cfd simulations of three-dimensional wall jets in a stirred tank

8/7/2019 CFD Simulations of Three-dimensional Wall Jets in a Stirred Tank

1/15

The Canadian Journal of Chemical Engineering, Volume 80, August 2002 1

Computational fluid dynamics (CFD) is increasingly being used forthe simulation of stirred tanks due to recent advances incomputer speed and efficiency of numerical schemes. While

simulations involving both steady state (Kresta and Wood, 1991; Fokemaet al., 1994; Harvey et al., 1995; Ranade and Dommetti, 1996 andothers) and time varying (Perng and Murthy, 1992; Derksen and Van den

Akker, 1999) methods have been reported in the literature, emphasis hasmainly been on the quantitative validation of flow close to the impeller.The main objective of this study is to extend the existing protocols forsimulating time-averaged velocity fields to flow near the tank wall and inthe bulk of the tank.

Studies with axial impellers (Bittorf and Kresta, 2001) show that walljets form in the region between the baffles and the tank wall and drivethe bulk flow, imposing a single characteristic velocity scale on both theupflow at the wall and the recirculation in the center of the tank.Accuracy in prediction of the mean flow characteristics of the wall jetsand the simulation of the bulk flow in the tank are therefore intimatelylinked. Moreover, Bittorf (2000) has shown that the velocities in thethree-dimensional wall jets, in balance with the settling velocities of thesolids, determine the cloud height of suspended solids at high solids

concentration. The cloud height of the suspended solids along with thejust suspended speed (Njs) of the impeller determines the uniformity ofsolids distribution in a stirred tank. While explicit relations between Njsand the fluid/particle properties and the impeller type are available, thecloud height model proposed by Bittorf (2000) requires an accuratedescription of the effect of size, off-bottom clearance and speed of theimpeller on the core velocity or source velocity of the jet. This study aimsto address these issues by developing a low-cost CFD protocol which canbe used to obtain geometry dependent parameters to the degree ofaccuracy required, without having to resort to scale model experiments.

While a detailed description of the wall jets is available in Bittorf andKresta (2001), the key results are restated here to facilitate comparisonwith CFD simulations in later sections. Figure 1a shows one of the walljets formed between the baffle and the tank wall. The expansion of the

jet and the decay of axial velocity in a vertical plane close to the baffleare shown for different axial positions in the tank. At any axial location,the axial velocity (U) increases rapidly from the no slip condition at thewall to its maximum value (Um), and then decreases with a smallergradient. At y~ 1.7b, the axial velocity reverses direction due to recircula-tion and asymptotically approaches the recirculating velocity (UR). The

*Author to whom correspondence may be addressed. E-mail address:[email protected]

CFD Simulations of Three-dimensional Wall Jetsin a Stirred Tank

Sujit Bhattacharya and Suzanne M. Kresta*

Department of Chemical and Materials Engineering, University of Alberta, Edmonton AB, T6G 2G6

The flow near the tank wall in a stirred tank drivenby a 45 pitched-blade turbine is simulated withMultiple Reference Frames, the k-e turbulence modeland standard wall functions. The results are comparedto the three-dimensional wall jet identified in aprevious paper. The self-similar velocity profiles in the

jet are predicted satisfactorily, but the decay of thelocal maximum velocity and jet expansion areunderpredicted. The underlying physical reasons forthis failure are investigated. The effect of impeller sizeand position on the impingement point of theimpeller discharge and the jet core velocity are wellpredicted by the simulations. The results provide abenchmark for CFD/MRF in the bulk of a stirred tank,identifying where CFD over- or underpredictsperformance.

Lcoulement engendr par une turbine palesinclines 45 prs de la paroi dun rservoir estsimul par la mthode des rfrentiels multiples(MRF), le modle de turbulence k-? et des lois de paroi

standards. Les rsultats sont compars ceux du jetde paroi tridimensionnel dcrit dans un articleantrieur. Les profils de vitesse auto-similaires du jetsont prdits convenablement, mais la dcroissance dela vitesse locale maximale et lexpansion du jet sontinsuffisamment prdites. Les raisons physiquessous-jacentes de cette lacune sont tudies. Leffet dela taille et de la position de la turbine sur le pointdimpact de lcoulement de refoulement de laturbine et la vitesse du noyau de jet sont bien prditspar les simulations. Les rsultats fournissent un bancdessai pour la CFD/MRF dans le cur dun rservoiragit, ce qui permet de dterminer les limites deperformance de la CFD.

Keywords: stirred tank, PBT impeller, CFD,turbulence, three-dimensional wall jets, mixing,multiple reference frames.

characteristic length scale, b, is the half-width of thejet and corresponds to the radial distance (y) from thewall, where U= Um/2. Um is located at ym, a distanceof about 0.15bfrom the wall. The region 0 < y ym isthe inner region of the wall jet, where the flowcharacteristics are similar to those in a simple


2/15

2 The Canadian Journal of Chemical Engineering, Volume 80, August 2002

boundary layer. The region y> ym is the outer region, where theflow characteristics are similar to a free jet. Interactions betweenthese two regions project the anisotropic influence of the wallto the outer (free shear) region of the jet and make the flow fieldsomewhat more difficult to model. Thus, the accurate prediction

of jet decay is intimately linked to the successful modeling ofturbulence. In Figure 1a (D = T/3 and C/D ~ 1.0), the globalmaximum velocity along the tank wall or the core velocity(Ucore) occurs at z/T~ 0.2. As the jet moves upwards, it expandsin the y direction with a simultaneous decay in Um. The jetfinally collapses near the top of the tank (z/T~ 0.7 forD = T/3and C/D ~ 1.0). Between these two limits of formation anddisintegration, the three-dimensional wall jet is self-similar. Thismeans that the radial profiles of axial velocity collapse onto asingle similarity profile when Uis scaled with Um and yis scaledwith b (h = y/b), as shown in Figure 1b.

For a free jet with zero free stream velocity the similarityprofile follows the similarity solution of Goertler for a round jet(Rajaratnam, 1976). By modifying the boundary conditions,Kresta et al. (2001) extended the Goertler solution to includethe effect of the recirculating flow outside the jet region:

where

In Equation (1), Bis used to capture the effect of the recirculatingvelocity (UR) and is given by:

F is needed to force the solution through U/Um = 0.5 when h = 1.The h d term in Equation (1) accounts for the displacement ofthe outer region of the jet by the inner boundary layer. Theboundary layer thickness grows linearly with the jet half-width(boundary layer thickness = db and d 0.15 for wall jets). Thevalues ofB, F and dfor a three-dimensional wall jet are definedusing the velocity data of Bittorf and Kresta (2001). The resultingsimilarity profile is:

This equation, in conjunction with expressions for themaximum velocity decay and jet expansion, provides a completecharacterization of the vertical flow. Bittorf and Kresta (2001)relate Um and b in two regions of the jet. In the characteristicdecay region a classical wall jet is still responding to the geometryof the nozzle. In the stirred tank the wall jet is responding to thecombined effects of tangential flow and circulation.

In radial decay, the jet is fully formed and has no recollectionof the initial effects of geometry. In cross-section it forms a quarter

circle with expansion in the radial direction. The maximumvelocity Um is inversely proportional to the distance traveled:

The jet half-width expands as:

(6)

b

T

z z

To=

-

0 38.

(5b)U

U

z

Tm

core

=

-

0 571 15

..

(radial decay)

(5a)

U

U

z

Tm

core

=

-

1 350 49

..

(characteristic decay)

(4)

U

Um= - -( )[ ]1 1 58 0 78 0 152. tanh . .h

(3)

BU

UR

m

= -

1

(2)

h

d

d

=

= +

= +

y

by y b

b b b

jet

jet

(1)U

UB

m

= - -( )[ ]1 2tanh F h d

Figure 1a. Radial velocity profiles in the wall jet at different axial

locations in a stirred tank.

Figure 1b. Similarity profile of mean axial velocity in the three-dimensional wall jet.


3/15

zo in Equation (6) denotes the virtual origin of the jet. Bittorfand Kresta (2001) modeled the three-dimensional jets driven byaxial impellers as originating from a point source located belowthe bottom of the tank. The position of this virtual point sourceis described as the virtual origin. Equations (4), (5) and (6) arevalid for PBT, HE3 and A310 impellers (Bittorf and Kresta, 2001).This point is important because it means that the vertical flowfield for all of these commonly used impellers can be character-ized using these three equations. The reader will note that whileEquations (4), (5) and (6) are universal for these axial impellers,the quantities zo and the core velocity (Ucore) are dependent onthe impeller and tank geometry and need to be determinedseparately. This provides an ideal opportunity for the use of CFD: thegeometry dependent parameters are predicted using simulationsand are then used in Equations (4) through (6) to obtain thecomplete flow field. This work will show that CFD simulationspredict core velocities which are in reasonable agreement withexperimental data. However, simulations are not able to predictzo because of its dependence on the decay ofUm and expansionofb, two aspects of jet simulation where CFD commonly fails.

The location of the wall jet in stirred tanks is determined bythe point of impingement of the impeller discharge stream. Theimpingement point can be on the bottom of the tank or on

the tank wall, depending on the impeller type, diameter andoff-bottom clearance. The impingement point has an effect onthe core velocity in the wall jet and the position of the jet in thetank (Bittorf and Kresta, 2001). While experimental datatypically lack the required resolution, detailed velocity profilesobtained from CFD simulations are suitable for identifying thelocation of the impingement point and can be used to studythe effect of Reynolds number, off-bottom clearance and tanksize on the formation and decay of the jet. In this paper, CFDresults are used to arrive at a qualitative understanding of therelationship between the geometry of the vessel, the corevelocity, and the impingement point.

The simulations in this work have been carried out using thesteady state, Multiple Reference Frames approach which

requires that there be negligible interaction between the bafflesand the impeller. If baffle-impeller interactions are large or iflarge-scale transient structures (macroinstabilities) are present,unsteady state modeling using the sliding mesh approachand/or large eddy simulation (LES) is advisable. However, thecomputational resources and post-processing required for steadystate simulations are only a fraction of those required for transientsimulations, and jets are, by definition, time-averaged reducedmodels of the flow. The MRF method has therefore been adoptedin this study with a view to identifying: (i) the useful informationthat can be obtained from this modeling approach, and (ii) theconditions under which this approximation fails.

The main objectives of this study can now be summarized asfollows:

the development of protocols for MRF simulation of flow nearthe tank wall and in the bulk of the tank, and a determinationof the limitations of this approach;

the prediction of the key geometry dependant characteristicsof the jet: virtual origin, core velocity and impingementpoint, for use in applications such as estimating the solidscloud height in solid suspensions;

the prediction of the decay ofUm in the radial decay regionof the jet.Once these objectives are met, the simulation results will be

used to gain further insight on the flow field and on the

development of the jet. This extension of CFD results representsa new generation of investigations, moving beyond validationand code development to a point where researchers can obtainnew information and insights from CFD results.

Experimental DataExperimental data from Bittorf and Kresta (2001) has beenextensively used in this study to illuminate and validate results of theCFD simulation. Since a detailed explanation of the experimentalsetup is available elsewhere (Bittorf and Kresta, 2001; Bittorf,2000), only a brief outline is presented here. Of the various casesstudied in the original experiments, only the geometries andoperating conditions used in the simulations are described.

Axial velocities were measured using a Laser DopplerVelocimeter (LDV). The important geometric variables for thestirred tank are shown in Figure 1a, b. The 0.24 m diameter tankhad a flat bottom with an aspect ratio (H/T) of 1 and was sealedat the top surface. The impeller was a standard, 4-bladed, down-pumping, 45 pitched-blade turbine (PBT). Experiments usedimpeller diameters of T/3 and T/2. For the T/3 impeller,experimental data is available for two clearances (C/D = 0.4 and 1.0)and two impeller speeds (N= 500 and 1000 rpm).

CFD SimulationsCFD simulations were carried out in two geometrically similarstirred tanks with diameters of 0.24 and 1.00 m. The effect ofimpeller clearance on the velocity field was studied by varying theclearance from 0.4 < C/D < 1.9. For the smaller tank, simulationswere carried out at impeller speeds of 500 and 1000 rpm(Re= 5.3 104 and Re= 1.1 105, respectively). The effect ofscale was tested using an impeller speed of 60 rpm in a T= 1.00 mtank. Water was used as the working fluid in all cases.

The fluid properties, tank geometry, impeller type anddimensions (shaft diameter, hub size, blade width, and bladethickness), and the rotational speed, were set using MixSim

version 1.5, a pre-processing software specifically designed forCFD simulation of stirred tanks. PreBFC, a mesh generator,

uses the parameters defined in MixSim

to create a structuredgrid, and subsequently CFD calculations are launched inFLUENT version 4.5.2. Iterations were continued until the sumof the normalized residuals of all variables (pressure, axial, radialand tangential components of velocity) was less than 5 104.Both the geometry of the impeller and velocity data at theimpeller are available for the PBT in the MixSim library;however, Fokema et al. (1994) observed that the boundaryconditions at the impeller are strongly affected by the off-bottomclearance, and recommended extreme caution when selectingdata for this approach. In the present study, the geometry of thepitched-blade turbine was defined in the grid and the velocityprofile near the impeller was calculated directly in the simulation.This approach has been validated by Harvey and Rogers (1996).

Simulations were carried out using the Multiple ReferenceFrames (MRF) scheme. In the MRF formulation, the grid is dividedinto two regions: a rotating volume is associated with the impeller,while a stationary volume is associated with the baffles and thetank wall, as shown in Figure 2. Although the grid in the impellerregion is stationary, the rotation of the impeller is modeled directlyby solving the transport equations in a reference frame whichrotates at the angular velocity of the impeller. The transportequations in the outer region are solved in the stationary reference frame of the tank. Since the grid in the impeller region is keptstationary, it is fixed relative to the location of the baffles. The most



4/15

important assumption made in the MRF model is that there is veryweak, if any, interaction between the impeller and the baffles. Atthe boundary between the two reference frames, the terms in thetransport equations in one frame require values from the adjacentreference frame. FLUENT enforces continuity of the absolutevelocity across the boundary, to provide the correct neighborvalues for the region under consideration. Note that velocities arenot averaged over all tangential positions of the reference frameboundary to obtain a full time-averaged solution (the mixing-planeapproach) but are passed point by point across it to give asnapshot of the flow-field. The velocity in the rotating reference frame, which has been calculated relative to the motion in thisregion, is converted to the absolute inertial reference frame usingthe following definitions for the position vector (r), etc.

the absolute velocity vector

and the velocity gradient

Since the velocity is now defined in terms of the stationaryreference frame, it can be readily used in the transport equations onthe stationary side of the boundary. For steady state calculations,the tank is considered symmetrical, and hence, for the four-bafflecase, only one quarter of the tank is simulated.

Grid ResolutionInsufficient grid resolution can give rise to errors in simulationeven when there are no deficiencies in the physical modelsapplied to the system. Thus, an appropriate grid must bedefined before any further simulation is undertaken. Afterreviewing numerous CFD studies on stirred tanks, Kresta (1996)

(9) = + ( )v v rr w

(8)v v r= + r w

(7)r x x= - o


Figure 2. Isometric view of tank configuration and grid geometry.Cylindrical co-ordinate equivalents to the FLUENT (i, j and R) co-ordinates are q = i, r=jand z= k.

Figure 3. Grid refinement in a single i-plane.

Figure 4. Effect of grid on maximum jet velocity: D = T/3, C/D = 0.4,N= 1000 rpm, x = 3.6 mm, y~ 5 mm.


5/15


suggested that a minimum grid should consist of at least about20 cells in the tangential (i), 20-25 cells in the radial (j) and 40cells in the axial (k) direction. In this survey, cell sizes rangedfrom 4.5 to 5 mm while the tank diameters varied from 0.15 to0.44 m. Fokema et al. (1994) used a grid consisting of 43 axial,20 radial and 20 tangential cells for the simulation of flowfieldwith a PBT. While simulating the flow for dual Rushton turbines,Jaworski et al. (2000) recommended that a reasonable degreeof grid independence could be obtained with a mesh densityfactor of at least 32 cells per tank diameter.

In the present study, five grids were investigated todetermine the effect of grid resolution on the computed results.A structured grid was generated using the fine grid (at least 60cells per tank diameter) option in MixSim. A view of the gridin the r-z plane is shown isometrically in Figure 2 and isreproduced as a plane in Figure 3. This mesh density produceda grid with 50 cells in the tangential, 32 cells in the radial and63 cells in the axial direction (50 32 63). It is the default grid(Figure 3a). To improve grid resolution near the tank wall the last four cells in the radial direction were halved to increase thenumber of near-wall cells while maintaining a cell-aspect ratioof two (Figure 3b). This grid had 50 36 63 cells. A third gridwas obtained by further refining the near-wall cells in the

second grid by halving the last four cells in the radial directionto obtain 50 40 63 cells (Figure 3c). The fourth grid (Figure3d) was created by adding cells near the tank bottom, giving50 40 71 cells. Finally (Figure 3e) the last four cells near thetank bottom in Figure 4e were halved to give a higher axialresolution (50 40 75 cells).

The effect of the grid resolution on an axial traverse of themaximum streamwise velocity is shown in Figure 4. The axialvelocity is grid independent beyond 36 cells in the radial and71 cells in the axial direction. Surprisingly, forz/Tless than 0.2, themodified grids all show similar velocities. There is no effect of gridresolution in this region after the initial refinement in the radialdirection (changing from 32 to 36 cells). Similar results wereobserved for traverses in the radial direction. A grid with 50 cells in

the tangential, 40 cells in the radial and 71 cells in the axial direction,using the near wall refinement shown in Figure 3d, was selected forall remaining simulations in this study.

Turbulence ModelingAlthough the most rigorous turbulence model is the instantaneousNavier-Stokes (NS) equation, direct numerical simulation (DNS)of this equation requires computational resources of the orderof the cube of the Reynolds number (Bradshaw, 1999). Analternative method, Reynolds-averaged turbulence modeling, isused extensively in engineering applications involving highReynolds numbers. However, the averaged models suffer froma lack of universality, and successful numerical modeling of aturbulent flow field strongly depends on the choice of an

appropriate turbulence model. In the current study a preliminaryinvestigation was conducted to select a suitable turbulencemodel based on its ability to:1. accurately reproduce the self-similar axial velocity profile;2. accurately reproduce the development and decay of the

maximum local axial velocity (Um); and3. attain stable convergence.The version of FLUENT used in this study (4.5.2), supports twowidely used and very successful turbulence models:1. two equation, k-e model; and2. Reynolds stress model (RSM)

It may be noted that the k-e model has several variants. Apart from its standard form (Jones and Launder, 1972), FLUENT

4.5.2 also supports the Renormalization Group (RNG) k-emodel, which is derived by the application of a rigorous statisticaltechnique (RNG), to the instantaneous Navier-Stokes equation.

The standard k-e model is based on the Boussinesq hypothesisthat the Reynolds stress is proportional to the mean velocitygradient, with the constant of proportionality being theturbulent or eddy viscosity (mt). The eddy viscosity is thendefined in terms of two variables: the turbulent kinetic energy(k) and the rate of dissipation of turbulent kinetic energy perunit mass (e). The full k-e model is given in Table 1. Detaileddiscussions of the terms in the k-e model are available in theliterature (Launder and Spalding, 1972, 1974; Mohammadi andPironneau, 1994). The k-e model has proven useful for manyturbulent flows (jet flow, duct flow etc); however, like othermodels based on Boussinesq viscosity approximations, the k-emodel has a number of shortcomings. The effect of thesedeficiencies on the simulation of complex flow fields has beendiscussed in numerous articles (Launder and Rodi, 1983;Launder, 1995; Launder, 1999; Bradshaw, 1999). A shortsummary of the problems associated with the eddy-viscosityassumption in simulating certain classes of jets is given below:

1. The k-e model does not predict the significantly larger lateralexpansion in three-dimensional wall jets. This is primarily dueto its inability to account for the generation of streamwisevorticity by turbulent stresses. This occurs in addition to thecontributions by the mean velocity gradients.

2. The model is unable to simulate the correlation betweenpressure and stress transport. Thus, it cannot account for thedamping of normal velocity fluctuations even in the outerfree-shear region due to pressure reflections from the wall.The overestimation of the rate of spread of plane wall jetsnormal to the wall (Launder and Rodi, 1983) has been attrib-uted to this deficiency.

3. The k-e model is unable to account for the generation ofadditional turbulence due to the effect of streamwise

curvature and is therefore very unreliable in simulating flowswith sharp curvature4. The spread of an axisymmetric round jet is overestimated by

the standard k-e turbulence model. Various modelers havetried to adjust the transport equation fore to account for thisshortcoming, with varying degrees of success (Wood andChen, 1985).The RNG k-e model aims to develop a more general and

fundamental model of turbulence. In this model formulation,renormalization group theory is first used to resolve the smallesteddies in the inertial range. Once these small-scale eddies havebeen resolved, a small band of the smallest eddies is eliminatedby representing them in terms of the next smallest eddies. Thisprocess of successively removing a band of the smallest modes

is continued until a modified set of Navier-Stokes equations isobtained, which can then be solved using a relatively coarse grid.The equations constituting the RNG k-e model are shown in Table 1,and further details of the model are available elsewhere.

The full differential Reynolds Stress Model (RSM), also calledthe stress transport or second-moment model, involves solvingthe transport equations for the six individual Reynolds stressesin the Reynolds-averaged Navier-Stokes equation (RANS). Thesetransport equations can be obtained directly from themomentum equation, and contain terms for the pressure-straindistribution, turbulent diffusion, and generation and dissipation


6/15


Table 1. Simplified Equations fork-e and RSM turbulence models.

Reynolds Averaged Navier-Stokes Equation (RANS) (for steady, incompressible flow without body-force*)

where the Reynolds stresses are and viscous stresses are negligible.

(Mean values are represented by capital letters, and fluctuating quantities by lowercase letters).

Turbulence Models*

i) k-e model Boussinesq hypothesis(isotropic stresses)

Turbulent viscosity

Cm = 0.09

A) Standard Form Transport equation for

turbulent kinetic energy (k)

sk = 1.0

Transport equation for rateof dissipations ofk(e)C1e = 1.44

C2e = 1.92

se = 1.3

B) RNG k-e Transport equation forturbulent kinetic energy (k)

Transport equation for rate of

dissipations ofk(e)

ii) RSM

rUU

x

P

x

U

x x

R

xk

i

k i

i

j j

ij

j

= -

+

+

2

R u uij i j = -r

R u u k U

x

U

xij i j ij t

i

j

j

i

= - = - +

+

r r d m2

3

m re

e n

mt

i i i

j

i

j

j

i

Ck

ku u u

x

u

x

u

x

=

= =

+

2

2where and

r mms

reUk

x

U

x

U

x

U

x x

k

xi

it

j

i

i

j

j

i i

t

k i

=

+

+

-

Convection Production Diffusion Dissipation

r m a m re

m m m

Uk

xS

x

k

x

S S S S U

x

U

x

ii

ti

k e ff i

ij ij ij j

i

i

jeff t

= +

-

+

= +

2

21

2


where,

and,

re e

m a me

re

e e eUx

Ck

Sx x

Ck

Rii

ti

effi

=

+

-

-1

22

2

Convection Production Diffusion Dissipation Other terms related

to mean strain and

turbulence quantities

Reynolds stress transport equations: , where the terms can be expanded as follows:

Production: calculated

Pressure train distribution: (modeled)

Dissipation:

r e

r

e m

Uu u

xP

J

x

P u uU

x u u Ux

s pu

x

u

x

ki j

kij ij ij

ijk

k

ij i k jk

j k ik

iji

j

j

i

ij

= + - +

-

+

-

+

F

F

( )

2uu

x

u

x

J u u u p u u

i

k

j

k

ijk i j k jk i ik j

= - + +( )

(calculated from

Turbulent diffusion: (modeled)

e

r d d

)

re e

mms

er

ee

eeU

xC

k

U

x

U

x

U

x x xC

ki

it

j

i

i

j

j

i i

t

i

=

+

+

-

1 2

2



7/15

of Reynolds stresses. The formulation of the RSM used in thiswork is also given in Table 1. More detailed discussion of theindividual terms and assumptions in this formulation is given byLaunder et al. (1975) and Launder (1989). The RSM avoids theisotropic eddy-viscosity assumption of the k-e models, andtherefore has very good potential for accurately predictingcomplex flows with streamwise curvature, swirl, rotation andhigh strain rates. However, these models require wall-reflectionterms which may need to be redesigned for a particulargeometry. The improvement in prediction accuracy must be

substantial to justify the increase in computational requirementsdue to the larger set of highly coupled differential equations.

Integrating the turbulence models (eitherk-e model or RSM)through the near-wall region and applying the no-slip conditionat the wall requires some additional consideration, since theturbulent quantities in the near-wall region are significantlyaffected by the proximity of the wall. Turbulence is heavilydamped very close to the wall; however, in the outer part of thenear-wall region, it is rapidly augmented by the production ofkinetic energy, due to Reynolds stresses and the large gradientin the mean velocity. To overcome these difficulties, twoapproaches can be used: Application of wall-functions to bridge the viscosity-

affected region between the wall and the fully turbulentregion of the flow.

Resolving the near-wall viscosity affected region (includingthe viscous sub-layer) with extra-fine mesh all the way to thewall. This near-wall modeling approach is implemented inFLUENT using a two-layer zonal model. This method doesnot rely on the semi-empirical log-law, but requires muchhigher CPU time.Since the grid resolution appropriate for the latter approach

requires substantial computational resources, wall-functions

were used in all simulations in this work. Various combinations of the three turbulence modelsdescribed above and the two wall-functions were tested. The mostimportant mean characteristic of the wall jet is the self-similar axialvelocity profile. In Figure 5a, the computational results for asingle traverse location (z = 13.6 mm above the bottom of the tank,x= 3.6 mm from the baffle) are compared with the experimentaldata at the same location and the universal similarity profile for three-dimensional wall jets in a stirred tank. The standard wall-function andthe pressure-sensitized non-equilibrium wall-function gave verysimilar results and in fact overlap. Since adverse pressuregradients are not important in this flow field, this result is notsurprising. Both the k-e model and the RSM with the standardwall-function are able to reproduce the similarity profile of the

wall jet. The RSM gives exact agreement with experimentalresults at this location. The time required for convergence withthe RSM was about nine times that for the k-e model. The RNGk-e model with the standard wall-function was not able toreproduce the similarity profile, overestimating the size of theinner region by a factor of four, underpredicting the point offlow reversal, y0, and overpredicting UR.

A more demanding test of turbulence models is the developmentand decay of the maximum velocity in the jet, Um, shown inFigure 5b. All velocities were measured at a distance of 5 mmfrom the tank wall and 3.6 mm from the baffle. The k-e modelwith the standard wall-function and the RSM with the standardwall-function come closest to reproducing the magnitude ofUm. The RNG k-e model does not give the correct magnitude ofUm,

but the growth, location of maximum velocity, and the decaywith distance match the experimental decay almost perfectly.The k-e model is able to capture the similarity profile and matchthe magnitude of Um, while being computationally the leastdemanding. This model is used for all remaining simulations.

Results and DiscussionIn the first part of this section, the similarity profile and theoverall flow field obtained from CFD simulations are comparedwith experimental results from previous studies. This part simplyconfirms that CFD is able to generate circulation patterns


Figure 5b. Effect of turbulence models on dimensionless axial velocityprofile: T= 0.24, C/D = 0.4, N= 1000 rpm, x= 3.6 mm, y= 5 mm. Standardwall functions were used for all cases. The results for the non-standard andstandard wall functions overlap for the test case with the k-e model.

Figure 5a. Comparison of turbulence models at z/T~ 0.56: T= 0.24 m,D = T/3, C/D = 0.4, N= 1000 rpm, x= 3.6 mm, z~ 13.6 mm.


8/15

observed in experiments and that the similarity profilesobtained from CFD conform to Equation (4). As discussedearlier, the three-dimensional wall jet equations (4, 5 and 6) canbe used to completely characterize the vertical flow field for allcommonly used axial impellers if the geometry dependentparameters, Ucore and zo , are known. The second part of thissection considers the ability of CFD to adequately estimatethese parameters. The reader may note that zo is obtained byextrapolating the loci of b backwards until the resulting lineintersects the plane of the wall. Accurate prediction of z

ois

therefore dependent on the rate of decay of Um and theexpansion ofb. Finally, the impingement point of the impeller

discharge at the tank wall and the vertical distance in the tank overwhich the jet exists are estimated from CFD simulations. Thesequantities cannot be readily obtained from experiments becauseof the requirement of high data resolution. The results deepen andconfirm our physical understanding of the flow field, providinginsights which are not available from experiments alone.

Circulation Pattern and Similarity ProfileFigures 6a to 6d show the predicted circulation patterns in anr-zplane for two impeller sizes (D = T/3 and T/2), and two off-bottomclearances. The discharge from the impeller flows downwardsand impinges either on the bottom or on the wall of the tank.If the impeller discharge stream impinges on the bottom of thetank (Figures 6a and 6c) it forms a single, upwards circulationloop at the wall. If, however, the discharge impinges on the wall(Figures 6b and 6d) it splits in two. Most of the flow movesupwards into the wall jet but below the point of impingement,a portion of the stream flows downwards into the weakersecondary circulation. Comparison of Figures 6b and 6d shows thatthe secondary circulation is much stronger for the larger (D = T/2)impeller. There is good qualitative agreement between theseresults and flow patterns observed in the flow visualization and LDVexperiments of Kresta and Wood (1993), and Bakker et al. (1996).

The MRF formulation correctly predicts the off-bottomclearance where the flow shifts from a single upwards circulationto wall impingement. This point is at C/D = 0.67 for the D = T/2impeller (Kresta and Wood, 1993).

As shown in Figure 7, there is also good agreement betweenthe similarity profiles obtained from CFD simulations andEquation (4). Beyond h = 1.7, the velocity profile is affected byrecirculation and similarity is lost at larger h. For h < 1.7 thesimilarity profile is immune to changes in C/D, N(not shown inthe figure) and scale of the vessel. These results support theexperimental observations of Bittorf and Kresta (2001), whichshow that the similarity profile is unaffected by tank geometry.


Figure 6.Velocity vector fields showing the interacting effects of impellerdiamter and off-bottom clearance: D = T/3, T= 1.00 m, N= 60 rpm, i= 47.Note the impingement on the tank wall at higher off-bottom clearances.

Figure 7. Effect of tank size and off-bottom clearance on similarityprofiles: D = T/3, Re= 105, x= 3.6 mm, z~ 136 mm forT= 0.24 mand 558 mm forT= 1.00 m.


9/15

Geometry-Dependent Parameters Ucoreand zoBefore starting a discussion of the geometry-dependentparameters, the important characteristics of the flow near thetank wall are identified using the development and decay ofUmalong the tank wall at different impeller clearances forT= 0.24 mand D = T/3, as shown in Figure 8. Starting from the bottomof the tank, the reader should note the regions of reverse flow,the point of impingement, and Ucore labeled in the figure. Notethat the small region of reverse flow near the tank bottom atlow impeller clearances (C/D = 1.2 and 1.4) is not due to wallimpingement but to a small eddy which forms in the cornerbetween the baffle, tank bottom and the wall (observe thebottom left corner in Figures 6a and 6c). A region of mildreverse (downward) flow is also seen in the top 20% of thetank. This is present at all clearances. The core velocity, Ucore, isidentified as the global maximum of Um, and is one of twoparameters needed to characterize the vertical flow in a newgeometry. The relationship between Ucore and the tankgeometry is discussed in the next section.

Core VelocityReynolds scaling dictates that the velocity field in a stirred tankscales with Vtip under fully turbulent conditions as long as

geometric similarity is maintained. Hence, Ucore is scaled withVtip in Figures 9a and 9b to compare values obtained for differenttank sizes (T= 0.24 and T= 1.00 m for D = T/3 impeller). Anumber of important observations can be made from the figures. First, Ucore/Vtip is unaffected by tank size, confirmingthat Ucore scales with the tip speed according to Reynoldssimilarity. Secondly, the Ucore obtained from simulations iswithin 14% of the experimental data for the T/3 impeller inFigure 9a. Figure 9b shows that the deviations are higher forthe T/2, impeller but the experimental trends are followed bythe simulations. Thirdly, the core velocities predicted using theRSM and RNG k-e models underestimate the experimental resultsby 16% and 23%, respectively, in Figure 9a. Predictions based onthe k-e model follow the trend defined by the experimental

data (Bittorf and Kresta, 2001) more closely, with smaller meandeviations, as given in Table 2.

For both impeller sizes, the core velocity initially drops withan increase in C/D and then starts increasing. Thus, for the T/3impeller in Figure 9a, Ucore/Vtip reduces with increasingclearance forC/D less than 1.4. At higher clearances there is aslight increase in the core velocity due to compression of flowtowards the tank wall by the secondary circulation loop. For theT/2 impeller in Figure 9b, Ucore/Vtip decreases rapidly when

the clearance is increased from 0.33 to 0.6. Thereafter, the corevelocity increases with an increase in C/D due to the influenceof the strong secondary flow.

CFD simulations using the k-e turbulence model provideestimates of Ucore which are in sufficient agreement withexperimental data to be used as estimates for Equation (5). Thisis especially true for the smaller (T/3) impeller. Since only twoexperimental data points are available for the T/2 impeller, aconclusive judgement about the accuracy of the results for thisimpeller size cannot be made, but the predicted values matchthe experimental trends.

Decay of Maximum Velocity, Expansion of Jet Half-widthand Virtual Origin

As explained earlier, the virtual origin, zo , is obtained byextrapolation of the loci of the jet half-width, b, back to thepoint where they intersect the tank wall. The expansion ofb isrelated to the decay of the maximum velocity by themomentum integral constraint, which requires the momentumin the jet to be conserved (Rajaratnam, 1976). Hence, the decayofUm is closely tied to both the expansion ofb and the locationofzo. Using C/D = 0.4 and 1.0, Bittorf and Kresta (2001) foundthat Um/Ucore varies with z/T

0.59 in the characteristic decayregion and with z/T1.15 in the radial decay region. Their resultsare compared with the Um/Ucore values obtained from CFDsimulations in Figure 10. Three regions can be identified in thefigure, corresponding to the potential core, characteristic decayand radial decay regions in a three-dimensional wall jet. The

simulated values ofUm are closer to the experimental values inthe characteristic decay region, but the slope is much smallerthan the experimental results in both regions. Consequently,the jet half-width obtained from simulations expands moreslowly than the experimental results.

The flow field along the wall of the tank provides further insightinto the development and growth of the wall jet along the baffles.Figure 11a shows the contours ofUm/2 at different axial locationsin the 0.24 m tank, with C/D = 0.4 and D = T/3. The filled regionsrepresent the locations on horizontal slices (r-q plane), where theaxial velocity is greater than Um/2. The impeller discharge


Figure 8.Variation of maximum axial velocity (Um) with axial position:T= 0.24 m, D = T/3, N= 1000 rpm, x= 3.6 mm, y= 0.56 mm.

Table 2. Comparison ofUcorefrom CFD simulations with data from

Bittorf and Kresta (2001). (D = T/3, T= 0.24 m)

Turbulence C/D Ucore/Vtip Ucore/Vtip %

model (Experimental) (CFD) Difference

k-e 0.4 0.32 0.3003 6.2k-e 0.8 0.28 0.2576 8.0k-e 1.0 0.27 0.2315 14.3k-e 1.5 0.17 0.1868 -0.1RNG 0.4 0.32 0.27 15.6

RSM 0.4 0.32 0.248 22.5


10/15

impinges on the bottom of the tank and then swirls outwardstowards the tank wall. On reaching the wall, the fluid isdeflected upwards and the radial swirling flow is transformedinto an upward flowing column of fluid with a strong tangentialmovement towards the baffle. Evidence of the tangential motionis seen in Figure 11a, in the shrinkage of the contours atlocations far from the baffle and the simultaneous radial growthat locations close to the baffle. Note also the velocity field (insetin Figure 11a). Velocity vectors show the movement towards

the baffle and provide further support for the flow patterndescribed earlier. The tangential movement along the curvedwalls of the tank compresses the fluid towards the baffles andaffects the growth of the three-dimensional wall jet in front ofit. Thus, the prediction of the wall jet characteristics along thebaffle depends on the ability of the numerical scheme andthe turbulence model to accurately simulate flow along thecurved tank surface.

If the impeller-baffle interactions are significant, then thesnapshot approach used in the MRF formulation will itself lead to

inaccurate results. In the presence of impeller-baffle interactions,the impeller will influence the tangential component of thevelocity near the wall throughout the 90 sweep, while the MRFformulation will not be able to capture this effect. Figure 11bshows the axial velocity profiles in the tangential direction atdifferent axial positions, as obtained from CFD simulations andfrom the experimental data of Bittorf and Kresta (2001). At lowaxial positions the flow is reproduced fairly well by CFD.However, higher up along the tank wall, CFD simulations showonly a slow decrease in velocity as the angular distance from thebaffle is increased, while a much sharper reduction in velocity isapparent in the experimental data. The larger axial velocities inthe CFD results imply that the tangential flow towards the baffleis smaller in the simulations than in the experiments because of

continuity and the two-dimensional nature of the near-wall flowfield. The smaller tangential flow is due to the inability of theMRF formulation to account for the impeller-baffle interactionand results in less compression of the fluid onto the baffle.

Figure 12 is a log-log plot of the dimensionless maximumaxial velocity (Um/Ucore) versus dimensionless axial distance(z/T) for various impeller clearances. The slopes of these curvesgive the decay of the maximum velocity (Um) with distance.Two regions can be defined for all clearances. These correspondto the characteristic and radial decay regions first shown inFigure 10. ForC/D < 1.4, decay ofUm in both regions is much


Figure 9a. Effect of off-bottom clearance and tank size on dimensionlesscore velocity: D = T/3.

Figure 9b. Effect of off-bottom clearance and tank size on dimensionlesscore velocity: D = T/2.

Figure 10. Regions in the wall-jet: T= 0.24 m, D = T/3, C/D = 0.4,N = 1000 rpm. The simulated slopes for the decay of Um/Ucore andexpansion ofb are smaller than the experimental results of Bittorf andKresta (2001).


11/15

smaller than the experimental results as detailed in Table 3. Theslower decay in Um also leads to a slower expansion of b andlargerzo , as shown in Tables 4. An increase in tank size from0.24 m to 1.00 m results in a very small improvement inperformance, which is similar to the improvement obtained byusing the RSM turbulence model. The RNG k-e model is able tocapture the slope of the velocity decay very accurately but thevalues are underestimated by more than 20%. Similar performanceis observed for high off-bottom clearance (C/D ~ 1.6), but inthis case the maximum velocity is overestimated.

Since CFD is unable to accurately predict the decay of Um ,reliable estimates ofzo cannot be obtained. The failure of CFDcalculations to properly predict the decay ofUm can be linkedto a failure to properly predict the tangential flow along thecurved tank surface. The marginally higher velocity decay forthe larger tank indicates a small effect of tank curvature, so partof the failure in predicting the flow along the tank wall lies inthe inability of the standard k-e model to properly simulate flowalong curved streamlines. Another important factor contributingto the failure in predicting the wall jet velocity decay is theassumption of no impeller-baffle interaction in the MRFformulation. Averaging of the outwards flow over the tangentialplane may provide a more accurate representation of the time-

averaged flow field. Simulations using transient methods, likethe large eddy simulation, should also be explored.

Impingement Point (zI) and Axial Distance over

which Jet ExtendsWhen the impeller discharge stream impinges on the tank wall,the axial position at which this impingement occurs is known asthe impingement point, z

I. The three-dimensional wall jet starts

to form between the tank wall and the baffle a short distancefrom the impingement point; hence z

Iinfluences not only the

position of the virtual origin, zo, but also the location andmagnitude ofUcore. Moreover, zI determines the axial position


Figure 11a. Contours of axial velocity (Um/2) at different axiallocations showing the transition from characteristic decay to radialdecay: D = T/3, C/D = 0.4, T= 0.24 m, N= 1000 rpm. Insert showsvelocity vectors at the wall of the tank where the flow is convectedtowards the baffle.

Figure 11b. Tangential profile of axial velocity at the tank wall: T= 0.24 m,D = T/3, N= 1000 rpm, C/D = 0.4, y= 5 mm.

Figure 12. Decay of maximum velocity: T= 0.24 m, D = T/3,N= 1000 rpm,

x= 3.6 mm (T= 0.24 m) and 11.5 mm (T= 1.00 m), y= 0.56 mm. Unlessstated otherwise all data refer to T= 0.24 m.


12/15

of the wall jet and hence the location of the region which isactively involved in the mean circulation within the tank. Thisregion was described as the active zone by Bittorf and Kresta(2000). Figure 13 shows the impingement points obtained fromCFD simulations for different tank and impeller geometries. Forthe smaller (D = T/3) impeller and both tank sizes, the dischargestream impinges on the tank bottom when C/D < 1.2, whileimpingement is on the tank wall when C/D > 1.5. It may benoted that Bittorf and Kresta (2000) found bottom impingementof the impeller discharge stream forC/D < 1.5. Since clearanceswith C/D > 1 are highly unusual, it may be concluded that forall practical purposes impingement is on the tank bottom for aT/3 impeller and the active volume extends only to the lower

two-thirds of the tank (Bittorf and Kresta, 2000).ForD = T/2 the impeller discharge stream impinges on the

tank wall forC/D > 0.55 (extrapolated value, Figure 13). Thisagrees with the observation of Bittorf and Kresta (2000) that thelimit ofC/D below which the impeller discharge impinges onthe tank bottom is 0.5. Since the impeller discharge impingeson the tank wall forC/D > 0.55, the active zone is in the middleof the tank with the zones of least activity distributed betweenthe bottom and the top sections. The relationship between thepoint of impingement and the location of the active zone ismade clearer in Figure 14, which will be discussed shortly.

The point of impingement of the impeller discharge on thetank wall may also be estimated from purely geometricalconsiderations, assuming that the surrounding flow field doesnot cause any deflection of the impeller discharge stream. If the

discharge leaves the impeller at an angle of 45, the axialdistance between the lower surface of the impeller and theimpingement point will be equal to the radial distance betweenthe tank wall and the point of discharge. Since the peakvelocity in the stream is at r= 0.4Dfrom the axis of rotation, theimpingement point (measured from the bottom of the tank) isestimated as:

(10)

z

T

C

T

D

TI

= - -

0 5 0 4. .


Table 3. Decay of maximum velocity in the characteristic and radial

decay regions.

T C/D Slope in Slope in

(m) characteristic radial decay

decay region region

(FLUENT) (FLUENT)

0.24 0.4 0.2664 0.5164

0.8 0.3278 0.6097

1 0.3707 0.7147

1.2 0.2391 0.7732

1.4 0.6235 1.1104

1.5 0.5973 1.1552

1.6 0.6819 1.3102

1.7 0.7474 1.3852

1.9 1.02 1.9243

Averages from FLUENT 0.3755 1.0555

(for all C/D given above) (s = 0.49) (s = 0.45)

Averages from FLUENT

(forC/D of 0.4 and 1) 0.3186 0.6156

Averages from experiments

(forC/D of 0.4 and 1) 0.59 1.13

1 0.4 0.3104 0.5618

0.8 0.5954 0.94820.9 0.4421 0.6062

1 0.457 0.7598

1.1 0.4838 0.7787

1.4 0.6153 0.9369

1.5 0.7878 1.1529

1.7 1.1043 1.8316

1.9 1.4384 0.999

Averages from FLUENT 0.3837 0.6608

(for all C/D given above. (s= 0.36) (s= 0.38)

Experimental data not available.)

Table 4. Rate of expansion of jet and virtual origin: T= 0.24 m and

D = T/3.

C/D Slope Virtual origin Experiment*

(mm)

0.4 0.1327 75

0.8 0.137 78

1 0.1265 108

1.2 0.1424 97 Slope = 0.38

1.4 0.1792 39 Virtual origin at

20 1.5 0.188 21 20 mm

1.6 0.1947 7

1.7 0.2021 9

1.9 0.1444 27

* Based on C/D of 0.4 and 1.0.

Figure 13. Impingement point of impeller discharge stream on the tank wall:Re= 1 105. At high off-bottom clearances, the point of impingement ofthe impeller discharge stream is deflected upwards by the secondarycirculation. This effect is much larger for the T/2 impeller.


13/15

In Figure 13, the impingement points calculated from CFDare compared with geometric estimates from Equation (10). Forthe T/3 impeller, the deflection of CFD data points fromEquation (10) increases at high clearances (C/D 1.4), indicatingthat the secondary flow must deflect the impeller stream moreand more as the impeller is lifted up. For the T/2 impeller,deviations from Equation (10) are larger as compared to the T/3case because of the stronger secondary flow. ForC/D > 0.7, theCFD data points are parallel with Equation (10), indicating awell-developed secondary loop at higher clearances. Thisagrees with Kresta and Wood (1993) who reported strongsecondary circulation at C/D > 0.667. For both impeller sizes,impingement point predictions match physical understandingwell and shed more light on limited experimental data.

The active zone in the stirred tank can be directly related tothe position of the three-dimensional wall jets driven by theimpeller. Figure 14 shows the axial distance over which thethree-dimensional wall jet is self-similar for a range of impellerclearances. As expected, Ucoreforms at a higher axial location asthe impeller clearance is increased. Moreover, at higherclearances this location increases faster with an increase in C/D,due to the deflection of the impeller discharge towards the wallby the secondary circulation loop. The point at which the jet

disintegrates is close to z/T= 0.7 for all C/D less than 1.4. Thisresult is in good agreement with the experimental results ofBittorf and Kresta (2000), who observed that the three-dimensional jet ended its upward climb at a height equivalentto two-thirds of the tank diameter (T). Moreover, forC/D > 1.5,when the impeller discharge stream impinges on the tank wall,the jet region is found to exist only in the middle of the tank, aswas also observed by Bittorf and Kresta (2000). At smallclearances, the impeller is located near the lower extremity ofthe jet, while at higherC/D ratios its location is well within thejet decay regions. Figures 13 and 14 represent a significantextension of our understanding of the active zone in the tank.

ConclusionsBittorf and Kresta (2001) showed that the flow field close to thetank walls can be completely characterized by the wall jetEquations (Equations 4, 5 and 6), if the geometry dependentvariables, Ucore and zo, are known. In this study, a CFDbenchmark is reported for simulation of the flow at the wall ina stirred tank. The simulation, which uses the k-e turbulencemodel with standard wall functions, a grid refined at locationsclose to the tank walls, and a steady state MRF approach, buildson previous studies that focused on the flow near the impeller.The CFD simulations were compared with the experimentaldata of Bittorf and Kresta (2000) to validate the results and tohighlight areas where the simulations failed.

The main conclusions are as follows:(i) The circulation patterns and the similarity profiles of the

three-dimensional wall jets (Equation 4) extracted from CFDsimulations match the experimental data well. This confirmsthe ability of steady state simulations to capture the overallcharacteristics of the near-wall flow field.

(ii) Accurately predicting the geometry dependent parameters,Ucoreand zo, is a more demanding test. Steady state simulationswith the k-e model were able to predict Ucore with goodaccuracy, giving results which were within 14% of the

experimental values forD = T/3 and within 20% for the D = T/2impeller. The effect of the impeller off-bottom clearance onUcorewas also captured accurately by the simulations.

(iii)The present simulations failed to predict the rate of decay ofUm and the expansion of the jet half-width, b, recorded inexperiments. Since the virtual origin, zo, is dependent onthese quantities, the steady state, snapshot simulations alsofailed to predict zo.The simulations were also used to determine two quantities

which provide important information about the flow field, butwhich are not readily available from experiments:(i) the impingement point of the impeller discharge on the tank

wall; and(ii) the axial distance over which the three-dimensional wall jet

extends.Both these quantities need high resolution for accuratedetermination and are therefore resistant to experimentalapproaches. The results presented here show that CFD provides anefficient method of obtaining consistent values for the impingementpoint and the extent of the jet, quantities which deepen ourunderstanding of the complex flow field in a stirred tank.

AcknowledgementsThe initial simulations and analysis were carried out by Mike Serink

(Serink, 1999; Kresta and Serink, 1999; and Kresta et al., 1999). The

financial support provided by NSERC and the software provided by

FLUENT are gratefully acknowledged.

Nomenclatureb radial distance to Um/2, (m)bjet distance from edge of boundary layer to Um/2, (m)

B constant in Equation (3) which is a function of UR(1B= UR/Um)

C impeller off-bottom clearance, (m)

Cm constant used in defining mtC1e, C2e constants in the k-e model (Table 1)D impeller diameter, (m)

H tank height, (m)

Jijk turbulent diffusion term in RSM equation (Table 1), (kg/ms3)


Figure 14.Axial distance over which jet extends: T= 0.24 m, D = T/3, N= 1000 rpm.


14/15

k turbulent kinetic energy ( ), (m2/s2)

N impeller speed, (rps)

Njs just suspended speed of impeller for solids suspension, (rps)

p fluctuating component of pressure, (N/m2)

P mean pressure, (N/m2)

Pij production term in RSM equation (Table 1), (kg/ms3)

r position vector relative to the origin of rotation given in

Equation (7), (m)

r,q,z cylindrical co-ordinates; also denoted as FLUENTco-ordinates j, iand k(Figure 2) where r=j, q = iand k= z

Re Reynolds number (ND2/n)Rij Reynolds stress, (Pa)

s standard deviation based on sample

S modulus of mean rate of strain, , (s1)

Sij mean rate of strain tensor, (s1)

t time, (s)

T tank diameter, (m)

ui ith component of fluctuating velocity, (m/s)

Ucore jet core velocity, (m/s)

U axial component of velocity, (m/s)

Ui ith component of the mean velocity, (m/s)

Um local maximum velocity in the jet, (m/s)

UR recirculating velocity, (m/s)

v absolute velocity vector, (m/s)vr relative velocity vector, (m/s)

Vtip impeller tip speed, (m/s)

x position vector in Cartesian coordinates, (m/s)

xo origin vector of the rotating frame, (m/s)

xi Cartesian coordinates, (m)

y radial distance from tank wall, (m)

yjet radial distance in outer region of jet, (m)

ym radial distance to Um, (m)

y0 radial distance at which U= 0, (m)

z axial distance from tank bottom, (m)

zo virtual origin, (m)

zI

impingement point of impeller discharge on tank wall, (m)

Greek Symbolsae, ak constants in diffusion terms of RNG k-e transport equationsforkand e (Table 1)

d proportionality constant relating boundary layerthickness to b

dij Kronecker delta function, dij = 1 when i=j; d = 0 when i je dissipation of turbulent kinetic energy (k) given by terms in

Table 1, (m2/s3)

eij dissipation term in RSM equation (Table 1), (kg/ms3)

h dimensionless radial distance given as y/bm dynamic (molecular) viscosity, (Pas)mt turbulent or eddy viscosity (Table 1), (Pas)meff effective viscosity (meff= m + mt), (Pas)n kinematic viscosity (m/r), (m2/s)

r density, (kg/m3

)sk, se constants in k-e modelF similarity constant in Equation (1)Fij pressure-strain distribution in RSM equation (Table 1), (kg/ms

3)

w angular velocity vector of reference frame, (s1)

ReferencesBakker, A., K.J. Myers, R.W. Ward and C.K. Lee, The Laminar and

Turbulent Flow Pattern of a Pitched Blade Turbine, Trans. IChemE, ,

Part A 74, 485491 (1996).

Bittorf, K.J. and S.M. Kresta, Active Volume of Mean Circulation for

S S Si j i j 2

ku ui i=2

Stirred Tanks Agitated with Axial Impellers, Chem. Eng. Sci., 55,

13251335 (2000).

Bittorf, K.J. and S.M. Kresta, 3-D Wall Jets: Axial Flow in a Stirred Tank,

AIChE J., 47, 12771284 (2001).

Bittorf, K.J., The Application of Wall Jets in Stirred Tanks with Solids

Distribution, PhD thesis, University of Alberta, Edmonton, AB (2000).

Bradshaw, P., The Best Turbulence Models for Engineers, in

Modeling Complex Turbulent Flows, M. Salas, J.N. Hefner and

L.Sakell, Eds., ICAS/LaRC Interdisciplinary Series in Science and

Engineering, Kluwer Academic Pub., The Netherlands (1999), pp. 928.

Derksen, J., and H.E.A. Van den Akker, Large Eddy Simulations on the

Flow Driven by a Rushton Turbine, AIChE J. 45, 209221 (1999).

Fokema, M.D., S.M. Kresta and P.E. Wood, Importance of Using the

Correct Impeller Boundary Conditions for CFD Simulations of Stirred

Tanks, Can. J. Chem. Eng. 72, 177183 (1994).

Harvey III, A.D., C.K. Lee and S.E. Rogers, Steady-state Modeling and

Experimental Measurement of a Baffled Pitched-Blade Impeller

Stirred Tank, AIChE J. 41, 21772186 (1995).

Harvey III, A.D. and S.E. Rogers, Steady and Unsteady Computations

of Impeller-Stirred Reactors, AIChE J, 42, 27012712 (1996).

Jaworski, Z., W. Bujalski, N. Otomo and A.W. Nienow, CFD Study of

Homogenization with Dual Rushton Turbines: Comparison with

Experimental Results. Part I: Initial Studies, Trans. IChemE, Part A

78, 327333 (2000).Jones, W.P. and B.E. Launder, The Prediction of Laminarization with a

Two-equation Model of Turbulence, Int. J. Heat Mass Transfer 15,

301314 (1972).

Kirkpatrick, A.T. and A.E. Kenyon, Flow Characteristics of Three-

dimensional Wall Jets, ASHRAE Trans. 104, Part 1B, ASHRAE, GA,

17551762 (1998).

Kresta, S.M. and P.E. Wood, Prediction of the Three-Dimensional Flow

in Stirred Tanks, AIChE J. 37, 448460 (1991).

Kresta, S.M. and P.E. Wood, The Mean Flow Field Produced by a 45

Pitched Blade Turbine: Changes in Circulation Pattern Due to Off

Bottom Clearance, Can. J. Chem. Eng. 71, 4253 (1993).

Kresta, S.M., Boundary Conditions Required for the CFD Simulations

of Flows in Stirred Tanks, in Multiphase Reactor and

Polymerization System Hydrodynamics, Advances in EngineeringFluid Mechanics Series, N.P. Cheremisinoff, Ed., Gulf Pub. Co.,

Houston, TX (1996), pp. 297316.

Kresta, S.M. and M. Serink, Use of Computational Fluid Dynamics

(CFD) in Conjunction with Experiments to Determine the Jet

Impingement Point in a Stirred Tank, AIChE Annual Meeting,

Dallas, TX, Oct. 31-Nov. 5 (1999).

Kresta, S.M., M. Serink and K. Bittorf, Use of Computational Fluid

Dynamics (CFD) in Conjunction with Experiments to Determine the

Jet Impingement Point in a Stirred Tank, 49th Canadian Chemical

Engineering Conf., Oct. 36, 1999, Saskatoon, SK (1999).

Kresta, S.M., K.J. Bittorf and D.J. Wilson, Internal Annular Wall Jets:

Radial Flow in a Stirred Tank, AIChE J. 47, 23902401 (2001).

Launder, B.E. and D.B. Spalding, Lectures in Mathematical Models of

Turbulence, Academic Press, London, UK (1972).Launder, B.E. and D.B. Spalding, The Numerical Computation of

Turbulent Flows, Computer Methods Applied Mech. Eng. 3,

269289 (1974).

Launder, B.E., G.J. Reece and W. Rodi, Progress in the Development of

a Reynolds-Stress Turbulence Closure, J. Fluid Mech. Part 3 68,

537566 (1975).

Launder, B.E. and W. Rodi, The Turbulent Wall Jet: Measurements and

Modeling, Ann. Rev. Fluid Mech. 15, 429459 (1983).

Launder, B.E., Second-moment Closure: Presentand Future?,

Interm. J. Heat Fluid Flow 10, 282300 (1989).



15/15

Launder, B.E., Modelling the Formation and Dispersal of Streamwise

Vortices in Turbulent Flow, 35 th Lanchester Lecture, Aeronaut. J. 99,

(990), 419431 (1995).

Launder, B.E., The Modeling of Turbulent Flows With

Significant Curvature or Rotation, in Modeling Complex

Turbulent Flows, M. Salas, J.N. Hefner and L. Sakell, Eds., ICAS/LaRC

Interdisciplinary Series in Science and Engineering, Kluwer Academic

Pub., The Netherlands (1999), pp. 2951.

Mohammadi, B. and O. Pironneau, Analysis of k-epsilon Turbulence

Model, John Wiley and Sons, Paris, France (1994).

Perng, C.-Y. and J.Y. Murthy, A Moving Mesh Technique for Simulation

of Flow in Mixing Tanks, AIChE Meeting, Miami Beach, FL (1992).

Rajaratnam, N., Developments in Water Science: Turbulent Jets,

Elsevier Scientific Publishing Co., New York, NY (1976).

Ranade, V.V. and S.M.S. Dommeti, Computational Snapshot of Flow

Generated by Axial Impellers in Baffled Stirred Vessels, Trans

IChemE 74, Part A, 476484 (1996).

Serink, M., The Impingement Point of the Discharge of a Pitched Blade

Turbine in a Stirred Tank, Senior Research Project Report, University

of Alberta, Edmonton, AB (April 1999).

Wood, P.E. and C.P. Chen, Turbulence Model Predictions of Radial Jet

A Comparison ofk-e Models, Can. J. Chem. Eng. 63, 177182(1985).

Manuscript received August 7, 2001; revised manuscript received

March 20, 2002; accepted for publication May 8, 2002.


cfd simulations of three-dimensional wall jets in a stirred tank

Documents