Chemical Engineering Science 63 (2008) 2436–2449www.elsevier.com/locate/ces

Flow field and residence time distribution simulation of a cross-flowgas–liquid wastewater treatment reactor using CFD

Yann Le Moullec, Olivier Potier, Caroline Gentric∗, Jean Pierre LeclercLaboratoire des Sciences du Génie Chimique, UPR 6811, CNRS-ENSIC-INPL, 1, rue Grandville BP 20451, 54001 Nancy, France

Received 16 January 2007; received in revised form 22 January 2008; accepted 24 January 2008Available online 6 February 2008

Abstract

A three-dimensional Eulerian–Eulerian two-phase approach has been used for the simulation of a cross-flow gas–liquid wastewater treatmentreactor. Two different turbulence models have been tested: the k.� and Reynolds Stress Model (RSM) models. Bubble induced turbulencesource terms have been added to these models. Numerical results have been validated using Laser Doppler Velocimetry (LDV) measurements.Simulations with both turbulence models successfully predicted the hydrodynamics of the reactor. Then particle tracking with a stochasticapproach has been used to calculate residence time distributions (RTD) with the flow previously simulated. It has been shown that dispersionin the reactor is primarily due to turbulence. Results have been compared with experimental RTD for various liquid and gas flowrates both ona bench scale and full scale plant. The RSM model accurately predicted the dispersion whereas the standard k.� model slightly underestimatedthe dispersion.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Dispersion; Multiphase flow; RTD; Simulation; Turbulence; Wastewater treatment

1. Introduction

The pollution removal by microorganisms is an essentialstep in the biodegradable wastewater treatment plants. Thesebiological reactions often take place in gas/liquid reactorswith a very long length compared to their height and width(channel reactors). In this type of reactors, the global water flowis along the length of the reactor and gas is sparged at the bot-tom. In wastewater treatment reactors, apparent reaction ordersare often greater than zero; therefore the efficiency of the pol-lution removal reaction depends on the hydrodynamics (Levinand Gealt, 1993). Therefore, in order to accurately predictconversions, the kinetics modelling must be coupled with thehydrodynamics description. But the prediction of bubbly flowhydrodynamics is quite difficult due to the complex phenomenainvolved.

Nevertheless, with the improvement of computationalpower and the availability of computational fluid dynamics(CFD) codes adapted to such two-phase flows, gas–liquid

∗ Corresponding author. Tel.: +33 3 83 17 53 38; fax: +33 3 83 32 29 75.E-mail address: Caroline.Gentric@ensic.inpl-nancy.fr (C. Gentric).

0009-2509/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2008.01.029

reactors have been studied more extensively. In particularbubble column reactors have been extensively investigatedeither with the Euler–Euler approach (Jakobsen et al., 2005;Pfleger and Becker, 2001; Sokolichin et al., 2004) or withthe Euler–Lagrange approach (Lain et al., 2002; Lapin andLübbert, 1994). Two- or three-dimensional (2D and 3D)and stationary or transient simulations have been performed.Although transient 3D simulations are necessary to representthe complex flow structure of bubble columns, 2D and station-ary simulations allow correct representations of the averageflow field and can be sufficient in certain cases (Sokolichinet al., 2004). But simulation of bubbly flows using CFD is notstraightforward. Closure terms, in particular interaction forcesbetween phases and their impact on the simulation resultshave been the subject of numerous studies (Jakobsen et al.,1997). Another critical point widely discussed, is the turbu-lence closure terms. Mainly three modifications of single phaseturbulent viscosity models have been proposed and comparedin the literature: the addition of a bubble induced turbulenceviscosity term (Sato and Sekoguchi, 1975), a pseudo-bubbleinduced turbulence tensor (Arnold, 1988) and the addition ofbubble induced turbulence source terms in the k.� transport

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2437

equations (Gu and Guo, 2005; Mudde and Simonin, 1999;Olmos et al., 2003; Pfleger and Becker, 2001). It seems that theaddition of bubble induced turbulence source terms is the bestcompromise between precision and complexity (Sokolichinet al., 2004). Recently, efforts have been made to simulate localand global mass transfer in bubble columns, air-lift and tricklebed reactors (Ekambara et al., 2005; Gunjal et al., 2003; Vialet al., 2005). CFD simulations of liquid phase residence timedistribution (RTD) in these gas–liquid reactors have also beenperformed with good results (Ekambara et al., 2005; Gunjalet al., 2003). With these improvements in gas–liquid simula-tions, some other industrial reactors can be investigated andspecifically wastewater treatment reactors. The previously pre-sented studies are limited to vertical liquid flow with counter orco-current gas flow. Wastewater treatment reactors by activatedsludge are mostly cross-flow reactors in which liquid flow ishorizontal and gas flow is vertical. Two types of reactors areused: closed-loop oxidation ditch reactors, also called carrouselreactors, and channel aerators. In carrousel reactors, the liquidhorizontal velocity is such that gas has no real impact on thehydrodynamics and mixing is sometimes performed with im-pellers (Roustan and Line, 1996). CFD modelling of the veloc-ity field and mass transfer has been carried out for these reactors(Cockx et al., 2001). In channel aerators, the liquid horizontalvelocity is very low compared to carousel reactors and mixingis performed by gas injection in the reactor. Therefore an accu-rate simulation of the interaction between bubbles and liquid isof primary importance. The purpose of this work is to obtain anaccurate and predictive hydrodynamics model of a channel aer-ator with respect to velocity, turbulence and RTD which is, asin most gas–liquid flows, governed by local scale phenomena(Vial et al., 2005). Validation is performed with Laser DopplerVelocimetry (LDV) measurements and RTD experiments.

2. Experimental setup and preliminary results

2.1. Bench scale reactor

Experiments have been carried out in a bench scale channelreactor filled with tap water. It is built in transparent material

Fig. 1. Schematic representation of the bench scale reactor.

(Plexiglas) to allow LDV measurements. Its total length is 3.6 mwith a rectangular section of width and height, respectively,equal to 0.18 and 0.2 m (Fig. 1). One side of the walls of thereactor is fitted with stainless-steel tubes where 1 mm holeshave been drilled every centimetre for air sparging. Gas andliquid flowrates are the variable parameters.

2.2. LDV measurements

2D LDV measurements have been performed to obtain ax-ial (Ux), lateral (Uy) and vertical (Uz) time-averaged velocityfields (see Fig. 1 for directions) as well as an estimated valueof the turbulent kinetic energy (k). An acquisition time of 180 sis required in order to obtain statistically meaningful results forall measurement positions.

Photographic technique has shown that the gas hold-up alongthe reactor length is inhomogeneous though the variations aresmall. In order to verify that these small variations of gas hold-up are of no importance for the LDV measurements, thesemeasurements have been carried out at two different planeswith different apparent gas hold-ups (planes 1 and 2: one islocated between two sparger tubes, the other is located at themiddle of a sparging device, see Fig. 1).

2.3. RTD experiments

Tracer experiments used for simulation validation have beenperformed in a previous study in the same bench scale reactor(Potier et al., 2005). Approximately 80 RTD experiments havebeen carried out by injecting pulses of a solution of sodiumchloride (NaCl) at the inlet and monitoring its concentrationwith a conductimetric probe at the outlet. A plug flow withaxial dispersion model (Villermaux, 1993) allows a very pre-cise representation of the data. Mean residence times and axialdispersion coefficients have been determined by model fittingwith the software DTSPRO 4.2 (PROGEPI, France). This pre-vious study has shown the existence of a correlation betweenthe axial dispersion coefficient on the one hand and the gasflowrate and the geometry of the reactor on the second hand:

D = (0.2032.w − 0.008569)

√Qg

L(100h)0.00476.w−1.99

(1)

2438 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

with h, w and L, respectively the height, width and lengthof the reactor and Qg the gas flowrate. This correlation hasbeen established for Qg (15.65 L min−1), Ql (1.5.5 L min−1),h (0.08.0.2 m) and w (0.05.0.2 m). Besides, this correlationhas been validated on an industrial reactor successfully. In thepresent work, results from simulation and experiments havebeen compared for different gas and liquid flowrates for a givengeometry: length of 3.6 m, height of 0.2 m and width of 0.18 m.

3. Hydrodynamics modelling

Because of the high number of bubbles (approximately 7000and more in an industrial size reactor) in the reactor and despitethe low gas fraction (< 10%), an Euler–Euler approach hasbeen used. CFD simulations have been carried out with theCFD software FLUENT.

3.1. Euler–Euler equations

The mass transfer between phases has been neglected there-fore the continuity equation is expressed by

�

�t(�q�q) + ∇ · (�q�qUq) = 0 with q = g or l (2)

besides:

�g + �l = 1. (3)

The momentum conservation equation for multiphase flows isgiven by

�

�t(�q�qUq) + ∇ · (�q�qUqUq)

= −�q∇p + �q�qg + ∇ · �q(�q + �t,q ) + Iq

with q = g or l, (4)

where Iq is the interphase momentum exchange term and �q and�t,q are, respectively, the viscous stress tensor and the turbulentstress tensor, defined as follows:

�q = �q(∇Uq + ∇Utq ) + (�q − 2

3�q)∇.UqI (5)

and

�t,q = �t,q (∇Uq + ∇Utq ) − 2

3 (kq + �t,q∇.Uq)I (6)

with

�t,q = �qC�k2q

�q. (7)

In most gas–liquid flows virtual mass force is necessary toreproduce transient flow characteristics (Leon-Becerril et al.,2002), nevertheless this force has no influence on stationarycalculation results (Sokolichin et al., 2004; Zhao et al., 2006)therefore it has been neglected and only the drag force has beenconsidered significant. Iq is thus defined as follows:

Il = 3

4Cd

�g�l

dg

|Ug − Ul |(Ug − Ul ) and Ig = −Il . (8)

Table 1Constants used in the two-phase k.� model

C� C�1 C�2 Ck C� �k ��

0.09 1.44 1.92 1.44 1.0 1.0 1.3

The drag coefficient is calculated from the following equation(Jamialahmadi et al., 1994):

Cd = 4

3

(�l − �g)

�l

gdg

u2∞with

u∞ =√

2�l

dg(�l + �g)+ gdg

2for dg > 3 mm. (9)

3.2. Two-phase k.� turbulence model

In order to model turbulence, a two-phase k.� turbulencemodel can be used (Elghobashi and Abou-Arab, 1982; Muddeand Simonin, 1999). Only the continuous phase turbulence istaken into account and is considered isotropic. The dispersedphase influences the continuous phase through bubble inducedturbulence terms. For the liquid phase, the transport equationsfor the turbulent kinetic energy k and the turbulent dissipationrate � are

�

�t(�l�lkl) + ∇ · (�l�lUlkl) − ∇ ·

(�l

(�l + �t,l

�k

)∇kl

)

= �l (�t,l∇Ul[∇Ul + (∇Ul )T] − �l�l ) + k,l , (10)

�

�t(�l�l�l ) + ∇ · (�l�lUl�l ) − ∇ ·

(�l

(�l + �t,l

��

)∇�l

)

= �l

�lkl

(C�1�t,l∇Ul[∇Ul + (∇Ul )T] − C�2�l�l ) + �,l .

(11)

The turbulent viscosity of the continuous phase is calculatedby Eq. (7).

The bubble induced turbulence source terms are calculatedfrom (Pfleger and Becker, 2001)

k,l = �lCk|Il | · |Ug − Ul |, (12)

�,l = �l

�lkl

C�1C�|Il | · |Ug − Ul |. (13)

All the constants of the two-phase k.� turbulence model aresummarized in Table 1.

3.3. Two-phase RSM turbulence model

The Reynolds Stress Model (RSM) (Daly and Harlow, 1970)does not assume the isotropy of turbulence and is more adaptedto swirling flows than the k.� turbulence model and its variants(Ranade, 2002). In this model, the required computational timeis only 60% higher than with a k.� model. As for the k.�model, turbulence is only considered for the continuous phase.

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2439

Table 2Constants used in the two-phase RSM model

C� C�1 C�2 Ck C� ��

0.09 1.44 1.92 1.44 1.92 1.0

Bubble induced turbulence source terms are added to the trans-port equations. The transport equation for the Reynolds stresstensor component Rij is

�

�t(�l�lRij ) + �

�xk

(�l�lUkRij )

= −�l�l

(Rik

�Uj

�xk

+ Rjk

�Ui

�xk

)+ �

�xk

(�l�l

�Rij

�xk

)

− �

�xk

(�l�lu′iu

′j u

′k) + �lp

(�u′

i

�xj

+ �u′j

�xi

)

− �l�l�ij + R,ij (14)

with

R,ij = 23ijk,l with k,l from Eq. (12) (15)

and

�ij = 23ij �l with �l from Eq. (17). (16)

The third and fourth term on the right-hand side of Eq. (14) aremodeled (see Fluent Inc., 2005).

The turbulence dissipation rate �l is calculated using thefollowing transport equation:

�

�t(�l�l�l ) + �

�xi

(�l�lUi�l ) − �

�xj

(�l

(�l + �t,l

��

)�

�xj

�l

)

=−�l

�lkl

(C�1�l

(Rik

�ui

�xk

+Rik

�ui

�xk

)+C�2�l�l

)+�,l (17)

with

kl = 12 (Rii + Rjj + Rkk) (18)

and �,l is given by Eq. (13) and �t,l by Eq. (7).All the constants of the two-phase RSM turbulence model

are summarized in Table 2.

3.4. Lagrangian particle motion model

In order to perform the simulation of RTD, particle track-ing has been used (Stropky et al., 2007; Thyn et al., 1998).RTD is deduced from the simulation of a pulse injection of asufficient number of particles. The trajectory of a particle ispredicted by integrating the force balance on the particle in aLagrangian reference frame (Fluent Inc., 2005). The character-istics of these particles have been chosen such as they can beconsidered as perfect tracers of the fluid flow. Their diameter is10−6 m and they have the same density as water. This ensuresthat they can perfectly follow all the simulated timescales. The

Table 3Constants used in the stochastic model

CL k.� model CL RSM model

0.15 0.3

turbulent dispersion of particles is also taken into account: par-ticles in a turbulent flow are submitted to a randomly varyingflow field. This phenomenon is modelled by a stochastic ap-proach, assuming that the particles interact with a sequence ofturbulent eddies, the fluctuating velocity within each eddy be-ing isotropic and obeying a Gaussian probability density func-tion. The interaction time �I is assumed to be sufficiently shortso that the fluid velocity in a given eddy is constant duringthis process. The particle trajectory is therefore calculated asfollows:

dup

dt=3

4Cd

�l

dp�p

|up−ul |(up−ul ) (19)

with

ul=Ul+u′l and u′

l composed of u′x,l=u′

y,l=u′z,l=�

√2

3kl .

(20)

� is a normally distributed random number and Cd is calculatedby the Morsi and Alexander correlation (1972).

A new value of � is applied each time the interaction time isreached. This interaction time is taken as the minimum of theeddy lifetime �e and the eddy crossing time �ct .

�e = −CL

kl

�llog(�), (21)

where � is a uniform random number between 0 and 1.

�ct = −� ln

(1 −

(Le

�|up − ul |))

(22)

with the particle relaxation time:

� = 4

3

�2pdg

�2l Cd |up − ul |

and the eddy lifetime:

Le =√

3

2C�

k3/2l

�l.

Different values for CL are recommended depending on theturbulence model (Fluent Inc., 2005) (Table 3).

3.5. Transport equation of a tracer

Another approach used to simulate RTD is the solution of atransport equation for a passive tracer with the same physicalproperties as the continuous phase (Talvy et al., 2007; Zhanget al., 2007). The transport equation for the concentration of a

2440 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

tracer in a turbulent flow is

�

�t(�lCtr) + ∇.(�lUlCtr) = ∇.

((�lDm + �t

Sct

)∇Ctr

)(23)

with Sct , the turbulent Schmidt number considered constant andequal to 0.7. The simulation of the RTD is obtained followingthe outlet concentration of the tracer injected as a pulse at theinlet.

3.6. Grid, boundary conditions and discretization scheme

It is commonly admitted in bubble induced flows that thereis no need for boundary layer or advanced wall treatment(Troshko, 2006). In our case, bubbles are only located near onewall which induces a high velocity gradient at this location.

Fig. 2. Projection of the grid on a vertical plane (y, z).

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

y (m)

Tim

e-av

erag

ed li

quid

ver

tical

vel

ocity

Uz

(m/s

)

129600 cells grid

345600 cells grid

1036800 cells grid

Fig. 3. Simulated average liquid vertical velocity on a transversal line (x = 1.8 m, z = 0.1 m) for three different grid sizes.

Therefore a refined mesh is necessary near that wall. Differ-ent grid sizes have been tested: cubic cells of 1 cm3 (130 000cells) or 0.125 cm3 (1 000 000 cells) and cells of 1 cm3 with arefined mesh near the walls (350 000 cells) (Fig. 2). Grid in-dependency has been verified (Fig. 3) and finally this last gridoffers the best compromise between precision and computa-tional effort.

Inlets of gas and liquid have been modelled with a velocityinlet boundary condition. The turbulence boundary conditionsat the inlets are given through the turbulence intensity (10%)and inlet hydraulic diameter. The outlet of liquid has also beenmodelled with a velocity inlet boundary condition; it allowedus to impose the outflow velocity in order to avoid backflowwhich causes mass balance problem when RTD simulations areperformed. In order to simulate the degassing boundary condi-tion at the top of the reactor, an outlet boundary condition hasbeen used for the gas phase and a symmetry boundary condi-tion has been used for the liquid phase, this degassing conditionhas been implemented through a user defined function.

A second order discretization scheme (QUICK: QuadrativeUpwind Interpolation for Convective Kinematics) has been cho-sen for the turbulent dissipation rate and void fraction equationsin order to limit their numerical diffusion (calculation errorsinduced by grids and discretization schemes). A first orderdiscretization scheme has been tested for the solution of the mo-mentum equations but it leads to unstable results; consequentlya second order discretization scheme (QUICK) has been usedto solve all equations. The SIMPLE (Semi-Implicit Method forPressure-Linked Equations) pressure–velocity coupling schemehas been used. The under relaxation parameters used are givenin Table 4.

Table 4Under relaxation parameter used for each variable

Pressure Momentum Volume fraction k � Rij

0.3 0.7 0.2 0.8 0.8 0.5

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2441

4. Results

This CFD study will be divided into three main parts. First,a simulation of the complete turbulent hydrodynamics for dif-ferent liquid and gas flowrates is carried out. The second stepis the simulation of the RTD. And finally, the numerical param-eters used for the bench scale reactor are validated on a plantscale reactor.

A test case experiment is chosen as a reference for the vali-dation of the simulations. It is carried out for a liquid flowrateQL of 3.6 L min−1 and a gas flowrate QG of 15 L min−1. Thebubble diameter for this air–water system has been estimatedusing a photographic technique: an average value of 4 mm hasbeen used and since the gas hold-up is very low (< 10%),bubble break-up and coalescence are neglected.

4.1. Velocity profiles and turbulence characteristics

Simulations have been performed with the k.� and RSM tur-bulence models. Fig. 4 presents the projection of the liquid ve-locity field on a vertical plane for the experimental data andthe results of simulations with the k.� and RSM turbulencemodels. Simulated velocity fields with the k.� and RSM mod-els are superimposed, both models giving similar results. Theoverall agreement is good, however, some discrepancies can beobserved near the bubble injection position and near the freesurface. Discrepancies observed near the sparger are probablydue to the simplification made for the gas inlet boundary con-ditions (velocity magnitude of 0.162 m s−1 and void fractionof 1) whereas those observed near the free surface are probablydue to the simplified representation of the surface which is notplanar in the experimental setup.

Fig. 5 compares the measured and simulated profiles ofthe vertical liquid velocity at three different heights. Thesequantitative comparisons show some reasonable agreement be-tween experiments and simulations (an average difference of0.035 m s−1 is observed which represent 18% of the maximumliquid velocity) whereas the two simulations results are veryclose. The LDV measurements on the two measurement planes(defined in Section 2.1) show that there is no significant differ-ence, with respect to the vertical liquid velocity, between thesetwo positions despite their apparent different gas hold-ups (ex-perimental data planes 1 and 2). As it will be discussed later,despite the discrepancies between experimental and simulatedresults, the liquid velocity fields calculated are satisfying forRTD calculations.

LDV measurements revealed a non-uniform time-averagedaxial velocity profile along the length of the reactor (Fig. 6).Simulations predict these spatial oscillations (Fig. 6) but failto predict the period and amplitude of the phenomenon. Themaximum experimental value of these oscillations (0.02 m s−1)

is ten times the mean axial velocity (0.0017 m s−1). Becauseof these very small velocity values, measurement uncertaintyis high. But, at the same time, the maximum axial velocityis still ten times smaller than the maximum vertical veloc-ity (0.2 m s−1). The amplitude of these fluctuations is thusnegligible compared to the highest velocities encountered but

Fig. 4. Overall representation of the experimental and simulated averagevelocity fields on a vertical plane (y, z) for both turbulence models.

may have an impact on dispersion; this will be investigated inSection 4.2.

Table 5 presents results relative to turbulence. The experi-mental results for turbulent kinetic energy k are surface aver-ages on the measurement plane (plane 1). It can be highlightedthat experiments have shown an isotropic turbulence. The tur-bulent dissipation rate � of the liquid phase can be estimatedwith an energy balance (Eq. (24)), assuming that all the energybrought by the isothermal expansion of the gas is dissipated byturbulence:

〈〈�l〉〉theoretical = PatmQG

�lVln

(1 + �lgh

Patm

). (24)

Simulation results for the turbulent kinetic energy k are alsosurface averages on plane 1 and turbulent dissipation rate � isa volume average over the whole reactor. The turbulent kineticenergy k is estimated by simulation with an underestimationof around 26% by the k.� model and a 33% overestimationby the RSM model. The turbulent dissipation rate � obtainedwith the k.� model is also underestimated by 20% whereas theturbulent dissipation rate � obtained with the RSM model ismore importantly underestimated by 25%. The underestimationof � could be explained by the energy dissipation at the freesurface due to bubble break-up. This dissipation can reach 30%of the total energy for some configurations (Desai et al., 1995).

4.2. Residence time distribution

All our experimental RTD can be described by the plugflow with axial dispersion model. Therefore the dispersioncoefficient is a good estimation of the overall mixing withinthe reactor. The simulated dispersion coefficient can bedescribed as the sum of four contributions, three of them are

2442 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

Fig. 5. Comparison between experimental and simulated average vertical velocities at three different heights: 5, 10 and 15 cm.

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 5 10 15 20 25 30 35 40 45 50

x (cm)

Ux

(m/s

)

Experimental data

RSM simulation

k-eps simulation

Fig. 6. Comparison between experimental and simulated average axial velocities along a fraction of the length of the reactor (at y = 4 cm, z = 10 cm).

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2443

physical: molecular diffusion, dispersion due to velocity field(convection) and turbulent dispersion, and the fourth term isthe numerical diffusion (Eq. (25)).

Dtotal = Dmolecular + Dconvection + Dturbulent + Dnumerical. (25)

There are two main methods to simulate a RTD with CFD:

• Particle tracking: the motion equations for numerous parti-cles are solved from the entrance of each particle in the reac-tor to its escape from the reactor (Section 3.4). A significantstatistical RTD can be obtained as long as enough particlestrajectories are calculated (approximately 4000 in our case).Using this method avoids solving a transport equation and,therefore, eliminates numerical diffusion because numericaldiffusion is due to the discretization of a transport equation(numerical schemes and mesh size). However, it does not al-low to take the molecular diffusion of a tracer into account.This has no consequence here since the molecular diffusionof our experimental tracer is negligible compared to the dis-persion coefficient. This will be verified later.

• Tracer transport: the transport equation for a passive tracer issolved (Section 3.4). A pulse of tracer is injected at the inletand its concentration is followed at the water outlet. Thismethod includes theoretically all the dispersion sources. Itsimplementation can be performed in two steps in order toreduce the calculation time: first the solution of the stationaryflow field and second the solution of the transient transportequation for the tracer.

Both methods have been tested in this work.

Table 5Comparison between experimental and simulated turbulence variables

Experimental/ k.� RSM

theoretical simulation simulation

k (10−3 m2 s−2) 2.42 (exp) 1.78 3.31� (10−3 m2 s−3) 8.75 (Eq. (25)) 6.84 6.10

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0 1000 2000 3000 4000 5000 6000Time (s)

E (t)

Experimental RTD

Simulated RTD realised with 100 ssample count (particles tracking)

Simulated RTD realised with 200 ssample count (particles tracking)

Simulated RTD fitted with DTS PRO(particles tracking)

Simulated RTD with RSM model(transport equation)

Fig. 7. Comparison between experimental and simulated RTD obtained with the RSM turbulence model and the transport equation method or the RSMturbulence model and the particle tracking method for different sample times for a liquid flowrate of 3.6 L min−1 and a gas flowrate of 15 L min−1.

4.2.1. RTD simulation with scalar transportAn integration time step of 4 s (0.2% of mean residence time)

has been selected. Fig. 7 presents the RTD obtained with theRSM turbulence model and the experimental one. Simulationresults obtained with the k.� turbulence model are not pre-sented because they are similar to the ones obtained from RSMsimulations.

The mean residence time is correctly simulated whereas thedispersion coefficient is 30% underestimated. The simulateddispersion coefficient is 4.26×10−4 m2 s−1 whereas the exper-imental one is 6.01 × 10−4 m2 s−1.

4.2.2. Influence of the different dispersion sources on theRTD simulated via tracer transport

Talvy et al. (2007) proposed a method to identify the partof the two different aspects of the dispersion: Dconvection andDturbulent (the two other aspects are negligible in our case:Dmolecular = 1.48 × 10−10 m2 s−1 and an estimation of thenumerical dispersion gives Dnumerical = 5.0 × 10−6 m2 s−1).Eq. (26) proposes an estimation of the turbulent dispersion(Talvy et al., 2007):

Dturbulent = �t

�lSct

. (26)

In our case this turbulent dispersion is equal to 3.81 ×10−4 m2 s−1, approximately 90% of the total dispersion.

The convective dispersion can be estimated from CFDsimulation with Eq. (27) (Talvy et al., 2007).

Dconvection =⟨⟨ 〈〈ul,xCtr〉〉y,z − 〈〈ul,x〉〉y,z〈〈Ctr〉〉y,z

�〈〈Ctr〉〉y,z

�x

⟩⟩

z

. (27)

This equation estimates the convective dispersion to 1.14 ×10−5 m2 s−1, approximately 2% of the total dispersion. Theseresults indicate that the convective dispersion is negligiblecompared to the turbulent dispersion. This conclusion will bediscussed later. Thus the main dispersion source is turbulence.

2444 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

Consequently the turbulent Schmidt number allows to adjustthe dispersion coefficient. It has been verified that a turbulentSchmidt number of 0.44 would give a perfect match betweenexperimental and simulated RTD.

Table 6Numerical experiments: Comparison of the influence of the main parameterson the simulated dispersion coefficient with the k.� turbulence model

Simulation method Ref Characteristics D (10−4 m2 s−1)

Particle tracking 1 Reference 2.512 2.k 5.023 2.� 3.184 2.uy, 2.uz 3.165 �ux/�xi = 0 2.37

Transport ofpassive tracer

6 Dm = 10−20 m2 s−1 3.34

7 Dm = 14.8 10−10 m2 s−1 3.35

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0Time (s)

E (t)

Experimental RTD

Simulated RTD with the RSMmodel fitted with DTS PRO

Simulated RTD with the k-epsmodel fitted with DTS PRO

1000 2000 3000 4000 5000 6000

Fig. 8. Comparison between experimental and simulated RTD obtained with the RSM and the k.� turbulence models and the particle tracking method for a

liquid flowrate of 3.6 L min−1 and a gas flowrate of 15 L min−1.

0

2

4

6

8

10

12

0 10 20 30 40 50 60 70Gas flowrate (l/min)

D (1

0-4 m

2 /s)

0

5

10

15

20

25

30

35

40

45

Res

iden

ce ti

me

(min

)

Simulated dispersion coefficientExperimental dispersion coefficientCorrelated dispersion coefficient (Potier 2005)Simulated residence timeExperimental residence time

Fig. 9. Experimental and simulated dispersion coefficients and mean residence times for different gas flowrates with a constant liquid flowrate of 3.6 L min−1

with the RSM model and the particle tracking method.

In order to verify the negligibility of the molecular diffusion,two simulations have been performed: two RTD simulationswith tracer transport method, the first one with a moleculardiffusion coefficient of 10−20 m2 s−1 (in order to simulate anon-diffusive tracer) and the second one with a molecular dif-fusion coefficient of 1.48 × 10−10 m2 s−1 (NaCl in pure waterat 20 ◦C). The results of these two experiments are presentedin Table 6 (Ref 6 and 7) and they show that the molecular dif-fusion has no significant impact on the overall dispersion.

4.2.3. RTD simulation with particle trackingExperimental molecular diffusion is negligible compared to

the turbulent and convective dispersions therefore particle track-ing can also be used to estimate RTD. Because of the discretenature of particle tracking, particles must be sampled in orderto exploit data: particles escaping the reactor during the sametime period are counted; RTD is obtained from the curve giving

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2445

02468

101214161820

2Liquid flowrate (l/min)

Dis

pers

ion

coef

fient

(10-4

m2 /s

2 )

0

10

20

30

40

50

Mea

n re

side

nce

time

(min

)

Simulated dispersion coefficientExperimental dispersion coefficientSimulated residence timeExperimental residence time

2.5 3 3.5 4 4.5

Fig. 10. Experimental and simulated dispersion coefficients and mean residence times for different liquid flowrates with a constant gas flowrate of 65 L min−1

with the RSM model and the particle tracking method.

the number of particles counted during one period as a func-tion of time. Fig. 7 shows the curves obtained with the RSMturbulence model for two different sampling times: 100 and200 s and an experimental curve fitted with the plug flow withclosed–closed axial dispersion model. As the sampling time in-creases, fluctuations decrease because of an integration effect.The mean residence time and dispersion coefficient D are de-termined using the raw data: first, the mean residence time isdetermined and in a second step the dispersion coefficient is op-timized in order to fit a plug flow model with axial dispersion.

Comparison between experimental and simulated RTD isshown in Fig. 8 for the k.� and RSM turbulence models. TheRSM turbulence model simulation leads to very good resultsand the k.� turbulence model leads to an underestimated valueof the dispersion coefficient of around 50%. These results aredue to the stochastic model and especially the constant CL

(Table 3). In fact, in the simulation of turbulent dispersion,there is a direct linear relation between CL and the disper-sion coefficient D. Consequently the value of CL can be opti-mized to obtain a good dispersion coefficient. Empirical valueof CL = 3C� = 0.27, which is close to the one used for theRSM model (Table 3), can be found in literature for the k.�turbulence (CD-Adapco-Group, 2003); with this value of CL

both the k.� and RSM turbulence models give the same results(not shown here).

Additional experiments and simulations have been performedfor different gas and liquid flowrates and similar good agree-ment between experimental and simulated RTD with the parti-cle tracking method have been obtained. All simulations werecarried out with the RSM turbulence model. Comparison be-tween experimental and simulated dispersion coefficients andmean residence times for different gas flowrates are presentedin Fig. 9. The dispersion coefficient seems to depend on Qg

according to a square root law, the same as in correlation(Eq. (1)). It can be highlighted that the calculations divergewhen the gas flowrate increases beyond 70 L min−1.

The same comparison, with different liquid flowrates, is pre-sented in Fig. 10. A good agreement can also be observed for

the dispersion coefficient even if it does not vary much withthe liquid flowrate in this domain. The mean residence time isalso correctly simulated.

4.2.4. Influence of the different dispersion sources on theRTD simulated via particle tracking

Simulated RTD with transport equation solution shows that avery high proportion of the axial dispersion is due to turbulentdispersion (90%). But the solution of the transport equation failsto accurately simulate experimental RTD. On the other hand,particle tracking allows a better RTD estimation, but the deter-mination of the different dispersion sources is not as straightfor-ward. With CFD it is possible to investigate the influence of oneof the dispersion source at once, with the other parameters keptconstant. This is impossible with physical experiments. These“numerical experiments” do not have real physical meaning butthey allow a better understanding of the physical phenomenaresponsible for the dispersion in the reactor. These numericalexperiments have been performed based on our reference case(Sections 4.1 and 4.2).

Our main objective, here, is to determine which of turbu-lence, convection or molecular diffusion is responsible for theobserved dispersion. All the following “numerical experiments”are performed with the k.� turbulence model in order to avoidtoo long calculations.

To study the impact of the velocity field, two simulationshave been performed: the first one with an axial velocity setto a constant value of 1.667 mm s−1 (i.e. the average valueof the axial liquid velocity) in all grid cells (�ux/�xi = 0)

in order to evaluate the impact of axial velocity oscillations,the second one with the velocity of the gas induced loopdoubled (see Fig. 3) in order to evaluate the contribution ofthese loops in the dispersion (i.e. uz = 2uz,ref , uy = 2uy,ref ).Results, presented in Table 6 (Ref 5 and 4), show that theaxial velocity oscillations (see Fig. 6) have negligible influenceon D and gas induced loops have a quite significant impacton D (increase of approximately 25% by doubling the loopvelocity).

2446 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

Fig. 11. Projection of the simulated liquid velocity field on a vertical plane for the industrial reactor (x = 51 m).

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9

y (m)

Tim

e-av

erag

ed v

ertic

al li

quid

vel

ocity

Uz

(m/s

)

91936 cells grid151776 cells grid194208 cells grid

Fig. 12. Simulated average vertical liquid velocity on a transversal line (x = 51 m, z = 1.8 m) for three different grid sizes.

Finally, in order to study the impact of turbulence, a simu-lation with a doubled value of � and another with a doubledvalue of k has been performed. Results are also presented inTable 6 (ref 2 and 3). The modification of � induces a smalldifference on D (increase of around 25%) whereas doubled thevalue of k doubles the value of D (impact of 100%). D seemsto be strongly linked to turbulence and especially to the valueof the turbulent kinetic energy k in accordance with Eq. (26)where �t is given by Eq. (7).

With all these “numerical experiments”, we validate the factthat turbulence is the main cause of the overall dispersion. Tur-bulent kinetic energy k is the variable which mainly impacts onthe overall dispersion. But the particle tracking method showsthat convective dispersion is not negligible, contrary to the con-clusion of the tracer method.

We can conclude that the RSM turbulence model coupledwith the particle tracking method gives very good RTD results.As it has been said in the RTD simulation part of Section 4.2,CL is the preponderant constant in the modelling of turbulentdispersion therefore with an empirically deduced CL such asCL = 3C�, good RTD results can also be achieved with the k.�turbulence model.

4.3. Scale up on industrial size reactor

In order to validate the chosen hydrodynamics model, RTDsimulations of a plant scale reactor have been carried out. RTDexperiments (Potier et al., 2005) have been conducted on a3300 m3 (102 m long, 9 m wide, 3.6 high) channel reactor, atthe Nancy Maxéville plant (France). Lithium chloride has beenused as an inert tracer and injected as a pulse near the liquidinlet. Lithium chloride concentration was measured by atomicabsorption. It was verified that the tracer was not absorbedon solid matter. The liquid flowrates were 1280, 1650 and2200 m3 h−1 and the linear gas flowrate was approximately0.0175 m3 m−1 s−1; aeration takes place on both sides of thereactor which causes a double gas induced loop, as seen in thesimulated flow field presented in Fig. 11. The vertical plane(x, z) in the middle of the reactor is a symmetry plane there-fore a symmetry boundary condition is used in order to shortencalculation time. The grids for this kind of wastewater plantscale reactor simulations are very coarse (average cell volumeof 0.016 m3) but give satisfactory results (Glover et al., 2006;Stropky et al., 2007). Grid independency has been verified(Fig. 12) and a 151 776 cells grid has been used with

Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449 2447

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0 4000 8000 12000 16000 20000Time (s)

E (t)

experimental low flowratesimulated low flowrateexperimental medium flowratesimulated medium flowrateexperimental high flowratesimulated high flowrate

Fig. 13. Comparison between experimental and simulated RTD obtained with the RSM model and the particle tracking method on a full scale reactor forthree different liquid flowrates.

boundary conditions similar to those used for the bench scalereactor. The physical properties of the mixture of wastewaterand sludge are approximated to those of water, which is ac-ceptable because of the low concentration of sludge (less than8 g L−1) in the reactor and the bubble diameter is approximatedto 1 cm as observed in experiments.

Experimental results have been treated by curve fitting witha plug flow with axial dispersion model. Simulated results havebeen obtained as in Section 4.2 with a particle tracking method.This method avoids possible numerical diffusion due to thecoarse grid. Fig. 13 shows very good agreement between exper-imental and simulated data. There is less than 5% of error onthe mean residence time and the dispersion coefficient. It canbe highlighted that the greater discrepancy is observed with thelowest flowrate, which is due to the difficulty to keep constantthe inlet flowrate of a real wastewater treatment plant duringsuch a long period. In fact, some small flowrate variations havebeen noticed.

5. Conclusion

3D Euler–Euler numerical simulations of a cross-flowgas–liquid reactor have been presented. These simulations havebeen carried out using the CFD code FLUENT. Grid depen-dence, discretization schemes and different turbulence modelshave been tested. It has been found that the use of the RSMturbulence model coupled with a second order discretizationscheme allows to successfully predict the dispersion coefficientof the reactor. RTD simulations with particle tracking methodwith standard parameters give more accurate results than thosewith transport equation solution. But it is also possible to ad-just the simulation to the experiment with the modification ofkey model parameters (Sct or CL). CFD allowed to investi-gate the different effects responsible for the dispersion. It hasbeen shown that the dispersion is mainly due to turbulence butthe dispersion due to convection is not negligible. The poorly

estimated axial velocity does not have significant effects andthe molecular diffusion is negligible.

The hydrodynamics model obtained can predict accuratelythe dispersion for different gas and liquid flowrates in the benchscale reactor and can predict the dispersion in an industrialsized wastewater reactor with the same precision than the pre-viously established empirical correlation (Potier et al., 2005).This correct estimation of the global hydrodynamics allowsthe future implementation of a kinetic model coupled with thehydrodynamics model.

Notation

C concentration, kg m−3

Cd drag coefficient, dimensionlessCL constant in the Lagrangian particle motion

model, dimensionlessC�, C�1,C�2, Ck , C�

constants in the k.� model or RSM model,dimensionless

d diameter of the dispersed phase (bubble orparticle), m

D dispersion coefficient, m2 s−1

Dm molecular diffusivity, m2 s−1

g gravity acceleration, m s−2

h reactor height, mI momentum transfer term between phases,

kg m−2 s−2

I unit tensor, dimensionlessk turbulent kinetic energy, m2 s−2

L reactor length, mLe characteristic length of turbulent eddies, mp pressure, PaPatm atmospheric pressure, 101 300 PaQg gas flowrate, m3 s−1

Ql liquid flowrate, m3 s−1

Rij Reynolds tensor component, m2 s−2

2448 Y. Le Moullec et al. / Chemical Engineering Science 63 (2008) 2436–2449

Sct turbulent Schmidt number, dimensionlesst time, su velocity, m s−1

U statistical average velocity, m s−1

u∞ terminal vertical velocity, m s−1

V reactor volume, m3

w reactor width, mx, y, z spatial coordinates

Greek letters

� volume fraction, dimensionlessij Kronecker factor, dimensionless� turbulent dissipation rate, m2 s−3

〈〈�l〉〉 volume average of turbulent dissipation rate,m2 s−3

� normally distributed random number, dimen-sionless

� dynamic viscosity, Pa s� bubble induced turbulent of dissipation rate

source term, kg m−1 s−4

k bubble induced turbulent kinetic energy sourceterm, kg m−1 s−3

� density, kg m−3

� surface tension, N m−1

��, �k constants in the k.� model or RSM model,dimensionless

�ct eddy crossing time, s

�e eddy lifetime, sSubscriptsg gasi, j, k spatial coordinatesl liquidp particleq phase index: l or gt turbulenttr tracer

Operator〈〈a〉〉z averaged value of a on a line along z〈〈a〉〉x,y averaged value of a on a (x, y) plane

Acknowledgements

The authors would like to thank Richard Lainé for his helpwith the LDV experimental setup, Benoit Fiers for his helpin the experimental works and Noël Midoux for his scientificcontribution.

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