Huina Wang,Xiaoxia Duan,*,Xin Feng,Zai-Sha Mao,Chao Yang,*

1 CAS Key Laboratory of Green Process and Engineering,Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China

2 School of Chemical Engineering,University of Chinese Academy of Sciences,Beijing 100049,China

Keywords:Particle image velocimetry Shear rate distribution Constant power consumption per unit volume Computational fluid dynamics Scale-up

ABSTRACT The main spatial distribution features of shear rate in a stirred tank operated with five different radial and axial flow impellers were presented with particle image velocimetry (PIV) experiments.Not only the average shear rate in the whole tank but also the local value in the vicinity of impeller increases linearly with impeller speed.Furthermore,the shear coefficient(Ks,imp)at the impeller outlet is linearly related to the impeller flow number (Nq) and decreases with the increase of Nq in general at the constant power consumption per unit volume (Pv).During scale-up based on the constant Pv and geometric similarity,CFD results show that the volume-averaged shear rate (γavg) for RDT decreases faster than that of other impellers with the impeller tip velocity (Utip).The novel multi-blade combined (MBC) impeller with the increased height-to-diameter ratio of the stirred tank is able to more effectively improve the distribution uniformity of shear rate at the same Pv after scale-up.These studies provide a data basis for selecting the impeller types and improving the shear rate environment in the large-scale stirred tank.

1.Introduction

As one of the important hydrodynamic parameters in industrial mixing,shear rate is related to the impeller type,geometric structure of the stirred tank,fluid rheology and impeller speed[1].Shear rate influences the apparent viscosity of the non-Newtonian fluid,thereby affecting the power consumption,mixing characteristics and mass transfer phenomena [2].It also affects the activity of enzyme [3] as well as size and structure of aggregates [4,5].For instance,shear rate can control crystallization by managing the primary and secondary nucleation rates,thus better promoting the more stable polymorphs [6].The requirements on shear rate characteristics produced by impellers are decided by the diversified mixing systems and mixing goals.For example,the animal cell cultivation system needs uniformly distributed low shear rates[7,8],because excessive shear rate can cause irreversible damage to cells[9].While in the gas-liquid stirred tank reactors,high shear rates are demanded to disperse the bubbles and promote mass transfer between phases [10],as reported by Dickey and Fasano[11]that the maximum shear rate value multiplied by fluid viscosity could control the final size of the dispersed gas/solid/liquid particles in continuous liquid phase.Therefore,the spatial distribution characteristics of shear rate for different impellers,such as average shear rate and its trend of change after scale-up,are of great significance to better understanding of the mixing process and guiding the application of impellers in different industrial occasions.

The concept of average (effective) shear rate first came from Metzner and Otto[12],who presumed that for non-Newtonian fluids,the average shear rate near the impeller had a linear relationship with impeller speed (N).Some authors have further considered shear rate characteristics of non-Newtonian fluids,that is,the relationship between the fluid rheological index and average shear rate [13,14].Wichterleet al.[15] adopted electrochemical methods to measure the maximum shear rate of Rushton impeller under turbulent flow conditions and found that the maximum shear rate was several orders of magnitude larger than the average shear rate.Dependent on the theoretical derivation,Sánchez Pérezet al.[16] also obtained the linear relationship between average shear rate and impeller speed in the laminar regime for both Newtonian fluid and non-Newtonian power-law fluid in a stirred tank,but the average shear rate was related toN3/(1+n)in turbulent conditions (nis the rheological index).Although these studies show the quantitative relationships between the characteristic shear rate and related system parameters(such as impeller speed,rheological index,etc.),they cannot give the shear rate distributions over the whole tank.

The development of advanced measurement technique represented by laser Doppler velocimetry (LDV) and particle image velocimetry (PIV) provides favorable conditions for experimental detection of flow field in stirred tanks[17-19].Some authors used LDV or PIV to study the shear rate distribution in stirred tanks equipped with a single impeller.Wuet al.[10] obtained the average shear rate at the impeller outlet plane by LDV,and found that the impeller shear coefficient was related to the flow number.Storyet al.[20] used PIV to study the effect of rheological properties(Newtonian fluid and non-Newtonian carboxymethyl cellulose solution(CMC))and impeller speed on the flow field and shear rate distribution when agitated by a Prochem Maxflo T(PMT)impeller.They found that the maximum shear rate appeared in the zone below the impeller,and at the same impeller speed,the maximum shear rate in the CMC system was approximately twice that of in water.With the assistance of PIV,Ramsayet al.[21] obtained the flow field and shear rate in a Newtonian fluid and a viscoelastic fluid by using a‘‘butterfly”impeller under laminar flow conditions.

In addition,the shear rate distributions for impellers with the same flow pattern have been analyzed by other authors.Collignonet al.[22] used PIV to investigate the shear rate distributions of axial flow impellers at their respective critical suspension speeds,and determined the impeller type that was more suitable for cell culture.Sossa-Echeverriaet al.[23] used both PIV measurement and CFD simulation to explore the shear rate distributions of three eccentrically placed axial flow impellers at the same speed in a non-Newtonian fluid.The results indicated that shear rate distribution of the impeller depended on the discharge flow structure of the impeller,which was in turn affected by the operating conditions and the blade angle.Both radial and axial flow impellers are widely applied in industrial projects,while there are few studies to compare their shear rate distributions in a stirred tank at the constantPv.As we all know,power consumption affects the heat and mass transfer processes in the stirred tank,and the cost of power-related input accounts for a large proportion of the overall operating cost of an industrial project[24],so it is worth studying shear rate distribution in the stirred tank at the samePv.

Scale-up of reactors is a necessary step for applying laboratory research results to industry,and it is also a key for large-scale cell cultivation.For cell cultivation,the common problems in scale-up include the slower cell growth and largely reduced survival rate.The fundamental reason is the high sensitivity of animal cells to shear stress [25].Traditional scale-up methods consist of theoretical,semi-theoretical,dimensional analysis and empirical methods[26].Due to the complexity of biological systems,the current commonly used method is still the empirical scale-up method.Parameters,such as impeller tip velocityUtip(πDN,Dis the impeller diameter),power consumption per unit volume (Pv),energy dissipation rate (ε),maximum shear rate (γmax) or average shear rate(γavg) [27-29],are usually used to describe the shear environment and selected as the scale-up criteria for bioreactors.Siecket al.[29]developed a scale-down model of hydrodynamic stress to investigate the performance of CHO cells under simulated production bioreactor conditions.They observed a slight decrease of monoclonal antibody production for cells because of the DNA damage by the increased hydrodynamic stress.In order to predict the impeller speed required to maintain the aggregate sizes in stem cell culture,Boryset al.[30] used CFD to evaluate seven scale-up parameters (i.e.,Uavg,γavg,ε,Re,Utip,P,and γmax),and concluded that if the energy dissipation rate or maximum shear stress was kept constant in from 10 ml to 100 ml stirred suspension bioreactors,the aggregate size could be maintained the same after scaleup.Obviously,there are many factors that affect the performance of scaled-up bioreactors.Most studies are focused on how to maintain the similarity of relevant parameters after scale-up,but sometimes it is not enough to ensure a satisfactory result because the local hydrodynamics including shear intensity may also play an important role [31].So the trend of change in volume-averaged shear rate with impeller tip velocity and the shear rate distributions in a scaled-up reactor are worth of attention.

In the present work,the PIV technique was used to measure the shear rate distribution in a stirred tank with Rushton disk turbine(RDT),45° pitched blade turbine pumping down (PBTD),45°pitched blade turbine pumping up (PBTU),centripetal turbine(CT) and elephant ear impeller (EE) at the constantPv.The spatial distribution characteristics of shear rate for different impellers were analyzed combined with discerning the differences of flow fields.Both the average and local shear rates were correlated with the impeller speed or the flow number.At the same time,by means of CFD simulation,the trends of change in average shear rate and shear rate distribution in a stirred tank were focused on during scale-up based on constantPvand/or geometric similarity.In addition,the dual-impeller combination and the novel multi-blade combined (MBC) impeller methods [32] were investigated to adjust shear rate distribution in the larger stirred tank.

2.Materials and Methods

2.1.Stirred tank and impeller configurations

The experiments are carried out in a flat-bottomed cylindrical stirred tank with internal diameter ofT=0.234 m.There are four wall baffles equally spaced in the tank with a width ofB=T/10.The liquid height level isH=T.The structure of the stirred tank is shown in Fig.1.The deflection angle of the blades relative to the disc for CT impeller is 45°,and the inclination angle of the blades for EE impellers is also 45° (pumping down).The shaded area in Fig.1 represents the measurement plane.In order to minimize the refraction caused by the laser irradiation on the curved surface of the stirred tank,it is placed in a 285 mm ×285 mm×300 mm tank filled with deionized water.Five impellers with the diameter ofD=T/3 are studied in the experiment,namely RDT,PBTD,PBTU,CT and EE impellers.The impeller clearance off the tank bottom(C)is equal toT/3.The detailed structure diagrams and geometric parameters of impellers are shown in Fig.2 and Table 1.The power consumption for stirring is measured by a torque sensor (SN-1050,Beijing Senso Zhongheng Technology Development Co.).The rotation direction of the impeller is clockwise.

Table 1Impeller geometric parameters

Fig.1.Schematic diagram of stirred tank.

2.2.PIV system

The 2D PIV system used in this work includes a dual head Nd:YAG laser (Leibao Optoelectronics,pulse energy 120 mJ,pulse width 6 ns,laser wavelength 532 nm),a high-speed digital charge coupled device (CCD) camera (Imperx,2048 × 2048 px),a synchronization controller and the commercial software MicroVec-V3(to collect experimental data and control laser/camera synchronization).The adopted tracer particles are polyamide seeding particles (PSP) with a diameter of about 10 μm and the density of 1030 kg·m-3,quite close to the liquid density used in this work,to track the liquid velocity.In order to prevent the reflection of laser from damaging the high-speed camera CCD,impellers and the shaft are painted to black.

Due to the mismatch of refractive index between the experiment fluid (water) and Perspex baffle,distortions in images by the baffle can be observed[33].So,the vertical laser plane is placed by Gabrieleet al.[33]at the angle of 5°from the adjacent baffle to give a clear and undistorted image of fluid flow.In this work,the laser plane through the tank axis is also place an angle of 5° off a baffle,as shown in Fig.3(b).The measurement area is the half of the tank,covering the plane clipped with 0 ＜r/T＜0.5 and 0 ＜z/H＜1,as shown in Fig.1.Fast Fourier transform (FFT) crosscorrelation method is used to interrogate the two successive images to determine the velocity vector field.The selected interrogation area is 16 × 16 px,and the overlap area is 50% during the correlation.Depending on the impeller types and the corresponding impeller speed (N),the time interval Δtis set to 400-1100 μs.Considering the accuracy of experimental data and the efficiency of image processing,215 pairs of the collected images are handled to acquire the time-averaged flow field for each condition.

In the Cartesian coordinate system,the shear rate in the stirred tank [34] is expressed as

While in the cylindrical coordinate system,the shear rate is

Fig.2.Schematic diagram of impellers.

Fig.3.Setup of PIV experiment.

Because the experiments are performed using a 2D PIV instrument,the third component of velocity,that is,the tangential velocity cannot be measured.Eq.(2) needs the approximate estimation of the 3rd component.As reported by Storyet al.[20],Eq.(2)can be approximated to

2.3.CFD simulation

The commercial CFD software ANSYS Fluent is employed for simulation.The rotation of the impeller is dealt by the multireference frame(MRF)method.The standardk-ε turbulence model and the standard wall function are used to simulate the turbulent flow field.Momentum,turbulent kinetic energy and energy dissipation rate equations are solved using the second-order upwind difference scheme,and the SIMPLE algorithm is used for the pressure-velocity coupling.

Four different sizes of stirred tanks are investigated with diameters of 0.234 m,0.72 m,1.0 m and 1.5 m in the CFD simulation.The whole stirred tank is selected as the computational domain.The rotating zone including the impeller is divided by tetrahedral mesh,and the outer zone which enclosed the rotating zone is divided by hexahedral mesh.The details about grid independence are described in Supplementary Material.

3.Experimental Results and Discussion

3.1.Impact of impeller types on power consumption

The power consumptions of different impeller types are measured by the shaft-torque method.Fig.4 shows the variation of power consumption per unit volume (Pv) versus impeller speed.It shows that for RDT,the power consumption is the maximum,and the counterpart of PBTU is close to that of EE at the same speed.The sequence by decreasingPvvalues at the sameNis:RDT ＞CT ＞PBTU≈EE ＞PBTD.

The scale-up rule based on the constant power consumption per unit volume is tested in this work,which is expressed as [35]

Fig.4.Power consumption per unit volume versus impeller speed (T=0.234 m,D= T/3, C= T/3).

whereNPSandNPL,NSandNL,DSandDL,andTSandTLare the impeller power numbers,impeller speeds,impeller diameters and stirred tank diameters before and after scale-up,respectively.Since the flow regime during scale-up in this study is fully-developed turbulent,the power number tends to be constant.Eq.(4) can be simplified as

The power consumption per unit volume is specified to be 178.0 W·m-3.The corresponding impeller speeds of different impellers for the stirred tank(T=0.234 m)are shown in the second column of Table 2,and the air is not drawn into the free surface of liquid phase under these conditions.The volume of the stirred tank(T=0.234 m)is enlarged to 30,80 and 260 times based on the constant power per unit volume rule.And the corresponding impeller speeds of five different impellers are calculated with Eq.(5).As shown in Table 2,the operating speeds are all within a reasonable range after scale-up.The variations of shear rate on scale-up are analyzed with the numerical simulation in Section 4.

3.2.Time-averaged flow field

Fig.5 shows the time-averaged flow fields of different impellers in the stirred tank (T=0.234 m).For RDT (Fig.5(a)),the fluid stream is ejected from the impeller to the tank side-wall,and then the radial jet goes up and down forming the upper and lower flow loops.Considering that the top of the tank is a free liquid surface,the distribution of two loops is not completely symmetrical on both sides of the impeller.

Compared with RDT,CT takes advantage of the deflection angle of the blades relative to the disc to push fluid to produce centripetal flow [36].Although the globally similar flow patterns are observed in Figs.5(a)and(d),the velocity fields of CT and RDT present some local differences,as for instance,the centers of the upper and lower vortices are closer for CT.As shown in Fig.5,at the samePv,the maximum velocity of CT is different from that of other impellers due to the unequal impeller speeds.Liet al.[37] found that the macro-mixing time of CT was shorter than that of RDT at the same relatively low speed,and tended to be similar with the increase of impeller speed.CT’s speed is larger than RDT’s at the samePv,suggesting that its macro-mixing time is shorter.In addition,the flow numberNq,which indicates the ability of axial recirculation,is calculated for different impellers (Table 3).As against RDT,the flow number of CT increases by 5.17%.

Table 2Impeller speeds under the designated specific power consumption (T1=0.234 m,T2=0.72 m, T3=1.0 m, T4=1.5 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

Table 3Flow number Nq,shear number Cs and average shear rate in the whole tank γavg for different impeller types (T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3)

The equations of the flow numberNqare as follows:

Fig.5.Time-averaged velocity fields of different impellers (unit:m·s-1, T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

whereQrvis the radial impeller flow rate,Qzvis the axial impeller flow rate,z1is the lowest height of blades,andz2is the highest height of blades.

Under the operation of PBTD,as shown in Fig.5(b),the fluid is discharged from an angle of about 45°below the impeller at a maximum speed of about 0.47Utip,and turns upwards after encountering the bottom of the tank and the baffles,forming the main loop.In addition,there is a small secondary induced circulation region between the impeller and the bottom of the tank.The flow direction of the main circulation loop formed by PBTU is opposite to that of PBTD,i.e.,PBTU develops an upward jet above the impeller.Besides,compared with the main circulation,the speed of upper flow circulation loop is smaller for PBTU,typically～0.1Utip.The time-averaged flow fields of PBTD and PBTU are consistent with the results of Gabrieleet al.[33] What’s more,from Figs.5(b)and(c),it can be seen that at the samePv,the velocity near the bottom of the stirred tank of PBTD is significantly greater than that of the PBTU.

Fig.5(e)is the flow field of EE,which is similar to that of PBTD.However,the downward velocity of EE is greater,showing obvious axial mixing characteristics,which is more conducive to suspend solid particles,such as the micro-carriers (the optimal densities of micro-carriers are 1.03-1.05 g·ml-1[38],slightly greater than that of water) from the bottom of the tank.And the flow number of EE is also greater than PBTD’s(Table 3).Evaluated from the comparison of the overall velocity flow field and flow numbers for the single-impeller at the samePv,the superiority of studied impellers is as follows:EE ＞PBTD ＞PBTU ＞CT ＞RDT.

3.3.Evaluation index for shear performance of impeller

Based on the relationship between the shear rate experienced by the fluid in the stirred tank and the power consumption per unit volume,the dimensionless shear numberCsis defined as [39]

where μ is the fluid viscosity (mPa·s-1).Table 3 shows the shear number of impellers atPv=178.0 W·m-3.For the five types of impellers studied in this work,the sequence ofCsvalues is:RDT ＞CT ＞PBTU≈EE ＞PBTD,which is inverse to the impeller speed.As listed in Table 3,the shear numberCsof radial flow impellers(i.e.,RDT and CT) is larger than that of axial flow impellers.If the shear performance is evaluated withCs,the shear level of RDT is the strongest,and that of PBTD is relatively weak.

3.4.Shear rate distribution

Fig.6.Contour plots of shear rate for five impellers (unit:s-1, T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

Fig.6 shows the contour plots of shear rates for different impellers under the samePv.As demonstrated in Figs.6(a) and (d),for RDT and CT,the higher shear rates are located at the blade edges,impeller discharge region,baffle and the tank wall.In the discharge region near the blade tips,shear rates are asymmetrically distributed along the mounted height of the impeller;in the upper circulation loop,high shear rates prevail over more area.For axial flow impellers(PBTD,PBTU and EE),the change of shear rate distributions is attributed to the variation of flow patterns.Such high shear rates are mainly generated at the edges of the blade,impeller swept zone,the discharge flow region above or below the impeller and the tank wall.Further,in the vicinity of the baffle edges,shear rates are relatively small compared with the radial impellers.

Stathopoulos and Hellums [40] reported that application of mechanical shear stress of 2.6 N·m-2or more would reduce the viability of flat monolayers of human embryonic kidney cells (the density ρ of the culture being close to that of water).While Wolfet al.[41] found that when the shear stress was less than 0.092 N·m-2(corresponding to a shear rate of about 90 s-1),the baby hamster kidney (BHK-21) cells could proliferate;when the shear stress was 0.051 N·m-2(corresponding to a shear rate of about 50 s-1),it could provide a good shearing environment for BHK-21 cells growth.Therefore,the contour plots of shear rate distribution displayed in Fig.6 are divided into four parts according to the local shear rate,namely 0-30 s-1,30-50 s-1,50-90 s-1and ＞90 s-1.It can be seen that most of the shear rate values in the stirred tank for five impellers are concentrated in the range of 0-30 s-1.For RDT and CT impellers,the higher shear rate values are distributed near the blades(r/T=0-0.36,z/H=0.31-0.37);for PBTD and EE,the shear rate values in the area below the blades(r/T=0-0.26,z/H=0-0.43) are higher,and the closer the region is to the blades,the greater the shear rate is;while for PBTU,the zone of higher shear rate values is located above the blades(r/T=0-0.26,z/H=0.25-0.6).In addition,for the studied impellers,the areas with shear rates more than 90 s-1are mostly located at the edge of blades,baffles,and tank wall or bottom,occupying a relatively small area,and for EE,shear rates near the tank bottom are significantly higher than the other four impellers.

As mentioned above,most of shear rates in the stirred tank for impellers are less than 30 s-1.The probabilities of shear rates less than 30 s-1are 87.2%,87.0%,86.3%,85.5% and 83.2% for CT,RDT,PBTD,PBTU and EE impellers respectively.As animal cells have no cell wall and are very sensitive to shearing,impellers with low and moderate shear should be selected in the animal cell bioreactor[42].In order to analyze the shear environment in more detail,the probability distributions of shear rates less than 30 s-1for five impellers are plotted in Fig.7.The more the probabilities of low shear rate occupies,the more moderate shear environment is.It can be seen from Fig.7 that when shear rate is less than 10 s-1,the probabilities of shear rate for RDT and CT are relatively close;when shear rate is between 10 and 20 s-1,the probabilities of shear rate for RDT,PBTD and EE are similar.Compared with RDT and PBTD,the shear rate of CT accounts for a higher probability in the range of 10-20 s-1.In addition,the probabilities of shear rate over 30 s-1for all the impellers are also discussed in Fig.8.It can be seen that in the high shear rate range,the probabilities of shear rate over 90 s-1for the radial impellers are smaller than the corresponding values for the axial impellers.

Therefore,comprehensively considering shear rate distributions of impellers in different ranges and average shear rate(Table 3)at the samePv,the superiority of impellers is ranked as CT ＞PBTD ＞RDT ＞PBTU ＞EE for the shear-sensitive systems.This result is different from the evaluation by shear numberCs,which indicates that in actual operating conditions,Cscannot be used solely to evaluate the shear performance of impeller.

Besides,the correlations of shear coefficientKs,impwith impeller flow numberNqare presented in Fig.9.The data are obtained by calculating the area-weighted shear rate γavg,sat the flow exit plane of impellers,within the impeller diameter periphery,soKs,impcould be used to quantitatively describe the shear environment of the impeller zone.The calculations ofKs,impandNqare performed as follows:

Radial flow impellers

Fig.7.Probability distributions of shear rate less than 30 s-1 for five impellers (T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

Fig.8.Probability distributions of shear rate over 30 s-1 for five impellers(T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

Fig.9.Variation of shear coefficient with impeller flow number (turbulent flow)(T=0.234 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

Axial flow impellers

wherejis thej-th measuring point,andtis the number of measuring points at the flow exit plane of impellers.And the summations are carried out within the periphery of the impeller blades.

It is obviously concluded thatKs,impis linearly related to the impeller flow number.For radial impellers,Ks,imp≈14Nq;for axial impellers,Ks,imp≈7Nq,which are in good agreement with data from Wuet al.[10] And we could also find that at the samePvfor five different impellers studied in this work,Ks,impdecreases with the increase of impeller flow number in general.

3.5.Effect of impeller speed on shear rate

Fig.10.Evolution of average shear rate in the whole tank with impeller speed(T=0.234 m, D= T/3, C= T/3).

Fig.10 shows the variation of average shear rate in the whole tank with the increase of impeller speed.The linear relationships between average shear rate and impeller speed are listed in Table 4.It can be seen from Fig.10 and Table 4 that at the constant impeller speed,the average shear rate of RDT is the largest;while the average shear rate of CT impeller is relatively close to that of PBTU and EE.Different fromKs,imp,Ks,avgrepresents the average shear environment of the whole tank.Besides,the effect of impeller speed on the probability distributions of shear rate for CT impeller is also concentrated on.As the probabilities of shear rates over 90 s-1for all impeller speeds are no more than 3.0%,the range of shear rates less than 90 s-1is only plotted in Fig.11.It can be seen from Fig.11 that the maximum probability of shear rate is decreased while the corresponding value of shear rate shifts towards the higher one with the increase of impeller speed.

For CT impeller,the variations of the radial profiles of shear rate with the impeller speed at different axial positions are also extracted.As shown in Fig.12,in the region close to the impeller(z/H=1/6,1/3 and 1/2),small peaks appear in the vicinity of baffle edges because the high speed impeller stream impinges on the baffles;while in the region away from the impeller (z/H=2/3),shear rate is hardly affected by the baffles.In addition,at the impeller plane (z/H=1/3),shear rate quickly decreases until peaks appear near the baffle and the tank wall.However,the shear rate is generally low away from the impeller,and the effects of radial distance(r)and impeller speed (N) on the shear rate are not obvious in the bulk.

Furthermore,the axial distribution characteristics of shear rate at different radial positions are also investigated.It can be seen from Fig.13,atr/T=0.19,0.22 and 0.28,all the variations of shear rate exhibit the bimodal distribution,and the lower peak values are slightly smaller than the upper ones.With the increase ofrandN,the axial positions of upper peaks gradually moves above the impeller blades,but the axial positions of lower peaks are always inside the blade range (as the shaded rectangle shown in Fig.13).The local minimum points,i.e.,the valleys of shear rate are slightly above the impeller center plane(z/H=0.33).This phenomenon is consistent with what found by Wuet al.[43] that the axial position of the maximum average velocity operated by RDT appears above the height of impeller center.This may be caused by the fact that the impeller is not installed symmetrically in the stirred tank and the tank top is a free surface.After further research,it is found that the peak and valley values of shear rate have the linear relationship with impeller speed.The variations of local shear-related coefficientKswith the radial distance are illustrated in Table 5.It can be concluded that the coefficients decrease as the radial positions are away from the impeller.Besides,the coefficients of the upper peak values are a little larger than that of the lower ones.

Table 4Relationship between average shear rate in the whole tank and impeller speed (T=0.234 m, D= T/3, C= T/3)

Table 5Relationship between the peak and valley values of shear rate at different radial positions and impeller speed (CT impeller, T=0.234 m, D= T/3, C= T/3)

4.Simulation Results and Discussion

In order to investigate the change trends of volume-averaged shear rate and shear rate distribution after scale-up,CFD simulation is applied in this work.The validation of CFD model is in Supplementary Material.

4.1.Scale-up based on geometric similarity

Previous work by Wuet al.[10] found that the average shear rate at the 3-blades PBTD impeller outlet decreases with the increase of impeller diameter on scaling-up by the rule of constantpower consumption per unit volume.However,for the full-scale stirred tank design and selection of impeller types,it is also interesting to estimate and compare the overall average shear rate for different impellers.

Fig.11.Variation of probability distributions of shear rate with impeller speed (CT impeller, T=0.234 m, D= T/3, C= T/3).

Fig.12.Variation of the radial profiles of shear rate with impeller speed at different axial positions (CT impeller, T=0.234 m, D= T/3, C= T/3).

In this work,the change trends of volume-averaged shear rate in the whole tank and the shear rate distributions of RDT,PBTD,CT and EE impellers are first focused on after scale-up by the constantPvrule.When scaling-up stirred tanks with geometric similarity,impeller speed generally decreases and the impeller tip velocity increases.Fig.14 demonstrates the variation of volumeaveraged shear rate in the whole tank with the impeller tip velocity based on the constantPv.It can be seen from Fig.14 that the volume-averaged shear rate for RDT decreases faster than that of other impellers with the impeller tip velocity (Utip).This provides guidance for estimating the volume-averaged shear rate in the whole tank for different impellers after scale-up by the rule of constantPv.Besides,the volume-averaged shear rate based on the sameNscale-up rule remains almost constant as displayed in Fig.14.Hence,during the geometric similarity scale-up,the volume-averaged shear rate is mainly related to the impeller speed and impeller type,not to the scale.

As reported by Liet al.[34] and Wuet al.[10] the maximum shear rate in the whole tank mainly locates at the impeller blades tip and increases withUtip.Therefore,a wider distribution of shear rates is inevitably generated after scale-up by the constantPvrule.Some studies show that the maximum shear rate is not the key factor causing the cell damage because of the rare chance of a cell encountering the maximum shear rate zone [28,44],the overall distribution of shear rate should be paid attention to.As for the distributions in the high shear rate range are not analyzed in consideration of the accuracy of standardk-ε turbulence model and the great proportion of low shear rate.Taking CT and EE impellers for example,the shear rate distributions in the range of 0-30 s-1during scale-up are displayed in Fig.15.It could be seen that a higher probability of shear rate shift towards the low shear rate range.The effect of the change of shear rate environment on the cell growth and the size distributions of dispersed phase drops,crystallization etc.should be further investigated combining with the local mixing intensify and circulation rate of flow.

4.2.Improvement of shear rate distribution after scale-up

Fig.13.Variation of the axial profiles of shear rate with impeller speed at different radial positions(CT impeller,T=0.234 m,D=T/3,C=T/3,the dotted line represents the height of the impeller center,and the shaded rectangle covers the height range of the blades).

Fig.14.Variation of volume-averaged shear rate with impeller tip velocity after scale-up based on constant Pv or constant N (geometric similarity)(T=0.234,0.72,1.0 and 1.5 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

In view of the significant changes in shear rate distributions after scale-up based on both the constantPvand geometric similarity,the height-to-diameter of stirred tank and impeller types are then adjusted to increase the average shear rate and narrow the distribution of shear rate as shown in Fig.16.In the case of a large-scale stirred tank withT=H=1.0 m,Tis reduced to 0.874 m when the height-to-diameter ratio of stirred tank is increased to 1.5 but the tank volume remains constant.Besides,except the single CT impeller,the dual-CT impellers and a novel MBC[32]are used to maintain the similarity of shear rate distribution.The detailed structure of MBC is illustrated in the previous work by Xuet al.[32] The impeller speeds for dual-CT impellers and MBC are 170 and 57 r·min-1based on the constantPv,respectively,according to the power consumption from corresponding simulations.The distributions of shear rate less than 30 s-1for these systems are compared in the Fig.16.It is clear that the probability distributions of 2CT-L and CT-L systems tend to be same.While the shear rate distribution of the novel MBC-L system is closer to that of CT impeller of small scale.This attributes that the blades of MBC are arranged as evenly as possible in the tank leading to an obvious improvement in the turbulent kinetic energy distribution uniformity with higher turbulent intensity [32].Furthermore,the maximum shear rate for MBC in the whole tank does not exceed that of CT-S,because the sequence of impeller tip velocity at the samePvfor four systems is as follows:single CT impeller after scale-up (CT-L) ＞dual-CT impellers combination after scale-up (2CT-L) ＞CT impeller before scale-up (CT-S) ＞MBC impeller after scale-up(MBC-L).As shown in Fig.17,the probability of shear rate over 90 s-1of MBC-L is considerably reduced by contrast with 2CT-L and CT-L after scale-up.In general,the phenomenon,that the lower volume-averaged shear rate and higher maximum shear rate after scale-up based on the constantPvand geometric similarity,is improved with the assistance of MBC.

Fig.15.Variations of shear rate distribution after scale-up (T1=0.234 m,T2=0.72 m, T3=1.0 m, T4=1.5 m, D= T/3, C= T/3, Pv=178.0 W·m-3).

5.Conclusions

Fig.16.Probability distributions of shear rate less than 30 s-1 of different scale-up improvement methods at the constant Pv (CT-S, H= T=0.234 m, D= T/3;CT-L,H= T=1.0 m, D= T/3;2CT-L, T=0.874 m, H=1.5 T, D= T/3, C= ΔC= T/3;MBC-L, T=0.874 m, H=1.5 T, D= T/2, Pv=178.0 W·m-3).

Fig.17.Probability distributions of shear rate over 30 s-1 of different scale-up improvement methods at the constant Pv (CT-L, H= T=1.0 m, D= T/3;2CT-L,T=0.874 m,H=1.5 T,D= T/3,C=ΔC= T/3;MBC-L,T=0.874 m, H=1.5 T,D=T/2,Pv=178.0 W·m-3).

In this work,the shear rate distribution characteristics in the stirred tank equipped with five different impellers at the samePvare focused on by using PIV,which are meaningful to understand the magnitude and location of shear rate in the stirred tank.The general distribution characteristics of shear rate are related to flow patterns.For radial impellers (RDT and CT),the larger shear rates are distributed around the blades;for PBTD,PBTU and EE,larger shear rates are located in the impeller discharge region.These higher concentrated shear rate zones provide reference locations of the feed points for fast-reacting systems.Considering the shear rate spatial distribution and the average shear rate in the whole tank at the samePv,CT impeller with higherNqand more moderate shear environment is superior than RDT for shear-sensitive systems.Besides,in the vicinity of CT impeller discharge zone,the axial profile of local shear rate exhibits the bimodal distribution at different radial positions,and the peak and valley values of shear rate all have the linear relationship with the impeller speed.Among the investigated axial impellers,EE owns the strongest circulation ability than PBTD and PBTU.Despite the similar axial-flow patterns are generated,the local high shear rates are localized in the impeller swept region rather than the impeller blades for PBTD and PBTU.And at the samePv,shear coefficient at the impeller outlet decreases with the increase of impeller flow number in general for studied impellers.

After scale-up by the rule of constant power consumption per unit volume and geometric similarity,it is found that the volume-averaged shear rate for RDT decreases faster than that of other impellers with the impeller tip velocity.Besides,the probabilities of shear rate in the low shear rate range increase,meanwhile,the maximum shear rate around the blade tips also increases with the increase of impeller tip velocity.The dual-CT impellers and a novel MBC are further adopted with the increase of the height-to-diameter ratio of the stirred tank.We find that the novel MBC impeller is contributed to improving the distribution uniformity of shear rate after scale-up based on the constantPv.

These results are expected to provide a reference for selecting the impeller types and matching the shear rate environment in large-scale reactors in different applications.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Financial support from National Key Research and Development Program (2020YFA0906800),the National Natural Science Foundation of China (21808221,91934301,21961160745),External Cooperation Program of BIC,Chinese Academy of Sciences(122111KYSB20190032),the Key Research Program of Nanjing IPE Institute of Green Manufacturing Industry (No.E0010719),and Innovation Academy for Green Manufacture,Chinese Academy of Sciences (IAGM2020C06) is gratefully acknowledged.

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2021.03.004.

Nomenclature

Bbaffle width,m

Cimpeller clearance off the tank bottom,m

ΔCdistance between two impellers,m

Csshear number (=)

Dimpeller diameter,m

ddisk diameter,mm

Hheight of liquid,m

jj-th measuring point

Kslocal shear coefficient in vicinity of impeller

Ks,avgshear coefficient in the whole tank

Ks,impshear coefficient at the flow exit plane of impeller

kturbulent kinetic energy,m2·s-2

lblade length,mm

Nimpeller speed,rpm

NPpower number (P/(ρN3D5))

Nqflow number (Q/(ND3))

nrheological index

Ppower consumption,W

Pvpower consumption per unit volume,W·m-3

Qflow rate,m3·s-1

Qrvradial impeller flow rate,m3·s-1

Qzvaxial impeller flow rate,m3·s-1

ReReynolds number (ρND2/μ)

rradial diatance,m

Ttank diameter,m

ttotal number of measuring points

Δttime interval,μs

Uavgvolume-averaged velocity,m·s-1

Utipimpeller tip velocity,m·s-1

wblade width,mm

zaxial distance,m

z1lowest height of blades,m

z2highest height of blades,m

γavgvolume-averaged shear rate,s-1

γavg,saera-averaged shear rate at the flow exit plane of impeller,s-1

γjlocal shear rate value of thej-th measuring point,s-1

γmaxmaximum shear rate,s-1

δ blade thickness,mm

ε energy dissipation rate,m2·s-3

θ circumferential direction in the cylindrical coordinate

μ fluid viscosity,mPa·s-1

ρ fluid density,kg·m-3

ψ sector width,(°)

Subscripts

L after scale-up

S before scale-up