Meng Li,Yangbo Tan,Yufeng Liu,Jianglong Sun,2,3,De Xie,2,3,Zeng Liu,2,3,*

1 School of Naval Architecture and Ocean Engineering,Huazhong University of Science and Technology,Wuhan 430074,China

2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration(CISSE),Shanghai 200240,China

3 Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics(HUST),Wuhan 430074,China

Keywords:Agitation characteristic curve Geometrical and physical factors Light particles Solid-liquid mixing Stirred tank

ABSTRACT In this study,the effects of geometrical and physical factors on light particles dispersion in stirred tank were investigated by agitation characteristic curve.The experiments and CFD simulations with discrete phase model(DPM)and volume of fluid model(VOF)were conducted in this paper.Five factors,which include four geometrical factors(submergence,impeller-to-tank ratio,number of impeller blades and baffling mode)and a physical factor(liquid viscosity)were considered.For each factor,the power consumption curve and agitation characteristic curve were drawn to compare the power consumption and mixing results in the stirred tank.Characteristics of the agitation characteristic curves were compared with the previous published literatures and theories.It is found that the agitation characteristic curves reflect the tendency of power consumption and particles distribution well in stirred tank.The good agreement indicates the applicability of the agitation characteristic curves for the study of light particles distribution in stirred tank.

1.Introduction

Mechanical stirring is used in many fields including chemical process,biological field,pharmaceuticals industry and steelmaking process.Tanking the KR mechanical stirring process as example,the melted iron is stirred by impeller to react with the sorbent particles to reduce the sulfur content in the melted iron.Usually,the four-blade impeller is used in KR(Kanbara Reactor)mechanical process.

To improve the efficiency of desulfurization in the KR process,various studies have been carried out.The KR desulfurization waste slag is a typical solid waste generated in the integrated steel company.Aiming at the effective utilization of KR slag in the iron ore sintering process,the characteristic of KR slag[1,2]and a novel valorization process of KR desulfurization waste slag[3]was studied.Ji et al.[4]paid attention to the rotation speed during the KR process and proposed a variable-velocity stirring method to improve the desulfurization efficiency of highsulfur hot metal for KR desulfurization process.And Ouyang et al.[5]developed a new type WG-3Y impeller stirrer to draw down light particles more efficiently than the traditional ones.

The KR process is a typical light-particle mixing process in stirred tank,as the melted iron(7000 kg·m-3)is heavier than the desulfurizer(3500 kg·m-3).Therefore,higher mixing efficiency of light particles in the stirred tank means more efficient KR desulfurization process.To improve the mixing efficient of light particles,many researchers have studied the drawdown mechanisms in the stirred tank.Three different mechanisms,single vortex,turbulent fluctuations and mean drag,were analyzed by Khazam and Kresta[6].Gül ?zcan-Taskin[7]reported the three drawdown modes,recirculation loop,surface aeration and main circulation loop,for both up and down pumping modes.Besides,the configurations of the stirred tanks and properties of the liquid and particles have also been considered.Based on the CFD methodology,Qiao et al.[8]studied the effects of geometrical design and physical property on the mixing efficiency of floating solids in stirred tanks with an up-pumping pitched blade turbine.Waghmare et al.[9]proposed a scale up correlation using CFD simulations for floating solids drawdown operation in stirred tanks.

Not only the mixing efficiency but also the power consumption in the stirred tank was concerned by many researchers.Obviously,less power consumption and better mixing results is preferred in the stirred tank.Scargiali et al.[10,11]studied the influence of the Reynolds and Froude numbers on the power consumption characteristics for unbaffled stirred tanks.And Taghavi[12]studied the local and total power consumption of a single phase and gas-liquid phase system in a fully baffled stirred tank.

The particle dispersion curve and the power consumption curve of light particles[13]in stirred tank were studied.And the drawdown mechanism of light particles for KR impeller[14]was analyzed.In this paper,the agitation characteristic curve that is made up of the Np curve and particles dispersion curve is considered for different geometrical and physical factors.For the geometrical factors,the submergence,impeller-to-tank ratio,number of impeller blades,and baffling mode are considered.The change of submergence[14]changes the circulation pattern in the stirred tank.And impeller-to-tank ratio,number of impeller blades and baffling mode affect the power consumption in the stirred tank.For the physical factor,the liquid viscosity is considered since the stirring state changes with the stirring liquid.The main purpose of this paper is to verify that the agitation characteristic curve can be used to predict the power consumption and particles dispersion in the stirred tank.

In Sections 2 and 3,the water model set-up and the computational fluid dynamics(CFD)methods were introduced.In Section 4,the power consumption and particles distribution of five factors were analyzed and compared with the results in other literature,respectively.Finally,the conclusion was shown in Section 5.

2.Experimental

In this paper,the purified water with density ρ=997 kg·m-3and dynamic viscosity μ=0.001 Pa·s was used to simulate the melted iron and 1000 EVA(ethylene-vinyl acetate copolymer)particles(density ρp=920 kg·m-3and diameter dp=0.003 m)were used to simulate the desulfurizer.The experimental set-up is shown in Fig.1.Experiments were carried out in a cylindrical plexiglass stirred tank with a diameter(T)of 0.29 m and height(H)of 0.40 m.The liquid height hlis 0.29 m(equals to T).Two kinds of stirred tanks were studied:zero and four baffling mode with the width of T/10.The submergence(S)was defined as the distance from the liquid surface to the centerline of the impeller in axial direction and S=1/3 T and 2/3 T were considered.To have a contrast in the impeller-to-tank ratio and number of impeller blades,three impellers were used in the experiments.The scale drawings of three and four blade impellers were shown in Fig.2.The detailed dimensions of all impellers are given in Table 1.The impeller shaft was driven by the LB2 L2Y15N Mixer from Lightnin.

The power consumption in the stirred tank can be calculated by:

Fig.1.The experimental setup:(1)flat bottom stirred tank,(2)impeller,(3)baffle,(4)electromotor,(5)self-propulsion apparatus,(6)computer,(7)high definition camera.

Fig.2.Scale drawing of the impellers.(a)4 blade impeller,(b)3 blade impeller.

where M is the torque on the impeller.Taking the accuracy into consideration,we measure the torque(M0)when the shaft rotated without impeller.And the final torque was calculated by:

where the MEis the torque on the impeller which was measured by selfpropulsion apparatus(CUSSONS)in the experiments.

The non-dimensional power number(Np),which depends on fluid properties and on the geometrical parameters of the impeller,is calculated by:

where ρ is the density of the fluid;N is the impeller rotational speed and D is the diameter of the impeller.

The curves of non-dimensional power number and particles dispersion were described by the Reynolds number in the agitation curves.And the Reynolds number was calculated by:

Table 1 Details of impellers

where μ is the fluid viscosity.

Following the previous work of Liu et al.[13],we take a photo of the mixing results in the stirred tank with a high definition camera.Then,we divide the liquid phase in the photos into three parts uniformly in axial direction and count the number of particles in each part.The particles dispersion was calculated by[13]:

where n is the number of layers;i is the i-th layer;xiis the number of particles in the i-th layer;viis the volume in the i-th layer;X is the number of total particles;and V is the volume of the liquid phase.

3.Numerical Simulation

The ANSYS(Pittsburgh,Pennsylvania,USA)Fluent(version 16.1)CFD commercial software package was used in the numerical simulations.In the stirred tank,there are two main phases:air and liquid.All the particles are immersed into the liquid through the liquid surface.Therefore,the liquid surface is important in the simulations.To simulate the liquid surface,the volume of fluid(VOF)method which can simulate two or more immiscible fluids was adopted.Based on the conservation principles,the continuity equation can be written as follows:

where αjstands for volume fraction,ρ anddenote density and velocity,respectively.In every control volume,the volume fractions of all the phases sum to one.And the variables and properties in any given cell represent either single phase or mixed phases,depending upon the

volume fraction values.The momentum equation is given below:

where the p and μ are pressure and viscosity,respectively.

To simulate the movement of the particles in the stirred tank,the discrete phase model(DPM)was adopted.Not only buoyancy force and gravitational force but also added mass force and pressure gradient force were considered in our simulations.Besides,the surface tension force which keeps the particles on the liquid surface was added to the simulation process.The trajectory of the discrete phase particle was predicted by integrating force balance and this force balance equates the particle inertia with the forces acting on the particles and can be written as:

the particle relaxation time[15],is the particle velocity;is the liquid velocity;μ is the fluid viscosity;ρ is the liquid density;ρpis the particle density and dpis the particle diameter.The relative Reynolds number Re′is defined as:

The added mass force which accelerates the fluid around the particle can be written as:

Fig.3.The instantaneous velocity magnitude along the diameter line near the liquid surface for 300 k,500 k,800 k and 1000 k grids.

Table 2 The torque on the four blade impeller at S=1/3 T in four baffled stirred tank

where Cvmstands for the virtual mass factor with a value of 0.5 in Fluent.And the pressure gradient force can be expressed as:

3.1.Solution procedure and boundary conditions

Based on the size of the model in the experiments,a threedimensional model was drawn in CFD.To solve the nonlinear governing equations,a finite volume based on fluid dynamic analysis program is used in Fluent.The grids were generated by ICEM.Grid independence was conducted at S=1/3 T and N=300 r·min-1in four baffled stirred tank.Fig.3 shows the instantaneous velocity magnitude along the diameter line near the liquid surface with 300 k,500 k,800 k and 1000 k grids considered.It is found that the velocity magnitude profile along the diameter line for 800 k and 1000 k grids match well with each other.Therefore,800 k grids was chosen in this work.The sliding mesh method was used to simulate the impeller rotation.The full domain is divided into two blocks:an inner one moving with the impeller and an outer motionless one.The two subdomains share the same inertial frame of reference with the inner domain rotating with time.

The no-slip boundaries with a standard wall function were assumed for all the tank walls.And reflecting mode was chosen in DPM boundary condition for all parts of the model.The standard k-ε turbulence model was chosen to simulate the turbulent flows in the stirred tank.The time step was set as 0.001 s.The first-order upwind scheme for governing equations was applied and the residual for each time step was set to 10-4.Besides,the torque on the impeller was monitored to decide whether the steady state was reached.

3.2.Power measurement

The power consumption is one of the focuses in our simulations.Therefore,an accurate CFD model is necessary to predict the power input in the stirred tank.The shear stress and the pressure distribution on the impeller blade are resolved in Fluent.Then power consumption can be calculated from the total torque required to rotate the impeller.The torque on each blade can be calculated by[16,17]:

where m stands for the m-th control cell on each blade;Δp is the pressure difference between the front and back sides of the blade at the m-th cell;rmis the radial distance from the shaft to the m-th cell.Then the power consumption in Fluent is calculated by:

Table 2 shows the comparison of torques on the four blade impeller between the experimental and numerical results at S=1/3 T in four baffled stirred tank.The relative error(δ)is defined as:

where the Tnand Testand for the torque on the impeller in numerical simulations and experiments,respectively.As shown in Table 2,the relative errors are small than 10%.Therefore,the simulation torques agree well with the related experimental records.

4.Results and Discussion

4.1.Geometrical factors

4.1.1.Submergence

The change of impeller submergence has an influence on the power consumption and particles dispersion.Qiao et al.[8]studied the effect of impeller submergence on particles dispersion and reported that a decrease in submergence decreased the difficulty in complete suspension of floating solids.

Fig.4.Power consumption of four blade impeller with r=48.5 mm at S=1/3 T and 2/3 T in the unbaffled stirred tank in the water.

Fig.5.The agitation characteristic curves of four blade impeller with r=48.5 mm at S=1/3 T and 2/3 T in the unbaffled stirred tank in the water.

The experiment of four blade impeller with r=48.5 mm at S=1/3 T and 2/3 T in the unbaffled stirred tank in the water was conducted.Fig.4 shows the power consumption with the rotation speed in the stirred tank.The curves of power consumption for S=1/3 T and 2/3 T are close to each other.And the curve of S=2/3 T is above the curve of S=1/3 T for all rotation speeds.It means the power consumption of large submergence is larger than that of small submergence.As for the critical agitation speed reported in the work of Scargiali et al.[10],it can't be observed evidently in Fig.4.The reason is that the number of sample point around the critical agitation speed is not dense enough.

In Fig.5,the agitation characteristic curves for the four blade impeller at S=1/3 T and 2/3 T in the unbaffled stirred tank were drawn.The Np curves of S=1/3 T and 2/3 T decrease in parallel and the curve of S=2/3 T is above that of S=1/3 T.Notably,at Re=47045.0(N=300 r·min-1)in the Np curve of S=1/3 T,the decreasing tendency of the curve changes suddenly.The same phenomenon can be observed at Re=54885.8(N=350 r·min-1)in the curve of S=2/3 T.According to the study of Scargiali et al.[10],there are two reasons for that.First,with the increase of rotation speed,the air will be digested into the liquid phase,as confirmed by visual observation in our experiments.Second,with the impeller rotation in the unbaffled stirred tank,there is an evident liquid vortex and the bottom will reach to the impeller with the increase of rotation speed.Then,the large portion of the impeller blades is no longer in contact with the liquid phase and unable to contribute to torque.

Fig.6.Power consumption of four blade impeller with r=48.5 and 72.5 mm at S=1/3 T in the unbaffled stirred tank in the water.

Fig.7.The agitation characteristic curves of four blade impeller with r=48.5 and 72.5 mm at S=1/3 T in the unbaffled stirred tank in the water.

The particles dispersion curves of S=1/3 T and 2/3 T were shown in Fig.5.The curve of S=1/3 T keeps steady from Re=15681.7(N=100 r·min-1)to Re=23522.5(N=150 r·min-1).Then it decreases from 0.8 to 0.4 between Re=31363.3(N=200 r·min-1)and Re=47045.0(N=300 r·min-1).Finally,the curve keeps steady at 0.4 after Re=47045.0(N=300 r·min-1).The curve of S=2/3 T keeps steady from Re=15681.7(N=100 r·min-1)to Re=31363.3(N=200 r·min-1).Then,it decreases from 0.8 to 0.2 between Re=39204.2(N=250 r·min-1)and Re=54885.8(N=350 r·min-1).At last,the curve fluctuates around 0.2 after Re=54885.8(N=350 r·min-1).Seeing from the value of the dispersion curves at high Reynolds number in Fig.5,the particles distribution of S=2/3 T is better than that of S=1/3 T.When the impeller was placed at S=2/3 T,the particles will be drawn down to lower position with the flow field circulation which is incurred by the rotating impeller than that at S=1/3 T.Therefore,at large submergence,the particles distribution will be better for high Reynolds number.Besides,the dispersion curve of S=1/3 T decreases to lower value than that of S=2/3 T at Re=31363.3(N=200 r·min-1)and Re=47045(N=300 r·min-1).When the impeller is placed at small submergence,the flow path length from the impeller to the liquid surface decreases,leading to a dual effect:the decay of turbulence eddies reduces,and the liquid velocity increases near the liquid surface[10].Therefore,when the impeller is placed at small submergence,it is easier to draw down light particles.

Fig.9.The agitation characteristic curves of four blade and three blade impeller with r=48.5 mm at S=1/3 T in the unbaffled stirred tank in the water.

Then we connect the dispersion curves with the Np curve for S=1/3 T and 2/3 T in Fig.5,respectively.For S=1/3 T,the sudden change of the Np curve occurred at Re=47045.0(N=300 r·min-1)and the dispersion curve keeps steady after Re=47045.0(N=300 r·min-1)as well.For S=2/3 T,the sudden change of the Np curve occurred at Re=54885.8(N=350 r·min-1)and the dispersion curve keeps steady after Re=54,885.8(N=350 r·min-1)as well.Therefore,the correction and relevance of the Np curve and dispersion curve are checked for different submergence.And the characteristics of the agitation characteristic curves for S=1/3 T and 2/3 T reflect the difference about the power consumption and particles dispersion caused by submergence in the stirred tank.

4.1.2.Impeller-to-tank diameter ratio

The effect of impeller-to-tank diameter ratio(λ)was discussed in this part.We conduct the experiments of the four blade impeller with r=48.5 and 72.5 mm,corresponding to the impeller-to-tank diameter ratio:λ=1:3 and λ=1:2,at S=1/3 T in the unbaffled stirred tank in the water.For the small impeller,the rotation speed ranges from 100 to 500 r·min-1.To have a contrast in the Reynolds number,the rotation speed of large impeller ranges from 50 to 300 r·min-1.

Fig.6 shows the power consumption of the four blade impeller with r=48.5 and 72.5 mm at S=1/3 T in the unbaffled stirred tank in the water.The power consumption curve of impeller with r=72.5 mm is above the curve of impeller with r=48.5 mm.Namely,at the same rotation speed,the power consumption increases with the impeller-totank diameter ratio.

Fig.10.Power consumption of four blade impeller with r=48.5 mm at S=1/3 T in the unbaffled and four baffled stirred tanks in the water.

Fig.11.The agitation characteristic curves of four blade impeller with r=48.5 mm at S=1/3 T in the unbaffled and four baffled stirred tanks in the water.

In Fig.7,the agitation characteristic curves of the four blade impeller with r=48.5 and 72.5 mm at S=1/3 T in the unbaffled stirred tank in the water were shown.Obviously,the Np curve of impeller with r=72.5 mm decreases in two steps:from Re=17520.8(N=50 r·min-1)to Re=52562.5(N=150 r·min-1);from Re=70083.3(N=200 r·min-1)to Re=105125(N=300 r·min-1).The particles dispersion curve of impeller with r=72.5 mm keeps steady from Re=17520.8(N=50 r·min-1)to Re=35041.7(N=100 r·min-1).Then,the curve decreases from Re=35041.7(N=100 r·min-1)to Re=70083.3(N=200 r·min-1).The curve fluctuates around 0.35 after Re=70083.3(N=200 r·min-1).Connecting the Np curve with the particles dispersion curve of impeller with r=72.5 mm,the particles dispersion curve keeps steady after Re=70083.3(N=200 r·min-1)which corresponds to the second parts in the Np curve.

By contrast,at the same Reynolds number,the value of the Np curve of impeller with r=72.5 mm is smaller than that of impeller with r=48.5 mm.So,to reach the same turbulent flow field in the stirred tank,the power consumption of impeller with r=72.5 mm is much smaller than that of impeller with r=48.5 mm.And,the particles dispersion curve of impeller with r=48.5 mm starts to decrease at smaller Reynolds number than that of impeller with r=72.5 mm.Therefore,although the speed required to draw down solids is lower for the larger diameter impeller,the power requirement is higher[18].And the characteristics of the agitation characteristic curves for λ=1:3 and 1:2 reflect the difference about the power consumption and particles dispersion caused by impeller-to-tank diameter ratio in the stirred tank.

4.1.3.Number of impeller blades

Different numbers of impeller blades will have different mixing results and agitation curves.In this part,we compare the power consumption curves and agitation curves of four blade and three blade impellers with r=48.5 mm at S=1/3 T in the unbaffled stirred tank in the water.

The power consumption curves of three and four blade impellers with r=48.5 mm at S=1/3 T in the unbaffled stirred tank in the water were shown in Fig.8.The power consumption curve of three blade impeller is below that of the four blade impeller.Therefore,at the same rotation speed,the power consumption of three blades will be smaller than four blades because the stress surface of the three blade impeller is smaller than that of four blades.

The agitation curves of three and four blade impellers with r=48.5 mm at S=1/3 T in the unbaffled stirred tank in the water were compared in Fig.9.The Np curve of three blade impeller decreases from Re=15681.7(N=100 r·min-1)to Re=47045(N=300 r·min-1)and decreases from Re=47045(N=300 r·min-1)to Re=78403.8(N=500 r·min-1)with a sudden slope change at Re=47045(N=300 r·min-1).Therefore,the decreasing tendency of the Np curve of the three blade impeller is the same with that of the four blade impeller.

Fig.12.The initial state of the particles(the red points)and liquid surface(the gray surface around the particles).

The particles dispersion curve of the three blade impeller has three stages as shown in Fig.9.The particles dispersion curve of the three blade impeller keeps steady from Re=15681.7(N=100 r·min-1)to Re=23522.5(N=150 r·min-1)and decreases from Re=31363.3(N=200 r·min-1)to Re=47045(N=300 r·min-1).Finally,the curve keeps steady after Re=47045(N=300 r·min-1).Besides,comparing the particles dispersion curve of the three blade impeller to that of the four blade impeller,they are nearly coincident.Hence,the influence of number of impeller blades on the power consumption and particles dispersion is not evident in this paper.And,the agitation curves of the four blade and three blade impellers reflect the results of power consumption and particles dispersion curves excellently.

4.1.4.Baffling modes

In the work of Khazam and Kresta[6],it reported that the fully baffled configuration has good solid distribution and the absence of a single stable surface vortex which exists in the unbaffled stirred tank.Therefore,the agitation curves in four baffled and unbaffled stirred tanks are compared in this part.

In Fig.10,the power consumption of the four blade impeller with r=48.5 mm at S=1/3 T in the unbaffled and four baffled stirred tanks in the water are presented.The power consumption curves in the four baffled stirred tank is above the curve in the unbaffled stirred tank.Because of the baffles in the stirred tank,the impeller torque will increase and the power consumption will be augmented.Therefore,the power consumption in four baffled stirred tank will be much larger than that in the unbaffled stirred tank.

Fig.13.The central vertical plane of the stirred tank at different time(red part:the air;blue part:the water).

Fig.14.Power consumption of four blade impeller with r=48.5 mm at S=1/3 T in the four baffled stirred tank in the water and high viscosity liquid.

The agitation curves of four baffled and unbaffled stirred tanks are shown in Fig.11.The Np curve in four baffled stirred tank keeps constant at 7.2 from Re=15681.7(N=100 r·min-1)to Re=78403.8(N=500 r·min-1).The Np curve decreases in the unbaffled stirred tank because the flow field in the unbaffled stirred tank is akin to that of fluid flow into smooth pipes,for which the friction factor is steadily decreased at all Re numbers.However,the baffled stirred tank is similar to rough pipes,where the friction factor eventually settles on a steady value[10].The value of the Np curve in the four baffled stirred tank is larger than that in the unbaffled stirred tank and it reflects the large power consumption in the four baffled stirred tank.

The particles dispersion curve in the four baffled stirred tank decreases from Re=15681.7(N=100 r·min-1)to Re=47045(N=300 r·min-1)and keeps steady at 0.2 after Re=47045(N=300 r·min-1).Compared to the particles dispersion curve in the unbaffled stirred tank,the particles dispersion curve in the four baffled stirred tank decreases at smaller Reynolds number and keeps constant at smaller value finally.Therefore,the particles distribution in the four baffled stirred tank is more efficient and better than that in the unbaffled stirred tank.And the agitation curves in the four baffled stirred tank and unbaffled stirred tank reflect the influence of the baffles in the stirred tank correctly.

4.2.Physical Factors Liquid viscosity

Fig.15.The agitation characteristic curves of four blade impeller with r=48.5 mm at S=1/3 T in the four baffled stirred tank in the water and high viscosity liquid.

To study the adaption of the agitation curve in different liquid in the stirred tank,we set the liquid in the stirred tank as the tap water(ρ=998 kg·m-3and μ=0.001 Pa·s),high viscosity liquid(ρ=998 kg·m-3and μ=0.0041 Pa·s)and low viscosity liquid(ρ=998 kg·m-3and μ=0.0008 Pa·s)in the CFD simulations.Fig.12 shows the initial state of the particles(the red points)and liquid surface(the gray surface around the particles).The central vertical planes of the stirred tank at different time are shown in Fig.13.Because of the effect of baffles,the fluctuation of liquid surface is not intense in Fig.13.

Fig.14 shows the power consumption of the four blade impeller with r=48.5 mm at S=1/3 T in the four baffled stirred tank in the water,high and low viscosity liquid.Three power consumption curves are close to each other.Therefore,the difference of the power consumption between water,the high and low viscosity liquid is not evident because the difference of the liquid viscosity is not large.

The agitation characteristic curves of the four blade impeller with r=48.5 mm at S=1/3 T in the four baffled stirred tank in the water,high and low viscosity liquid were shown in Fig.15.The value of the Np curve in the high viscosity liquid keeps constant at 6.8 because the flow field in the stirred tank is in the state of turbulent flow.And the value of the curves in the tap water and low viscosity liquid fluctuates around 7.1.Three Np curves are nearly constant straight line.

The particles dispersion in the high and low viscosity liquid decreases from 0.8 to 0.1 from Re=3824.8(N=100 r·min-1)to Re=19124.0(N=500 r·min-1)and Re=19602.1(N=100 r·min-1)to Re=98010.4(N=500 r·min-1),respectively.Compared with the particles dispersion curve in the water,the basic tendency of the particles dispersion curves is similar.However,at the same Reynolds number,the value of particles dispersion curves decreases with the increase of liquid viscosity.Because,at the same rotation speed,the Reynolds number decreases with the increase of liquid viscosity.The agitation curves in the water,high and low viscosity liquid reflect the tendency of the Np curve and the particles distribution in the stirred tank accurately.

5.Conclusion

To study the adaption of agitation characteristic curves of light particles in the stirred tank,the experiments and CFD simulations were conducted in this paper.Five factors,which include four geometrical factors(submergence,impeller-to-tank diameter ratio,number of impeller blades and baffling mode)and a physical one(liquid viscosity)were considered.Both the power consumption curves and the agitation characteristic curves that are made of Np curve and particles dispersion curve are analyzed.It is found that the agitation characteristic curves represent well the power consumption and particles dispersion for different factors.In our following work,more samples and factors will be considered to study the agitation characteristic curve.

Nomenclature

amarea,m2

Cvmvirtual mass factor

D diameter of the impeller,m

dpdiameter of the particle,m

F additional force,N

Fppressure gradient force,N

Fvadded mass force,N

H height of the stirred tank,m

h blade height,mm

hlheight of still liquid,m

i i-th layer

k turbulent kinetic energy,m2·s-2

M torque on the impeller,N·m-1

MEtorque measured by experiments,N·m-1

M0unloaded torque,N·m-1

m m-th control cell on blade

N rotation speed,r·min-1

Npnon-dimensional power number

n number of layers

P power consumption,W

p pressure,Pa

Δp pressure difference,Pa

r radium of the impeller,mm

rmradial distance,m

Re Reynolds number

Re' relative Reynolds number

S submergence,m

T diameter of the stirred tank,m

Tetorque in experiments,N·m-1

Tntorque measured by simulations,N·m-1

ujvelocity of phase in simulation,m·s-1

upparticle velocity,m·s-1

V volume of the liquid phase

vivolume in i-th layer

w blade width,mm

X number of total particles

xinumber of particles in i-th layer

αjvolume fraction

δ relative error

ε turbulent dissipation rate,m2·s-3

θ intersection angel of impeller,(°)

λ impeller-to-tank diameter ratio

μ viscosity of the liquid,Pa·s

ρ density of the liquid,kg·m-3

ρpdensity of the particles,kg·m-3

σ particles dispersion

τ torque on each blade,N·m-1

τrparticles relaxation time,s