Yuyun Bao,Bingjie Wang,Mingli Lin,Zhengming Gao,Jie Yang*

State Key Laboratory of Chemical Resource Engineering,School of Chemical Engineering,Beijing University of Chemical Technology,Beijing 100029,China

Keywords:Gas holdup Mixing Multiphase reactors Relative power demand CFD Multi-impeller stirred tank

ABSTRACT The impeller configuration with a six parabolic blade disk turbine below two down-pumping hydrofoil propellers,identified as PDT+2CBY,was used in this study.The effect of the impeller diameter D,ranging from 0.30T to 0.40T(T as the tank diameter),on gas dispersion in a stirred tank of 0.48 m diameter was investigated by experimental and CFD simulation methods.Power consumption and total gas holdup were measured for the same impeller configuration PDT+2CBY with four different D/T.Results show that with D/T increases from 0.30 to 0.40,the relative power demand(RPD)in a gas–liquid system decreases slightly.At low super ficial gas velocity V S of 0.0078 m·s-1,the gas holdup increases evidently with the increase of D/T.However,at high super ficial gas velocity,the system with D/T=0.33 gets a good balance between the gas recirculation and liquid shearing rate,which resulted in the highest gas holdup among four different D/T.CFD simulation based on the two- fluid model along with the Population Balance Model(PBM)was used to investigate the effect of impeller diameter on the gas dispersion.The power consumption and total gas holdup predicted by CFD simulation were in reasonable agreement with the experimental data.

1.Introduction

Gas–liquid dispersion in mechanically agitated vessels is a common operation used in many industrialprocesses,such as chemicalengineering,mineralprocessing,and wastewater treatment,because itoffers unmatched flexibility and control to tailor the fluid dynamics.During the last two decades,more and more researchers have devoted to the study of gas dispersion in stirred tanks[1–3].In industrial applications,the increase of the reactor scale causes that the height-diameter ratio of vessels could be as large as 2 or 3,which requires multi-impeller configuration for mixing and gas dispersion.Thus,more and more studies were carried out in systems with multiple impellers in recent years[4–9].Moreover,the optimum design of multi-impeller agitation becomes more and more important for the large scale industrial reactors.

Because of the interactions between fluid and impellers,the flow pattern caused by an impellercan be significantly in fluenced by another one in a multi-impeller stirred tank.These interactions are greatly affected by the parameters like impeller diameter and spacing[10],leading to different gas dispersion performances in stirred tanks.In large scale industrial gas–liquid reactors,the manufacturing and operational costs are closely related to the optimal design of the impeller,especially the diameter of impellers.Given the same power input into a reactor,the impellers with smaller diameter have higher rotational speed and less torque compared with the impellers of larger diameter.As a result,the sizes of the gearing box,mechanicalseals,and impeller shaft are determined after the impeller diameter is specified,which becomes very important in large scale reactors with volume up to 800 m3.Thus,more research is needed to obtain the relationship between gas holdup and agitator geometrical parameters for the purpose of optimized design of agitators.

During the last two decades,computational fluid dynamics(CFD)techniques have been used to simulate gas–liquid flows in agitated tanks[11–14].Due to the lack of experimental data,numerical investigation on multiple-impeller systems is less popular than that on single-impeller tanks[15,16].Moreover,CFD technique can also reveal the detailed information that cannot be easily obtained from experiments,especially the information near the impeller discharging region.Therefore,the application of CFD has obvious advantages in the investigation of gas–liquid flow in stirred tanks.

In the present study,a turbine-propeller combination consisted of a parabolic blade disk turbine below two down-pumping hydrofoil propellers,identified as PDT+2CBY,was used to investigate the effect of D/T(the ratio of impeller diameter to tank diameter)on power consumption and total gas holdup.Furthermore,the CFD simulation was also used to predict the gas dispersion properties in the multiimpeller stirred tank.

2.Experimental

All the experiments were carried out in a stainless steel dishedbottom cylindrical tank with internal diameter T=0.48 m and a filled aspect ratio H/T=1.66,as sketched in Fig.1.Four baf fles each 0.045 m wide were mounted 0.005 m away from the wall.The impeller configuration with a six parabolic blade disk turbine below two downpumping hydrofoil propeller,identified as PDT+2CBY(Fig.2)was used in this study.Impellers with diameters of 0.30T,0.33T,0.37T,and 0.40T were used.The structural parameters of CBY impeller are shown in Table 1.The distance between neighbor impellers was keptconstantly at 0.48T.The clearance between the lowest impeller and the base of the tank was 0.33T.A ring sparger of 0.8D was located 0.25T above the tank bottom and with 27 holes whose diameter was 2 mm.

Fig.1.Schematic of the experimental setup.

Fig.2.Impellers.

Air and deionized water were used as the gas and liquid phase in all experiments.Airpassed through three stage filtersbefore being sparged into the tank in order to get rid of the impurities in gas.The total gas rates ranged from 5 to 59 m3·h-1,and the corresponding super ficialvelocities VSwere from 0.0078 to 0.092 m·s-1.The liquid bulk temperature was kept at 24°C.

Table 1 Structural parameters of CBY impeller

The power consumption was calculated from the torque and rotational speed of the shaft measured with a torque transmitter and a portable tachometer,respectively.The gasholdup wascalculated fromthe changes in liquid levelmeasured by a calibrated radar probe(Krohne Re flex-Radar BM100A,Germany).The total gas holdup is defined as

where Hg,H0and H′are respectively the liquid level of air-sparged tank,static liquid level and elliptic tank bottom correction coefficient which is equal to 0.04 m.

3.Computational Model

In an aerated stirred tank,the Eulerian–Eulerian approach is a reasonable choice for its ability to handle the system with large and low volume fractions with acceptable accuracy.The standard k-ε model was applied with its standard constants to simulate the continuous liquid-phase turbulence.For the dispersed gas phase,the zero equation model was used with the Sato enhanced eddy viscosity model[17]which included the in fluence of large bubbles on the liquid-phase turbulence.

The total interfacial force between the two phases may arise from several independent physical effects,such as the interphase drag,lift,wall lubrication,virtual mass forces.According to the analysis of Khopkar et al.[18]some forces such as lift force and virtual mass force can be neglected in the case of stirred vessel.So we only considered the drag and turbulence dispersion forces.The modified Grace model[19],which used a simple power law correction to amend the single bubble Grace drag coefficient for high bubble volume fractions,was used to calculate the interphase drag force.The model of Lopez de Bertodano[20]was used to compute the turbulent dispersion force.

According to the analysis of Min et al.[16]the single average bubble diameter(SABD)approach could not correctly predictthe gas void fraction whether for the high-shear-rate region near the lowest impeller or the low-shear-rate region near the free surface.In order to predict the performance of gas–liquid stirred tank accurately,the distribution of bubble size should be considered.The population balance(PB)equations could be used to simulate the non-uniform bubble size distribution in a stirred vessel.In this paper,multiple size group(MUSIG)Model with PB model in CFX were used to handle poly-dispersed multiphase flows.The MUSIG model was a framework in which PBs were solved simultaneously with the Navier–Stokes equations for Eulerian gas and liquid phases.Population balances including breakup and coalescence effects provided a well-established method to calculate the changing size distribution of a poly-dispersed phase.The MUSIG model assumed that all bubble size groups shared a common velocity field but with different slip velocities for different size bubbles.The drag force on bubbles was calculated based on the local Sauter mean bubble diameter.

The general form of the population balance equation is

where BB,DB,BC,and DC,respectively,represent the birth rate due to breakup of larger particles,the death rate due to breakup into two or more smaller particles,the birth rate due to coalescence of smaller particles,and the death rate due to coalescence with other particles.Bubbles from 1 to 10 mm in diameter were divided into 10 classes.The diameter of size group i is calculated from

The break-up model,which was based on the theory of isotropic turbulence and probability and assumed binary breakup,was taken from Luo and Svendsen[21].The breakup kernel is modeled as follows:

where g(Vi；fBVVi)is the breakup rate per unit volume of the continuous phase(m-3·s-1)of a parent bubble with volume Viinto a daughter bubble with volume(fBVVi)(where fBVis the volume fraction of the parent bubble that constitutes the volume of one daughter bubble)and ξ is the dimensionless size of eddies in the inertial sub-range of isotropic turbulence.In addition,εcis the continuous-phase eddy dissipation rate,σ is the interfacial tension,and FBis a calibration coefficient,and β=2.

The Prince and Blanch model[22]was used as the coalescence model.This model describes the coalescence process as occurring in three steps: first,the bubbles collide and trap a layer of liquid between them；second,this liquid layer drains untilitreaches a criticalthickness；and third,this liquid film disappears and the bubbles coalesce.The collisions between bubbles may be caused by turbulence,buoyancy,or laminar shear.The coalescence kernel is,therefore,modeled by the collision rate of two bubbles with a collision efficiency related to the time required for coalescence:

The collision efficiency is modeled by comparing the time required for coalescence tijwith the actual contact time during the collision τij:

where h0is the initial film thickness,hfis the critical film thickness when rupture occurs,and rijis the equivalent radius:

The turbulent contributions to collision frequency are modeled as

where Sijis the cross-sectional area of the colliding particles and utiis the turbulent velocity.

The buoyancy contribution to collision frequency is modeled as

The shear contribution to collision frequency is currently neglected.

4.Numerical Methods

The set of equations were solved numerically using the commercial code CFX.The Multiple Frames of Reference(MFR)method was used to simulate the impeller rotation in the stirred tank.After grid independence test as shown in Table 2,the computational cells were about 1.6 million；tetrahedrons and mixed elements were chosen for the entire calculated region.Several criteria,such as reduction of the residuals,gas mass flow rate through various horizontal planes,variation of overallgasholdup and energy dissipation rates,were used in orderto ensure adequate convergence.The convergence criteria were all set at 10-4.A scalable wallfunction was used to specify the wallboundary conditions.The boundary condition was free slip wall for the gas phase and no slip wallforthe liquid phase.The “opening”boundary condition for the tank outletallowed the fluid to cross the boundary surface in eitherdirection.We set the value of the interfacial tension between the two phases to 0.073 N·m-1.A step function was used to set the initial water level.

Table 2 Grid independence test result

5.Results and Discussion

5.1.Flow field

The predicted vectorplots forthe liquid flow field are shown in Fig.3 for D/T=0.30,0.33,0.37 and 0.40,respectively.The flow patterns are similar at all different D/T.The bottom radial- flow impeller PDT pumps the fluid to the wall,and the jetting fluid returns over the elliptical bottom towards the impeller.Two large-scale axial circulations are shown in the centralregion ofthe tank,caused by the combined effectof the two down-pumping axialimpellers CBY.Harvey etal.[10]suggested that a change in impeller size in multiple impeller configurations could lead to variations in flow interactions between multiple impellers,greatly affecting the fluid flow in the vessel.As D/T increases from 0.30,0.33,0.37 to 0.40,the corresponding fluid circulation flow rates for the axial impeller CBY are 0.012,0.146,0.184 and 0.250 m3·s-1,respectively.It should be noticed that the results were obtained for constant agitation speed so that the power consumption increases with the increase of D/T.

Fig.3.Liquid velocity vectors(ungassed,N=9 s-1).

5.2.Relative power demand

5.2.1.Effect of impeller diameter

As we know,the introduced gas causes the gassed power consumption decreases compared with the ungassed power input.Define the relative power demand(RPD,η)as the ratio of the gassed power input to the ungassed one.Fig.4 shows the measured RPD curves for impellers with different D/T.In all impeller configurations,the RPD decreases with the increase ofeither gas flow number(Qg/ND3)or Froude number(N2D/g).Both the cavities behind the impeller blade and the gas recirculation can affect the power consumption in the aerated stirred tank.After the gas is introduced into the tank,it is dispersed by mainly the bottom impeller and sent to the other regions.Part of the gas may flow with the liquid and re-enterthe impellerregion instead ofescaping from the liquid surface.As the impeller diameter increases,the blockage area of the impeller increases,forcing more gas to re-circulate to the impeller region and leading to the decrease of power consumption at a given gas flow rate and agitation speed.

Fig.4.In fluence of D/T on RPD.

5.2.2.Correlations for RPD

Because the gas flow number,Froude number and the impeller diameter are three important factors to in fluence on the power consumption,we use Eq.(16)to correlate the experiment data of η:

The resulted regressive correlation is

The negative exponent of D/T reflects the reduced aerated agitation power at larger impeller diameter as discussed above.Moreover,the absolute exponent of D/T is larger than that for gas flow number and Froude number,which means that D/T has greater effect on RPD.Fig.5 shows the comparison between the measured RPD and the correlation.

Fig.5.Comparison between the correlated RPD and experimental data.

The reasonable agreement between Eq.(17)and the experimental data could be got with deviation within±5%in our experimental range(0.015＜Flg＜0.721,1.01＜Fr＜2.31,and 0.30≤D/T≤0.40).

5.3.Gas holdup

5.3.1.Effect of impeller diameter

The gas holdup in stirred tanks is an important factor for the design and scale-up of gas–liquid stirred-tank reactors.For its strong in fluence on the mass transfer,gas hold-up has been a very important research subject in an aerated stirred tank.

Fig.6 shows the in fluence of impeller diameter on the gas holdup.When the power input and super ficial gas velocity VSincrease,the total gas holdup increases accordingly.However,the effects of D/T varies with VS.As shown in Fig.6(a),when the super ficial gas velocity is low(VS=0.0078 m·s-1),an increase of D/T leads to an obvious increase of total gas holdup at a given power input.The total gas holdup of system with D/T=0.40 is obviously higher than others which are almost the same at given power inputs.However,when the super ficial gas velocity becomes higheras VS=0.025 m·s-1,the totalgas holdups in all systems are almost the same.When the super ficial gas velocity reaches the high level as VS=0.05 and 0.092 m·s-1,as shown in Fig.6(b),the total gas holdup of the system with D/T=0.33 becomes higher than other D/T.It should be noticed that all experiments were carried out above the complete dispersion impeller speed corresponding to specific gas rate.

Fig.6.In fluence of impeller diameter on total gas holdup.

In the stirred tank,the residence time ofbubbles is a dominantfactor to the gasholdup.The gas recirculation in the tank can enhance the bubble residence time,leading to a higher gas holdup.Atlow super ficialgas velocity VS=0.0078 m·s-1,the gas recirculation in fluences the gas holdup directly.The amount of gas in recirculation increases with the increasing impeller diameter because of the larger blockage area of a bigger impeller,leading to the increases of gas holdup.However,with the increase of VS,the gas recirculation effect becomes less dominant as before.For a given power input,the smaller impeller combination can have higher agitation speed and higher shear rate,leading to more chances to break up bubbles.Fig.6 shows that the system with smaller impeller diameter has higher gas holdup athigh super ficial gas velocity.However,as also shown in Fig.6,the effect of the impeller diameter on the gas holdup is notmonotonic.Athigh super ficialgas velocity,the system with D/T=0.33 gets a reasonable balance between the gas recirculation and shear rate,leading to the highest gas holdup eventually.Li et al.[23,24]also concluded that the impeller with D/T=0.33 could have a better micromixing performance than either D/T=0.25 or D/T=0.50 in an aerated stirred tank.The effect of impeller diameter on the micromixing performance in gas–liquid stirred tanks was also non-monotonic.

5.3.2.Correlations for gas holdup

The power input and super ficial gas velocity are the important factors in all cases with different impeller diameters.The gas holdup could be correlated by

The results for different D/T are listed in Table 3.

Table 3 Regression results of total gas holdup based on Eq.(18)

In order to show the effect of D/T on the gas holdup,Eq.(18)is extended to

and the resulted quantitative equation is

Fig.7 showsthe relationship ofgas holdup between the experiments and the correlated data.

It should be noted that Eq.(20)is suggested to be used in the range of 0.30≤D/T≤0.40,0.20≤Pm≤4.52 W·kg-1and 0.0078≤VS≤0.092 m·s-1.

5.4.Comparisons of simulations with experiments

5.4.1.Gas holdup

Fig.7.Comparison of the correlated gas holdups by Eq.(20)with experimental data.

Table 4 Simulation conditions(V S=0.0078 m·s-1)

Table 5 Total gas holdup comparison(V s=0.0078 m·s-1,N=9 s-1)

Table 4 shows the simulation conditions and the results are compared with experimental results in Fig.8.The predicted gas holdups in all cases are in reasonable agreement with the experimental data.The quantitative comparisons of the experimental and simulation gas holdups at N of 9 s-1and VSof 0.0078 m·s-1are shown in Table 5.A reasonable agreement can be seen between the experimental and simulated results,indicating that at low VS,a satis fied prediction of gas holdup can be obtained by CFD method used in this paper.However,when the super ficial gas velocity is high,it is not easy for the simulation to converge,and the break-up and coalescence modelneed to be further researched to satisfy this condition.So the simulation of the cases with high aeration conditions is still open for our future investigation.

6.Conclusions

The effect of impeller diameter on gas–liquid flow in multiphase stirred tank was investigated by experimental and computational methods in a stirred vessel of 0.48 m diameter,equipped with a multiimpeller configuration named PDT+2CBY.The following conclusions could be made,

(1)As the ratio of impeller diameter to tank diameter D/T increases from 0.30 to 0.40,the relative power demand RPD in a gas–liquid system decreases.Satisfactory predictions of RPD could be obtained as

with 0.015＜Flg＜0.721,1.01＜Fr＜2.31 and 0.30≤D/T≤0.40.

(2)At low super ficial gas velocity VS=0.0078 m·s-1,the total gas holdup increases with the increases of D/T.However,at high VSof 0.05 or 0.092 m·s-1,the system with D/T=0.33 obtains a good balance between the gas recirculation and shear rate level,leading to the highest gas holdup among four different D/T.Satisfactory predictions of gas holdup can be obtained as

with 0.30≤D/T≤0.40,0.20≤Pm≤4.52 W·kg-1and 0.0078≤VS≤0.092 m·s-1.

(3)A CFD computational model was established to study the gas–liquid flows generated by a multi-impeller configuration with different D/T.The model can help us understand the effect of the impeller size on the gas–liquid flows.The predicted gas holdups at VSof 0.0078 m·s-1are in reasonable agreement with the experimental data.

Nomenclature

D diameter of impeller,m

Flgflow number(=Qg/ND3)

Fr Froude number(=N2D/g)

H height of liquid in tank(general),m

N agitation speed,s-1

Pmmean total specific energy dissipation rate,W·kg-1

Qginlet gas flow rate,m3·s-1

R radius of the tank,m

T diameter of the tank,m

VSsuper ficial gas velocity,m·s-1

ε total gas holdup