ATOMIZAÇÃO_phd7

8/13/2019 ATOMIZAO_phd7

1/22

Chapter 7

DROP FORMATION IN LIQUID-LIQUID SYSTEMS

BEFORE AND AFTER JETTING

I do not suppose there is anyone in this room who has not occasion-ally blown a common soap bubble, and while admiring the perfectionof its form, and the marvelous brilliancy of its colours, wondered howit is that such a magnificent object can be so easily produced. I hope

that none of you are yet tired of playing with bubbles, because, asI hope we shall see, there is more in a common bubble than thosewho have only played with them generally imagine.

C. V. Boys, Soap bubbles(1931)

7.1 Background

In chapter 6 we observed the breakup of liquid-liquid jets. In this chapter

we examine the size of the drops produced. When a liquid is injected into another

liquid at low velocities, drops are formed that detach and break off from the nozzle,

and at a velocity above a certain critical value, the jetting velocity, a jet forms,

rises to a certain length, and then breaks up into drops. These drops result in

the creation of large new surface area, which leads to enhanced heat and mass

transfer (Skelland and Walker, 1989). The study of the drop formation dynamics

is necessary for prediction of mass transfer rates in these processes. The VOF-CSF

technique is used in this chapter to predict drop formation before and after jetting.

The implementation details of the numerical method have already been presented

in chapters 3, 5 and 6.

Bogy (1979) has reviewed drop formation for liquid jets into air, with more

recent work done by Mansour and Lundgren (1990), Vassallo and Ashgriz (1991),

and Orme et al. (1993). Drop formation in liquid-liquid systems has been studied

149


2/22

150

both experimentally and theoretically for the regimes before and after jetting

occurs. In a series of publications, Meister and Scheele (Meister, 1966; Meister

and Scheele, 1967, 1968, 1969a, 1969b) worked to develop an understanding of

the jet and drop formation based on experiments obtained with 15 liquid-liquid

systems. They described the general behavior of the jets as follows. For low flow

rates drops form, grow, neck and break off from the nozzle at regular intervals.

Above a certain critical velocity a jet is formed that rises to a certain length from

the nozzle, at which point it breaks up into drops. At still higher nozzle velocities,

the jet disrupts into small drops. Much of the earlier literature on liquid-liquid

drop and jet formation has been reviewed by Meister and Scheele (1968, 1969b)

and more recently by Kitamura et al. (1982) and Skelland and Walker (1989).For drop formation at low velocities, Scheele and Meister (1968) have

presented an improved correlation over previous theories, starting with the work

of Harkins and Brown (1919), who balanced surface tension and buoyancy forces

at the time of necking and breakoff. The general approach of Scheele and Meister

(1968) was to perform an overall force balance on the drop as it detached from

the nozzle, taking into account surface tension, buoyancy, inertia, and drag forces

and additionally correcting for flow into the drop during the necking process. In

addition, they presented a correlation for prediction of the jetting velocity.

Tyler (1933) used Rayleighs (1879) instability theory for liquid into air jets

assuming that the disturbance responsible for breakup travels at the jet average

velocity. Tyler assumed that if waves were formed on the jet due to disturbances

at the nozzle, the volume of the resulting drops would be equal to a cylinder

of radius equal to that of the jet and length equal to that of the disturbance

wavelength. For liquid-liquid drop formation above jetting, Christiansen (1955)

modified the Tyler (1933) analysis by calculating the wavelength based on an

instability analysis for two inviscid liquids. Meister and Scheele (1969b), using

the Tomotika (1935) stability equation, developed an expression for the drop size

that was an improvement over the equation used by Christiansen (1955). The


3/22

151

improvement resulted from the greater generality of the Tomotika (1935) stability

analysis as well as taking into account that the disturbance on the jet travels at

the interfacial velocity and the phenomenon of wave-splitting that results from

flow of material through the nodes of the jet.

Kitamura et al. (1982) experimentally varied the motion of the continuous

phase to be either faster, the same as, or slower than the jet down to the case of a

stagnant continuous phase. They found that the jet shortened as the absolute value

of the continuous phase velocity relative to the jet increased from zero. Moreover,

the condition of a zero relative phase velocity was found to give the best agreement

between Tomotikas stationary jet solution (translated at the average velocity of

the jet) and the experimental measurements of jet length and drop size. Theyused the Tyler (1933) analysis for drop size using Tomotikas (1935) analysis to

calculate wavelength and found the best agreement when the relative velocity of

the phases was zero.

Bright (1985) attempted to perform a linear viscous stability analysis to

calculate the disturbance wavelength assuming a constant, but unequal, velocity

in each liquid phase and to predict drop volumes using the Tyler (1933) analysis.

One difficulty with this study rests on the inconsistency of having a discontinuity

of velocity at the interface for fluids that are not assumed to be inviscid. Even if

the equations are formulated correctly, we have found that the particular solution

provided by Bright (a nonhomogeneous modified Bessel ODE) is incorrect; its

correct form is much more complex than the one provided by his equation (18).

Details of the derivation can be found in appendix H.

Thus, all the previous liquid-liquid theories, based on simplifying assump-

tions, have limited validity. The difficulties are associated with the various effects

(such as viscous, buoyancy, surface tension, inertial forces, jet contraction, veloc-

ity profile relaxation, and relative motion of the continuous phase) that are not

all fully accounted for in any of the previous theories. This lack of an adequate

theoretical description that is driven by the inherent complexity of the problem


4/22

152

led us to the present direct numerical simulation of the drop formation process

as discussed in chapter 6. The only assumptions involved with the present ap-

proach are those of laminar flow of the two Newtonian fluids with constant bulk

densities and viscosities, constant interfacial surface tension, and axisymmetric

disturbances. Of these the latter assumption has been shown experimentally to

fail in the region above a certain critical velocity where the jet lengthens appre-

ciably to a maximum length (Meister and Scheele, 1969a). However, in the region

below this critical velocity, experimental observation (Meister and Scheele, 1969a)

shows that axisymmetric disturbances dominate, and in consequence, this is the

region of validity of the present study.

The objectives of the present study are to use the VOF-based numerical

method that we have developed in this dissertation for calculating liquid-liquid

drop dynamics, from the startup of their time evolution to their apparent (or

pseudo) steady-state involving the region from the nozzle exit to their breakup

through necking and detachment of drops. We compare the results with the

available experimental data of Meister and Scheele (1968, 1969b) and the available

linear stability theory of Meister and Scheele (1967, 1969a, 1969b).

In section 6.2 we have presented the definition and formulation of the

governing equations for the solution to the problem of a Newtonian liquid injected

vertically into another Newtonian quiescent liquid. In section 6.3 we have discussed

the numerical implementation of the governing equations to the dynamic liquid-

liquid drop and jet problem. In section 7.2 we discuss the Meister and Scheele drop

formation theory used in this chapter. In section 7.3, the numerical simulation is

compared with the inviscid liquid-liquid cylinder breakup solution. The present

numerical simulation of free surface dynamics and drop sizes is compared to

the existing experimental data (Meister and Scheele, 1969b) and the Meister

and Scheele (1969b) drop formation analysis. Finally, in section 7.4 we reach

conclusions based on the results of the present work.


5/22

153

7.2 Meister and Scheele Drop Formation Theory

In this section we briefly present the results of the Meister and Scheele

drop formation theory (Meister and Scheele, 1968, 1969b). The flow configuration

investigated here is shown in Figure 6.1. It corresponds to the experimental setup

of Meister and Scheele (1966, 1968, 1969b), details of which can be found in those

references. Briefly, the problem investigated involves the jet of a liquid (in this

chapter n-heptane) injected vertically from a circular nozzle upwards into a tank of

stationary mutually saturated immiscible water. Figure 6.1 shows the stationary

tank of fluid 1 (water) with density 1 and viscosity 1 with the jet of fluid 2 (n-

heptane), with density 2 and viscosity 2, flowing upward with average velocity

v from a nozzle of inner radius R and outer radius Ro. The constant interfacialsurface tension is , and the gravitational acceleration, g, is directed downward.

The distance from the nozzle tip to the top of the tank is L1 and to the bottom

is L2. The distance from the centerline of the nozzle to the outer tank wall is L3

and the nozzle is of total length L4 with the velocity profile fully developed at

length L5 from the nozzle tip. Disturbances originating in the nozzle propagate

with the jet flow and eventually lead to breakup of the jet, with the length of the

jet given byL, and the wavelength of the growing disturbance given by . Drops

are produced which can be roughly characterized by the diameter of a sphere of

equivalent volume,Dd= 2Rd (Meister and Scheele, 1968, 1969b).

The Meister and Scheele (1969a) calculation of the jet length is discussed in

chapter 6. The Meister and Scheele (1968, 1969b) equations used in this chapter

to calculate drop volume below and above the jetting velocity are now discussed.

There appears to be some differences between the work presented in Meisters

(1966) dissertation and the papers that appeared later; the papers are used as

the final standard for the MS model. Below the jetting velocity, the mechanism

for drop formation was found by Meister and Scheele (1968, 1969b) to be that of

a force balance, while above the jetting velocity for a long jet, amplification of

disturbances on the jet were assumed to lead to droplet formation. If the jet was


6/22

154

short, a combination of these two mechanisms was postulated to occur.

At low nozzle velocities Scheele and Meister (1968) considered four major

forces acting on the drops. For liquids injected vertically upwards and2< 1, the

buoyancy force due to the density difference between the liquids and the inertial

force associated with the fluid flowing out of the nozzle act in the upward direction,

while the interfacial tension force and the drag force associated with the continuous

phase act in the downward direction. When these forces balance, the drop begins

to break away from the nozzle. Scheele and Meister arrived at the following final

correlation with parabolic flow in the nozzle for droplet volume Vd:

Vd= FHBDg + 201QDD2

dg

42Qv3g

+ 4.5

Q2

D2

2(g )2

13 (7.1)

where FHB is the Harkins-Brown (1919) correction factor taking into account

that part of the drop remains on the nozzle after detaching, D = 2R is the nozzle

diameter, Q = vR2 is the nozzle volumetric flow rate, and 1 2 is the

density difference between the two fluids. The Harkins-Brown correction factor is

obtained from a plot ofFHB vs. D (FHB/Vd)1

3 (Figure 3 of Scheele and Meister,

1968). The first term on the right-hand-side represents the surface tension force,

the second term the Stokes drag, the third term the inertia, and the fourth term

represents the increase in volume due to flow of fluid into the drop during necking,

all ratioed to buoyancy. This equation involves an iteration on Dd that generally

converges quickly. A first guess may be obtained by assuming v= Q= 0, reducing

equation (7.1) to the Harkins-Brown equation for drop volume at low velocities.

The jetting velocity was obtained by the assumption that a jet will form if

there is sufficient upward force at the nozzle to support it. At this critical point,

the surface tension force, D, is assumed to be equal to the excess pressure force

necessary to sustain a spherical drop, D2/Dd, plus the inertial force of the fluid

leaving the nozzle, assuming a parabolic velocity profile, 2V2

JD2/3. With these


7/22

155

assumptions the jetting velocity may then be estimated from:

VJ= 1.73

2D 1 D

Dd1

2

(7.2)

Above the jetting velocity, Meister and Scheele (1969) assumed that for long

jets the size of drops formed is controlled by the instability mechanism starting

with the analysis of Tyler (1933). Tyler assumed that if waves are formed on the

jet the volume of the resulting drop would be equal to the volume of a cylinder

having the radius of the jet a(z) and length equal to the disturbance wavelength

= 2/k:

Vd= a2=

22a3

ka

(7.3)

where ka is the dimensionless wave number of the most dangerous disturbance

evaluated at the nozzle radius, a = R. This equation has been very successful

in predicting drop sizes of jets injected into air (Tyler, 1933; Merrington and

Richardson, 1947). Meister and Scheele modified this analysis to take into account

the continuous phase and the velocity profile inside the jet. If the jet was long

(L >2), they assumed that the drop volume was the product of the volumetric

flow rate Q and the time interval between initiation of successive nodes of the

most unstable wave which travel at the interfacial velocity VI evaluated near the

nozzle at z= /2:

Vd= Q

vI

z=

2

=22

a3z=

2

N(kR)vIvA

z=

2

(7.4)

wherevA= vR2/a2 is the jet average velocity andNis a factor which depends on

the wave-splitting ratio:

Rc=(vI)z=L (a)z=

2

(vI)z=2

(a)z=L

2

(7.5)

which takes into account that as the volume of the nodes increases a new node

can be formed which splits the original node in half if the node grows longer than


8/22

156

2. The wave splitting factor is computed as:

N=

1, if 0 Rc< 22, if 2 Rc< 4

4, if 4

Rc

< 8

(7.6)

where the jet length L and kR are calculated from the equations of chapter 6.

If the jet length is small (L < ) drop formation is assumed to be controlled

by the force balance equation (7.1) with the jet radius (a)z=LDd

being the basis

for the force balance:

Vd= FHB

2a

g +

401Qa

D2d

g

42Qv

3 aR2

g

+ 7.15Q2a22

(g )

2 1

3

(7.7)

If the jet length is in between the short and long limits ( L 2) they assume

an arithmetic average of the drop volumes calculated by the two mechanisms.

7.3 Results and Discussion

Meister and Scheele (1969b) measured the jet length and drop diameter

vs. the dispersed-phase Reynolds number, Re2, for five nozzle diameters. Their

dimensionless jet length (L/R) results, obtained by averaging the lengths of jets

measured from four photographs per point (which contained several drops), are

summarized in Figure 7.1 (Figure 1 of Meister and Scheele, 1969a), and the

dimensionless drop radius (Rsv/R) is given in Figure 7.2. If the drop was deformed

from a spherical shape, the drop diameter was recorded as the arithmetic average

of the major and minor diameters, which was expected to yield an error of less than

5% of drop volume if the diameter ratio was less than 1.7 (Christiansen, 1955).

Most of their drops were reported to have diameter ratios less than 1.2. They then

reported their results in terms of a Sauter average diameter Dsv = 2Rsv:

Dsv =

i

niD3

dii

niD2di(7.8)


9/22

157

and a specific surface area per unit volume:

Ss= 6

Dsv(7.9)

presumably since the diameters of drops in liquid-liquid systems are not as uniform

as in liquid-air systems.

0 500 1000 1500 2000 2500 3000 3500

Re2

0

20

40

60

80

100

120

140

L/R

R = 0.04065 cm

R = 0.08 cm

R = 0.127 cm

R = 0.166 cm

R = 0.344 cm

Figure 7.1: Dimensionless jet lengthvs.Reynolds number, for n-heptane injected

into water for five nozzle sizes. Data of Meister and Scheele (1969a).

We have used Meister and Scheeles data on n-heptane injected into water

for comparison purposes. The parameters corresponding to the five base cases


10/22

158

0 500 1000 1500 2000 2500 3000 3500

Re2

0

1

2

3

4

5

6

Rd

/R

R = 0.04065 cm

R = 0.08 cm

R = 0.127 cm

R = 0.166 cm

R = 0.344 cmJet

Jet

Jet

Jet

Jet

Figure 7.2: Dimensionless drop radius vs. Reynolds number, for n-heptane in-jected into water for five nozzle sizes. Data of Meister (1966).

that we have used in this work are summarized in Table 6.1, encompassing five

nozzles. The systems were allowed to become mutually saturated (to avoid mass

transfer between the phases) before the experiments, and the physical parameters

for the saturated systems are given in Table 6.2. In Table 6.2 note the substantial

differences between the experimentally measured and the literature values for the

interfacial surface tension. The most probable cause for this difference is the

presence of (unknown) contaminants. Since the emerging drops and jet form a

new, clean surface that under the experimental circumstances is not anticipated

to have had enough time to become saturated with the contaminant, the literature


11/22

159

values for pure components were used for the interfacial surface tension in this work

(see discussion in section 6.4).

In the numerical simulations, the flow rate at the nozzle was instantaneously

increased from zero to its final value with the n-heptane filling the nozzle and

water filling the tank, in much the same way as the physical system was started.

Then the simulation followed the time evolution of the flow until the drops and

jet length vs. time profile reached pseudo-steady behavior. A typical (coarse)

mesh is shown in Figure 6.2, with the increased mesh refinement confined, as

illustrated, to the dispersed phase. A locally refined, but uniform, mesh is used

inside the jet and drop formation region, whereas a variable, coarser, mesh is

employed in the much more slowly moving outer region. The mesh size that wasused for the runs involved 60 cells stacked in the radial direction (with 40 cells

being confined to the interval 0 r 2) and 55 80 cells in the axial direction.

This cell distribution was optimized through mesh sensitivity studies. In addition,

the effects of the effluent boundary distance L1, the bottom distance, L2, the

fully developed velocity profile distance, L5, and the outer wall distance L3 were

investigated separately by sensitivity studies. The actual tank dimensionsL1/R

and L3/R are given in Table 6.1. The bottom distance was not given in the

reference, but was estimated as at least L2/R > 10.

For the conditions of Table 6.1, it was found that L1/R= 1015,L2/R= 2,

L3/R = 6, and L5/R = 2 were large enough to establish insensitivity of the

results to the actual values ofL1/R, L2/R, and L3/R. For example, comparing

results obtained with L3/R = 6 to 12, no significant difference was found in the

breakup length. Likewise, because of the high speed of the jet, comparing results

corresponding toL1/R= 10 to 25 for nozzle 5, no significant difference was found

in the dispersed phase solution as long as the drop formation occurred far from

the outflow, as was mostly the case.

The breakup of the jets occurs experimentally as a result of naturally

occurring perturbations. However, below the jetting velocity, no perturbation


12/22

160

is necessary to produce the drops since they are formed by the force balance

mechanism. Given the uncertainty of its exact form in the reported experiments,

we added a numerical perturbation to the free surface at the nozzle lip for the jets

as discussed in section 6.4. The angular frequencies of the perturbations applied

are given in Table 6.1. The amplitude of the perturbation for the jet simulation

cases, a0/R = 0.01, was fixed as discussed in 6.4. Its variation has a varying

influence on the results, depending on the Reynolds number.

7.3.1 Drop Formation Dynamics

The results presented in this chapter are comparisons between the experi-

ments of Meister and Scheele (1966, 1968, 1969b) and our numerical simulations.

Figure 7.3 represents a simulation of drop formation for nozzle 5 (see Table 6.1)

at low Re2. Each frame of the figure is a contour of the interface (F = 1/2) at

equal time intervals, starting from startup att = 0. Notice the trend to oscillating

ellipsoid-like drops. As was mentioned in chapter 6, there is some evidence of

non-axisymmetric drops in the experimental frame on Figure 6.5; this is obviously

not captured by the simulations, but appears to be a minor feature in the overall

picture.

The jet length,L/R, or the height of the forming drop if the nozzle velocity

was below the jetting velocity, was calculated as a function of time for each

simulation case. A typical result for low velocity drop formation is shown in

Figure 7.4. As can be seen in Figure 7.4, the general pattern is a rise to a certain

length, detachment of a drop, with a buildup in length to form the next drop.

This results in a sawtooth pattern as seen in the figure. It can be seen that

at long times the drop height is periodic, but complex. We have found that the

variation in amplitude and period generally increases with increasing Reynolds

number, Re2, with the fluctuations becoming more random. We thus conjecture

that the aperiodicity is due to the increasing importance of the nonlinear terms

in the momentum equations, causing a chaotic response. Drop volume and radius


13/22

161

Figure 7.3: Drop formation sequence the present simulation. Heptane intowater, nozzle 5,R = 0.344 cm, v = 2.52 cm/s.


14/22

162

(assuming a spherical shape) can be calculated using this type of data from the

simple material balance:

Vd= R2vtd=

4

3R3d (7.10)

where td is the drop formation time, which can be found from Figure 7.4 as the

time between successive peaks.

0 1 2 3 4

t (s)

0

2

4

6

8

10

L/R

Figure 7.4: Drop height vs. time. Heptane into water, nozzle 5, R = 0.344 cm,v = 2.52 cm/s.

7.3.2 Drop Size Comparisons and Discussion

The average, minimum, and maximum drop radii predicted by the numer-

ical simulation are shown in Figures 7.5 to 7.10 for the five cases. Also shown are


15/22

163

the experimental drop radii data and the numerical results of Meister and Scheele

(1968, 1969b) (MS) from equations (7.1) to (7.7). Here the actual experimental

jetting velocity was used to determine the appropriate drop volume equations for

the MS model instead of equation (7.2), as this equation deviated as much as 28%

in jetting velocity for nozzle 5. Figures 7.5 to 7.9 show the predicted drop radii for

nozzles 1 to 5 as a function of Reynolds number, Re2. Figure 7.10 shows predicted

drop radii vs. experimental radii for all five nozzles.

Several points regarding the origin and nature of the experimental data

should be noted in assessing the agreement of theory with experiment. One is

that Meister and Scheele (1969b) took four photographs containing several drops

each and averaged them to obtain their experimental drop sizes. No informationis available on the accuracy of this approach, except the regularity and frequency

of the data points themselves. One can imagine minimum and maximum bars on

the experimental data about as large as those included in Figures 7.5 to 7.10 for

the numerical simulations.

Although the predictions from the MS model come close to the data it can

be seen from Figures 7.57.10 that their slope, particularly in the jetting region, is

much less than the data would indicate. On the other hand, the predictions from

the present simulation are within experimental error, both above and below the

jetting velocity. This is true even though the results from the present simulations

for jet length presented in chapter 6 show a small bias to over-prediction of the

jet length; this is attributed to the presence of numerical viscosity (Hirt, 1968),

which tends to make the numerical solution more stable than the physical one (see

section 6.4.4).

The MS model has many assumptions, as mentioned in sections 6.2.4 and

7.2. As the MS model deviates the most from experiment above the jetting velocity,

we focus on this region. As can be seen by examining the drop volume equation

(7.4), three of the critical steps are the calculation of the most dangerous wave

number kR, the velocity at which the disturbance is propagated, the interfacial


16/22

164

0 500 1000 1500 2000

Re2

0

1

2

3

4

5

6

Rd

/R

MS model

This workMS Data, R = 0.04065 cm

Jet

Figure 7.5: Drop radii vs. Reynolds number data of Meister and Scheele (1968,1969b) (n-heptane into water) and numerical results of Meister andScheele (1968, 1969b) (MS) compared to the present simulation fornozzle 1.

velocityvI, and the jet radiusa. As discussed in section 7.1, Kitamura et al. (1982)

experimentally varied the relative motion of the continuous phase. They found

that the drop size varied as the absolute value of the continuous phase velocity

relative to the jet increased from zero, with the condition of zero relative phase

velocity giving the best experimental agreement with Tomotikas stationary jet

solution translated at the average velocity of the jet. They concluded that the

discrepancy between experiment and Tomotikas analysis in the nonzero relative


17/22

165

0 500 1000 1500 2000

Re2

0

1

2

3

4

5

6

Rd

/R

MS model


Jet


velocity cases was due primarily to the relative motion of the continuous phase,

and not to other factors such as velocity profile relaxation. In our case, the outer

phase is relatively stagnant away from the interface, and this could result in a

shearing type instability (Russo and Steen, 1989) that would result in an incorrect

kR, since the Tomotika equation (6.10) assumes that the two phases have the

same axial velocity. To correct for this, an analysis similar to that of Sterling

and Sleicher (1975), Bright (1985), or Russo and Steen (1989) would have to be


18/22

166

0 500 1000 1500 2000

Re2

0

1

2

3

4

5

6

Rd

/R

MS model


Jet


performed, but since gravity (buoyancy) and the outer phase viscosity would have

to be included, the modifications would be nontrivial and, if feasible, would result

in a rather complex solution procedure. The value of the Tomotika analysis is that

it is currently the only approach that incorporates in the outer phase the physical

properties of density and viscosity.

Another source of error in the MS model is the calculation of the axial

interfacial velocity, vI, as discussed in section 6.4. To test these ideas we com-


19/22

167

0 500 1000 1500 2000

Re2

0

1

2

3

4

5

6

Rd

/R

MS model


Jet


pared the MS model calculation of jet radii a/R (equation (6.27)) and interfacial

velocitiesvI/v (equation (6.26)) with the current simulation for nozzle 3; the re-

sults are shown in Table 6.3. The comparison is quite good until just before jet

breakup, which is consistent with the experimental checks by Meister and Scheele

(1969b). Therefore, we conclude that the error in the calculation of the interfa-

cial velocities is not a major contributor to the overall discrepancy between the

MS model and the data. The prediction of the jet radii a in Table 6.3 and in


20/22

168

0 500 1000 1500 2000

Re2

0

1

2

3

4

5

6

Rd

/R

MS model


Jet


Figures 5.45.6 shows that the Addison and Elliott (1950) equation (5.11) gives a

reasonable rough estimate. Thus, given these facts, coupled with the experimental

observations by Kitamura et al. the most likely reasons for the MS disagreement

are (1) the maximum growth rate s and the most dangerous wavenumber k are

assumed independent of the jet velocity v, and (2) the entire analysis is a local

one, not taking into account the velocity profile relaxation in the axial direction

away from the nozzle exit. Still, the MS analysis can be used as a first estimate, its


21/22

169

0 1 2 3 4 5 6

Rd/R Experimental

0

1

2

3

4

5

6

Rd

/RP

redicted

MS model, MS Data

This work, MS Data

Figure 7.10: Drop radii predicted vs. experimental data of Meister and Scheele(1968, 1969b) (n-heptane into water) and numerical results of Meis-ter and Scheele (1968, 1969b) (MS) compared to the present simu-lation for nozzles 15.

major drawback being its relative insensitivity towards Reynolds number changes

in the jetting region.

7.4 Conclusions

The robust and stable VOF-based numerical method developed in this work

has been used here successfully to predict drop sizes for low to high Reynolds num-

ber flows, before and after the jetting velocity, encompassing all of the physics of

the previous models. With our method, there is no need to switch between various


22/22

170

mechanisms depending on the flow regime, as the correct physics is embedded in

the full equations of motion and continuity. We have compared the predicted drop

sizes with the experimental data of Meister and Scheele (1966, 1968, 1969b). We

conclude that the present simulation agrees satisfactorily with the data, within

experimental error. The use of this direct numerical simulation allowed us to as-

certain the accuracy of the approximate numerical model of Meister and Scheele

(1968, 1969a, 1969b) and to examine the validity of the key underlying assump-

tions.

Documents

ATOMIZAÇÃO_phd7