Yongjun Wu, Pan You, Peicheng Luo

School of Chemistry and Chemical Engineering, Southeast University, Nanjing 211189, China

Keywords:Stirred tank Computational fluid dynamics (CFD)Turbulent flow Long-short blades (LSB) impeller Pitched blades

ABSTRACT This work focuses on the design improvement of the long-short blades (LSB) impeller by using pitched short blades (SBs) to regulate the flow field in the stirred vessel.After mesh size evaluation and velocity field validation by the particle image velocimetry, large eddy simulation method coupled with sliding mesh approach was used to study the effect of the pitched SBs on the flow characteristics.We changed the inclined angles of the SBs from 30° to 60° and compared the flow characteristics when the impeller was operated in the down-pumping and up-pumping modes.In the case of down-pumping mode, the power number is relatively smaller and vortexes below the SBs are suppressed, leading to turbulence intensification in the bottom of the vessel.Whereas in the case of up-pumping mode,the axial flow rate in the center increased significantly with bigger power number, resulting in more efficient mass exchange between the axial and radial flows in the whole vessel.The LSB with 45° inclined angle of the SBs in the up-pumping mode has the most uniform distributions of flow field and turbulent kinetic energy compared with other impeller configurations.

1.Introduction

Stirred tanks are widely used in many process industries, such as chemical engineering, petrochemical, papermaking, sewage treatment, food, material blending, and metallurgy,etc.[1-3].The required mixing quality is basically determined by the impeller configurations [4].For a specific mixing purpose, the impeller should be well designed to achieve either effective mass exchange rate or high turbulence intensity, or the both.Conventional axialmotion impellers,e.g.propeller, can only generate large axial flow rate,and have very poor mass exchange rate between the axial and radial flows [5].Moreover, turbulence intensity in the tank with the propeller is relatively low.Conventional radial-motion impeller,e.g.Rushton turbine, can generate enhanced radial flow in two separate loops with high turbulence intensity [6].However,there is still the disadvantage of poor mass exchange rate, in particular, between the two loops.An improved impeller configuration is so-called mixed-motion impeller,e.g.pitched-blades impeller, which combines the advantages of the axial-motion impeller and radial-motion impeller[7].The pitched-blades impeller always operates in two modes,i.e.up-flow mode and downflow mode.

In general,the turbulence intensity in the lower part of the tank can be intensified in the case of down-flow mode, whereas the axial flow rate increases when the agitator operates in the upflow mode, which always improves the mixing performance in the whole tank,in particular,for an agitator installed near the bottom of the tank[8].For example,Taghaviet al.[9]investigated the reactive mixing process in stirred tank reactors with pitched blade turbine,and found a pitched blade up-flow turbine could make the concentration distribution more uniform, while the concentration distribution became uneven in the case of down-flow mode.Liet al.[10]investigated the solid suspension performance in a stirred tank with a down-pumping pitched-blade turbine impeller,and observed that most of the particles were suspended throughout the whole tank.Chenet al.[7]studied the optimization of dual-impeller configurations in a gas-liquid stirred, and revealed that the combination of pitched concave blade disk turbine(PCBDT)and pitched turbine disk(PTD)was superior to any others,owing to the strong axial circulation ability of PTD and the excellent gas handling capacity of PCBDT.Buffoet al.[11]used Rushton turbines and elephant ear(EE)impellers in down-pumping and uppumping modes in a stirred tank bioreactor and found that the combination of two EE impellers provided very satisfactory oxygen transfer efficiency.

The above conventional impellers have been widely used in practical applications.However, the turbulent kinetic energy(TKE)is usually larger around the impeller,and most regions have less distribution,in particular,in an industry-scale bigger size stirred tank.An improved strategy is to combine multiple impellers,or use large cross-section impellers,e.g.Maxblend impeller and Fullzone impeller [12-14].Recently, our group has developed longshort blades (LSB) impeller and multi-blade combined impeller,which both arrange blades relatively evenly in the tank [15,16].Previous studies indicate that the TKE distribution becomes much more uniform than that of multiple impellers or Maxblend impeller[17].Although an obvious improvement of the TKE distribution uniformity has been made,there still exists some stagnant area,e.g.the middle position of the upper part of the tank equipped with the LSB impeller.This is because partial axial upflow generated by the short blades (SBs) will change to radial flow due to the radial pumping of the long blades (LBs).Large eddy simulation predictions show that the axial flow rate in the upper part of the tank is only about half of that in the lower part.Thus, it is necessary to modify the impeller design to avoid the occurrence of the stagnant area as much as possible.

The objective of this work is to modify the LSB impeller design by using pitched SBs.We intend to increase the axial flow rate further, thus reducing the low velocity area fraction and improving the uniformity of the velocity distribution and the TKE distribution.The angles between the SBs and the horizontal plane are 30°, 45°,60° and 90°(i.e.the configuration in our previous studies),respectively.The clockwise and counterclockwise rotation of the impeller yield axial up-flow intensification and axial down-flow intensification, respectively.Large eddy simulation using a dynamic kinetic energy sub-grid scale model was firstly assessed by the dimensionless wall distance,Kolmogorov scale and Taylor microscale calculation.Then the predictions were validated by particle image velocimetry (PIV) measurement of the flow field.The flow characteristics and hydrodynamics,including velocity distribution,power consumption, TKE distribution, axial and radial flow rates, were finally discussed to evaluate the effect of the pitched SBs on the mixing improvement in the stirred tank with the modified LSB impeller.

2.Experimental

2.1.Tank and impeller configurations

The flows generated by four pitched SBs (30°, 45°, 60° and 90°)of LSB impeller were measured in a flat-bottomed cylindrical Plexiglas vessel,which was symmetrically equipped with four equallyspaced baffles.The internal diameter of the tank(T)is 200 mm and the aspect ratio of the liquid level (H) to the diameter,H/T, equals 1.Water was used as the liquid working fluid, with ρl= 998.2 kg·m-3and μl= 1.003 mPa·s.The baffles have 20 mm width and adhere to the tank wall.Fig.1 shows the operation modes (uppumping mode and down-pumping mode) and structure diagram of SBs of LSB impeller.The characteristic diameter of the LSB impeller (D) isT/2.It consists of one fixed bracket, three LBs, six SBs and two connected rings.The length and height of the SBs are 25 and 20 mm, respectively.The same off-bottom clearance(C=T/4)was used.The angle(β)between the SBs and the horizontal plane is 30° (Fig.1(c)), 45° (Fig.1(d)), 60° (Fig.1(e)) and 90°(Fig.1(f),i.e.the configuration in our previous works [15,18]),respectively.When operating in the up-flow intensification mode,the impellers are represented by 30U, 45U, and 60U.Whereas in the down-flow intensification mode,the impellers are represented by 30D, 45D, and 60D.90V represents vertical SBs (as shown in Fig.1(f)).

2.2.Particle image velocimetry measurement

The used PIV system is the same as that in our previous work[16,17], as shown in Fig.2(a).It consists of a continuous solid Nd:YAG laser system (KSPL-05, Kingder, China), a high-speed CMOS camera (CAMMC1362, Mikrotron, Germany), an image acquisition card and an image post-processing system.The resolution of the camera is 1280×1024 pixels,and the highest recording frequency is 506 frames per second.The PIV system is equipped with 850 MB·s-1image acquisition card and the computer memory is 32 GB.The images collected by the high-speed camera are transmitted to the image acquisition card with a speed of 850 MB·s-1.The continuous semiconductor laser has a power of 5.27 W with the wavelength of 532 nm.The thickness of the laser sheet is less than 1 mm.The planer laser sheet is located on the plane ofPπ/4.The seeding particles used are 6 μm diameter hollow glass beads.The cylindrical tank was placed inside a square tank filled with water to reduce the optical refractive index effects on the cylindrical surface of the tank.The LSB impeller is also sprayed black to prevent reflection of the laser.

In the PIV experiments, the temperature of the water was controlled at(21±0.5)°C and the Reynolds number ofRe=ND2ρ/μ is 17425.By changing the wiring mode, the impeller was driven to rotate clockwise or counterclockwise by the motor.The plane in the middle of two adjacent baffles (Pπ/4) was chosen as the sampling plane and a rectangular area of 61 mm×134 mm(as shown in Fig.2(b))was recorded by the high-speed CMOS camera.Crosscorrelation analysis was performed on two adjacent images to obtain the transient velocity distribution.The images were analyzed with the fast Fourier transformation (FFT) in 2 passes using interrogation area (IA) [19].The camera frequency was fixed at 430 Hz (Δt= 2.33 ms), which can ensure that the maximum inplane and out-of-plane displacements of the seeding particles are less than one quarter of the size of IA and the diameter of the light sheet[20].The PIV-lab in the commercial software of MATLAB tool was used to analyze the velocity field quantitatively [21].In a 2D PIV system, the tangential component of fluctuating velocity can be estimated using a pseudo-isotropic assumption and the turbulent kinetic energy [22]can be calculated by

whereandare fluctuating radial and axial velocities,respectively.

3.Numerical Simulation Approach

3.1.Governing equations

In recent years, large eddy simulation (LES) has been widely used to predict turbulent flows [23].The filtered Navier-Stokes equations are the fundamental governing equations for simulation of turbulent flow [24].The following equations are formally derived by applying a low-pass filter to the incompressible continuity and to the Navier-Stokes equations in three-dimensional forms [25]:

Fig.1. Operation modes and structure diagram of SBs of LSB impeller.(a)Up-pumping mode;(b)Down-pumping mode;(c)30°pitched SBs(30D and 30U);(d)45°pitched SBs (45D and 45U); (e) 60° pitched SBs (60D and 60U); (f) Vertical SBs (90V).

Fig.2. Schematic diagrams of the experimental setup.(a)PIV test system,(b)Side view of the plane Pπ/4.(1)Stirred tank;(2)Jacket;(3)Torque meter;(4)Motor;(5)Speed controller; (6) Torque recorder; (7) Thermostat water system; (8) CMOS camera; (9) Continuous laser; (10) Post-processing system.

where μ is the kinematic viscosity of the fluid,uis velocity,pis pressure andis the large scale strain-rate tensor,which is defined as

In this work, dynamic kinetic energy (DKE) model was used as SGS model.The DKE model solves the transport of the subgrid scale turbulence kinetic energy,which has been successfully applied to the simulation of turbulent wall bounded flows.The SGS eddy viscosity, μtis computed usingksgs,, where Δfis the filter size computed asV1/3.

Then, the SGS stress can be given as:

The transport equation for SGS kinetic energy is written as:

In the above equations, the model constants,CkandCε, are determined dynamically [23].σkis equal to 1.0.

3.2.Numerical simulation parameters

In the simulations,the geometric parameters used are identical to those in the PIV experiments.The detailed simulation operation conditions are listed in Table 1.The volume of the vessel was divided into a rotating part and two stationary parts, according to the requirements of the sliding mesh (SM) method.According to the previous work [16,17], the number and size of grids used in this paper can give quite reasonable results.The rotating and stationary zone are connected by mesh interfaces.The free water surface was set symmetric boundary.The second-order implicit method was used in time discretization.The second-order upwind scheme was applied for spatial discretization.The coupling between the continuity and momentum equations was realized by using the pressure implicit with splitting of operators (PISO)algorithm.The steady standardk-ε model with multiple reference frame(MRF)was used to obtain initial values for transient simulation.The time step was set to 0.005 s.In order not to affect the results, the time averaging must begin when a pseudo-stationary state has been achieved [27].

4.Results and Discussion

4.1.Prediction accuracy assessment

LES requires that the flow adjacent to the wall is laminar flow,which means that the dimensionless wall distance,Y+,is generally lower than 5.In this study,theY+values of all cases are calculated.Fig.3 shows theY+values are less than 2 in most regions and less than 5 in all regions, which implies that good simulation accuracy can be expected.

In addition,the Taylor microscale and Kolmogorov length scale are calculated to evaluate the grid resolution[16,27,28].The Taylor microscale,LT=(15νu2/ε)1/2, is a measure of the eddy size in the inertial subrange.The Kolmogorov length scale,LK=(ν3/ε)1/4, is the size of the smallest eddies present in the flow.The calculated results are listed in Table 1.The Taylor microscale is 3-5 mm,which is close to the maximum grid size, and the Kolmogorov length scales are all larger than the corresponding minimum grid size.Therefore, it is convinced that the grid resolution can meet the requirements of LES predictions.

The predicted time-averaged radial and axial velocities are then normalized by the impeller tip velocity,Utip, and compared with the experimental values at two selected radial positions,r/R= 0.7 and 0.9, as shown in Fig.4.It is seen that the predicted profiles are in good agreement with the experimental results.

Fig.4. Axial profiles of the dimensionless radial (Ur) and axial (Ua) time-averaged velocity at two radial positions on the plane of Pπ/4.(a, c) r/R = 0.7; (b, d) r/R = 0.9.

In addition to the time-averaged velocities, the comparison of the turbulent fluctuations can better demonstrate the prediction accuracy of the adopted LES approach.Fig.5 compares the predicted radial and axial velocity fluctuations in terms of normalized RMS velocities,Urms,r/Utip, andUrms,a/Utip, with the experimental results.Although the velocity fluctuations are slightly over predicted, the predicted profiles are basically consistent with those from the experimental results.The mean deviation between the prediction and the experiments is of about 20%.This should be due to the difference between the impeller rotation frequency and the image acquisition frequency,as well as the limited resolution capabilities and interrogation area averaging implicit to the PIV measurements,i.e.the filtering out of the small-scale fluctuations leads to underestimation of the TKE by time-resolved PIV[29].

4.2.Power consumption

The power input is an important index to evaluate the performance of the impeller, which can be obtained by torque measurements or CFD predictions.Effective power consumption of the impeller is related to the energy dissipated in the liquid.Thus, in CFD simulations, the power input can be calculated by volume integration of turbulence energy dissipation rate [30], in which the thickness of the blade should be accounted for and resolved properly by the mesh [31].Normalized power input is generally represented by the power number,Np, which is defined as

wherePis the power input,which can be calculated from the torque acted on the impeller,i.e.

whereMis the torque.

When the quasi-steady state was achieved in the simulation,the torques acted on the impeller and the wall of the tank were recorded continuously, respectively.After time averaging operation,the power inputs for all the impellers are obtained and listed in Table 2.It is seen that the difference of torque values on the impeller and on the wall of the tank is no more than 2%.Moreover,the toque was also experimentally measured by a torque meter,and the averaged error between the predictions and experimental results is 2.3%.From Table 2 we can see that the configuration of the SBs has a significant effect on the power number,Np.For a fixed configuration, the impeller operated in an up-pumping mode has anNpvalue of 8%-15%larger than that in a down-pumping mode.When the angle between the SBs and the horizontal plane, β,decreases,i.e.inclination degree of the pitched SBs becomes larger,theNpvalue decrease significantly,e.g.theNpvalue decreases from 8.94 of 90V impeller to 4.95 of 30U impeller.This implies that it provides an important method to regulate the flow field in the tank by designing pitched SBs.

Table 1Grid parameters, operating conditions, Taylor microscale (LT) and Kolmogorov length scale (LK)

Table 2Predicted torques,power number and power consumption per unit liquid volume for different configurations of the impeller

Table 3Axial and radial pumping number for all the impeller configurations under the same power consumption

4.3.Flow patterns

To have a thorough understanding of the effect of impeller configuration on the flow patterns,the time-averaged velocity vectors of the pitched SBs impeller are compared with that of the vertical SBs impeller,as shown in Fig.6.As discussed in our previous study,for a vertical SBs impeller,two radial vortexes are generated by the SBs and then impinge into each other, leading to forming a strong axial upflow in the center.From Fig.6 we can see that these two vortexes are suppressed when the LSB impeller with pitched SBs operated in the down-pumping mode.It is obvious that the impeller of 30D (Fig.6(a)) has the smallest velocity magnitude.Moreover, impingement intensity becomes weaker, leading to less upward flow in the center.Whereas in the cases of up-pumping mode, the axial upflow in the center is intensified by the pitched SBs.One can see that the axial upflow gets to a higher position,e.g.for the impellers of 30U (Fig.6(e)) and 45U (Fig.6(f)), the strong axial upflow gets to the height ofz/H= 0.6-0.8.Under the combined radial pumping action of the LBs, velocity distribution becomes more uniform for the impellers of 30U and 45U.

Fig.6. Time-averaged velocity vectors colored by the contours of the velocity magnitude on the plane of Pπ/4.

Fig.3. Contours of the instantaneous Y+ values.

Fig.7 provides a more intuitive comparison of the axial flow intensity in terms of 3D contour plots of the dimensionless mean axial velocity at different axial positions.In the down-pumping mode, the axial velocities near the wall under the SBs are higher.Compared with the up-pumping mode, the velocity distribution at the bottom of the vessel is more uniform, which indicates that the turbulence will be intensified and the stagnation area reduces at the bottom of the vessel in the down-pumping mode.In the uppumping mode, the axial velocities in the center increase significantly in the region above the pitched SBs, which should be beneficial to the mixing improvement in the whole tank.For example,for the impeller of 30U,significant axial flow can be observed even near the top of the vessel.

To have an overview of the radial and tangential flows in the vessel,the velocity vectors on ther-θ plane ofz/H=0.5 in the vessel with different impellers are compared in Fig.8.There exists an obvious low-speed flow region in the center for the impellers of 30D, 45D, 60D, 90V and 60U.Whereas for the impellers of 30U and 45U, the low-speed flow region disappears and the velocity distributions are more uniform, which indicates that an appropriate inclined angle of the SBs is beneficial to the fluid mixing in the vessel in the case of up-pumping flow mode.

Fig.8. Velocity vectors colored by the dimensionless time-averaged velocity magnitude at a position of z/H = 0.5.

4.4.Turbulent characteristics

Turbulent kinetic energy (TKE) and its dissipation rate are two other important indexes to evaluate the turbulence intensity in the tank.According to Kolmogorov theory, TKE is dissipated from large scale vortices to small scale vortices [32].Fig.9 shows the distributions of the TKE scaled byUtip2on the sampled plane ofPπ/4.It is obvious that the TKE has a maximum value adjacent to both the LBs and the SBs, which is due to the strong shear action by the blades.Moreover, the TKE distribution in the tank with the LSB impellers is more homogeneous than that of RT agitator with the same geometry size [17].In addition, the TKE distributions of the impellers of 30D and 30U are more uneven, which implies that more inclined SBs have less effect on the fluid flow in the tank.For other configurations of the LSB impellers with inclined SBs, regions with higher TKE in the up-pumping modes(e.g.45U or 60U) are larger than those in the corresponding down-pumping modes (e.g.45D or 60D).This indicates that the axial upflow is strengthened by the up-pumping action of the inclined SBs.Furthermore, the TKE distribution in the tank with 45U impeller in the whole tank is the most uniform.

Fig.5. Axial profiles of radial (Urms,r) and axial (Urms,a) velocity fluctuations at two radial positions on the plane of Pπ/4.(a, c) r/R = 0.7; (b, d) r/R = 0.9.

4.5.Flow rates and pumping capacity

To have a quantitative comparison of the axial and radial flows in the tank,the axial flow rates and the radial flow rates are calculated by integrating the axial and radial profiles of the timeaveraged mean axial and radial velocity, respectively.Fig.10(a)shows the averaged axial flow rates upwards and downwards throughout the horizontal cross sections of the vessel at different liquid height,z.It is seen that there is a maximum value atz/T= 0.25-0.3 on the curves of the axial flow rates for all impeller configurations, where the bottom edge of the SBs are located.A second hump appears atz/T= 0.6 for the impellers with inclined SBs operated in the up-pumping modes, which provides a direct proof that the inclined SBs intensify the axial flow in the uppumping modes.

Fig.7. 3D contour plots of the dimensionless mean axial velocity at different axial positions (z/T = 0.1, 0.3, 0.5, 0.7, 0.9, respectively).

To evaluate the relative strength of the axial flow and radial flow in the tank, the ratios of axial flow rate to radial flow rate are calculated and plotted against axial position,z/T, in Fig.10(b).We can see that the ratios at 0.05 ＜z/T＜0.95 for the configurations of 45U and 90V are close to or greater one.Whereas for the other impeller configurations, the axial flow are relative weak, particularly at 0.45 ＜z/T＜0.8.The fluid pumped by the pitched SBs of 45U impeller still has a strong axial flow ability in the upper part of the tank, which is better than 90V impeller.Moreover, the fluid pumped by the pitched SBs of 45U impeller can well penetrate the whole vessel,forming a large axial circulation in the tank.In addition,different from the traditional RT impeller,the generated axial flow interact with radial flow pumped by the LBs,leading to effective mass exchange between the axial and radial flows in the whole tank.

Fig.9. Distributions of the turbulent kinetic energy (TKE) on the plane of Pπ/4.

Fig.10. Time-averaged flow rates of the LSB impeller.(a)Axial direction flow rates per unit area;(b)The ratios of axial flow rate to radial flow rate at different cross sections in the axial direction, y.

This also explains that the 45U impeller has less stagnant area in the middle position of the upper part of the stirred tank compared with other impeller configurations, as sketched in Fig.6(f).One can see that the stagnant area in the middle region of the tank with 30U impeller is also small(Fig.6(e)).However,the ratio of the radial flow to the axial flow is relatively low for the 30U impeller at 0.45 ＜z/T＜0.8,implying that the mass exchange between the axial flow and the radial flow is less effective than that of 45U impeller.Therefore,the impeller of 45U is an appropriate design from a comprehensive consideration of reducing stagnant area,increasing turbulent energy distribution uniformity and improving the mass exchange efficiency in the whole tank.

The flow rate through the sweeping circle of the SBs just below the impeller is generally used to calculate the axial pumping number,NQa[33,34],i.e.

whererrepresents the radius of the impeller.

The radial pumping number,NQr,is calculated from the integral of radial velocity in a cylindrical areaArad[35], where the inner boundary is the sweeping circle of the SBs and the outer boundary is the radial profile where the mean axial velocity changes its direction.

wherehrepresents the height of the impeller.

The axial and radial pumping numbers of all the impeller configurations are listed in Table 3.The results show that the axial flow rate in the up-pumping mode is larger than that in the down-pumping mode for the fixed inclined angle of the SBs.However, the operation mode has less effect on the radial pumping capacity represented by the pumping number.This quantitative comparison gives a further proof that the pitched SBs plays a vital role in intensifying the axial flow in the tank.This intensified axial flow also improves the mass exchange efficiency between the radial and axial flows in the tank.All these are due to the combination of the pitched SBs and the LBs.

5.Conclusions

The flow characteristics in the turbulent stirred tank equipped with improved LSB impellers have been investigated by PIV measurement and LES simulation.The numerical approach using the DKE SGS model and sliding mesh method was first assessed by the analysis ofY+values, Taylor microscale, Kolmogorov length scale, and the prediction results were validated by the velocities and turbulent fluctuations obtained by the PIV experiments.The power consumption,flow fields,turbulent kinetic energy distribution as well as the flow rates and pumping capacity were then discussed to evaluate the effect of the pitched SBs with different inclined angles operated in the up-pumping and down-pumping modes.Main findings are as follows:

(1) For a fixed LSB impeller with pitched SBs,the power number in the up-pumping mode is larger than that in the downpumping mode.When the inclination degree of the pitched SBs becomes larger, theNpvalue decrease significantly.

(2) The two vortexes generated by the SBs are suppressed when the LSB impeller with pitched SBs is operated in the downpumping mode.The turbulence at the bottom of the vessel is intensified, whereas in the up-pumping mode, the axial upflow gets to a higher position in the tank,leading to more uniform distributions of velocity and TKE in the whole vessel.

(3) The prediction results show that the impeller of 45U has the maximum values of axial and radial pumping numbers, and has the most uniform distributions of flow field and TKE,which should be attributed to the intensification of the upflow in the center by the pitched short-blades and effective mass exchange between the axial and radial flows.

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

We acknowledge the financial support from the National Natural Science Foundation of China(22078058).We also thank the Big Data Center of Southeast University for providing the facility support on the numerical simulations in this work.

Nomenclature

Coff bottom clearance, mm

Ddimeter of the agitator, mm

Hliquid height, mm

hthe height of the impeller, m

LKKolmogorov length scale, m

LTTaylor microscale, m

Mtorque, N m

Nrotation speed, s-1

Nppower number

NQpumping number

kturbulent kinetic energy, m2·s-2

ksgssubgrid scale turbulent kinetic energy, m2·s-2

Ppower input, W

Pπ/4the plane in the middle of two successive baffles

Qflow rate, m3·s-1

Rradial of the tank, mm

rradial coordinate, mm

Tdiameter of the tank, mm

Utime-averaged velocity, m·s-1

UrmsRMS velocity, m·s-1

Utipimpeller tip velocity, m·s-1

uradial velocity, m·s-1

u’radial fluctuating velocity, m·s-1

Vliquid volume, mm3

vaxial velocity, m·s-1

Y+dimensionless wall distance

ythe ratio of axial flow rate to radial flow rate

zaxial coordinate, mm

β the angle of pitched short blades

ε the rate of kinetic energy dissipation, m2·s-2

μ viscosity, mPa s

vkinematic viscosity, m2·s-1

ρ density, kg·m-3

Subscripts

i, jx, y, zcoordinate direction

im impeller of the tank

w wall of the tank