Three Phase Holdup

Chemical Engineering Science 61 (2006) 75357550www.elsevier.com/locate/ces

Measurement of phase holdups in liquidliquidsolid three-phase stirredtanks and CFD simulation

Feng Wang1, Zai-Sha Mao, Yuefa Wang, Chao YangInstitute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China

Received 9 November 2005; received in revised form 3 August 2006; accepted 13 August 2006Available online 5 September 2006

Abstract

Phase holdup is an important hydrodynamic characteristic of multiphase systems relevant to optimization and scale-up of related processequipment. In the present article, measurements of phase distribution of solid particles and oil droplets are conducted in a lab-scale stirredtank by sample withdrawal under various operating conditions. A EulerianEulerian three-fluid model is established for the prediction of phasedistribution of two dispersed phases in the agitated liquidliquidsolid dispersion system. The turbulence structure in the system is describedby an extension of the standard k. turbulence model to three-phase flow including the influence of presence of two dispersed phases as anadditional source of turbulent kinetic energy. Momentum exchange between continuous and dispersed phase as well as between the two dispersedphases are incorporated into the model formulation. Comparison of model predictions with experimental data suggests reasonable agreementfor the dispersed oil phase. The predicted distribution of solid particles shows some discrepancies in comparison with the measurements, butthe agreement is significantly improved for higher impeller speeds. 2006 Elsevier Ltd. All rights reserved.

Keywords: Stirred tank; Three-phase flow; Turbulence; Numerical simulation

1. Introduction

Liquidliquidsolid three-phase stirred tanks are common inprocess industry. Typical applications include reactive floccula-tion and solid catalyzed liquidliquid reaction, etc. The knowl-edge of the hydrodynamic characteristics, such as suspensionof solid particles, dispersion of dispersed liquid phase and theirspatial distribution in the stirred tank, is essential in determi-nation of rates of heat/mass transfer and desired chemical re-actions, and consequently of vital importance for the reliabledesign and scale-up of such chemical reactors.Design and scale-up of multiphase stirred tanks are mainly

based on empirical and semi-empirical correlations gained fromexperimental data so far. Extrapolating use of those empiricalcorrelations beyond the original operating conditions is highly

Corresponding author. Tel.: +86 10 6255 4558; fax: +86 10 6255 1822.E-mail addresses: [email protected] (F. Wang),

[email protected] (Z.-S. Mao).1 Present address: Process & System Department, China HuanQiu Con-

tracting & Engineering Corp., Beijing 100029, China.

0009-2509/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.08.046

risky. The strategy of stagewise scale-up is costly and time-consuming, and the satisfactory scale-up to large-scale reactoris not guaranteed.Scientific and reliable design of a liquidliquidsolid reac-

tor requires full understanding of the hydrodynamics and trans-port properties in such multiphase systems, including the phaseholdups and spatial distributions. This demands sufficient ex-perimental and theoretical investigations, but up to date littlehas been reported in the open literature.Among three-phase systems in stirred tanks, the gasliquid

solid system is the most investigated one. Chapman et al.(1983ac) investigated the effects of the presence of solid par-ticle on gasliquid hydrodynamics as well as the aeration onthe suspension of particles. More recently, Dohi et al. (2004)performed experimental measurements on the power consump-tion and solid suspension in gasliquidsolid stirred tanks.However, other hydrodynamic characteristics, in particularthe distribution of two dispersed phases in three-phase stirredtanks, are scarcely reported in the literature.The computational fluid dynamics (CFD) approach has

attracted intensive attention in recent years for its powerful

http://www.elsevier.com/locate/cesmailto:[email protected]:[email protected]

7536 F. Wang et al. / Chemical Engineering Science 61 (2006) 75357550

capacity for understanding the physical reality of multiphaseflows. Nevertheless, CFD simulation of the hydrodynamics instirred tanks is very difficult because the flow field in the tankis complex and highly unsteady due to the interaction of therotating impeller with multiphase dispersion. Most of the liter-ature on numerical simulation of flow field in stirred tanks islimited to single-phase and two-phase flows. When it comes tothree-phase flow, the interactions between two dispersed phasesand the contribution of dispersed phases on the turbulence ofthe continuous phase make the numerical solution of the gov-erning equations more challenging.A few papers have been published to date on the nu-

merical prediction of three-phase flow, mostly focused ongasliquidsolid three-phase flow in the bubble column reac-tors, e.g. Krishna et al. (2000), Michele and Hempel (2002),Mitra-Majumdar et al. (1997, 1998), Padial et al. (2000). Ingeneral, the EulerianEulerian multifluid approach was em-ployed unanimously to formulate the turbulent gasliquidsolidthree-phase flows. The k. two-equation turbulence model wasadopted to represent the turbulence of the continuous phase.For modeling the momentum exchange between continuousand dispersed phases as well as between two dispersed phases,very limited information is available in the literature and noconvergent opinion on the expression of the inter-phase in-teraction has been reached, yet. To our best knowledge, nothree-dimensional numerical simulation of liquidliquidsolidthree-phase flow field in stirred tanks has been reported to date.In addition, experimental data is also scarce for the purposeof validation of numerical simulation of liquidliquidsolidthree-phase flow.The objective of the present study is to provide experimental

measurements of axial and radial variations of phase holdups oftwo dispersed phases in a stirred tank under different operatingconditions. Also a computational approach is established basedon the EulerianEulerian multi-fluid approach to describe themotion of each phase, and then to predict the phase distributionof two dispersed phases in the three-phase stirred tank by nu-merical solution of Reynolds time-averaged mass and momen-tum conservation equations (RANS). The experimental dataobtained in this work are compared with the model predictionsto validate the computational approach. Preliminary effort ofthis work has been reported previously in a CFD conferencepresentation (Wang et al., 2005).

2. Experiments

Experiments are carried out in a cylindrical, flat-bottomedstirred tank made of Perspex with the inner diameter T =0.154m, equipped with four equally spaced baffles with thewidth B = 0.1T . The impeller is a six flat blade disc (Rush-ton) turbine with the diameter D = 0.062m. The configura-tion of the impeller and the stirred tank are shown in Fig. 1. Itis noted that a relatively large impeller (D = 0.4T ) is chosenin this study in comparison to the standard configuration withD = T/3, in order to provide good suspension and dispersionof the two dispersed phases. The clearance between the cen-ter plane of the impeller disc and the tank bottom is set to be

B

H

T

DC

d

D

b

w

Fig. 1. Configurations of stirred tank and Rushton impeller: T = 0.154m,H = T , B = 0.1T , D = 0.062m, d = 0.75D, b = 0.25D, w = 0.2D.

21

3

45

6

7

Fig. 2. Experimental setup. 1. Stirred tank, 2. impeller, 3. motor, 4. samplingtubes, 5. graduated cylinder, 6. peristaltic pump, 7. valve.

C=T/3. Nine copper sampling tubes with 4.5mm inner di-ameter ( 6.0mm outside diameter) are mounted midway be-tween two baffles vertically along the tank wall from the bottomto the free liquid surface (z = 0.1H0.9H ). Sample tubes canreach the tank interior from r/R= 0.1 to 0.9, if not prohibitedby the sweeping impeller blades.The experimental setup is illustrated in Fig. 2. The stirred

tank is charged with tap water as continuous phase and n-hexaneas oil phase. Glass beads with a density of s=2550 kgm3 anda mean diameter of ds = 110 m are the solid phase. The totalheight of three-phase mixture is always equal to the diameter ofthe tank (T=H). The impeller speed is measured using a digitallaser tachometer and is accurate to 5 rpm. In all experimentalruns, 1 h is needed to achieve a steady state of dispersion in thepresent investigation, although Armenante and Huang (1992)claimed that the equilibration period was only 1015min for ag-itated liquidliquid dispersion. To avoid the entrainment of gasinto the liquidliquidsolid system, the impeller speed testedis between N = 300 and 500 rpm, which is still below that thecritical speed for full suspension of solid particles.A peristaltic pump is activated after the steady dispersion

state is achieved and the dispersion is partly circulated through

F. Wang et al. / Chemical Engineering Science 61 (2006) 75357550 7537

the sample tube back to the tank (to ensure the representativesamples of tank holdup being collected). Then, the flow is di-rected to the graduated cylinder and about 15ml of three-phasedispersion is collected. Since there is significant difference involatility between oil and water phases, the sample is evapor-ized at 80 C after centrifuged and weighted to determine theamount of oil phase, given the lighter phase (n-hexane) whichboils at 78 C under atmospheric pressure and always acts asdispersed phase in all experimental runs. The remained sam-ple (water and glass beads) is dried in the oven and weightedto obtain the local holdup of solid phase. The total quantity ofthe samples in each run is about 3% of the total system and thebulk flow is deemed to not being changed significantly by thesample withdrawal. Adequate amounts of water, oil and glassbeads are added into the tank to keep the individual phase in-ventory of the system at the original values. The reproducibilityof the measurements is about 3.3%. The same procedure is re-peated for various axial and radial positions, impeller speeds,and phase fractions of two dispersed phases.

3. Mathematical model

In the present study, the mathematical model is formulatedbased on the EulerianEulerian multi-fluid model. Water, oiland solid phases are all treated as different continua, inter-penetrating and interacting with each other everywhere in thedomain under consideration. The oil and solid phases are inthe form of spherical-dispersed droplets and particles, respec-tively. The effect of breakup and coalescence of droplets is ig-nored. The pressure field is presumed to be shared by threephases, which are exerted, respectively, by the pressure gradi-ent in proportion to their volume fraction. Motion of each phaseis governed by respective mass and momentum conservationequations.The RANS version of the governing equations for three-

phase flow is derived from averaging the instantaneous conser-vation equations of physical variables over the space or time.In the present work, the time averaging method of Ishii (1975)is followed. Reynolds averaging procedure leads to the meanflow equations by decomposing the instantaneous variables intoa mean value and a fluctuating component since the multiphaseflow in stirred tanks is usually turbulent. The resulted massconservation equation for phase k is written as

t(kk)+

xj(kkukj )=

xj(k

ku

kj ) (1)

and the momentum conservation for phase k reads

t(kkuki + kkuki)+

xj(kkukiukj )

=k Pxi

xj

(kkukiu

kj + kukikukj + kukjkuki

+ kkukiukj )

+ xj

kk

(ukixj

+ ukjxi

)+ kk

(ukixj

+ ukj

xi

)

23

xj

(kkij

ukjxj

+ kijkukjxj

)

+ kkgi + Fki . (2)The fluctuating pressure P and the fluctuating inter-phase

momentum exchange term F ki as well as other third-order termsare neglected in time-averaging process. The viscous shearstress is also omitted as compared with large magnitude of tur-bulent shear terms.Several turbulent fluctuation correlation terms with second

order appear in Eqs. (1) and (2), namely the Reynolds stresses,are modeled by invoking the Boussinesq gradient transport hy-pothesis:

ukiukj =kt(

ukixj

+ ukjxi

)+ 2

3kij , (3)

where kt is the turbulent kinematic viscosity.The correlation of the fluctuating velocity and fluctuating

holdup, kuki appearing in both mass and momentum conser-vation equations, represents the transport of mass and momen-tum by dispersion. The simplest method to evaluate the cor-relation is to assume gradient transport as (Elaghobashi andAbou-Arab, 1983):

kuki =ktt

kxi

, (4)

where t is the turbulent Schmidt number for phase disper-sion, which is set to 1.0 in the present work after preliminarynumerical trials.The inter-phase momentum exchange term Fki represents

the interaction between phases, which is composed of a lin-ear combination of different momentum exchange mechanisms:the drag force, the added mass force and the lift force, etc.Only the contribution of drag force is taken into account in thisstudy since other forces are testified to be less significant forboth the liquidliquid and liquidsolid flows in stirred tanks(Ljungqvist and Rasmuson, 2001; Wang and Mao, 2005). Thedrag force between continuous phase and two dispersed phasesis expressed following the practice of two-phase flows:

Fci,drag =k

Cdrag(udi uci), (5)

where Cdrag is the momentum exchange coefficient:

Cdrag = 3ccdCD|ud uc|4dd . (6)

For determination of the drag coefficient CD , the followingcorrelation is used as Clift et al. (1978) recommended:

CD =

24

Red(1+ 0.15Re0.687d ), Red 1000,

0.44, Red > 1000,


where

Red = cdd |ud uc|c,lam

.

Different from two-phase flows, there are still inter-actions between dispersed droplets and solid particles inliquidliquidsolid three-phase flows, which have to be takeninto account as well. However, this factor has not been modeledso far in the literature. In the computational model developedby Padial et al. (2000) for gasliquidsolid three-phase bubblecolumn, the drag between solid particles and gas bubbles wasmodeled identically to drag between liquid and gas bubblesbased on the notion that particles in the vicinity of gas bubblestend to follow the liquid, while in the model used by Micheleand Hempel (2002), the momentum exchange terms betweenthe dispersed gas and solid phases are expressed as

Fgs,i =Fsg,i =3gscgs |us ug|(usi ugi)

4dp. (7)

The combination of cgs |us ug| was defined as a fitting pa-rameter, whose value was determined by fitting model predic-tions to measured local solid holdups.Since two dispersed phases are assumed to be continua as

mentioned above, it is reasonable to expect the drag betweensolid particles and droplets behaving in a similar way as thedrag force between the continuous and the dispersed phase:

Fos,drag,i = Fso,drag,i

= 3oosCD,os |us uo|(usi uoi)4ds

. (8)

The modeling of a multiphase turbulent flow is basically anunresolved problem as predicting the hydrodynamic character-istics of multiphase flow is concerned, mainly because the in-fluence of the presence of dispersed phases on turbulence of thecontinuous phase is difficult to model. In the context of CFDmodeling, it is assumed that the turbulence in multiphase stirredtanks is dominated by the continuous phase and in multiphaseflows it is more often described using an extension of the widelyaccepted standard k. two-equation turbulence model in anal-ogy to the single-phase one. The effects of dispersed phases onturbulence structure are thought to generate additional turbu-lent kinetic energy. In this work, the conservation equations forthe turbulent kinetic energy of the continuous phase is writtenas follows:

t(cck)+

xi(ccucik)

= xi

(c

ctk

k

xi

)+ c[(G+Ge) c] (9)

and the turbulent energy dissipation rate is expressed as

t(cc)+

xi(ccuci)=

xi

(c

ctk

xi

)

+ c k[C1(G+Ge) C2c]. (10)

Here, G is the turbulent kinetic energy production term givenby

G=cuciucjucixj

(11)

and Ge is an extra production source term representing the in-fluence of the dispersed phase, which is taken to be propor-tional to the production of drag force and slip velocity betweentwo phases as Kataoka and Serizawa (1989) suggested:

Ge = Cb|Fdrag|[

(udi uci)2]1/2

, (12)

where Cb is an empirical constant with the value of 0.02 in thisstudy.The model parameter C1 for the impeller region is modified

as follows to describe the effect of strongly swirling flow (Zhou,1993):

C1 = 1.44+ 0.8Rf cG+Ge (13)

with

Rf =

1

[urur

r

(ur

)], 1.5w


where t1 = 0.41k/ is the mean eddy lifetime and td is theparticle response time given by

td = 4ddd3cCDd |ud uc|

.

It is well known that flow in the stirred tank is always un-steady due to interaction of the rotating impeller blades with thestationary wall baffles. However, the flow pattern will becomeaxisymmetrically repeating once it is fully developed. Ranadeand van den Akker (1994) suggested to ignore the time deriva-tives in the governing equations without introducing much er-ror in most part in the tank except in the impeller swept vol-ume. A snapshot of the flow can describe the flow within theimpeller region at a particular instant. In the present work, theflow field in the impeller swept volume is simulated in a non-inertial reference frame rotating with the impeller, where flowmay be steady and the time-dependent terms disappear. Thus,the resulting formulation of the mass and momentum conser-vation equations for phase k in general form in a cylindricalcoordinate system reads

1

r

r

(rkkukr

)+ 1r

(kkuk

)+ z

(kkukz

)

= 1r

r

(rkeff

r

)+ 1

r

(keff

r

)

+ z

(keff

z

)+ S, (18)

where

eff = lam + t .For the complete set of conservation equations and corre-

sponding source terms it may be referred to the counterpartsfor the two-phase turbulent flow as tabulated in the authorsprevious papers (Wang and Mao, 2002; Wang et al., 2004).It is still necessary to specify the laminar viscosity of the

solid phase in the momentum conservation equations. In thepresent study, the solid laminar viscosity is set equal to that ofthe continuous phase as s,lam=103 Pa s. This is considered asa reasonable approximation since test calculations showed thatvariation of solid laminar viscosity between 101 and 104 Pa sdo not yield significant difference in the simulation results. Thiswas also validated by the examination of Michele and Hempel(2002).It is worthy to note that the computational model developed

in this study does not involve any adjustable parameters tomatch the model predictions with the experimental data, andall relevant constants are cited from the literature as is.

4. Numerical procedure

The phase distributions of two dispersed phases is numeri-cally predicted as a part of the solution of three-dimensionalturbulent flow in a liquidliquidsolid three-phase stirred tank

Fig. 3. The computational grid of inner and outer domains.

under the flow conditions in coincidence with the experimentin the present study. The computational domain is chosen as ahalf of the tank since the time-averaged flow field is periodicalrepeating in the azimuthal direction. The computational gridadopted for the simulations is depicted in Fig. 3, which con-sists of 36 36 90 elements in radial, azimuthal and axialdirections, respectively. The action of impeller is modeled us-ing a modified innerouter iterative procedure that has beendetailed in Wang and Mao (2002) and is not described here. Itsmain advantage is that the calculated flow parameters on thesurface of the inner and outer regions are not averaged inthe present procedure, differing from the original innerouterapproach developed by Brucato et al. (1998), to preserve thepseudo-periodical turbulent properties.The governing equations of mass and momentum of each

phase together with the turbulent kinetic energy and energydissipation rate transport equations of the continuous phaselisted above are solved by a finite volume technique (Patankar,1980). The discretization of the equations is implemented usinga power-law differencing scheme in a staggered grid system.The most important concern in the numerical procedure is

the manner in which the pressure field is computed and its ef-fect on the continuity. The extensively used SIMPLE algorithm(Patankar, 1980) is adopted here to solve the pressure field,although other modified algorithms like SIMPLER (Patankar,1980), SIMPLEST (Van Doormal and Raithby, 1984) and SIM-PLEC (Spalding, 1980) give faster convergence than the SIM-PLE algorithm at the cost of increased computational load assuggested by Ekambara and Joshi (2003).The pressure field shared by three phases is obtained using

a pressure-correction formula derived by combining the threecontinuity equations together after being normalized with the


respective phase density as Carver (1984) suggested:[(ap

)c

+(ap

)o

+(ap

)s

]P p

=nb

[(anb

)c

+(anb

)o

+(anb

)s

]P nb

+(f

)c

+(f

)o

+(f

)s

+(D

)c

+(D

)o

+(D

)s

, (19)

where fk is the turbulent diffusive term due to the asymmetryof the mass flow, and Dk is the mass imbalance term indicatingthe extent that the continuity equation is not satisfied since thepressure field is updated gradually from the initial guess in theSIMPLE algorithm.For the numerical simulation of liquidliquidsolid three-

phase flow, at least two continuity equations have to be solved toobtain the spatial distribution of the phase holdup in the wholeflow domain. In this article, the continuity equations of thetwo dispersed phases are modified by subtracting the continuityequation of the continuous phase and using the relation c =1 o s . This gives

o

[i

(Foi

o+ Fci

c

)]out

={

i

[o

(Foi

o+ Fci

c

)]}in

+(

i

(1 s)Fcic

)out

(

i

(1 s)Fcic

)in

+(f

)o

(f

)c

(20)

and similarly,

s

[i

(Fsi

s+ Fci

c

)]out

={

i

[s

(Fsi

s+ Fci

c

)]}in

+(

i

(1 o)Fcic

)out

(

i

(1 o)Fcic

)in

+(f

)s

(f

)c

. (21)

The trial calculation shows faster convergence in this man-ner as compared with directly solving the original continuityequation of the two dispersed phases.The sets of discretized equations for physical variables are

solved iteratively through an ADI algorithm. The non-linearityin the phase momentum and turbulence equations is tackledwith suitable under-relaxation. The convergence of the solu-tion is assessed by relative residuals R(). For momentum andturbulence transport equations

R()= |app (aEE + aWW + aNN + aSS + aT T + aBB + b)|[|app| + |aEE | + |aWW | + |aNN | + |aSS | + |aT T | + |aBB | + |b|] (22)

while for the continuity equation,

R(m)= |b|

0.5cDHT, (23)

where 0.5cDHT is a reference liquid mass flux in propor-tion to the pumping capacity of the impeller. The solution isdeemed convergent when the relative residuals of the continu-ity equation is less than 104 and for others less than 106.The boundary conditions are:

(1) Symmetry axis (r=0): ucr=uor=usr=0, uc=uo=us=0,/r = 0 ( = ucr , uor , usr , uc, uo, us).

(2) Free surface: The free surface of the stirred tank is assumedto be flat, then ucz = uoz = usz = 0, /z = 0 ( =ucz, uoz, usz).

(3) Solid surface: The wall function is applied to all solidsurface of the stirred tank, including the tank wall, bottom,and the impeller.

5. Results of measurement and modeling

5.1. Experimental results of phase distribution

For liquidliquidsolid three-phase flow in stirred tanks, itis desired to gain information on the state of dispersion of theoil phase and solid particles for diagnosis and optimization ofoperation of such reactors. Effects of impeller speed on the ax-ial and radial variation of local holdup of solid and oil phasesare investigated. The present measurement results normalizedwith the respective phase volume fraction are presented inFig. 4 for the system with phase volume fractions of oil andsolid phases being o,av = 0.10 and s,av = 0.10. It can be seenthat, at relative lower impeller speed (see Fig. 4(a)), the lo-cal holdup of the solid phase shows a maximum near the tankbottom, while for the dispersed oil phase, the maximum lo-cal holdup appears at the top of the tank. It implies that bothsolid and oil phases are not sufficiently dispersed at such im-peller speed. Increasing the impeller speed will significantlypromote the dispersion of the oil phase, as seen from Fig. 4(b).It is also observed that the local holdup of the oil phase be-low the impeller plane is larger than those at the upper section,which can be attributed to the fact that some droplets adhereto the surface of solid particles as a result of better wettabilityto oil, and thus move downwards under gravity. This indicatesthat the introduction of a solid phase will affect the disper-sion characteristics of the oil phase. However, the variation ofimpeller speed has only marginal effect on the suspension ofsolid particles within the speed range employed in the presentstudy.The axial distribution of solid and oil phases is shown in

Fig. 5 for o,av= 0.30 and s,av= 0.10. Experiments show that


00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

s/s,av

z/H

0 40.0

0.2

0.4

0.6

0.8

1.0

z/H

0.2(r/R)

0.3

0.5

0.7

0.9

(a)

1 2 3 4 5 1 2 3 5 6

o/o,av

0 40.0

0.2

0.4

0.6

0.8

1.0

z/H

0.2(r/R)

0.3

0.5

0.7

0.9

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

(b)

1 2 3 5 6 7

o/o,avs/s,av

Fig. 4. Axial profiles of normalized holdup of solid and oil phases at different impeller speed (s,av = 0.10 and o,av = 0.10). Left: solid; right: oil.(a) N = 300 rpm, (b) N = 500 rpm.

the increase of volume fraction of the oil phase yields morehomogeneous distribution of solid particles, especially athigh impeller speeds, presumably because the adherenceof droplets to solid particles gives more buoyancy to solidparticles.An increase in the amount of solid particles tends to en-

hance the homogeneity of the oil phase dispersion, as shownin Fig. 6. The axial distribution of the oil phase becomes moreuniform at s,av = 0.10, compared to the experimental resultsat s,av = 0.05.The variation of phase holdups of two dispersed phases with

influencing factors is more easily shown by the phase holdupcontour maps drawn from the experimental data across a ver-tical section mid-way between two wall baffles with the reso-

lution of 10mm 20mm (Figs. 79, impeller shaft at the leftside is not marked).

5.2. Model prediction of mean flow field and phase distribution

Numerical simulations are conducted to examine the hydro-dynamic characteristics in the liquidliquidsolid three-phasestirred tank under the flow conditions identical to the presentexperiment. The model prediction of the mean velocity fieldsfor the continuous and the dispersed phases are presented inFig. 10 as velocity vector plots corresponding to differ-ent impeller speeds. The phase volume fractions of oil andsolid phases are all 0.10. The well-documented flow pattern


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

s/s,av

s/s,av

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2 (r/R)

0.3

0.5

0.7

0.9

z/H

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

(b)

o/o,av

o/o,av

Fig. 5. Effects of the volume fraction of oil phase on the axial profiles of normalized local holdup of solid and oil phases at different impeller speeds(s,av = 0.10 and o,av = 0.30). Left: solid; right: oil. (a) N = 300 rpm, (b) N = 500 rpm.

generated by a disc turbine in stirred tanks is clearly illustrated,that is, two large ring vortices exist, respectively, above andbelow the impeller plane, and a high-velocity radial impellerstream is also predicted. Overall, the flow fields of three phasesare very similar to each other in most parts of the domain.The velocity field of the oil phase shows a trend of driftingupwards to the top of the tank at lower impeller speed. Thismight be attributed to the fact that the oil phase with lowerdensity accumulates easily in the top section of the tank. Thetime-averaged flow field of the solid phase reveals a smallrecirculation zone above the center of the tank bottom, indi-cating that the solid particles tend to settle down in this zone.

With the increase of impeller speed, the mean velocity flowfields of three phases become more consistent.The calculated normalized local holdups of two dispersed

phases are illustrated using the contour plots as shown inFig. 11. It is easy to observe that the distributions of oil andsolid phases are all less homogeneous at low impeller speeds(N = 300 rpm). The maximum oil phase holdup is locatedin the center of the free surface due to the ring eddy in theupper bulk zone, in qualitative agreement with the experi-mental observations. The distribution of all phases becomesmore uniform at higher impeller speed. The maximum solidconcentration occurs at the center of the tank bottom due to


0.0 0.4 0.8 1.2 1.6 2.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

s/s,av

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

o/o,av(a)

00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

s/s,av

0.0 0.4 0.8 1.2 1.6 2.00.0

0.2

0.4

0.6

0.8

1.0

0.2(r/R)

0.3

0.5

0.7

0.9

z/H

o/o,av(b)

1 2 3 4

Fig. 6. Effects of the volume fraction of solid phase on the axial profiles of solid and oil phases normalized local holdups at different impeller speeds(s,av = 0.05 and o,av = 0.10). Left: solid; right: oil. (a) N = 300 rpm, (b) N = 500 rpm.

the density difference and the ring eddy at the bottom, whichis confirmed by the circulation flow depicted in the velocityvector plot.The numerically computed axial profiles of dispersed oil

and solid phases are compared in Fig. 12 with the presentmeasurements. The simulation results are generally in rea-sonable agreement with the experimental data, especially forhigher impeller speeds. The comparison indicates that thecomputational approach adopted here is suitable for predict-ing the dispersed phase distribution in the liquidliquidsolidthree-phase stirred tank. The model prediction for the solidphase, however, is notably above the experimental data es-pecially at low impeller speeds. With the increase of N , theagreement improves slightly. The discrepancy between theprediction and the experimental results is probably because

the model of inter-phase momentum exchange employed inthe present work is too simple to describe the real com-plex inter-phase interaction coupling in liquidliquidsolidthree-phase flows. Additionally, the isotropic k. two-equation turbulence model is deficient in describing thewell-recognized anisotropic nature of turbulent flow in stirredtanks.Fig. 13 shows the comparison between the simulated axial

profiles of the holdup of two dispersed phases and the experi-mental results obtained in the present work with s,av = 0.05and o,av= 0.10. Also, the numerical prediction agrees reason-ably well with the experimental data. Decreasing the amountof solid particles leads to more accurate quantitative agreementwith the experimental results, as can be seen from Figs. 3 and4. A possible reason may be that the lower volume fraction of


(b)

3.50 2.50

2.00 1.50 1.00 0.50

0.02 0.04 0.06

0.15

0

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1.00

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2.50

3.00

3.50

4.00

4.50

r (m)

z (m

)

0.501.00

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2.002.50

3.003.50

3.504.00

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r (m)

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)

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0.09

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3.30 3.002.70

2.10

1.801.501.20

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00.300.600.901.201.501.802.102.402.703.003.303.50

r (m)

z (m

)

1.00

1.05

0.95

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1.10

1.000.95

1.051.20

1.15

0.00 0.02 0.04 0.06

0.03

0.06

0.09

0.12

0.15

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

r (m)

z (m

)

0.00

Fig. 7. Experimental results of solid and oil phase normalized holdup at different impeller speeds (s,av = 0.10 and o,av = 0.10). Left: solid; right: oil.(a) N = 300 rpm, (b) N = 500 rpm.

4.504.00

3.502.50

2.001.50

1.00 0.50

0.00 0.02 0.04 0.060.00

0.03

0.06

0.09

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00.501.001.502.002.503.003.504.004.505.005.506.00

r (m)

z (m

)

0.95

1.00

1.00

1.00

1.05

0.90

1.05

0.00 0.02 0.04 0.060.00

0.03

0.06

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0.15

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

r [m]

z [m

]

Fig. 8. Experimental results of solid and oil phase normalized holdup at N = 500 rpm (s,av = 0.10 and o,av = 0.30). Left: solid; right: oil.


3.002.101.50

1.80

1.20

0.90

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00.300.600.901.201.501.802.102.402.703.003.303.50

r (m)

z (m

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0.770.84

0.910.98

1.05 0.98

1.12

1.12

1.05

1.05

1.12

0.00 0.02 0.04 0.060.00

0.03

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0.70

0.77

0.84

0.91

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1.12

1.19

1.20

r (m)

z (m

)

Fig. 9. Experimental results of solid and oil phase normalized holdup at N = 500 rpm (s,av = 0.05 and o,av = 0.10). Left: solid; right: oil.

Fig. 10. The velocity vector plots of the continuous, solid and oil phases. Impeller shaft at the left side is not marked. Left: continuous; middle: solid; right:oil. (a) N = 300 rpm, (b) N = 500 rpm.


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r (m)

z (m

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0.65

0.80

0.95

1.10

1.25

0.9

0.95

0.80

0.800.65

1.40

0.00 0.02 0.04 0.060.00

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0.65

0.80

0.95

1.10

1.25

1.40

1.55

1.60

r (m)

z (m

)

(b)

Fig. 11. Model predicted contour plots of normalized holdup of solid and oil phases (s,av = 0.10 and o,av = 0.10). Impeller shaft at the left side not shown.Left: solid; right: oil. (a) N = 300 rpm, (b) N = 500 rpm.

solid phase makes the complete suspension of solid particleseasier and the interaction between solid particles and dropletsbecomes less important.Fig. 14 shows the model predicted holdup axial profiles cor-

responding to the oil and solid phase with o,av = 0.30 ands,av = 0.10. The numerical simulation reproduces quite wellthe general trend of phase distribution of measurements.

6. Conclusions

Experimental measurement and numerical simulation areconducted to investigate the phase distribution of dispersedphases in the liquidliquidsolid three-phase stirred tank. Thesample withdrawal method is employed in the experimentalstudy. It is found that the presence of solid particles has signif-icant influence on the liquidliquid dispersion, and vice versa,the introduction of the lighter oil phase eases the suspension

of solid particles. In addition, the measured local holdup ofthe oil phase at high impeller speed below the impeller planeis larger than that in the upper section of the tank. This isdue to adherence of the oil droplets to the surface of solidparticles as a result of better wetting. Wettability is a uniquefactor of dispersion of the liquidliquidsolid three-phasesystem.A three-fluid model is developed based on the Eulerian

Eulerian approach for numerical simulation of a liquidliquidsolid three-phase flow. The inter-phase momentum exchangebetween continuous and dispersed phases as well as betweentwo dispersed phases is modeled using a unified drag forceexpression. Furthermore, an extension of the standard k. tur-bulence model including the influence of the presence of twodispersed phases is suggested to describe the turbulent effects.The computational model is implemented to predict the three-dimensional turbulent flow field in the three-phase stirred tankas well as the phase holdup distribution of two dispersed phases.


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

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z/H

SimulationExperiment.

r/R=0.5

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SimulationExperiment.

r/R=0.9

z/H

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Experiment.r/R=0.5

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s/s,av

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0.2

0.4

0.6

0.8

1.0

z/H


r/R=0.9

o/o,av

Fig. 12. Comparison of simulated normalized holdup profiles of oil and solid phases with the experimental data for different impeller speeds. (s,av = 0.10and o,av = 0.10). Left: solid; right: oil. (a) N = 300 rpm, (b) N = 500 rpm.

In general, the numerical simulation is capable of reproducingthe trend of the axial holdup profile of the oil and solid phasesand of giving an approximation of the phase holdup valuescomparable to the experimental results.Further improvement of the present model is necessary

so as to obtain quantitative agreement with measurements,

including refining the interaction model for inter-phase mo-mentum exchange and incorporating the breakup and coales-cence of droplets. A well-grounded anisotropic multiphaseturbulence model for three-phase flow is also desired fora more accurate description of the turbulent flow in stirredtanks.


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

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0.2

0.4

0.6

0.8

1.0

z/H


r/R=0.9

r/R=0.5

Fig. 13. Comparison of simulated normalized holdup profiles of solid and oil phases with the experimental data at s,av=0.05 and o,av=0.10 (N=500 rpm).Left: solid; right: oil.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

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s/s,av o/o,av

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r/R=0.5

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r/R=0.5

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r/R=0.9

z/H

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0


r/R=0.9

z/H

Fig. 14. Comparison of simulated normalized holdup profiles of solid and oil phases with the experimental data at s,av=0.10 and o,av=0.30 (N=500 rpm).Left: solid; right: oil.


Notation

a coefficient in algebraic equationb source term in algebraic equationB width of baffle, mC clearance of impeller plane to bottom of

stirred tank, mC1, C2, C constants in k. modelCb constant in extra turbulent kinetic energy gen-

eration termCD drag coefficientd diameter of droplet, m

diameter of disc of impeller, mD diameter of impeller, mDk mass imbalance term in continuity equation,

kg s1fk turbulent diffusive term in continuity equa-

tion, kg s1F interphase force, NFdrag drag force, NFcen centrifugal force, NFcor Coriolis force, Ng gravity acceleration, m s2G turbulent generation term, kgm1 s3Ge extra turbulent generation term, kgm1 s3H height of liquid in stirred tank, mk turbulent kinetic energy, m2 s2K proportional factorN impeller rotation speed, rpmp pressure correction, PaP pressure, PaP fluctuating correction, Par radial coordinater radial vectorR radius of stirred tank, mRf Richardson number defined by Eq. (14)R() relative residualRed particle Reynolds number (Red = cdd|ud

uc|/c)S source termt time, st1 mean eddy lifetime, std particle response time, sT diameter of stirred tank, mu mean velocity component, m s1u fluctuation of velocity component, m s1u velocity vector, m s1w height of impeller blade, mz axial coordinate starting from the tank bottom

Greek letters

holdup

diffusion coefficient, Pa sij Kronecker delta turbulent energy dissipation rate, m2 s1

azimuthal coordinate, rad viscosity, Pa s kinematic viscosity, m2 s1 density, kgm3k , constants in k. modelt Schmidt number shear stress, Pa general variable angular speed, rad s1

Subscripts

av averagedc continuous phased disperseddrag drag forceeff effectiveg gasi, j coordinate axesk kth phaselam laminarnb neighboring nodeo oil phasep,E,W, S, center, east, west, south, north, top andN,T,B

bottom neighboring nodes of a cellr, , z radial, azimuthal and axial directions solid phaset turbulence

Acknowledgments

Financial support from the National Natural Science Founda-tion of China (Nos. 20006015, 20236050) and the National Ba-sic Research Priorities Program (No. 2004CB217604) is grate-fully acknowledged.

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Measurement of phase holdups in liquid--liquid--solid three-phase stirred tanks and CFD simulationIntroductionExperimentsMathematical modelNumerical procedureResults of measurement and modelingExperimental results of phase distributionModel prediction of mean flow field and phase distribution

ConclusionsAcknowledgmentsReferences

Three Phase Holdup

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