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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.


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


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.


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


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


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:


])()([ 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


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).


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,


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


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°)


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


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°).


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


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


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


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.


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


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.


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


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.


(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.


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.


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


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.


(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.


(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


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.


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.


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.


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.


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]


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 [-]

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