Asme Dist f - Ectc 2012 Journal - Section 6

5/20/2018 Asme Dist f - Ectc 2012 Journal - Section 6

1/32

SECTION 6

FLUIDS ENGINEERING

ASME 2012 Early Career Technical Journal - Vol. 11


2/32



3/32

ASME Early Career Technical Jour2012 ASME Early Career Technical Conference, ASME EC

November 2 3, Atlanta, Georgia U

MIXING TIME DETERMINATION OF STEADY AND PULSE JET MIXERS

Ibraheem R. Muhammad and John P. KizitoDepartment of Mechanical Engineering

North Carolina A&T State UniversityGreensboro, NC, USA

ABSTRACTThe present study focuses on the performance of continuous

and pulsing jet mixers by experimental and computational fluid

dynamics (CFD). Pulse jet mixers have not been studied

extensively and it is necessary to provide further insight intothe performance of pulse jet mixers compared to steady jet

mixers. A jet pulse is created by turning the inlet jet velocity onand off in a cyclic manner. The mixing time and flow patterns

of different configurations of jet mixers are studied for single

and multiple jet configurations. Experimentally, the flow

patterns and mixing time are studied using a dye tracer. Theconcentration at the outlet of the mixing time is measured using

a spectrophotometer. CFD methods are used to visualize the

flow patterns created in the tank as well. Results show that themixing time decreases as the jet Reynolds number increases

and increases the momentum flux entering the mixing tank.

Mixing time is affected by the orientation of the jets and the

ability of the jet to recirculate off the walls, which can also

eliminate low mixing zones. As a free jet turns into a wall jet,

mixing is diminished. The current results give some insight intothe potential for pulse jet mixers for mixing in various

processes.

NOMENCLATUREC Concentration (g/mL) of dye tracer at time, t

Co Initial concentration (g/mL) of dye tracerCf Final concentration (g/mL) of dye tracer

C Heat capacity (J/kgK)

D Tank diameter (m)

Dnozzle Jet nozzle diameter (m)

Dov Overflow port diameter (m)D ipe Pipe outside diameter (m)

DC Duty cycle (DC = tD/tC)g Gravity acceleration constant (m/s2)

H Tank height (m)Hfluid Liquid level height (m)

k Thermal conductivity (W/(mK)

L Characteristic length scale (m)m Mixing time parameter

P Pressure (Pa)

Pe Peclet number (Pe = LV/kC )

Q Jet volumetric flow rate (m3/s)

Rej Jet Reynolds number (Rej= VDnozzle/)

t Time (s)

tC Pulse cycle time (s)tD Jet discharge time (s)

T Temperature (K)

TM Mixing time (s)

TM* Dimensionless mixing time (TM

* = TM/(D2/(QV)1

V Jet velocity (m/s)

Greek symbolsmax Maximum wavelength (nm) Dynamic viscosity (Ns/m2) Kinematic viscosity (m2/s) Liquid density (kg/m3)

INTRODUCTIONJet mixers are common mixing devices used in nume

processes. They are used in liquid blending, solid suspensand gas/liquid contacting [1], chemical reactions [2, 3], stor

tank homogenization [4], controlling process parameters

reducing thermal stratification [5], and nuclear wprocessing [6]. Jet mixers operate by withdrawing fluid fthe mixing tank and supplying the fluid back to the t

through a nozzle at high velocities. As the jet is discharge

expands and the relative velocity between the jet and the b

fluid causes the bulk fluid to get entrained by the jet. Theyadvantageous compared to other mixing devices as they h

the ability to provide adequate mixing, high turbulence

shear rates, while operating with no moving parts and e

installation.Quantitative and qualitative measurement of mixin

important in mixing processes. The most common paramete

estimate the mixing performance of jet mixers is mixing ti

or blend time. Patwardhan and Gaikwad [3] has summarmixing time correlations that have been developed for pasmixing studies. Most of the correlations have been develo

using different measurement techniques and are only applic

to a limited range of parameters.Jet mixing has been studied experimentally [7-14] and

computational fluid dynamics (CFD) [15-20] by m

researchers. Mixing time is usually measured experimentall

monitoring a scalar quantity of some tracer (dye, electrol

hot water) as a function of time. The mixing time is measu



4/32

as the time it takes for the tracer to reach a certain amount of

homogeneity in the vessel, usually 90 or 95%.

In the past decade, several CFD studies on jet mixing have

been completed which focuses on the actual flow patterns

created in mixing vessels [17, 19, 21, 22]. All the studies havebeen run using different jet mixer configurations, but some

general results have been found. The jets create circulatorypatterns in the tank as they interact with the walls of the tanks.Different patterns have been noticed depending on the shape of

the tank, length of the jet, jet angle, and number of jets.

However, there have not been many studies that focus on

pulsing jet mixers.

Pulsing, or pulse, jet mixers are mainly used for nuclearwaste remediation in which pulsing is created using

compressed air to vent and discharge contents within a tank.

Pulsing jets have been found to increase entrainment close tothe jet due to increased vortices [23, 24]. Zhang and Johari [23]

studied accelerating jets and found that there was a decrease in

entrainment rate due to the acceleration. Anders et al. [25] used

large eddy simulations (LES) and Reynolds-averaged Navier-

Stokes (RANS) simulations to study the interaction between aninitial pulse followed by a subsequent pulse. Results of LESsimulations showed that there was a reduced strength of the

vortex head from the first pulse. The results of the RANS

simulations showed that the turbulent kinetic energy of thesecond pulse decreased due to the first pulse. Ranade [21]

studied alternating jet sequences and noticed that though there

is not an overall increase in mixing, initial dispersion of a tracer

is enhanced. Muhammad and Kizito [15] studied mixing time

for different number of jets for both continuous and pulsing jetmixers. It was shown that mixing times for pulsing jets and

continuous jets are similar. However, pulsing jets can provide

increased local vortices and be useful for mixing in certain

applications.The current study focuses on the mixing performance of jet

mixers using both experimental and CFD techniques. The

mixing time and flow patterns for continuous and pulsing,

single and multiple jet mixer configurations are experimentallystudied. A dye is used to calculate mixing time experimentally

and imaging techniques are used to observe flow patterns. Flow

patterns are further visualized using CFD. The results can be

used as tools for application to a variety of processes, includingliquid blending and solid suspension.

EXPERIMENTAL METHODSFigure 1 shows a schematic of the jet mixing system used

in the current study. The driving force for the jet is a centrifugalpump, which suctions fluid from a holding tank. Figure 2

displays the actual jet mixing system used for experimentalstudies. The jet mixing apparatus included a cylindrical,

polycarbonate tank (D = 0.305 m and H = 0.610 m) enclosed in

a rectangular tank, which was used to correct any optical

distortion. There was an overflow port (Dov = 19.05 mm)located at half of the tank height, which keeps the liquid level,

Hfluid, at about 0.305 m. The non-dimensional nozzle diameter,

Dnozzle/D, for the studies were 14.62 and the fill height aspect

ratio, Hfluid/D, was 1. The jet nozzles were made from cop

tubing (Dpipe = 12.7 mm) with 45 degree elbows with s

stream nozzles (Dnozzle = 4.32 mm) attached. The jet noz

were located 0.07625 m from the bottom of the tank. Fluid

supplied to the tank by a 0.3 hp centrifugal pump (McMasCarr). A solenoid valve was used to control the pulsing (on/

action of the jet.To create a pulsing jet, a solenoid valve was used. valve was able to cycle the fluid momentum through the

nozzle in an on/off manner. One complete cycle, known as

pulse cycle time (tC), is 5.5s long. The pulse consisted

discharge time (tD) of 5 s and an off time of 0.5 s. this spec

was chosen because it was previously shown to give the

results [15].

Mixing time was calculated using a blue dye tracer, wh

was injected at the bottom of the tank. The concentration of

was monitored at the outlet as a function of time usin

spectrophotometer (Milton Roy SPECTRONIC 20D). wavelength of the blue dye was not known before hand and

measured by plotting the absorbance as a function

wavelength. Figure 3 displays the absorbance as a function

wavelengths for a sample. The maximum wavelength (

Figure 1: Schematic of jet mixer system.

Figure 2: Actual jet mixing system used forexperimental studies.



5/32

was found to be 610 nm. Mixing time was based on a 95%

homogeneity criterion, or the time it takes for the concentration

in the tank to reach 95 % of the fully mixed concentration.

Mathematically, the mixing time can be expressed as

m =C C

C C

< 0.05 (1)

Figure 3: Maximum wavelength (max) determined byabsorbance vs. wavelength.

The dye was also used to monitor mixing behavior and

flow profiles within the mixing tank. Video and snapshots werecaptured using a Basler acA2040-180 km CMOS camera

attached to a PIXC1-E8 frame grabber housed in a computer,

all purchased from Epix, Inc.

COMPUTATIONAL METHODSThe basic equations which describe the flow of an

incompressible, Newtonian fluid with constant properties arethe conservation of mass and conservation of momentum. The

conservation of mass and momentum are expressed,

respectively, as

divV = 0 (2)

DV

Dt= g P + V (3)

Figure 4 shows the mixing tank model created forsimulations to mimic the actual mixing tank used in

experiments. ANSYS Fluent was used for all of the CFDsimulations. Single and dual jet arrangements are simulated in

the current study and the flow patterns of each are determined.

Figure 5 displays an example of a meshed grid with labeledboundary conditions. Tetrahedral element types were used for

meshing. Mesh intervals of 10 20 mm was used. The velocity

at the bottom wall was monitored for the different meshintervals. An interval of 15 mm was chosen because it was the

largest interval size in which resulted in a grid independent

solution.

The mixing tank was modeled as a tank with a free sur

where the liquid level was the same as the tank height. The

surface was modeled as a free shear surface. Outlet conditi

were set as an outflow. The no slip condition was set at the t

walls and the jet walls. The inlet velocity from the jet nowas varied to simulate the same jet Reynolds numbers (

used for experiments.

RESULTSFlow patterns created by the jet mixers were visuali

using dye tracers and CFD results. Figure 6 display

schematic of the flow patterns created by single and dual j

As the jet is discharged, part of it is recirculated off the bot

wall and creates a semi-rollover effect. Some of the jet tu

into a wall jet and where it travels up the tank to the surfacethen a portion of it rolls over and is recirculated through

tank. As the jets are directed downwards, the most prominlow mixing zones are at the top of the tank. Most of

momentum of the jet roll overs due to the side walls and th

is not adequate force to create substantial mixing near

surface.

Figure 7 displays the flow patterns of dye at capturedsnapshots at different times for a single jet directed away f

0

0.02

0.04

0.06

0.08

0.1

0.12

500 525 550 575 600 625

Absorbance

Wavelength (nm)

(a) (b)

Figure 4: Jet mixer models for (a) single jet and (b)dual jet arrangements.

Figure 5: Example of meshed geometry with labeleboundar conditions.



6/32

the tank outlet. The dye was injected slowly, so that the

momentum of the dye injection did not greatly affect the

results. The most prominent region of low mixing occurs in the

area behind the direction of jet discharge. As the fluid circulates

off the bottom and side walls and travels to the top of the tank,some of the fluid exits through the overflow port, which further

inhibits the region behind the jet to become well mixed. Someof the dye motion is due to diffusion, rather than solely,convection.

Figure 8 shows the pathlines found from CFD simulations.

The pathlines shows the jet circulating as it hits the bottom and

side walls and then travelling to the outlet. Although initiallythe middle of the tank is not well mixed, after some time, the

middle of the tank starts to get mixed more due to the

circulatory patterns created in the tank. The mixing in the

center of the tank increases as the flow deviates from beinguniform to more chaotic. The model however does not do a

very good job of capturing the effect of the wall jet. This effectcan most like be improved by varying the turbulence model

and/or the turbulent parameters used in simulations.

Figure 9 displays the pathlines for the dual jet mixers. Asthe jet is injected, it hits the bottom wall and travels towards the

surface. Most of the flow travels towards the outlet, but some

of it recirculates near the middle of the tank. The low mixing

zone is located between the two jets at the bottom of the tank.

This was noticed during dye studies as well, though the

snapshots are not shown. Similar to the single jet, after aperiod of time, increased mixing occurs in the middle of the

tank as the flow becomes more chaotic and the interactionbetween the circulatory patterns increases.

Figure 6: Schematic of flow pattern created by (a)

single jet and (b) dual jets.

a b

a b

c d

Figure 7: Snapshots of dye mixing in the jet system adifferent times.

(a) (b)

(c) (d)

Figure 8: Pathlines of single jet mixer.



7/32

MIXING TIME RESULTS

The mixing time was measured using experimental

methods of turbulent, submerged, jet mixer nozzles. Studies

were run for single and dual jet nozzles inclined at 45 from the

horizon, directed towards the bottom of the tank. Studies wererun for Rej ranging from about 6,000 to 22,000. The

experimental results are somewhat limited due to the number of

sampling locations. Samples of the bulk fluid were taken at thetank outlet. This was done as the jets were directed towards the

bottom of the tank, where the initial and most rapid mixing

would occur. The outlet region is one of the low mixing zones

found for the jet mixers in the configurations used in the current

study, as shown by Figures 7-9, but this may not be the most

accurate location.Figure 10 displays an example of dimensionless

concentration as a function of time for a single jet and dual jet.

The mixing time was considered as the time in which thedimensionless concentration deviated less than 5% of the final

concentration difference. The peak height represents the

distribution of the dye in the tank to the outlet [3]. The dye wasdispersed to the outlet for the dual jets more rapidly than in the

single jet configuration.Figure 11 displays mixing time as a function of jet

Reynolds number for the steady and pulse single jet. As Rejincreases, the normal mixing time decreases. This is expected

as increasing Rej, can be due to an increase of jet momentum

force into the mixing time, allowing for quicker circulation of

fluid throughout the tank.

Figure 10: Comparison of dimensionless concentration

a function of time for single and dual jets.

There is very little difference between the values of mix

time for the steady jet and the pulse jet, though the pulse

mixing time is slightly lower. At Rej= 19160, the mixing t

of the steady jet was about 9% higher than that of the pulse

At Rej= 5625, the mixing time was about 12% higher for

steady jet compared to the pulse jet. Muhammad and Ki

[15] reported that the mixing time was not lower for the p

jet compared to the steady jet, but a much lower homogencriteria was used for mixing time which could account for

for the discrepancy.

Figure 11: Mixing time as a function of jet Reynoldsnumber single jet configuration.

DUAL JET MIXING TIME

Figure 12 shows mixing time as a function of jet Reyno

number for steady and pulse dual jets. Similarly to the sin

jet, the mixing time for the steady and pulse jet does not v

0.0

0.2

0.4

0.6

0.8

1.01.2

1.4

1.6

1.8

2.0

0 50 100 150 2

DimensionlessConcentration

TM (s)

Single Jet Dual Jets

100

1000

1000 10000 1000

MixingTime(s)

Rej

Steady Jet Pulse Jet

(a) (b)

(c) (d)

Figure 9: Pathlines of dual jet mixers.



8/32

much. At a combined Rej of about 20000, the mixing time of

the steady dual jets was about 9% higher than the pulse jets. At

a combined Rej of about 7000, the steady dual jets had about a

3% higher mixing time than the pulse jet.

Figure 12: Mixing time as a function of jet Reynoldsnumber for dual jet configurations

Figure 13: Mixing time comparison of single and dual

steady jets.

Figure 13 compares the mixing time for steady single jet

and dual jets. The mixing time is significantly reduced with the

addition of another jet. At Rej of 16030, the mixing time was

reduced by about 47% with the addition of another jet. Thiscoincides with results of previous studies, as it was found that

by doubling the number of jets, the mixing time was decreased

by half [15]. Besides adding more momentum in the tank andthe additional jet helps eliminates the low mixing zones.

Elimination of the low mixing zones is one of the mimportant factors for enhancing mixing performance in

mixed tanks.

CONCLUSION

The flow patterns of single and dual jet mixers w

studied using experimental and computational methods. major low mixing zone for the single jet mixer is located byjet, opposite of the injection location. The most prominent

mixing zone for the dual jets is between the jets at the bot

of the tank, as the jets are directed outward from the center.

The patterns in the tank are greatly influenced by thelocation in reference to the tank walls. The walls and surf

help create circulation patterns which enhance mixing and

wall jets, which are not ideal for mixing. The CFD f

patterns results were able to model the actual patterns createthe tank using dye fairly well. Further improvements can

made to the model to enhance the effect of the wall jet.

The mixing time was experimentally studied by injecti

dye tracer. The dye concentration was monitored at the ouThe mixing time decreased as the jet Reynolds num

increased, due to an increase in velocity. The mixing time

decreased for the dual jet mixers compared to the single jet

to an increase in momentum flux and elimination of low mixzones. Decreasing or eliminating the low mixing zone

possibly the biggest factor influencing the mixing t

Optimization of the jet mixing tanks (i.e. jet location, hei

etc.) can lead to a further reduction in mixing time and sho

be explored in further studies.

ACKNOWLEDGMENTS

The authors would like to acknowledge the finan

support of the Title III Program at North Carolina A&T S

University, which is administered by the U.S. DepartmenEducation, Institutional Development and UndergradEducation Services.

REFERENCES

[1] Bathija, P.R., Jet mixing design and applications. Chemic

Engineering 1982: p. 89-94.

[2] Simon, M. and C. Fonade, Experimental study of mixing

performances using steady and unsteady jets. The CanadianJournal of Chemical Engineering, 1993. 71(4): p. 507-513.

[3] Patwardhan, A.W. and S.G. Gaikwad, Mixing in Tanks

Agitated by Jets. Chemical Engineering Research and Desig

2003. 81(2): p. 211-220.

[4] Rahimi, M. and A. Parvareh, Experimental and CFDinvestigation on mixing by a jet in a semi-industrial stirred

tank. Chemical Engineering Journal, 2005. 115(1-2): p. 85-9

[5] Breisacher, K. and J. Moder, Computational Fluid

Dynamics (CFD) Simulations of Jet Mixing in Tanks ofDifferent Scales. 2010 NASA: Cleveland, OH.

[6] Powell, M.R., Onishi, Y., and Shekarriz, R., Research on

mixing of settled sludges in nuclear waste tanks at Hanford other DOE sites: A historical perspective. 1997, Pacific

Northwest Laboratory: Richland, Washington. p. 90.

10

100

1000

1000 10000 100000

MixingTime(s)

Rej

Continuous Jet Pulse Jet

10

100

1000

1000 10000 100000

MixingTim

e(s)

Rej

Single Jet Dual Jets



9/32

[7] Zughbi, H.D. and I. Ahmad, Mixing in Liquid-Jet-Agitated

Tanks: Effects of Jet Asymmetry. Industrial & Engineering

Chemistry Research, 2005. 44(4): p. 1052-1066.

[8] Lane, A.G.C. and P. Rice, Comparative assessment of the

performance of the three designs for liquid jet mixing.Industrial & Engineering Chemistry Process Design and

Development, 1982.21

(4): p. 650-653.[9] Grenville, R.K. and J.N. Tilton, Turbulent Flow or Flow as aPredictor of Blend Time in Turbulent Jet Mixed Vessels.

Proceedings of 9th European Conference on Mixing, 1997: p.

67 - 74.

[10] Fox, E.A. and V.E. Gex, SINGLE-PHASE BLENDING

OF LIQUIDS. AIChE Journal, 1956. 2(4): p. 539-544.[11] Grenville, R.K. and J.N. Tilton, A New Theory Improves

the Correlation of Blend Time Data from Turbulent Jet Mixed

Vessels. Chemical Engineering Research & Design, 1996.74(3): p. 390-396.

[12] Fossett, H. and L.E. Prosser, The Application of Free Jets

to the Mixing of Fluids in Bulk. Proceedings of the Institution

of Mechanical Engineers, 1949. 160: p. 224-232.

[13] Maruyama, T., Y. Ban, and T. Mizushina, Jet Mixing ofFluids in Tanks. Journal of Chemical Engineering of Japan,1982. 15(5): p. 342-348.

[14] Tatterson, G.B., Fluid mixing and gas dispersion in

agitated tanks. 1991: McGraw-Hill.[15] Muhammad, I.R. and J.P. Kizito, Evaluation of Pulse Jet

Mixing Using a Scalar Quantity and Shear Stress. ASME Early

Career Technical Journal, 2011. 10: p. 45-51.

[16] Tian, X. and P.J.W. Roberts, Mixing in Water Storage

Tanks. I: No Buoyancy Effects. Journal of EnvironmentalEngineering, 2008. 134(12): p. 974-985.

[17] Furman, L. and Z. Stegowski, CFD models of jet mixing

and their validation by tracer experiments. Chemical

Engineering and Processing: Process Intensification, 2011.50(3): p. 300-304.

[18] Zughbi, H.D., Numerical simulation of mixing in a jet

agitated horizontal cylindrical tank. International Journal of

Computational Fluid Dynamics, 2006. 20(2): p. 127 - 136.[19] Jayanti, S., Hydrodynamics of jet mixing in vessels.

Chemical Engineering Science, 2001. 56(1): p. 193-210.

[20] Zughbi, H.D. and M.A. Rakib, Investigations of mixing in

a fluid Jet agitated tank. Chemical EngineeringCommunications, 2002. 189(8): p. 1038-1056.

[21] Ranade, V.V., Towards better mixing protocols by

designing spatially periodic flows: The case of a jet mixer.

Chemical Engineering Science, 1996. 51(11): p. 2637-2642.

[22] Parvareh, A., et al., Experimental and CFD study on theeffect of jet position on reactant dispersion performance.

International Communications in Heat and Mass Transfer,

2009. 36(10): p. 1096-1102.

[23] Zhang, Q. and H. Johari, Effects of acceleration onturbulent jets. Physics of Fluids, 1996. 8(8): p. 2185-2195.

[24] Crow, S.C. and F.H. Champagne, Orderly structure in jet

turbulence. Journal of Fluid Mechanics, 1971. 48(03): p. 547-591.

[25] Anders, J.W., V. Magi, and J. Abraham, A Computational

Investigation of the Interaction of Pulses in Two-Pulse Jets.

Numerical Heat Transfer: Part A -- Applications, 2008. 54(1p. 999-1021.



10/32

ASME Early Career Technical Jour

2012 ASME Early Career Technical Conference, ASME EC


COMPARATIVE EFFECTS OF FORCES ACTING ON SWIRLING ANNULAR LIQUID

SHEETS

Mohammed Ali

Department of TechnologyJackson State University

Jackson, Mississippi, USAPhone: (601) 979-0327

Fax: (601) 979-4110Email: [email protected]

Essam A. Ibrahim

Department of Mechanical EngineeringThe University of Texas of the Permian Basin

Odessa, Texas, USAPhone: (432) 552-3217

Fax: (432) 552-2433Email: [email protected]

ABSTRACTThe respective effects of the multiple forces that control the

development of swirling liquid sheets injected from an annular

nozzle into quiescent surrounding medium are studied. These

forces include inertia, viscous, gravity, pressure, surface tension,

centrifugal and Coriolis forces. In order to simplify the

mathematical formulation of the inherently complex transient,

three-dimensional problem considered, a body-fitted coordinate

system is employed. Use of these coordinates enables the

transformation of the system partial differential equations,

consisting of mass and momentum conservation equations with

appropriate boundary conditions, into ordinary differential

equations. These equations are then solved numerically to yield

sheet trajectory, thickness, and velocity, for a given set of massflow rate and liquid-swirler angles. By eliminating any of the

acting forces from the governing equations, one at a time, the

individual influence of each force on sheet evolution

characteristics is isolated and evaluated. It is found that

centrifugal and Coriolis forces play significant roles in

determining the resulting configuration and flow velocities of a

developing swirling annular liquid sheet. Whereas centrifugal

forces act to increase the developing sheet radius and angle,

Coriolis force has opposite effects. The sheet thickness variation

is independent of Coriolis force, but sheet thickness increases

significantly if the centrifugal force is not taken into account.

Neglecting either of the centrifugal or Coriolis forces causes the

sheet stream-wise velocity to decrease. In the absence ofCoriolis force, the sheet swirl velocity remains constant at its

initial value while the centrifugal force has the tendency to

diminish swirl velocity. For the range of parameters investigated,

gravitational acceleration, surface tension, and interfacial friction

forces exhibit minimal impact on the formation of a swirling

liquid sheet. The present assessment of the influence of various

forces on the injected sheet behavior may be applied to guide

efficient design of swirl injectors.

INTRODUCTIONTo date, most of the transportation, industrial, and po

combustion applications of fuel atomizers use either press

pressure-swirl, or air-blast atomizers [1, 2]. The formatio

thin sheets and the conical nature of the liquid surface emerg

from swirl atomizers ensure more efficient breakup of the li

into droplets owing to the larger surface energy of the ho

cone. Therefore, for a given liquid supply pressure, the qu

of atomization from a swirl atomizer is superior to

produced by a conventional pressure atomizer. Enhan

atomization leads to a more intimate fuel-air mixing, fa

evaporation, and hence higher combustion efficiency, w

results in reduced fuel consumption and pollutant emiss

Despite its practical significance, the fundamental mechaniof liquid fuel sheet injection and atomization are not w

understood. In particular, analytical/computational models

accurately predict injection parameters and spray characteris

such as cone radius and angle, sheet breakup length, drop s

and velocity are lacking.

A number of research articles have been published on

theoretical and experimental aspects of an annular s

emanating from a nozzle, relevant to different prac

applications, though mostly not for injectors. Water bells h

been considered by Taylor [3] and Baird and Davidson

Research on converging non-swirling annular sheets w

reference to Inertial Confinement Fusion (ICF) reactors

been performed by Hoffman et al. [5], Ramos [6, 7], and Haet al. [8]. Sivakumar and Rghunandan [9, 10] stu

converging swirling annular liquid sheets produced by liq

liquid coaxial swirl atomizers used in bipropellant rockets

elsewhere. The transition of a converging (bell or tulip-shap

to a diverging (cone-shaped) swirling annular sheet

investigated, both experimentally and theoretically,

Ramamurthi and Tharakan [11, 12].



11/32

Mao et al. [13], Chuech [14, 15], and Przekwas [16]

advanced an analytical/computational model to study the

evolution of non-swirling and swirling annular liquid sheets.

Their technique is based on the solution of the continuity and

momentum equations in a curvilinear co-ordinate system

conforming to the sheet boundaries. They reported predictions

of the spray angle, film thickness and spray velocities that werein general agreement with their experimental measurements.

The modeling effort of Chuech [14, 15] was extended by

Ibrahim and McKinney [17] who presented a clarified version of

Chuechs model that incorporated interfacial friction effects.

Ibrahim and McKinneys model [17] accounted for most of the

various forces that control the progress of a swirling annular

liquid sheet emanating from a nozzle: inertial, viscous,

gravitational, surface tension, centrifugal and Coriolis forces.

The results of this model provided useful information on the

liquid sheet trajectory, thickness, and flow velocity for given

nozzle configurations and mass flow rates.

The present work is aimed at analyzing the comparative

effects each of the multiple acting forces has on the developmentof a swirling annular liquid sheet issued from an injector nozzle.

The present analysis makes use of the formulation of Ibrahim

and McKinney [17] in performing the necessary computations.

Such an analysis has not been attempted before, although its

results have the potential of improving our understanding of

annular liquid sheet formation. Evaluating the relevant role each

of these many forces plays in the evolution of annular liquid

sheets is essential to advancing accurate models of fuel injection

and atomization processes. The evolution of the liquid sheet

emanating from the injector nozzle predetermines its subsequent

atomization characteristics, hence fuel-air mixing, evaporation,

and ultimately, combustion efficiency and stability. Therefore, it

is important to improve our understanding of the forces thatinfluence annular liquid sheet formation in order to be able to

optimize fuel injector design for more complete combustion.

MODEL FORMULATIONAs alluded to earlier, Ibrahim and McKinneys [17] model

is adopted in the present effort. The basic features of the model

are summarized here for easy reference. Ibrahim and McKinney

[17] have shown that an annular liquid sheet injected into a

quiescent gaseous environment assumes a bell-shape in the

absence of swirl but takes the form of a diverging hollow-cone

due to swirl. Only the case of a swirling annular liquid sheet is

studied in the present work, owing to its relevance to fuel

injection applications.A curvilinear coordinate system --, as shown in Fig. 1,

is utilized as a non-inertial reference frame to analyze the liquid

flow in a swirling axi-symmetric hollow-cone sheet emerging

from a nozzle at an initial stream-wise velocity uf0 , tangential

velocity wf0, and cone angle 0 in a surrounding gas. The co-

ordinates , , are perpendicular to each other and coincide

with the liquid stream-wise, tangential, and normal to the

streamline directions, respectively. The choice of a curvilinear

coordinate system that conforms to the sheet bounda

simplifies the mathematical analysis because only the stre

wise and tangential velocity components, ufand wf, respectiv

survive while the normal velocity component, vf, vanishes.

liquid flow is assumed to be Newtonian, incompressible

inviscid, in the sense that viscous stresses are negligible rela

to liquid-gas interfacial friction. Since in practical spapplications, the sheet thickness is usually much smaller than

cone radius, variations of stream-wise velocity across the s

thickness may be neglected. Mathematically, the govern

equations describing conservation of mass and momentum

unit volume at steady state may be expressed as:

Continuity:

0)( =

ruff

(1)

Momentum in the stream-wise -direction:

cossin

gSr

wuu

uffff

f

ff

+=

(2)

Momentum in the normal -direction:

sin

pcos f gr

wwuu fffffff

=

(3)

Momentum in the tangential -direction:

S

rwu

wu fff

f

ff =+

sin (4)

where r and z are the cylindrical radial and axial coordina

respectively, f is the liquid density, is the local s

thickness, and is the cone/spray angle, defined as the an

between the nozzle axis and the tangent line at

corresponding spray edge location. The first term in each

Eqs. (2), (3), and (4) represents the directional component

the inertia forces in the stream-wise, normal, and tange

directions. The second term in each of Eqs. (2) and (4) den

ds

ds

r

fu ds

uu ff

dsr

r

z

r

Figure 1. Schematic diagram of an annular liquid sh



12/32

the directional components of the Coriolis force. The second

term in Eq. (3) relates to centrifugal force. The terms f g cos

and f g sin in Eqs. (2) and (3) designate the directional

components of the gravity force. The terms Sand Sin Eqs. (2),

(3) account for the liquid-gas interfacial friction forces in the

stream- wise and tangential directions, respectively. The use of

primitive variables in the present formulation is intended to addto the clarity of the models equations.

The pressure gradient in the normal direction can be

approximated by its integrated form as a function of the gas

pressure difference across the liquid gas interface and surface

tension forces:

2 cosf f gp p p

r

=

(5)

where p is pressure and is surface tension. Subscripts f and g

denote liquid and gas quantities, respectively. The terms inside

the bracket represent the axial and meridian radii of curvature,

respectively [3].

Following Chuech [14, 15], the viscous forces in the stream-wise and tangential momentum equations are accounted for

through the interfacial friction forces acting on the inner and

outer liquid-gas interfaces. Therefore, the viscous forces may be

written, respectively, in terms of Rizk and Lefebvres [18] gas-

liquid interfacial friction factors representation as:

1/ 40.79(1 150 / )(Re ) ( )

2

g

g f g f S r u u u u

= + (6)

1/ 40.79(1 150 / )(Re ) ( )

2

g

g f g f S r w w w w

= + (7)

where Re and Re are Reynolds numbers based on the orifice

diameter, gas properties, and the absolute value of the difference

between the gas and liquid velocities in the stream wise andtangential directions, respectively. The terms between square

brackets in Eqs. (6) and (7) designate Rizk and Lefebvres [18]

interfacial friction factors.

For pressure swirl injectors, the gas velocities in Eqs. (6) and

(7) vanish, since there is no gas flow involved. Air flow in air-

blast/air-assist injectors maybe assumed to be at a constant

velocity through the entire domain, depending on the air

pressure drop across the nozzle [13].

Due to the simplifying assumptions and the use of

conforming curvilinear coordinates in the present model, all the

dependent variables have gradients only in the stream-wise

direction, . Therefore, the governing equations (1), (2), (3), (4)

subject to Eqs. (5), (6), (7) may be simplified to a system ofnonlinear first-order ordinary differential equations in the form

0=++

d

dru

d

dru

d

dur ff

f (8)

(9)

cossin

gSr

wuu

u fffff

ff +=

(9)

sin)

cos(

2pcos gg

d

d

rrwwuu fffffff +

=

(10)

S

rwu

wu fff

f

ff =+

sin (11)

Since the system of Eqs. (1), (2), (3), (4) subject to Eqs.

(6) and (7) include four equations and five unknowns, r,,

wf , an additional equation is needed to make the sys

determinate. Such an equation may be derived from geometr

considerations of the median streamline as shown in Fig. 1.

sin=d

dr (12)

A set of five boundary conditions are needed to bring clo

to the model. Since the hollow-cone liquid flow is bounde

the nozzle orifice, the boundary conditions to be coupled w

the system of differential equations may be stated as

2/000 DRr === (13)

00

==

(14)

00

==

(15)

00

00 D

muu

f

f

ff === (16)

tan000 fff

uww ===

(17)

where subscript 0 denotes initial quantities, R0 and D0 are

respective initial sheet radius and diameter, and fm is the kn

fuel mass flow rate, and is the fuel port/swirler angleenable tracking of sheet trajectory, its axial coordinate

evaluated in reference with Fig. 1 as

cos=d

dz (18)

subject to the boundary condition,

00=

=z (19)

RESULTS AND DISCUSSIONFor the present computations the flow conditions are tato be similar to those used by Ibrahim and McKinney [17

permit direct comparison to their results. Therefore, the ann

sheet diameter at the nozzle orifice is D0= 6.63mm. The in

sheet thickness is taken to be, 0 = 0.1524 mm, whic

equivalent to 1/4 of the pre-filmer width and is taken to

invariable with other flow conditions. The water shee

assumed to be injected vertically downward so that initial s

cone angle, 0= 0. To study the effects of liquid-port angle



13/32

liquid-swirler angle, , and liquid mass flow rate, m f, on theircalculations, Ibrahim and McKinney [17] varied the port angle

between 0 and 60 and considered mass flow rates of 17.76,40.82, and 79.13g/s. Since the present study is concerned with

examining the contribution of various acting forces to liquid

sheet behavior at the same flow conditions, a representative port

angle of = 30 and mass flow rate of mf = 17.76 g/s are

selected for the numerical simulations. The liquid fuel properties

are density, f = 765 kg/m3, dynamic viscosity, f = 9.2x10

-4

kg/m.s, and surface tension, = 0.025 N/m. The surrounding

gas is assumed to be air at atmospheric conditions with density

of g = 1.22 kg/m3 and dynamic viscosity of g = 17.9x10

-6

kg/m.s. The initial sheet velocities in the stream-wise and

tangential directions, uf0 and wf0, are calculated from nozzle

geometric parameters, liquid properties, and mass flow rate.

The system of nonlinear first-order ordinary differential

equations given by (8), (9), (10), (11), (12), and (18), subject to

the boundary conditions expressed as (13), (14), (15), (16),

(17), and (19) is solved using a fifth order Runge-Kutta Verner

method to yield solutions for r, , z, , uf, wf. In the present

model, since the surrounding gas is assumed to be quiescent,

quantities representing gas velocity, ugand wg, vanish from Eqs.

(6) and (7), following Chuech and co-authors [13-16] and

Ibrahim and McKinney [17]. It is also assumed that the outer

and inner gas pressures are equal, so that, pg = 0. Note that

results for a non-swirling annular sheet may be obtained when

the liquid swirl velocity, wf, is set to zero.

The effects of any one of the acting forces on the

configuration and velocities of a developing swirling liquid sheet

may be extracted by setting terms corresponding to a particular

force to zero in the governing equations. Therefore, the gravity

force contribution is cancelled by substituting g= 0, and surfacetension is disregarded by using = 0, and interfacial friction is

eliminated by dropping corresponding terms for Sand Sfrom

Eqs. (9) and (11), respectively. The centrifugal force is ignored

by deleting the second term in Eq. (10), and the Coriolis force is

not considered if the second term in each of Eqs. (9) and (11)

are removed. As will be expounded later, it turns out that

neglecting either of gravitational acceleration, surface tension,

or gas-liquid interfacial friction forces, produces only slight

modification of the swirling annular sheet evolution attributes.

Therefore, the effects of centrifugal and Coriolis forces are

discussed first.

Figures 2 and 3 portray the respective variations of the

dimensionless sheet radius and cone angle with dimensionlessaxial distance measured from the nozzle exit. Numerical

solutions are presented for three cases: (1) all forces in the

governing equations are taken into account, (2) centrifugal force

is neglected; and (3) Coriolis force is neglected. It can be seen

from Figs. 2 and 3 that excluding the centrifugal force from the

governing equations would lead to the production of an annular

sheet with a radius and angle that slightly increase in the axial

direction, but are significantly below their corresponding values

if the centrifugal force is accounted for. This may be explained

by the fact that the centrifugal force acts to deform the she

the direction perpendicular to the axis of rotation. Theref

sheet expansion is substantially hindered by the absence of

centrifugal force. Hence, if a specific application necessitat

wider hollow-cone sheet, the centrifugal force could

increased to meet that requirement, for example, by increa

the swirler angle in accordance with Eq. (17).

Figs. 2 and 3 also indicate that, unlike the centrifugal fo

excluding Coriolis force promotes the growth of the sheet ra

and angle in the axial direction, with magnitudes that are m

greater than when these forces are accounted for. This trendconsequence of the reduction in the sheet stream-wise velo

that is experienced when the inertial action of the Coriolis f

is disregarded, as will be explicated later. The sheet radius

angle gradually increase in the axial direction, signifying

formation of a diverging hollow-cone sheet, in accordance w

the observations of Ibrahim and McKinney [17].

Dimensionless axial distance, z/D0

Dimensionlesssheetradius,r/D0

Figure 2. Effects of neglecting centrifugal or Coriolforce on axial variation of dimensionless sheet

0

2

4

6

8

0 1 2 3 4

All forc es

No centrifugal

No Coriolis

Figure 3. Effects of neglecting centrifugal or Corioliforce on axial variation of sheet angle


Sheetangle,

-10

10

30

50

70

90

0 2 4 6 8 10 12

Al l forces

No centrifugal

No Corioli s



14/32

Figure 4 displays the dimensionless sheet thickness variation

versus dimensionless axial distance along the nozzle axis. It is

noted in Fig. 4 that disregarding the centrifugal force favors a

larger annular sheet thickness. This is expected since the sheet

radius is smaller than that yielded from including the centrifugal

force in the analysis, as revealed earlier in Fig. 2. Hence, to

satisfy conservation of mass, the sheet thickness must increase

as its radius is reduced. Interestingly, Figure 4 demonstrates thatneglecting Coriolis force doesnt generate much change in the

sheet thickness, apart from that when it is included. With and

without Coriolis force in place, the sheet thickness tends to

decrease in the axial direction in a similar fashion. This is may be

surprising given that the sheet radius is much larger for the case

of negligible Coriolis force compared to when all modeled

forces are represented in the analysis, as the results in Fig. 2

confirm. However, the sharp rise in sheet radius, which occurs

in the absence of Coriolis force, is compensated for by a s

drop in the stream-wise velocity (as will be discussed la

conserving mass in a manner that preserves sheet thick

variation and that mirrors the one dictated by the comp

governing equations.

The variation of the non-dimensionalized stream-w

velocity variation with the dimensionless axial distancedepicted in Fig. 5. It is evident from Fig. 5 that, when Cori

force is absent, the stream-wise velocity decreases rapidly in

axial direction and is considerably lower than that produ

when Coriolis force is present. This behavior is due to

reduction in sheet inertia associated with subtracting the Cor

force. The centrifugal force deduction has a somewhat sim

but much smaller effect. Solving the full governing equat

results in a stream-wise velocity that is greater than wha

observed if Coriolis or centrifugal forces are not included

this case, the stream-wise velocity exhibits an initial incre

followed by a gradual decrease in the axial direction. The in

increase in the stream-wise velocity corresponds to the in

rapid drop in sheet thickness remarked in Fig. 4. As the sthickness reduction levels off, the stream-wise velocity start

decrease because of the increase in sheet radius in the a

direction, as noted in Fig. 2, becomes the dominant fac

Therefore, the behavior of the stream-wise velocity is consis

with mass conservation.

Figure 6 illustrates the variation of dimensionless tange

velocity against the dimensionless axial distance in the upstr

direction. It is clear from Fig. 6 that eliminating ei

centrifugal or Coriolis forces leads to an increase in s

tangential velocity. This observation is the opposite of what

been mentioned for the stream-wise velocity in relation with

5. It is therefore deduced that inertial effects accompan

centrifugal and Coriolis forces act to enhance sheet stream-wvelocity at the expense of tangential velocity. It is worth no

that the sheet tangential velocity remains constant at its in

magnitude if Coriolis force is nonexistent, as delineated in

6. Therefore, it is envisaged that modulations of sheet

Figure 4. Effects of neglecting centrifugal or Coriolis forceon axial variation of dimensionless sheet thickness

Figure 5. Effects of neglecting centrifugal or Coriolisforce on axial variation of dimensionless sheet

stream-wise velocity

Dimensionlesssheetstream-wise

velocity,uf/uf0


0

0.4

0.8

1.2

1.6

0 2 4 6 8 10 12

Al l forces

No centrifugal

No Coriolis

Figure 6. Effects of neglecting centrifugal or Corioliforce on axial variation of dimensionless sheet

tangential velocity


Dimensionlesssheettangential

ve

locity,wf/wf0

0

0.3

0.6

0.9

1.2

1.5

1.8

0 2 4 6 8 10 12

Al l for ces

No centrifugal

No Corioli s


0

0.4

0.8

1.2

0 2 4 6 8 10 12

Al l forces

No centrifugal

No Coriol is

Dimensionlessshee

tthickness,

/0



15/32

tangential velocity are imparted solely by Coriolis force. As

noticed in Fig. 6, solutions of the full governing equations

disclose a monotonic decrease in sheet tangential velocity in the

axial direction. The reduction in tangential velocity is related to

the increase in sheet radius and angle reported in Figs. 2 and 3.

Table 1. Numerical simulation results consideringall modeled forces

z/D0 r/D0 /0 uf/uf0 wf/wf0

0 0.5 0 1 1 1

0.7344 0.6533 19.9512 0.6839 1.1191 0.7654

1.4281 0.9484 24.9898 0.4243 1.2426 0.5272

2.1069 1.277 26.4215 0.2993 1.3081 0.3915

2.7805 1.6162 26.9554 0.2304 1.3425 0.3094

3.4519 1.9595 27.1805 0.1876 1.3603 0.2552

4.1225 2.3047 27.2728 0.1586 1.368 0.2174.7927 2.6504 27.2988 0.1378 1.3691 0.1886

5.4629 2.9963 27.2876 0.1222 1.3655 0.1669

6.1332 3.3418 27.2539 0.1101 1.3585 0.1496

6.8038 3.6869 27.2053 0.1005 1.3488 0.1356

7.4746 4.0313 27.1464 0.0928 1.3369 0.124

8.1459 4.3751 27.0799 0.0864 1.3233 0.1143

8.8176 4.718 27.0075 0.081 1.3083 0.106

9.4897 5.06 26.9303 0.0765 1.2921 0.0988

10.1623 5.4011 26.8491 0.0726 1.2749 0.0926

10.8354 5.7412 26.7643 0.0693 1.2569 0.0871

11.5091 6.0803 26.6763 0.0664 1.2382 0.0822

Tables 1-4 document numerical solutions of the governing

equations for four cases: (1) all modeled forces are considered;

(2) gravity force is neglected; (3) surface tension force is

neglected; and (4) interfacial viscous forces are neglected,

respectively. As can be seen in Tables 1-4, only minute

differences exist between the solutions for these four scenarios.

So, these results are mainly presented here for completeness.

However, some minor differences in some of these results

warrant comment. For example, comparison of the results in

Tables 1 and 3 point to a slightly larger sheet radius and coneangle, or curvature, paralleled by a little reduction in sheet

thickness and velocities, to conserve mass, in the absence of the

contracting action of surface tension force. In addition,

contrasting the results of Tables 1 and 4 exposes that, to a small

extent, sheet velocities are increased while sheet thickness,

radius, and angle are moderated, due to the lack of the

dissipating, i.e. decelerating, effects of interfacial friction force.

Hence, it is concluded that, for the range of parameters

scrutinized, the forces of gravity, surface tension, and interfacial

friction wield only minimal deviations in sheet evolu

characteristics.

Table 2. Numerical simulation results neglecting gravforce


0 0.5 0 1 1 1

0.7344 0.6533 19.9549 0.684 1.1189 0.7653

1.4281 0.9485 24.9969 0.4243 1.2424 0.5272

2.1068 1.2772 26.4314 0.2994 1.3077 0.3915

2.7803 1.6165 26.9679 0.2305 1.342 0.3093

3.4517 1.96 27.1954 0.1876 1.3597 0.2551

4.1222 2.3053 27.2902 0.1586 1.3673 0.2169

4.7923 2.6513 27.3186 0.1378 1.3683 0.1886

5.4623 2.9974 27.3098 0.1222 1.3646 0.1668

6.1325 3.3432 27.2785 0.1102 1.3575 0.1496

6.8029 3.6886 27.2323 0.1006 1.3477 0.13567.4736 4.0334 27.1759 0.0928 1.3357 0.124

8.1447 4.3775 27.1119 0.0864 1.322 0.1142

8.8162 4.7207 27.0421 0.081 1.3069 0.1059

9.4881 5.0632 26.9676 0.0765 1.2906 0.0988

10.1605 5.4047 26.889 0.0727 1.2734 0.0925

10.8333 5.7453 26.807 0.0693 1.2553 0.087

11.5067 6.0849 26.7218 0.0665 1.2365 0.0822

CONCLUSIONSThe present work sheds a light on the contribution eac

the various forces acting on a swirling annular liquid s

makes to its evolution characteristics. The use of a body-fitte

Table 3. Numerical simulation results neglectingsurface tension force


0 0.5 0 1 1 1

0.7343 0.6538 20.0223 0.6832 1.1194 0.764

1.4275 0.9501 25.1275 0.4233 1.2432 0.5262

2.1052 1.2808 26.6338 0.2983 1.3087 0.3904

2.7773 1.6229 27.2458 0.2294 1.3433 0.308

3.4468 1.9701 27.5506 0.1865 1.361 0.253

4.1148 2.3201 27.7239 0.1575 1.3687 0.2155

4.7821 2.6716 27.8318 0.1366 1.3698 0.1872

5.4487 3.0241 27.9035 0.121 1.3662 0.1653

6.115 3.3773 27.9535 0.1089 1.3591 0.148

6.7811 3.7311 27.9898 0.0993 1.3492 0.134



16/32


7.4469 4.0852 28.0168 0.0915 1.3373 0.1224

8.1126 4.4395 28.0374 0.0851 1.3235 0.1126

8.7782 4.7941 28.0534 0.0797 1.3083 0.1043

9.4437 5.1489 28.066 0.0752 1.2919 0.0971

10.1092 5.5038 28.0759 0.0713 1.2745 0.0908

10.7745 5.8587 28.0838 0.0679 1.2562 0.0853

11.4399 6.2138 28.0901 0.065 1.2373 0.0805

Table 4. Numerical simulation results neglectinginterfacial friction


0 0.5 0 1 1 1

0.7347 0.652 19.73 0.6758 1.1348 0.7668

1.43 0.9436 24.6499 0.4167 1.2717 0.5299

2.1109 1.2679 26.0353 0.2921 1.3501 0.39442.7868 1.6024 26.5454 0.2232 1.3977 0.312

3.4607 1.9408 26.7565 0.1803 1.4292 0.2576

4.1338 2.2809 26.8405 0.151 1.4515 0.2192

4.8066 2.6216 26.8617 0.1299 1.4681 0.1907

5.4794 2.9623 26.8486 0.114 1.4809 0.1688

6.1524 3.3027 26.8153 0.1015 1.491 0.1514

6.8256 3.6426 26.7691 0.0916 1.4993 0.1373

7.499 3.982 26.7146 0.0834 1.5061 0.1256

8.1729 4.3207 26.6543 0.0765 1.5119 0.1157

8.8471 4.6586 26.5899 0.0708 1.5169 0.1073

9.5216 4.9958 26.5225 0.0658 1.5211 0.1001

10.1966 5.3321 26.4529 0.0615 1.5249 0.0938

10.872 5.6677 26.3816 0.0577 1.5282 0.0882

11.5478 6.0023 26.309 0.0544 1.5311 0.0833

non-inertial reference frame, in which centrifugal and Coriolis

forces manifest themselves, enabled a deeper insight about their

important role in determining the outcome of the developing

swirling annular sheet profile and directional velocities. Thus,

these forces could be manipulated to induce desired outcomes.

The present results indicate that Coriolis force promotes

sheet stream-wise velocity while simultaneously diminishing

sheet radius, angle, and tangential velocity. The centrifugal forceacts to reduce sheet thickness and velocities while supporting a

more pronounced sheet radius and angle. Whereas it might be

obvious to most researchers that the centrifugal force is essential

to modeling the behavior of injected annular swirling liquid

sheets, the present study proves beyond doubt that Coriolis

force is also indispensable to ensuring a models physical

integrity. For the range of liquid properties and flow conditions

investigated, gravity, surface tension, and gas-liquid interfacial

viscous forces exhibit undetectable influence on the swir

annular sheet developmental features.

The plausibility of the predictions of the model employe

the present numerical simulations casts confidence on

models accuracy. Since in fuel injection applications the s

eventually disintegrates into drops, model predictions of s

trajectory, thickness, and velocities resolve resultant ligamand drop sizes, orientation and velocities. Therefore, this m

may be linked to a sheet breakup model to seamlessly tie

injector geometrical and operating conditions to the final up

of liquid atomization processes. Thus, the influence of

injectors design parameters on its functionality can be extra

and exploited in enhancing fuel injector design and he

combustion performance.

REFERENCES

[1] Lefebvre, A., H., 1989, Gas Turbine Combust

Hemisphere Publishing, New York.

[2] Bayvel, L., and Orzechowski, Z., 1993, Liquid Atomizat

Taylor and Francis, New York.[3] Taylor, G. I., 1959, The Dynamics of Thin Sheets of Fl

I. Water Bells, Proceedings of the Royal Society of Lon

Series A: Mathematical and Physical Sciences, 253(1274),

289-295.

[4] Baird, M. H. I., and Davidson, J. F., 1962, Annular Jet

Chemical Engineering Science, 17(10), pp. 467-472.

[5] Hoffman, M. A., Takahashi, R. K., and Monson, R.

1980, Annular Liquid Jet Experiments, ASME Journa

Fluids Engineering, 102(9), pp. 344-349.

[6] Ramos, J. I., Liquid Curtains: I. Fluid Mechanics, 19

Chemical Engineering Science, 43(12), pp. 3174-3184.

[7] Ramos, J. I., 1990, Analytical, Asymptotic and Numer

Studies of Liquid Curtains and Comparison with ExperimeData, Applied Mathematical Modeling, 14(4), pp. 170-183.

[8] Hasan, M. Z., Mitsutake, Y., Monde, M., 1997, Shape o

Annular Liquid Jet, ASME J. Fluids Engineering, 119(9),

591-596.

[9] Sivakumar, D., and Raghunadan, B. N., 1997, A Study

Converging Thin Annular Jets, ASME Journal of Fl

Engineering, 119(12), pp. 923-928.

[10] Sivakumar, D., and Raghunadan, B. N., 2002, Conver

Swirling Liquid Jets from Pressure Swirl Atomizers: Effec

Inner Air Pressure, Physics of Fluids, 14(12), pp. 4389-439

[11] Ramamurth, K., and Tharakan, T. J., 1995, Experime

Study of Liquid Sheets Formed in Coaxial Swirl Injecto

AIAA J. Propulsion and Power, 11(6), pp. 1103-1109.[12] Ramamurthi, K., and Tharakan, T. J., 1998, F

Transition in Swirled Liquid Sheets, AIAA Journal, 36(3),

420-427.

[13] Mao, C. P., Chuech, S. G., and Przekwas, A. J., 1991,

Analysis of Pressure Swirl and Pure Airblast, Atomizat

Atomization and Sprays, 1(2), pp. 215-235.

[14] Chuech, S. G., 1992, Numerical Simulation

Nonswirling and Swirling Annular Liquid Jets, 30thAerosp



17/32

Sciences Meeting, Reno, NV, 6-9 January 1992, AIAA Paper

92-0464.

[15] Chuech, S. G., 1993, Numerical Simulation of

Nonswirling and Swirling Annular Liquid Jets, AIAA Journal,

31(6), pp. 1022-1027.

[16] Przekwas, A. J., 1996,Theoretical Modeling of Liquid Jet

and Sheet Breakup Process, Recent Advances in SprayCombustion: Spray Atomization and Drop Burning Phenomena,

AIAA Inc., 1, pp. 211-239.

[17] Ibrahim, E. A., and McKinney, T. R., 2006, Injection

Characteristics of non-swirling and Swirling Annular Liquid

Sheets, IMechE Journal of Mechanical Engineering Science,

220(2), pp. 203-214.

[18] Rizk, N. K., and Lefebvre, A. H., 1980, The Influence of

Liquid Film Thickness on Air Blast Atomization, ASME

Journal of Engineering for Power, 102 (7), pp. 706-710.



18/32

ASME Early Career Technical Jour2012 ASME Early Career Technical Conference, ASME EC


CLOUD HEIGHT MEASUREMENTS IN JET MIXED TANKS

Ibraheem R. Muhammad and John P. KizitoDepartment of Mechanical Engineering

North Carolina A&T State UniversityGreensboro, NC, USA

ABSTRACTJet mixers can be used as an alternative to conventional

mechanical mixers for solid suspension processes. In the

present study experiments were run to evaluate the suspension

of solid particles in a jet mixed tank. The cloud height, or thedistinct interface at which no solids are suspended beyond, is

measured using three different silica dioxide particles. Theeffect of jet nozzle clearance from the bottom of the tank and

the effect of jet Reynolds number (Rej) is studied for the

different particles. Results show that the cloud height increases

as the Rej is increased. As particle size increased, the

dimensionless cloud height decreased as the drag force is

dominated by the weight of the particle. As the jet nozzle

clearance was lowered, the cloud height decreased slightly. Foran average particle size of 120 m with the jet positioned

0.07625 m from the bottom of the tank, about 90%

homogeneity was achieved. A physical model was developed topredict the cloud height based on a force balance of a single,

spherical particle. The model was able to predict the particle

rise fairly well at Rej greater than 25000. Severalrecommendations for improvements in the model and future

studies were made.

NOMENCLATUREA Area of spherical particles (m2)

Ar Archimedes numberCD Drag coefficient

C Constant used for geometrical conditions

d Particle diameter (m)

D Tank diameter (m)

Dj Jet nozzle diameter (m)Dov Overflow port diameter (m)

f Drag correction coefficientFAM Added mass force (N)

FD Drag force (N)FB Buoyancy force (N)

FG Gravity force (N)

g Gravity acceleration constant (m/s2)H Tank height (m)

Hc Cloud height (m)

Hc* Dimensionless cloud height (Hc

*= Hc/H)

Hfluid Liquid level height (m)

Hj Jet nozzle clearance (m)

m Mass of spherical particle (kg)

Rej Jet Reynolds number (Rej= VDnozzle/)Uf Velocity of bulk fluid (m/s)

U Particle velocity (m/s)

V Jet velocity (m/s)

V Terminal velocity (m/s)

ws Solids weight percentz Distance to any location along the path of the jet (

Greek symbols

Dynamic viscosity (Ns/m2) Kinematic viscosity (m2/s)

L Liquid density (kg/m3)

Particle density (kg/m3)s Solids volume fraction

INTRODUCTIONThe suspension of solid particles is an important proces

many applications, including chemical reactions, biolog

processing, and environmental remediation (i.e. slu

removal). The most common mixers are mechanical agitatsuch as impeller mixed stirred tanks. However, jet mixers

be used as an alternative and have even been reported to be

as efficient as impeller mixed systems while using less ene

[1]. They operate by withdrawing fluid from the mixing t

and discharging it back into the tank through a nozzle at hvelocities. They are especially appealing as they operate w

no moving parts, they are easily installed, and they prov

high turbulence and shear rates, which are advantageous

mixing processes. Since the jet mixers operate without moparts, they are especially useful for processes in w

maintenance of the mixing equipment can be hazardous

Jet mixers have been studied for decades now [2-8],

there is not a lot of literature on the use of jet mixers for ssuspension processes. Bathija [1] studied jet mixapplications and reported a design process for jet mixer

solid suspension processes. Shamlou and Zolfagharian

determined the just suspension velocity of the jets as a funcof jet nozzle diameter, height, solid particle size, and part

density. Kale and Patwardhan [10] studied the effect of no

diameter, nozzle clearance, particle size, nozzle angle,

solids loading on the power necessary for solid suspens

They also provided a semi-empirical model for prediction.



19/32

The performance of solid suspension processes is usually

measured by cloud height, particle dispersion, or cleared area of

the bottom of the tank, or amount of solids suspended. The

current study focuses on cloud height. The cloud height can be

defined as the level at which most of the particles getssuspended. Some authors define this as the point at which no

further particles are suspended upon. It can also be viewed asthe height at which there is a sharp change in the solidsconcentration [11]. This is more likely to occur in systems with

a high concentration of solids.

The purpose of the current study is to determine cloud

height in jet mixed tanks. The jet Reynolds number is varied

along with particle size. The height of the jet nozzle from thebottom of the tank is also varied. The results can be used to

further enhance solid suspension processes using jet mixers. A

physical model is developed based on the particle motion of asingle particle.

SOLID SUSPENSION MECHANISM IN JET MIXERSSolid suspension in jet mixers occurs differently than in

mechanical mixers. As the jet is discharged, a free jet forms,

which expands until it impinges on the solids bed, or tank

bottom. A wall jet is formed and results in solid particles rollingover outwardly from the impingement location. If the velocity

is high enough, this creates a region on the bottom of the tank

which is free of solids, known as the effective cleaning radius

[12]. This also creates a mound of particles along the outer edge

of the tank. At a higher velocity, instead of just rolling, solids

begin to start suspending in the bulk fluid.Once a particle gets suspended, the flow field of the liquid

jet and its interaction with the particle determines how the

particle behaves. The flow fields of liquid jets are known soprediction of particle behavior once suspended can be

predicted. If the upward velocity of the fluid throughout thetank is not sufficient, the terminal velocity of the particle will

cause the particle to fall. The particle will either fall back to the

bottom of the tank or to a point in which the drag overcomesthe weight of the particle.

EQUATIONSFor off-bottom suspension to occur, the hydrodynamic

force of the jet must overcome the weight of the particles. Once

suspended, weight, drag, and buoyancy forces all become very

important. The terminal velocity of a particle is determinedfrom a force balance and is written as

=4 3 (1)The drag coefficient can be easily measured from experiments.

In the Stokes regime, the terminal velocity can be expressed as

= 18 (2)

An important dimensionless parameter is the Archime

number, Ar, which is expressed as [13]

=

The Ar increases as the particle diameter increases. For larg

(Ar >100), the weight of the particle tends to become m

dominant than the drag force and homogeneity in the system

not created [13].

Other important dimensionless quantities include particle Reynolds number and the Froude number, shown be

respectively.

=

=

NUMERICAL MODELA physical model was developed to estimate the height

of particles. The model was based on the suspenmechanism previously described in which the motion

suspension of particles is due to the wall jet that is created o

the jet impinges on the particle bed. The model assumes that

particles are spherical and particle-particle interaction is

considered. For initial development of the model, a sin

particle is used. It is also assumed that the particle reacequilibrium when the settling velocity of the particle

balanced by the upward velocity of the jet.

Rajaratnam [14] found that a three-dimensioNewtonian, turbulent jet velocity can be expressed as

= The jet velocity at point z, u(z), does not depend on whether

jet is impinging on walls or boundaries or just a wall jet.

constant, Cj, is a parameter which accounts for geometry an

usually between 5-6 for turbulent, circular jets. For the cur

study, equation (6) will be used to represent the velocity at

point along the primary travelled path of the jet. This velowill represent the velocity of jet that is responsible

suspending the particles. The initial particle velocity assumed to be equal to the jet velocity at the bottom cornethe tank base or solids bed. So the initial location, zo, was se

Hj+ D/2.

A force balance was completed on a single particle to f

an expression to measure the motion of a single, spherparticle. The steady forces acting on a single particle inc

drag, weight, and buoyancy. Also, a term for added mas

added to account for the inertia added due to a part



20/32

accelerating through the bulk fluid and displacing the bulk fluid

as it travels. The drag force is expressed as

=12 (7)where the drag coefficient, CD, depends on Rep. Forintermediate flow outside of the Stokesian regime, CD can be

expressed as

= 24 (8)where f is a drag correction coefficient. The Rep in the model

was slightly different than the one presented previously in

equation (4). The Repin the model is based on the slip velocity

as

= (9)The correction coefficient used for the model is based on the

widely used Schiller Naumann drag coefficient [15] which is

written as

= 1 + 0.15. (10)The buoyancy and gravity force is expressed, respectively, as

=16 (11)

=16 (12)The added mass term is written as

= 112 (13)The added mass is an important term as it gives an inertial masswhich is different than the gravity mass. This values are much

different when the density of a particle is close to that of the

fluid [16]. For the study at hand, the density of the particles andfluid are of the same magnitude.

By combining all of the forces, the balance on a single

particle becomes

=12 + 16 + 1

12

The resulting equation for particle motion is based on instantaneous velocity of the bulk fluid. The velocity of

particle is solved for and subsequently, the height of the par

is obtained from

= where x is the position of the particle.

All numerical methods were completed using MATL

Equation (14) is discretized using an Euler iterative processaccount for the non-linearity of the first term on the right h

side of equation (14), one of the slip velocities was solve

the present time step (i.e. t + dt), while the other slip velowas set to the previous time step (i.e. t). the iteration pro

continued until the convergence criterion was met. the distinterface between fluids and particles is formed due to

balancing of the downward velocity of the particles and

upward velocity of the fluid at the wall, which is a result ofjet [17]. So for the current study, iterations were run until

upward velocity equaled the terminal velocity of the particle

EXPERIMENTAL METHODSA schematic of the experimental tank is shown in Figur

All experiments were run in a 0.305 m (12) cl

polycarbonate, and cylindrical tank. The tank was equip

with an overflow port (Dov = 19.05 mm) such that the liqheight, Hfluid, remained at 0.305 m. The diameter of the

nozzles used for experiments was 4.32 mm. The nozzle centered in the tank and directed downward at a cleara

height (Hj) of 0.07625 and 0.038 m from the bottom of the t

The orientation of jets were used in previous jet mixing stu

[5]. The jet orientation is such that the jets are able to crea

circulatory pattern due to jet impinging on the tank walls. Oone nozzle was used in the present study. A 0.3 hp centrif

pump was used to supply the fluids to the mixing tank. W

was used as the working fluid. The velocity of the jet varied from about 1.9 6.5 m/s. The jet Reynolds number (

varied from about 8300 28000.

Various silicon dioxide particles (U.S. Silica) were used

experiments. For each sample, the size distribution of particles varied. The d50 particle size, or the diameter at w

50% of the solids are finer, was used. Figure 2 shows the

distribution for the solids used. Table 1 summarizes

properties of the particles. A solids volume fraction, s0.045 and weight percent of solids, ws, of 10.6% was usedall tests.



21/32

Figure 3 shows microscopic images of the different

particles used in experiments. The images show that particles

are not spherical. The d50_120 and d50_265 particles areclassified as subangular and the d50_700 particles are angular.

Figure 2: Size distribution for solid particles

The cloud height was measured as Rej was varied. The

steady jet was initiated and as the solids suspended, a distinct

interface was observed. The entire process was recorded using a

Basler acA2040-180 km CMOS camera attached to a PIXC1-E8 frame grabber housed in a computer, all purchased from

Epix, Inc. A solid state green laser (MGH-H-532, Opto Engine,

UT, USA) was used to illuminate regions of the tank. The cloud

height was considered the maximum formed interface after a

period of time. The interface was not always constant and so ahigh and low value was recorded for each run.

Table 1: Properties of solid particles

Microsil

CGS

Mapleton #1

Glass

Mystic W

II

Sample No. d50_120 d50_265 d50_70

Mineral Quartz Quartz Quartz

d50 (m) 120 265 700Specific

Gravity2.65 2.65 2.65

Grain Shape Subangular Subangular Angula

pH 6.5 7 6.5

Vt (m/s) 0.077 0.117 0.186

Ar 28 336 5551

Rep* 9 32 130

RESULTS

The observed cloud height was measured using n

intrusive optical techniques. Figure 4 shows a snapshot ofobserved cloud height. It can be seen where the dist

interface is shown and how the high and low cloud he

values were calculated. Some additional particles were not

above the line, but there were not included in determiningcloud height. Those outlier particles were due to the rang

particle sizes in each run. An actual scale was placed in

image to accurately measure the height. Though, not the foof the current study, the gray scale intensity decreases as w

increased height, due to the gradients in particle size.

0

10

2030

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

%P

assing

thruSieve

Particle Diameter (m)

MicrosilCGS

Mapleton#1 Glass

MysticWhite II

0.07625 m

0.1525 m

Figure 1: Schematic of mixing tank used inexperiments.

(a) (b)

(c)

Figure 3: Microscopic images for (a) Microsil CGS(d50_120), (b) Mapleton #1 Glass (d50_265), and (c

Mystic White II (d50_700) particles



22/32

calibration of the gray scale intensity, an estimate of axial

concentration can be determined through analysis.

The maximum cloud height was determined with the jet

nozzle positioned at 0.07625 m and 0.038 m from the bottom of

the tank. The cloud height was non-dimensionalized using theliquid fill level height. Figure 5 displays the maximum cloud

height measured for the three different particle samples at a

nozzle height of 0.07625 m as a function of Rej. Overall, therewas an increase in cloud height with increase in jet velocity and

ultimately Rej. Once a jet Reynolds number of about 23800 was

reached, there was a steeper increase in cloud height. Since thed50_120 particles had the smallest diameter and the lowest

terminal velocity, they were easily suspended. About 90%homogeneity was achieved using the d50_120 particles at Rej=

28500. At the same Rej, only 19% homogeneity was achieved

using the d50_700 particles. This can be due to the largerparticles colliding with one another and decreasing the kinetic

energy of the particles [18]. More suspension can be achieved

by providing more kinetic energy from the jet. Much

suspension was not expected for the large particles as the Ar is

so high, meaning the gravity force is more dominate thandrag force.

Figure 6 shows the minimum dimensionless cloud he

for the three particle sets at a nozzle height of 0.07625 m. A

Rejof about 8300, the jet force was not sufficient to creauniform radial cloud height for the d50_265 and d50_

particles. The minimum cloud height was measured as 0 wthe cloud height did not extend out to the complete diametethe tank.

Figure 6: Minimum dimensionless cloud height at a jenozzle height of 0.07625 m

Figure 7 displays the dimensionless cloud height a

function of Rej for a jet nozzle positioned 0.038 m from

bottom of the tank. The maximum and minimum valuecloud height are displayed. The minimum results for

d50_700 particles are not shown because they were all equa

0, as the cloud did not extend the full diameter of the tank

the nozzle was closer to the solids bed, the maxim

homogeneity did not exceed 25%. For the largest partic

d50_700, a jet Reynolds number of 8300 was not ablesuspend the particles.

With the jet positioned lower, the homogeneity in the

was lowered. The jet was absorbed by the particles and thwas not enough force to suspend the particles. When the fre

impinged on the solids, it turned into a wall jet. Even for

largest particles, when no suspension occurred, particles w

dispersed outward from the center and created mounds aro

the edges of the tank. If the velocity of the jet is increased,

particles would then be suspended [10].

As the jet is discharged, it expands radially, but whenjet is closer to the bottom of the tank, its expansion is limAlthough a wall jet is then created, if the velocity of the wal

is not sufficient, suspension will not occur. The lowe

expansion also was a factor in the cloud height not being a

to expand the complete diameter of the tank.

Figure 8 shows a comparison of the dimensionless cl

height for both jet heights as a function of Fr. At Fr = 514both heights, more than 60% homogeneity was achieved. A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.9

1

5000 10000 15000 20000 25000 30000

Hc

*

Rej

d50_120 d50_265 d50_700

00.10.2

0.30.40.50.6

0.70.8

0.91

5000 10000 15000 20000 25000 300

Hc

*

Rejd50_120 d50_265 d50_700

Hc,low

Hc,high

Figure 4: Snapshot of cloud height

Figure 5: Maximum dimensionless cloud height at a jetnozzle height of 0.07625 m



23/32

= 44, the dimensionless cloud height did not exceed 10% of the

total liquid height. The minimum cloud height value does not

vary much at the two different heights. At higher Fr, the lower

nozzle clearance provided cloud heights greater than that at the

higher clearance level.

The results of the physical model and experimental resultswere compared to determine the validity of the model. Figure 9

shows the comparison between the experimental and physical

model results for the dp50_265 particles. At Rej greater than25000, the model predicts the cloud height well. At the two

highest Rej, the maximum deviation from the experimental

values was only 4.6%. However, there is larger error from theexperimental results at a Rejof 23800 and 8300.

Figure 10 shows a comparison of experimental and model

results for the d50_700 particles. The model predicts the cloud

height very accurately at Rejof 2800 and 28500. The maxim

error from the experimental results is only 2%. The e

increases as the Rej decreases. This is similar to the result

the d50_265 particles. This could be due to the drag mo

which was used. The model for drag coefficient is a functioRep, but it is most accurate at intermediate Rep values.

model could be improved by incorporating different dcoefficient models for the various Repranges.

Figure 10: Comparison of experimental and physicamodel results for the d50_700 particles

The results of the physical model for the d50_120 parti

are shown in Figure 11. The model predicts that the tank wo

be fully mixed at Rejgreater than 23800. Further improvemin the model should be made, but some of the error can

attributed to the drag model used. However, at a Rejof 83

the model predicts the non-dimensional cloud height v

accurately. There was only 0.2% error from the experimeresults at a Rejof 8300.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5000 10000 15000 20000 25000 30000

Hc

*

Rej

Max, d50_265 Max, d50_700Min., d50_265 Min., d50_700

0

0.10.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400 500 600

Hc

*

Fr

Max., Hjet = 0.07625 m Max., Hjet = 0.038 m

Min., Hjet = 0.07625 m Min., Hjet = 0.038 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5000 10000 15000 20000 25000 3000

Hc

*

Rej

Experimental Model

0

0.05

0.1

0.15

0.2

5000 10000 15000 20000 25000 3000

Hc

*

Rej

Experimental ModelFigure 8: Comparison of dimensionless cloud height

as a function of Fr for d50_265 particle sample

Figure 9: Comparison of experimental and physicalmodel results for the d50_265 particles

Figure 7: Dimensionless cloud height for jet nozzle ata height of 0.038 m



24/32

During experiments at low Rej (i.e. 8300), the cloud did

not extend the full diameter of the tank for the d50_265 and

d50_700 particles. To account for this in the model, multiple

particles should be used. For example, a statistical approach

should be used to account for the varying particle height.However, the model does predict the height of a single particle

correctly. The model predicted that at a Rejof 8300, the particlerose very slightly with a diameter of 265 and 700 m. Since,experiments showed that the cloud height did not extend the

full diameter of the tank, the model is actually accurate. For the

d50_120 particles, the model predicted that the particle forces

were not able to equilibrate within the height of the tank and

thus it rose to the liquid height of the tank. When measuring thecloud height, there may be outlier particles which extend above

the measured cloud height. So the model does predict the single

particle well.

The model presented in the present study is just an initial

model and further improvements should be made to increase itsaccuracy in predicting cloud height. For instance, the model is

based on a single particle, whereas in the actual experiments, a

solids weight percent of 10% was used. At such solids loading,

other phenomena like particle-particle interaction should be

accounted for. Another assumption used in the current modelwas the particles were spherical. The actual particles used in

experiments are clearly not spherical as shown in Figure 3. It is

expected that spherical particles and non-spherical particlesbehave differently. Further studies should include model

parameters to account for geometrical differences in the

particles. Also, the model should be tested using sphericalparticles.

Though equation (6) given by Rajaratnam has been tested

and well documented, a more accurate model for the wall jetcreated by impinging liquid jets should be used. Another

suggestion is to improve the drag function used. Not only

should the drag be a function of the different ranges of Rep, but

it should account for the change in drag as an effect of

proximity to boundaries and additional particles.

CONCLUSIONStudies were run studying the effect of Rej on the cl

height achieved in jet mixed tanks. The jets were placed at

different heights, 0.07625 and 0.038 m, from the bottom of

tank. The percent of homogeneity was increased as the increased. The level of solids suspension was lowered when

jet was placed closer to the solids bed. The jets did a goodof suspending particles with a d50 size distribution of 120as 80% homogeneity was achieved.

The model that was developed was able to predict

cloud height well for Rejgreater than 23800 for the parti

with a mean diameter of 265 and 700 m. The error in thsystems increased as Rejdecreased. For the smallest partic

the model was able to accurately predict the cloud height at

of 8300, but there was error for the larger Rej. The m

actually predicted the entire tank to be homogenized aboveof 8300.

Several recommendations for improvements in the m

were made including modifying the drag coefficient for a wrange of Rep and including effects of walls and particles on

drag. Also recommendations for the wall jet that is responsfor particle suspension should be improved. Since the mo

was developed for a single, spherical particle, modificat

should be made to account for differences in particles andparticle-particle interaction phenomenon.

The results from the present study can be used to

Asme Dist f - Ectc 2012 Journal - Section 6

Documents

Transcript of Asme Dist f - Ectc 2012 Journal - Section 6