CFD simulation of flow and mixing characteristics in a stirred tank agitated by improved disc turbines

Zhuotai Jia,Lele Xu,Xiaoxia Duan,Zai-Sha Mao,Qinghua Zhang,*,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

3 School of Chemical Engineering,Nanjing Tech University,Nanjing 211816,China

Keywords:Flow regime Mixing time Stirred vessel Computational fluid dynamics Improved disc turbine

ABSTRACT To reduce the power consumption and improve the mixing performance in stirred tanks,two improved disc turbines namely swept-back parabolic disc turbine (SPDT) and staggered fan-shaped parabolic disc turbine(SFPDT)are developed.After validation of computational fluid dynamics(CFD)model with experimental results,CFD simulations are carried out to study the flow pattern,mean velocity,power consumption,pumping capacity and mixing efficiency of the improved and traditional impellers in a dished-bottom tank under turbulent flow conditions.The results indicate that compared with the commonly used parabolic disc turbine (PDT),the power number of proposed SPDT and SFPDT impellers is reduced by 43% and 12%,and the pumping efficiency is increased by 68% and 13%,respectively.Furthermore,under the same power consumption (0-700 W·m-3),the mixing performance of both SPDT and SFPDT is also superior to that of Rushton turbine and PDT.

1.Introduction

Radial flow impellers are widely used for mixing in chemical production,very important to stirred equipment.The efficiency of a mixing system is influenced by the flow pattern therein,which is in turn significantly dependent on the geometry of agitator [1].Therefore,development of new impellers with low power consumption and high mixing efficiency is of economic value in industrial production.

A typical radial flow impeller is the Rushton turbine(RT),which is effective in mixing but has the disadvantage of high power consumption and large power drop when aeration is significant.According to the cavitation theory [2],the concave blades are investigated [3] and subsequently many new configurations/designs are proposed,such as the semi-circular tubular disc turbine paddle CD-6 [4],asymmetric parabolic disc turbine (BT) [5],Scaba 6SRGT impeller[6],fan-shaped turbine(FT)[7],semi-elliptic tubular disc turbine (HEDT) and parabolic blade turbine (PDT) [8,9].These improvements are all in line of further increasing the curvature of blades to produce a ‘‘deeper” blade profile [10].Some researchers also introduce hollow blades to decrease power consumption.Raoet al.[11] and Ameur [12] made the cuttings in blades to save the impeller energy.Ameur [13] made perforations in the blades of Scaba 6SRGT according to different patterns.Curved blade edge is also proved beneficial to improving pumping efficiency by reducing the length of resistance arm[14].Moreover,many previous studies of reducing power consumption and improving mixing efficiency are mainly in flat bottom stirred tanks,while few research works on improving mixing efficiency in dished bottom tanks with baffles not extended to the bottom dish are reported.

CFD methods are now playing an important role in predicting the flow and mixing performance of stirred tanks,and different turbulence models are used to predict the turbulent flow structure in the tank,such ask-ε model[15-18],large eddy simulation(LES)[8] and direct numerical simulation (DNS) [19].Among these turbulence models,while LES and DNS are quite time-consuming for simulation jobs,the standardk-ε model is still extensively applied for simulating the stirred tanks with a best balance between simulation accuracy and computational load.

To further reduce the power consumption and improve the mixing performance of stirred tanks,two improved disc turbines,swept-back parabolic disc turbine (SPDT) and staggered fanshaped parabolic disc turbine (SFPDT),are developed in this work,and their flow and mixing characteristics in a dished bottom stirred tank at various Reynolds numbers(Re=3.95×104-9.89×104)are investigated by CFD simulation.The CFD models are firstly assessed against the experiments of Wu and Patterson [20].Then the flow pattern,mean velocity,turbulent characteristic,power number,flow number,pumping efficiency and mixing time of the dished bottom stirred tank agitated by RT,PDT,SPDT and SFPDT are systematically investigated.

2.Stirred Tank and Impeller Geometry

The geometrical details of the stirred tank reactor are shown in Fig.1.The dished bottom tank has diameter ofT=0.282 m,height ofH=Tand elliptical head height ofh=0.0705 m.Four baffles ofB=T/10 are placed at equal peripheral distance on the tank wall,and the baffle thickness is 0.002 m.The shaft diameter is 0.016 m.Four different turbine impellers with the same diameter ofD=0.5Tare placed at the clearance ofC=T/3.

Figs.2-4 show the geometrical structure of PDT,SPDT and SFPDT,respectively.The two improved turbine impellers SPDT and SFPDT are designed based on PDT.The blades of SPDT are curved backwards by 15° in the disk plane,and more details are provided in Fig.3.SFPDT is formed by the interlacing of two different blades (Blade 1,2) as shown in Fig.4,the width of the upper part of a blade(F1,F3)is greater or less than the width of the lower part (F2,F4).For SFPDT,the height of upper and lower portions of the blade,hi,and the blade widthFiconform with equation (1).Regarding to PDT and SPDT,the height of the blade,w,and the width of the bladeFconform with Eq.(2).

The dimensions of the four turbine impellers are listed in Table 1.All turbine impellers rotate counter-clockwise (the concave surface of blade pushing the fluid).

3.Numerical Approach

3.1.Formulation

3.1.1.Governing equations

The simulation is carried out by ANSYS FLUENT 2020R2.The Reynolds-averaging approach is generally used to turbulence modeling,and the Reynolds-averaged Navier-Stokes (RANS) equations are as follows:

Employing the Boussinesq hypothesis [21] to relate the Reynolds stresses to the mean velocity gradients,the Reynolds stresses are modeled as follows:

In this work,the standardk-ε model is applied to turbulent flow,it has been widely used for fully developed turbulent flow conditions [22-24].The turbulence kinetic energykand dissipation rate ε are obtained from the following transport equations[25]:

Gkrepresents the generation of turbulence kinetic energy due to the mean velocity gradients:

The turbulent (or eddy) viscosity μtis

Gbis the generation of turbulence kinetic energy due to buoyancy.YMrepresents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate,for incompressible fluid μ=0,Gb=0,YM=0.The model constantsC1ε,C2ε,Cμ,σkand σεhave the following default values:C1ε=1.44,C2ε=1.92,Cμ=0.09,σk=1.0 and σε=1.3.SkandSε are the source terms [26].

3.1.2.Species transport model

Mixing time is obtained by solving the transient transport equation of a non-buoyant tracer as given by.

Γeffis the effective diffusion coefficient.Dmis the molecular diffusion coefficient and taken as 0.00005 m2·s-1[26].σtis the turbulent Schmidt number and set as 0.7 [27].

3.2.Mesh construction

Computational grid is shown in Fig.5.The geometry models of all cases are built by ANSYS SCDM 2020R2.The entire stirred tank is spilt into a rotating region(the region around the impeller)and a stationary region (the rest of the stirred tank).The rotating region are bounded byr/R=1.3,-35.25 mm ≤z≤35.25 mm.Two regions are meshed by the method of poly-hexcore using ANSYS Fluent Meshing 2020R2.Poly-hexcore meshing is a hybrid meshing scheme that generates axis-aligned Cartesian cells inside the core of the domain and polyhedral cells close to the boundaries [28],and can deal with complex geometry and generate a lower quantity and higher quality cells than tetrahedral meshes [29,30].

Taking SFPDT as research object,the mesh sensitivity study is performed using three generated meshes with 9.9 × 105,2.05×106and 3.14×106cells respectively atRe=9.89×104.Liquid phase velocity components,turbulent kinetic energy and stirrer shaft torque are used to assess the solution independence from the mesh refining.As it can be seen in Fig.6,little difference is found between 2.05 × 106and 3.14 × 106cells on liquid phase velocity components and turbulent kinetic energy.The result of torque for the 2.05 million cells is almost the same as for the 3.14 million one (the difference is below 1%) as shown in Table 2.Therefore,the 2.05 million mesh is chosen for further simulations.By the same meshing criterion,the discretized domain containing 1.87×106,1.94×106and 2.13×106elements for the stirred tank equipped with RT,PDT and SPDT is used,respectively.

Table 1 Dimensions of four turbine impellers

Table 2 Independence of torque on Mesh

Table 3 Mixing time,power consumption and homogeneity energy (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104, M1)

3.3.Simulation details

3.3.1.Discretization of governing equations and boundary conditions

Fig.1.Configuration of a stirred tank equipped with SFPDT.(a) Schematic diagram;(b) Geometry definitions.

Fig.2.Geometry structure of PDT: (a) top view;(b) front view.

Fig.3.Geometry structure of SPDT: (a) top view;(b) front view.

The finite volume method(FVM)is used to discretize the continuity,momentum,turbulence equations and tracer mass balance by ANSYS Fluent 2020R2.Reynolds-averaged Navier-Stokes(RANS)equations with thek-ε turbulence model are discretized by the second-order upwind finite volume scheme and solved on the computational domain.The rotation of the impeller is achieved using a multiple reference frame(MRF).Pressure-velocity coupling is resolved with the semi-implicit method for the pressure linked equations (SIMPLE) algorithm.The wall boundaries are all no-slip boundary condition,and the stirred tank liquid surface is set as a symmetric boundary.The 3D coordinate origin is chosen at the center of the impeller,with thez-axis pointing towards the liquid surface and the stirrer rotating counterclockwise around thez-axis.The convergence criteria are:residual error of algebraic matrix solver is below 10-4and the variation of the torque over 1000 iterations is less than 1%.

Water is used as the working fluid with properties of ρ=998.2 kg·m-3and μ=1.003 × 10-3Pa·s.Four impellers are operating at atmospheric pressure and room temperature with stirring speed ofN=2-5 r·s-1.

3.3.2.Mixing time simulation

Simulation of mixing time is conducted by introducing an inert tracer into the primary liquid in the stirred tank through coupiling a user defined function (UDF).The concentration and location of the tracer are defined in the stirred tank using the patch function in Fluent.Mixing time is determined by analyzing the tracer concentration response curve at a monitoring point.The mixing time θmis defined as the time from the tracer releasing until the tracer concentration enters the 95%-105% interval of the averaged concentration in the tank and then never escaped the bounds.The time step is 0.001 s and the residual error of algebraic matrix solver is set at 10-4for the species transport equation at each time step.

Fig.4.Geometry structure of SFPDT: (a) (d) top view;(b) (c) (e) (f) front view.

Fig.5.Schematic of computational grids for SFPDT impeller.

The tracer is released into the fully developed flow field near the liquid surface and then the monitoring started simultaneously at two monitoring points.The injected tracer is modeled as a sphere with the center pointp(x=-84.5 mm,y=-84.5 mm,z=172 mm)and a radius of 13.4 mm.The monitoring pointM1(90 mm,90 mm,176.5 mm) is near the liquid face contrasting topandM2(0,0,-88.125 mm) is near the bottom of tank.The locations ofM1,M2andpare shown in Fig.1.

4.Results and Discussion

4.1.Assessment and validation of standard k-ε predictions

The experiment data of Wu and Patterson [20] are used to validate the accuracy of simulation.Based on the physical geometry parameters in Wu and Patterson [20],a fine grid verified by mesh sensitivity is chosen for the numerical simulations.The simulated axial,radial and tangential velocities at radial positionsr=5 cm andr=6 cm are all normalized by the tip velocity of blade,Utip(Utip=πDN),and compared with the literature data in Fig.7.The fluctuations in the axial profile of axial,radial and tangential normalized velocities at two radial positions are in good agreement with the experiment results,except for under-prediction of velocity components peak.The under-prediction result may be due to the difference in the impeller geometry (blade and disc thickness)[31] and the standardk-ε model under-prediction of the turbulent intensity around impeller tip region due to the assumption of turbulence isotropy[32].The average deviation of velocity com-ponents between experimental and numerical results is below 10%.To further evaluate the model accuracy,NpandNqare calculated for comparison with experimental results of the literatures.The predicted power numberNpis 5.00 with a deviation of 7.5% compared with the experiment result of Smith and Gao’s [33],and the predicted flow number (Nq) is 0.885 with a deviation of 3.8%compared with Wu and Patterson’s experimental result[20].These results indicate that the standardk-ε model can have a good prediction on turbulent flow in the stirred tank.

4.2.Flow pattern

The velocity distribution for different impellers at the vertical mid-plane underRe=9.89×104is shown in Fig.8,all the velocity components are normalized byUtip(Utip=πDN).Two circulation loops are obviously observed above and below the disc for four impellers,which conforms to the radial flow pattern [34].It is noted that the radial jet stream of four impeller is inflected downward,especially for RT,which is contrary to the results of Wu and Pattern [20].Deenet al.[35] also observed this phenomenon in a dished bottom tank with diameter ofT/3 and attributed it to the dished bottom shape.In addition,the downward inclined jet stream is also observed due to the strong tangential cyclonic flow in a flat bottom tank without baffles [36].So the elliptical bottom structure without baffles extended to the dish bottom results in the downward inclination of radial jet stream.

Fig.6.Mesh independence tests via velocity components,turbulent kinetic energy and mixing time (SFPDT, T=282 mm, D= T/2, N=300 r·min-1, r/R=0.57).

The shape of recirculation loop is also resultant from the impeller type.As shown in Fig.8(a),due to the mostly downward inclination of radial jet stream,the upper and lower circulation loops of RT are shifted downwards,and the upper circulation loop range is reduced greater in the tank than that of other impellers.Moreover,high velocity area is concentrated at the bottom of the mixing tank.In addition,two separate secondary loops are formed near the bottom of the stirring shaft,which is not conducive to mixing at the bottom of the tank below the stirring shaft.Regarding to PDT and SPDT,the left and right cycles are symmetrical with the two symmetrical sub-cycles below the stirrer shaft.It is indicated that more fluid exchange is in the lower sub-cycle of the PDT,while the distribution of the looped streamlines above the SPDT is more evenly distributed.For SFPDT,the circulations on the left and right sides of impeller are asymmetrical and greater range of circulation loops are observed in comparison to other impellers.The staggered blades facilitates the disruption of the circulation loops near the impeller,meanwhile the asymmetrical sub-cycles of SFPDT also facilitates fluid exchange in the dead zone below the impeller.

The distributions of normalized velocities at the horizontal midplane for different impellers are presented in Fig.9.The high speed regions(U/Utip＞1.0)of PDT,SPDT and SFPDT are all located outside of the end of the blade,while RT is located behind the blade,and the high speed region of RT is the largest while SPDT is the smallest.For low speed region(U/Utip＜0.2),it is mainly located near the baffles and between the blades and the baffles.Among the four impellers,SPDT can achieve more even velocity distribution.

4.3.Mean velocity

The axial distribution profiles of the mean axial,radial and tangential velocities normalized with the impeller tip velocity for four impellers at two radial positions underRe=9.89 × 104are shown in Figs.10-12,respectively.Slightly difference is found in the mean axial,radial and tangential velocities for the three concave impellers (PDT,SPDT,SFPDT) at the region near the impeller (r/R=1.15).At different axial positions,velocity components of SPDT are the smallest and little difference is observed between PDT and SFPDT except for the position near the liquid face (3 ＜2z/w＜15)where SPDT has the largest axial velocity in Fig.10(a).This indicates that the swept-back blades of SPDT can enhance the axial flow near the liquid face.In Fig.11(a),little difference in radial velocity also indicates that the swept back blades of SPDT and the staggered blades of SFPDT have little effect on the pumping flow rate.Fig.12 shows that the tangential velocity magnitude below the impeller is larger than that above of the impeller for four impellers,which means the tangential motion is concentrated on the region below the impeller.Below the impellers (-6 ＜2z/w＜0),the tangential velocity magnitude of SPDT is the lowest.This result may be contributed to that the swept back blades of SPDT resulted in reducing positive contact area between blades and the fluid,and then weakened the tangential motion below the impeller.

Regarding to RT,due to the severe downward inflection of the radial jet stream,the peak velocity components shift and the directions of velocity components are changed,which are opposite to concave impellers(PDT,SPDT,SFPDT).The position of radial velocity peak inflected downward to position of 2z/w=-1.5 with the smallest peak in four impellers indicating the reduction of pumping flow rate for RT.As shown in Fig.12,it is also noted that the tangential velocity of RT is the greatest at different axial positions,particularly below the impeller,where the tangential velocity of RT is nearly as three times as that of the other impellers,resulting in severe tangential motion below the impeller.

Fig.7.Axial profiles of the simulated and experiment mean normalized velocity components at two radial positions.Solid lines: CFD predictions.Filled squares: the experimental data reported by Wu and Patterson [20] (T=270 mm, D= T/3, N=200 r·min-1).

Fig.8.Predicted normalized velocity plots with streamline in a vertical plane midway between two baffles (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

Fig.9.Predicted normalized velocity plots in a horizon plane midway at z=0 (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

Fig.10.Axial profiles of axial velocity at two radial positions (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

Fig.11.Axial profile of radial velocity at two radial positions (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

Fig.12.Axial profile of tangential velocity at two radial positions (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

At the position near the baffle and tank wall (r/R=1.45),the trends of velocity components for four impellers are similar to those at position ofr/R=1.15,while all the velocity magnitudes decrease with increasing radial distance.It can be seen that for three concave impellers,two axial velocity peaks are below the impeller (2z/w＜0) in Fig.10(b),which is contrary to the results of Wu and Pattern [20] and Deenet al.[35].This may be a consequence of dished bottom and impeller diameter of 0.5Tthat fluid flow is intensified near the dished bottom walls.Regarding to tangential velocity in Fig.12(b),the decrease is not significant unlike the axial and radial velocities with increasing the radial distance.And for RT,the peak position shifted downward to position of 2z/w=-4.5.Furthermore,at different axial positions,the tangential velocity magnitude is still larger than that of other impellers which means that strong tangential motion and spins are also formed near the tank-walls.Strong tangential motion and spins can result in bad mixing which should be minimized by using improved impellers or mounting baffles in the dished bottom [36].

4.4.Turbulent kinetic energy (k)

Fig.13 shows the predictedk/distribution at the horizontal mid-plane (z=0) for four impellers underRe=9.89 × 104.The results show that thek/region distribution of concave impellers (PDT,SPDT,SFPDT) is more uniform than that of RT.It is noticeable that highk/region is mainly located at the tip of the concave impellers,while it is mainly located behind blades for RT.The highk/region of SFPDT is the largest among the three concave impellers.

Fig.14 shows axial profiles ofk/for four impellers at two radial positions (r/R=1.15,1.45).k/magnitude reaches two peaks in the jet region(-1.5 ＜2z/w＜1.5)for all the three concave impellers,and then decreases at below and above of the impeller along with the height of the stirred tank.As shown in Fig.14(a),at different axial positions,the value ofk/of SPDT is the minimum and PDT is the maximum,except at jet region (-1.5 ＜2z/w＜1.5),where thek/of SFPDT is larger than that of PDT.It is noticed that only a peak is observed in the jet region for RT,and RT has the highest turbulent energy in the four impellers at different axial positions.This result is determined by the characteristics of the flat blades of RT and also consistent with result that the power number of RT is the largest.

The values of turbulent kinetic energy for four impellers have been decreased with increasing the radial distance atr/R=1.45.In Fig.14(b),the peak magnitude reduced to one in the jet region for three concave impellers and peak of RT shifted downward.It can be seen that thek/of PDT is larger than that of SFPDT,which is contrary to the situation ofr/R=1.15.This result indicates that at jet region thek/is intensified when equipped with SFPDT.Combined with the previous analysis of tangential velocity(Fig.13),it can be concluded that the highk/of RT is caused by the severe tangential velocity and may have an unusual influence on axial mixing of the full tank.

4.5.Power characteristic

The power number (Np) is an important parameter for the performance of turbine impellers,and can be calculated by the shaft torque exerted on the impeller.It is defined as follows:

Fig.13.Contour of Predicted k/ at the horizontal mid-plane of z=0 (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

Fig.14.Axial profile of k/in the full tank at two radial positions (T=282 mm, D= T/2, N=300 r·min-1, Re=9.89 × 104).

wherePis the power consumption of the stirrer(W),Dis the diameter of the turbine impeller(m),Mis the torque(N·m),Nis the turbine impeller speed(r·s-1),ρ is the density of water(kg·m-3)and μ is the viscosity (Pa·s).Reis the Reynolds number.When the flow field is in a fully turbulent state,the power number (Np) tends to be stabilized with further increasingRe.

In Fig.15,power consumption as a function of Reynolds number is shown for stirred tanks equipped with different impellers.Compared to RT and PDT,the power consumptions for both newly designed impellers are lower at different Reynolds numbers.

Fig.16 shows the power numbers(Np)of four turbine impellers at different Reynolds numbers.The results show that the power number remains essentially constant with increasing Reynolds number(Re=3.95×104-9.89×104).Compared to PDT,the value ofNpfor SFPDT and SPDT decreases 12%and 43%,respectively.TheNpreduction for SFPDT is due to the curved modification on outer and inner edges of blades,which reduces the length of resistance arm,and then decreases power consumption.While for SPDT,the reason is the reduction of the positive contact area between blades and the fluid,and subsequently the resistance is reduced.The results indicate that the modifications of positive contact area between blades and fluid have greater effect on power consumption than modifying the edges of blades.

Fig.15.Effect of Re on power consumption for different impellers.

Fig.16.Effect of Re on power number for different impellers.

4.6.Flow number and pumping efficiency

The pumping capacity of the impeller is also an important index to characterize the mixing performance in a stirred tank.The pumping flow rate,Qv,is defined as the flow rate through the sweeping circle of the radial impeller turbine [37,38] and calculated as follow:

Fig.17.Flow number versus Reynolds number curves.

Fig.18.Pumping effectiveness versus Reynolds number curves.

whereris the radial distance from shaft,r+is the boundary of the sweeping circle,zis the axial height,z1andz2denote the axial height of the highest and lowest points of the paddle respectively,vris the radial velocity component.2r/D=1.1 andz1-z2=1.1w.The pumping capacity is generally expressed in terms of the normalized power number,Nq,and can be calculated as Eq.(16).The pumping effectiveness,ηE,gives the pumping rate of the impeller turbine per unit power consumption [37] and is defined as follow:

Figs.17 and 18 show the effect ofReon flow number and pumping effectiveness for four different impellers,respectively.It can be seen thatNqand ηEof the four impellers remain almost invariant with increasingRe(Re=3.95 × 104-9.89 × 104).Moreover,it is found that the concave blades of PDT,SPDT and SFPDT can pump more fluid than the flat blades of RT.Compared to PDT,with little reduction of flow number,the pumping efficiency (ηE) of SPDT and SFPDT increased by 68% and 13%,respectively.These results indicate that SPDT and SFPDT have higher efficiency in pumping capacity compared with PDT and RT in a dished bottom tank.

4.7.Mixing time

To evaluate the mixing performance of different stirrer systems at the same Reynolds number,the homogenization energy,η,is calculated as follows [39,40]:

The trend of the correlation of mixing timeversuspower consumption for the four turbine impellers in fully turbulent condition(Re=3.95×104-9.89×104)is shown in Fig.19.It can be seen that the mixing time decrease sharply with increasing impeller speed(in the range of 0-200 W·m-3).With further increasing impeller speed,the decreasing trend of mixing time becomes slower.

Fig.19.Mixing time versus power consumption curves at two points.

In Fig.19,both of the two new turbine impellers have less mixing time than RT and PDT at the same power consumption (0-700 W·m-3),which indicates that the SPDT and SFPDT have higher mixing efficiency than RT and PDT under the same power consumption.The largest mixing time for RT may be due to the severe tangential motion and spins in the tank,which is bad for axial mixing.In contrast,the staggered blades of SFPDT can facilitate mixing in the full tank and local dead zones by a greater range of asymmetrical circulation in the tank.The swept-back blades of SPDT can enhance the axial flow near liquid face and suppress the tangential motion near the bottom of tank,which results in a uniform velocity distribution in the tank.

Homogenization energy for all impeller turbines atN=5 r·s-1is shown in Table 3 (MornitorM1).The simulatedTmof RT is 21.99 which is within 5% deviation in comparison to 23.03 calculated from the empirical equation of Ochieng and Onyango[39],indicating that the CFD simulation predicts correctly the mixing time in our work.As illustrated in Table 3,both SPDT and SFPDT turbines have less homogenization energy than RT and PDT,indicating that the SPDT and SFPDT have higher mixing efficiency than PDT and RT at the same Reynolds number.Combining mixing time per power consumption and homogenization energy,both SPDT and SFPDT have higher mixing efficiency than RT and PDT.

5.Conclusions

Two improved radial concave impellers called SPDT and SFPDT are proposed in this work,the performance of the SPDT and SFPDT impellers including flow pattern,mean velocity components,power number and mixing time in a dished bottom tank with wall baffles ending above the bottom are investigated in comparison to PDT and RT by numerical simulation.

It is demonstrated that in the dished bottom tank,the jet stream of RT shifts downwards severely resulting in uneven mixing throughout the tank.Among the four turbines,SPDT has the most evenly velocity distribution in the tank,and the SFPDT can facilitate mixing in the full tank.For turbulent kinetic energy,the turbulent kinetic energy of SPDT has a great reduction in the jet region in comparison to PDT,while SFPDT is superior to PDT.

Compared to PDT,the power numbers of SPDT and SFPDT decrease 43% and 12%,respectively,and the pumping efficiencies increase by 68% and 13%,respectively.The modifications on positive contact area between blades and fluid have greater effect on pumping efficiency than modifying the inner and outer edges of blades.According to the comparison of mixing time per power consumption,both SPDT and SFPDT have higher mixing efficiency than RT and PDT.

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

This work is supported by the National Key Research and Development Program(2020YFA0906800),the National Natural Science Foundation of China (22078325,21938009),the NSFC-EU project(31961133018),the Special Project of Strategic Leading Science and Technology CAS (XDC06010302),Chemistry and Chemical Engineering Guangdong Laboratory,Shantou (No.1922006) and the Li Foundation Fellow Program.

Nomenclature

Bwidth of baffles,m

Cclearance height,m

C1ε,C2ε coefficient ink-ε equation

Dimpeller diameter,m

Dmmolecular diffusion,m2·s-1

Fwidth of blade,m

Hliquid depth,m

hiheight of different portion blade,m

kturbulent kinetic energy,m2·s-2

llength of blade,m

Mtorque,N·m

M1,M2monitor position

Nimpeller speed,r·s-1

Nppower number

Nqflow number

Ppower consumption,W

pfeed position

Qvpumping flow rate

Rradius of tank,m

ReReynolds number

rradial coordinate,m

Sk,Sε source term

Ttank diameter,m

Tmmixing time number

Uvelocity,m·s-1

Utiptip velocity of blade,m·s-1

wheight of the blade,m

zaxial coordinate,m

ε turbulent energy dissipation rate,W·kg-1

ε-mean specific kinetic energy dissipation rate,W·kg-1

η homogenization energy

ηEpumping efficiency

θmmixing time,s

μ viscosity,Pa·s

μtturbulent viscosity,Pa·s

ξ thickness of baffle,disk,blade,m

ρ liquid density,kg·m-3

σk,σεcoefficient ink-ε equation

Subscripts

E efficiency

r,θ,zradial,tangential,axial