Synergistic and interference effects in coaxial mixers:Numerical analysis of the power consumption☆

Juan Huang *,Gance Dai

1 School of Perfume and Aroma Technology,Shanghai Institute of Technology,Shanghai 201418,China

2 State Key Laboratory of Chemical Engineering,East China University of Science and Technology,Shanghai 200237,China

1.Introduction

Mixing tanks equipped with rotating impellers exist extensively in chemical,food,cosmetic,pharmaceutical industries etc.Among them,there are some complex mixing problems,such as the mixing of highly viscous and variable viscosity fluids[1],mixing with multi-objects[2],mixing in multifunctional reactors[3–6].Single impeller may not be suitable for the above situations because each has its own specific function( flow to shear ratio)in the fluid mixing.Every impeller design creates some balance of flow and shear.It is almost impossible to have strong flow circulation and high shear rate in a single impeller stirred tank for a given power consumption.Coaxialmixers with two separately driven impellers in one vessel can offer additional adjustable parameters which makes the design of this kind of impeller flexible.The adoption of the composite or coaxial mixers may be bene ficial for solving the aforementioned problem,or even enhancing the mixing efficiency[7].

Intensive studies on this subject have been reported from Tanguy and his co-workers[8–14],then Liu et al.[15,16]and Bao et al.[17]in China have done some work.The majority of the studies have been on the generalization of the coaxial mixer power number correlation[4,9–14,18].Pakzad et al.[19]reviewed the development and scope of application of the power correlations from the early age systematically.The establishment of the generalized power curve poses an elegant solution for the power consumption of the composite impeller;there is still a significant gap in knowledge regarding the reliability of the master curve approaches proposed above[20].Besides,it is apparent that the total power obtained by the generalized power number equation could not be used to determine the separate power of the inner and outer impellers in coaxial mixing systems.The overall power input has to be divided into two parts:the power input of the inner impeller and outer impeller.Therefore,it might be a good idea if we change the way to the power data processing.

With regard to the effect of each impeller on the other one,the results got from the literature were mainly that the outer impeller speed does not have any effect on the power drawn of the central impeller,while the central impeller speed was shown to affect the inner impeller power consumption for both co-and counter-rotating modes[11,18,20].The topic of interaction between the inner and outer impellers was hardly covered.

Up to now,the anchor was mostly used as the outer impeller.The helical ribbon and its modi fied family(Paravisc)have been employed by a few researchers[8,21].Coaxial mixing systems equipped with pitched blade turbine(PBT)-helical ribbon(HR)and inner-outer helical ribbon(IOHR)were studied in this paper,especially,the latter one which has not been reported in the literature.The separate power characteristics of each individual impeller and the partition of the total power input between inner and outer agitator were investigated.Furthermore,some important parameters were suggested and the interaction between component impellers,i.e.the synergistic and interference effects were revealed quantitatively.

2.Experiments

2.1.Experimental setup

The experiments were carried out in a tank of 400-and 200-mm diameter,respectively.A schematic diagram of the setup is shown in Fig.1.The coaxial mixers were composed of a central impeller and an outer impeller supported by two shafts rotating independently at different speeds and directions.Pitched blade turbine(PBT)and inner helical ribbon(IHR)were utilized as the central impeller.Helical ribbon(HR)was employed as the outer impeller.The specifications of these two coaxial mixing systems are listed in Table 1.A 2.2-kW gear-drive motor and a 2.2-kW direct drive motor were employed to rotate the outer and central impellers,respectively.Two frequency inverters(Great wall)controlled these variable speed motors.

2.2.Experimental fluid and methods

The working fluids were various corn syrup solutions at different concentrations.The viscosity was varied in the range of 2–11 Pa·s which was measured by the Brabender rheometry(Brabender Germany);the density was in the range of 1350-1416 kg·m-3.The temperature of the syrup was measured by the mercury thermometer after each run of the experiment in order to determine the accurate viscosity of the fluid.

Fig.1.Schematic of the coaxial mixers.(a)PBT-HR coaxial mixer;(b)HR impeller;(c)PBT impeller(d)IOHR coaxial mixer 1-electric motor for outer impeller;2-torque meter for outer impeller;3-HR or OHR;4-PBT or IHR;5-torque meter for inner impeller;6-electric motor for inner impeller;7-stirred tank.

Table 1 Geometric parameters for helical ribbon and pitched blade turbine

The torque exerted on the shafts was measured by non-contact torque meter(HX-901,Huaxin Electromechanical Ltd.Beijing,China).The measurement range was 0-20 N·m and 0-10 N·m for the upper and lower torque meter,respectively.The speed was measured by the online tachometer.The following relation gives the power consumption:

Here,M isthe torque,N·m,itwas calculated by subtracting the friction torque from the displayed torque.To measure the friction torque,the shafts were rotated at different rotational speeds in the empty tank,subsequently,the torque of the central and anchor impellers were obtained atdifferentspeed ratios.ω=2πN,N is the rotationalspeed,r·s-1.The rotational speed of the HR and PBT was in the range of 15–70 r·min-1and 30–900 r·min-1,respectively.The rotational speed of the outer HR and inner HR was in the range of 10–50 r·min-1and 20–700 r·min-1.The speed ratio was between 0 and 20.The impellers were operated in both co-and counter-rotating modes.

3.Numerical Simulation

The movement of the inner and outer impellers in the coaxial mixer was simulated by the sliding mesh method(SM)technique;the whole coaxial mixer was divided into three regions:two rotating grid zones(one around the inner impeller(zone 1)and one around the outer impeller(zone 2))and a stationary zone,as shown in Fig.2.

The rotating zones can rotate independently with different rotational speeds and their sizes were kept constant for all numerical simulations.The central impeller zone and the outer impeller zone were implicitly coupled by the interface separating the moving zones via a mesh where the required interpolations were performed due to the relative motion between the subdomains.To solve the conservation equations of mass and momentum,the system was discretized into small volumes by means of discretization grid.In this study,the stationary zone was discretized with structured hexahedron grid,the inner and outer impeller zones were discretized with unstructured tetrahedralgrid.To capture the high velocity gradientnearthe impeller,smaller grid elements are required near the impeller.The optimal number of nodes was determined by conducting a grid independence test.The numerical grid employed to carry out the veri fication is shown in Table 2.The comparison of the radial distribution of the velocity magnitude at different number of nodes is given in Fig.3.It can be seen from Table 2 and Fig.3 that with the increase of the number of nodes,the change ratio of the power consumption,and the velocity magnitude was less than 0.5%and 3%,respectively,when the numerical node number changed from case 2 to case 3,so 610937 and 331928 nodes were applied for the simulation of the flow domain generated by the coaxial mixer for the PBT-HR and IOHR coaxial mixing systems,respectively.

Simulations were carried out using the laminar model based on the calculated Reynolds numbers.No-slip boundary condition was assumed on the tank wall,bottom and shaft.Free slip boundary condition was assumed on the free surface of the fluid,and the tangential velocity was applied on the tip of each impeller.High resolution for advection terms and non-staggered grid algorithmwas used for velocity–pressure coupling[22].The time step 0.001 s was employed to make sure thatthe RMS Courant number was less than unity in the simulation(0.1–0.3)[6].The convergence was achieved for each transport equation with the scale residuals below 10-6.The CFD model in this work was validated by the experimentally determined power consumption for corn syrup at different speed ratios.Equations used in this work are as follows.

Table 2 Number of grids used for the grid independence test

3.1.Governing equation

The numerical models describing the flow of the fluid include:

Continuity equation:

Momentum equation:

For simulations with rotating body,

Here,Scoris Coriolis force;Scfgis centripetal force.Scor=-2ρω×U,Scfg=-ρω×(ω×r);U is the velocity relative to the rotating reference frame.

The torque on the impeller was calculated using the pressure force and the viscous force from the results file.The average value over the time of one revolution cycle was used.The result can be positive or negative,indicating the direction of the force.

Fig.2.Schematic of the grids.

Fig.3.In fluence of mesh size on the axial velocity distribution.

The kinetic energy dissipation rate(ε)can be expressed in terms of the principle eigenvalues()as shown in Eq.(5)[23],

The axial circulation rate was computed by the following equation,

The average circulation rate was calculated by,

Where the+stands for the positive axial velocities and-stands for the negative axial velocities.Since the obtained velocity field ensures mass conservation,the corresponding values ofis of the same magnitude as.

4.Results and Discussion

4.1.Veri fication of the numerical simulations

The torque M,exerted by the HR was calculated by a method mentioned in Section 3.Values ofthe corresponding power number,de fined as NP=(2πNM)/(N3D5ρ)from the simulation was compared to experimental results atdifferentspeed ratios(RN=Ninner/Nouter).The coaxial mixer was operated in both co-and counter-rotation modes.The result is shown in Fig.4,the investigated rotational speed of the HRwas in the range of 15–60 r·min-1and the viscosity of the working fluid was 7 Pa·s.The Reynolds number of the impeller was de fined by Eq.(8).

where ρ,kg · m-3and μ,Pa ?s is the density and viscosity of the fluid,respectively.d is the diameter of the impeller,m;N is the rotational speed of the impeller,r·s-1.The Reynolds number investigated was in the range of 2–192 and 13–81 for the PBT and HR impeller,0.2–22 and 1.6–8.1 for the IHR and OHR,respectively.

Itcan be seen thatthe difference between the powerconsumption in the simulation and that in the experiment was less than 5%.The figure shows good agreement between the numerical and experimental results.

4.2.Synergistic effects characterization:circulation rate and energy dissipation rate

The mixing performance of the mixer can be typically expressed in terms of the axial circulation rate(QZ)and energy dissipation rate(ε),so they were investigated for each individual impeller and coaxial mixer at constant power consumption,respectively.The average axial circulation rate and energy dissipation rate are shown in Tables 3 and 4,respectively.The powergiven in the tables were obtained fromnumerical simulations.

Itcan be seen from Table 3 thatatconstantpowerconsumption 17 W,the coaxial mixer has the largest average axial circulation rate 15.83 m3·h-1,the HR impeller has the smallest QZave9.21m3·h-1and the QZaveof the PBT impeller was a bit higher than that of HR,that is 12.08 m3·h-1.These results mean that there existed synergistic effect in coaxial mixers.The combination of inner and outer impellers strengthens the axial circulation in the tank.

Table 4 gives the volume average and maximum energy dissipation rate in component and coaxial mixers.It can be seen from Table 4 that at constant power consumption the volume average energy dissipation rate εvolavewas almost the same for the above three mixing conditions,but the maximum energy dissipation rate was different.The PBT has the largest εmax=956.33 m2·s-3,followed by the coaxial mixer,εmax=705.15 m2·s-3,and the HR has the smallest εmax=243.17 m2·s-3.

In stirred tanks,the axial circulation accounts for the macro mixing and the energy dissipation rate accounts for the micro mixing.It can be seen from the above results that there existed synergistic effect in coaxial mixers;the coaxial mixer both have stronger axial circulation rate and higher energy dissipation rate atconstant power consumption,which is very bene ficial to the control of macro and micro mixing in stirred tanks.

The interaction between the inner and outer impeller and the partition of power consumption between the inner and outer impeller in coaxial mixers were investigated in the following.

4.3.Quantitative analysis of power consumption

Various mixing performances are related to the power consumption of the mixer to some extent,so power consumption was chosen as a basic characteristic to analyze the interaction between the outer and inner component impellers.The variation of the inner and outer impeller power consumption with the speed ratio was investigated separately first,then the contribution of each individual impeller power consumption to coaxial mixer power was given and finally a simple method to calculate the power consumption of the coaxial mixing system was given.

Fig.4.Comparison ofexperimentaland simulated results in coaxialmixers(error bar:5%)(a1)PBT-HRco-rotating mode(a2)counter-rotating mode(b1)Inner-outer HRco-rotating mode(b2)Inner-outer HR counter-rotating mode.

Table 3 Average axial circulation rate in component and coaxial mixers

Table 4 Volume average and maximum energy dissipation rate in component and coaxial mixers

4.3.1.Impact of impeller speed ratio on power number

The power demand of PBT-HR and IOHR coaxial agitation systems were investigated.The variation of power number with the speed ratio RNis shown in Fig.5.

From Fig.5(a1)(a2),it can be seen that in the co-rotating mode,the power number of the outer impellers NpHRor NPOHRdid not change almost at lower speed ratio(0＜RN＜2);the variation of the power number was less than 5%as indicated by the error bars in the figures;while at higher speed ratio(RN＞2),the power number of the outer impellers decreased significantly with the increase of the speed ratio.So,it can be taken that in coaxial mixing systems with both PBT-HR and IOHR configurations the in fluence of inner impeller on the outer impeller power consumption was weak at lower speed ratio(0＜RN＜2),while it was strong at higher speed ratio(RN＞2).The critical speed ratio was RN=2 under conditions investigated.

In the counter-rotating mode,the trend of power variation was reverse,that is,the power number of the outer impellers increased with the increase of the speed ratio.The critical point was the same as that of the co-rotating mode,which can be observed from Fig.5(b1)(b2).

The in fluence of speed ratio on the inner impeller power number in both the co-and counter-rotating modes were also studied,Fig.6 presents the results.

In contrast to the power number variations of the outer impeller power with the speed ratio,the power number of inner impeller was affected by the outer impeller at lower speed ratio(RN≤2),as Fig.6(a1)(a2)shows,the power number of the inner impeller increased with the speed ratio in the co-rotating mode.With the speed ratio increased further(RN＞2),the power number of inner impeller was unaffected by the outer impeller both for the PBT-HR and IOHR configurations.

It should be noted that the effect of the outer impeller on the power of the inner one was more significant at speed ratio less than 1.As indicated in Fig.6(a1),NPPBTincreased obviously as the speed ratio increased from 0.5 to 1.

The same conclusion can be obtained in the counter-rotating mode,as shown in Fig.6(b1)(b2);the outer impeller had a significant impact on the inner one at lower speed ratio(RN≤2).Contrary to the trend in co-rotating mode,the power number of the inner impeller decreased with the speed ratio.This can be explained with the fact that an increasing outer impeller rotation the inner impeller has to work against a stronger countercurrent.

Fig.5.In fluence of the speed ratio on the power number of the outer impeller in coaxial mixers.

In the mixing of variable viscosity mixing in industrial mixing,such as the simultaneous sacchari fication and fermentation of corn stover at high solids loading,there existed a reaction stage in which the viscosity of the fluid was high;at this stage,the coaxial mixing system should be operated at lower speed ratio[1].

In the literature,mostresearchers concluded thatthe power number of the inner impeller did not change with the speed ratio;in other words,the outer impeller did not have an impact on the power of the inner one.One reason for the difference between the conclusions of the literature and that of this paper was because of the higher speed ratio investigated(larger than 2)in the former[11,15,17,18].The speed ratio less than 2 was just the range in which the outer impeller had an effect on the power consumption of the inner one.

Besides,the flow regime and impeller configuration may also have an impact.Liu et al.[15]observed that the power consumption of the inner impeller increased with the decreased speed ratio in transitional regime for counter-rotating mode.Heiser etal.[24]investigated the performance ofa coaxialmixerconsisting ofa double helicalribbon and a centralscrew impellerand concluded thatthe power consumption ofeach impellerwas affected by the other regardless of the chosen rotating mode.

In conclusion,at lower speed ratio(RN≤2),the outer impeller had a remarkable impact on the power of the inner one;the lower the speed ratio,the more significant the effect was;at higher speed ratio(RN＞2),the inner impeller had a significant in fluence on the power of the outer one;the higher the speed ratio,the more obvious the effect was.There existed a criticalpointin the speed ratio,underthe conditions studied in this paper;it was RN=2 both for the PBT-HR and IOHR configurations in co-and counter-rotating modes.

4.3.2.Rate of variation in power consumption of each impeller

A new parameter which expresses the relative variation of power draw of impeller with the rotational speed ratio is de fined as,

where P0is the power consumption of the inner and outer impellers when each of them is rotating alone,and PRNis the power consumption at the corresponding rotational speed when the inner and outer impeller are rotating together.

The relative variation of power consumption of the inner and outer impeller with the speed ratio in co-and counter-rotating modes with the PBT-HR and IOHR coaxial mixers were both investigated.Fig.7 shows the results.

Take the PBT-HR coaxial mixing system operated in co-rotating mode as an example(Fig.7(a1)),it can be seen that with the increase of the speed ratio in co-rotating mode,the power drawn by the inner and outer impeller increased and decreased,respectively.The negative symbol represents the decrease of impeller power comparing to that when the component impeller was rotating alone.De fine 10%as a criterion,when the value of R was less than this value,we deemed that the power was almost unchanged.Based on this criterion,it can be seen from Fig.7(a1)(a2)that for the PBT-HR coaxial mixing system both in co-and counter-rotating modes,the variation was significant for the PBT and HR when RN≤2 and RN＞4,respectively.

Fig.6.In fluence of speed ratio on the power number of the inner impeller.

Take the variation of R in co-rotating mode as an example,when RNwas increased from0.5 to 3,the value of R ofthe centralPBT impellerdecreased from(-)75%to(-)25%;the decrement was very large;in contrary,the R ofHRwas less than(-)10%.This phenomenon means thatat lower speed ratio,the outer impeller had a strong impact on the inner one,but the inner impeller did not in fluence the outer one,the interaction between these two component impellers was one-way.At the same way,as the speed ratio increased higher than 4,the R of the central PBT is less than(-)10%,while the R ofthe outerimpeller increased from(-)10%to(-)65%,the inner impeller had a great impact on the outer one when RN＞4.The interaction between these two impellers was also one-way.There existed a narrow speed ratio range of 3–4 where the value of R both for the PBT and HR was less than(-)10%;the interaction between these two impellers was weak.This narrow range did not exist in counter-rotating mode because the R curves of PBT and HR impeller intersected at 10%(Fig.7(a2)).

So it can be concluded that for the PBT-HR coaxial mixing system,the interaction between the two component impellers was mainly one-way.The HR impeller had a significant in fluence on the PBT impeller at lower speed ratio(RN＜2),while the PBT impeller had a strong impact on the HR at higher speed ratio(RN＞4).

For the IOHR coaxial mixing system,Fig.7(b1)(b2)depicts the results.Take the co-rotating mode as an example,as shown in Fig.7(b1),in the speed ratio range of 0–2,the value of R of the IHR decreased from(-)115%to(-)40%and that of the OHR was less than(-)10%,which indicates that the OHR had a great impact on the IHR,while the IHR had a little in fluence on the outer one;the interaction between the inner and outer HR was one-way.When RNwas higher than 6,the value of R of the inner HR was less than(-)10%and that of the outer HR increased from(-)40%to(-)75%.This means that the interaction between the outer and inner HR impellers at higher RNwas also one-way.

While in the speed ratio range of 2–6,the value of R of the IHR decreased from(-)40%to(-)10%and that of the OHR increased from(-)10%to(-)40%with the increase of the speed ratio.The interaction between these two impellers was strong,and it was two-way,that is,both of the inner and outer HR had in fluence on each other in the speed ratio range of 2–6.This was different from the PBT-HR coaxial mixing system.

In counter-rotating mode,the same conclusion was obtained,as Fig.7(b2)shows.In the speed ratio range of 0–2 and 6,the interaction between the inner and outer HR impellers was one-way,while in the range of 2–6,the interaction between the component impellers was two-way.

4.3.3.Ratio of power consumption

The variation ofthe proportion ofthe outer impeller power in thatof the coaxial impeller(PHR/Ptotalor POHR/Ptotal)with the speed ratio is shown in Fig.8.The rotational speed of the HR and the OHR was in the range of 20–120 r·min-1and 10–50 r·min-1,respectively.

Fig.7.The relationship between the relative variation of the power consumption and speed ratio in coaxial mixers.

For PBT-HR coaxial mixing system,as indicated in Fig.8(a1)(a2),as RN≤2,PHR/Ptotal≥90%,that is,the power input by the HR occupied a large part in the total power.The mixing performance was usually controlled by the level of power consumption which significantly in fluenced the flow pattern and flow field in the tank.

It is meaningful to infer that the impeller which provides the most power plays a dominant role in the mixing performance of the coaxial mixing system and it was called the dominant impeller.This conception is bene ficial to make the operational strategy and to estimate the power consumption of component impeller.It can be seen from Fig.8(a1)(a2)that RN=2 was the critical speed ratio distinguishing the HR dominate region for the HR-PBT coaxial mixing system,it was de fined as the first critical speed ratio RNcri1.When the speed ratio was lower than this critical value,the HR impeller played a dominant role in the coaxial mixing system.

As RN≥12 in co-rotating mode,PHR/Ptotal≤10%,or PPBT/Ptotal＞90%,which indicating that the power of PBT impeller occupies a large amount in the total power,so the PBT was the dominant impeller under this speed ratio range.RN=12 was de fined as the second critical speed ratio RNcri2.When the speed ratio is higher than RNcri2,the PBT impeller played a dominant role in the coaxial mixing system;while when the speed ratio was in the range of RNcri1and RNcri2,the total power of the coaxial mixer was both determined by the central and outer impeller,the performance of the coaxial mixer was determined by both of the component impellers.

In counter-rotating mode for PBT-HR coaxial mixing system,as shown in Fig.8(a2),due to the interference effect between the central and outer impellers,the power drawn by the outer HR increased with the speed ratio.RNcri2increased to higher value compared to that in co-rotating mode.

For IOHR coaxialmixing system,as can be seen from Fig.8(b1)(b2),the relative power input by the outer HR collapsed on the same curve at different outer HR rotational speed both in co-and counter-rotating modes.RNcri1was 1.5 both in co-and counter-rotating modes.RNcri2were 7 and 11 in co-and counter-rotating modes,respectively.The RNcri2in counter-rotating mode was higher than that in co-rotating mode also.The reason for the increase of RNcri2in counter-rotating mode may be due to the interference effect in the coaxial mixing system.

4.4.New suggestion for estimation of power demand in a coaxial mixer

Comparing the power consumption of the coaxial mixing system to that of the sum of the power consumption when the inner and outer impellers were rotating alone,Fig.9 presents the results.The rotational speed of the outer impeller was kept constant for the given results,when the inner impeller rotational speed was kept unchanged the same results could be given which was not shown here.

In co-rotating mode,it was interesting to find that in the speed ratio range of 0–18,the power consumption of the coaxial mixing system and the sum of each individual impeller power consumption was very close to each other,the difference between these two curves was less than 10%as indicated by the error bars in Fig.9(a1)(a2).This finding inspired us that the power consumption of the coaxial mixer can be computed by the sum of the power of each individual impeller when it was rotating alone.As the speed ratio increased beyond the range illustrated in the figures,the relative power consumption of the HR impeller in the total power of the coaxial mixing system was less than 10%which was given in Fig.8(a1),and the power of the coaxial mixing system can also be computed by the sum of power consumption of the component impeller when they are rotating alone.

Fig.8.Effect of speed ratio on the relative power input by the outer impeller.

In counter-rotating mode,the power consumption of the coaxial mixer can also be computed by the sum of the power consumption of the component impeller and the error was less than 10%,as indicated in Fig.9(b1)(b2).

Besides,itcan be seen from Fig.9 thatthe powerconsumption ofthe coaxial mixer was slightly less or more than the sum of power consumption when the component impellers were rotating alone.The slight decrease of the power consumption of the coaxial mixer was due to the drag effectof the PBT on the HRwhen the individualimpeller is brought into a single tank,while the bit increase of power of the coaxial mixer was because of the interference force between the component impellers.

On the basis of the invariance of the coaxial mixer power consumption,the power of the component impellers can be computed by Pcomponent=Ptotalα.Ptotalis the sum of power consumption when the component impeller was rotating alone at the corresponding rotational speed and α is the weighting factor which can be obtained from Fig.8.This equation can be used to determine power partition for the inner and outer impellers based on the operating strategy of the coaxial mixer.The determination of the operating strategy should be in accordance with the engineering characteristics of the reaction such as higher rotational speed of the outer HR at high fluid viscosity and lower rotational speed of the inner PBT at lower fluid viscosity in the simultaneous sacchari fication and fermentation(SSF)of corn stover at high solids loading which has been reported in Huang et al.[1].

5.Conclusions

Power consumption is the basic characteristic of the mixer.For coaxial mixers,the speed ratio should be the most important factor which has in fluence on the powerconsumption.The power characteristics at various speed ratios were investigated first and the quantified parameters characterizing the interactions between the impellers were putforward.The connotations ofthe power curve and the interactions were further explored.The research results were bene ficial to the effective design of the coaxial mixers,such as the determination of the operating strategy,the calculation of the power consumption and the impeller selection.

Main conclusions are as follows:

(1).There existed synergistic effects between the inner and outer impellers in coaxial mixing system.Two important parameters:axialcirculation and energy dissipation rate were used to characterize the improvement of the mixing performance in coaxial mixing systems.The coaxialmixing systems have the highestaverage circulation rate at constant power consumption.

Fig.9.Comparison of power consumption between coaxial and the summation of component impeller(error bar represents±10%).

(2).The factors including the speed ratio,rotation mode and impeller configurations were investigated systematically,in which speed ratio was the strongest.The criticalspeed ratio that distinguishes the strong and weak interaction between the inner and outer impellers both for the PBT-HR and IOHR coaxial mixing systems in co-and counter-rotating modes was proposed.Three parameters describing quantitatively the interaction were put forward;they are rate of variation in power consumption,multiplicity of interactive mode and ratio of power consumption.

(3).The results mentioned above presented the dynamic characteristics of the power consumption of the coaxial mixing system,besides,there existed the invariance ofpower consumption,thatis,the powerconsumption ofthe coaxialmixercan be calculated by summing up the power of the component impellers when each of them was rotating alone under the speed ratio and impeller configuration investigated.The difference was less than 10%.So,the power assignment of total power consumption in coaxial mixing system can be calculated by Pcomponent=Ptotalα.Ptotalis the sum of power consumption when the component impeller was rotating alone at the corresponding rotational speed,and α is the weighting factor which can be obtained from Fig.8.

Nomenclature

C the distance between the PBT and the bottom of the tank

c the clearance between the outer edge of the HR and the tank wall

D tank diameter,mm

d impeller diameter,mm

dsshaft diameter,mm

H liquid height in the tank,mm

h impeller height,mm

M torque,N·m

N impeller rotational speed,r·min-1or r·s-1

Nppower number

P power consumption,W

Ptotaltotal power,W

p pressure,Pa

Qzaxial circulation rate,m3·h-1

R relative variation of power consumption

Re Reynolds number

RNspeed ratio(Ninner/Nouter)

r location vector

S impeller pitch,mm

S*iirate-of-strain along the ith principal axis,s-1

ScorCoriolis force,Pa

Scfgcentrifugal force,Pa

U fluid velocity

Uzaxial velocity,m·s-1

W impeller width,mm

Z axial height,mm

α weighting factor of the component impeller power

δ Kronecker Delta function

ε energy dissipation rate,m2·s-3

μ viscosity,Pa·s

ν kinematic viscosity,m2·s-1

ρ density,kg·m-3

ω angular velocity,s-1

Subscripts

HR helical ribbon

IHR inner helical ribbon

OHR outer helical ribbon

PBT pitched blade turbine

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