Deyu Luan*,Yiming Chen,Hong Wang,Yue Wang,Xing Wei

College of Electromechanical Engineering,Qingdao University of Science and Technology,Qingdao 266061,China

Keywords:Chaos Pseudoplastic fluids Largest Lyapunov exponent(LLE)Kolmogorov entropy(K entropy)Particle image velocimetry(PIV)6PBT impeller

A B S T R A C T Xanthangumsolutionswithdifferentmassconcentrationswereusedtostudythechaoticcharacteristicsinduced by the impeller of perturbed six-bent-blade turbine(6PBT)in a stirred vessel.Based on the velocity time series obtainedby the experiment of particle image velocimetry(PIV),with thesoftware MATLAB(R2016a),thedistributions of the largest Lyapunov exponent(LLE)and Kolmogorov entropy(K entropy)of the system,as two important parameters for characterizing the chaotic degree,were investigated respectively.Results showed that both of the LLE and K entropy increased with the increasing speed at the beginning.As the speed was up to 200 r·min?1,thetwoparametersreachedthe maximalvaluesmeanwhile,correspondingto 0.535and0.834,respectively,which indicated that the chaotic degree of the flow field was up to the highest level.When the speed wasincreasedfurther,bothoftheLLEandKentropydecreasedonthecontrary,whichmeantthatthechaoticdegreewasdecreasing.Itwasalsoobserved thatthechaoticcharacteristicsofflow field werehardly affectedby thefluid rheology and the detecting positions.The research results will enhance the understanding of the chaotic mixing mechanism and provide a theoretical reference for optimizing impeller structure.

1.Introduction

Pseudoplastic fluids possessing a yield stress are an important class of non-Newtonian fluids,which are commonly encountered in many industrial mixing operations,such as polymer engineering and biopharmaceuticalengineering[1,2].Theshear-thinningproperties of fluids can cause the viscosity changes in different regions of the stirred tank.Mixing of such fluids results in the cavernformation around the impeller,which will greatly reduce the heat or mass transfer efficiency[3-5].The improvement of mixing efficiency is important for the pseudoplasticfluidstoachieveamosteconomicalway[6-8].Fromtheliteraturereview,theflowcharacteristics especiallythechaotic mixingperformances ofthe pseudoplastic fluids with the complex rheological properties,need to be investigated as a few publications mainly focused on the flow fields of non-Newtonian fluids with the flow behavior index in the narrow range[9-13].It has been found that the mixing efficiency is closely related to the structure of flow field,and the most economical way of improving mixingefficiencyistoinducethechaoticflowandenhancechaoticdegree in a stirred vessel[14-17].Several studies have indicated that the largest Lyapunov exponent(LLE)and Kolmogorov entropy(K entropy)are both key parameters for determining the chaotic degree of the mixing system[18-20].For instance,Liu et al.[21,22]studied the chaotic mixing characteristics in the stirred tank with the rigid-flexible combination impeller.TheyfoundthattheLLEandKentropyoftheflowfieldproducedbycombination impeller were obviously higher than those by the rigid impeller,which results in the higher mixing efficiency.Li Meimei[23]studied the chaotic characteristics of the power load time series through analyzing the distribution of the LLE and K entropy.

The impeller of the perturbed six-bent-blade turbine(6PBT)can generate the asymmetric flow field structure and induce the chaoticflowinthetank[24,25].Therefore,theaim ofthispaperistoinvestigate thechaoticcharacteristicsofpseudoplasticfluidsinastirredtankwitha 6PBT impeller.The velocity time series measured by particle image velocimetry(PIV)test are used to calculate LLE and K entropy by programming with the software MATLAB(R2016a).These data can be used to explore the mechanism of the chaotic mixing in pseudoplasticfluids.The research results provide the useful information regarding the optimal design of the impeller structure.

2.Experimental

2.1.Experimental apparatus

Fig.1.Schematic of the stirred experimental apparatus.1.Stirred tank;2.6PBT impeller;3.encoder;4.laser transmitter;5.synchronizer;6.computer;7.CCD camera.

Theexperimentalapparatus is shown in Fig.1.TheStereo PIV(2D3C PIV)measurement system produced by Dantec Dynamics A/S company inDenmarkisselectedinthisexperiment,whichmainlyincludesalaser transmitter,cross-frame CCD camera,synchronizer,computer and image data processing system.The laser transmitter is a DualPower series Nd:YAG dual cavity pulsed laser for image measurement,which consists of three parts:laser head,power supply and remote control panel.The laser transmitter's maximum laser power is 1200 mJ.It can emit green light with a wavelength of 532 nm.The pulse duration is 4 ns and the maximum repetition frequency is 200 Hz.Two crossframeCCDcamerasaretheFlowSenseEO4Mcameraswitharesolution of2048×2048pixels.Thecamera'scross-frametimeis200ns,thepixel size is 7.4 μm,and the camera speed in full resolution and overclocking mode is 16.3 fps and 20.4 fps,respectively.The mixing system mainly consists of a shaft,impeller,stirred tank,square glass trough,motor,torque sensor,speed control and so on.

2.2.Measuring method

Experiment was carried out in a cylindrical tank of inner diameter T=210 mm,height H=T.The flat-bottomed tank was fitted with four equally spaced flat baffles,each with a width w=T/10.The impeller of diameter D=T/2,thickness 2 mm,blade width b=T/10 and backswept angle θ =30°was mounted on a centrally positioned shaft of diameter d=16 mm and positioned at an off-bottomed clearance C=T/3.P1,P2 and P3 were three speed detecting points,their distances from the axis were all 57 mm and the distances from bottom of the vessel were 70 mm,110 mm and 30 mm,respectively.The stirred tank is shown in Fig.2.

Fig.2.Sketch map of stirred tank and detecting point positions.

The rheological parameters of xanthan gum solutions in water at 0.5%,0.75%and 1.0%mass concentrations were measured using DV3T rheometer under temperature 30°C,as shown in Table 1.

Table 1 Rheological parameters of xanthan gum solutions

The polystyrene microspheres of diameter equal to 1-5 μm were selectedastracerparticlesowingtoitsgoodfollowingbehaviorandimagingvisibility.Thelaser sheetthicknesswasadjustedto1mm.The signal interval of laser pulse was 220 μs.The CCD camera exposure time was calculated according to the biggest linear velocity of the impeller blade tip and the sampling frequency was all set on 8 Hz at different speeds.

3.Theoretical Analysis

The characteristic parameters of a strange attractor are usually used to reveal the chaotic phenomenon and characterize the chaotic degree ofthesystemaccordingtothechaotictheory.Itmainlyincludestwoimportant parameters:largest Lyapunov exponent(LLE)and Kolmogorov entropy(K entropy).

3.1.The largest Lyapunov exponent(LLE)

A definite property of chaos is of the sensitivity to the initial conditions.For example,the near two points in phase space move on their own orbits at the beginning,and then the distance between the two points will increase at an exponential rate overtime.If the initial distance between the two points on the phase trajectory is set to ∣δx(t0)∣,after the iterative movement for h times,the distance,∣δx(th)∣,can be calculated as follows:

where λ is defined as the Lyapunov exponent of the system,and its expression is as follows:

TheC-Cmethodusesa correlationintegraltosolvethetimedelayταand the embedding dimension m,and then the time window τwcan be obtainedbyτw=(m?1)τα.Basedonthevelocitytimeseriesofthedetectingpointsobtainedbytheexperimentofparticleimagevelocimetry(PIV),theembeddingdimensionmandtimedelayταofthesystemvariables are solved out using the C-C method through the programming calculation with the software MATLAB(R2016a).These data are reconstructed in the phase space according to the time series,and then arefitted using the least squares method to get the regression line whose slope is the LLE of the system.LLE is one of the important parameters to describe the chaotic degree of the system,for example,LLE＞0 means the system in chaos.

3.2.Kolmogorov entropy(K entropy)

Fig.3.The velocity time series at different speeds.

TheKentropycanreflectthemovingstateofthedynamicsystem,and it is the characteristic quantity for measuring the disorder degree in the system.ThechaostheorypointsoutthatKentropymaybeusedasajudgment standard to assess the chaotic characteristics of the system.When the system is in regular motion,the corresponding K entropy is equal to zero,while the K entropy will approach infinity in the random motion system.If K entropy of the motion system is a constant of more than zero,it means that the motion system is in a definitely chaotic state.The bigger the value of K entropy,the higher the information loss rate,i.e.,more complexity of the motion system,which indicates that the chaotic degree of the system is improved.The amount of information(Q)generated in a chaotic system increases exponentially over time(t),i.e.,Q∝ekt,where the coefficient K is the Kolmogorov entropy.In this paper,the Kolmogorov entropy is solved out based on the maximum likelihood algorithm through calculating the average period of the time series[26].

4.Results and Analysis

4.1.The velocity time series

Taking the xanthan glue solution in water at 0.5%mass concentration as an example,the radial velocity signals of the three detecting points were collected over 200 s record,respectively.Due to limited space,the time series of the radial velocity at P1 point are shown only in Fig.3.The recording shows a clear cyclic variation of mean flow superimposed on the velocity fluctuation.The higher fluctuation range of velocity is also observed with the increasing speed,which illustrates that the turbulent degree is enhanced.

4.2.Effect of impeller speeds on LLE and K entropy

The embedding dimension m,time delay ταand the average period P of the velocitytime series atdifferentspeedsare showninTables2 and3,respectively.In the meantime,the distribution of LLE and K entropy is shown in Fig.4.

Table 2 The delay time and embedding dimension of velocity time series at different speeds

Table 3 The average period P of velocity time series at different speeds

It can be seen that the LLE and the K entropy are all more than zero,which indicates that the system is of the chaotic characteristics.Moreover,both of the LLE and K entropy increase with the increasing speed atthebeginning.Whenthespeedis200r·min?1,theLLEandKentropy climb the highest value simultaneously,corresponding to 0.535 and 0.834,respectively,which means that the chaotic degree of the system isuptothesummitlevel.However,boththeLLEandKentropydecrease as the speed increases further,which is mainly due to the emergence of the repetitive quasi-ordered structures(e.g.,trailing vortex,columnar backflow)in the flow field under the high speed.These quasi-ordered structures will no doubt lead to the reduction of chaotic degree.

Fig.4.The LLE and K entropy distribution at different speeds.

4.3.The effect of detecting point positions on LLE and K entropy

Thedistributionof theLLEand K entropyat thethreedetecting points is shown in Figs.5 and 6,respectively.It can be seen that the corresponding difference of the LLE and K entropy obtained at the three detecting points is very small with the similar distribution as the increasing speed,which is due to the fact that the energy distribution and dissipation law of different positions in the flow field are similar.This may be interpreted that the effect of the different detecting positions on chaotic features can be neglected.So it canbe considered that flow field is of the same chaotic characteristics in a stirred vessel.

Fig.5.The LLE distribution at different detecting points.

4.4.The effect of fluid rheology on the LLE and K entropy

Table 4 summarizestheLLEandKentropyobtainedwiththexanthan gumsolutionsinwaterat0.5%,0.75%and1.0%massfractionsatthedifferent speeds.It can be seen that the effect of the rheological properties of thefluidonLLEandKentropyisslight.Thisisbecausethesmallertherheological index is,the worse the flow of pseudoplastic fluid is,which results in the reduction of LLE and K entropy.However,the reduction degree of the LLE and K entropy is very small.As a result,it is concluded that the chaotic characteristics of the flow field are essentially similar as the rheological index varies.

Fig.6.The K entropy distribution at different detecting points.

Table 4 The LLE and K entropy in different xanthan gum solutions

5.Conclusions

Based on the velocity time series of xanthan gum solutions at three detecting points obtained by PIV measurement,the LLE and K entropy ofthesystemweresolvedoutusingtheC-Cmethodandmaximumlikelihood algorithm,respectively,combined with the software MATLAB(R2016a).The effects of the impeller speeds,the detecting positions and fluid rheology on the LLE and K entropy were made the focus of an inquiry.The results are shown as follows:

(1)The impeller speed has a great effect on the LLE and K entropy of the system.Both of the LLE and K entropy increase with the increasing speed at the beginning.As the speed is up to 200 r·min?1,the LLE and K entropy of the system reach the highest value simultaneously,corresponding to 0.535 and 0.834,respectively,which indicates that the chaotic degree of the system is highest.As the speed increases further,both of the LLE and K entropy tend to decline instead.

(2)The corresponding difference of the LLE and K entropy at the different detecting positions is very small.So it can be considered that the flow field is of the same chaotic characteristics.

(3)Therheological properties of thefluid havelittle effectontheLLE and K entropy of the system.The chaotic characteristics of theflow field are essentially similar in stirring the fluids with different rheological properties.

Nomenclature

b blade width,mm

C off-bottom clearance,mm

D impeller diameter,mm

d shaft diameter,mm

dpparticle diameter,μm

H fluid height,mm

K consistency index,Pa·sn

m embedding dimension

N impeller speed,r·min?1

n rheological index

T inner diameter,mm

w flat baffles width,mm

wtmass concentration,%

θ backswept angle,(°)

ρ density,kg·m?3

ρppolystyrene microsphere density,kg·m?3

ταtime delay

τwtime window