius for given fluid andprocess properties such as surface tension, density, contact angle, and desired drop vol-ume. To avoid nozzle contamination, a nozzle radius has to befound at which the contact


608 Etzold et al.

angle during the formation is below the considered contact angle until the desired dropvolume is reached.

2.2 Experimental Results

To validate the computations in the previous section, measurements were carried outusing a DSA30 contact angle measuring instrument (KRUSS GmbH). It provides a sy-ringe pump, a camera to observe the drop, and special software for extracting the dropshape, and consequently the contact angle and the volume, from the camera picture. Theexperimental setup is shown in Fig. 5.

A downward-facing cannula made of steel with a very thin wallthickness was usedas a nozzle, so the inner and the outer diameters were nearly the same and the drop couldnot run out. The nozzle had a radius ofrD = 0.92 mm. By using two liquids (water anda water-soap solution) two different dimensionless radii were tested. Table 1 gives theproperties of the tested liquids.

The experimental results are shown in Fig. 6. It can be seen that the computation andthe measurement agree very well in the experiment with water. Even the point wherethe drop breaks up and falls from the nozzle is well predicted. It detaches when thedifferential equation provides no solution for a higher drop volume. This kind of dropdetachment mechanism was suggested by Durst and Beer (1969).

In Fig. 6(b), for water-soap solution, there is a small discrepancy at high volumes.This might be because of an uncertainty in the measurement ofthe surface tension of thesolution. Small changes in surface tension lead to a different Laplace constanta, whichresults in compression or stretching of the curve in theV ∗ direction. Nevertheless, theresults show that the theoretical considerations predict quite well the real behavior of thedrop formation concerning drop volume and contact angle.

cannula

drop

camera

syringe pump

FIG. 5: Experimental setup.

TABLE 1: Properties of the tested liquids (water andwater-soap solution)

Liquid a (mm) σ (N/m) ρL (kg/m3)Water 3.9 0.073 1000

Water-soap solution 2.9 0.042 1000



a) b)

[ °] [ °]

* *

computation

+ measurement

0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.80 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8

FIG. 6: Comparison of computed and measured contact angle as a function of dropvolume for (a)r∗D = 0.238 (water) and (b)r∗D = 0.316 (water–soap solution).

3. DYNAMIC DROP FORMATION

3.1 Limits of the Dynamic Drop Formation

In Section 2, the conditions for static drop formation were described and a representationinvolving Reynolds and Ohnesorge numbers was derived [see Eq. (6)]. This condition isthe lower limit of dynamic drop formation, because in its range the forces of inertia aredominant compared with gravity forces.

The upper limit of dynamic drop formation is jet formation. Aliquid jet from around-holed nozzle commonly forms when the following downward momentum andsurface tension force condition is fulfilled:

π

4D2ρLU2 > πDσ (22)

This means that a fluid jet cannot be formed if it does not fulfill the following condition:

m >D

2π√

ρLσD (23)

This relation is the upper limit of dynamic drop formation and can also be formulated asfollows:

Re· Oh > 2 (24)

Expressing this relation in terms of Weber number leads to

We > 4 (25)

This criterion was also obtained by other authors, e.g., Clanet and Lasheras (1999). Theyderived an equation for the transition from dripping to jetting by taking gravity, inertia,


610 Etzold et al.

and the nozzle wall thickness into account. Neglecting gravity and wall thickness influ-ences in their relation leads to the condition stated in Eq. (25). In Fig. 7, different rangesof drop and jet formation are shown. It can be seen that above Re· Oh = 2 jet formationtakes place. Below that condition, the theory of dynamic drop formation elaborated inthe next section can be applied. An additional computation of the shape of the drops withthe help of the theory of static drop formation can be carriedout for ReOh values below0.02.

3.2 Theoretical Considerations

If the momentum of a fluid flowing through a nozzle is lower thana certain value, itdisintegrates without external disturbance right at the nozzle outlet into single drops.This mechanism is not quasistatic as in Section 2, because the jet velocity cannot beneglected. In this section, dynamic drop formation, which lies below jet formation, isconsidered and a new relationship for the dripping frequency and its representation interms of Ohnesorge and Reynolds numbers is obtained.

For the formation of single drops the time derivative of the momentummU has tobe lower than the force due to the surface tension at the nozzle outlet:

0.001

0.01

0.1

1

10

0.01 0.1 1 10 100 1000

Oh

Re

0 ≤ h < 0.02

Static drop formation

∙ = 0 < h ≤ 2

Dynamic drop formation

∙ = .

h > 2

Jet formation

FIG. 7: Different ranges of drop and jet formation in Ohnesorge–Reynolds representa-tion.



π

4D2ρLU2 < πDσ (26)

Gravity is not considered here, since it does not come into play at the fluid exit throughthe nozzle. The above inequality leads to the minimal possible volume flow without dropformation, the so-called characteristic volume flowVc with characteristic velocityUc:

Vc =πDσ

ρLUc=

π

4D2

√

4 σ

ρLD(27)

where

Uc =

√

4 σ

ρLD(28)

If the aim is the formation of single drops, the actual volumeflow Vt has to be lowerthan the characteristic volume flowVc:

Vt < Vc (29)

If the drop diameterd is known, the drop frequency can be computed with the help ofthe volume flow rate:

Vc = fcπ

6d3 (30)

Vt = ftπ

6d3 (31)

The characteristic frequencyfc can be obtained by combining Eqs. (27) and (30):

fc =3

2

D2

d3

√

4σ

ρD(32)

The actual frequencyft depends on the actual volume flow and on the adjusted velocityat the nozzleUt:

ft =3

2

D2

d3Ut (33)

The ratio of the two frequencies is

ft

fc=

ρUtD

2√

σρD(34)

Introducing the viscosity and rearranging the equation yields a relation with Reynoldsand Ohnesorge numbers, and one can derive

Re· Oh = 2ft

fc(35)

This relation represents a straight line in the Oh–Re diagram. It is possible to express thecriterion for single-drop formation (and the jet criterion) with the characteristic and the


612 Etzold et al.

actual frequency by replacingVc andVt in Eq. (29) by the expressions of the Eqs. (30)and (31):

ft < fc (36)

Hence the criterion for drop formation can also be written with these frequencies:

Re· Oh < 2ft

fc(37)

In Fig. 8, an Oh–Re diagram is shown presenting different lines of different values offt/fc. They illustrate that, in the whole range of drop formation,the theory regardingdynamic drop formation of the upper section can be applied, even in the range of staticdrop formation. In the latter, it is possible to additionally compute the drop shape withthe help of the theory of static drop formation.

3.3 Experimental Results

Experiments were carried out to prove the conclusions in theprevious section. A cannulawas used as a nozzle. Pure water or glycerin–water solutionswith different viscositieswere conveyed through the cannula at different velocities.The experimental setup wasthe same as in Section 2.2, but here the drops were not observed by the camera. Thefrequency of the falling drops was measured manually by counting the number of dropsdetaching during one minute or with the help of a light barrier at higher frequencies. The

0.001

0.01

0.1

1

10

0.01 0.1 1 10 100 1000

Oh

Re

0.5

0.1

0.001

0.0001

h = 2;

jet criterion

h = 0.02 ;

crit . for static

drop formation

= 0.5⁄

= 0.1⁄

= 0.01⁄

= 0.001⁄

FIG. 8: Oh–Re diagram with different values offt/fc and the limiting line of dropformation (jet criterion) and the limiting line for the static drop formation.



measurements were repeated three times for each frequency,leading to values for thevariation coefficient below 2%.

Different volume flow rates (approximately 67, 140, 214, 365, 1072, 1510, 1832and 2260µL/min) and different viscosities (1, 22, 50, 108, and 220 mPa) were studied.The volume flow rate varied slightly with a change in viscosity. Therefore the flow ratevalues given above are mean values. Two of the fluids with the highest viscosities (108and 220 mPa) were conveyed through the nozzle at only 67, 140,214, and 365µL/min.Their properties are summarized in Table 2.

The cannula had a diameter ofD = 0.5 mm and a length ofL = 40 mm. Withthis length and with the maximum tested Reynolds number, Re≈ 100, the flow insidethe nozzle can be regarded as laminar and fully developed since the entrance length isLe = 0.035 D Re= 1.75 mm < L (Perry, 1950).

The theoretical value of the actual frequency could be calculated with Eq. (33), be-cause the nozzle diameter and the velocity were known. It is independent of the viscos-ity [see Eq. (33)]. The drop diameter was measured for each viscosity with the pendantdrop method before performing the experiment with a constant volume flow. It was as-sumed that the drop volume is controlled nearly exclusivelyby gravitation forces, be-cause Zhang and Basaran (1995) observed a drop volume changeof only 4% for anincrease in Reynolds number from 0.03 to 3.

In Fig. 9, the theoretical values of the actual frequency in this study are shown to-gether with the measured values for three chosen viscosities and with the frequenciescomputed with the model for the drop size of Scheele and Meister (1968):

VD = F

πσD

gρL−

16 V 2

3π gD2+ 4.5

[

V 2D2σ

g2ρL

]1/3

(38)

ft =V

VD(39)

whereF is the Harkins–Brown correction factor. Since the variation coefficient of themeasurements was below 2%, no error bars are depicted.

TABLE 2: Properties of the tested liquids (water and water–glycerin solutions)

Liquid µ (mPas) σ (N/m) ρL (kg/m3)Water 1 0.073 1000

Water–glycerin solution

22 0.064 118550 0.063 1207108 0.062 1227220 0.063 1239


614 Etzold et al.

0

50

100

150

200

250

300

350

0 1000 2000 3000

0

50

100

150

200

250

300

350

0 1000 2000 3000

0

10

20

30

40

50

60

0 100 200 300 400

a) = 1mPas (pure water) b) = 50mPas

c) = 220mPas

t[1

min

⁄]

10

15

20

25

t[1

min

⁄]

20

30

40

t[1

min

⁄]

t[µl min⁄ ] t[µl min⁄ ]

t [µl min⁄ ]

FIG. 9: Measured drop formation frequencies (points) and theoretical frequencies in thisstudy (solid lines) and values computed with the model of Scheele and Meister (1968)(dashed lines).

The theoretical lines in this study and those of Scheele and Meister (1968) lie closetogether, indicating that in the latter the gravity force isdominant since the solid line iscomputed with the measured drop volume for the static case.

The frequencies in the experiment approximate the theoretical lines very well, exceptfor the fluid with the lowest viscosity at the two highest flow rates. These points had thehighest Reynolds numbers in the experiments (approximately 80 and 100), indicatingthat the assumption of the independence of the drop size fromthe Reynolds numberis not valid at such high velocities. According to Zhang and Basaran (1995), the samebehavior of the drop size was observed at a Reynolds number ofapproximately 70.

The advantage of Eq. (35) developed here is the representation in a double-logari-thmic Oh–Re diagram as straight lines. The Reynolds number of the flow through thenozzle is



0.001

0.01

0.1

1

10

0.01 0.1 1 10 100 1000

Oh

Re

= 67µl/min = 140µl/min = 214µl/min = 365µl/min = 720µl/min

-- = 1072µl/min -- = 1510µl/min -- = 1832µl/min -- = 2260µl/min ∙ h = 2

FIG. 10: Theoretical (lines) and measured (points) drop-formationprocesses in Oh–Rerepresentation.

Re=ρLUtD

µ(40)

With the help of Eq. (35), the Ohnesorge number can be calculated:

Oh =ft

fc

2

Re(41)

With these two equations it is possible to show the same measurements as in Fig. 9 in anOh–Re representation (Fig. 10).

In Fig. 10, the lines represent one volume flow and hence one frequency. Thereforethe relation Re· Oh = 2ft/fc is illustrated, since every line stands for one particularvalue of2ft/fc. The product Re· Oh is independent of the viscosity. Therefore, mea-sured mixtures at the same volume flow rates with only different viscosities are locatedin the Oh–Re diagram on a straight line parallel to the jet criterion line (Re· Oh = 2).In the present case, the density and the surface tension of the glycerin–water mixturesand the flow rates changed slightly, so the different viscosities are not located on exactlystraight lines. Here, all tested velocities and viscosities are shown combined in one di-agram, and it can be seen that the theoretical predictions and the practical values agree


616 Etzold et al.

very well (except for Re= 80 and100; the discrepancy seems not to be high due tologarithmic scaling). No further information can be obtained from this diagram in com-parison with the plots in Fig. 9, but it is a different and demonstrative way to representthe relationships of dynamic drop formation.

4. CONCLUSIONS, FINAL REMARKS, AND OUTLOOK

In this study, static and dynamic drop formation were investigated both theoretically andexperimentally and a generalization was obtained by presentation of the data in the Oh–Re diagram. First, a short summary of the well-known theory of the static drop formationwas presented. The algorithmic approach to solving the differential equation taken hereled to a representation of the drop-formation process in onediagram with respect tocontact angle, nozzle diameter, and drop volume. The vertexradius is not present in thisrepresentation; hence it is very suitable for practical applications where the vertex radiusis usually not known or not of interest. The numerical results regarding the contact angleevolution during the drop formation with increasing drop volume could be verified byexperiments.

A criterion has been derived up to which the value of Oh· Re drop formation can betreated by the differential equation for static drop formation. Above this value (ReOh=0.02) the drop shape cannot be computed with the theory of static drop formation. Thenit is possible to apply the theory of dynamic formation of single drops from round-holednozzles. This theory is applicable in the whole range of dropformation, even in therange of static drop formation (from ReOh> 0 up to the jet criterion ReOh= 2). Bothlimits, for static drop formation and for dynamic drop formation, are shown together inan Ohnesorge–Reynolds number representation.

For the dynamic drop formation a characteristic frequency dependent on nozzle di-ameter and fluid properties was derived. It was shown that theactual frequency resultingfrom the actual volume flow rate has to be smaller than the characteristic frequency.Otherwise, a jet forms, since the characteristic frequencyrepresents the volume flowrate at the jetting point (Re· Oh = 2). These considerations could be expressed in termsof Reynolds and Ohnesorge numbers. The theoretical resultswere confirmed by ex-perimental results for different viscosities and different flow rates. The treatment of dy-namic drop formation in terms of characteristic and actual frequency and the Ohnesorge–Reynolds number representation is regarded by the authors as a recent aspect of thisstudy.

The results of dynamic drop formation can be used to design nozzles to producedrops of a certain diameter at a desired frequency, for example, for so-called drop-on-demand systems. For this and many other kinds of nozzles,the results regardingstatic drop formation are useful. In order to avoid contamination of the orifice outlets,the correct nozzle diameter can be chosen for favored drop and given material proper-ties.



REFERENCES

Bashfort, F. and Adams, J. C.,An Attempt to Test the Theory of Capillary Action, CambridgeUniversity Press, Cambridge, 1883.

Clanet, C. and Lasheras, J. C., Transition from dripping to jetting,J. Fluid Mech., vol. 383, pp.307–326, 1999.

Clift, R., Grace, J. R., and Weber, M. E.,Bubbles, Drops, Particles, Academic Press, New York,1978.

Durst, F. and Beer, H., Blasenbildung an Dusen bei Gasdispersionen in Flussigkeiten,Chem.-Ing.-Tech., vol. 41, pp. 1000–1006, 1969.

Fordham, S., On the calculation of surface tension from measurements of pendant drops,Proc.R. Soc. London, Ser. A, vol. 194, pp. 1–16, 1948.

Harkins, W. D. and Brown, F. E., The determination of surfacetension (free surface energy), andthe weight of falling drops the surface tension of water and benzene by the capillary heightmethod,J. Am. Chem. Soc., vol. 41, pp. 499–524, 1919.

Hartland, S. and Hartley, R. W.,Axisymmetric Fluid–Liquid Interfaces, Elsevier Scientific Pub-lishing Co., Amsterdam, 1976.

Lando, J. L. and Oakley, H. T., Tabulated correction factorsfor the drop-weight-volume deter-mination of surface and interfacial tensions,J. Colloid Interface Sci., vol. 25, pp. 526–530,1967.

Lohnstein, T., Zur theorie des abtropfens mit besonderer r¨ucksicht auf die bestimmung der kap-illaritatskonstanten durch tropfversuche,Annu. Phys., vol. 325, pp. 237–268, 1906.

Perry, J. H.,Chemical Engineers’ Handbook, McGraw-Hill, New York, 1950.

Siemes, W., Gasblasen in Flussigkeiten Teil I Entstehung von Gasblasen an nach oben gerichtetenkreisformigen Dusen,Chem.-Ing.-Tech., vol. 26, pp. 479–496, 1954.

Scheele, G. F. and Meister, B. J., Drop formation at low velocities in liquid-liquid systems, PartI. Prediction of drop volume,AIChE J., vol. 14, pp. 9–15, 1968.

Wilkes, E. D., Phillips, S. D., and Basaran, O. A., Computational and experimental analysis ofdynamics of drop formation,Phys. Fluids, vol. 11, pp. 3577–3598, 1999.

Zhang, X. and Basaran, O. A., An experimental study of dynamics of drop formation,Phys.Fluids, vol. 7, pp. 1184–1203, 1995.


STATIC AND DYNAMIC DROP FORMATION ON DOWNWARD-FACING NOZZLES

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