Chemical Engineering Science 66 (2011) 5976–5988

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

An assessment of different turbulence models for predicting flow in a baffledtank stirred with a Rushton turbine

Harminder Singh a, David F. Fletcher b, Justin J. Nijdam a,n

a University of Canterbury, Chemical and Process Engineering, Private Bag, 4800, Christchurch 8140, New Zealandb School of Chemical and Biomolecular Engineering, University of Sydney, NSW 2006, Australia

a r t i c l e i n f o

Article history:

Received 21 April 2011

Received in revised form

4 August 2011

Accepted 9 August 2011Available online 16 August 2011

Keywords:

Turbulence

Mixing

Mathematical modelling

Fluid mechanics

CFD

Turbulence energy dissipation rate

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.08.018

esponding author. Tel.: þ64 3 3642137.

ail address: justin.nijdam@canterbury.ac.nz (J

a b s t r a c t

This work presents a comprehensive study of different turbulence models, including the k–e, SST,

SSG–RSM and the SAS–SST models, for simulating turbulent flow in a baffled tank stirred with a

Rushton turbine. All the turbulence models tested predict the mean axial and tangential velocities

reasonably well, but under-predict the decay of mean radial velocity away from the impeller. The k–emodel predicts poorly the generation and dissipation of turbulence in the vicinity of the impeller. This

contrasts with the SST model, which properly predicts the appearance of maxima in the turbulence

kinetic energy and turbulence energy dissipation rate just off the impeller blades. Curvature correction

improves the SST model by allowing a more accurate prediction of the magnitude and location of these

maxima. However, neither the k–e nor the SST model is able to properly capture the chaotic and three-

dimensional nature of the trailing vortices that form downstream of the blades of the impeller. In this

sense, the SAS–SST model produces more physical predictions. However, this model has some

drawbacks for modelling stirred tanks, such as the large number of modelled revolutions required to

obtain good statistical averaging for calculating turbulence quantities. Taking into consideration both

accuracy and solution time, the SSG–RSM model is the least satisfactory model tested for predicting

turbulent flow in a baffled stirred tank with a Rushton turbine.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Stirred tanks are widely used in industry for dispersing gases orsolids in liquids, crystallization, polymerization, and to performchemical reactions and bioconversions. In the past, stirred tankdesign was confined to the use of empirical correlations becausemore detailed analysis of the flow fields by means of Computa-tional Fluid Dynamics (CFD) was not reliable due to limitationson computer power and memory. As a result of these limitedresources, CFD simulations of stirred tanks were restricted tocoarse grids and steady-state predictions, or in the case of transientsimulations, the use of large time-steps that did not properlyresolve the transient flow features. In addition, these limitedresources also constrained the use of turbulence models totwo-equation models, which assume an isotropic eddy viscosity,although the flow in the impeller region of the stirred tank hasbeen demonstrated to be anisotropic (Derksen et al., 1999; Escudieand Line, 2006; Lee and Yianneskis, 1998). These issues haveresulted in unsatisfactory agreement between experiment andCFD simulations (Hartmann et al., 2004a; Jones et al., 2001;

ll rights reserved.

.J. Nijdam).

Montante et al., 2001; Ng et al., 1998; Sheng et al., 1998). However,advances in computer technology and mathematical models haveenabled researchers to use finer computational grids and smallertime-steps, as well as more complex turbulence models (Delafosseet al., 2008). Large Eddy Simulation (LES) in particular has beenshown to perform well in capturing the complexity of turbulentflow in a stirred tank (Delafosse et al., 2008, 2009; Derksen and Vanden Akker, 1999; Hartmann et al., 2004a, 2004b; Murthy and Joshi,2008; Yapici et al., 2008; Yeoh et al., 2004). Unfortunately, thisapproach is out of reach in most practical settings due to the ratherintense computations required. The present work aims to test theaccuracy of recently developed turbulence models that are com-putationally less demanding than LES and are thus more practical.

Another aspect of concern in CFD modelling of stirred tankshas been the definition of the kinetic energy of fluctuatingmotions. Wu and Patterson (1989) showed that this kineticenergy consists of two components, a random component and aperiodic component, with the total kinetic energy being the sumof these two components. The random component is due toturbulent eddies, while the periodic component is due to non-random oscillations caused by the cyclic passage of the impellerblades. The periodic component appears in the turbulence energyspectrum as peaks at the impeller blade frequency and harmonicsthereof. Montante et al. (2001), Hartmann et al. (2004a), Murthy

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5977

and Joshi (2008) and Delafosse et al. (2008) have separated thesecomponents in the modelling of stirred tanks. However, manyother researchers have not made this distinction clear. Theturbulence kinetic energy predicted by the Reynolds-AveragedNavier–Stokes (RANS) based models, such as k–e, k–o, SST andReynolds stress models, corresponds to the random component ofthe kinetic energy of fluctuating motions (Montante et al., 2001).This modelled turbulence kinetic energy comes about through theReynolds-averaging of the Navier–Stokes equations, which iscarried out to avoid resolving unsteady turbulent eddies directly,thus reducing the computational effort. In the case of turbulencemodels that contain ‘‘LES content’’ for unsteady simulations,such as the SAS–SST (Menter and Egorov, 2005), large-scaleturbulence is resolved directly to a certain extent, while thesmaller-scale turbulence is dealt with through Reynolds-aver-aging. Thus, the predicted kinetic energy due to random fluctua-tions is the sum of the turbulence kinetic energy due to thedirectly resolved large eddies and that coming about throughReynolds averaging. This is made more complex, because theperiodic kinetic energy must first be extracted before the randomcomponent can be determined.

Transient simulations of stirred tanks have been well docu-mented for the k–e and LES models. Although the k–e model is themost commonly used model for stirred tank simulations, its useof the Boussinesq approach of modelling the Reynolds stressesusing mean velocity gradients and an isotropic eddy viscositymeans a limit is placed on the simulated development ofanisotropic turbulence (Bakker et al., 1996; Jahoda et al. 2007;Javed et al., 2006; Ng et al., 1998; Yeoh et al., 2004; Zhang et al.,2006). To overcome this limitation, many researchers havesuggested using the Reynolds Stress Models (RSM), in which caseeach Reynolds stress is modelled with a separate transportequation, which eliminates the assumption of the isotropic eddyviscosity. However, both steady-state and transient results of theReynolds Stress Model have shown an under-prediction of turbu-lence kinetic energy (Bakker et al., 1996; Montante et al., 2001;Murthy and Joshi, 2008; Sheng et al., 1998). Another approachthat can improve two-equation model predictions is the inclusionof a curvature correction term to sensitize the model to the effectsof streamline curvature and system rotation (Smirnov andMenter, 2009). As far as the authors are aware, this approachhas not been investigated for stirred-tank simulations.

Some researchers have ventured towards other turbulencemodels for simulating stirred tanks, such as the Shear StressTransport (SST) and Scale Adaptive Simulation (SAS–SST)(Hartmann et al., 2004a; Honkanen et al., 2007). The SST modelis a hybrid model that combines the best features of the k–e andk–o models, blending between the k–o model in the near-wallregion to the k–e model in the free stream using a smooth function.The SST model has been shown to accurately predict flows withadverse pressure gradients and separation. However, it is unable toresolve any details of unsteady turbulent structures, which is oftenrequired (at least to some extent) to increase the accuracy ofunsteady-flow predictions (Menter, 2009). The SAS–SST turbulencemodel is a recent development by Menter and Egorov (2005) toovercome this limitation. This model is formulated on the idea ofcapturing as much as possible of the turbulence field that an LESsimulation would resolve but making use of the RANS capabilitiesnear walls and regions of ‘‘steady’’ flow (Menter, 2009). Honkanenet al. (2007) have shown that the length and intensity of thetrailing vortices that are generated off the impeller blades in astirred tank are predicted well by the SAS–SST model, and areunder-predicted by the SST model, which shows that the SSTmodel over-estimates the dissipation rate of the turbulent struc-tures. However, in their study, Honkanen et al. (2007) did notprovide a detailed comparison between experimental data and

simulated results of periodic and turbulence (or random) kineticenergy and turbulence energy dissipation rate obtained using theSST and SAS–SST.

The aim of this paper is to conduct a detailed and compre-hensive assessment of the ability of the k–e, SST, RSM and SAS–SST turbulence models to predict important flow parametersassociated with stirred tanks, such as the periodic and turbulence(random) kinetic energy, turbulence energy dissipation rate,turbulence length-scale, trailing vortices and power number. Bestpractice numerical methods are employed, and care is taken toproperly separate the random and periodic components of thekinetic energy of fluctuating motions. The experimental data ofWu and Patterson (1989) for a standard configuration systemstirred by a Rushton turbine are used for validation. This data setis a standard used by many workers for validating simulations ofstirred tanks (Brucato et al., 1998; Coroneo et al., 2011; Deglonand Meyer, 2006; Derksen and Van den Akker, 1999; Zadghaffariet al., 2010; Zhang et al., 2006). The data of Wu and Patterson(1989) are supplemented with the experimental data of Ducci andYianneskis (2005), also for a standard configuration tank stirredby a Rushton turbine. As will be explained in more detail below,Ducci and Yianneskis (2005) have shown that the experimentalmethod used by Wu and Patterson (1989) results in a 40%underestimation of the local maximum turbulence energydissipation rate near the impeller, with the location of themaximum estimated to be too close to the impeller tip. Ducciand Yianneskis (2005) determined e through direct measurementof nine out of the twelve fluctuating velocity gradients comprisingthe turbulence energy dissipation rate using a four-channel laser-Doppler anemometer (LDA) and a fine spatial resolution of 4–5Kolmogorov length-scales. They demonstrated the accuracy oftheir measurements for e by evaluating every term in theturbulence kinetic energy transport equation, and showing that,with the substitution of the directly-measured turbulence energydissipation rate, the balance on this equation was acceptable.

2. Computational model

2.1. Turbulence modelling

In this section, the various turbulence closures used in thisstudy are described. The equations used for each model are thoseas coded in version 12.1 of ANSYS-CFX (2009).

All of the models used in this study are derived from Reynolds-averaging of the underlying Navier–Stokes equations. Thisprocess introduces Reynolds stresses that have to be modelledto form a closed system of equations. The manner in which this isdone impacts both the reliability of the modelled equations andthe computational cost required to solve them.

In two-equation models, the Boussinesq hypothesis is used tomodel the unknown Reynolds stresses by assuming they can beapproximated by the product of the mean strain rate multipliedby an isotropic turbulence viscosity. There are a variety of modelsthat have been formulated but here we focus on just two of them.Firstly, the k–e model, based on transport equations for theturbulence kinetic energy, k, and the turbulence energy dissipa-tion rate, e, is used. Whilst this model has known limitations,particularly its unsatisfactory performance in the near-wallregion, it is still very widely used and also provides a goodstarting point to generate an initial flow field for more compli-cated models.

The k–e model employs wall functions as the equations cannotbe integrated to the wall. In order to address this problem, modelsin which the turbulence energy dissipation rate is replaced by theturbulence eddy frequency, o, have been derived. These behave

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885978

well near the wall and allow the detailed behaviour inside theboundary layer to be calculated. However, they perform poorlyaway from walls due to a problem with over-sensitivity to thefree stream conditions. Menter (1994) blended the k–e and k–omodels and added a turbulence production limiter in the calcula-tion of the eddy viscosity to produce the now widely used ShearStress Transport (SST) model. It provides accurate predictions ofthe onset and the amount of flow separation under adversepressure gradients (Menter, 1994).

A key failing of unsteady two-equation models is that theycause excessive damping of turbulence and thus do not resolveany of the details of the turbulence directly (Menter, 2009).Menter and Egorov (2005) introduced the SAS–SST to overcomethis deficiency. The SAS–SST model (SAS stands for Scale AdaptiveSimulation) uses a turbulence length-scale, which induces ‘‘LES-like’’ behaviour in unsteady regions of the flow field. By adjustingthe turbulence length-scale to the directly resolved turbulentstructures, the eddy viscosity is reduced to the level of thelimiting LES model. In this way, the model does not dissipatedirectly resolved turbulent structures as classical RANS would(Menter and Egorov, 2010; Menter et al., 2010).

Another drawback of eddy viscosity RANS based models is theassumption of an isotropic turbulence viscosity, which limits thedevelopment of anisotropy in the Reynolds stresses. ReynoldsStress Models (RSM), which solve transport equations for each ofthe Reynolds stresses, remove this assumption. The model ismuch more computationally intensive than two-equation modelsand suffers from poor convergence behaviour (Aubin et al., 2004).Just as there are many variants of the two equations models, thereare a large number of Reynolds stress models that differ in theirclosure relationships, especially for the pressure–strain term.Here the variant of the model developed by Speziale et al.(1991), known as the SSG–RSM, is used. It is known to performwell in the simulations of strongly rotating flows, in, for example,cyclones.

A key limitation of two-equation models is their insensitivityto streamline curvature and rotation, which is the main motiva-tion for the use of Reynolds stress models. Recently, a modifica-tion to the SST model has been developed by Menter (Smirnovand Menter, 2009) based on the work of Spalart and Shur (1997).This has shown considerable benefit in applications where highlyswirling flows have been modelled, but to our knowledge it hasnot yet been applied to the simulation of stirred tanks.

2.2. Description of the test case

A standard tank and Rushton turbine configuration wasemployed experimentally by Wu and Patterson (1989). The tankhas a diameter and height of 270 mm with four equally spacedbaffles of width one-tenth of the tank diameter. The impeller hasa diameter of 93 mm with six blades and is located at one-third ofthe tank diameter from the base of the tank. The blades have awidth and height of one-quarter and one-fifth of the impellerdiameter, respectively. Water at 25 1C was used as the workingfluid and the impeller rotational speed was 200 rpm, whichcorresponds to a Reynolds number of 28,830.

2.3. Computational mesh and time-steps

Simulations were carried out using the sliding mesh techniquewith the impeller swept region as the inner rotating zone and restof the tank, including the baffles, as the stationary zone. Usingcyclic symmetry, only half of the geometry was modelled toreduce computational effort, which was possible due to thesymmetry of the six-blade impeller and four baffles. The thick-nesses of the impeller blades, impeller disc and baffles, which Wu

and Patterson (1989) did not specify in their paper, were assumedto be zero in order to reduce the mesh complexity. Rutherfordet al. (1996) have shown experimentally that the thickness of theimpeller blades significantly influences the power number, meanvelocities and turbulence levels. They measured a reduction in theimpeller power number of 33% with an increase in the ratio ofblade thickness to impeller diameter from 0.0082 to 0.0337, withthe impeller diameter remaining constant. Delafosse et al. (2008)showed a similar influence of the impeller thickness on the powernumber after comparing the power number of 5.5 measured byEscudie and Line (2003) for a blade thickness to impeller diameterratio of 0.0133 with the power number of 5 measured by Ducciand Yianneskis (2005) for a blade thickness to impeller diameterratio of 0.03, both cases using an impeller of the same diameter.

In addition to the effect on power number, blade thickness alsoaffects the mean and fluctuating velocities, mostly around thecentre of the discharge stream of the impeller (at the impeller discelevation). Rutherford et al. (1996) observed experimentally anincrease in the maxima of the radial velocity of 26% and 11% atdimensionless radial distances rR from the impeller axis of 1.048and 1.752, respectively, as the blade thickness to impellerdiameter ratio decreased from 0.0337 to 0.0082. Moreover, theyobserved an increase in the maxima of the radial fluctuatingvelocity of 15% and 25% at dimensionless radial distances rR of1.048 and 1.752, respectively, as the blade thickness to impellerdiameter ratio decreased from 0.0337 to 0.0082. CFD simulationsof blades with a given thickness are often avoided because thethickness of the blade cannot be meshed sufficiently well whilehaving a mesh that is practical from a computational point ofview. Any differences between the prediction and experimentaldata in this paper may be due to the neglect of the impellerthickness in the simulations, as well as weaknesses in theturbulence models tested.

The sliding interface was located at a radius corresponding tothe tip of the impeller. No discontinuities in velocity, pressure,turbulence kinetic energy or turbulence energy dissipation ratewere observed across the sliding interface with the mesh even-tually adopted in the simulations (Fig. 1), which indicates a goodresolution of the mesh across the interface. Grid independencewas tested with the SST turbulence model. In this case, theaverage percentage difference in the solution between meshescontaining 1.644 and 2.138 million nodes for the mean totalkinetic energy of fluctuating motions and mean turbulenceenergy dissipation rate was 7.2% and 3.7%, respectively, over theregion examined (shown in Fig. 2b and c). Moreover, the averagepercentage difference in the solution between these meshes forthe power number based on torque (defined below) was only1.5%. Further mesh refinement from 2.138 to 3.127 million nodeshad very little effect on the numerical solution for the mean totalkinetic energy of fluctuating motions, mean turbulence energydissipation rate and power number. For the SAS–SST predictions,the average percentage difference between the meshes containing1.644 and 2.138 million nodes for mean total kinetic energy offluctuating motions and mean turbulence energy dissipation ratewas 6.1% and 11.3%, respectively, over the region examined(shown in Fig. 2d for turbulence energy dissipation rate). Thevalue of 11.3% for the turbulence energy dissipation rate wasconsidered to be too high to regard the SAS–SST predictions asbeing grid independent. For this turbulence model, a grid inde-pendent solution is more difficult to achieve, especially in theturbulence quantities, because further refinement of the meshwill result in smaller turbulent fluctuations (or eddies) beingresolved. In this work, a mesh containing 2.138 million nodes,shown in Fig. 1, was adopted for all of the turbulence modelstested. Refining the mesh further to resolve even finer turbulentstructures and obtain more accurate solutions for the SAS–SST

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0Ur/ Utip

z w

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

1.0

k t/U

tip2

0

2

4

6

8

10

12

14

16

18

20

1.0

SST 0.5°, 1.027×106

SST 1°, 1.027×106

SST 1°, 1.644×106

SST 1°, 2.138×106

SST 1°, 3.127×106

ε/(N

3 D2 )

0

2

4

6

8

10

12

14

1.0

10 rev SAS 1°, 2.138×106

20 rev SAS 1°, 2.138×106

30 rev SAS 1°, 2.138×106

30 rev SAS 1°, 1.644×106

ε/(N

3 D2 )

rR

0.5 1 1.5 2.0

rR

1.5 2.0rR

1.5 2.0

Fig. 2. Grid and time-step independence check. (a) Mean radial velocity at a dimensionless radial distance rR of 1.07 from the impeller axis, (b) total kinetic energy of

fluctuating motions and (c) turbulence energy dissipation rate for the SST turbulence model and (d) turbulence energy dissipation rate for the SAS–SST turbulence model at

the impeller disc elevation. The legend for (c) applies to (a) and (b).

Fig. 1. Views of the 2.138�106 node mesh for the tank and impeller zones.

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5979

was found to be prohibitively expensive computationally. Note,there were 30 nodes across the blade width in the mesh adoptedin this work. Derksen et al. (1999) have stated that at least8 nodes are required to properly resolve the trailing vortices thatform behind the blades of the impeller.

Fig. 2 shows that little improvements in accuracy can begained by moving from one degree resolution for the rotatingimpeller (equivalent to 8.3�10�4 s per time-step) to a 0.5 degreeresolution using the same mesh (containing 1.027 million nodes)and the SST model. In this comparison, the average percentage

error for both the mean total kinetic energy of fluctuating motionsand the mean turbulence energy dissipation rate was about 0.4%and 0.3%, respectively. For the SAS–SST model prediction with amesh containing 1.027 million nodes and for 20 simulatedrevolutions, the average percentage difference in solutions for atime-step of 11 and 0.51 resolution for both the mean total kineticenergy of fluctuating motions and mean turbulence energydissipation rate was about 2.0% and 6.4%, respectively, over theregion examined. As with mesh refinement, further time-steprefinement would result in better resolution of turbulence when

Table 1Mesh size and time-step used by various authors for transient simulations of flow in stirred tanks.

Author Modelled tank geometry Mesh size: In millions Time-step: resolution in degrees

Ng et al. (1998) Half 0.239 (cells) 1

Montante et al. (2001) Half 0.080 4

Hartmann et al. (2004a) Full 0.242 (nodes) 4

Yeoh et al. (2004) Full 0.49 (cells) 1.67

Bakker and Oshinowo (2004) Full 0.763 (cells) 6

Zhang et al. (2006) Not specified 1.728 (cells) 2.4

Honkanen et al. (2007) Full 0.542 (nodes) 2.5

Jahoda et al. (2007) Full 0.615 1.8

Murthy and Joshi (2008) Not specified 0.575 Initially 0.16; gradually increased to 1.6

Delafosse et al. (2008) Full 1 (cells) 0.5

Coroneo et al. (2011) Full 6.6 (cells) 3.75

Present study Half 2.138 (nodes) 1

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885980

using the SAS–SST model. However, such refinement is con-strained by the computational effort that can be afforded.Table 1 shows a comparison of various mesh sizes and time-stepsused in recent studies. It can be seen that some researchers haveused coarser meshes and longer time-steps than used in thiswork, perhaps due to limitations in computer power available butat the expense of a certain degree of accuracy.

2.4. Computational aspects

The advection terms were modelled using the high resolutionscheme for all equations and the second order backward Eulerscheme was used for the transient terms. The convergence criteriaused for the continuity, momentum and turbulence quantitieswas an RMS scaled residual of 10�5. This was achieved in amaximum of 10 iterations per time-step for all the turbulencemodels presented in this study. Double precision arithmetic wasused in all simulations. In these simulations, an un-convergedsteady-state simulation was used as the initial guess for thetransient simulations. Collection of transient data was startedonly after a pseudo-steady state was achieved for all turbulencemodels. Only one revolution of data was collected for the k–e, SST,and SSG–RSM models since repeat revolutions produced repeatdata. For the SAS–SST model, data representing a sufficientnumber of revolutions is required to ensure good statisticalaverages (Frank et al., 2010). In this case, data was collected for30 revolutions equivalent to around 9 s of real time. Fig. 2d showsthat 20 revolutions suffice to achieve time-averaged turbulencequantities that are independent of the number of further simu-lated revolutions.

The simulations have been performed using eight 1.9 GHzparallel processors each with 2 GB of memory. No significantcomputing time difference was observed between the k–e, SST,SST–CC and SAS–SST models, with the simulation of one revolu-tion with one degree resolution in time-step and with a meshhaving 2.138 million nodes being achieved in about 55 h com-pared with 90 h for the SSG–RSM model.

2.5. Data post processing

2.5.1. Kinetic energy of fluctuating motions for the k–e, SST and RSM

models

In the case of the RANS turbulence models (k–e, SST and RSMmodels), the instantaneous velocity for a given point in the tankcan be defined as the sum of the mean velocity, the periodicvelocity and a random velocity, as defined in Eq. (1) below

uins ¼ uþupþur ð1Þ

Only the mean and periodic velocity components can beresolved directly in RANS simulations, provided these simulations

are transient. In this case, the velocity at a point will oscillate in aregular manner around the mean due to the cyclic passage of theimpeller blades. For example, with six blades, one revolution ofthe impeller will result in six oscillations. The transient velocityfield in RANS simulations, ui, can be decomposed as follows:

ui ¼ uþup ð2Þ

The time-averaged periodic velocity component in a givendirection can be extracted as follows:

upup ¼1

n

Xn

i ¼ 1

ðui�uÞ2 ð3Þ

where n is the number of time-steps over which the time-averaging takes place. The components in the other two ortho-gonal directions are extracted in the same manner. The periodickinetic energy is then given by

kp ¼12ðuPuP þvPvP þwPwP Þ ð4Þ

The random velocity component comes from RANS modellingthrough the turbulence kinetic energy (kr). Thus, the total kineticenergy of fluctuating motions is given by

kt ¼ krþkp ð5Þ

2.5.2. Kinetic energy of fluctuating motions for the SAS–SST model

In the case of the SAS–SST model, the instantaneous velocitycan also be described by Eq. (1). However, the random velocitycomponent, ur, now becomes the sum of a random velocity due tothe large eddies, ule�r, which are resolved directly by the SAS–SSTmodel and the random velocity due to the smaller eddies, use�r,which come from the RANS content of the model through theturbulence kinetic energy (k)

uins ¼ uþupþule�rþuse�r ð6Þ

For SAS–SST simulations, the transient velocity at a point isresolved as follows:

ui ¼ uþu0 ð7Þ

where u0 ¼ upþule�r , which implies that any simulated fluctua-tions are caused by periodicity and large eddies. The time-averaged component in each direction can be extracted separately

u0u0 ¼1

n

Xn

i ¼ 1

ðui�uÞ2 ð8Þ

where n is the number of time-steps over which the time-averaging takes place. The kinetic energy due to these fluctuations(periodic plus the large eddy) is given by

kpþ le�r ¼12ðu0u0 þv0v0 þw0w0 Þ ð9Þ

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5981

The total kinetic energy of fluctuating motions is therefore thesum of the kinetic energy due to the periodic and large eddyfluctuations and kinetic energy due to the small-eddy fluctuations(calculated as the turbulence kinetic energy, k, from the SAS–SSTmodel):

kt ¼ kpþ le�rþk ð10Þ

The periodic and large eddy fluctuations are separated byeffectively averaging out the fluctuations due to the large eddies.One impeller rotation is resolved using mtotal angular positions.For example, for a 11 resolution, there are 360 angular positionsbefore the impeller completes a full cycle and returns to its initialposition. For a given repeat angular position of the impeller m, thevelocity at a point can be averaged over a number of full turns

~um ¼1

nTS

XnTS

j ¼ 1

uj,m ð11Þ

where uj,m is the transient velocity at a point in the tank when theimpeller has an angular position corresponding to m, and nTS isthe number of full turns of the impeller for which data areavailable. Clearly, the more full-impeller turns that have beensimulated the better the average. Eq. (11) can be used to calculatethe periodic fluctuation (due to the cyclic passage of the impellerblades) once the effect of the large eddy fluctuations have beenaveraged out. The effect of this averaging can be seen clearlyin Fig. 3. Thus, the time-averaged periodic velocity components ineach direction can be derived from the averaged periodic fluctua-tions over a full turn, such as shown in Fig. 3, using Eq. (3). Theperiodic kinetic energy is then given by

kp ¼12ðuPuP þvPvP þwPwP Þ ð12Þ

The random component of the total kinetic energy can now beextracted as follows:

kr ¼ kt�kp ¼ kpþ le�rþk�kp ð13Þ

2.5.3. Calculation of the power number

The power number can be obtained in two different ways fromthe numerical simulations: (1) using the torque applied on theimpeller (Eq. (14)) and by integrating the turbulence energy

0.0

0.5

1.0

1.5

2.0

2.5

0.0

Averaged over 30 revolutions

One revolution only

Real time (s)

Tra

nsie

nt v

eloc

ity-

u i (

m/s

)

0.1 0.2 0.3

Fig. 3. Effect of averaging on the periodic velocity calculated using the SAS–SST

turbulence model with 11 time-step and a mesh with 2.138�106 nodes at a

dimensionless radial distance rR of 1.07 from the impeller axis at the axial height

of the impeller disc.

dissipation rate e over the tank volume (Eq. (15)), as follows:

NPt ¼ 2pNt=rN3D5 ð14Þ

and

NPe ¼

ZredV=rN3D5 ð15Þ

where N is the rotational speed in rev/s, t is the torque applied onthe impeller, r is the fluid density and D is the diameter of theimpeller.

2.5.4. Calculation of the turbulence length-scale

The turbulence length-scales used in this study are given byEq. (16) for the k–e and SSG–RSM model and by Eq. (17) for theSST and SAS–SST models.

l¼ C3=4m k3=2=e ð16Þ

l¼ffiffiffikp

=ðC1=4m oÞ ð17Þ

The constant Cm takes a value of 0.09.

3. Results and discussion

3.1. Mean velocity field

Fig. 4a and b shows that all of the turbulence models, exceptfor the k–e model, slightly over-predict the maxima of the meanradial and tangential velocity profiles at a dimensionless radialdistance rR of 1.07 from the centre of rotation. In addition, allmodels slightly over-predict the magnitude of the mean axialvelocity below and above the blades of the impeller at this sameradial location (Fig. 4c). Fig. 4d and e shows that the SAS–SSTmodel predicts the decay of the mean tangential and axialvelocities radially away from the impeller reasonably well,although the mean radial velocity decays too slowly (Fig. 4f).Similar comparisons as seen in Fig. 4d, e and f have beenconducted between experiment and prediction for all otherturbulence models tested; however, for the sake of brevity,figures showing these comparisons are not provided here. Ingeneral, all turbulence models tested modelled the decay of themean tangential and axial velocity radially away from theimpeller reasonably well, but they under-predicted the decay ofmean radial velocity. Nevertheless, all these models at leastqualitatively predict the radial jet-like flow away from theimpeller and the entrainment of fluid into the jet from the bulkflow, which causes the jet to broaden and decrease in speed.Finally, using curvature correction with the SST turbulence modeldid not appreciably change the mean velocity predictions com-pared with using the SST turbulence model alone.

3.2. Kinetic energy of fluctuating motions

Fig. 5a, b and c presents axial profiles of the periodic, randomand total components, respectively, of the kinetic energy of thefluctuating motions at a dimensionless radial distance rR of 1.07from the centre of rotation. All of the turbulence models, exceptfor the k–e model, over-predict the total kinetic energy offluctuating motions. This over-prediction of the total kineticenergy, which only occurs in the vicinity of the impeller(Fig. 5c), is directly related to the over-prediction of the periodickinetic energy (Fig. 5a). Away from the impeller, the agreement isbetter for all the turbulence models, except for the SSG–RSMmodel, as shown in Fig. 5d and f. The k–e model predicts therandom, periodic and total components of the kinetic energyreasonably well away from the impeller. However, closer to the

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Fig. 4. Mean radial, tangential and axial velocity profiles for different turbulence models at a dimensionless radial distance rR of 1.07 from the impeller axis (a), (b) and

(c) and for the SAS–SST model at various radial locations (d), (e) and (f). The legend for (c) applies to (a) and (b). The legend for (f) applies to (d) and (e).

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885982

impeller, it severely under-predicts the periodic component andover-predicts the random component, which Delafosse et al.(2008) also found. Conversely, the SSG–RSM model severelyover-predicts the periodic kinetic energy and under-predicts theturbulence (or random) kinetic energy. Murthy and Joshi (2008)have found a similar under-prediction of the turbulence (random)kinetic energy for the SSG–RSM.

None of the turbulence models is able to properly predict thepeaks and local minimum in the turbulence (random) kineticenergy profile in the axial direction, as shown in Fig. 5b. Thesepeaks and local minimum are predicted only weakly by the SSTand SAS–SST model and not at all by the SSG–RSM, SST–CC andk–e models. Wu and Patterson (1989) state that the peakscorrespond to the location of a pair of vortices, one vortex above

and one vortex below the disc of the impeller, that originate frombehind each blade and trail out into the bulk of the flow. Suchtrailing vortices, which will be visualized later in this paper, arepredicted by all turbulence models tested.

Regarding the turbulence (random) kinetic energy profile in theradial direction shown in Fig. 5e, the SST, SST–CC and SAS–SSTturbulence models predict the maximum at a dimensionless radialdistance rR of about 1.4 in agreement with the experimental results.The k–e model predicts the maximum at the impeller tip, while theSSG–RSM model predicts the maximum at a dimensionless radialdistance rR of 1.6. Of all the turbulence models tested, the SST–CCpredicts the periodic, random and total kinetic energy of fluctuatingmotions most satisfactorily. These predictions mirror the experi-mental observations of Wu and Patterson (1989), who found that

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1.07 from the impeller axis and for radial profiles of (d) periodic, (e) random and (f) total kinetic energy of fluctuating motions at the impeller disc elevation. The legend for

(f) applies to (a), (b), (c), (d) and (e).

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5983

periodic fluctuations dominate the turbulence field close to theimpeller, as can be seen by the high values of kp compared with kr inthis region, but decay rapidly away from the impeller in the radialdirection. Meanwhile, the random turbulent fluctuations develop inthe near-impeller region, dominate at a dimensionless radial dis-tance rR of 1.4 and then decay further away.

3.3. Turbulence energy dissipation rate

Figs. 6 and 7 present the turbulence energy dissipation ratepredictions of the different turbulence models tested in thispaper. For comparison, two sets of experimental data are pre-sented, the first by Wu and Patterson (1989) and the second byDucci and Yianneskis (2005). The experimental turbulence energydissipation rate determined by Wu and Patterson (1989) wascalculated from the turbulence (random) kinetic energy by afitted correlation rather than being measured directly. Ducci and

Yianneskis (2005) have shown that this ‘‘dimensional’’ methodtends to underestimate the local maximum turbulence energydissipation rate near the impeller by about 40%. In addition, thelocation of this maximum is found to be too close to the impellerusing this method (Fig. 6). Ducci and Yianneskis (2005) measurede directly and more accurately using a four-channel laser-Doppleranemometer (LDA) and a fine spatial resolution to properlyresolve the dissipative scales. They found that the turbulenceenergy dissipation rate normalized with N3D2 was approximatelyconstant between Reynolds numbers of 20,000 and 40,000. Giventhat the same standard tank configuration and Rushton turbinewere used by both Wu and Patterson (1989) and Ducci andYianneskis (2005), and that the Reynolds number in the experi-ments of Wu and Patterson (1989) was 28,830, then a directcomparison of the predictions of the turbulence models presentedin this paper with the experimental data of Ducci and Yianneskis(2005) is possible. Note that in the paper of Wu and Patterson

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885984

(1989), profiles of turbulence energy dissipation rate normalizedwith an average turbulence energy dissipation rate eavg arepresented. In order to estimate eavg so that these profiles could

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and (c).

be re-dimensionalised here, Eq. (15) was used with a powernumber of 5, which Wu and Patterson (1989) assumed on thebasis of measurements conducted by Bates et al. (1963). Anotherpoint is that Ducci and Yianneskis (2005) only provide one axialprofile of turbulence energy dissipation rate at a radial location of1.35 rR. This position lies between the two radial locations (at 1.29rR and 1.5 rR) of the turbulence energy dissipation rate axialprofiles measured by Wu and Patterson (1989). In order to allow agood comparison, the axial e profile of Ducci and Yianneskis(2005) at 1.35 rR is overlaid on both the axial e profiles of Wu andPatterson (1989). An axial profile of turbulence energy dissipationrate at 1.07 rR is shown in Fig. 7c, which will be discussed inrelation to predicted power numbers further below.

Fig. 6 shows that the k–e model predicts the location ofmaximum turbulence energy dissipation rate to be near theimpeller tip, in agreement with the k–e model predictions ofDelafosse et al. (2008). This does not agree with the experimentalresults, which suggest that the location of the maximum is at adimensionless radial distance rR of around 1.4. The SST model isable to predict the appearance of a maximum and its magnitude,although its location does not align with that of the experimentaldata. Curvature correction helps to improve this prediction.Indeed, the SST–CC model accurately predicts both the increasein turbulence energy dissipation rate radially away from theimpeller tip in the near-impeller region, where turbulence isdeveloping, and the decrease of the turbulence energy dissipationrate further away from the impeller, where turbulence decays.

Neither the location of the maximum nor its magnitude ispredicted well by the SAS–SST and SSG–RSM models, as shown in

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cations for different turbulence models. The legends for (a) and (b) apply to (a), (b)

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5985

Fig. 6. The SSG–RSM is the least accurate model tested. Given thatthis model also requires more than one-and-a-half times thecomputational time of the k–e and the SST-based models, it wouldbe a poor choice of turbulence model for predicting flow in baffledstirred tanks. Unfortunately, as indicated by the numerical-solu-tion verification check detailed above (illustrated in Fig. 2d),further refinement of the mesh is required before any conclusionscan be drawn about the accuracy of the SAS–SST turbulence modelfor predicting the turbulence energy dissipation rate. Such refine-ment proved to be impractical from a computational perspectivein this work due to the significant number of impeller revolutionsrequired to gain good statistical averages of the turbulencequantities and the prohibitively long time needed to achieve this.

A comparison with the data of Ducci and Yianneskis (2005)indicates that all turbulence models underestimate the width ofthe turbulence energy dissipation rate profile in the axial direction(Fig. 7). Thus, it is reasonable to assume that, in general, theturbulence models under-predict the volume-integral of theturbulence energy dissipation rate throughout the tank. As willbe discussed below, this has repercussions for the accurateprediction of power number based on the volume-integral of theturbulence energy dissipation rate (Eq. (15)). Note that the experi-mental data of Ducci and Yianneskis (2005) at 1.35 rR are moreappropriately compared with the predictions shown in Fig. 7a,which apply at a location of 1.29 rR. Further away radially, thepredicted turbulence energy dissipation rates decrease and becomeeven lower relative to the experiment measurements at 1.35 rR,which further highlights the underestimation of the turbulenceenergy dissipation rate by all turbulence models tested.

3.4. Trailing vortices and turbulence length-scale

The vortical structures in a flow can be visualized in a numberof different ways. Here we have used the swirling strength, based

(a) k-� (

(c) SSG-RSM (

Fig. 8. Trailing vortices visualized using a swirling strength of 0.1 for

on the computation of the eigenvalues of the velocity gradienttensor. A threshold value of 0.1 was found to be a goodcompromise between missing structures if the value was toohigh and masking the structures if the value was too low.

Unsteady two-equation models, such as k–e and SST, are well-known for excessively damping turbulence so that any detail ofthe turbulent structure (even on the larger scales) cannot beresolved directly. This is reflected in Fig. 8a and b, which showsthat these models predict very small and hence dissipativetrailing vortices with no secondary vortex motion apparent.Nevertheless, both the k–e and SST models predict the appearanceof the pair of vortices, one vortex above and one vortex belowthe disc of the impeller, which originate from behind each bladeand trail out into the bulk of the flow. The upper and lowertrailing vortices on each blade have slightly different lengthsbecause the impeller is located at one-third of the tank heightfrom the bottom of the tank, which results in a slightly asym-metric flow field above and below the mid-plane of the impeller.Note that Honkanen et al. (2007) also predicted smooth, overly-dissipative trailing vortices with hardly any secondary motionusing the SST turbulence model. In the k–e and SST models,all turbulence scales are modelled through Reynolds-averaging,and hence a relatively large length-scale of turbulence wascalculated using Eqs. (16) and (17), as can be seen in Fig. 9aand b. Curvature correction in the SST model had a minor effecton the length of the trailing vortices and the length-scales ofturbulence – this comparison is not shown here – and therefore,from this perspective, it did not improve the SST predictionsappreciably. Interestingly, the length-scales of turbulence ofup to 3 mm predicted by the k–e and SST models in the regionof the impeller (shown in Fig. 9) are similar to the experimentalturbulence length-scales of between 0.3 and 3.0 mm measured byvarious researchers and compiled together by Wu and Patterson(1989).

b) SST

d) SAS-SST

(a) k–e, (b) SST, (c) SSG–RSM and (d) SAS–SST turbulence models.

(a) k-� (b) SST

(c) SSG-RSM (d) SAS-SST

Fig. 9. Predicted turbulence length-scales for (a) k–e, (b) SST, (c) SSG–RSM and (d) SAS–SST turbulence model simulations at the impeller disc elevation.

Table 2The effect of the turbulence model on power number calculated using the torque

and volume-integral of the turbulence energy dissipation rate methods.

Turbulence model Power number from torque Power number from e

k–e 5.770.2 5.170.1

SST 6.570.2 5.170.04

SAS–SST 6.970.7 5.270.4

SSG–RSM 6.570.3 5.570.2

SST–CC 6.670.3 4.570.05

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885986

For the SAS–SST model, some details of the turbulence struc-tures – the larger scale structures – can be directly resolved, whileReynolds-averaging accounts for the smaller-scale turbulencestructures. Fig. 8d shows that the SAS–SST model predicts longertrailing vortices and secondary vortex motions, in agreement withHonkanen et al. (2007), who found that the SAS–SST model is ableto predict vortices of comparable length and intensity to thosefound experimentally. The turbulence length-scales predicted bythe SAS–SST model (Fig. 9d) are much shorter than thosepredicted by the k–e and SST models, because the larger scaleturbulence is now being directly resolved through the ‘‘LES’’content of the model. Each mesh volume in the vicinity of theimpeller has a dimension of 0.62 mm in the axial direction,0.72 mm in the radial direction and 1.46 mm in the tangentialdirection. These dimensions represent the absolute minimumlength-scales of turbulence that are resolved directly by theSAS–SST model using the mesh adopted in this work. Thelength-scales of turbulence structures being modelled indirectlyby the SAS–SST model through Reynolds-averaging are up toaround 1 mm in size, as shown in Fig. 9d. Figs. 8c and 9c showthat the SSG–RSM model is able to predict trailing vortices moresimilar to those predicted by the SAS–SST model, although nosecondary motions are evident.

3.5. Power number

Table 2 summarizes the predictions of the power number forthe different turbulence models, calculated by the two differentmethods described in Section 2.5.3. By comparing these valueswith the power number of 5, which was assumed (not measureddirectly) by Wu and Patterson (1989) on the basis of measure-ments conducted by Bates et al. (1963), it can be seen that thepower number determined from the torque on the impeller

over-predicts the experimental data by 14% for the k–e modeland at least 30% for the SST, SST–CC, SAS–SST and SSG–RSMmodels. Delafosse et al. (2008) found an over-prediction of thepower number determined from the torque of 20% for the k–emodel. The predicted power number determined by integratingthe turbulence energy dissipation rate throughout the fluidvolume is close to the power number of 5 for nearly everyturbulence model tested. However, there is a question of whetheran assumed power number of 5 is reasonable, given that therewere no data provided on the thickness of the impeller blades inthe paper of Wu and Patterson (1989). In fact, the predictedpower numbers determined from the torque in this paper arecloser to the value of 6 measured by Rushton et al. (1950). Asdescribed above, both Rutherford et al. (1996) and Delafosse et al.(2008) have shown that the power number increases withdecreasing blade thickness. In the simulations presented here,an impeller with blades of negligible thickness was modelled andthus predicted power numbers at the higher end of the spectrumcan be expected. Furthermore, Fig. 7 clearly shows an under-estimation of e throughout the tank by all the turbulence modelstested, and it is therefore reasonable to assume that the simulated

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–5988 5987

power numbers based on the volume-integral of e are also under-predicted. Therefore, it is assumed in this paper that the predictedpower number based on torque is more indicative of the actualpower number than the predicted power number based onturbulence energy dissipation rate. It became clear during thiswork that CFD simulations of a stirred tank using a mesh thatproperly resolves the blade thickness would be impractical withthe computing power available.

The width of the axial profile of turbulence energy dissipationrate at 1.07 rR predicted by the SST–CC model is narrower than thatof nearly every other turbulence model tested, as shown in Fig. 7c.This lower turbulence energy dissipation rate in the vicinity of theimpeller explains the relatively low power number (based on thevolume-integral of e) of 4.5 predicted by the SST–CC modelcompared with the other turbulence models tested, as shown inTable 2. The predicted power number based on the volume-integral of e for the SSG–RSM model is highest at 5.5, becausethe predicted turbulence energy dissipation rate for this turbulencemodel becomes higher than for the other models radially awayfrom the impeller tip, as shown in Fig. 6. Note that the powernumber fluctuates as the impeller rotates, and the predicted rangeof these fluctuations is presented in Table 2.

3.6. Computational cost

Approximately five impeller revolutions were required to obtaina pseudo-steady-state simulation for the k–e, SST, SSG–RSM andSST–CC models. Only one further revolution was required foraveraging purposes after the pseudo-steady state had been reachedfor these models. For the SAS–SST model, a large number ofrevolutions were required to obtain good statistical averages. Thirtyrevolutions worth of data were collected in this study. Given that atleast 20 revolutions were required for the SAS–SST model to obtaingood statistical averages, the computational time required to obtainmeaningful results from the SAS–SST model was at least twentytimes greater than that required for the k–e or the SST models. LESmodels are computationally even more intensive since they aremore sensitive to the grid and therefore would require even moregrid refinement (Menter and Egorov, 2005) and more impellerrevolutions for good statistical averages. In an industrial setting,the solution time is likely to be an important factor affecting thechoice of turbulence models in the simulation of turbulent flow instirred tanks. From this point of view, the SST model with curvaturecorrection is the most attractive choice, since it provides a reason-ably accurate solution in the lowest computational time.

4. Conclusions

In this study, the k–e, SST, SST–CC, SAS–SST and SSG–RSMturbulence models were used to simulate flow in a cylindrical tankagitated by a Rushton turbine. All the turbulence models predictthe mean axial and tangential velocities reasonably well, but theytend to under-predict the decay of mean radial velocity away fromthe impeller. The k–e model predicts the random and periodiccomponents of the kinetic energy of fluctuating motions poorly inthe vicinity of the impeller. Furthermore, the predicted trailingvortices are too short and there are no secondary vortex motions.This model also wrongly predicts the location of the maxima ofboth the turbulence (random) kinetic energy and turbulenceenergy dissipation rate to be at the impeller tip, whereas experi-mental results show that the maxima occurs at dimensionlessradial distance of around 1.4 from the impeller axis. The SST modelpredicts the magnitude of these maxima in reasonable agreementwith experiment, and curvature correction helps to improve thepredicted location of the maximum. However, both the SST and

SST–CC models predict overly-dissipative trailing vortices that aretoo short and without the presence of secondary vortex motions,similar to the k–e model.

The SAS–SST model predictions of periodic and turbulence(random) kinetic energy provide qualitative and quantitativeagreement with experiment similar to both the SST and SST–CCturbulence models. Moreover, this model predicts long trailingvortices with secondary motions, as observed experimentally byother researchers. However, the predictions of the turbulenceenergy dissipation rate are not as good as those of the SST andSST–CC turbulence models, although further refinements of themesh are required to make a proper assessment of the accuracy ofthe SAS–SST model. Such refinement proved impractical in thiswork due to limitations on the computer power available and thesignificant number of revolutions required to obtain good statis-tical averages of the turbulence parameters. The SSG–RSM modelreproduces the experimental velocity flow fields well and predictstrailing vortices similar in length to that of the SAS–SST model;however, the prediction of the periodic and turbulence (random)kinetic energy and the turbulence energy dissipation rate arepoor, and no secondary motions are predicted in the trailingvortices. Overall, the SST model with curvature correction is themost satisfactory turbulence model tested for predicting turbu-lent flow in baffled stirred tanks, in terms of both computing timeand accuracy.

For all turbulence models tested, the predicted power numberdetermined by integration of the turbulence energy dissipationrate over the tank is likely to be under-predicted and is signifi-cantly lower than that calculated using the impeller torque. Inorder to accurately predict the power number using the impellertorque, the thickness of the blade must be accounted for andresolved properly by the mesh in the simulations. However, thiswas impractical in this work due to limitations on the computerpower available to produce such a fine mesh.

Nomenclature

Cm A constant in the definition of the turbulence length-scale

D Diameter of tankk Modelled turbulence kinetic energykp Periodic kinetic energykpþ le�r Periodic kinetic energy plus the turbulence (random)

kinetic energy due to large eddies in SAS–SST modelkr Turbulence (random) kinetic energykt Total kinetic energy of fluctuating motionsl Turbulence length-scalen Number of time-stepsN Rotational speedNPt Power number determined from the torque on the

impellerNPe Power number determined by integrating turbulence

energy dissipation rate over the tank volumenTS Number of full turns of the impellerr Radial distance from the centre of rotationR Impeller radiusrR r/R

u0 Velocity fluctuations in the radial directionu Mean velocityui Transient velocityuins Instantaneous velocityule�r Random velocity due to the large eddiesup Periodic velocityur Random velocityUr Radial velocity

H. Singh et al. / Chemical Engineering Science 66 (2011) 5976–59885988

use�r Random velocity due to the smaller eddiesUtip Impeller tip velocityUz Axial velocityUy Tangential velocityv0 Velocity fluctuations in the tangential directionW Width of the bladew0 Velocity fluctuations in the axial directionz Axial distance from the centre of impellerzw 2z/W

Greek Letters

e Turbulence energy dissipation rateeavg Average turbulence energy dissipation rater Densityt Torqueo Turbulence eddy frequency

References

ANSYS-CFX, 2009. 12.1 ed: /http://www.ansys.com/S.Aubin, J., Fletcher, D.F., Xuereb, C., 2004. Modeling turbulent flow in stirred tanks

with CFD: the influence of the modeling approach, turbulence model andnumerical scheme. Experimental Thermal and Fluid Science 28 (5), 431–445.

Bakker, A., Myers, K.J., Ward, R.W., Lee, C.K., 1996. The laminar and turbulent flowpattern of a pitched blade turbine. Chemical Engineering Research and Design74 (A4), 485–491.

Bakker, A., Oshinowo, L.M., 2004. Modelling of turbulence in stirred vessels usinglarge eddy simulation. Chemical Engineering Research and Design 82 (A9),1169–1178.

Bates, R.L., Fondy, P.L., Corpstein, R.R., 1963. An examination of some geometricparameters of impeller power. Industrial & Engineering Chemistry ProcessDesign and Development 2 (4), 310–314.

Brucato, A., Ciofalo, M., Grisafi, F., Micale, G., 1998. Numerical prediction of flowfields in baffled stirred vessels: a comparison of alternative modellingapproaches. Chemical Engineering Science 53 (21), 3653–3684.

Coroneo, M., Montante, G., Paglianti, A., Magelli, F., 2011. CFD prediction of fluidflow and mixing in stirred tanks: numerical issues about the RANS simula-tions. Computers & Chemical Engineering 35 (10), 1959–1968.

Deglon, D.A., Meyer, C.J., 2006. CFD modelling of stirred tanks: numericalconsiderations. Minerals Engineering 19 (10), 1059–1068.

Delafosse, A., Line, A., Morchain, J., Guiraud, P., 2008. LES and URANS simulationsof hydrodynamics in mixing tank: comparison to PIV experiments. ChemicalEngineering Research and Design 86 (12), 1322–1330.

Delafosse, A., Morchain, J., Guiraud, P., Line, A., 2009. Trailing vortices generated bya Rushton turbine: assessment of URANS and large Eddy simulations. Chemi-cal Engineering Research and Design 87 (4), 401–411.

Derksen, J., Van den Akker, H.E.A., 1999. Large eddy simulations on the flow drivenby a Rushton turbine. American Institute of Chemical Engineers AIChE Journal45, 209–221.

Derksen, J.J., Doelman, M.S., Van den Akker, H.E.A., 1999. Three-dimensional LDAmeasurements in the impeller region of a turbulently stirred tank. Experi-ments in Fluids 27 (6), 522–532.

Ducci, A., Yianneskis, M., 2005. Direct determination of energy dissipation instirred vessels with two-point LDA. American Institute of Chemical EngineersAIChE Journal 51 (8), 2133–2149.

Escudie, R., Line, A., 2003. Experimental analysis of hydrodynamics in a radiallyagitated tank. American Institute of Chemical Engineers AIChE Journal 49 (3),585–603.

Escudie, R., Line, A., 2006. Analysis of turbulence anisotropy in a mixing tank.Chemical Engineering Science 61 (9), 2771–2779.

Frank, T., Prasser, H.M., Menter, F., Lifante, C., 2010. Simulation of turbulent andthermal mixing in T-junctions using URANS and scale-resolving turbulencemodels in ANSYS CFX. Nuclear Engineering and Design 240 (9), 2313–2328.

Hartmann, H., Derksen, J.J., Montavon, C., Pearson, J., Hamill, I.S., van den Akker,H.E.A., 2004a. Assessment of large eddy and RANS stirred tank simulations bymeans of LDA. Chemical Engineering Science 59 (12), 2419–2432.

Hartmann, H., Derksen, J.J., Van Den Akker, H.E.A., 2004b. Macroinstabilityuncovered in a Rushton turbine stirred tank by means of LES. AmericanInstitute of Chemical Engineers AIChE Journal 50 (10), 2383–2393.

Honkanen, M., Vaittinen, J., Saarenrinne, P., Korpijarvi, J., 2007. Time-resolvedstereoscopic PIV experiments for validating transient CFD simulations. In:Proceedings of the Seventh International Symposium on Particle ImageVelocimetry. Roma, Italy, September 11–14. p 1–13.

Jahoda, M., Mostek, M., Kukukova, A., Machon, V., 2007. CFD modelling of liquidhomogenization in stirred tanks with one and two Impellers using large eddysimulation. Chemical Engineering Research and Design 85 (5), 616–625.

Javed, K.H., Mahmud, T., Zhu, J.M., 2006. Numerical simulation of turbulent batchmixing in a vessel agitated by a Rushton turbine. Chemical Engineering andProcessing 45 (2), 99–112.

Jones, R.M., Harvey III, A.D., Acharya, S., 2001. Two-equation turbulence modelingfor impeller stirred tanks. Journal of Fluids Engineering 123, 640–648.

Lee, K.C., Yianneskis, M., 1998. Turbulence properties of the impeller stream of aRushton turbine. American Institute of Chemical Engineers AIChE Journal 44(1), 13–24.

Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineer-ing applications. AIAA Journal 32, 1598–1605.

Menter, F.R., 2009. Review of the shear-stress transport turbulence modelexperience from an industrial perspective. International Journal of Computa-tional Fluid Dynamics 23 (4), 305–316.

Menter, F.R., Egorov, Y., 2005. A scale-adaptive simulation model using two-equation models. In: Proceedings of the 43rd AIAA Aerospace Sciences Meet-ing and Exhibit, January 10–13. Reno, NV, United states: American Institute ofAeronautics and Astronautics Inc. pp. 271–283.

Menter, F.R., Egorov, Y., 2010. The scale-adaptive simulation method for unsteadyturbulent flow predictions. Part 1: theory and model description. Flow,Turbulence and Combustion 85 (1), 113–138.

Menter, F.R., Egorov, Y., Lechner, R., Cokljat, D., 2010. The scale-adaptive simula-tion method for unsteady turbulent flow predictions. Part 2—application tocomplex flows. Flow, Turbulence and Combustion 85, 139–165.

Montante, G., Lee, K.C., Brucato, A., Yianneskis, M., 2001. Numerical simulations ofthe dependency of flow pattern on impeller clearance in stirred vessels.Chemical Engineering Science 56 (12), 3751–3770.

Murthy, B.N., Joshi, J.B., 2008. Assessment of standard k–e, RSM and LES turbulencemodels in a baffled stirred vessel agitated by various impeller designs.Chemical Engineering Science 63 (22), 5468–5495.

Ng, K., Fentiman, N.J., Lee, K.C., Yianneskis, M., 1998. Assessment of sliding meshCFD predictions and LDA measurements of the flow in a tank stirred by aRushton impeller. Chemical Engineering Research and Design 76 (6), 737–747.

Rushton, J.H., Costich, E.W., Everett, H.J., 1950. Power characteristics of mixingimpellers. Chemical Engineering Progress 46 (9), 467–476.

Rutherford, K., Mahmoudi, S.M., Lee, K.C., Yianneskis, M., 1996. The influence ofRusthon impeller blade and disk thickness on the mixing charasteristics ofstirred vessels. Trans IChemE 74(A), 369–378.

Sheng, J., Meng, H., Fox, R.O., 1998. Validation of CFD simulations of a stirred tankusing particle image velocimetry data. Canadian Journal of Chemical Engineer-ing 76, 611–625.

Smirnov, P.E., Menter, F.R., 2009. Sensitization of the SST turbulence model torotation and curvature by applying the Spalart–Shur correction term. Journalof Turbomachinery 131 (4), 1–8.

Spalart, P.R., Shur, M., 1997. On the sensitization of turbulence models to rotationand curvature. Aerospace Science and Technology 1 (5), 297–302.

Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling the pressure–strain correla-tion of turbulence: an invariant dynamical systems approach. Journal of FluidMechanics 227, 245–272.

Wu, H., Patterson, G.K., 1989. Laser-doppler measurements of turbulent-flowparameters in a stirred mixer. Chemical Engineering Science 44 (10),2207–2221.

Yapici, K., Karasozen, B., Schafer, M., Uludag, Y., 2008. Numerical investigation ofthe effect of the Rushton type turbine design factors on agitated tank flowcharacteristics. Chemical Engineering and Processing: Process Intensification47 (8), 1340–1349.

Yeoh, S.L., Papadakis, G., Yianneskis, M., 2004. Numerical simulation of turbulentflow characteristics in a stirred vessel using the LES and RANS approaches withthe sliding/deforming mesh methodology. Chemical Engineering Research andDesign 82 (7), 834–848.

Zadghaffari, R., Moghaddas, J.S., Revstedt, J., 2010. Large-eddy simulation ofturbulent flow in a stirred tank driven by a Rushton turbine. Computers &Fluids 39 (7), 1183–1190.

Zhang, Y., Yang, C., Mao, Z., 2006. Large eddy simulation of liquid flow in a stirredtank with improved inner–outer iterative algorithm. Chinese Journal ofChemical Engineering 14 (3), 321–329.