asme dist f - ectc 2012 journal - section 6
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ASME hournal- ECTC 2012TRANSCRIPT
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SECTION 6
FLUIDS ENGINEERING
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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
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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.
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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.
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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.
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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.
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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.
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ASME Early Career Technical Jour
2012 ASME Early Career Technical Conference, ASME EC
November 2 3, Atlanta, Georgia U
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].
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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
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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
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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
Dimensionless axial distance, z/D0
Sheetangle,
-10
10
30
50
70
90
0 2 4 6 8 10 12
Al l forces
No centrifugal
No Corioli s
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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
Dimensionless axial distance, z/D0
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
Dimensionless axial distance, z/D0
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
Dimensionless axial distance, z/D0
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
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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
z/D0 r/D0 /0 uf/uf0 wf/wf0
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
z/D0 r/D0 /0 uf/uf0 wf/wf0
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
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z/D0 r/D0 /0 uf/uf0 wf/wf0
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
z/D0 r/D0 /0 uf/uf0 wf/wf0
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.
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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.
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ASME Early Career Technical Jour2012 ASME Early Career Technical Conference, ASME EC
November 2 3, Atlanta, Georgia U
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.
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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
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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.
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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
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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
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= 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
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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