3D hydrodynamics involving multiple eccentric impellers in unbaffled cylindrical tank

Houari Ameur

Institut des Sciences et Technologies,Centre Universitaire Salhi Ahmed CUN-SA,BP 66,Naama 45000,Algeria

1.Introduction

Stirred vessels are commonly used in many industrial processes as well as in laboratory study.They are usually equipped with baffles in order to promote small scale mixing by suppressing the primary vortex.However,there are applications where unbaffled vessels are preferred,as for instance in food and pharmaceutical industry where cleaning is a major issue[1],or as in the crystallization because baffles can damage growing particles,or as in the laminar regime where baffles can cause the formation of dead regions.

Lamberto et al.[2]and Alvarez et al.[3]pointed out that the unbaffled stirred vessels were not optimized with a large waste of power,because of the presence of large poor mixing regions with a typical toroidal shape,located both above and below the impeller plane.Therefore,two different strategies have been suggested in order to improve mixing in unbaffled vessel in the laminar regime:the use of variable agitating speed[2]and the eccentric stirrer shaft[3].

Up to now,only few studies on the mixing with eccentrically located impellers have been reported.For the laminar regime,Alvarez et al.[3]have experimentally investigated effects of the shaft position on the flow fields.They found important changes in the flow structure and great enhancement in the mixing efficiency even at a low eccentricity.

Other studies[4,5]showed the impeller eccentricity effect on the heat transfer process in a vessel stirred by axial flow impellers.Karcz et al.[6]analyzed the mixing time,power consumption and momentum distribution in a tank equipped with an eccentrically located propeller or a HE3 impeller.These authors showed a decrease in the mixing time with the increase in impeller eccentricity.

Using PIV,Hall et al.[7,8]studied the hydrodynamic of gas-liquid in a vessel stirred by a pitched blade turbine.Ascanio and Tanguy[9]determined experimentally the mixing time for inelastic shear-thinning fluids in stirred tanks by a combination of two off-centered impellers(Rushton and pitched blade turbines).They found that mixed flow impellers are less efficient in terms of mixing time than radial flow impellers.

Other researchers[10-12]interested to the hydrodynamicsinvolved by a Rushton turbine.Galleti et al.[13]found that high flow instabilities are generated by the vortex formed near the free surface of liquid.The characteristic frequency of such instabilities decreases with the increasing eccentricity or with increasing impeller blade thickness.

With three different impellers(Rushton turbine,A315 impeller and pitched blade turbine),Karcz and Cudak[14]studied the impeller eccentricity effect on the heat transfer.The radial flow turbines were found to be more efficient than the axial flow impellers.For radial flow impellers(Rushton and A315 turbines),Cudak and Karcz[15]found that the heat transfer rate is higher by 40%comparing eccentrically(e/R=0.53)and centrically located agitators.

Within the transitional regime,Szoplik and Karcz[16-19]studied experimentally the mixing of viscous Newtonian and non-Newtonian fluids by a propeller.They found that the mixing time decreases with the increase of the propeller eccentricity.For the same type of impellers,Cudak and Karcz[20]reported that the axial and angular distributions of effective viscosity on the vessel walls increase with the increase of impeller eccentricity.

Fig.1.Example of the stirred systems simulated.

The multiple impeller systems present many advantages such as a lower decrease of heat exchange area in the scale-up treatment,a longer residence time and lower power consumption per impeller as compared to single impeller systems.These systems are encountered in the biochemical industry,since they provide usually higher mass transfer coefficient in comparison to single impeller vessels[21].

A thorough search in the literature suggests that a very limited effort has been devoted to the study of viscoplastic fluid flows in a vessel stirred by multiple impellers with an eccentric shaft.The purpose of the present work is to investigate the 3D hydrodynamics in a cylindrical tank stirred by multiple eccentric impellers for viscoplastic fluids.The impeller used has six-curved blades(Scaba 6SRGT).We note that this configuration has not been studied previously.We focus on the effects of stirring rate, fluid rheology,impeller number and its clearance from the tank bottom.

2.Stirred System

Simulations were performed in a vessel with diameter D=0.3 m,filled with the liquid up to the height(H)equal to the vessel diameter(Fig.1).The vessel has a flat bottomed cylindrical shape.The impeller consists of six curved blades fixed on a disc with 8 mm of thickness(dt),which is attached on a cylindrical eccentric shaft of diameter ds/D=0.05.The impeller blade diameter(from tip to tip)is d/D=0.25.Each impeller is located at an eccentric position e/D=0.25.

The effect of impeller clearance(c/D)from the tank bottom is investigated by realizing six geometries.Table 1 gives the values of c/D for these cases.

3.Theoretical Background

Aqueous solutions of xanthan gum used in this study have a viscoplastic behavior modeled by the Herschel-Bulkley model[22].Thus,its apparent viscosity(η)is given by

Table 1 Values of the impeller clearance(c/D)for all geometries studied

whereis the yield stress,is the shear rate,and K and n are the consistency index and the flow behavior index,respectively.

Table 2 summarizes the rheological properties of the different xanthan gum solutions used in this work,which are based on measurements conducted by Galindo and Nienow[23].

Table 2 Rheological properties of xanthan gum solutions

The Metzner-Otto correlation[24]was employed to calculate the modified Reynolds number for the Herschel-Bulkley fluids:

4.Numerical Method

To perform the simulations,the commercially available computer code(CFX 13.0)developed by AEA Technology,UK,was used.In the following part,only the main points about the mixing system modeling options and numerical procedures will be given.Further details are found in a previous paper[26].

CFX is a computer code based on the finite volume method to solve the equations of momentum.The Navier-Stokes equations are solved in a rotating,cylindrical frame of references.Because of the choice of a rotating frame,two terms are added to the equations:centrifugal and Coriolis accelerations.The equations are written in terms of pressure and velocity components.A pressure-correction method of the type Semi-Implicit Method for Pressure-Linked Equations-Consistent(SIMPLEC)is used to perform pressure-velocity coupling.

A pre-processor(ICEM CFD 13.0)was used to discretize the flow domain with a tetrahedral mesh.Mesh tests were performed by increasing the number of cells by a factor of about2 as used by other researchers in CFD simulation of the agitation systems[27,28].When the number of cells was increased from 112243 cells to 224486 cells,the power number and the velocity magnitude at the blade tip changed by more than 3%.When the number of cells was changed from 224486 cells to 448972 cells,the corresponding changes in the power number and the velocity magnitude were about 2.5%.Therefore,the mesh with 224486 cells was used for subsequent computation.

Simulations were considered converged when the scaled residuals for each transport equation were below 10-7.Most simulations required between 100 and 200 iterations for convergence.Computations were carried out using a 2.20 GHz Pentium(R)i7 Core CPU with 8.0 GB of RAM and convergence was typically achieved after 4-5 h.

5.Results and Discussion

5.1.Validation of the numerical results

Before our main investigation,it has been necessary to validate the CFD model.For this purpose,we referred to the experimental work of Pakzad et al.[29].

Pakzad and her co-authors used one Scaba-6SRGT impeller for mixing yield stress fluids.In the present study,we are interested in the same impeller for agitating the same fluids,but we investigate the effects of multiple Scaba-6SRGT impellers.

Fig.2 presents the variation of the radial velocity along the vesselradius for an angular position θ=-45°(see Fig.1).As is observed,the comparison between our numerical results and the experimental data given in[29]shows a satisfactory agreement.

Fig.2.Radial velocity for Re y=21.5,No.1 solution,Z*=0.21,θ=-45°.

5.2.Effect of Reynolds number

The viscosity of non-Newtonian fluids depends on the shear rate or on the shear stress.For the laminar and transitional regimes of fluid flows,the homogenization of non-Newtonian liquids is slower than that of Newtonian fluids[30,31],as the shear rates existing in an agitated vessel cause strong viscosity differences.

In this work,shear thinning fluids with yield stress are used for a range of Reynolds number covering the laminar and transitional regimes.For the geometrical configuration named case 1(i.e.two eccentric impellers placed at c/D=0.5),the effect of Reynolds number has been investigated.From the vessel center until the vertical wall and for an angular position θ=90°,variations of the tangential velocity are followed for different values of Rey(Fig.3).As remarked,the tangential velocity increases continuously until a certain maximum and then deceases until becoming negligible at the vessel wall.Higher values of the velocity are present in the area swept by the impeller.Another remark is that,the increase of the impeller rotational speed gives higher shear rates and a wide cavern.

The streamlines for different Reyare presented for both vertical and horizontal planes(Fig.4).As showed,the flow generated by the Scaba 6SRGT impeller is radial(Fig.4a),the flow impinging from the blades goes horizontally towards the vertical wall of the vessel,then it will be divided in two streams,one towards the vessel base and the other to the free surface of liquid.

Fig.3.Tangential velocity for No.2 solution,Z*=0.5,θ=90°(the values of Reynolds number Re y=10,200,600,1200 and 2000 correspond to N=39,207,375,549 and 728 r·min-1,respectively).

At a low Reynolds number,four vortices appeared in the space between impellers because of the interaction of radial flows impinging from the two agitators.The two small vortices located at each blade tip are formed because of the blade shape(the curvature).These vortices can be eliminated by increasing Rey.

The formation of vortices in the space between the blade and the vessel sidewall is due to the wall effects,since there is a little distance between the impeller blade tip and the vertical wall of the tank.Even at low Reynolds number,these vortices are present at the blade near the wall.However,no vortices are formed behind other blades,even at the highest Rey.

Another remark is,for a higher impeller rotational speed,the two vortices formed at the external part of the blade(i.e.the space between the blade and the vessel sidewall)can be detached and reduced in size.The axial circulation is then enhanced.

At the middle high of the vessel,the streamlines presented on horizontal planes give other insight on the flow structures generated(Fig.4b).The insufficient stirring rates permit the formation of dead zones between the impeller and the vessel wall,because of the wall effect.This problem is solved by increasing Rey.

Another point that should be explained,is the structures generated between the two agitators.Since the Scaba has curved blades,the increasing Reyyields high interactions in this space.These toroidal regions are present just in this area,i.e.the axial circulation is better and the agitation is ensured until the free surface of liquid for great Rey(Fig.5).

Fig.5.Axial velocity for No.2 solution,R*=0(at the centerline).

5.3.Effect of fluid rheology

The fluid rheology is another parameter which can affect strongly the flow fields.The testis performed with two cases of xanthan gumconcentrations(Table 2).The flow fields generated for both cases are presented on Fig.6.The streamlines given on Fig.6a show that the vortices formed at the blade tip can be detached and reduced in size with the increase of the flow behavior index n.On the other hand,the well-stirred region will be enlarged(Fig.6b),and this is due to the viscous forces.

Fig.4.Flow fields for No.2 solution.

Fig.6.Flow fields induced for N=250 r·min-1,a)streamlines,b)cavern size.

The profiles plotted on Fig.7 give another illustration of this phenomenon,where variations of the axial velocity are followed along the vessel height.The minus sign of the velocity indicates the existence of recirculation loops.As shown,the size of these loops increases with respect to n.

Fig.7.Axial velocity for N=250 r·min-1,R*=0.

5.4.Effect of impeller number

By the following,we try to examine the effect of the mixing system design.At a first step,the impeller number effects are investigated with the help of four geometrical configurations with the same offbottom clearance(c/D=0.5).For a location below the impeller(Z*=0.45)and along the vessel radius(Fig.8),the tangential velocity is plotted for the four cases studied.As illustrated,the increase of the impeller number can agitate the motion of fluid particles,but the difference between α=3 and 4 is slight.

Fig.8.Tangential velocity for No.2 solution,Re y=20,c/D=0.5,θ=0°,Z*=0.45(below the impeller).

As for the flow structures generated,Fig.9 indicates that due to the vessel wall effect and the lower Reynolds number,a dead zone appeared between the impeller and the vessel sidewall.With the same Rey(Rey=30),these vortices can be eliminated by increasing α.

On the other hand,when using just one eccentrically located impeller,another vortex appears behind the blade due to the impeller eccentricity.Even the rotational speed is increased(Rey=2000),this vortex is still present.For α=2 and Rey=2000,two vortices appeared in the space between the two impellers because of the interaction of radial flows induced.These structures can be eliminated with three or four impellers,the fluid circulation is then enhanced and the free surface of liquid can be reached(Fig.10).

Fig.10.Axial velocity for No.2 solution,Re y=30,R*=0(at the centerline),c/D=0.5.

The well-agitated region size is plotted on Fig.11 for the four cases.As clearly illustrated,the increasing impeller number enlarges the well stirred region and ensures mixing in the whole vessel volume,but the power required is greater(Fig.12).

Fig.12.Power number for No.2 solution,Re y=20,c/D=0.5.

From these results in Figs.8 and 12,the configuration which gives the best performance is the 3rd case,since the difference between the 3rd and 4th cases is slight(in terms of well-stirred region and power consumption).

5.5.Effects of impeller clearance

Effects of the impeller clearance from the tank bottom are investigated using four cases as shown in Fig.13.

Fig.11.Well-stirred region size for different stirring systems,No.2 solution,c/D=0.5.

Fig.13.Flow fields for Re y=2000,No.2 solution.

For case 1 where two impellers are mounted at the middle height of the vessel on the same level,the lower part of the vessel is not well agitated.This is due to the radial flows induced from the curved blade.We have the same problem with case 2.When an impeller is located at the middle height of the vessel and the other impeller is placed very close to the vessel base(case 3),that yield spoor mixing at the upper part of the vessel for this kind of impellers in this range of Reynolds number.Thus,to ensure mixing in the whole vessel volume,it is recommended to place one impeller at the lower part of the vessel and the second at the upper part as in case 4.

Concerning the power number(Table 3),case 2 requires the least power consumption.In case 1,the interactions of fluid motions are very intense.Case 3 requires more power consumption than case 1 because of the vessel base effect.From these results,we can judge case 4 as the best configuration.

Table 3 Power number for Re y=20,No.2 solution

Fig.14.Stirred system with 4 impellers.

5.6.Another mixing system design

From the previous results,using two impellers with the size of d/D=0.25 is found inefficient.Thus,the use of four impellers with different locations is suggested as case 6 in Fig.14 to ensure mixing in the whole vessel volume(Figs.15 and 16).The power required for case 6 is slightly greater than case 5 because of the wall effect(Fig.17).

6.Conclusions

The 3D flow fields generated in a vessel stirred by Scaba 6SRGT impellers with viscoplastic fluids have been investigated.The key

findings and conclusions based on this work are listed as follows:

·Using one impeller with a small size(d/D=0.25)at an eccentric position is found inefficient.A dead zone was formed behind the blade near the vessel centerline,and another one near the cylindrical vessel wall.Thus,to solve this problem,it is necessary to increase the stirring rates and to add impellers with the same eccentricity.

·Just two agitators placed at the same level were found also to be less efficient because smaller well-stirred region was obtained.

Fig.15.Axial velocity for Re y=20,No.2 solution,R*=0(at the centerline).

Fig.16.Flow fields for Re y=3000,No.2 solution.

Fig.17.Power number for No.2 solution.

·The impeller clearance from the tank bottom plays also an important role.For multiple impeller systems,the agitators should be placed equally spaced at the middle height,near the vessel base and the free surface of liquid in order to ensure mixing in the whole vessel volume and to reduce the power consumption.

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