1-s2.0-s0009250905006214-main

Chemical Engineering Science 61 (2006) 1217–1236www.elsevier.com/locate/ces

CFD simulations of two stirred tank reactors with stationarycatalytic basket

P. Magnicoa,∗, P. Fongarlandb

aLaboratoire de Génie des Procédés Catalytiques, CNRS-ESCPE Lyon, 43 bd du 11 Novembre 1918, B.P. 2077, 69616 Villeurbanne Cedex, FrancebLaboratoire de Catalyse de Lille, USTL, 59655 Villeneuve d’Ascq Cedex, France

Received 24 September 2004; received in revised form 29 April 2005; accepted 8 July 2005Available online 3 October 2005

Abstract

Among the different systems used for laboratory kinetic investigation, stationary catalytic basket stirred tank reactors (SCBSTRs) allowone to study triphasic reactions involving shaped catalyst with large size. The hydrodynamics of these complex reactors is not well knownand has been studied experimentally in only a few cases. Despite the difference in the design of two commercial SCBSTRs reported inthese works, the local measurements of the liquid–solid mass transfer coefficient inside the catalytic basket revealed the same velocityprofile. The aim of the present work is therefore to investigate more accurately the hydrodynamics of the two reactors by means of CFDin order to compare the effect of the blade/baffle hydrodynamic interaction on the flow pattern. Owing to the geometrical complexity ofthe reactors, the hydrodynamic investigation is based on the k–� model and the Brinkman–Forsheimer equations. The agreement at thelocal level with the experimental data (PIV and mass transfer measurements) validates this preliminary work performed with the standardvalues of the parameters present in the turbulent model and the Brinkman–Forsheimer equations. The simulations reveal in both reactorsa ring-shaped vortex around the impeller in the agitation region. The high axial location of its centre induces a reverse flow at the tips ofthe basket. Owing to the fluid friction in the porous medium, the azimuthal flow in the core region is transformed into a radial flow inthe basket where the flow decreases abruptly. Vertical vortices are located at the blade tips and at the downstream face of the baffles orthey are located in the basket on both sides of the baffles, depending on the design and the location of the baffles. At the inner radiusinterface of the basket, the vertical blade impeller induces a rather homogeneous velocity profile, but the pitched blade impeller imposes ahigh velocity at the plane of symmetry. Therefore the simulations demonstrate that two different local velocity patterns and two differentporous media may induce the same mass transfer properties.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Computational fluid dynamic; Particle image velocimetry; Stationary catalytic basket; Continuous stirred tank reactor; Mass transfer

1. Introduction

Several kinds of laboratory reactors are used to study thekinetics of triphasic reactions catalysed by a solid phase:triphasic fixed bed reactors (trickle-bed or packed bed),structured reactors like monolith or microreactors, and alsoperfectly mixed reactors with catalyst in suspension or main-tained in a basket (Perego and Peratello, 1999; Dudukovic

∗ Corresponding author. Tel.: +33 04 72 43 17 63;fax: +33 04 72 43 16 73.

E-mail addresses: [email protected], [email protected](P. Magnico).

0009-2509/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.07.025

et al., 2002). In petroleum engineering, and especially inthe case of hydrotreatment reaction, trickle-bed reactors aregenerally employed for kinetic investigations because theyare used in industrial processes. But these integral reactorsare not the most adapted for kinetic measurement. Continu-ous stirred tank reactors may be preferred since they give theapparent rate directly by the measurement of inlet and out-let concentrations. Furthermore, detection of mass or heattransfer limitations is easier in this kind of reactor, whichare a major concern to measure intrinsic kinetic.

Contrary to slurry perfectly mixed reactors where cata-lyst is suspended, a catalytic basket stirred tank is set upin order to use shaped catalytic particles with a size greaterthan 1 mm. The solid particles, too heavy to be suspended,

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1218 P. Magnico, P. Fongarland / Chemical Engineering Science 61 (2006) 1217–1236

are fixed in a static annular basket around an impeller(Arroyo et al., 2000, 2001; Goto and Saito, 1984) or in a ro-tating basket moved by a shaft (Goto and Saito, 1984; Turekand Winter, 1990; Teshima and Ohashi, 1977; Warna et al.,2002). For the case of the stationary catalytic basket stirredtank reactor (SCBSTR), also called Robinson–Mahoneyreactor (Mahoney et al., 1978), the hydrodynamics of theliquid phase is a major concern even for these laboratoryreactors. Since the stationary basket is placed around theimpeller, a significant effect of the basket on the flow pat-tern induced by the impeller can be expected. This mayhave an important impact on the liquid–solid mass transferall around the basket.

Few experimental hydrodynamic characterizations havebeen carried out until now for these reactors (Pavko et al.,1981; Mitrovic, 2001; Fongarland, 2003). The reason lies inthe opacity of the fixed bed inside the catalytic basket, whichprevents one investigating close to the turbine. Two SCB-STRs commercialized by Autoclave Engineers (AE) and ParrInstruments have been studied. The velocity field of the AEreactor has been investigated in the region between the bas-ket and the external tank wall by means of the particle im-age velocimetry (PIV) technique (Mitrovic, 2001). In orderto investigate the velocity field inside the basket, the localliquid–solid mass transfer has also been studied by means ofdissolution of naphthol pellets. The second reactor has beeninvestigated as well by the mass transfer characterization(Fongarland, 2003). The PIV data reveal a high heterogene-ity of the velocity field inside the basket. Despite the differ-ence on the impeller and the baffle design, the experimentalresults also show that the two reactors have similar localproperties of mass transfer. In fact the mass transfer measure-ments are not accurate enough to estimate the superficial ve-locity profile because (a) it is necessary to determine a Sher-wood correlation to link the velocity to the liquid–solid masstransfer coefficient (ks), (b) it allows one to investigate thefield of the superficial velocity module only, and (c) the spa-tial resolution of the experiments is the basket width whichis much larger than the scale of the velocity heterogeneity.

Therefore the interest of the present work is to use thecomputational fluid dynamic (CFD) approach to go deeperin the investigation and to describe the hydrodynamics inthe catalytic basket and in the region round the impeller.Moreover, until now no hydrodynamics study has been car-ried out by means of CFD in fixed basket stirred tank re-actors. Only a brief study has been performed in the caseof a spinning basket reactor (Warna et al., 2002). We showin the present paper that (a) the CFD approach allows oneto quantify the hydrodynamic interaction of the blades withthe baffles, (b) the same liquid–solid mass transfer proper-ties can be obtained with two different flow patterns and twodifferent porous media, and (c) the velocity field inside thebasket cannot be compared with the outside velocity evenclose to the outer basket surface.

The simulations are carried out by means of the commer-cial software package FLUENT for three rotation speeds.

The use of the numerical approach in order to characterizesuch a complex reactor (presence of a heterogeneous porousmedium, complexity of the impeller design, close distancebetween the blades and the baffles, multiphase transport)represents a challenge, particularly at the interfaces of thecatalytic basket and in the region close to the tank wall wherethe turbulence is not well established. Therefore the goal ofthe present work is to sketch the mean flow pattern insidethe catalytic basket and around the impeller. In order to suc-ceed, several simplifications are carried out on the reactordesign and on the hydrodynamic model (assumption of flowperiodicity, stationary flow, use of the k–� model) despitethe non-periodic and non-stationary flow and despite theanisotropic turbulence close to the impeller. But, the agree-ment at the local level with the experimental data (hydrody-namics and mass transfer) validates this preliminary workperformed with the standard values of the parameters presentin the turbulence model and the Brinkman–Forsheimerequations.

The paper is organized as follows. In Section 2 wedescribe briefly the geometry of the two reactors and theexperimental methodologies. In Section 3, the computa-tional domains are described. The modelling approach andits limitations are also explained. In Section 4, the meshsensitivity is checked. The simulations are also validatedagainst the PIV data. The velocity field in the basket andclose to the impeller is described in Section 5 for the twoSCBSTRs. In a final section, the agreement of the computedks with experimental data confirms the validation of thenumerical approach. The superficial velocity profiles insideand outside the basket of the two reactors are compared. Inthis section, the discussion sheds light on the inadequacy ofthe solid dissolution rate measurements to characterize thevelocity field in the basket.

2. Reactor geometry and experimental investigations

2.1. Reactor geometry

The AE reactor, which has a volume of 0.9 l, was inves-tigated by Mitrovic (2001) (see Fig. 1). It has an inner di-ameter of 8 cm and a height of 18 cm. The fluid is stirredby a six-blade radial-type turbine and a pair of four 45◦pitched blade axial-type turbines (Fig. 1a). The diameter ofthe blades is 3.2 cm. The radial turbine is 5.5 cm high and0.2 cm thick; the axial turbine is 1 cm high and 0.2 cm thick(Fig. 1b and c). The catalytic basket has an inner diameterand an outer one of 4.25 and 5.75 cm, respectively. The bas-ket height is 9 cm. Two horizontal rings (0.9 cm width) closethe top and the bottom of the basket. Two kinds of bafflesequip the reactor. Four outer baffles (9 cm × 0.4 cm) are lo-cated along the catalytic basket at the outer interface. Fourinner baffles (9 cm × 1.6 cm) are inside the basket and inthe stirred region. The basket and the turbines are located at11.4 cm respectively from the top of the reactor. The gas is

P. Magnico, P. Fongarland / Chemical Engineering Science 61 (2006) 1217–1236 1219

(4)

(5)

(3)

(1)

(2)

(1)

(2)

(6)

15˚

(7)

(a) (b) (c)

Fig. 1. Sketch of the AE reactor: (1) six-blade radial-type turbine; (2) four pitched blade turbine; (3) catalytic basket; (4) internal baffle; (5) externalbaffle; (6) ring closing the basket at the top and at the bottom.

injected from the bottom of the reactor, but at the top thereis no gas/liquid interface.

The Parr reactor investigated by Fongarland (2003) hasa volume of 300 ml (Fig. 2). It has a diameter of 6.4 cmand a height of 10 cm. The interface gas/liquid is located at8.2 cm from the reactor base. The fluid is stirred by a gas-inducing turbine which is composed of three 17◦ pitchedblades (Fig. 2a). The blade dimensions are 2.4 cm in diam-eter, 4.3 cm high and 0.2 cm thick. The gas is transferred tothe liquid phase through the shaft by means of two holeslocated above the gas/fluid interface and at the horizontalplane of symmetry of the blades. The basket has an innerdiameter of 3.4 cm, an outer diameter of 5 cm and a heightof 5.1 cm. As in the AE reactor, two horizontal rings (0.3 cmwidth) close the basket. The reactor has two kinds of baffles.Three inner baffles of cylindrical form are located inside thebasket and three outer ones are located along the tank wall(5.1 cm height, 4 mm width and 3 mm large). The bottom ofthe blades and of the basket are located at 3.9 and 1.3 cm,respectively, from the bottom of the reactor.

2.2. Experimental methodologies

The velocity measurements were performed by means ofthe PIV method in the region between the basket filled withPMMA cylinders (0.61 cm long and 0.217 cm in radius)and the vessel wall. Three vertical planes of measurement(6 cm×6 cm) were located at the azimuthal angle � of −35◦,0◦ and +35◦ from one of the inner baffles. To obtain a pro-file all along the basket at each angle, two planes definedby the axial location 0 cm < z < 6 cm and 3 cm < z < 9 cmfrom the bottom of the basket were necessary. In the sharedzone (3 cm < z < 6 cm), the data accuracy was evaluated andeach velocity component was averaged. In order to measurethe azimuthal components, a vertical plane tangent to thebasket grid is needed. The intersection between two perpen-

dicular planes forms a straight line along which the velocityfield was analysed. The axial component (Z direction) mea-sured in the two planes was also averaged. The measure-ment frequency was 10 s−1. The statistical average was per-formed over 200 measurements which were not locked withthe impeller orientation. Therefore the velocity data are av-eraged over all the impeller orientations. The agitation speed� ranged from 500 to 1500 rpm.

In order to estimate the performance of the SCBSTR,Mitrovic and Fongarland measured the liquid–solid masstransfer coefficient with the dissolution of naphthol grainsin water and in n-Heptane at standard conditions of temper-ature and pressure. Two approaches were used: the globalapproach consisting in filling all the basket with naph-thol pellets (see Tables 1 and 2) and the local approachconsisting in filling at an axial position only a 1 cm thickslice, the remainder of the basket being filled with inertgrains of PMMA. The local experiments in the AE reactorgave a characteristic time of dissolution in water rangingfrom 9 × 103 to 1.7 × 104 s and a time of dissolution inn-Heptane ranging from 1600 to 3100 s, depending on theagitation speed. In the Parr reactor filled in n-Heptane, thecharacteristic time ranged from 300 to 1200 s. These timesare much greater than the macromixing one (few seconds)measured experimentally. To prove the coherence betweenthe hydrodynamic measurement outside the basket and themeasurements of the local dissolution rate, Mitrovic hasdetermined, for both fluids, a Sherwood correlation whichlinks the local superficial velocity and the liquid–solid masstransfer coefficient of naphthol. The parameters were de-termined by means of a cylindrical fixed bed reactor, thelength of which was equal to the basket width (Mitrovic,2001). The Reynolds number Reks ranged from 2 to 55. Thecorrelation used in the AE reactor, after a more accurateanalysis, seems to give good predictions of the liquid–solidmass transfer coefficient of naphthol in n-Heptane, but alarge deviation is observed for ks values in water.


(6) (5)

(4)

(1)

(2)

(1) (1c)

(1b)

(3)

(a) (b) (c)

Fig. 2. Sketch of the Parr reactor: (1) three pitched blade gas-inducing-type turbine; (1b) gas suction hole; (1c) gas ejection hole; (2) catalytic basket;(3) ring closing the basket at the top and at the bottom; (4) external baffle; (5) internal baffle; (6) liquid/gas interface.

Table 1Geometrical characteristics of the grains and values of the permeability of the catalytic basket inside the AE reactor

dcyl (m) Lcyl (m) dp (m) dks (m) � � (1/m2) � (1/m)

PMMA 0.00217 0.0061 0.002765 0.0033 0.47 5.3 × 107 3233Naphthol 0.002 0.0058 0.00256 0.00259 0.47 6.193 × 107 3490

Table 2Geometrical characteristics of the grains and values of the permeability of the catalytic basket inside the Parr reactor

dcyl (m) Lcyl (m) dp (m) dks (m) � � (1/m2) � (1/m)

PMMA 0.00217 0.0061 0.00276 0.0033 0.6 1.453 × 107 1172Naphthol 0.002 0.0058 0.00256 0.00259 0.6 1.695 × 107 1266

3. Numerical methodology

The interaction of the baffles with the blades makes theflow non-stationary. Macroinstabilities coming from the in-teraction of the impinging jet with the tank wall, the swirlingflow at the tank bottom or the shedding of the trailing vortexfrom the impeller, for example, also generate non-stationaryflows. The macroinstabilities have a frequency much lowerthan the impeller rotation one, depending on the tank andthe impeller design. Therefore the flow pattern cannot beperiodic. Hydrodynamics is also characterized by complexturbulent properties such as anisotropy, unclear frequentialseparation in the power spectrum between the macroinsta-bilities and the turbulent fluctuations (Galletti et al., 2005).The complete study of the flow pattern requires outstand-ing numerical tools and large computational requirements(Roussinova et al., 2003; Hartmann et al., 2004). To the bestof the authors’ knowledge, most of these studies are carriedout in simple reactors such as the standard mixing vesselstirred by a Rushton impeller. Owing to the complexity ofthe SCBSTR geometry and therefore of the flow, we decidedto simplify the design and to use an unsophisticated hydro-

dynamic approach in order to study the mean flow patternonly.

3.1. Geometry of the computational domain

In order to reduce the computational requirements, geo-metrical simplifications are carried out in the following way.In the AE reactor, we replace the conical-shaped base by ahorizontal one and we do not take into account the basketsupport ((7) in Fig. 1c). In the Parr reactor, we do not takeinto account the gas volume and the free surface gas/liquidat the top of the reactor. We assume that the impeller, thebasket and the baffles are located at the centre of the tank.Therefore, the domain in the two reactors can be divided intotwo symmetrical parts by a central horizontal plane, and thesimulations are performed in the lower half of the reactor.Owing to the geometrical periodicity in the azimuthal direc-tion, the computational domain is reduced to a sector withperiodic boundary conditions set at the two opposite verticalplanes. The computational domain of the Parr reactor hasdimensions of 6.4 cm, 4.1 cm and 60◦ in radial, axial and


azimuthal directions, respectively. The dimensions of the AEreactor domain are 4 cm, 9 cm and 180◦ in radial, axial andazimuthal directions, respectively.

Versicco et al. (2004) showed that the periodic assump-tion cannot be applied to the fluid flow and gives inaccurateresults. In particular, the authors found that the radial veloc-ity is overestimated if the simulations are performed on adomain lower than 1/4th of the tank in the azimuthal direc-tion. They used an immersed boundary method coupled withthe direct numerical simulation (DNS) and a Reynolds av-erage Navier–Stokes (RANS) approach to compute the flowfield at a Reynolds number (Retip) of 1636. In the past, sev-eral authors used the periodic boundary conditions in orderto reduce the computational cost (Ranade, 1997; Brucatoet al., 1998; Sheng et al., 1998). The numerical results atReynolds numbers higher than 9000 do not show any over-estimation of the radial component of the mean velocity. Inthe present study, half and 1/3rd of the vessel volume in theazimuthal direction are used for the AE and the Parr reactor,respectively. Therefore the recommendation made by Vers-icco et al. is respected. If the periodic boundary conditionsare not used, the computational requirement would be toolarge: 2 × 250 000 and 3 × 300 000 cells are needed for theAE and the Parr reactor respectively, in this case (see below).

From now, we define the inner zone as the cylindricalregion delimited by the inner radius face of the catalyticbasket. Its bottom is located at the base of the ring belowthe basket. In the AE reactor the bottom is at z = 4.5 cm,and in the Parr reactor it is at z = 1.5 cm. The outer zonedefines the volume of the reactor minus the inner zone andthe porous medium. The basket faces defined by the innerradius and the outer one are named inner boundary and outerboundary, respectively. The bottom of the basket defines theorigin of axial coordinate.

3.2. Turbulence model

In the experimental works described in Section 2, the im-peller rotation speed ranged from 500 to 1500 rpm, giving aReynolds number Retip running from 13 000 to 40 000 andfrom 9000 to 25 000 for the AE and the Parr reactor, respec-tively. Therefore the turbulence can be assumed well estab-lished in most of the reactor volume if the reactor is suitablybaffled and if the free gas/liquid surface is not too deformed(Nagata, 1975). But even if the Reynolds number range ishigh enough around the impeller, we will see that the smallpermeability of the catalytic basket prevents the fluid fromtravelling through with a sufficiently high velocity in accor-dance with the experimental observations. The turbulencecannot be well established in the outer region and it shouldbe relevant to use a turbulent model available at moderateReynolds number.

In the zone around the impeller the anisotropic turbulencewould be suitably modelled by large eddy simulation (LES)(Hartmann et al., 2004; Revsteldt et al., 1998; Sheng et al.,

2000). The Reynolds stress model (RSM) has the advan-tage of being simpler and taking into account the turbulenceanisotropy. But, as in the k–� model, the turbulence scalesare not resolved properly. Moreover, all the RANS modelsgive the same mean flow field accurately (Campolo et al.,2003; Sheng et al., 1998), but underestimate the turbulentkinetic energy process and do not model accurately the mix-ing (Yeoh et al., 2005). The interest of LES is to solve bymeans of DNS the large scales of the anisotropic motions.The small scales are supposed to be universal and are mod-elled by a sub-grid model. The cut-off length scale lies inthe inertial sub-range of turbulence and the two scales areclearly separated by means of a filter, the width of whichis of the order of the grid size (Germano, 1992). But in thepresence of a porous medium, in which the flow is laminar,only the k–� and k–� RNG models, used outside the basket,are implemented in the software FLUENT.

3.3. Hydrodynamic model in the catalytic basket

The catalytic basket is filled with cylinders. The annularfixed bed is assumed to be an effective medium characterizedby its porosity and permeability. The laminar flow field iscomputed with the Brinkman–Forsheimer equations whichcouple the Navier–Stokes equations with the Ergun ones inthe following way (Ergun, 1952; Macdonald et al., 1979):

�

�t�U + �U �∇ �U = − �∇P

�− �

K�U − �� U , (1)

where

1

K= � + 1

��| �U | (2)

with

� = 150(1 − �)2

�3d2p

and � = 1.75(1 − �)

�3dp

. (3)

In expression (3), dp is the equivalent diameter of a spher-ical particle. For a cylinder the diameter has the followingexpression:

dp = 6Lcyl

2 + 4Lcyl/dcyl. (4)

Tables 1and 2 show the characteristic parameters of theporous medium inside the catalytic basket for each SCB-STR. The empirical relation (2) is available in the range ofReynolds number Rep less than 1000. In the simulations,the Reynolds number reaches a maximum value of 340 at1500 rpm. Therefore the use of the Brinkman–Forsheimerequations (1) is justified.

Expressions (2) and (3) imply the assumption of isotropyand homogeneity of the medium. The hypothesis of effectivemedium is justified if the size of the fixed bed is muchlarger than the grain size (Magnico, 2003). In the SCBSTR,the basket thickness to the cylinder length ratio is around


4. Therefore the fixed bed should be considered as a two-phase structure in which the flow field is computed at thepore scale. Considering the pellet size and their randomspatial arrangement, the basket boundaries cannot be planeand must present an important roughness. Moreover the poremouths constitute the holes at the interfaces. Therefore, at theouter boundary, the flow would be constituted of jets comingfrom the pores and in the inner boundary where the flowis essentially azimuthal, the roughness would influence theflux through the boundary. These considerations are all themore important because the anisotropic shape of cylindersincreases the heterogeneity of the packing close to the basketboundaries. The above assumptions (effective medium withhomogeneous boundaries and isotropy) must be consideredas very rough. But separating the two phases in the catalyticbasket would require too large resources. The anisotropicassumption may be introduced by defining the parameters �and � as second-order tensors. The problem is therefore toguess the value of each term of the tensors.

3.4. Numerical method

Two numerical methods are commonly used in stirred re-actors in order to reproduce the effect of the impeller rota-tion: the multiple reference frame (MRF) approach (Luo etal., 1994) and the sliding mesh approach. In the two meth-ods, the reactor is divided into two regions: (1) a cylindricalone containing the impeller and (2) the remainder of the re-actor. The momentum equations are solved in the rotatingframe reference in region (1) and in the laboratory framereference in region (2). The MRF method solves the steadystate momentum equations, considering the impeller positionas fixed. In the case of a strong impeller baffle interactionor a mixing structure analysing, the sliding mesh approachis the most appropriate because informations are exchangedthrough the sliding interface at each time step. In the reac-tors studied here, the inner baffles and the blades are closeto each other. Therefore, the sliding mesh approach is nec-essary to compute the velocity field inside the inner region.But with the software version 6.1, the sliding mesh fails inthis context producing overflows at the first iterations what-ever the value of the time step. When periodic surfaces andnon-conformal interfaces are present in the computationaldomain, the computation of the new position of the nodesduring the sliding mesh simulation may fail (private com-munication with FLUENT France). In the region betweenthe catalytic basket and the tank wall, we will see that thevelocity field is not very sensitive towards the blades ori-entation, allowing one to assume that in this zone the MRFmethod is accurate enough. The interface position betweenthe two regions (1) and (2) must be far from the blade tipsin order to minimize its effect on the numerical solution(Oshinowo et al., 2000). No static solid must be present inregion (1) and the interface must be of cylindrical symme-try. Therefore, in order to move the interface away from the

blade tips, the interface is located along the inner radius ofthe basket in the Parr reactor and at the inner baffle edges inthe AE reactor. But the interface position remains too closeto the blade tips. For the two reactors, the base of region (1)is located at the bottom of the tanks.

The mesh adopted in the AE reactor is prismatic aroundthe radial turbine, tetrahedral around the pitched blade tur-bine and hexahedral elsewhere. The cell number is 247 362.Along the three vertical straight lines where the PIV dataare analysed, the mesh is refined in order to have a betternumerical accuracy locally. The refined volume is a cylin-der 0.6 cm in radius and 4.5 cm high. In the Parr reactor,the mesh is tetrahedral around the blades and the inner baf-fles, and hexahedral elsewhere. The cell number is 300 000.No-slip conditions are defined at the blades wall and thestandard wall treatment used in turbulent flows is imposedat the tank wall. The momentum equations are solved withthe coupling SIMPLE algorithm and the second upwind dis-cretization scheme. The pressure is computed by means ofthe PRESTO scheme.

4. Hydrodynamic validation

In this section, the numerical methodology is validatedwith the AE reactor by checking the mesh sensitivity andby comparing the numerical results against the PIV data.Therefore the basket and the reactor are assumed to be filledwith PMMA cylindrical particles and water, respectively.The values of the parameters defining the permeability ofthe porous medium are reported in Tables 1 and 2.

4.1. Mesh sensitivity

In order to check the mesh independence, the refinementis performed in three ways: (a) in regions where the velocitygradients are large, (b) around the surface separating the twoMRF zones, and (c) along the straight lines where PIV mea-surements were performed. In the last case the refinementregion is a cylinder 6 mm in radius and 4.5 cm high. Fig. 4displays the radial profiles of the three components of thenormalized velocity at different axial locations in a verticalplane defined by �=−60◦, midway between two blades (seeFig. 3a). In the domain of computation, the inner baffles arelocated at −80◦ and +10◦, and the outer baffles are locatedat −35◦ and +55◦. The impeller rotation speed is 1500 rpm(Utip = 2.51 m/s). The axial locations of the radial profilesare shown in Fig. 3b which displays the velocity vector fieldin the vertical plane (2a). Owing to the high velocity in theinner region, the vector length is set constant all over theplane. Every other vector is skipped. The profiles obtainedwith refinement (a) and (b) are compared with the profilescomputed without any refinement.

The refinement (a) takes place along the blades, the innerbaffles and the wall at the bottom of the tank. The final cellnumber is 397 000. This refinement has an effect on the radial


Fig. 3. (a) Sketch of the blades/baffles configuration in the computationaldomain of the AE reactor: (1a) plane of periodicity; (2a) vertical planelocated at −60◦. (b) Vector velocity field in the plane (2a). The rotationspeed is 1500 rpm. The vector length is set constant. Every other vectoris skipped. (1) Radial location of the blade tips, (1′) axial location of theblade bottom, (2) inner boundary, (3) outer boundary, (4) reactor wall, (5)ring closing the basket tips. (a) z = 0.65 cm; (b) z = 1.5 cm; (c) z = 2 cm;(d) z = 2.5 cm; (e) z = 4.5 cm (symmetry plane location).

component near the symmetry plane only in the inner region.At the surface separating the two MRF zones, the azimuthalcomponent displays a small discontinuity in the vicinity ofthe plane of symmetry. As in (a), the refinement (b) has nomajor effect on the velocity. It decreases the discontinuity ofthe azimuthal component a little. In the outer region and inthe porous medium, the velocity profiles remain unchanged.Along the PIV measurement lines, the velocity is sensitiveto the local refinement (c) if the lines are located in front ofthe inner baffles. The sensitivity is less than 10% dependingon the blade orientation. In the k–� model, the production ofturbulent energy and the turbulent viscosity are expressed interms of the velocity gradient and are more sensitive to themesh refinement. Therefore, refinement (b) is chosen to de-scribe the spatial properties of the turbulence (see Section 5).

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04r (m)

U' r

(1) (4)(3)(2)

(e)

(d)

(c)

(b)

(a)

adaptation inside the inner regionadaptation vs velocity gradients

-0.95

-0.85

-0.75

-0.65

-0.55

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

r (m)

U' t

(1) (4)(3)(2)

(e)

(d)

(c)(b)

(a)

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

r (m)U

' z

(1) (4)(3)(2)

(e)

(d)(c)

(b)

(a)

(a)

(b)

(c)

Fig. 4. Radial profile of the normalized velocity in the vertical planelocated at −60◦ at several axial locations displayed in Fig. 3b (AEreactor). The rotation speed is 1500 rpm. (a) Radial component; (b) axialcomponent; (c) azimuthal component. (1), (2), (3), (4), (a), (b), (c), (d),(e): see Fig. 3.

4.2. Validation using PIV data

The stationary hypothesis of the flow imposed by theMRF approach can be circumvent by computing the flowfield for different angular orientations of the impeller. Theflow field computed at each orientation is considered as in-stantaneous. In the AE reactor, an accurate computation isnecessary to compare the results of the simulations againstexperimental data. The flow field is computed for four im-peller orientations �(blade/baffle) : 0◦, −7.5◦, −15◦, −22.5◦where �(blade/baffle) is the azimuthal angle between a verticalblade and an inner baffle arbitrarily chosen together. Eachorientation corresponds to an instantaneous configuration.We do not take into account the azimuthal orientation of thepitched blades because we assume that they induce a globaleffect of fluid pumping and do not influence the flow locallyin the porous medium and in the outer region. Therefore the


-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Ur (

ms-1

)

Z (m)

1000 rpm 1500 rpm

500 rpm

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Uz (

ms-1

)

Z (m)

1000 rpm

1500 rpm

500 rpm

(a)

(b)

Fig. 5. Axial profile of the velocity averaged over four blade/baffle ori-entations in the AE reactor at 4 mm from the catalytic basket in front ofthe inner baffles. (a) Radial component; (b) axial component. Lines withsymbols: experiments; full lines: numerical simulations.

impeller has an azimuthal periodicity of 30◦. The experi-mental measurements were located at −35◦, 0◦ and +35◦from an inner baffle. The two inner baffles present inside thecomputational domain are geometrically equivalent. There-fore, the stationary velocity profiles along the catalytic bas-ket are averaged over height instantaneous ones.

In Fig. 5, the computed stationary velocity profiles, alongthe basket in front of an inner baffle and at 4 mm from theouter boundary, are compared to the experimental ones. Theradial and the axial components of the velocity are repre-sented for the three agitation speeds. The origin of the axialcoordinate is the bottom of the basket. The vertical bars rep-resent the variation amplitude of the instantaneous velocityat z = 0, 2.25, 4.5, 6.75 and 9 cm. Fig. 5b also reports thedifference between the axial component measured in the ra-dial plane and in the azimuthal one by means of vertical barsat z = 1.5, 3.5, 5.5 and 7.5 cm.

In Fig. 5a, the experimental curves display local maxima,the axial positions of which are mostly independent of theimpeller rotation speed. This observation means certainlythat the flow at the outer boundary is not homogeneousowing to the presence of pore mouths, which would induceflow jets. Flow disequilibrium between the lower and theupper part of the basket can be observed in this figure and inFig. 5b. The flux is more important in the upper part. Thedifference of the radial velocity between the two parts com-pared to the mean value is 74% (500 rpm), 4% (1000 rpm)and 38% (1500 rpm). In Fig. 5b, the flux disequilibriumalso induces a smaller axial velocity at the top of the basket

-0.05

-0.025

0

0.025

0.05

0.075

0.1

0.125

0.15

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Ur (

ms-1

)

Z (m)

+35˚-35˚

Fig. 6. Axial profile of the radial velocity in the AE reactor at 4 mmfrom the catalytic basket and at −35◦ and +35◦ from the inner baffles.Rotation speed: 1500 rpm. Lines with symbols: experiments; full lines:numerical simulations.

compared to the velocity at the bottom and an offset ofthe experimental curves of 1 cm to the bottom. This fluxinhomogeneity can be explained by the heterogeneity of thepacking which would be more permeable in its upper part.But this cannot explain the erratic variation of the disequi-librium with the agitation speed because the same packingwas used. The other explanation is the discrepancy of themeasurements between the upper and the lower verticalplane of PIV measurement as observed in the dissertation(Mitrovic, 2001).

In the case of the radial velocity, a good accordance be-tween the simulations and the experimental data is displayedat the centre of the porous medium. Close to the tips of thebasket, the computed radial velocity curves decrease morerapidly than the experimental ones. On the contrary, the axialcomponent displays a discrepancy in the region z = 4.5 cmowing to the dissymmetry of the experimental curves and agood accordance at the tips of the basket. But at 1500 rpm,a discrepancy appears also in these regions. The numericalsimulations have shown that the flow field in the outer re-gion is sensitive to the blade orientation despite the porousmedium. The amplitude of variation of the radial compo-nent ranges from 13% to 46% at z = 4.5 and 2.25 cm andfrom 160% to 450% at z=0 cm in the range of the impellerrotation speed investigated in this work.

Fig. 6 displays the profiles of the radial component of thevelocity at � = −35◦ and +35◦ from an inner baffle. Theagitation speed is 1500 rpm. As in Fig. 5, the vertical barsrepresent the variation amplitude of the instantaneous veloc-ity. The profiles show that the amplitude of the maximumis slightly dependent on the azimuthal location. But, at thetips of the basket, the negative values of the radial velocityare greater than the ones near the inner baffle. This meansthat far from the inner baffle, the fluid enters the basket fromits outer boundary at its tips. The radial velocity is posi-tive only close to the inner baffle. Therefore the simulationsagree with the experimental observations but with a higheramplitude especially in the case of � = −35◦. Two reasonscan explain this observation:

(a) The numerical simulations locate the main vortex at2.5 cm above the bottom of the basket (and 2.5 cm below


the top of the basket in the symmetrical part of the reactor)in the inner zone. The axial location may be too high. Itdepends on the permeability of the packing (see Section 5.1for the discussion).

(b) The PIV measurements were performed in coinci-dence with an angular orientation of the blades. At 1500 rpm,the impeller performs two and a half rotations during 0.1 s.Therefore, if we neglect azimuthal orientation of the pitchedblades, the velocity measurements were carried out withthe same impeller position. This observation is also validfor the two other rotation speeds. The comparison with thesimulations at 1500 rpm gives an optimal agreement for the�(blade/baffle) value of 15◦. But this angle cannot be the sameat each impeller rotation speed. Unfortunately, this cannotbe checked because the experimental data at +35◦ and −35◦are not available at the other agitation speeds.

5. Description of the flow field

5.1. Autoclave engineers reactor

Inside the inner region two kinds of vortex characterizethe fluid flow: (a) a main ring-shaped vortex around the im-peller (Fig. 3b), with a radial location around 1.6–1.7 cmand a mean axial one around 2 cm (i.e., 1.35 cm above thebottom of the vertical blades), and (b) secondary vorticesfollowing the blade tips. No trailing vortices take place be-hind the blades as in the case of the Rushton impeller (Leeand Yianneskis, 1998).

The radial position of the main vortex is independent ofthe azimuthal angle �. In Fig. 4a (� = 1500 rpm, Utip =2.5 m/s), at z = 2 cm the sign of the radial velocity changesat r =1.6 cm. Above and below this axial position the curveshave opposite sign, confirming the main vortex position. Atthe inner boundary, the radial velocity decreases abruptly inthe axial region 2 cm < z < 2.5 cm. Above the main vortex,the radial velocity ranges from 0.08 Utip to 0.055 Utip. Be-low, the velocity is constant and equal to −0.02 Utip andremains negative inside the catalytic basket. This means thatthe fluid goes through the basket from the outer boundary.The vortex size is important enough to induce a rotationalmovement to the fluid inside the porous medium so that atthe outer boundary the radial component velocity is negativefor z < 2 cm.

The size and form of the vortex depend on its azimuthalposition towards an inner baffle. Fig. 7 displays the colourmap of the radial velocity at the outer boundary in thecase of the instantaneous blade/baffle orientation shown inFig. 3a. Upstream the inner baffle located at � = +10◦, thevortex size decreases as the azimuthal position is closer tothe baffle so that at the bottom of the basket the radial veloc-ity is less and less negative. Just upstream the inner baffle(+10◦ < � < + 20◦), the radial velocity is positive all alongthe axial position because the vortex is so small that allover the porous media thickness (2.13 cm < r < 2.88 cm) the

Fig. 7. Colour map of the radial component of the velocity normalizedby Utip at the outer interface (r = 2.88 cm) in the AE reactor in thecase of the blades/baffles configuration sketched in Fig. 3a. The rotationspeed is 1500 rpm. (1) Bottom of the interface; (2) top of the interface;(3) angular position of the blade at � = −30◦; (4) angular position ofthe blade at � = +30◦; (5) outer baffle at � = −35◦; (6) outer baffle at� = +55◦; (7) inner baffle at � = +10◦.

radial velocity remains positive. This is visualized in Fig. 8,which displays the velocity vector field in three horizontalplanes axially located on both sides of the main vortex. Onlythe vectors located at r > 1.7 cm are reported. In front of thebaffle (rung with dotted lines), the vortex disappears. In theregion z < 2 cm close to the baffle (Fig. 8a and b), the fluid,pushed azimuthally to the upstream face of the inner baffles,is deviated to the porous medium. Therefore the flow runsthrough the basket radially along these baffles. But the pres-sure downstream the inner baffles is so low that the fluid,going outside the basket, moves round these baffles and en-ters the basket through the outer boundary. The low pressureinduced by the baffles amplifies the fluid flux at the tips ofthe basket and increases the vortex intensity (see Fig. 7). Italso induces an important azimuthal flow close to the innerbaffles in the outer region.

The position of the vortex depends on the permeabilityof the packing as displayed in Fig. 9. Simulations with dif-ferent values of grain size reveal that the axial and the ra-dial locations increase with the packing permeability, i.e., itdisplaces away from the blade tips to the symmetry plane.Fig. 9a displays the velocity vector field in the case of agrain size dp equal to 8 times the size of the PMMA parti-cles. The position of the vortex is at the inner boundary andat 3.5 cm above the bottom of the basket. Therefore, the neg-ative magnitude of the radial velocity at the outer boundaryincreases with the permeability. If the grain size is 2 timeslower than the PMMA size (Fig. 9b), the vortex is displaceddown to the vertical blade bottom. The low axial position


Fig. 8. Vector velocity field on three horizontal planes (with r > 1.7 cm) inside the AE reactor with the blade orientation displayed in Fig. 3a. The rotationspeed is 1500 rpm. The vector length is set constant. Every other vector is skipped. (a) z = 1 cm; (b) z = 2 cm; (c) z = 4.5 cm.

of the vortex induces a positive radial velocity at the tips ofthe basket. But the vortex interacts with the flow induced bythe pitched blades so that its form and its position dependon the azimuthal position.

In order to decrease the flow from the outer boundary atthe tips of the basket, we have to slide the vortex down, i.e.,we must decrease the permeability without decreasing theflux through the basket. Two trials were carried out. First,the tips of the basket (0 cm < z < 1.5 cm) were assumedless permeable (�′ = 2�, �′ = 2�). The numerical resultsshowed that the velocity field remains identical. Second, theporous medium filled with cylindrical particles should beanisotropic especially close to the tips of the basket. Simu-lations were performed assuming that the second-order ten-sors �ij and �ij are diagonal in the cylindrical reference.Using �rr = �, �rr = �, �� = �zz = 2�, �� = �zz = 2� inthe entire porous medium, the computed velocity profile re-

mained also unchanged all along the basket. Therefore thenumerical simulations are sensitive to �rr and �rr only. Butchanging their value also changes the flux through the porousmedium.

Secondary vertical vortices are located in front of theblades close to the inner boundary (rung with full lines inFig. 8). But their axial location depends on the blade az-imuthal position towards an inner baffle. These vortices areproduced at the downstream face of the inner baffles at thehorizontal plane of symmetry (Fig. 8c). When a blade passesin front of one of these baffles, it catches the vortex fromthis baffle. During the blade displacement, the vortex fol-lows the blade tip and moves down in the axial directionuntil it reaches the blade bottom. When the blade reachesthe following inner baffle, the vortex disappears. Owing tothe small distance between the blade tips and the basket, thevortex induces an azimuthal flow inside the porous medium.


Fig. 9. Velocity field at 1500 rpm for two particle diameters. The vectorlength is set constant. Every other vector is skipped. (a) dp = 2.2 cm; (b)dp = 0.138 cm; (a) axial location of the main vortex centre in the caseof PMMA particles; (b) symmetry plane; (1), (1′), (2), (3), (4), (5): seeFig. 3.

The production of the turbulent kinetic energy and theturbulent dissipation is induced by the local shear stress.Fig. 10 shows the radial variation of the normalized tur-bulent kinetic at several axial positions at 1500 rpm in thesame vertical plane as in Fig. 3 and with the same bladesand baffles locations. The curves display two sources of tur-bulent kinetic energy: the main vortex centre and the innerboundary. In the range of agitation speed studied, the max-imum value of k′ is around 0.2 and around 0.1 at the innerboundary and in the main vortex, respectively. At the innerboundary, the maximum is located at z ∼ 2–2.5 cm whichcorresponds to the axial position of the main vortex. At thisposition the velocity field is azimuthal and axial inside theinner zone and becomes suddenly radial and axial inside theporous medium. This sudden change of direction creates alarge velocity gradient, which induces a transfer of the ki-netic energy from the mean flow to the turbulent one. Onboth sides of this axial location, the normalized kinetic en-ergy decreases because the flow field is more and more ra-dial in the inner region as the axial position deviates fromthe maximum location. In the catalytic basket, the turbu-lent kinetic energy vanishes because the flow is assumed tobe laminar. In the outer zone, k′ remains null meaning thatthe porous medium makes the flow laminar in this zone,i.e., all the turbulent energy is dissipated inside the turbulentzone.

5.2. Parr reactor

The main vortex is located at z=1.3 cm (i.e., 0.8 cm abovethe blade bottom) and at r = 1.4 cm. In Fig. 11 the radialprofiles of the radial component velocity are displayed atseveral axial positions on both sides of the vortex location.In the domain of computation, the pitched blade is orientedat � = +30◦, the baffles are located at 0◦ and the verti-cal plane is midway between two blades, i.e., at � = −30◦(Fig. 11a). The curves at z = 2.6 and 0.45 cm representthe profile at the symmetry plane and at the blade bot-tom, respectively. The impeller rotation speed is 1500 rpm(Utip =1.88 m/s). Fig. 11b displays the velocity vector fieldin the plane (2a) and shows the axial position of the curvesin Fig. 11c with respect to the main vortex. Fig. 11c dis-plays the vortex position by the change of the sign of theradial velocity at z = 1.5 cm and r = 0.15 cm and by theopposite sign of the profiles at z = 2 and 1 cm, i.e., onboth sides of the vortex centre. The curves (a) and (b) showthat below the main vortex (0 cm < z < 1 cm), the fluid ispumped through the porous medium to the impeller. Con-trary to the AE reactor, above the main vortex (curves (d)and (e)), the radial velocity increases rapidly with the ax-ial position and reaches a maximum in the axial directionat the symmetry plane. This maximum at the radial position(1) (Ur(r = 1.2 cm)= 0.43 Utip) is much greater than in thecase of the AE reactor. The computation confirms that thefluid is radially ejected along the symmetry plane owing to


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04r (m)

k'

(1) (4)(3)(2)

(e)

(d)

(c)

(b) (a)

Fig. 10. Radial profiles of normalized turbulent kinetic energy at several axial positions in the plane (−60◦), midway between two vertical blades insidethe AE reactor (see Fig. 3a). The rotation speed is 1500 rpm. (1), (2), (3), (4), (a), (b), (c), (d): see Fig. 3.

Fig. 11. Flow pattern in the plane (� = −30◦), midway between two pitched blades inside the Parr reactor. The rotation speed is 1500 rpm. (a) Sketch ofthe blades/baffles orientation. (1a) Planes of periodicity, (2a) vertical plane location (� = −30◦). (b) Vector velocity field in the plane (2a). The vectorlength is set constant. Every other vector is skipped. (1) Radial location of the blade tips, (1′) axial location of the blade bottom, (2) inner boundary,(3) outer boundary, (4) reactor wall, (5) ring closing the basket tips, (a) z = 0.5 cm; (b) z = 1 cm; (c) z = 1.5 cm; (d) z = 2 cm; (e) z = 2.5 cm. (c) Radialprofile of the normalized velocity at different axial locations displayed in (b).


Fig. 12. Vector velocity field on three horizontal planes (with r > 1.6 cm) inside the Parr reactor with the blade orientation displayed in Fig. 11a. Therotation speed is 1500 rpm. (a) z = 0.2 cm; (b) z = 0.5 cm; (c) z = 1.5 cm.

the design of the blades and proves the efficiency of the im-peller to introduce gas bubbles in a stirred reactor. The sud-den decrease is also observed along the inner boundary. Atthe symmetry plane the radial velocity is equal to 0.3 Utipand decreases to 0.05 Utip at z= 2 cm. The velocity profilesinduced by the two impellers are very different at the innerboundary. But we will see that the liquid–solid mass transferis the same in the two reactors.

Fig. 12 displays the velocity vector field in three planesbelow the main vortex and for a radial distance higher than1.5 cm. In the inner zone, the flow field is mostly azimuthal.But close to the bottom of the basket the flow is radialand azimuthal (Fig. 12a). As we move closer to the mainvortex centre along the axial direction, the flow becomesazimuthal and axial. Inside the porous medium, the flow ismostly radial and axial. Below the main vortex, two vorticestake place on both sides of each inner baffle in the region0.5 cm < z < 1.25 cm (Fig. 12b). They induce an azimuthalflow over the width of the porous medium. But the flow

remains mostly radial at the inner boundary. Above the mainvortex (z > 1.25 cm), the change of the flow direction occursinside the porous medium over a depth much smaller thanthe grain size (Fig. 12c). Around the plane of symmetry, theimpeller imposes a radial flow in the inner region and in theporous medium (see Fig. 11c).

The important change of flow direction at the inner bound-ary induces an intense shear stress and therefore an impor-tant turbulent production as shown in Fig. 13. In this figurethe vertical plane is the same one as in Fig. 11a. All the tur-bulence is dissipated inside the inner region. But contraryto the AE reactor, there is no other maximum of turbulentenergy at the main vortex location. In the range of agitationspeed investigated, the power number Np is computed as-suming that the dissipation takes place mainly in the innerzone. Np is defined as

Np = N

∮Vrz

�� dV

�(2�/60)3d5imp

,


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.004 0.008 0.012 0.016 0.02 0.024 0.028 0.032r (m)

k'

(1) (4)(3)(2)

(e) (d)

(c)

(b)

(a)

Fig. 13. Radial profiles of normalized turbulent kinetic energy at several axial positions in the plane (−30◦), midway between two vertical blades insidethe Parr reactor. The rotation speed is 1500 rpm. (1), (2), (3), (4), (a), (b), (c), (d): see Fig. 11.

where N is the ratio of the reactor volume to the computa-tional domain one and Vrz is the volume of the inner zone.N = 4 for the AE reactor and N = 6 for the Parr reactor.For the two reactors Np is found to be constant and equal to9.0 ± 5% for the three agitation speeds. Observing the highintensity of the turbulent kinetic energy and its high gradi-ent close to the inner boundary, a refinement of the meshingis carried out as mentioned in Section 4.1. The k–� RNGmodel is also used at 1500 rpm. This turbulent model com-putes the same flow field as the k–� model. On the contrary,the magnitude of the turbulent quantities k and �, averagedover the inner zone, decreases by a factor of 2.2 and 3.2,respectively. But these quantities cannot be measured insidethe inner zone. For a more detailed analysis, a robust modelof turbulence and the sliding mesh approach must be used.

6. Liquid–solid mass transfer properties

6.1. Profiles computed by CFD and comparison withexperimental data

The local measurement of ks allows one to estimate avelocity profile inside the basket and to compare it to theprofile computed by CFD. However, the velocity deducedfrom the Sherwood correlation is very sensitive to the Sher-wood number (U ∼ Sh2) and to experimental uncertainties.Therefore, it is relevant to compare the measured and thecomputed ks profiles.

In order to compute the dissolution time in each slice andto compare this time to the experimental data, the dissolvedspecies transport over the entire reactors must be simulatedwith the liquid–solid mass transfer. But the MRF approachdoes not conserve the mass of dissolved species and the localvelocity used by the software inside the MFR region (1) is therelative velocity and not the absolute one. Therefore the ks

Table 3Physical properties of water of n-Heptane at standard condition

(Pa s) � (kg/m3) D (m2/s)

Water 0.001 1000 7.1 × 10−10

n-Heptane 0.0004 684 2.56 × 10−9

profiles are computed by means of the local velocity moduleinside the porous medium and the Sherwood correlation.Then the ks profiles are compared to the experimental ks

estimated from the time to reach 95% of the equilibriumconcentration (Fongarland, 2003) or from the time evolutionof the dissolved naphthol concentration (Mitrovic, 2001).In the numerical simulations, the permeability differencebetween the slice of naphthol and the remaining packingfilled with PMMA is not taken into account.

The liquid–solid mass transfer coefficient is determinedby means of the Sherwood number, which has the generalexpression

Sh = ksdks

D= a + bRec

ksScd , (5)

where Sc is the Schmidt number, Reks is the Reynolds num-ber defined with dks , the diameter of a sphere which has thesame surface as a naphthol cylinder. The values of the fourparameters a, b, c and d determined by Mitrovic are 0, 1.1,0.48 and 0.33, respectively, for Reynolds numbers Reks rang-ing from 2 to 55. These values were computed with diffusioncoefficient values of 1.17 × 10−9 and 3.44 × 10−10 m2 s−1

in water and n-Heptane, respectively. But these are not thevalues computed by means of the Wilke and Chang correla-tion under standard conditions of temperature as mentionedin the dissertation (see Table 3). Therefore another expres-sion of the Sherwood number was found (Fig. 14):

Sh = 1.1Re0.41ks Sc0.4. (6)


0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

n-heptane

water

U (ms-1)

Ks (

ms-1

)

Mitrovic's correlation (2001)Present work

Fig. 14. Variation of the mass transfer coefficient with the superficial velocity. Comparison of Mitrovic’s correlation and of the corrected one againstexperimental data (Mitrovic, 2001).

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

n-heptane

water

ks

(ms-1

)

Z (m)

1500 rpm 1000 rpm

500 rpm

1500 rpm

1000 rpm

500 rpm

0.0E+00

2.5E-05

5.0E-05

7.5E-05

1.0E-04

1.3E-04

1.5E-04

0 0.01 0.02 0.03 0.04 0.05

k s(m

s-1)

Z (m)

1500 rpm 1000 rpm

500 rpm

(a)

(b)

Fig. 15. Axial profiles of the liquid–solid mass transfer coefficient ks of naphthol for three rotation speeds: (a) in n-Heptane and water in the AE reactor;(b) in n-Heptane in the Parr reactor. Symbols: experiments; lines: simulations.

The ks profiles inside the catalytic basket in the AE re-actor are displayed in Fig. 15a. for water and n-Heptane.The experimental accuracy of ks was estimated at ±12%.The simulations overestimate the mass transfer coefficientin water but a good agreement can be observed in the caseof n-Heptane. Mitrovic compared the velocity measured byPIV in the vicinity of the symmetry plane to the interstitialvelocity deduced from the Sherwood correlation and theporosity. The interstitial velocity was used because the ve-locity oscillations along the basket were assumed to be gen-

erated by flow jets, i.e., the PIV velocity is the velocity in-side the pores. So only the maxima have been taken intoaccount. But Mitrovic did not succeed in matching the PIVvelocity to the velocity deduced from ks . The PIV veloc-ity was always greater by a factor ranging from 1.7 to 3.2,depending on the agitation speed. In fact the velocity mea-sured outside the catalytic basket is not representative of theinside velocity owing to the high heterogeneity of the flowfield in the basket and especially at the inner boundary asexplained below in Section 6.2.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Blade axial position

inner boundary

outer boundaryS(r = 0.02733 m)

porous medium

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

U (

ms-1

)U

(m

s-1)

Z (m)

Z (m)

inner boundary

outer boundaryS(r = 0.0335 m)

porous medium

Blade axial position

(a)

(b)

Fig. 16. Axial profiles of the velocity module in the Parr reactor. Rotation speed: 1000 rpm. (a) AE reactor; (b) Parr reactor.

In the simulations the velocity outside the basket has tobe compared to the superficial one inside the basket becausethe packing is an effective medium. Using relation (6) withthe global experimental values of ks , the mean superficialvelocity inside the basket is 0.0115, 0.0145 and 0.05 m/s foragitation speeds of 500, 1000 and 1500 rpm, respectively.At 1000 rpm, the simulations give a mean value of 0.07 m/s.Such a difference (a factor of 5) should induce a large dis-agreement of the numerical results with the PIV data.

Therefore the ks measured in water is far too small thanwhat is expected by the correlation used in the simula-tions. Several reasons can be mentioned to explain this dis-crepancy: (a) The correlation was determined in a range ofReynolds number smaller than the range investigated in theAE reactor. The correlation must be extrapolated in orderto be used in the SCBSTR. (b) Mitrovic observed a drift ofconcentration during the experiments of dissolution in waterin the AE reactor. This drift came from the increase of thesolvent temperature. Each of these experiments took severalhours and a mean value of the equilibrium concentrationwas used to estimate the dissolution rate. No drift was ob-served in the case of dissolution in n-Heptane, and the sim-ulations predict a mass transfer coefficient in n-Heptane inaccordance with the experiments.

It must be mentioned that the computed velocity in n-Heptane is only 5% upper than in water, even if n-Heptane

has a dynamic viscosity 41% greater than water. Hydrody-namics depends on the physical properties of the fluid ifthe Reynolds number is small. Therefore in the outer re-gion and in the porous medium, the flow should depend onthe solvent. But in the first region where the flow is weaklyturbulent, the k–� model gives a turbulent viscosity value10 times the laminar one even at 500 rpm. In the porousmedium, the physical properties of the fluid are introducedby Darcy’s term � and by the local pressure gradient term.The inertial term has a greater contribution than Darcy’s oneif the velocity module is greater than 0.016 m/s in water and0.01 m/s in n-Heptane. Therefore Darcy’s term should haveeffects only at the tips of the basket, as we will see belowand in Fig. 16. It was also observed that the global pressuredrop through the basket divided by the density depends onthe impeller rotation speed only. Therefore the computationis not globally sensitive to the nature of the solvent.

Fig. 15b displays the ks profiles in the Parr reactor. Eachdot represents a value averaged over three experimentsand the standard deviation, displayed by vertical bars at1000 rpm, is globally less than 25% of the mean value. Animportant disequilibrium between the upper and the lowerpart of the basket characterizes the experimental profilesfor the three impeller rotation speeds, so that half of theexperimental data at 1500 rpm have a smaller value than at1000 rpm. This disequilibrium cannot be simulated because


Table 4Computed and measured mass transfer coefficient ks

Watera Waterc n-Heptanea n-Heptanec n-Heptaneb n-Heptaned

500 rpm 1.15 × 10−5 2.96 × 10−5 5.73 × 10−5 6.3 × 10−5 6.55 × 10−5 6.52 × 10−5

1000 rpm 1.84 × 10−5 3.85 × 10−5 8.3 × 10−5 8.5 × 10−5 9.28 × 10−5 8.85 × 10−5

1500 rpm 2.53 × 10−5 4.55 × 10−5 10 × 10−5 9.98 × 10−5 9.37 × 10−5 10.3 × 10−5

aMitrovic (2001).bFongarland (2003).cThis work in the AE reactor.dThis work in the Parr reactor.

the computational domain is assumed symmetric. If wemake the experimental profiles symmetric by averaging theks values of two slices at equal distance from the symmetryplane, the computed profiles fit much better the modifiedexperimental ones. These two observations explain the dis-crepancy between the predictions of the simulations and theexperimental data. Table 4 reports the values of ks averagedover the basket (predicted and measured). The simulationsagree with the experiments in the case of dissolution inn-Heptane. In the two reactors, the predicted ks in the twosolvents varies as �0.41. This means that the mean velocityinside the basket is linearly dependent on �. This is alsoobserved experimentally in the two reactors if the solvent isn-Heptane. But in the AE reactor, the ks measured in watervaries as �0.48, and is systematically less than the simu-lated one. Despite this discrepancy, the simulations confirmthe experimental observations which show that the twoSCBSTRs have the same global solid–liquid mass transferproperties.

The question is now: why the mass transfer properties ofthe two reactors are similar even though the flow pattern iscompletely different?

6.2. Comparison of the velocity profiles inside and outsidethe basket

Fig. 16 displays the axial profiles of the velocity modulein the porous medium of the two reactors. The velocity isaveraged over 5 mm height slices and all over the azimuthalangle �. In Fig. 16a, the curve ‘S(r=0.0335 m)’ is the profileat 4 mm from the outer boundary. In Fig. 16b, the curve‘S(r = 0.0273 m)’ is the profile at 2.33 mm from the outerboundary, i.e., at 1/3rd the distance between the basket andthe tank wall as in the AE reactor. The curves named ‘porousmedium’ represent the velocity profiles averaged also overthe basket width. The agitation speed is 1000 rpm. At theinner boundary, the velocity values are multiplied by 1/4thowing to their large magnitude.

At this boundary of the AE reactor, the velocity remainsconstant along most of the blade height and has a valueequal to 0.64 m/s(=0.4 Utip). On the contrary, in the Parrreactor the velocity profile has a maximum at the symme-try plane equal to 0.32 m/s (=0.25 Utip). This maximum

means that the fluid is ejected from the pitched bladescentre in the radial direction (see Fig. 11c). But averagedover the inner boundary, the velocity represents 15% and17% of Utip for the Parr and the AE reactor, respectively(i.e., 0.192 and 0.27 m/s). As explained in Section 5, fromthe inner boundary to the outer one, the azimuthal flow istransformed into a radial one. The azimuthal velocity andthe velocity module drop abruptly with r over a distancemuch smaller than the grain size. But the radial velocity de-creases slowly as 1/r owing to the conservation of the fluidflux through the porous medium. At the outer boundary, theParr reactor has a constant velocity of 0.073 m/s (0.06 Utip).In the case of the AE reactor, the profile has a max-imum located at the symmetry plane with a value of0.09 m/s(0.05 Utip) and the velocity, averaged over the outerboundary, is equal to 0.06 m/s (0.037 Utip). The permeabil-ity of the AE reactor porous medium is smaller than theParr reactor one owing to the smaller value of the porosity(see Tables 1 and 2). The consequence is a higher decreaseof the velocity from the inner to the outer boundary in thecase of the AE reactor despite the smaller width of the cat-alytic basket. The other consequence is the smaller valueof the velocity averaged over the porous medium width inthe AE reactor (curves ‘porous medium’). It can also benoticed that the mean velocity over the basket is equal to0.07 and 0.09 m/s in the AE reactor and in the Parr reac-tor, respectively. The Sherwood correlation and the masstransfer coefficient ks are weakly sensitive to the velocity(Sh ∼ U0.41). So the difference of the Sherwood coeffi-cient between the two reactors is of the order of magnitudeof the experimental accuracy, i.e., 12%. Outside, the tworeactors have the same profile: a minimum at the symme-try plane where the velocity is radial and a maximum atthe tips of the basket where the flow is axial. These pro-files do not look like the profiles of the velocity averagedover the basket width even at the plane of symmetry. Inthe case of the AE reactor the gap between the velocityoutside the basket and the velocity average over the bas-ket width is 43% near the symmetry plane whatever theagitation speed. The gap is around 50% in the case of theParr reactor.

Therefore the velocity outside the basket is not represen-tative of the inside velocity owing to the great change ofthe flow direction at the inner boundary even close to the


symmetry plane. The permeability is much smaller in the AEreactor. Even if the velocity at the inner boundary is greaterin this reactor than in the Parr reactor, it becomes smallerwhen it is averaged over the basket width. But the velocitydifference between the two SCBSTRs cannot be measuredowing to the lack of accuracy of the experimental method.Moreover, the abrupt drop of the velocity through the bas-ket width involves a spatial resolution of the order of thegrain size which cannot be reached by the liquid–solid masstransfer experiments. Therefore the mass transfer approachis not adapted to local velocity investigations.

Finally, in both reactors, the average of the flow veloc-ity profile inside the basket gives close mean velocity andconsequently the liquid–solid mass transfer coefficients ex-perimentally measured are quite the same.

7. Conclusion

The flow field is studied in two fixed basket reactors bymeans of the k–� model and the Brinkman–Forsheimer equa-tions, assuming that the packing inside the basket is a ho-mogeneous porous medium. In the two sets of equations,standard values of the parameters are used. The geometry ofthe simulated reactors is simplified: no gas/liquid interfaceat the top of the Parr reactor, flat bottom of the AE tank, andassumption of a horizontal plane of symmetry midway fromthe top and the bottom of the tank. Despite the use of theMRF approach and the azimuthal periodicity assumption ofthe flow pattern, the numerical results in the AE reactor arevalidated by PIV measurements carried out along the cat-alytic basket for three impeller rotation speeds (500, 1000and 1500 rpm). Extending the hydrodynamic study to theliquid–solid mass transfer characterization inside the basket,the validation of the numerical approach is confirmed onemore time by the agreement with the measurements of thelocal ks in the two SCBSTRs.

The simulations show that the flow structure has similarcharacteristics in the two reactors. The structure does notdepend on the impeller rotation speed but its intensity islinearly dependent on �. A ring-shaped vortex around theimpeller is located in the inner zone. Its position depends onthe permeability of the packing inside the basket. Using theexperimental conditions, the simulations localize the vortexat 2 cm above the bottom of the basket in the AE reactor andat 1.5 cm in the Parr reactor. The axial position of the vortexinduces a flow from the outer boundary to the inner one atthe tips of the basket. This was also observed by the PIVmeasurements but with a lower amplitude. This differencemay be due to the high axial position of the main vortexcomputed by CFD. Owing to the friction of the fluid withthe packing, the porous medium induces a great change offlow between the inner zone and the outer one. In the innerzone the flow is mostly azimuthal and the turbulent energyis dissipated at the inner boundary. In the outer zone, thelaminar flow is radial and axial. The porous medium, in

which the flow is also laminar, represents a transition region.Close to the inner boundary the azimuthal component of thevelocity decreases abruptly over a distance much smallerthan the particle size and becomes negligible at the outerboundary.

But the design of the blades and of the inner bafflesinduces differences in the inner zone and in the porousmedium. In the AE reactor, vertical vortices are located atthe blade tips and at the downstream face of the baffles. Thevertical vortices induce locally an azimuthal flow over thepacking width. At the inner boundary, the vertical blade im-peller induces a rather homogeneous velocity profile. At theouter boundary, the velocity profile displays a maximum atthe plane of symmetry, i.e., the flow is much smaller at thebasket tips. In the Parr reactor, the vertical vortices are lo-cated in the basket on both sides of the baffles. The pitchedblade impeller imposes a high velocity at the plane of sym-metry. But compared to the AE reactor the flow seems to bemore homogeneous at the outer boundary and the reverseflow is located in a thinner region at the basket tips.

In order to compare the mass transfer efficiency of thetwo reactors, the liquid–solid mass transfer was also inves-tigated by Mitrovic. Coupling the ks measurements with aSherwood correlation, the velocity field inside the basketwas also studied. But the experimental results display simi-lar profiles despite the difference of the two reactor designs.In fact the geometry acts on the flow field but also on thecompacity of the packing. At the inner boundary, the veloc-ity in the AE reactor is greater than in the Parr reactor. Butowing to the smaller permeability of the AE reactor porousmedium, the velocity decreases more rapidly from the innerto the outer boundary. The simulations show that the profileof the velocity averaged over the basket width is identical inthe two reactors, demonstrating that two different local ve-locity patterns with two different porous media may lead tothe same liquid–solid mass transfer properties. The simula-tions also show that the high decrease of the velocity field atthe inlet boundary imposes a spatial resolution smaller thanthe particle size and that the ks measurement is inappropri-ate to velocity investigations.

The geometrical complexity of the SCBSTR induces spe-cific problem for the numerical model: low Reynolds tur-bulence in the outer zone, high hydrodynamic interactionbetween the inner baffles and the blades, and high velocitygradient in the porous zone especially at the inner bound-ary. Despite several numerical approximations, the simula-tions describe the essential part of the hydrodynamics andallow one to consider an optimization of the SCBSTR ge-ometry in order to increase the mass transfer coefficient andto decrease its radial and axial distribution. In order to godeeper into hydrodynamic study, it will also be necessary touse the sliding mesh approach, a more realistic turbulencemodel and to suggest a better boundary condition at the in-ner and the outer boundary. This will allow one to study thedynamics of the hydrodynamics and of the chemical speciesmixing.


Notation

dcyl cylindrical diameter, mdimp impeller diameter, mdks diameter of a sphere which has the same

surface as a cylinder, mdp diameter of a sphere which has the same

specific surface as a cylinder, mD naphthol diffusion coefficient, m2/sk turbulent kinetic energy, m2/s2

ks liquid–solid mass transfer coefficient, m/sk′ = k/U2

tip normalized turbulent kinetic energyK permeability, m2

Lcyl cylindrical particle length, mP pressure, Pa

Rep = Udp

� Reynolds number

Retip = Utipdimp� Reynolds number

Sc = �D

Schmidt numberSh = ksdks

DSherwood number

Utip tip velocity, m/sU ′ = U/Utip normalized velocity

Greek letters

� Darcy permeability, 1/m2

� inertial term of the Ergun correlation, 1/m� turbulence dissipation rate, 1/m2s3

dynamic viscosity, Pa s� kinematic viscosity, m2/s� fluid density, kg/m3

� porosity� impeller rotation speed, rpm

Subscripts

r radial coordinatet azimuthal coordinatez axial coordinate

References

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Sharp, K.V., Adrian, R.J., 2001. PIV study of small-scale flow structurearound a Rushton turbine. A.I.Ch.E. Journal 47, 766–778.

Wilke, C.R., Chang, P., 1955. Correlation of diffusion coefficients in dilutesolutions. A.I.Ch.E. Journal V1 (N2), 262.

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