Hydrodynamics of Liquids in Mechanically Agitated Vessels 13
3Hydrodynamics of Liquid
in Mechanically Agitated VesselsA detailed knowledge of hydrodynamics existing in stirred vessels, especially is key to
understand mixing process occurred in the system. In this chapter, the flow liquid in both
single impeller and multiple impeller systems with various type of impeller will be discussed
and the formation of the trailing vortex behind the impeller blade, which plays an important
role in dispersion of gas is examined in depth.
3.1 Flow Patterns in A Single Impeller System
Due to its complexity, studies in early years on hydrodynamic characteristics of the
stirred vessel have been conducted mostly by experimental techniques such as hot film
anemometry, laser Doppler anemometry and laser sheet illumination for determination of
fluid velocity or flow patterns inside the vessel. All these approaches require not only
expensive investment but also needs laborious time consuming efforts. Recent advances in
Computational Fluid Dynamics (CFD) enable us to simulate these hydrodynamic
characteristics with considerable accuracy.
3.1.1 Brief description of simulation method
Numerical simulation of a single-phase turbulent flow in a stirred vessel with a single
impeller has been studied extensively in recent decades. Applying Fluent, Myers et al. (1994)
have calculated 2-D flow pattern for a multiple impeller system. Desouza and Pike (1972)
solved mathematically the flow of a fluid in a stirred vessel quantitatively. Thiele (1972)
solved Navier Stokes Equation for an unbaffled system under Re<20. Placek et al. (1978)
used an algebraic turbulence model to solve the 2-D flows in a stirred vessel. Most
approaches to solving flows in stirred vessels have substituted the time average quantities into
a continuity equation and turbulent equation of motion: i.e.,
+=∂∂
+∂∂
φφρρφ Duxt i
i
)()( φS (3.1.1-1)
accumulation convection diffusion source
where
14 MULTIPLE IMPELLER GAS-LIQUID CONTACTORS
'φφφ += (3.1.1-2)
For an appropriate time, Eq. (3.1.1-1) can be integrated to give
φφφρφρρφ SDux
uxt i
i
++∂∂
=∂∂
+∂∂ )'()()( (3.1.1-3)
where )'( φρu− is known as the Reynolds stress. FLUENT provides three kinds of models to
describe the Reynolds stress, i.e., (1) the κ-εmodel, (2) the differential Reynolds stress
model and (3) the renormalization group theory.
In theκ-εmodel, it is assumed that turbulent stress follows Newton’s law of viscosity,
i.e., that the turbulent eddy viscosity, μt, can be expressed as
εκρµ µ
2
Ct = (3.1.1-4)
where
κ= turbulent kinetic energy
ε=dissipation rate ofκ
Substitution of Eq.(3.1.1-4) into the conservation equation leads to
⎪⎪⎭
⎪⎪⎬
⎫
−−++∂∂
∂∂
=∂∂
+∂∂
+++∂∂
∂∂
=∂∂
+∂∂
κερ
κε
σεµ
ερρε
ρεσ
κµκρρκ
εεκεκ
κκ
2
231 ])1([)()()(
)()()(
CGCGCxx
uxt
GGxx
uxt
bi
t
ii
i
bi
t
ii
i (3.1.1-5)
where
j
i
j
i
i
jt x
uxu
xu
G∂∂
∂∂
+∂
∂= )('µκ
ih
tib x
gG∂∂
−=ρ
ρσµ
Equations (3.1.1-4) and (3.1.1-5) can be solved simultaneously with the equation of continuity
and the momentum equations.
Commercially available soft, such as FLUENT or CFX, which were often used in
simulation to solve a set of partial differential equations generated from discretized Navier-
Stokes equation using a control-volume difference technique. The two-equation turbulence
κ-εmodel, the SIMPLE algorithm with the power law different scheme, and the single
direction sweep solution method are used in the simulation. Figs. 3.1-1 give examples the
computational grids of the different systems which will be discussed.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 15
For system discussed in this study is a standard geometry, with T=0.288 m, C’=T/3,
C=2T/3, D=T/3 and b=0.1T, displaced with respect to each other by 90 degrees. For each
impeller, the blade height w=D/5, and the width L=D/4. The calculation domain comprises
1/4 of the tank volume. The nonuniform grids used are 27×30×30 for the single Rushton
impeller system, 21×25×47 for the dual Rushton impeller system, 28×20×44 for a combination
of the A310 and Rushton impeller system, and 21×20×59 for the triple-Rushton impeller
system in (γ×θ×z) coordinates. Here, γdenotes the radial distance from the tank axis, θis
the tangential coordinate, and z is the distance from the liquid surface. The V, W, and U-
velocities, respectively, denote the radial, tangential and axial velocities. The sum of the
normalized residuals of all variables converged to less than 1×10-3 within 1,500-1,200
iterations.
Fig. 3.1-1 Computational grid, side view and top view.(a) Single Rushton turbine impeller(b) Dual-Rushton turbine impellers(c) Dual impeller system with different impeller combination(d) Triple Rushton impellers
Physical properties
FLUENT provide a data base to set the physical properties of the working fluids. Under
this situation, the only thing that the user has to do is to choose the working fluids. If the
wanted working fluids are not included in the data base, FLUENT offers another facility to
16 MULTIPLE IMPELLER GAS-LIQUID CONTACTORS
input the physical properties of the working fluids by the user themselves, including the
temperature, density, viscosity and surface tension, etc.
Boundary conditions for simulation
Computational Fluid Dynamics has not yet reached the stage where flow patterns in the
baffled cylinder vessel can be calculated without any empirical input experiments because the
impeller generates a highly complex periodic flow in its vicinity and due to the existence of a
pair of trailing vortices behind the impeller blades on either side of the impeller disk. For
simplicity, the boundary conditions used in these simulations are set as follows.
Boundary conditions of the impeller region
The approach used here is to apply experimental velocity data measured with the LDA to
prescribe profiles of both the mean velocity and the turbulent kinetic energy κ along the
vertical periphery of the volume swept out by the impeller; the turbulent kinetic dissipation
rate ε has been imposed on the vertical sweeping surface of the impeller according to the
data presented by Ranade and Joshi(1990). The radial and tangential velocity can be specified
on the vertical surface of the impeller using parabolic type axial profiles with a maximum 0.7
impeller tip velocity ( )NDVtip π= .In this text, for the Rushton turbine, the mean velocity
profile on the vertical surface of the impeller was imposed at radius 5 cm. For the axial flow
impeller (A310), the axial velocity profile 2.8-cm below the impeller was imposed.
Cyclic cell
The three-dimensional calculations were done for a 90-degree sector of the vessel as a
result of the cyclic repetition of the vessel with four identical regions in order to reduce the
volume, which had to be included in the simulation. The r-z planes between the baffles, the
mid-baffle plane, were so-called cyclic planes to satisfy continuity of all values and gradients
condition.
Solid wall
The no-slip boundary condition was imposed on all solid surfaces. These boundary
conditions were specified using wall functions at the tank wall, the baffles, the bottom of the
tank, and the impeller shaft.
Free surface
The liquid surface was assumed to be flat, and there was no momentum transport at the
air-liquid interface.
Symmetry at the tank axis
The systems discussed here were all symmetry at the tank axis.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 17
Simulation by using CFX
Multi-block facilty
Using the multi-block facility of CFX, the stirred vessel system was divided into some
simple configurations, such as disks and rectangle under the geometries of impeller and tank
can be depicted precisely. The “Matched Grid” was set at the interface between adjacent
blocks to link each block to form the desired geometry. With this facility, the effects of some
geometrical variations, e.g. blade size, blade number, impeller type and baffle design, on the
flow field in stirred vessels can be examined accurately.
Sliding grid facility
CFX provides the “sliding grid” facility to cover the whole computational domain. The
computational domain was divided into the impeller rotational core and the stationary part of
the vessel. During the simulation, the whole impeller rotational core was set to rotate with the
input rotational speed of the impeller N and the rest is motionless. The “Unmatched Grid”
was set at the interface between the rotary part and motionless part. The calculated data were
transferred over the interface between the impeller core and the stagnant part. The sliding grid
was set to rotate one grid cell in azimuthal direction per unit time step, and the simulation
continues to proceed until a steady state is reached, i.e. five rotations at least, under which all
the residual are less than 10-4.
Grid distribution for the flow field simulation
Finer grids were adopted in the regions where the velocity gradient and the variation of
turbulent kinetic energy and energy disspation rate are larger, namely, the impeller region,
liquid surface and the places close to the wall boundary, etc. Figure 3.1.1b shows an example
of the grid layout of the computational domain used in this study.
Boundary condition
As applying the sliding grid facility of “CFX” to the single-phase flow field simulation,
the rotational speed was the only extra-needed boundary conditon. No slip conditions were set
to the solid boundaries and the intrinsic wall functions were adopted at the boundaries of the
computational domain. The followings describe the boundary conditons used in this
calculation.
Shaft
The “shaft” was set as symmetry to match actual condition.
Periodic Plane
According to the impeller blade number and baffle number, the computational sector was
determined. For examples, a 1/4-tank simulation is sufficient for the system equipped with a
18 MULTIPLE IMPELLER GAS-LIQUID CONTACTORS
4-blade impeller and for baffles, while a 1/2-tank simulation is needed for a complete
simulation in the system with 6-blade impeller and four blades. For a fractional sector of a
computational domain, the two ending planes in azimuthal direction were set as “ periodic
plane” to get the repetition for the integrity of whole tank.
R θ z Total Grid No.Stagnant Part 10 40 41 16,400
Rotational core 20 40 41 32,800An example of grid layout for the single impeller system used in CFX
Fig. 3.1-1b An example of grid layout for the single impeller system used in CFX.
Free Surface
The liquid surface was set as “ free surface”, under which it was seen as a flat plane, and
no mass or momentum transport occurs at the liquid surface between air-liquid.
Thin Surface
Apart from the “Solid wall”, CFX provides a patch named as “ thin surface “ to describe
the solid wall, which has one dimension approached zero, such as baffle plate, impeller blade
and impeller disk, etc. No slip conditions were still set to thin surface.
Mass transfer boundary
For a continuous or a semi-continuous system, e.g. gas continues to feed to a stirred
vessel, the “ mass transfer boundary” or “inlet” was set at the cells matching the sparger. The
gas superficial velocity was set as the axial velocity at these cells.
Physical properties
Just like FLUENT, CFX also provides a data base to set the physical properties of the
working fluids. With the fluids not included in the data base, the user also can input the
Hydrodynamics of Liquids in Mechanically Agitated Vessels 19
related physical properties of the working fluids by themselves. For the gas-liquid flow field
simulation, CFX offers another facility to set different fractions of gas and liquid at various
locations within the agitated system. The user can divide the whole system into several blocks
to suit the experimental section and set the fraction of gas and liquid based on the
experimental data of local gas holdup values. Then, the physical properties of the mixture of
gas and liquid were calculated according to the weight fraction of the working fluid at each
block.
Models for the flow field calculation
Applying the Reynolds average approach to the continuity, momentum and conservation
equations of a quantity φ , and integrating them over a long period of time, these three
equations can be rewritten as:
SuUt
uuBUUtU
Ut
+−∇Γ⋅∇=⋅∇+∂∂
⊗−⋅∇+=⊗⋅∇+∂∂
=⋅∇+∂∂
)()(
)()(
0)(
φρφφρρφ
ρσρρ
ρρ
Where U denotes the mean velocity, S is a source term and B is the body force. uu⊗ρ and
φρu are known as the Reynolds stress and Reynolds flux, which can be solved by using the
eddy-viscosity concept or solved for individual components according to the turbulent model
adopted.
Five different turbulent models were provided by “ CFX”, namely, k-εmodel, low
Reynolds number k-εmodel, RNG k-εmodel, Reynolds stress model and Reynolds flux
model. All these models were tried to calculate the flow field within the stirred vessel and
compared with the experimental data obtained from the LDA experiment( Lu and Yang, 1995).
It was found that the calculated results produced by the Reynolds stress model are always
more accurate than those given by other models. It took the anti-isotropic distributions of
turbulent characteristics within the stirred vessels into account, and was adopted to calculate
the flow field within the stirred vessels throughout this study. For the Reynolds stress model,
uu ⊗ satisfies the following equation:
IGFuuuuC
Uuut
uu T
DE
s ρεϕρεκ
σρρρ
32])([)( −++=⊗∇⊗⋅∇−⊗⊗⋅∇=
∂⊗∂
Here F and G are the shear stress and buoyancy stress production tensors. F is given by:
20 MULTIPLE IMPELLER GAS-LIQUID CONTACTORS
])()([ uuUUuuF T ⊗∇+∇⊗−= ρ
In a rotating system the above equation should be modified as:
uuuuUUuuF T ⊗⊗−⊗∇+∇⊗−= ρωρ ])()([
In a buoyant flow variations in density are expressed in terms of variations in enthalpy (h)
and scalar mass fraction(s). For an incompressible flow using the Boussinesq approximation.
G can be written as:
])([])([ TT usgusguhguhgBG ⊗+⊗+⊗+⊗−= ραρ
φ is the pressure-strain correlation given by:
φ=φ1+φ2+φ3
where )32(11 IuuC S κ
κερϕ −⊗−=
)32(22 IMC S κϕ −−=
)32(33 LIGC S −−=ϕ
J and L denote the shear anad buoyancy production of turbulence kinetic energy, which can be
consulted in the CFX flow solver user guide(AEA technology).
Numerical schemes of the simulation
The “finite volume approach” was used to integrate the differential equations within each
control volume to result in the finite difference equation, then these differential equations
were discretized through the “ General Version of Algebraic Multi-grid” facility. Finally, by
using SIMPLE scheme to link the relationship between velocity and pressure, the flow field
within the stirred vessel was calculated. It is assumed that the variable of each cell has the
same value initially. With the obtained discretized continuity and momentum equations, the
values of variables of each cell at any position of the computational domain was calculated
step by step. Using the calculated results of the past time as the input of the next time step, the
value of variables at any time step were obtained. Since the calculated result of the past time
step always affect the result of the current time step, it is impossible to compute variables of
each grid node at the same time. Therefore, they are calculated iteratively until the
convergence criterion was matched, i.e. all the residuals are smaller than 10-4.
Validation of the CFX simulated results
To examine the accuracy of the simulated single-phase flow field obtained from CFX,
the calculated tangential and axial velocities by using the Reynolds stress model were
Hydrodynamics of Liquids in Mechanically Agitated Vessels 21
compared with the LDA experimental value of Lu and Yang(1995). Figures 3.1.1c and 3.1.1d
show such comparisons of the tangential and axial velocity in the mid-plane with N=4.17rps
and Z*=2Z/W=0, respectively. From the data points shown in these two figures, it is found
that the CFX simulated results agree with the LDA experimental data very well.
Fig. 3.1-1c Comparison of dimensionless computational and LDA experimentaltangential velocities in various locations(Z*=2Z/W=0, θ=450 and N=4.17rps).
Fig. 3.1-1d Comparison of the dimensionless computational and LDA experimentalaxial velocity in various locations(Z*=2Z/W=0, θ=450 and N=4.17 rps).
22 MULTIPLE IMPELLER GAS-LIQUID CONTACTORS
Calculation of the gas-liquid flow field based on the single-phase flow field
Once the single-phase flow field was determined, the liquid velocity with aeration can be
evaluated based on the results of Bakker and Akker(1994) as follow:
β)/( 0,1,1 PPUU gug ×=
where U1,g and U1,u denote the liquid-phase velocity with and without aeration, Pg and PO are
the power drawn by the impeller under gassed and ungassed condition, respectively. The most
suitable exponent “β” was determined by comparing the simulated mixing times with
experimental data, which was described in detail in Chapter 5.
Using the multi-phase simulation facility of CFX to calculate the flow field of gas and
liquid phases within the stirred vessel, the dispersed phase and continuous phase should be
differentiated, and the averaged size of gas bubble is needed to know. From the correlations
proposed by Moo-Young and Blanch(1981) and Fukuda et. al.(1968),i.e.
427.017.032 109)/27.0 −− ×+= sg VVPD
39.008.0320 )/(35.6 QNDPPg =
the mean bubble size within the gas-liquid stirred vessel could be evaluated through the
original operating conditions (i.e. the rotational speed and sparged gas rate). Take liquid as the
calculated mean bubble size, the gas-liquid flow velocity could be calculated by using the
numerical theories described in above sections.
3.1.2 Flow patterns in the Single Impeller Systems
As shown in Fig. 3.1-2, different type of impeller has different pumping characteristics,
and the discharge flow will impinge to the wall (or bottom) surfaces of the vessel, which will
generate different flow pattern for system in the single impeller system which are generated
by three major types of impeller. The Rushton turbine impeller (disk turbine impeller)
discharge fluid radially and creates four independent loops which pitched blade or propeller
impeller discharges fluid in axial direction and results two large loops at both sides of the axis.
Fig. 3.1-3 and Fig. 3.1-4 gives typical velocity vector plots of Yamamoto’s at mid plane for
the system equipped with Rushton turbine impeller and pitched blade impeller respectively.
As seen in Figs. 3.1-3 and -4 there are two circulating loops above and below the disk of the
Rushton turbine system while there is only one large loop circulating the entire half plane of
the vessel for the system equipped with pitched blade impeller. In the both systems, the largest
velocity vectors appear in discharge stream and decrease as it departs from the impeller. Since
there exists a segregation plane in the central line of the blade of the Rushton turbine impeller,
Hydrodynamics of Liquids in Mechanically Agitated Vessels 23
the interaction of the mixing between the upper part and lower part of the stirred vessel is
worse than that of the pitched blade impeller system, which makes the mixing rate for the
single Rushton impeller may be lower than that of the pitched blade impeller system.
(a) Pitched paddle (b) Axial flow impeller (c) Rushton turbine
Fig. 3.1-2 Discharge flow pattern of different type impellers.
3.1.3 Effect of baffle design on the liquid flow pattern
The role of baffles in a mechanically agitated vessel is to promote the stability of the
power drawn by the impeller to suppress the swirling and vortexing of liquid and to eliminate
the stratification within a vessel.In commercial scale tanks, insertion of extra baffles to have
cfully babbfled environment or to secure more heat transfer area is a very common practice.
However, excessive baffling may induce a reduction in mass flow and cause a serious
localizing flow within the system, which should be avoided . Therefore, the well baffle design,
including selection of optimum number and width of baffle is necessary to provide the
appropriate liquid flow within the stirred vessel to result in a better mixing.
Fig. 3.1-3 Velcity vector plots of a singleRushton turbine impeller system(Yamamoto and Nishino,1992).
Fig. 3.1-4 Velcity vector plots of a singlepitched paddle impeller system.(Yamamoto and Nishino,1992).
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS24
Fig. 3.1-5 The calculated flow pattern of the single Rushton turbine impeller agitatedvessel at various azimuthal slices (a) mid-plane (45°); (b) baffle plane (0°); (c)in front of baffle (2.44°) and (d) behind the baffle (2.45°).
Fig. 3.1-5 shows the calculated flow patterns at different azimuthal planes for the single
Rushton turbine impeller system. From this figure, it can be seen that since the Rushton
turbine impeller is a radial impeller system, it always pumps the liquid radially. After
colliding with the tank wall, the flow direction of liquid changes into axially and separates
into two individual circulating loops above and beneath the impeller at the impeller plane
respectively. As shown in this figure, although the flow pattern at each R-Z plane looks very
similar, the liquid velocity at 2.45° behind the baffle plate is larger than those at other planes.
In addition, the stagnant point of the circulating loop at plane is closest to the center of the
vessel. However, in the plane in front of the baffle plate (i.e. 2.44° in front of the baffle plate),
the flow intensity is smallest and results in a larger stagnant zone.
3.2 Flow Patterns in the Multiple Impeller System
3.2.1 The system with dual Rushton turbine impellers
Dual Rushton Impeller System
The effect of additional impeller on flow field as well as total power drawn by impellers
attracts the interest of the researches recently, Giorgio et al. (1999) have presented their
simulated results of velocity vector plots for a stirred vessel with two identical Rushton
turbine impellers. Fig. 3.2-1 depicts how the additional impeller and the impeller distance will
affect the flow patterns of fluid in the system. From these figures, it can be seen that the
impellers apart a distance less than 1.5 impeller diameter tends to interact each other and
construct a merging flow patterns, and each impeller will performs independently if the pitch
of the impeller is greater than 1.5 D. The Fig. 3.2-1 (c) indicates that the height of the bottom
Hydrodynamics of Liquids in Mechanically Agitated Vessels 25
impeller also affects the flow pattern, if the height is less than 0.15T or 0.5D the flow patterns
becomes a diverging flow.
(a) Mering Flow (b) Parallel Flow (c) Diverging Flow
Fig. 3.2-1 Flow patterns generated by dual Rushton turbine impeller with variousimpellerdistance(Giorgio et al.,1999).
(a) θ=45∘ (b) θ=-2.44° (c) θ=2.45°
Fig. 3.2-2 Velocity vector plot for a dual Rushton turbine impeller system at rotationalspeed=350 rpm.
(a) Simulated result between the baffles (θ=45°)(b) Simulated result in front of the baffle (θ=-2.44°)(c) Simulated result behind the baffle (θ=2.45°)
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS26
Effect of Baffles on flow patterns in a dual Rushton turbine impeller system
Figs. 3.2-2(a)-(c) give velocity vector plots obtained from the numerical simulation for
the mid-plane, the plane in front and the plane behind the baffle, respectively. From the plots
shown in Fig. 3.2-2(b), the boundary line between the two loops in the middle circulation
zone lies below approximately 10% pitch distance the center of this zone. Comparing the
plots shown in Figs. 3.2-2(a)-(c) at the vertical plane at the front of the baffle (Fig. 3.2-2), the
profile is almost the same as seen at the 45°mid-plane except that the shapes of the loops near
the discharge zone are not as smooth as the loops seen at the mid-plane (Fig. 3.2-2(a)). At the
plane behind the baffle (Fig. 3.2-2(c)), it is interesting to note that the velocity vectors along
the wall are much larger than the same location vectors seen at the other planes, and the
boundary line between the two circulation loops lies far below the center of this region.
Figure 3.2-3 shows that a profile of the velocity vectors lies on a horizontal plane near the
lower impeller, and a cleared difference in the size between the vectors at the front of the
baffle and the vectors behind the baffle can be observed.
Fig. 3.2-3 Top view of velocity vector plot at z=0 of the Rushton impeller.
The system with Dual impeller of various types
In Fig. 3.2-4 the simulated results of velocity vector plots of different θ locations for a dual
impeller system which has a combination of an axial flow impeller, A 310 and a Rushton
turbine impeller at bottom. Comparing the result shown here with the result of the dual
Rushton turbine impeller, no independent loop can be found above the disk of Rushton
turbine impeller and it has a single large circulatory loop to the upper free surface due to the
axial flow characteristics of A 310 impeller. The size of the velocity vectors of the lower
circulation region seems to be almost similar to that of the dual Rushton turbine system and
seems not to be affected by the strong downward axial flow generated by the A 310 impeller.
The effect of the baffle on the velocity vectors can be also found by comparing the profiles
shown in Fig. 3.2-4(c) with (a) and (b), where the strong up flow is generated behind the
Hydrodynamics of Liquids in Mechanically Agitated Vessels 27
baffle.
(a) (b) (c)
Fig. 3.2-4 Velocity vector for dual impeller of different impeller combination atrotational speed=350 rpm. (a) mid plane (θ=45°); (b) In front of the baffle (θ=-2.44°); (c) θ=2.45.
(a) (b) (c)
Fig. 3.2-5 The calculating flow patterns of the triple Rushton turbine impeller agitatedvessel at various azimuthal slices (a) mid-plane (45°); (b) baffle plane (0°); and(c) in front of baffle (2.44°).
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS28
Triple Rushton turbine impeller system
Figure 3.2-5 shows the calculated flow patterns at different azimuthal planes for a triple
Rushton turbine impeller system. From this figure, it can be seen that there is a pair of
circulating loops above and below to disk for each impeller. As seen in Fig. 3.2-5, the flow
velocity in the plane behind the baffle is also the strongest and the stagnant point of the
circulating loops at this plane is closer to the center of the vessel. Comparing the circulating
loops of each impeller, it can be found that the lower circulating loop of the uppermost
impeller is larger than the upper circulating loop of the middle impeller, i.e. the separation
plane between two circulating loops is closer to the middle impeller.
(a)N=13.3rps (b)N=13.3rps (c)N=13.3rps (d)N=13.3rps (e)N=14.7rps (f)N=16.0rps (g)N=18.7rps
Pg/V=580W/m3 Pg/V=700W/m3 Pg/V=851W/m3 Pg/V=1004W/m3 Pg/V=1004W/m3 Pg/V=1004W/m3 Pg/V=1004W/m3
Fig. 3.2-6 The flow patterns of the various multiple impeller systems in the mid-plane(45°)
with N=13.3rps and Pg/V=1004.4W/m2.
Triple Impeller System with Various Impeller combinations
In Fig. 3.2-6, the calculated flow pattern in mid-plane(45°) for the RRR,PRR, PPR and
PPP impeller systems with constant rotational speed N=13.3 rps as well as the same power
input Pg/V=1004.4 W/m2 are shown. It is worthy to note that the liquid flow from the upper
pitched blade impeller tends to enforce the circulating flow of the adjacent lower impeller,
especially when Rushton turbine impeller was located beneath a pitched blade impeller. This
enforcement of liquid flow carries the dispersed bubbles around the whole vessel, and may
induce a higher mass transfer rate as a result. If Pg/V values are set as identical to the RRR
Hydrodynamics of Liquids in Mechanically Agitated Vessels 29
system at N=13.3rps, tip velocity of each impeller in PRR, PPR and PPP systems would be
increase to about 1.1, 1.2 and 1.4 times comparing to each system at it was under the same
rotational speed condition. Their pumping rates also increase to 1.4, 1.8, and 2.1 times under
this situation, respectively. These facts indicate that with a given Pg/V, the multiple impeller
system with a hybrid of the Rushton turbine impeller and pitched blade impeller can offer a
stronger liquid flow to provide a more uniform distribution of KLa and give a high overall
average <KLa> value with a relatively lower shear level.
3.3 Structure of the Trailing Vortex behind the Impeller Blade
Due to the versatile applications of mechanically stirred vessels, many researchers have
been interested in the gas dispersion phenomena within stirred vessels. Prior to 1970s,
researchers recognized that gas was dispersed by the impeller blade itself and postulated that
the impeller with more blades can perform a better gas dispersion. However, Van’t Riet and
Smith (1973) pointed out that the trailing vortexes behind impeller blades dominate the gas
dispersion within the vessel. Therefore, it is exigent to investigate the vortex structure prior to
understand the gas dispersion mechanism around the impeller. Many researchers had paid
attention to the characteristics of the trailing vortex behind the blade of the Rushton turbine
impeller in a single-phase system; however, they only emphasized the turbulent kinetic energy
and the flow field around the impeller region and depicted the vortex locus accordingly. Very
little information about the conformation of the trailing vortex is seen in the literature.
Yianneskis et a1. (1987) used the laser-slit photography method to measure the single-phase
flow field within the stirred vessel equipped with a single Rushton turbine impeller and
determined the locus of trailing vortex by connecting the zero axial velocity point at each
azimuthal slice between two neighboring blades. Shoots and Calabrese (1995), Lu and Yang
(1998) and Lee and Yiannekis (1998) applied LDA to measure the velocity distribution in the
single Rushton turbine impeller and also used the same approach to determine the vortex
locus.
Although the aforementioned approach can depict the vortex locus approximately, there
are three main disadvantages stemming from this method. First, the criteria used for the zero
axial velocity point are different from person to person, which makes the stretched angle of
the vortex under the same experimental condition inconsistent from each other. Secondly, as
mentioned by Van’t Riet and Smith (1973, 1975), an abrupt turn appears to the vortex locus as
it immediately departs from the leading blade, while the locus depicted by this method cannot
show this trend. Finally, nothing as far was proven that this method could be used to depict
the conformation of the vortex. To avoid these shortcomings , a method named “the minimum
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS30
pressure point connecting method” was developed by Lu and Wu (2001) and was used to
depict the locus and conformation of the vortex.
In this section, the commercial software ‘CFX’ (AEA technology company) and the LDA
anemometry are used to determine flow fields and the pressure distributions around impeller
region. The loci of the trailing vortexes were delineated by using the zero axial velocity
connecting method and the minimum pressure point connecting method simultaneously. The
results obtained by different approaches are compared with each other to highlight the
advantages of the minimum pressure point connecting method. The conformation of the
trailing vortex was depicted based on the shape change in pressure contour at each azimuthal
slice between two neighboring blades. Effects of the number and size of impeller blade on the
structure of trailing vortex are discussed in detail.
3.3.1 Formation of the trailing vortex
The complexity of hydrodynamics of fluids in mechanically stirred gas-liquid vessel
makes theoretical analysis difficult and the theoretical basis to scale-up such a system is still
lacking. Since Rushton published his pioneer design of six bladed disk turbine impeller in
1946, it has been used almost exclusively in situations that require high shear to disperse gas
or liquid. Numerous works have been done to comprehend the phenomena of gas dispersion
and turbulence near the impeller region. The rotation of a disk turbine impeller generates a
pair of rolling vortices behind each blade, which forms low pressure regions and sucks the
sparged gas into the trailing vortex to disperse it through the high shear force. Since the
introduction of the concept of the trailing vortex by van't Riet and Smith (1973, 1970, it is
well understood that the trailing vortices which cling behind blades play a very important role
in the dispersion of gas to mechanically stirred vessels and have also attracted the attention of
many researchers in this field.
3.3.2 Effect of the blade number on the loci of trailing vortex
In this section, structure of the vortex and the mechanisin of gas dispersion in an aerated
stirred vessel are discussed by using LDA to conduct a three-dimensional measurement of the
velocity field around 2, 4, 6 and 8 bladed disk turbine impellers. These results is coupled with
the data obtained from a bubble size measurement in the impeller discharge region to discuss
how the structure of the vortex affects the gas dispersion under a low gassing rate for a
straight blade disk turbine impeller.
Determination of Fluid Flow Characteristics by Laser Doppler Anemometry
A flat bottom transparent acrylic glass cylindrical vessel of 0.288m in I.D. equipped with
four full equally spaced baffles of width 0.029m in was used as the mixing tank. To minimize
Hydrodynamics of Liquids in Mechanically Agitated Vessels 31
the effect of reflection of the laser beam at the curved surface of the cylinder, the tank was
placed within a larger square plexiglass vessel filled with the same fluid.
To allow for easier observation, the impeller was installed at a height of T/3 or 0.144 m
from the bottom. Fig.3.3-1 depicts a schematic diagrain of the experimental set-up while
Fig.3.3-2 shows the detailed dimensions of various straight blade turbine impellers used here.
The working fluid used was filtered tap water while filtered air was fed through a pipe sparger
beneath the disk of the impeller.
Fig.3.3-1 The schematic diagram of the equipment setup.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS32
Fig. 3.3-2 The major dimensions of various impellers.
To achieve a appropriate comparison, all the measurements were conducted under the
same basis. Table 3.3-1 lists the experimental conditions employed. However, no air was fed
to the system during LDA measurements. The rotational speed of 4.17rps was chosen to avoid
the interference of surface aeration during the LDA measurement.
In the top of Fig. 3.3-1 the coordinate system adopted here is shown and in the following
section,~
iV , Vi, Ui, and ui’ denotes the instantaneous velocity, mean velocity (under the fixed
coordinates), mean velocity (under the rotating coordinates) and r.m.s fluctuation vclocity to
the i direction (i = r, θ, z ) respectively. Transferring a value from a fixed coordinate to a
value under a rotating coordinate, requires only the transfer value of the tangential component
of tile velocity by using Uθ =Vθ-ωr, where ω=the angular velocity of the impeller, and r = the
radial distance of the measured point. To express the locus of the vortex axis more clearly, the
dimensionless coordinate r*(=2r/D) and z*(=2z/w) were adopted.
The LDA used in this study was a 3w DISA 55N two components forward LDA. It was
coupled with a frequency shifter and a frequency tracker. To measure the tangential
component of fluid velocity, a pair of mirrors was assembled as shown in Fig.3.3-1
Table 3.3-1 The experimental conditions.
(1) Constant Rotational Speed (250 rpm)Number of Blades
2 4 6 80.52 0.55 0.52 0.508Power input per Blade (W)
Gas Flow Rate (L/min) 0.2 0.4 0.6 0.8(2) Constant Power Consumption and Air Flow Rate/Blade
(0.52 W/Blade)Number of Blades
2 4 6 8250 245 250 253Impeller Rotational Speed(rpm)
Gas Flow Rate (L/min) 0.2 0.4 0.6 0.8
Hydrodynamics of Liquids in Mechanically Agitated Vessels 33
During the measurement, the laser beam was focused at a measurement point, and the
light refracted by the tracer particles at the measurement point was received by the
photo-detector and was sent to the frequency tracker to transform the Doppler frequency
signal into an electrical voltage signal, then it was transmitted to a P.C. for further data
processing. Fig.3.3-3 shows the measurement points for LDA measurement. The plane was
set at the middle plane between two baffle plates. The signal obtained through the photo
detector was coupled with the pulse signal from the optical shaft encounter to detect the fluid
velocities of 360 azimtitltal positions. To minimize the errors occurring in the measurements,
extraordinary care was taken for positioning, and calibration of calibration coefficient time by
time. Details of measuring technique are available in Yang's work (1995) or in the recent
articles of Stoot and Calabrese (1991) and Yiannekis and Whitelaw (1993).
Mean and RMS velocity
Knowing each component of the fluid velocity, the mean velocity and RMS velocity are
calculated by
∑=
=n
jiji V
nV
1
~1 (3.3-1)
∑=
−=n
jiiji VV
nu
1
22' )~(1 (3.3-2)
where ijV~ denotes the jth instantaneous velocity in i direction and n is the number of sampling
velocities. More than 200 sets of data were collected for each measurement point.
Fig. 3.3-3 The distributions of measuring points for LDA and bubble sizemeasurements.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS34
Average deformation rates
The motion of gas bubble to liquid can be characterized into the following four parts:
(l) Purely translation motion : which can be expressed by mean velocities to each
direction such as Vr, Vθ, Vz.
(2) Expansion or elongation which can be described by the normal strain rates in
three directions.
(3) Shear deformation
(4) Rotation
Among these, elongation and shear deformation play active role in dispersion of gas near
the impeller. The combination of these two terms as mean rate deformation tensor can be
written asTVV ∇+∇=∆ (3.3-3)
In cylindrical coordinates, the mean rate defomation tensor can be expressed as:
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
+∂∂
∂∂
+∂∂
θ∂∂
+∂∂
+θ∂
∂θ∂
∂+
∂∂
∂∂
+∂∂
θ∂∂
+∂∂
∂∂
=∆ θθθ
θ
zV2
zV
rV
zV
rV
Vr1
zV
)r
VVr1(2
Vr1)
rV
(r
r
zV
rVV
r1)
rV
(r
rr
V2
zrzrz
zrr
rzrr
(3.3-4)
In this matrix, the sum of the diagonal components denotes the normal strain rates while
non-diagonal terms denote the shear deformation rates. Therefore the average deformation
rate at the measured point can be given as
∆∆= :2/1δ (3.3-5)
Turbulent kinetic energy
The other quantity used to describe the intensity of the turbulent flow field is the value of
turbulent kinetic energy (TKE) per unit mass of fluid, which can be given as
[ ]2z
22r 'u'u'u
21TKE ++= θ (3.3-6)
It is often expressed in a dimensionless form as
[ ]2tip
2z
22r2
tip V'u'u'u
21V/TKEk ++
== θ (3.3-7)
Loci of the trailing vortices
Figure 3.3-4(a) and Fig.3.3-4(b) show the top view and the font view of the loci of the
axis of the trailing vortices which cling behind the blades of the disk turbine impellers having
Hydrodynamics of Liquids in Mechanically Agitated Vessels 35
2, 4, 6 and 8 blades. Fig. 3.3-4(c) shows a comparison of the result of this study with the locus
of the axis of the trailing vortex of the other researchers. [Yiannekis et al (1987). Stoot &
Calabrese (1995)]. The locus can be found by determing the point where its axial velocity
component is zero along each azimuthal angle, while the point of teriminal is also located by
finding the point which Vz on the locus diverges or is not zero. From the top view of these
loci, it can be seen that the vortex first flows along the blade then turns into a tangential
direction and diverges at r*= 1.22~1.41 as it merges with the tangential flow stream at outside
of the impeller. For the impellers having more blades, its vortex tends to flow more apart from
the impeller. This is probably because the impeller with the narrower blade pitch could not
provide sufficient space for the vortex to develop, and the radial pumping fluid font the next
blade also forces the vortex to flow outward. By comparing the length of these loci, it is also
interesting to note the ratio of the azimuthal angle of each locus of axis of the trailing vortex
to the azimuthal angle occupied by neighboring blades is 0.56, 0.89, 0.91 and 0.67 for the
impeller having =2, 4, 6 and 8 blades respectively. The value of this ratio can be seen as a
measure of the close connection of the neighboring vortices. The closer connection of the
neighboring vortices could result in stronger flow characteristics generated by the trailing
vortex and vice versa.
(a)
Fig. 3.3-4(a) Top view of the loci of the vortex axis for various impellers.
(b)Fig. 3.3-4(b) Front view of the loci of the vortex axis in azimuthal direction for various
impellers.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS36
(c)
Fig. 3.3-4(c) Comparison of the loci of the vortex axis between this work and otherresearchers.
Comparison of the mean deformation rates
Since the data of mean deformation rates enables us to estimate the value of viscous
energy dissipation and the production rate of turbulence energy, the estimation of mean
deformation rates from mean velocity gradients is an important approach in examining the
effect of the strength of the trailing vortex on gas dispersion to a mechanically stired
aeratedvessel. The magnitude of the mean deformation rate of each trailing vortex is a
measure to compare its strength to disperse gas in the impeller region because it involves not
only normal stress components and shear stresses, but also turbulence energy production.
Figure 3.3-5(a)~(d) show the contour maps of the mean deformation rates for the
impellers having 2, 4, 6 and 8 blades at Z*=0.63 where most the axes of the vortices flow
through or near this plane. These contour maps also show that the stronger contours generally
follow the vortex axis which agrees with the finding of Stoots and Calabrese (1995). These
maps also indicate that the impellers having 4 or 6 blades have larger mean deformation rates
between the neighbor blades while the impeller with 2 or 8 blades has rather weak mean
deformation rates. To grasp the difference more clearly, the average and the maximum values
of the rnean deformation rates are calculated from contours along the axis of the vortex and
are listed for further discussion. Table 3.3-2 for the same rotational speed and Table 3.3-3 for
the same power input per each blade.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 37
Fig. 3.3-5(a)~(d) The contour of the mean deformation rates for the impellers having 2,4, 6, and 8 blades at Z*=0.63.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS38
Table 3.3-2 Comparison of average and maximum mean deformation rate of trailingvortex with N=4.17 rps.
Comparison of Average and Maximum Mean Deformation Rate
(•
δ ) of Trailing Vortices under 250 rpm (unit: s-1)2-blade 4-blade 6-blade 8-blade
vortex avg. max. avg. max. avg. max. avg. max.upper 151 176 175 220 160 189 135 159lower 138 170 144 176 140 173 125 157
Table 3.3-3 Comparison of average and maximum mean deformation rate of trailingvortices with 0.52 w/blade.
Comparison of Average and Maximum Mean Deformation Rate
(•
δ ) of Trailing Vortices under the Power Consumption of0.52 W/Blade (unit: s-1)
2-blade 4-blade 6-blade 8-bladevortex avg. max. avg. max. avg. max. avg. max.upper 151 176 172 204 160 189 138 162lower 138 170 141 176 140 173 120 150
Comparison of turbulent kinetic energy
It is understood that the rate of turbulent energy dissipation is a measure of the strength
of break up or dispersal of fluids in a turbulence field. Wu and Patterson (1989) have
proposed that the local rate of turbulent energy dissipation can be estimated by
L)TKE(A
2/3
=ε (3.3-8)
where A=0.83 and L is turbulence macroscale which is the order of the width of the
impeller. TKE is turbulent kinetic energy. From Eq. (3.3-8), the local rate of turbulent energy
dissipation is proportional to 3/2 power of TKE and TKE can be estimated from and the
results of LDA measurements through Eq.(3.3-6).Using the same approach as described in the
last section, the average and the maximum values of dimensionless TKE for each different
impeller under two different bases are shown in Tables 3.3-4 and 3.3-5.
Table 3.3-4 Comparison of average and maximum dimensionless turbulent kineticenergy of trailing vortex with N=4.17rps.
2-blade 4-blade 6-blade 8-bladevortex avg. max. avg. max. avg. max. avg. max.upper 0.129 0.160 0.143 0.190 0.130 0.170 0.057 0.090lower 0.088 0.128 0.101 0.152 0.100 0.142 0.055 0.086
Hydrodynamics of Liquids in Mechanically Agitated Vessels 39
Table 3.3-4 Comparison of average and maximum dimensionless turbulent kineticenergy of trailing vortex with N=4.17rps.
2-blade 4-blade 6-blade 8-bladevortex avg. max. avg. max. avg. max. avg. max.upper 0.129 0.160 0.143 0.190 0.130 0.170 0.057 0.090lower 0.088 0.128 0.101 0.152 0.100 0.142 0.055 0.086
The values listed in these tables demonstrate the same trend seen in the case of mean
deformation rates. The impeller with four blades produces the highest TKE followed by the
impeller with six blades while the impeller with eight blades has the weakest TKE.
3.3.3 Pressure distributions behind blade of the disk turbine impeller
Prior to depict the locus and conformation of trailing vortex By using the minimum
pressure connecting method, it is necessary to calculate the pressure distribution around
impeller. Fig. 3.3-8 shows the top view of the calculated pressure distribution for impellers
having different blade sizes with z*=0 and N=4.17rps. From the pictures shown in this figure,
it is clearly found that regardless of the impeller blade size, negative pressure regions always
appear behind the leading blade. It expands with the increase in stretching tangential angle
initially and shrinks after passing a maximum negative pressure zone. Since the power drawn
and liquid pumping capacity of the impeller were approximately proportional to the impeller
blade size under the same rotational speed condition, the large-blade impeller always results
in a broader and stronger negative pressure zone. This result indicates that the large-blade
impeller can generate a stronger vortex system and may perform a better gas dispersion as a
result. Adjusting the rotational speeds of the large-blade and small-blade impellers from
4.17rps to 2.75 and 5.35rps to give the same energy dissipation density as the standard-blade
disk turbine impeller has. Similar pictures of the pressure distribution were drawn for
impellers with different blade sizes under the same energy dissipation density
Pg/V=577.8W/m3 at the z*=0 plane, and the results were shown in Fig. 3.3-7. Comparing the
plots shown in this figure to those in Fig. 3.3-6, it is found that almost contrary results are
obtained. In other words, the small-blade impeller has broader and stronger negative pressure
zone behind the leading blade, which implies that the small-blade impeller may possess a
better gas dispersion capability at a constant Pg/V.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS40
(a)Large blade
(b)Standard blade
(c) Small blade Unit: Pa
Fig. 3.3-6 Top view of the calculated pressure distribution for the impellers havingdifferent blade sizes with z*=0 and N=4.17rps.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 41
(a) Large blade N=2.75rps, Pg/V=577.8W/m
(b) Standard blade N=4.17rps, Pg/V=577.8W/m
(c) Small blade N=5.35rps, Pg/V=577.8W/m Unit: Pa
Fig. 3.3-7 Top view of the calculated pressure distribution for the impellers havingdifferent blade sizes with z*=0 and Pg/V=577.8W/m3.
From the calculated pressure distribution around impeller, the locus of trailing vortex
between two neighboring blades can be easily depicted by using the minimum pressure point
connecting method. Figure 3.3-8 shows the top view of vortex loci for the impellers having
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS42
different blade sizes with z*=2z/W=0 and N=4.17rps. There are three points to be noted: (1)
the trailing vortex originates from the leading blade and stretches back and outside from the
impeller disk initially, and finally terminates at certain tangential angles behind the leading
blade; (2) the larger the impeller blade is, the closer the vortex locus to impeller disk will be,
which implies that the vortex structure is less destroyed by the shear flow produced by the
impeller; (3) the vortex generated by the large-blade impeller stretches over a larger tangential
angle, which are about 55o/60o, 40o/60o and 30o/60o for the large-blade, standard-blade and
small-blade impeller respectively. Since the vortex system dominated the gas dispersion, it
can be concluded that the large-blade impeller can disperse gas more effectively than the
standard-blade and small-blade impellers under the same rotational speed condition.
Fig. 3.3-8 Top view of the vortex loci for the impeller having different blade sizes withz*=0 and N=4.17rps.
A similar plot was drawn in Figure 3.3-9 to show the azimuthal view of the vortex loci
for the impellers with different blade sizes under the same situations. It is found that there are
two symmetrical vortices clinging to each blade. However, the vortex pair of the small-blade
impeller tends to merge into a single vortex after a long distance from the leading blade,
which may reduce the strength of the shear stress of trailing vortex and promote the efficiency
of coalescence between dispersed bubbles. The plots depicted in this figure also show that the
vortex locus will change its direction from vertical to horizontal as it departs from the leading
blade, especially for the large-blade impeller. Due to this trend, it is unsuitable to depict the
vortex locus close to the leading blade by connecting the zero axial velocity point at each
azimuthal slice. However, once the vortex departs from the leading blade for a short distance,
the vortex locus becomes almost horizontal and this method can be applied.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 43
Fig. 3.3-9 Azimuthal view of the vortex loci for the impeller having different blade sizeswith the rotational speed N=4.17rps.
For the practical design, a certain value of Pg/V is adopted more often than to assign a
given rotational speed. Fig. 3.3-10 and 3.3-11 show the top view and azimuthal view of
vortex loci with a given Pg/V value of 557.8W/m3, respectively. By comparing these vortex
loci with those obtained under the same rotational speed condition, it is found that although
the vortex locus for the large-blade impeller is still the closest to the impeller center, its
stretched tangential angle has shrunk from 55o/60o to 40o/60o. While, for the small-blade
impeller, not only the vortex locus becomes closer to impeller center, but the stretched
tangential angle has expanded from 30o/60o to 53o/60o, which indicates that the small-blade
impeller possesses the stronger vortex system among these impellers under this situation.
Fig. 3.3-10 Top view of the vortex loci for the impeller having different blade sizes withz*=0 and Pg/V=577.8W/m3.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS44
Fig. 3.3-11 Azimuthal view of the vortex loci for the impeller having different bladesizes with z*=0 and Pg/V=557.8W/m3.
Vortex conformation of the impellers with different blade sizes
Fig. 3.3-12 shows the vortex conformation depicted according to the calculated pressure
contour at each azimuthal slice around the Rushton turbine impeller with N=4.17rps. From
the picture shown in this figure, it is found that: (1) a pair of vortices cling behind each blade
and stretch back and outside from the impeller. They expose intense but opposite rotary
motion, which can tear gas into small bubbles; (2) vortex develops close to the leading blade
and grows in diameter along vortex axis initially. It changes direction from axial to horizontal
and sweeps outside from impeller. After passing a maximum diameter, the diameter of the
vortex becomes smaller and smaller along the vortex axis and finally the vortex breaks into
small eddies. Table 3.3-6 lists the change in the vortex diameter along vortex axis. The trailing
vortex grows in diameter within 0o to 9o behind the leading blade. After passing the largest
diameter “2.01cm” at 9o, the vortex shrinks and finally disappears at 40o behind the leading
blade.
Fig. 3.3-12 Conformation of the trailing vortex for the Rushton turbine impeller withN=4.17rps.
Hydrodynamics of Liquids in Mechanically Agitated Vessels 45
Table 3.3-6 Variation of the vortex diameter along the vortex axis.
Location
r*(=r/R) 0.5 0.5-0.7 0.93 1.15
θ 0o 1o-3o 9o 40o
Z*(=2Z/W) 0 0±1 ±0.6 ±0.4
Vortex diameter 0.81cm 1.20cm 2.01cm
Similar plots are drawn for the vortex of the large-blade and small-blade impellers and
the results are shown in Fig. 3.3-13. It is found that all the characteristics of conformations of
large and small blade impellers are similar to those for the standard blade impeller as shown
in Fig. 3.3-12. However, the two symmetrical vortices of the small-blade impeller tend to
merge after a long distance from the leading blade, which may diminish its gas dispersion
capability. Comparing the diameters of vortices for various impellers, it is found that the
vortex of the large-blade impeller always have the largest diameter under the same rotational
speed condition.
(a) Large-blade impeller
(b) Samll-blade impeller
Fig. 3.3-13 Conformations of the trailing vortex of the large-blade and small-bladepellers with a constant rotational speed N=4.17rps.
MULTIPLE IMPELLER GAS-LIQUID CONTACTORS46
Based on the determined locus and conformation of trailing vortex, the pressure
distributions at the core and circumference of vortex could be plotted against the position
change along vortex axis. Figure 3.3-14 shows such a plot for the impellers having different
blade sizes with N=4.17rps. In this figure, Zv denotes the position coordinate along the vortex
axis and Zc is the full length of vortex axis, therefore the value of Zv/Zc is zero at the
commencement of vortex and equal to 1 at the vortex tail (Van’t Riet and Smith, 1973). From
the plots shown in this figure, it is seen that no matter at the core or edge of trailing vortex,
the maximum negative pressure values always appears at the backside of the leading blade,
and then the pressure increases with departing from the leading blade. Comparing the pressure
at the core and around the circumference of trailing vortex, it can be found that the larger
negative pressure always occurs at the vortex core close to the leading blade, where the gas is
sucked into to form a ventilated cavity. It also can be found that the larger the impeller blade
is, the more negative pressure values were produced at the backside of the leading blade,
which may collect more sparged gas there.
Fig. 3.3-14 Variation of the pressure along vortex axis for the impellers having differentblade sizes with N=4.17rps.
NOTATION
b Baffle width [m]
C Tracer concentration [M]
Hydrodynamics of Liquids in Mechanically Agitated Vessels 47
C’ The lowest impeller clearance [m]
C1ε 1.44; empirical constant [-]
C2ε 1.92; empirical constant [-]
C3ε Empirical constant [-]
Cd Drag force coefficient [-]
CL Life force coefficient [-]
Cμ Coefficient of μt [-]
D Impeller diameter [m]
D32 Sauter mean bubble diameter [m]
Di Bubble size for each measurement point [m]
g Gravitational acceleration [m/s2]
k Dimensionless turbulent kinetic energy [-]
L Turbulent macroscale [m]
r Radial coordinate [m]
t Time [s]
tM Mixing time [s]
T Tank diameter [m]
u’,v’ The fluctuation velocity [m/s]
U Mean axial velocity of liquid [m/s]
v Mean radial velocity of liquid [m3]
V Stationary frame of reference mean velocity [m/s]
vmax Maximum radial velocity [m/s]
vtip Impeller tip velocity [m/s]
W Mean tangential velocity of liquid [m/s]
w Impeller blade width [m]
z Axial coordinate [m]
<Greeks Letters>Δ Mean rate of deformation tensor [1/s]
δ Mean deformation rate [1/s]
ε Turbulent energy dispersion per unit mass of fluid [W/kg]
θ Tangential coordinate [radian]
κ Turbulent kinetic energy per unit mass [W/kg]
μt Turbulent eddy viscosity [kg/m2]
ρ Density of fluid [kg/m3]
σk 1.0; empirical constant [-]
σt 1.3; empirical constant [-]
σε 1.3; empirical constant [-]
φ Variable of flow [-]
- φρu Reynolds stress [-]