Mixing time in stirred vessels:A review of experimental techniques☆

Gabriel Ascanio*

Center of Applied Sciences and Technological Development,National Autonomous University of Mexico,Circuito Exterior,Ciudad Universitaria,DF 04510 Mexico

Keywords:Mixing time Stirred vessel Homogeneity

ABSTRACT Mixing time is defined as the time required for achieving a certain degree of homogeneity of injected tracer in a unit operation vessel.It has been used as a key parameter for assessing the performance of a mixing system.From an experimental standpoint,several techniques have been developed for measuring the mixing time.Based on the disturbances to flow,they can be classified into two groups:non-intrusive and intrusive.However,depending on the type of data generated,they can be also classified into direct measurements and indirect measurements(Eulerian and Lagrangian).Since the techniques available for measuring mixing times in an agitated tank do not provide the same information,its choice depends on several factors,namely:accuracy,reproducibility,suitability,cost,sampling speed,type of data,and processing time.A review of the experimental techniques reported in the literature in the last 50 years for the measurement of mixing time in stirred vessels under single and gas–liquid flow conditions with Newtonian and non-Newtonian fluids in the laminar and turbulent regime is made,and a comparison between these techniques is also presented.

1.Introduction

Mixing time is a key parameter used for analyzing the performance and the hydrodynamics of a stirred vessel.In general terms,the mixing time is defined as the time required for achieving a certain degree of homogeneity of tracer inserted in a stirred vessel[1].From a macro mixing standpoint,the bulk mixing time is the time required to get all points in the vessel uniformly distributed,while the local mixing time is the measure of how fast a material is distributed in a particular region of the vessel,which depends on the local turbulence.

Therefore,local measurements are time and space dependent,while bulk mixing time is based on time dependent measurements(temporal measurements)[2].Mixing time can be expressed in its nondimensional form:

where θmis the mixing time in s,N is the impeller speed in revolutions per second and K is a constant which depends on the size,geometry of the tank and the flow regime[3].Constancy of K is valid in the laminar and turbulent regimes but not in the transitional one.Moo-Young et al.[4],the mixing time can be correlated in a dimensionless manner as

where α and β are the adjustable parameters and Re is the Reynolds number,

where ρ is the fluid density in kg·m-3,D is the impeller diameter in m and μ is the fluid dynamic viscosity in Pa·s.

In the case of non-Newtonian fluids,such a number is defined by

where k and n are the consistency index and the flow behavior index,respectively,for fluids obeying the Ostwald-de Waele model(power law model),in which the rate of deformation has been taken as γ˙=N.However,the Reynolds number for non-Newtonian can also be defined in terms of the effective viscosity,which is a function of the shear rate.Nowadays,the most used definition of the shear rate in stirred vessels is the one developed by Metzner and Otto[5],which is calculated from the power-law model:

where the shear rate is proportional to the impeller rotational speed,being ksthe proportionality constant.Based on this approach,the Reynolds number is defined as

A number of mixing time correlations have been reported in the literature taking into account the effect of baffles on mixing time[6–8],as well as the relationship between circulation and mixing times[9].In the turbulent regime,the dimensionless mixing time is independent on the Reynolds number and it can be correlated by empirical expressions involving the size and geometry of the tank as well as some hydrodynamic parameters[10,11].Experimental techniques for measuring mixing time can be classified depending basically on two different scenarios,namely:(1)level of disturbance to flow and(2)type of data collected.Based on the first scenario,the techniques are classified as non-intrusive and intrusive.With respect to the intrusive techniques,they are based on local and global measurements.Although they are accurate,its use is limited because the flow pattern is modified by the presence of probes in the vessel.Mavros[12]reviewed the experimental techniques available for the study of flow patterns in stirred vessels.Although the aim of his paper was to give a general overview of intrusive and non-intrusive techniques for the flow visualization in tanks,some of them can be also applied also for estimating mixing times.

On the other hand,considering the type of data collected,these techniques are classified as direct measurements or indirect measurements(Eulerian data or Lagrangian data).In the case of indirect measurements,mixing time is to be inferred from physical measurements of conductivity or velocity for instance,while direct techniques provide real time results or at least requiring minimum processing.Table 1 summarizes the classification of experimental techniques as a function of both scenarios.

Table 1 Classification of measurement techniques of mixing times in stirred vessels

Nere et al.[8]performed a critical review of the literature dealing with liquid-phase mixing in stirred vessels in the turbulent regime.Although,the limitations of the techniques employed for measuring mixing times were described therein,they focused on the effects of different parameters such as impeller geometry,impeller position on liquid-phase mixing as well as the mathematical models developed for analyzing the hydrodynamics in stirred vessels.In this paper,the experimental techniques reported in the literature over last 50 years for the measurement of mixing time in stirred vessels have been reviewed by highlighting their advantages and drawbacks along with the most outstanding findings.This review covers single phase and gas–liquid mixing of Newtonian and non-Newtonian fluids in the laminar and turbulent regimes.

2.Experimental Techniques

2.1.Colorimetry

From a practical standpoint,colorimetry is by far the most common technique employed for measuring mixing times in stirred vessels[13–15].It is a non-intrusive technique extensively reported in the literature,which is used not only to determine the time required to achieve the desired degree of homogenization,but also to visualize qualitatively flow patterns and to reveal the presence of secondary flows generated under steady stirring such as well-mixed regions(caverns),islands and other segregated regions like stagnant of dead flow zones[16–20].The technique basically consists of injecting a liquid tracer and observing how it is dispersed in the fluid contained in stirred vessels.A variant of this technique is based on the color-decolorization approach by using a pH sensitive tracer.The reader is referred to Ascanio et al.[18]for a detailed description of the measuring protocol.Fig.1 shows an example of a typical mixing sequence by the colorimetric technique.

Norwood and Metzner[22]reported for the first time the use of an acid–base neutralization reaction for measuring mixing time in baffled stirred tanks equipped with turbines operating in the turbulent regime,which has been extensively adopted in the research.Hari-Prajitno et al.[23]used also this technique for measuring mixing time under aerated and unaerated conditions for dual and triple coaxial impeller configurations in the turbulent regime.The decolorization method has not only been used for measuring mixing time,but also for getting the insight of flow patterns generated by different impellers such as close-clearance impellers,multiple and coaxial arrangements,inclined impeller,etc.[24–34].

Although colorimetry is a very simple technique to implement,mixing time estimation depends strongly on the subjectivity of the human observer.For that reason,complementary techniques based on image processing have been developed[35–40].Kouda et al.[35]dissolved phenolphthalein as pH indicator in bacterial cellulose and CMC solution,which became dark purple when NaOH being present.Then,H2SO4was added at time zero and the images were captured every 0.5 s.Afterwards,the digital brightness of images was analyzed by image processing,and the mixing time was determined in terms of the decolorization ratio(output/reference).

Table 2 summarizes the operating principle,advantages and disadvantages of the estimation of mixing time by colorimetry.

2.2.Electrical resistance tomography

Electrical Resistance Tomography(ERT)is a non-intrusive and noninvasive technique from which cross-sectional images showing the distribution of electrical conductivity of gas–liquid flows in a stirred vessel from Lagrangian measurements taken at the boundary of the vessel.It can be used where the continuous phase is conductive and the other phases have different values of conductivity.A typical ERT setup consists basically of a set of electrodes,the data acquisition system and a user interface[41].The array of sensors are usually a series of rings in which the electrodes are equally spaced around the vessel circumference.Electrical measurements are made without affecting the flow in the vessel.A typical measurement protocol consists of applying the current to the next electrode pair and the resultant potential differences on all other electrode pairs are measured.This step is repeated until all electrode pair combinations have been used.A reconstruction technique converts the raw peripheral potential difference measurements into a two-dimensional electrical conductivity distribution of the measurement area(tomogram).Such a technique is based mainly on three different algorithms:linear back projection(LBP),a modified Newton–Raphson method(MNR)and a parametric model based technique(PM).The three algorithms use a single point calibration on a set of reference voltages from a sensor filled with a solution having a known conductivity[42].

Fig.1.Typical mixing evolution in a stirred tank using colorimetry[21].

Table 2 Operating principle,advantages and disadvantages of colorimetry

Fig.2 shows an example of a conductivity map showing the evolution of the mixing process in the stirred vessel.For that purpose a color scale is used for showing the variations in conductivity.For this example,blue regions indicate no mixing,green region partial mixing and red zone full mixing.

Although,ERT is a relatively new technology developed to get insight about the hydrodynamics inside stirred vessels,among other applications,its usefulness in such applications has been already reported[43–57]and validated with Computational Fluid Dynamics(CFD)[58].Wang et al.[48,59]reported the use of an ERT system for the gas–liquid mixing in a stirred vessel for discriminating the differences of three-dimensional gas hold up variations in the liquid phase viscosity.Simultaneously,the fluid dynamics was also investigated by following the spatiotemporal conductivity changes of a fluid tracer injected in the working liquid.Fig.3 shows the real time mixing curves of the liquid for the air-water mixing,which have been plotted from stack of tomograms obtained in the bottom plane and in the top plane of the stirred vessel.Looking at both curves,it is observed that the brine tracer is thoroughly mixed after 19 s and from this point the concentration in the top and the bottom remains constant,which can be confirmed by observing the corresponding tomograms.

Although most of the tomographic measurements have been done using sensors placed outside the tank,Abdullah et al.[60]reported the use of a series of conductivity electrodes place into the tank to study the nature of gas–liquid–solid mixing in a gas-induced stirred tank reactor.Edwards et al.[62]investigated the semibatch precipitation of aluminum hydroxide by the addition of concentrated sodium hydroxide(NaOH)to aluminum nitrate in a stirred vessel by using multimodal tomographic techniques.For that purpose ERT was used to analyze the mixing performance during precipitation,while positron emission particle tracking(PEPT)was used to study the flow field in the vessel as a function of the fill level,power input,etc.They found by PEPT that mixing time measured with ERT depends on the ratio of the feeding rate to the bulk velocity in the feed region.PEPT is a useful technique for studying three-dimensional flow phenomena in opaque systems[63].The operating principle is based on a particle tracer labeled with a radionuclide that decays via beta-plus decay generating two gamma rays.A PEPT camera detects the two gamma rays and defines a line along which the annihilation occurred.Fangary et al.[64]used the PEPT technique for determining the flow patterns in water and in viscous non-Newtonian CMC solutions inside a vessel agitated by axial flow impellers.Barigou et al.[65]validated the results obtained with a PEPT system with PIV and LDA in a stirred vessel containing a transparent fluid.Subsequently,the approach was applied to opaque systems containing low viscosity and high solid content suspensions obtaining the spatial distribution of suspended particles of two sizes throughout the vessel as well as flow fields for liquid and solid phases.Edwards et al.[62]combined ERT with PEPT to study the hydrodynamics of a stirred vessel of aluminum hydroxide.In that case the temporal and spatial distribution of fed ionic species provided by ERT was complemented by flow field information on the feed plume provided by PEPT.

Fig.2.Conductivity maps in a stirred vessel[41].

Fig.3.Liquid phase mixing curves for air–water mixing and the tomograms of the liquid phase mixing of a brine tracer pulse.Adapted from[48,59].

A variant of radioactive tracking consists of injecting a small volume of a radioactive liquid tracer and monitoring its concentration.This technique offers some advantages over optical methods such as the possibility of measuring over a wide temperature range(more than 300°C)and working with nontransparent vessels[8].

The operating principle as well as the advantages and disadvantages of ERT technique are summarized in Table 3.

2.3.Planar laser-induced fluorescence

Planar laser-induced fluorescence(pLIF)is a non-intrusive Eulerian technique suitable for the instantaneous measurement of concentration maps in liquid flows.Fig.4 shows a pLIF setup,which consists basically of the following parts:

·Light source:Typically a Nd:YAG laser(neodymium–doped yttrium aluminum garnet)or an argon-ion laser.Although a Nd:YAG laser emits light with a wavelength of 1064 nm,it is also operated in both continuous or pulsed mode,in which a high-intensity pulse can be generated at 532 nm or higher harmonics at 355 and 266 nm.

·During the experiments the dye is excited by laser light whose frequency closely matches the excitation frequency of the dye.It absorbs the laser light energy at short wavelength and re-emits light at a longer wavelength that can be detected by a photo detector.The most common dyes used in pLIF are fluoresce in and rhodamine 6G.

·A charge-coupled device camera(CCD)with a narrow-band filter for capturing fluorescent light only.

Since the dye fluorescence is strongly dependent on the concentration,the pLIF system must be firstly calibrated.If a dye is chosen for concentration measurements( fluoresce in or rhodamine 6G)the concentration signal(S)is calibrated with the following[61]:

where C is the dye concentration,E is the laser light intensity,Qλis the quantum efficiency of the dye(at the laser excitation wavelength),fopticcorresponds to optical factors,Vcis the sampling volume,and ACrepresents the absorption phenomena integrated on the light path(L)in the fluid characterized by the absorption index(ε),which is calculated by

For low concentration experiments,absorption phenomena are negligible,so that AC=1,leading to linear relationship between the signal(S)and(C,E,γ),being γ a constant that characterizes all experimental parameters.In such conditions,the concentration is accurately measured as the amount of light received by the detector.Many commercial LIF systems include a software capable of performing a quick calibration of concentration,as well as signal corrections and other operations.

Table 3 Operating principle,advantages and disadvantages of Electrical Resistance Tomography(ERT)

Fig.4.Basic setup of pLIF.Adapted from[61].

Once the system has been calibrated,the signals are processed by converting the fluorescence images into concentration or temperature fields.As the marked fluid flows through the light sheet,the dye is excited and re-emits light,so that the dye concentration is

This process is repeated for every interrogation zone in the whole region during a period of time,in such a way that the dye concentration can be correlated with time and the mixing time can be deducted.This technique allows simultaneously analyzing flow structures in the tank[19,66–72].Distelhoff et al.[67]reported for the first time the use of pLIF to determine mixing time in a baffled vessel containing water as working fluid.In this case,the mixing time was defined as the time required for the dye to be transported from the top of the vessel to the bottom plus the time required for a further two rotations of the bulk flow in the circumferential direction.The mixing time was based on the 99%concentration because large variations were found when using lower concentration criteria.Arratia and Muzzio[73]reported the use of the pLIF technique to measure the dye concentration in a laminar three-dimensional flow in a stirred vessel equipped with three coaxial impellers.Fig.5 shows an example of the concentration fields observed after 90 s,in which the effect of the Reynolds number on the mixing time is clearly observed.They stated that the tracer variability decayed exponentially,so that the mixing time can be obtained once the concentration in the vessel reaches a plateau with a minimum variability.

Hu et al.[74]developed a technique to quantify the reactive process in an unbaffled stirred tank using a novel reactive planar laser-induced fluorescence,which consists of visualizing two liquids mixed and reacted with each other.For that purpose the fluorescence signal of the dye,which varies due to the presence of a reacting material(H2O2),was continuously recorded along the reaction process.Fig.6 shows an example of the concentration evolution,in which the red color corresponds to a tracer concentration of100μg·L-1,while the blue color denotes 0μg·L-1.

PLIF has been also used to characterize reactive and non-reactive mixing processes in terms of timeθ99and timesτ95andτ99,respectively[75].They reported a time ratio κ99= θ99/τ99(99%mixing time to 99%non-reactive mixing time)ranging from 0.42 to 0.58,which indicates that reactive mixing is mainly controlled by the corresponding nonreactive mixing.As in colorimetry,simple digital image processing has been used with concentration maps obtained with pLIF for determining mixing times[68,75–77].

The advantages and disadvantages as well as the operating principle of the planar laser induced fluorescence are summarized in Table 4.

2.4.Thermography

Fig.5.Concentration fields observed at 90 s:(a)Re=40;(b)Re=60.Concentration scale bar in μg ·L-1.Reprinted with permission[73].Copyright(2004)American Chemical Society.

Fig.6.Normalized concentration fields[74].

Liquid crystal is an organic material in the amorphous solid form at a certain temperature and pure liquid beyond the upper limit.Due to its molecular structure it behaves as a crystal between these two phases.When an incident light is selectively scattered liquid crystals are the basis for temperature measurements.Based on the thermography principle,Lee et al.[78]and Lee and Yianneskis[79]developed a liquid crystal thermographic technique for the measurement of mixing time in a stirred vessel.For that purpose,thermochromic liquid crystals encapsulated as gelatin-shell micro-spheres having a mean diameter of20μm were injected at the top surface.Since the tracer and the working fluid had almost the same density the tracer remained suspended.Fig.7 shows the hue contours of one-half of the flow field in the vessel stirred by 2-Rushton turbine array rotating at540 r·min-1,200 ms after the tracer insertion.The tracer was injected at z/T=1.2 ata higher temperature than the working fluid,and the latter is hued by red color.

The operating principle,advantages and disadvantages of the thermography technique are summarized in Table 5.

2.5.Conductometry and pH

Physical measurements of conductivity and pH are made using probes placed at different points into the stirred vessel.Irrespective of their size,the presence of such probes perturb the flow,so that the measurement of mixing times should be considered as apparent,especially when using very small vessels.The conductivity technique is based on the measurement of the electrical conductivity of a solution over the time[80].For that purpose a conductivity probe is placed into the stirred vessel at a specific position.Fig.8(a)shows a typical setup of a conductivity measurement system,which consists basically of a conductivity probe and a conductivity meter connected to a personal computer through a data acquisition card.

Basically,there are two types of conductivity probes or cells,namely:2-electrode cells and 4-electrode cells.The firs tone is the most common conductivity probe made of glass with two electrodes made of platinum.However,this kind of probes is prone to polarization errors due to the electro-chemical reactions.To overcome this drawback,the 4-electrode probe is the best option.In such a design,alternating current is applied to the outer pair of rings and the voltage is measured on the inner rings avoiding polarization effects since there are no current flows in the measuring circuit.A typical conductivity probe is composed by anode and cathode,being the latter made from an inert met al.The probe is placed into the solution and it is activated when voltage is supplied.Then electron-carrying(negative)ions move towards the anode while electron-less(positive)ions move towards the cathode.It is important to point that conductivity solutions are sensitive to temperature changes,therefore the system must be temperature compensated or calibrated at the testing temperature.

The conductivity values are then converted into concentration data versus time scale by using the calibration factor of the conductivityprobe.In such a case,the initial concentration should be measured and the mixing time requires the final concentration,where the following expression must be small enough:

Table 4 Operating principle,advantage and disadvantage of Planar Laser Induced Fluorescence technique(pLIF)

Fig.7.Hue contours of the flow field observed with two-Rushton turbines 200 ms after tracer injection[79].

where C(0)is the initial tracer concentration,C(∞)is the final tracer concentration and C(t)is the tracer concentration at a certain time.Fig.8(b)shows a typical plot of concentration fluctuations of tracer as a function of time.As stated by Tatterson[3],the amplitude of the concentration fluctuations decays exponentially with e-Kat,where Kais a constant determined for various impellers and geometries.The data can be replotted in terms of probe log variance as a function of time[78].Under these conditions,the mixing time can be determined once the concentration fluctuations are smaller than 5%,which is known as the 95%mixing time(θ95).

Holmes et al.[81]reported for the first time the use of the conductivity technique to perform a study of mixing effectiveness in a baffled stirred vessel.Using a conductivity cell around the impeller,the average time required for a fluid element to complete one circulation around the tank,which can be used for estimating the mixing time.Bouwmans et al.[82]found that the probe position does not have any effect on the mixing time measurement in the turbulent regime.Although the conductivity technique has been used as an intrusive technique,Giona et al.[83]reported the use of impedance probes attached to the baffles and another one mounted on the shaft,so that the flow in the tank was not perturbed.

Fig.8.(a)Stirred vessel equipped with a conductivity measurement system;(b)tracer concentration fluctuating with time[3].

Most of the studies reporting the effect of different parameters on the measurement of mixing time coincide that the mixing time in the laminar and turbulent regime is strongly dependent on the following parameters[84–98]:

where D is the diameter of the impeller,T is the tank diameter,NQ/Np is the pumping effectiveness of the impeller,Vais the volume of the tracer added,V is the volume of the liquid in the vessel and ρ is the density.

The use of the conductivity technique has been also used for determining mixing times in several applications such as gas dispersion[92,99],steel converters[100],quality of mixing on animal cell culture[71],blending of fluids having different densities[101],gassed conditions[102,105],etc.

Table 5 Operating principle,advantages and disadvantages of thermography

On the other hand,Nagata[6]described a method based on local measurements of pH for the determination of mixing times in stirred vessels.Following that approach,Poulsen and Iversen[103]developed a system based on the signal of two pH sensors placed at the bottom and top of a stirred reactor and a conventional stimulus–response technique,in which the tracer signal was the difference between two pH measurements,so that the effect of the position of the probes in the tank was minimized and the initial and final values of the tracer became zero being easier the determination of mixing times.Fig.9 compares the determination of mixing time based on the measurement of pH at the top(pH1)and at the bottom(pH2)of the vessel and the difference between the two pH probe measurements(ΔpH).As mixing time is slightly shorter when using the difference of the two signals,the position of each probe in that case does not play an important role.

Fig.9.pH signal as a function of time:Signals from each probe(upper plot);difference between the two probes(lower plot).Adapted from[103].

Vallejos et al.[104]developed a confocal optical to study mixing times in a stirred reactor,whose results were validated by measuring pH at a specific position.Van der Gulik et al.[106]reported also the use of the pulse-response technique for the measurement of mixing time in a horizontal stirred reactor.For that purpose,they perform the measurements outside the tank by circulating the reactor contents through a spectrophotometer placed in an external loop.

The advantages,disadvantages and the operating principle of the technique based on the use of probes(conductivity and pH)are summarized in Table 6.

3.Empirical Correlations

From the findings reported previously,mixing time can be predicted as a function of the operating conditions.The mixing time correlations in Table 7 have been selected from mixing studies using differentmeasurement techniques as well as different conditions,namely: flow regime,type of impeller,baffle and fluid rheology.From such a table it is observed that the mixing time under the laminar regime correlates well with the hydrodynamic characteristics,mainly on the agitation regime or the power drawn by the impeller,which is true when mixing viscous fluids with close clearance impellers such as helical ribbon impellers.However,in the turbulent regime under ungassed conditions,mixing times is not only dependent on the agitation regime and the power consumption(Np=2πNTm/ρN3D5),but also on the tank geometry.Looking at the ratio of impeller size(D)to tank size(T),it is observed than bigger impellers are more effective than small impellers in the turbulent regime.However,in the same flow regime but under gassed conditions,mixing time correlates with a number of parameters,additionally to the latter case,the gas flow rate and the volume of liquid,among other parameters.In this case,the mixing time is strongly dependenton the ratio of the impeller size to the tank size being the exponent of the order of 2.On the other hand,the gas flow rate does not play a significant role on the mixing time.

Table 6 Operating principle,advantages and disadvantages of probes

4.Comparison Between Techniques

The selection of a specific technique for the measurement of mixing time in stirred vessels is dependent of several factors such as the following:

·Accuracy and reproducibility

·Type of information generated

·Suitability

·Costs

·Assembly of the experimental setup

·Calibration

·Sampling speed and data processing time

In general terms,the accuracy provided by techniques involving the use of tracers is higher compared to physical methods.Colorimetry is by far the simplest technique in which a color tracer is only needed;however,its accuracy and reproducibility is strongly dependent on the subjectivity of human observer.This drawback can be overcome by combining the technique with image processing methods.In that case,higher resolution images will provide better results,but the cost will increase and the data processing time could be longer.On the other hand,sophisticated techniques such as electrical resistance tomography(ERT)are very accurate methods for the measurement of mixing time.In such a case,the initial investment for acquiring those systems should be considered,as well as the time for assembling the experimental setup and the data processing time.

In regard to the type of information generated,it is important to point out that experimental techniques providing Eulerian or Lagrangian data do not give a direct measurement of mixing time,so the information obtained from these techniques must be related to the mixing efficiency.

Other techniques,such as positron emission particle tracking(PEPT),computer automated radioactive particle tracking(CARPT),particle image velocimetry(PIV),and laser Doppler anemometry(LDA)have been developed for visualization purposes.However,mixing times canbe indirectly inferred by following a sophisticated protocol.For instance,García-Cortés et al.[109]reported the use of LDA data,from which Eulerian information is generated and the mixing time can be deducted by means of a mass balance as follows:

Table 7 Mixing time correlations for stirred vessels

1.Radial profiles of the mean velocity(vr)in the vicinity of the impeller were obtained.

2.The pumping capacity(Qp)was calculated by integrating the mean radial velocity along the blade height times the cross sectional area.

3.The renewal time(tR)was determined as the ratio of the pumping capacity to the volume of liquid,which is equivalent to the circulation time(θm).

4.The mixing time was found to be proportional to the renewal time:

where m is the number of time the discharge liquid circulates in the vessel,which is a function of the tank diameter,the impeller geometry and the impeller blade number.

On the other hand,the Lagrangian approach follows a fluid particle over time as it moves through the flow field,then the mixing time can be determined from the flow number or the circulation time.Nienow[110]found that macromixing mixing time correlates well with the flow number,which is defined as the pumping capacity(Qp)divided by ND3,being N the impeller rotational speed and D the impeller diameter.The circulation time(θc)is here defined as the ratio of volume of liquid(V)to the pumping capacity.

In regard to the suitability,most of the techniques,especially those based on flow visualization(e.g.colorimetry,pLIF)allow determining not only mixing times but also obtaining 2-D and 3-D flow patterns in transparent vessels and fluids,which is a limitation at industrial scales.This drawback can be alleviated using other techniques like ERT,which can be applied to opaque systems.On the other hand,the use of probes can provide also information about the hydrodynamics with opaque fluids.Techniques involving the use of probes for the measurement of physical properties are in general affordable systems and easy to be implemented;however,since the measurements are local,a map showing the distribution of physical values could result in a time consuming and costly process.From a physical standpoint,the use of probes can disturb the flow,being this effect stronger when using small tanks.

Both calibration and data processing time are also important issues to consider when selecting a measurement technique for mixing times.Particularly,pLIF is a technique requiring calibration before use and sophisticated software is required for data processing.However,such a technique provides reliable information about flow patterns,which can be further used for the accurate estimation of mixing time.Besides calibration,some measurement techniques are sensitive to temperature;therefore they need to be compensated for these changes(e.g.pLIF).

On the other hand,techniques based on probes provide almost real time measurements.For instance,the measurements of conductivity or pH can be easily correlated with mixing time in a stirred vessel if the probes are connected to a data acquisition system and then such data is processed to determine the mixing time.However,response time should be also considered,especially when using probes for the measurement of conductivity or pH.

Mixing time has not only been determined by using experimental techniques but also by numerical simulation[111–113].All the techniques here described are useful methods also for validating mixing time predicted with numerical simulations,such as the one reported by Coroneo et al.[111],who performed simulations of fluid mixing in a single phase of a stirred vessel with different levels of spatial discretization by finding good agreement when comparing their results with pLIF results.

To summarize,besides the investment costs,the ideal experimental technique would be the one which does not disturb the flow,it can be applied over a temperature range,it can be used with opaque and transparent fluids and nontransparent tanks,and it can provide accurate and reproducible results.

5.Conclusions

The experimental techniques developed in the last50 years for measuring mixing times in stirred vessels have been reviewed and the most outstanding works reporting these techniques have been described.The applicability and limitations of all techniques as well as a comparison between them in terms of the accuracy,type of information generated,cost and processing time,among others,have also been described.Techniques such as colorimetry and pLIF have been supplemented in the last years with digital image processing resulting in more robust and accurate measurement methods.On the other hand,relatively new techniques based on radioactive tracking like positron emission particle tracking have emerged to provide information not only on the homogeneity level in the tank,but also on the flow fields in nontransparent systems.

Nomenclature

Acabsorption phenomena factor

b equation constant

C concentration

cTtemperature compensation slope of the solution

D impeller diameter,m

E laser light intensity

Eiimpeller efficiency(=(NQ3/Np)(D/T)4)

fopticoptical factor

H liquid height,m

J number of impellers

K mixing time constant

Kaconstant dependent on impeller type

k consistency index(power-law model)

ksMetzner–Otto constant

L light path

N impeller rotational speed,s-1

NGgyrational speed,s-1

Np Power number(=Tm/(ρN3D5))

NpGgassed power number(=Tm/ρN3D5)

NQflow or pumping number(=Q/ND3)

NQssecondary flow number(=Q/ND3)

NRrotational speed,s-1

n flow behavior index(power-law model)

Q gas flow rate,m3

Qλquantum efficiency of the dye

Qssparged gas flow rate,m3·s-1

R radius of vessel,m

Re Reynolds number(= ρND2/μ)

S concentration signal

T tank diameter,m

Tcalcalibration temperature,°C

Tliqtemperature of liquid,°C

Tmtorque,N·m

t time,s

Δt time difference,s

V volume of liquid,m3

Vavolume of tracer added,m3

Vcsampling volume,m3

vnnormal velocity,m·s-1

W mean tangential velocity,m·s-1

z axial position,m

δ spacing of interference light fringes,m

εdmean energy dissipation rate,W·s-1

θ9090%mixing time,s

θ9595%mixing time,s

θ9999%mixing time,s

θccirculation time,s

θmmixing time

κ9595%time ratio

κ9999%time ratio

μ dynamic viscosity,Pa·s

ρ density,kg·m-3

σ absorption index

σ conductivity,S·m-1

σTcconductivity at temperature Tc,S·m-1

σTcalconductivity at calibration temperature,S·m-1

τ variance

τ9595%non-reactive mixing time,s

τ9999%non-reactive mixing time,s

ω angular speed,rad·s-1