Qianqian Kang,Jinfan Liu,Xin Feng,Chao Yang,Jingtao Wang
1 School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China
2 Key Laboratory of Green Process and Engineering,Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China
3 School of Chemical Engineering,University of Chinese Academy of Sciences,Beijing 100049,China
4 Innovation Academy for Green Manufacture,Chinese Academy of Sciences,Beijing 100190,China
5 School of Chemical Engineering,Sichuan University,Chengdu 610065,China
Keywords:Laminar mixing Stirred tank Isolated mixing regions Particle clustering Solid-liquid separation
ABSTRACT Laminar mixing in the stirred tank is widely encountered in chemical and biological industries.Isolated mixing regions(IMRs)usually exist when the fluid medium has high viscosity,which are not conducive to mixing.In this work,the researches on IMRs,enhancement of laminar mixing and the phenomenon of particle clustering within IMRs are reviewed.For most studies,the aim is to destroy IMRs and improve the chaotic mixing.To this end,the mechanism of chaotic mixing and the structure of IMRs were well investigated.The methods developed to destroy IMRs include off-centered agitation,dynamic mixing protocol,special designs of impellers,baffles,etc.In addition,the methods to characterize the shape and size of IMRs as well as mixing effect by experiments and simulations are summarized.However,IMRs are not always nuisance,and it may be necessary in some situations.Finally,the present engineering applications are summarized,and the prospect of the future application is predicted.For example,particle clustering will form in the co-existing system of chaotic mixing and IMRs,which can be used for solid–liquid separation and recovery of particles from high viscosity fluid.
Most reactions and mixing are carried out in stirred tanks as a popular industrial option,and homogeneous mixing is generally required.Efficient mixing can be easily achieved by turbulence in bulk flow for low-viscosity fluid .However,polymerization reactions are usually carried out in the high-viscosity laminar system.It is difficult to achieve homogeneous mixing.Moreover,for some special materials/medium that are delicate under high shear rate (fungi,plant,insect,and animal cells) in biological industries,turbulence should be prevented and high-viscosity laminar systems need to be established[1–3].In general,two types of regions do coexist in a high-viscosity laminar stirred tank:isolated mixing regions (IMRs) first discovered by Norwood and Metzner  and chaotic mixing regions (CMRs)  (also called active mixing regions ).
IMRs are the invariant fluid tori that form beside impellers.They are separated from the main flow by Kolmogorov-Arnold-Moser (KAM) surfaces,which become isolated from each other.Fan et al.divided the IMR formation process into the axial flow stage,saddle vortex stage,annular vortex stage,and annular filament stage.The IMR formed and became stable in the last two stages.IMRs are usually formed when the Reynolds number of the whole tank was less than 500.Hashimoto et al.reported that IMRs formed at specific range of Re number(30 ＜Re ＜60).The stability of IMRs are good that can be maintained for several hours .There is no cross-streamline movement in high-viscosity laminar flow mixing.So as long as there is periodic/symmetric circulation,IMRs exist and they are poor in mixing .IMRs usually have a toroidal shape,elliptic shape or “Doughnut-shaped” consisting of several filaments(islands)and core torus[1,9,11–13],which are difficult to explain in detail [9,14,15].Two types of shapes of undulated IMR (UIMR) and ribbon-like IMR (RIMR) were found by Woziwodzki and Jedrzejczak,which were dependent on the eccentricity ratio .
The shape of IMRs is independent of impeller type,but the structure of IMRs will be affected by the number of blades,the type of impellers,and the rotational speed of impeller [18,19].Three stable island filaments were found surrounding the torus under a six-bladed turbine ,whereas Lamberto et al. found five filaments.Four filaments were observed for a four-bladed turbine.With the increase of Re,the filament number of an IMR decreases.For the same number of blades and Re,the filament number of the IMR produced by disk-turbine impeller is larger than that by the paddles .Only single filament wrapped around core of the IMR was observed at low eccentricity values.Hashimoto et al.[5,19] revealed the IMR structure as shown in Fig.1.The fluid particles moved in both vertical and horizontal directions of primary circulating flow (θ1) and secondary circulating flow (θ2) with the rotational angular velocity of ω1and ω2,respectively.The filament formed in right-handed spiral structures,which was interestingly opposite to the movements of fluid particles.The filaments did not wrap spirally around the IMR core region and they stood alone from each other and rotated along a spiral path.Filaments only existed when 20 ＜Re ＜94 found by tracer-particle method .Theoretical formulas were derived to predict its position,size and geometrical structure according to the Poincaré-Birkhoff theorem of resonance.
Fig.1.The schematic diagram of the IMR structure and the relationship between filaments and islands.Reproduced from Ref. with permission of Elsevier Ltd.,©2009.
The formation of CMRs is due to the stretching and folding of fluid elements repeatedly.Studies showed that the flow outside a torus could be chaotic,which depended on the symmetry of the flow,the rotational protocol of the impellers,and frequency of the periodical perturbations [16,20].Abatan et al. proposed that the chaotic flow was made up of nested island chains.Similarity,Inoue and Hirataand Christov et al.[22,23]proposed that “streakline” played an important role for the formation of chaotic mixing.The visualization of 3D streakline sheet was realized through injecting a tracer fluid (methylene blue) by Inoue and Hashimoto and Hashimoto et al.[24,25].Hashimoto et al.also put forward a new mixing index “streak lobe”.As shown in Fig.2,the perturbation wave,streakline and streak lobe were illustrated.The streakline is a travelling wave,which is caused by minute perturbations of the periodic rotation of the impeller blade (Fig.2(a)).The temporary variations of streakline and streak lobes were calculated from a simplified model of a 3D velocity field (Fig.2(b,c)).As the impeller blade rotating,the streaklines will be folded and expended near the tank wall,then it circulates in the whole tank and returns back to the impeller.Repeated folding and stretching of the streaklines result in the chaotic mixing.The streak lobe was defined as the region that surrounded by streaklines.The streak lobe can be used to indicate chaotic mixing,because the chaotic mixing will proceed from top to bottom regions when the streak lobe edges are between the top and bottom of the tank.
Fig.2.Conceptual pictures of chaotic mixing in a tank stirred by the two-bladed paddle impeller:(a) the perturbation wave caused by minute perturbations of impeller blade;(b)the change of streakline in vertical section that obtained numerically;and(c)schematic illustration of streak lobes.Reproduced from Ref.with permission of Elsevier Ltd.,©2011.
Researchers had investigated the single-phase mixing and particle clustering in the laminar stirred tank.The mechanisms of the chaotic mixing and the structure of IMRs were studied by experimental and simulated methods.Different ways of destroying the IMRs were developed to improve the effective mixing and reduce the mixing time.Moreover,the IMRs can be applied to the solid–liquid separation,because particles migrate and form clustering within the IMRs.All of the research works are very meaningful.
In this review,we will discuss the research progress on the methods for visualizing IMRs and CMRs,the mechanisms of chaotic mixing,as well as the structure and shape of IMRs.The effects of different internals of tank and operation mode of stirring on the damage,location and size of IMRs are reviewed.To separate solid particles from high viscosity liquid,the conditions for particle clustering were explored.The potential applications of IMRs in the future are predicted finally.
2.Visualization of IMRs
Decolorization technique is one of the most common IMR visualization methods.The schematic illustration of decolorization experiment apparatus is shown in Fig.3 .A typical example is the neutralization reaction of NaOH and HCl in the working fluid of glycerin [5,13,17,27].A passive tracer fluorescent dye may be used as a pH indicator,such as sodium fluorescein ,uranine and bromothymol blue .After a small amount of fluorescent tracer and NaOH(2 mol·L-110 ml)was poured into the tracer containing working fluid,the green colored working fluid was homogenized gradually.Then,equal amount and concentration of acid solution of HCl (2 mol·L-110 ml) was injected near the tip of the blade.Consequently,decolorization took place in the CMR,and green IMRs were stood out as shown in Fig.4 .The redox reaction of I2with Na2S2O3as another method was also used to observe the structure of IMRs [5,28,29].
Fig.3.Schematic illustration of decolorization experiment apparatus.Reproduced from Ref. with permission of Elsevier Ltd.,©2013.
Fig.4.The visualization of IMRs with decolorization experiment.Reproduced from Ref. with permission of Elsevier Ltd.,©2013.
2.2.Planar laser-induced fluorescence (p-LIF) technique
The 2D flow structure and mixing in time can also be described by using planar laser-induced fluorescence (p-LIF) experiments in laminar systems,where a plane of laser light was directed toward a transparent stirred tank to view the mixing structure on a vertical 2D plane.YAG-laser sheet is the only illumination source,rhodamine B or fluorescein as the fluorescent dye is dissolved in the working fluid (glycerin).A photographic camera is be used to capture the mixing patterns in the illuminated plane,which should be located perpendicularly to the laser plane [30,31].
UV fluorescence is another simple visualization technique to reveal 3D mixing patterns as shown in Fig.5.It is usually conducted in the dark room with a UV lamp.The flow field could be reproduced by the fluorescent dye,and non-diffusing passive tracer being injected into the stirred tank system [27,31,32].
Fig.5.The mixing structure observed with different techniques.(a) Planar laserinduced fluorescence (p-LIF) technique ;(b) 3D UV visualization .Reproduced from Ref. with permission of Elsevier Ltd.,©2004 and Ref. with permission of Wiley Ltd.,©2013.
2.4.Particle tracking by CFD simulation
2.4.1.Poincaré sections (2D)
The internal structure of IMRs was found to be similar to the Poincaré sections[33,34].It is used to reproduce the size and shape of IMRs,complex internal structure of IMRs and asymptotic mixing performance based on the Poincaré -Birkhoff theorem as a computational tool [12,35].In addition,3D features of concentric and eccentric flows could be well revealed by Poincaré sections calculation.The Poincaré section is generated by processing stroboscopic snapshots of particle trajectories and recording all of the intersections within a plane positioned at the vertical central plane of a tank on a single graph to reveal asymptotic mixing behavior[8,30].Zalc et al. computed Poincaré sections,and results showed that the sizes and locations of the computed IMRs agreed very well with the results obtained from p-LIF experiments.Lamberto et al. plotted Poincaré sections through tracking a small number of particles(O(100))in the flow for a large number of periods (O(1000)) as shown in Fig.6.Two IMRs were observed above and below the impeller.It can be concluded that Poincaré sections are valid as a diagnostic tool to visual the structure of IMRs and reveal the details of flow.
Fig.6.Poincaré section of the stirred tank flow (Re=17.28):(a) initial positions of particles,(b) after 400 periods (impeller rotations).Reproduced from Ref. with permission of Elsevier Ltd.,©2001.
2.4.2.Particle paths (3D)
CFD calculations of 3D trajectories of massless particles were carried out by Bulnes-Abundis and Alvarez .By interpolating particles location in time based on the velocity field solution,the trajectories of fluid particles were generated.It can be observed from Fig.7(a),(b)that fluid particles above and below the impeller would not mixed with each other.Two typical mixing pathologies coexisted in laminar and transitional stirred tanks.IMRs formed above and below the impeller (Fig.7(d)) with the impeller midplane as the separatrix.Fig.7(c),(e)revealed that eccentrical location of the impeller would trigger chaos.
Fig.7.CFD calculations of 3D particle trajectories in concentrically(a),(b),(d)and eccentrically(c),(e)agitated stirred tanks at 250 r·min-1(Re=416).25 massless particles tracers were released at different locations:(a)above the impeller plane;(b)below the impeller plane;(c)below the impeller plane,a 45°angled disc impeller;(d),(e)four initial locations above and below the impeller.Reproduced from Ref. with permission of Wiley Ltd.,©2013.
At present,the methods of visualizing IMRs include decolorization technique,p-LIF technique,UV fluorescence and simulation tools.Decolorization technology is the simplest method,which does not depend on any experimental equipment.Compared with p-LIF method,UV fluorescence method is more intuitive.Different from the experiments,the simulation method can not only reproduce the shape,size and location of IMRs,but it is most likely to reveal the mechanism of the formation and disappearance of IMRs through analyzing the details of the flow field.
3.Enhancement of Laminar Mixing
3.1.Characterization of mixing effects
3.1.1.Area and size of IMRs
The study of the IMR size is very important to predict whether good mixing is achieved.The definition of IMR size is different.The process of decolorization experiments is recorded with a photographic camera.The obtained pictured should be processed by changing the color mode from RGB 24 bits to 8 bits grayscale and converting image to a 2-bit-system.Then,the cross sectional area and the location of IMRs can be identified by using the ImageJ software .The area of every elliptical cross section S is calculated by Eq.(1):
where r1and r2are the major and minor axes of the ellipse measured directly from the photos.The size of IMRs is determined as the sum of the volume of every toroidal IMR.The volume of each IMR is defined as:
where R is the distance between the centre of a ellipse and the tank shaft.
Wang et al. also characterized the size of IMRs with S/AT,where S is the total cross-sectional area of IMRs and ATis the cross-sectional area of the tank.
The size and shape of IMRs depend highly on the rotational speed .It decreases as Re increases,since the magnitude of the perturbation increases .Hashimoto et al. had reported that the type of the impeller would affect the size of the IMR.The size of the IMR in the system stirred by a two-bladed paddle was slightly smaller than that of the six-bladed disk turbine.That is because the discharge flow of the disk-turbine impeller is stronger than that of the paddle impeller,causing shorter circulation pathway,which may determine the size of an IMR.For PBT impeller,the size of IMR is smaller above the impeller than that below the impeller,because the velocity in the higher part of the tank is larger.The distance between two adjacent parallel impellers will affect the size of IMRs for multi-impellers systems .
3.1.2.Area of the neutralized region
For acid-base neutralization decolorization experiments,the size of the neutralized region can be evaluated by the following generic curves :
where A is the area coverage,Amaxis the maximum achievable area coverage,m is the intensive rate of coverage,and t is the time.
The dimensionless mixing time (Ntm) was used to evaluate the overall mixing efficiency for different configurations.T(s) is the rotational period of the impeller.tm1(s) is the complete decolourization time.It means that the mixing time for decolorization of the whole region except for the IMR [29,36–38].
The energy efficiency is the specific energy(E,J·m-3)required to achieve complete mixing,which can be used to evaluate the performance of different types of impellers.It is defined as follows:
where P(V) is the power consumption,and tm2is the time to reach homogenous mixing in the laminar system.
The power number is used to compare the required power quantitatively for different impellers .For the steady mixing,the power number Ne is written as:
where P(W)is the mixing power,ρ(kg·m-3)is the density of working fluid,N(r·min-1)is the rotational speed,M(N·m)is the momentum of the shaft,D(m) is the diameter of the tank.
For the forward-reverse mixing mode (the rotating speed and direction of the impeller change according to triangular timecourse),the power number NeFRand the mixing power PFR(W)are defined as [17,39]:
where NFR(r·min-1) and MFR(N·m) represent the average value of rotational speed and torque over one cycle of revolution,respectively.They are calculated from the following equations:
where i is the number of cases.
3.1.6.Largest Lyapunov exponent (LLE)
Lyapunov exponent is also a very important parameter for evaluating mixing.The degree of chaotic mixing can be characterized according to the largest Lyapunov exponent (LLE) in the laminar stirred tank.The sufficient condition for the existence of chaos is that LLE ＞0.It is measured by recording the time series values of pressure fluctuation,then these values will be transferred to the computer and denoised by wavelet analysis.LLE is calculated with the Wolf method in MATLAB software .The core codes were detailed by .
The local rates of stretching can be used to estimate the overall efficiency of the mixing in the fluid domain (Fig.8).As early as 2 decades years ago,Muzzio et al. had proposed that the area of inter-material surface was proportional to the amount of stretching.Lamberto et al. carried out Lagrangian simulations of stretching fields and stretching distributions by considering a fluid element as an infinitesimal vectorI,the local values of stretching experienced by fluid elements could be calculated.At an arbitrary position in the flow domain,the vector can be located and it is stretched when convected by the flow .VectorIis expressed as the following Eq.(12).It was found that IMRs were surrounded by high stretching values.The stretching values increased exponentially with time corresponding to the chaotic region.Dynamic impeller speed would increase the stretching values and generate globally chaotic flow.
Fig.8.Stretching field plotted by releasing ≈10,000 particles and tracked for 200 periods at Re=17.28.The colors indicate stretching values,which increase from black to red.Reproduced from Ref. with permission of Elsevier Ltd.,©2001.
The magnitudes of the vector I at time t and initial time are written as Itand I0.Stretching(λ)is defined as the ratio of Itand I0.
Eq.(12)is calculated with fourth-order Runge-Kutta algorithm.Aiming to reveal the regions of non-chaotic and chaotic flow,stretching values are obtained for many vectors (O(104)) throughout the flow domain.To characterize the distribution of stretching values,a probability density function (PDF) is defined.
Parameter ξ has been defined and used to quantify the mixing performance in the simulated Lagrangian particle method.Normalized mixing rate MR of ξ was proposed and defined as follow[42,43]:
where n is the total number of the particles,nsis the number of mixed particle when their distance is smaller than a certain criterion rs(rs=1.4 l0,l0is initial particle spacing),ξ0is the initial ξ at time t=0.
3.2.Elimination of IMRs
Homogenous mixing in a reactor is very important for the yield and selectivity of desired reaction products.Poor mixing can result in undesirable by-products or incomplete extraction in leach processing,which is often unacceptable in most production processes.Because of IMRs will prohibit effective mixing greatly,there are many problems in the laminar mixing compared with turbulent mixing,such as segregation,compartmentalization,flow bypasses and cell focusing.The mixing time required to achieve homogenous blend is usually considered as “infinite”.Alvarez et al.have observed that silver-coated tracer particles(ρp=1300 kg·m-3,dp=10 μm)were concentrated within IMRs for even several hours,although their streamlines and velocities are expected to match those of fluid elements well.Higher frequency of visitation of the concentrated particles usually corresponds to the high shear areas.Cells are certainly focused within these high shear regions,which are not to be desired.Thus,it is necessary to eliminate the IMRs and enhance mixing of the laminar flow.Only eliminating IMRs and increasing CMRs in the laminar stirred tanks could effectively improve the mixing efficiency and shorten the mixing time.Noui-Mehidi et al. demonstrated that the mechanism of disappearance of IMRs in the system stirred by a Rushton turbine is the perturbations attributed to stretching and shrinking.Generally,mixing is generated by minute perturbations,such as creating time or space asymmetries.Researchers have been devoted much more efforts and developed many strategies to enhance mixing.
It was demonstrated that the period perturbation of the regular manifold of the flow caused by the impeller blade is the mechanism of enhance laminar mixing in the stirred tank [20,45].All of the factors affecting chaotic mixing include time periodicity,perturbations,stretching,reorientation and iteration.Each of them is related to the geometrical features of the stirred tank.Wider blades or asymmetric arrays of the blades are alternative optimization schemes .The passing of the blades would elongate the fluid element into a thin sheet.The elongated sheet is forced to be reoriented and reinjected to the center of the impeller repeatedly .Hence,it is easy to understand why chaotic mixing cannot be introduced when the impeller blades are replaced with concentric discs.The design of blade structures is significant.
Various types of impellers were designed to enhance fluid mixing and energy transfer[46,47].Large impellers,such as an anchor impeller,helical ribbon impeller,and the double helical screw ribbons impeller  were proved to be effective.Results showed that the rigid-flexible impeller could enhance the energy transfer process[49,50].Gu et al.applied six types of impellers for laminar mixing of non-Newtonian CMC solution,including a rigid impeller,a rigid impeller coupled with a chaotic motor,a rigidflexible impeller,a rigid-flexible impeller coupled with a chaotic motor,a punched rigid-flexible impeller,and a punched rigidflexible impeller coupled with a chaotic motor.The largest Lyapunov exponent (LLE) and dimensionless mixing time (Ntm) were used to characterize the mixing performance.Results showed that the chaotic mixing was improved by the punched rigid-flexible impeller obviously.The chaotic motor could further increase LLE value,and reduce Ntmas compared with the normal motor.
Radial,axial impeller,low-speed impellers and reciprocating impellers are valid for laminar mixing [51–56].For two-bladed paddle impeller,the size of IMR is slightly smaller than that for two-or six-bladed disk turbine,because paddle impellers have high shearing capacity,and disk-turbine impellers have high pumping ability and discharging fluid efficiency.The strong discharge flow will not always enhance the circulation of fluid particles.Wang et al.have investigated the effects of impeller types on the mixing efficiency,involving a three-bladed propeller(PR30,supplier:Heidolph),a 6-alternating pitched(45°)blade turbine(A-45°PBT6),a six-bladed disc turbine(DT6)and a disc impeller (DI).Results showed that the impeller type influenced the shape,size and location of IMRs because of their different flow patterns.For axial-flow propeller PR30,the IMR above the impeller is smaller than that located below the impeller.The IMR above and below the impeller are about the same for the other impellers.At a given Reynolds number,the Rushton turbine (DT6) produced the smallest IMRs,whereas the disk impeller has the largest IMRs.The asymmetry impeller blade such as A-45° PBT6 could generate chaotic mixing and improve energy efficiency.The extensive work conducted by Bulnes-Abundis and Alvarez  showed that an angled (45°) disc impeller and a round-bottomed tank were effective for improving the mixing performance.
By analyzing the streak sheet and cross sections generated by an ordinary paddle(OP)impeller and an alternative pitched paddle(AP) impeller,it was found that the AP impeller has better performance on accelerating homogeneous mixing.Because the asymmetric structure of the AP impeller led to smaller frequency and larger amplitude of the perturbation wave,smaller number and lager area of streak lobes covering the whole tank (Fig.9).Thus,the type of impeller blade is a very important factor for efficient mixing by affecting the number,size and dynamic behavior of the streak lobe .
Fig.9.Perturbation wave and vertical cross sections of streak sheets (Re=40,t=20 T)(a)OP impeller;(b)AP impeller.Reproduced from Ref.with permission of ACS Ltd.,©2012.
The combination of the radial(RT)and axial(PBT)causes stronger chaotic flow than 2-RT systems and 2-PBT.That means the axial flow is more efficient in mixing.It was also be found that the mixing efficiency was improved by 10%-15% generated by the axial impeller compared to the radial impeller when liquid viscosity was increased.The double-shaft mixing paddle undergoing planetary motion would break the IMRs efficiently.In addition,corotating and counter-rotating modes were implemented and indicated that mixing efficiency in counter-rotating modes was higher than that in co-rotating modes because of the stronger interaction between two axial flow.Moreover,other large object insertion and novel perturbation waveforms  are alternative valid impellers.
The IMR can be destroyed efficiently through breaking the resonance condition between circulation flow and impeller rotation.Eccentricity (e) could improve the axial mixing and shorten the mixing time by disruption of spatial symmetries.As eccentricity increasing,the 2D flow under concentric flow turns into 3D completely.The original IMRs are also deformed,the mid-plane separatrices are eliminated,and an extremely complex structure is formed .In addition,the destroy of IMRs by suitable eccentricity and conditions can avoid damage induced by mechanical stress .
In the eccentricity systems,the IMRs are inclined relative to the impeller plane[16,37,43,61–63].Karcz et al.reported that the blend time would decrease with the increase of the eccentricity.Ascanio et al. pointed out that the reason why the eccentric impeller can shorten the mixing time was that the speed of the eccentric impeller was 1.6 times than that of the center impeller.Eccentricity would bring an improvement of axial flow,leading to the deformation and reduction of the IMR volume.However,it was also been revealed that eccentricity may lead to stronger compartmentalization and the creation of RIMR in multi-impeller systems.The value of the eccentricity ratio is a critical parameter.Shaft eccentricity promotes mixing by generating a pair of counterrotating vortices,making the interaction between bulk and nearwall fluid more effective.In general,the mixing performance increases with the increase of the eccentricity,because it is closely related to the increase in the size of the secondary rotating flow,which is conducive to driving the near-wall fluid particles from the outer region to the inner region .Off-center located disc impeller promoted chaos even without blades,since symmetric regular regions were destroyed above and below the impeller.Different eccentricities of an angled (45°) disc impeller in a round-bottomed tank were proved more efficient (Fig.10) .Wang et al.also confirmed that a large angle of impeller inclination can enhance viscous mixing.
Fig.10.Mixing structure observed by 3D UV visualizations and Poincare sections in concentric and eccentric laminar stirred tanks agitated with a flat disc.(a) 3D UV visualizations in concentric stirred tank;(b)Poincare sections in concentric stirred tanks;(c)3D UV visualizations in eccentric stirred tank;(d)Poincare sections in eccentric stirred tanks.Reproduced from Ref. with permission of Wiley Ltd.,©2013.
It was revealed that IMRs could also been observed in continuous stirred tank reactors (CSTRs) under viscous and laminar flow.All the off-center location of inlet/outlet pipes and dynamic inlet flow conditions would produce asymmetric flow patterns.Then,the size of IMRs was reduced and chaotic flow was enhanced.In addition,higher rotational speed would stabilize the toroidal regions (Re=20–200),which was not good to destroyed IMRs and enhance mixing efficiency [27,35].
Unsteady rotation includes unsteady forward-reverse mixing[17,67] and time-depended rotational speed [11,43].Forwardreverse mixing can enhance laminar mixing through reducing the mixing time.The mixing time is dependent on oscillation frequency f of the impeller.Woziwodzki  proposed a correlation of the dimensionless mixing time based on experimental data,Eq.(17).The dimensionless mixing time decreased with the increase of oscillation frequency f.In the forward-reverse system,toroidal-shape regions are more complicated,because toroid surface is much more folded in comparison to unidirectional mixing.
where C5,C6,C7are constant number,they are different for different types of impeller.
Speed modulation protocols of forward-reverse unsteady mixing is expressed as the Fourier series for the triangle wave :
Time-depended rotational speed or direction will cause fluctuations of fluid and enhance the mixing .Triangular timecourse of impeller speed was used in previous studies[17,39].Also,the frequency of oscillation and amplitude may have an effect on the mixing efficiency .Takahashi et al. pointed that the increase of oscillation frequency would shorten the mixing time.However,it is also found that smaller oscillations frequency is more efficient .
Increasing the rotational speed to mix at a higher Reynolds number is effective to destroy IMRs.However,increasing the rotational speed will result in shaft torque requirements exceeding hardware capabilities.The increase of the rotational speed will lead to erosion damage on impellers or increase the shear rate,which will reduce the product quality of shear sensitive reaction [11,67,69,71,72].It was pointed out that the variable speed period of 10 s and constant speed period of 5 s of chaotic motor were particularly suitable for the laminar mixing process.Takahashi et al. found that the mixing time can be significantly reduced through the use of co-reverse periodic rotation.
Studies showed that the differences of power number between forward-reverse mixing and unidirectional mixing were negligible when Reynolds number was below 35 in laminar flow .In the turbulent flow regime,forward-reverse mixing and time periodic fluctuations could enhance the global mixing,but it will cause greater power consumption and its use becomes limited in industrial applications .Increasing the rotational speed or unsteady forward-reverse mixing are not always practical due to the increased power,erosion damage on impellers or damage to the shear-sensitive materials .It is usually expensive and complicated .
Fig.11.3D streak sheet consists of helical rounded surface spreading to 3D space in the stirred tank(Re=50,t=8 T):(a)unbaffled and(b)baffled stirred tanks.Reproduced from Ref. with permission of Elsevier Ltd.,©2011.
The placement of baffles is very important for the aim of removing IMRs.IMRs could be eliminated by installing baffles at the optimum location with clearance,which was investigated by decolorization experiments (oxidation–reduction reaction) .Particle image velocimetry (PIV) was used to measure flow velocity.Starch syrup was used as the working fluid (ρ=1300–1385 k g·m-3,μ=0.72–2.2 kg·m-1·s2).Baffling could enhance the radial momentum exchange between the near-wall and the bulk fluid revealed by the Lagrangian numerical analysis based on the Moving Particle Semi-implicit (MPS) method .Moreover,setting baffles could prevent the discharge flow from turn over.Then,the circulation flow is not formed and IMR is not generated .
Hashimoto et al. have revealed that the mechanism that baffles can enhance mixing by analyzing the streak cross-sections in the unbaffled and baffled stirred tank.As shown in Fig.11,these streaks refold and generate new steak lobes near the tip of the baffle,because the tangential flow is transformed to radial and/or axial flows by the baffle.As a result,the minute perturbation of the impeller blade is amplified and the renewed perturbation is created.The refolding of the streak will increase the number and size of streak lobes.Thus,the mixing is enhanced in the baffled stirred tank.Moreover,it can be seen from Fig.12 that IMRs in the baffled stirred tank become smaller than that of the unbaffled stirred tank.
Fig.12.Time variation of mixing patterns and IMRs (Re=50):(a) unbaffled,(b) baffled (fixed baffles),and (c) baffled (one baffle is rotated) stirred tanks .Reproduced from Ref. with permission of Elsevier Ltd.,©2011.
Baffles can eliminate the IMRs easily and are more energy efficient in contrast to systems at high Reynolds number.It was considered as more chaotic mixing from the disturbance generated by the baffles.The power input at critical condition where IMRs are eliminated is independent on the number of baffles.Using more than two baffles cannot improve mixing anymore.The critical Reynolds number required to destroy the IMRs is sensitive to the impeller type in the baffled system .
The impeller clearance to the tank bottom(c/H)has a significant effect on the IMRs,where c is the distance between the impeller to the tank bottom,H is the height of the liquid level.When c/H=0.5,two IMRs were observed above and below the Rushton turbine(DT6) impeller.As c/H is decreased to 0.05,only one IMR existed above the impeller.While c/H is increased to 0.72,the IMR above the impeller is destroyed by the fluctuation of liquid surface,resulting in a single IMR below the impeller .Kato et al.pointed out that the impeller moving up and down can eliminate IMRs effectively,but it will be very difficult in practice.
As the step temperature change,the IMR structure transformed from a combination of crescent shape and torus,to narrow torus,to a spiral structure of string IMR,and finally the IMR disappeared.The time to eliminate IMR decreases because a larger temperature difference enhances diffusion mechanism between IMR and CMR more easily .Takahashi et al. have demonstrated that the bottom shapes (a cone-fillet,a fully profiled and an ellipsoid bottom) have no significant influence on the mixing time.
The computational simulations could further reveal the mechanism of the laminar mixing in stirred tanks,that is,the impeller blades lead to the reorientation of fluid elements.The resulting mixing structures were deflected by the impeller blades and became elongated,which resulted in an iterative chaotic mixing process .Rice et al. performed CFD calculations to elucidate the mechanism of viscous mixing flow patterns by analysing the forces acting on a fluid element.Results showed that the balance between pressure and viscous forces resulted in small material derivation of radial velocity and negligible net pumping at the blade tip for the lowest Re.In addition,a simplified mathematical model was established to describe the flow qualitatively and quantitatively.Ng et al.studied the laminar mixing performance of various mixing procedures based on the Moving Particle Semiimplicit (MPS) method,involving baffling,shaft eccentricity and unsteady mixing (angularly-oscillating impeller) employed in a cylindrical tank agitated by a plate impeller.The role of eccentricity is still obvious without the perturbations of impeller blades even at Re=15,which is simulated and verified by replacing the impeller with a disc (Fig.13) .The overall mixing efficiency and power consumption of double helical and double helical screw ribbons impellers were calculated and compared.The double helical screw ribbons impeller has better comprehensive performance.Zhang et al. studied the mixing patterns stirred by double-shaft paddle under co-rotating and counter-rotating modes.It was found that axial flow was the main factor affecting mixing efficiency and the counter-rotating mode would provide stronger axial flow.CFD results indicated that homogenous mixing was achieved by using inclined-shaft in both liquid–liquid and liquid–solid stirred systems .
Fig.13.Evolution of tracer trajectories in stirred tanks (disc impeller,D=0.03 m,N=500 r·min-1,Re=15) at 0,2,5,and 10 impeller revolutions.The top and bottom row illustrate a concentric system and an eccentric system (e=0 0.42),respectively.Reproduced from Ref. with permission of Elsevier Ltd.,©2006.
Fig.14.Clustering process of inertial particle in a stirred tank.Reproduced from Ref. with permission of Elsevier Ltd.,©2016.
To conclude,this section summarizes different methods of characterizing mixing effects and eliminating IMRs.The area and size of IMRs is the simplest experimental method,but it is only suitable for laboratory scale studies.Energy efficiency and power number are more practical indexes for industrial situations.Since it is very difficult to judge IMRs according to flow field variables,stretching value has provided possibilities to characterize the mixing effect and guide industrial designs.At present,all the impeller designs,off-centered agitation,unsteady agitation,baffles and appropriate impeller clearance could eliminate IMRs well.However,offcentered agitation and unsteady agitation may bring instability to the system and increase the power consumption.Special impeller and baffles designs are more reliable.Despite the rapid development of numerical simulation technology,there is a lack of research on IMRs and high viscosity laminar stirring system.There are no CFD studies that can be compared with the experiment completely,and no one can use CFD to confirm the hydrodynamics mechanism of IMR formation and elimination theoretically.In the future,deep simulation research should be carried out from these aspects to provide guidance for the realization of efficient mixing of high-viscosity laminar flow system.
4.Particle Clustering within IMRs and its Applications
4.1.Particle clustering within IMRs
Segré and Silberberg [78,79] first found that neutrally buoyant particles(ρp=ρ)can migrate across streamlines in a unidirectional flow to a non-trivial equilibrium position for any initial positions.In Couette and Poiseuille flows,particle migration is a general phenomenon revealed by experiments and simulations [80,81].The mechanism of particle migration is the effect of buoyancy or non-linear fluid effects at medium Reynolds numbers[82,83].Particle migration was also observed in the ideal Stokes flow with dense multi-particles,because of particle interactions and shearinduced migration[84,85].It has been studied that the migration is as a function of particle density,sphericity,and deformability[86–89].It has also been predicted computationally that small particles lighter than the continuous phase only would move inwards to the torus center,while particles heavier than the continuous phase would move outwards [86,90].
Experiments proved that the most stable circulating orbit of the tracer particle occurred near the surface of core torus.It is due to the fact that this orbit corresponds to the one where the filaments exist .“Clustering” would occur when multiple particles are traced out within the torus and their circular paths are at the same radial positions.The locations of the clusters seem to coincide with IMRs .The migration and cellular orbitals of different particles were investigated by Abatan et al..The selected particles were large enough with diameter of 1.6 mm or 2 mm.The density is slightly smaller or slightly greater than glycerin.Results showed that both lighter and heavier particles moved inwards toward the torus.The rate of migration will increase with an increase of particle size,impeller spacing and rotational speed.The stability of the clusters also depends strongly on the impeller spacing and rotational speed.With an increase of impeller spacing,lowerperiod clusters move to lower radial positions,and higher-period clusters move to higher radial positions.Moreover,migratory competition was observed in the experiment when multiple particles were poured into the fluid.
All particles with density(ρp=1090–1320 kg·m-3)higher than the fluid could be drawn into IMRs even when Reynolds number is as low as 10.The secondary circulation loops influence the mixing and particle migration,which depends on the off-bottom clearance of the impeller.A nonlinear dynamic analysis of Poincaré sections was carried out.It was found that particles were captured in a phase-locked orbit within IMR.Nishioka et al. also showed that where particles were captured depended on the position where particles were released.In the beginning,particle orbit obtained by Poincaré section indicates that particle trajectories covered the full surface of the torus orbit.After a long time,the particle orbit converged on only three discrete points observed from the Poincaré section,which is called phase-locked orbit.Because particles always move within the IMR,the fluid around the particles will be accelerated and decelerated repeatedly.Thus,the drag force acts on the surface of the particle frequently.As a results,particle motion can enhance material exchange between IMRs and CMRs.
Fig.15.Particle trajectories within two IMRs.Reproduced from Ref. with permission of Elsevier Ltd.,©2016.
Due to the coexisting IMRs and CMRs in the laminar flow,special particles will move into the IMRs and clustering will be exhibited when they are poured into the stirred tank at a particular Reynolds number (Fig.14).Particle clustering in a stirred tank is necessarily dependent on IMRs and CMRs,which are considered as attractors and repellors,respectively.The repelling flow regions could convert the passive particle trajectories into non-passive particle trajectories,which are independent on the fluid trajectories.Meanwhile,the IMRs would attract particles into IMRs and prevent particles from entering the repelling regions again.Only particles with low enough inertia would behave passively.The instability of particle clustering depends only on material properties and the characteristic strain rate .If a particle is sufficiently small,it behaves like the fluid and moves passively.Thus,it cannot be constrained in IMRs.A noval technique,Capsule Tracking Velocimetry(CTV),was developed by Wang et al.and used to estimate a particle’s 3D position through an externally positioned camera along with a planar mirror and light-emitting capsules in a calibrated setup (Fig.15).
The inertial equation of particle motion was proposed by deriving the Maxey-Riley equation of motion as follows [95–97]:
where v and u are the velocity of particle and fluid,respectively.St is the Stokes number.φ is the inertia parameter,the larger the φ is,the less effect of inertia will be.The density ratio Rdbetween fluid and particles is used to distinguish neutrally buoyant particles(Rd=2/3) from bubbles (0 ＜Rd＜2/3) and aerosols (2/3 ＜Rd＜2).a is the radius of a spherical particle,and D is the diameter of the tank.As the neutrally buoyant particles can’t form clustering,the clustering criterions were only derived for non-neutrally buoyant particles.A necessary condition for the existence of an inertial particle attractor is the following :
Moreover,Wang et al. proposed a criterion to predict the formation of clusters:
The critical rotational speed for particle clustering can be derived by Eq.(24) and Eq.(25) as a function of particle and fluid properties.
where μ is the viscosity of the fluid,k is a constant number and different for specific impeller type,k=5 for Rushton turbine blade.It can be inferred from Eq.(26) that smaller particles needs higher critical rotational speed to cluster into the IMR.
Actually,non-Newtonian fluids are more commonly encountered in biorefineries and minerals industries,such as shearthinning fluids.The viscosity of shear-thinning fluids decreases with the increase of shear rate.High-viscosity carboxymethyl cellulose (CMC) fluid was used to investigate particle clustering .A phenomenon called “Pseudo-cavern” was observed in the CMC stirred system.Particle clustering only takes place inside the “Pseudo-cavern” incompletely,because the fluid velocity inside the “Pseudo-cavern” is far larger than that outside the “Pseudocavern”.Using multiple impellers may be effective to achieve complete clustering by maximize the CMRs.
Particle clustering within IMRs is a very interesting phenomenon.So far,some researchers had studied partial clustering conditions,but some of which are with different conclusions.For example,some people proposed that both lighter and heavier particles can cluster into IMRs,but some people thought only small particles lighter than the continuous phase.The only criterion proposed by Wang et al. to predict the formation of clusters has deviations from the experimental results.There is a lack of research on the critical conditions for particle clustering.Moreover,there are few reports on the dynamic characteristics of inertial particles in high-viscosity laminar stirring systems in terms of experiments and simulations.This will not only lead to insufficient understanding of the theory of particle clustering,but also limits its application.
The formation of particle cluster may have significant applications in different areas.Paireau and Tabeling  have pointed out that IMRs may not be troublesome.On the contrary,IMRs would be necessary as effective reaction-site in specific circumstances.Nishiokafound that particles move near the KAM surface,which will accelerate mass transfer of materials between IMRs and CMRs.Chaotic mixing performance was intensified by flexible impeller and floating particles greatly .Wang et al.applied the phenomenon of particle cluster to the solid–liquid separation of highly viscous systems,such as liquid-biomass separation in the biorefinery.It was found that the two IMRs above and below the impeller will become to one below the impeller when the impeller clearance to the tank bottom is small.Thus,all particles are concentrated and the clarified liquid above the impeller can be pumped away.In addition,continuous liquid–solid separation is possible using multi-stage tanks.
5.Conclusions and Outlook
The homogenous mixing has attracted much attention in turbulent and laminar stirred tanks in recent years.Turbulence itself is one of the effective means to enhance mixing,but the homogenization in high viscosity laminar stirred tanks is a challenge.This comprehensive review highlights different methods of observing IMRs and charactering the chaotic mixing in the laminar stirred tank.The mechanism of chaotic mixing and the structure of IMRs are elucidated to understand and strengthen the high viscosity laminar mixing.Previous studies focus on investigating the shape,size,location and elimination of IMRs.Different methods including different types of impellers,off-centered agitation,unsteady rotation,changing the impeller clearance and installing baffles can eliminate IMRs well.
Inertial particles may cluster to IMRs.For some situations,IMRs need to be eliminated to disperse solid particles homogenously.For some situations,IMRs could provide an effective reaction-site in particulate reactant mixing,enhance mass transfer and realize particle-liquid/particle–particle separation as the future possible applications,for example,separating the cell biomass in cell bioreactors or crystals in crystallization reactors.However,researches on the clustering conditions including critical rotating speed,clustering time particle capacity within IMRs,etc.are lacking for particles with different sizes and densities.In the future,it is possible to realize the separation of particle-liquid/particle–particle by adjusting rotational speed or the viscosity of fluid,etc.In addition,it is unknown whether particles will gather to be clusters after the stirred tank is scaled up.CFD works devote to interpret the doubts and reveal some mechanistic laws of high viscosity laminar mixing with flow variables.However,the behavior of particle clustering in high viscosity laminar flow is not explained by numerical simulation at present.
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.
Financial supports from National Key Research and Development Program (2020YFA0906804),the National Natural Science Foundation of China(21776282,21978296 and 22078229),the NSFC Key Program(21938009) and major project (91934301),the National Key R&D Program of China (2019YFC1905805),Chemistry and Chemical Engineering Guangdong Laboratory Shantou(1922006),Innovation Academy for Green Manufacture,Chinese Academy of Sciences(IAGM2020C06) and Youth Innovation Promotion Association CAS are gratefully acknowledged.Special thanks to Professor Jiayong Chen.
A size of the neutralized region,m2
Amaxmaximum achievable decolorized area coverage,m2
ATcross-sectional area of the tank,m2
a radius of particle,m
c distance between the impeller to the tank bottom,m
D diameter of the tank,m
dpdiameter of particle,m
E energy efficiency,J·m-3
e eccentricity of the shaft
f oscillation frequency,Hz
H height of the liquid level,m
I an infinitesimal vector
i number of cases
M momentum of the shaft,N·m
MFRaverage value of momentum of the shaft for the forward–reverse mixing mode,N·m
MR Normalized mixing rate
m intensive rate of coverage
N rotational speed of the impeller,r·min-1
NFRaverage value of rotational speed of impeller for the forward-reverse mixing mode,r·min-1
Nmaxmaximum rotational speed of the impeller for the forwardreverse mixing mode,r·min-1
Nmincritical rotational speed for particle clustering,r·min-1
Ntmdimensionless mixing time
Ne power number
NeFRpower number for the forward-reverse mixing mode
n total number of the particles
nsnumber of mixed particle
P mixing power,W
PFRmixing power for the forward–reverse mixing mode,W
R distance between the centre of IMR and the tank shaft,m
Rddensity ratio between fluid and particles
Re Reynolds number
ReFRReynolds number for the forward-reverse mixing mode
r1major axes of the elliptical IMR,m
r2minor axes of the elliptical IMR,m
S area of every elliptical cross section,m2
St Stokes number
T rotational period of the impeller,s
tm1complete decolourization time,s
tm2time to reach homogenous mixing,s
u velocity of fluid,m·s-1
V volume of the stirred tank,m3
VIMRvolume of each IMR,m3
v velocity of particle,m·s-1
μ viscosity of the fluid,kg·m-1·s2
ρ density of fluid,kg·m-3
ρpdensity of particle,kg·m-3
φ inertia parameter
ω rotational angular velocity
FR forward–reverse mixing mode
max maximum value
min minimum value
0 initial value
Chinese Journal of Chemical Engineering2022年1期
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