Zaisha Mao*,Chao Yang

Key Laboratory of Green Process and Engineering,Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China University of Chinese Academy of Sciences,Beijing 100049,China

1.Introduction

Mixing has been recognized for long as an important unit operation in process industries,and more thorough academic study of mixing in chemical reactors bears practical significance with respect to design,optimization and scale-up of chemical reactors.In particular,the impact of mixing is more obvious on fast complex reactions in either homogeneous or multiphase systems.

It is an acceptable consensus to subdivide the mixing process into a cascade of macro-mixing,meso-mixing and micro-mixing to address more easily the process on different spatial scales[1].Baldyga and Bourne[2]defined the micro-mixing as the diffusive mixing on the characteristic Batchelor scale and the macro-mixing above the Kolmogorov scale,with the intermediate meso-mixing between Batchelor and Kolmogorov scales.Villermaux and David[3]suggested that micro-mixing proceeded via three main mechanisms:laminar stretching to striation,turbulent erosion and shrinking,and molecular diffusion.Micro-mixing homogenizes the reacting species on the molecular scale so that chemical reactions may proceed to give the products we desire.In most works,the term of meso-mixing is not used and its effect is lumped into that of macro-mixing.Intuitively,macro-mixing on the process unit scale is realized mainly through macroscopic convective turbulent transport of different materials plus shear,chaotic flow and vortex motion.Thus,the segregation of two materials is gradually reduced to the Kolmogorov scale to facilitate the micro-mixing proceeding by molecular diffusion.Micro-mixed medium with uniform composition is probably the best environment for undergoing desired chemical reactions since it provides mostclosely the operation conditions when the chemist and chemical engineer optimize the targeted production technology.Therefore,the micro-mixing study has extremely important academic and commercial values.

In about 50 years of micro-mixing study,many informative monographs[2,4]and some recent review articles[1,5]appeared to address its different aspects.While tremendous achievements have been acquired in understanding the mechanisms underlying the micro-mixing phenomena,developing micro-mixing models,and simulating micro-mixing in chemical reactors,many issues towards the thorough formulation and reliable simulation are open for academic and applied resolutions.In the following sections,we will briefly review a few important aspects of micromixing study and present our perspective views on future tentative topics.

2.Measurement of Micro-mixing

2.1.Experimental techniques

Since micro-mixing is admitted to have direct impact on the conversion and selectivity of chemical reactions,researchers intuitively adopt a fast competitive chemical reaction system as the probe to test the effect of micro-mixing in a chemical reactor.These test reaction systems should have the following features:the characteristic time of reaction is shorter than or close to the micro-mixing time;the reaction mechanisms and kinetic equations are known;the amount of reaction products is easy to be accurately determined;the system is safe and cheap.Many test reaction systems were developed and their kinetics was determined,so that the micro-mixing experiments can give some quantitative in dices for evaluation and comparison between reactor configurations and operation modes.

The mostly used test systems are categorized into consecutive competitive reactions and parallel competitive reactions.A good guide in selecting the reaction systems for testing micro-mixing may refer to the papers by Guichardonet al.[6],Bourne[7],K?lblet al.[8],Unadkatet al.[9].Popular parallel competitive reactions include the neutralization and hydrolysis of ethyl chloroacetate developed by Bourne and Yu[10],and the borate/iodide/iodate system by Fournieret al.[11].Chuet al.[12]proposed a method for calculating the index of segregation from the experiment with an improved procedure for preparation of the testing iodide/iodate solutions.Habchiet al.[13]proposed a generic experimental procedure for the borate/iodide/iodate system to ensure better testing accuracy based on matching the reactor micromixing time with the optimal reagent concentrations.A typical serial competitive reaction system is the coupling of 1-naphthol with diazotized sulfanilic acid in dilute alkaline solution to produce two(monodiazo and bisazo)dyestuffs,developed by Bourneet al.[14].

Only chemical method is suitable for probing the micro-mixing performance in chemical reactors.However,chemical reaction could work together with other physical techniques,so that the micromixing efficiency may be visualized more easily and intuitively.For simultaneous measurement of macro-mixing and micro-mixing,it was demonstrated possible to use a fluorescent dye to respond to a fast neutralization reaction(for micro-mixing)together with another inert fluorescent dye for macro-mixing[15].

Lehwaldet al.[16,17]screened out a cheap fluorescent dye( fluoresce in disodium salt,C20H10Na2O5,also called Uranine)which responds to local H+-ion concentration change,and demonstrated its suitability in quantifying the micro-mixing performance.They reported an experimental Two-Tracer-PLIF(2TPLIF)procedure for simultaneous measurements of both macro-mixing and micro-mixing(another pH-inert fluorescent tracer,pyridine 2,for macro-mixing).From the PLIF-images,normalized pyridine 2 and H+concentrations are determined for the 2D measurement window.

This two-tracer method offers very intuitive visualization of macro and micro-mixing patterns in laminar and turbulent flow fields,but there is a great room for further development to get quantitative in dices to the performance of micro-mixing and its correlation with concurrent macro-mixing.It thus reminds us that other physical methods for macro-mixing may be advantageously combined with chemical methods for micro-mixing to give more clear and intuitive understanding of a total mixing process.

2.2.Index to micro-mixing efficiency

For micro-mixing experiments,it is the usual way to express the micro-mixing performance using the selectivity of byproduct(Q in the following 2 examples)generated from the slower competitive reaction.For example,the parallel competitive test reactions(instantaneous A+B→R,fast C+B→Q),the segregation index is defined as

whereas the definition for serial competitive reactions(instantaneous A+B→R,fast R+B→Q)is

using the product concentrations at the end of experiment.

The shortcoming of these definitions is the values are dependent on the concrete reaction systems,so that they have no universal basis for comparison of segregation index data on different reactor configurations reported in different works.Such in dices are obtained from the product concentrations at the reactor exit or at the end of experiment only.Just a numerical figure cannot reveal the volume fraction of efficient micromixing in the reactor,and the other temporal information of local micro-mixing during the process in a reactor is not embodied in the segregation index.This makes the voluminous data on micro-mixing information useless in scientific engineering design of chemical reactors for fast complex reactions.

In view of this defect,the procedure based on field measurements may lead to more thorough understanding of micro-mixing mechanisms and give a segregation index applicable to all test reaction systems in various kinds of reactor.The reaction-sensitive fluorescent tracer techniques reported by a few authors present good examples in this line[15–17].However,this technique is still under development,and some aspects need improvement:the way of tracer feeding may be optimized,a better method is needed to identify the active regions of micro-mixing,effective algorithms are to be developed to quantify the interaction of micro-mixing with macro-mixing,etc.

3.Classic Micro-mixing Models

For the engineering design and operation of commercial chemical reactors with dominant micro-mixing effects,the understanding of relevant macroscopic behavior is absolutely necessary.For this purpose,the modern CFD technique has become the powerful tools to present numerically such information with fine spatial and temporal resolutions.Nevertheless,chemical reaction is a process on the molecular scale,which remains difficult to be acquired from CFD even with the direct numerical simulation(DNS).In this case,a link of macroscopic hydrodynamics to the molecular kinetics is absolutely necessary for numerical simulation of fast complex reactive systems.Thus,the micro-mixing models must be built to account for these sub-grid phenomena of micro-mixing effects.In the last 50 years,many models have been developed[1].

Micro-mixing models may be divided into two categories:empirical and mechanistic ones.However,such a distinction is not sharply set,for no present model could comprehend all the physico-chemical mechanisms way down to the molecular scale.Recently,the research reports on empirical models are hardly seen,whereas the work on mechanistic ones remains booming,with a focus on improvement of model formulation for better match-up with the macroscopic scale simulation with significant micro-mixing.

3.1.Mechanistic models

Mao and Toor[18]proposed a simple slab diffusion(SD)model by representing the whole system with typical stagnant slabs,and the model parameter(the fixed slab thickness)had to be acquired from a calibrating fast neutralization reaction.It is commented that the model was too simple because the acceleration of diffusion process by fluid deformation due to viscous shear was not taken into account.

For better modeling micro-mixing,more realistic models were proposed in the last two decades of the 20th century.Ottinoet al.[19]proposed a lamellar model(LM)by perceiving that convective flow and turbulent vortex motion create a lamellar structure from the fluid streams to be mixed,and lamella are stretched to reduce its thickness to facilitate the micro-mixing by molecular diffusion(Fig.1(a)).The engulfment deformation diffusion(EDD)model by Baldyga and Bourne[20]was based on the mechanisms of engulfment,deformation and diffusion,and the striation of fluid elements are formed by engulfment due to motion of vortices,followed by fluid layers stretching and accelerated molecular diffusion(Fig.1(b)).

Liet al.[21,22]proposed a shrinkage slab model(SS)by arguing that the fluid elements to be mixed are stretching slices subjected to turbulent vortices as sketched in Fig.1(c).Their hydrodynamic experiment using high speed micro-photography indicated that a lump of tracer fluid was dispersed through the bulk flow in forms of slab and slice instead of being lamellar.The slab shrinks in its thickness due to the stretch by turbulent shear.Micro-mixing in the last stage is described in terms of diffusion in slabs accompanied by viscous deformation.

Bakker and van den Akker[23,24]proposed the mechanism of a cylindrical stretched vortex(CSV)model based on the physical picture of stretching and shrinking of turbulent vortex tubes,and the overall frame was a Lagrangian description of reaction zones in a Eulerian flow field.The key model parameter was the rate of strain in the vorticity direction,which decides the thinning rate of fluid layers by stretching(Fig.1(d)).Being related empirically to the Kolmogorov time scale,the model prediction was reasonable.

The above four works have different ways of formulating the micromixing process,but they all rely on the theory of turbulence to retrieve the model parameters.Moreover,the common basic component of modeling is a coupled,non-linear,parabolic partial differential equation describing diffusion and reaction in a slab-like domain for each chemical speciesci:

Fig.1.Schematic of mechanistic micro-mixing models.

but the parameters on the computational domain,the shrinking velocityu(x,t),the time period for integration,and the initial and boundary conditions are generally different in the literature.The reaction rateR(x,t)also needs additional modeling for non-linear chemical reactions.The interested readers may refer to their original papers.

Baldyga and Bourne[25]argued that the effects of deformation and molecular diffusion were negligible under the condition of the Schmidt number much lower than 4000 and more than two times of engulfment being necessary for achieving complete mixing.Thus they simplified the EDD model to the E model(only the rate-controlling engulfment included)and its formulation is reduced from a partial differential equation into an ordinary one without the second-order derivative of the diffusive term:

which describes the transfer of A from the surroundings near reaction zone with 〈cA〉to the reaction zone with the concentration ofcA,andEis the engulfment rate,

where ε is the energy dissipation per mass,and ν is the kinematic viscosity.Comment:It is somewhat puzzling that for theEmodel for micro-mixing is not dependent on the molecular diffusivity.It might have more relevance to mixing at a larger scale than the Batchelor one.If a complex reaction system is involved with two species having different diffusivities and competing for a common reactant,the model might generate certain error in its prediction of product yield and selectivity.If the goal is only to explore the effect of mass transfer resistance to the apparent rate of reaction,the prediction results are expected to be reasonable.

Though simple in mathematics,many reports con firmed that theEmodel works rather well in simulating the micro-mixing test reaction systems and fast precipitation of BaSO4in a variety of chemical reactors,just as well as LM,EDD,SS and CSV models based on more physical mechanisms.

3.2.Lagrange approach to micro-mixing

The above mechanistic models include a convective term to express the acceleration of interlayer diffusion due to slab shrinking in the normal direction(or stretching in the lateral direction)in Eq.(3),but the formulations ofu(x,t)differed from author to author.

Baldyga and Bourne[20]gave

wherexis measured from the central plane of the thinning fluid(Fig.2),the velocityuat any distancexis directed towards the symmetrical plane( fluid slab thinning)and is proportional tox.And earlier,Angstet al.[26]reported a derivation which reduces finally to

Indeed,it is quite correct that slab shrinkage accelerates the diffusion of species A and B into each other's domains.Exactly the enhanced diffusion rate is resulted from the shorter distance for diffusion or increased concentration gradient,and this factor is better expressed as a mobile boundary rather than the convective term in the mass transfer equation.

In a Eulerian description of a shrinking slab pair as Eq.(3),the negative-signedu(x,t)was presumed to mean the compressing action on the slabs so that the concentrations of species were steepened across the whole domain no matter whether there are reaction and diffusion or not.This description seems to be flawed because the imaginary flow ofu(x,t)∝?xdoes not assure the conservation of mass.If taking a glance at the small infinitesimal elementaat the boundary between A and B slabs(a filled black square)in Fig.2,its concentration of species A would change due to the diffusion of A,but not byu(x,t)ΔcA/Δxin the Eulerian frame.The boundary elementafeels the diffusion of A only,but nothing of convectional transport of A in the Lagrangian frame.Physically,when two slabs shrink in thexdirection,they must stretch laterally to keep the fluid incompressible.So the netaction of shrinkage is only the reduced slab thickness in thexdirection so as to ease the gradient-driven diffusion.The previous concentration pro filecA=f(x,t0)for 0≤x≤δ0will change tocA=f(x,t0+Δt)for 0≤x≤δ0?u(x,t)Δtat a moment later,and the topological shape of concentration pro file remains unchanged if there are no diffusion and reaction terms in Eq.(3).Comment:The popular mechanistic micro-mixing models are deemed not physically sound,and any simplifying assumption in modeling should be soundly made to comply with basic physical and chemical laws.For example,the turbulent force may possibly act in the direction perpendicular to thexaxis to make the slabs thicken and micro-mixing retarded.This possibility seems ironically not be mentioned in the micro-mixing study.

Suggestion:The micro-mixing models have to be built by a Lagrangian approach.Extending farther,the numerical simulation can only work by resolving the macroscopic phenomena in the Eulerian approach,with all sub-grid models being established with a Lagrangian approach.The slab shrinkage should be reformulated into a soundly based micro-mixing model,which probably is a formulation of reaction plus molecular diffusion over a pair of shrinking slabs:

Fig.2.Shrinking slab models in single-and two-phase systems.

with the model parameters soundly estimated from the turbulent flow theory.

Certainly,more factors should be considered when building a model of strong physical sense.Since the randomness of turbulence,a pair of slabs may be compressed in an interval,but it may contract backwards in the following interval to make the diffusion distance increased and thus micro-mixing retarded.Therefore,turbulence-induced shrinking of fluid slabs is not the only physical factor to be included in renovated micro-mixing models,and careful scrutiny of the distributed nature of strain rate tensor,its principal orientation and frequency of birth and deathetc.is necessary when developing new models.

4.Numerical Simulation of Micro-mixing in Chemical Reactors

4.1.Strategy of micro-mixing simulation

When the above-mentioned popular micro-mixing models were proposed,the proposers had given their validations against the corresponding experimental results.But only rather coarse methods of approximate estimation and averaging were available for figuring out the characteristic flow properties and in turn the parameters of micromixing models needed for subsequent simulation of fast reactions in a chemical reactor.When nowadays computational fluid dynamics(CFD)approach is to be incorporated into the simulation of micromixing effect in chemical reactors,a suitable CFD procedure and the micro-mixing modeling endeavor must be made mutually compatible.

The straightforward way of simulating micro-mixing effect in a reactor is the direct numerical simulation(DNS)based on the coupled first principle formulation(including Navier–Stokes equation,chemical species convective diffusion equation,energy conservation equation and reaction rate equations)with sufficiently fine spatial and temporal resolutions.However,the DNS of turbulent flow is at present not feasible for a chemical reactor,not to say that the DNS of mass transfer problem demands even finer resolution.For practical purposes,macroscopic flow has to be simulated by means of turbulence models,and hence the micro-mixing phenomena must be simulated with a modeling approach at the same time.

In the CFD frame for resolving the macroscopic flow and transport,the micro-mixing models become a sub-grid modeling component for all unclosed terms,no matter whether a RANS(Reynolds-averaged Navier–Stokes)or LES(large eddy simulation)approach is used for the macroscale phenomena.Thus,the 1st stage is the numerical simulation to get the mean velocity components,turbulent kinetic energy and turbulent energy dissipation distributions.From such information,the parameters in micro-mixing models can be evaluated,so that the chemical reactions in each control volume may be calculated in the 2nd stage.If the chemical reaction products and reaction heat do not change the physical properties of the reacting system,the 1st stage computation can be done only once to get the stabilized flow field,otherwise two stages of computation must be done in a timedependent way.

4.2.Progress in simulating micro-mixing

In the last three decades,more and more attention has been attracted to investigating the macro-mixing and micro-mixing in both non-reactive and reacting flow systems.Nevertheless,it is still challenging to simulate a chemical reactor with dominant micro-mixing effects.Many micro-mixing models have been incorporated with CFD simulations,including engulfment-deformation-diffusion(EDD),engulfment(E),multi-environment(ME),interaction by exchange with the mean(IEM)models,and direct quadrature method of moments(DQMOM)-IEMetc.Micro-mixing models based on the probability density function(PDF)methods are also getting popular,including the full PDF models and the presumed PDF ones.PDF methods are particularly attractive for simulating turbulent reacting flows wherein the non-linear chemical reaction rates appear in closed forms in the governing equations[27].The PDF is used to close the covariance of scalars involved in the transport equations of reacting species A and B,such as 〈cA′cB′〉,is treated easily as explained by Pope[27]and Fox[4].Molecular mixing is commonly described by the models extended from Lagrangian micromixing models,such as the multi-environment model.Conventional Lagrangian micro-mixing models like CD(coalescence-redispersion)and IEM have also been used to model this term.

Baldyga and Makowski[28]coupled CFD with micro-mixing models to simulate the test parallel chemical reactions(neutralization and hydrolysis of ethyl chloroacetate)in a stirred tank,and found that using the micro-mixing model with reactant concentration fluctuations being accounted for by presumedβ-PDF of the mixture fraction gave satisfactory simulation results,much better than the micro-mixing model using a spiked distribution( finite mode PDF model)and that neglecting concentration fluctuations.Hanet al.[29]proposed a CFD method combining the standard E model and the finite-rate/eddy-dissipation(FR/ED)model to study the micro-mixing effect in viscous fluids on a competitive parallel reaction system in a semi-batch stirred tank reactor with a Rushton turbine.The effect of viscosity was tested,but the difference in molecular diffusivity which will play a dominant role in micro-mixing was not addressed.Duanet al.[30]proposed a new modeling approach coupling CFD and the E-model for micro-mixing in stirred tanks with the parallel competitive reaction systems(hydrolysis of ethyl chloroacetate,borate/iodide/iodate competing for H+ions).They solved the transport equations of mixture fraction and its variance across the simulated macro- flow and calculated the representative turbulence kinetic energy and dissipation in the reaction zone to determine the E-model parameter.The predictions of segregation index compared with experimental results were satisfactory.

Akiti and Armenante[31]simulated numerically the micro-mixing effects in a fed-batch stirred tank reactor using the E model and modified E model.They used the VOF model to track the evolving volume of feed reactant pseudo-phase by solving a continuity equation,

for the volume fraction ? of the dispersed pseudo-phase(reactant)in each cell in the computational domain,andS?is the source term.In the micro-mixing model,the parameterEwas decided by the reaction zone averaged turbulent dissipation rate and turbulent kinetic energy from the simulated turbulent flow field.It was found that the simulated segregation index for the surface feeding agreed with their own experiments,but relatively poor when the feeding point was close to the impeller.Comment:This work may go on for a further step.It is straightforward to apply a micro-mixing model to each computational cell,in particular the cells where the fed reactant has not been exhausted.This will make the used micro-mixing model embedded in a more accurate environment of local strength of turbulence.Of course,this step costs more but worthwhile computational time.

Comment:So far some micro-mixing models have been developed and applied in numerical simulations,but another general observation seems that the above mathematical formulations of PDF approaches are not closely and explicitly correlated with the molecular diffusion of reactive species.Turbulence and vortices are indeed important factors in determining the micro-mixing efficiency in the PDF formulation of micro-mixing,but the last stage of uniform distribution of reactants on the molecularscale is accomplished only with molecular diffusion.It is felt strongly that the CFD coupled micro-mixing simulation should make molecular diffusion in explicit full play.

Although it is generally recognized that micro-mixing plays an important role,the present simulation reports seem not to be precise and sensitive enough to judge if the micro-mixing models used express the truth of mechanisms in fast reaction and chemical precipitation.Mixing sensitive reactions were modeled for a number of cases via CFD without a micro-mixing model by Brucatoet al.[32],but they found that the results showed good agreement with experimental data to their surprise.This superficial paradox needs further deep theoretical and experimental scrutiny.

4.3.Simulation of chemical precipitation

Micro-mixing is also an important mechanism in chemical precipitation.As two streams containing either barium hydroxide or sodium sulfate are mixed to give barium sulfate crystalline precipitate,the precipitation proceeds via macro-mixing,micro-mixing,supersaturation formation,nucleation,and crystal growth.The nucleation rate and the crystal growth speed are directly dependent upon the micromixing of reactants and the supersaturation of BaSO4.The supersaturation itself is in turn decided by the balance between micro-mixing influenced ion–ion reaction and consumption via nucleation and crystal growth.The overall process is very complicated,also due to quite different kinetics for nucleation and growth steps.It is generally accepted that chemical precipitation is also a good probe to the micromixing in chemical reactors.

To conduct numerical simulation of chemical precipitation,the mathematical model should cover both macro-mixing(viaCFD)and micro-mixing throughout a precipitating reactor.For precipitation,the micro-mixing model combining a multi-mode(or environment)probability density function(PDF)method and the concept of mixture fraction was developed[33–37].Baldyga and Orciuch [33,34]introduced a presumed β-distribution as the simplified PDF function to simulate the precipitation of barium sulfate in a tubular flow reactor.They used the finite environments PDF model for micro-mixing,in which each control volume is divided into several environments,and the micro-mixing status is represented using the mixing among these environments.Zhanget al.[38]extended the simulation with a multiple-time-scale turbulent mixer model to the precipitation in a continuous flow stirred tank with two feeding streams.Marchisoet al.used 3-mode PDF approach to the simulation of BaSO4precipitation a semi-batch Taylor–Couette reactor[36]and a tubular reactor[37],and the simulation gave satisfactory results as compared with experimental data.

However,Marchisio and Barresi[39]commented that the role of micro-mixing in fast reactions like precipitation varied depending on the operating conditions;for barium sulfate precipitation when the reactant rate was not very fast,the effect of micro-mixing can be neglected.In such cases,it seems that the selection of micro-mixing model did not matter much.Wanget al.[40]used CFD coupled with the finite mode-probability density function(FM-PDF)micro-mixing model and the method-of-moments population balance equation to simulate barium sulfate precipitation in a continuous stirred tank.The predicted particle size of barium sulfate precipitate was in good agreement with the literature measurements,but the prediction accuracy was roughly the same as that predicted using macro-scale CFD hydrodynamics and precipitation kinetic equations[41].Similar comments that no micro-mixing model tested was perfect to lead to perfect agreement with all experimental cases was made by ?ncület al.[42].These seemingly suggest that the present available micromixing models are insensitive to certain influencing factors,and the reason of their limited accuracy needs further investigation.

4.4.Closure of concentration fl uctuation term

When modeling the flow and transport of chemical species and reactions in a reactor,direct numerical simulation can be accurately done by solving the following species convective diffusion equation:

in combination with the continuity and Navier–Stokes equations.By the Reynolds averaging,the governing equation of momentum of the macroscopic flow field in terms of time-averaged velocity components contains extra terms of Reynolds stresses,and these second order terms are closed by the Boussinesq hypothesis based on known time averaged quantities.

The same time-averaging Eq.(12)will produce second-order turbulent transport terms and leads to[43]

The closure of the unresolved term is usually done using the Boussinesq hypothesis:

and the constant turbulent diffusivityDTmay be related linearly to the turbulent kinematic viscosity νT.

A classic and typical method was made by Bourne and Toor[44]to close the scalar covariance:

Fig.3.Sketch of concentration pro files of separate feeds of A and B for a 2nd order reaction.

Secondly,the closure models were established with a Eulerian approach by viewing at a fixed location,where the turbulence in a heterogeneous system makes local reactant concentrations fluctuate.Nevertheless,when following a fluid packet,the Lagrangian view of the fluctuation is quite another picture.A simple case may show that the above closure(Eqs.(16)and(15))seems non-physical.In Fig.4(a),a small isolated rotating vortex consisted of 2 segregated areas respectively for species A and B,and the net reaction rate is zero.In Fig.4(b),the reaction proceeds in a quarter of the rotating vortex and the apparent reaction rate iscA0cB0/4.The closed reaction rate from Eqs.(15)and(16),

Fig.4.Sketches of small vortex with segregated species A and B.

demands the value of?1 and 1 for these two vortices,respectively,to get the true reaction rates(as shown in Table 1).These distinct values can hardly be resolved from the concentration covariance model when these two reacting vortices may be present in the same reactor under the same hydrodynamic conditions.The available closure models suggest only that the valueISbe negative,if the segregated vortices are formed under the un-premixed feeds.

Table 1Reacting vortices with different segregation of reactants

Suggestion:The above discussion indicates that the Eulerian when a vortex containing A and B is eroded by other turbulent modulation,both A and B molecules in a fluid packet are subjected to the same motion and deformation.For the CFD simulation of micro-mixing and reacting flow,the Eulerian approach to the macroscopic flow field is a must,but the Lagrangian approach to the micro-mixing as the subgrid model seems also necessary for reliably evaluating the reaction rates in the environment of turbulent vortex motion.For this purpose,all the mechanistic micro-mixing models should be renovated to suit the reactor-scale frame of numerical simulation.

4.5.Strategy of CFD-coupled simulation

How to make full use of resolved macro-scale hydrodynamics in the simulation of micro-mixing phenomena with less empiricism remains for long a major task of academic chemical engineering community.In this line,it is better to develop a new numerical scheme for micromixing simulation in a more straightforwardly way with a realistic physical picture(Fig.5).The total concentration of a chemical species,C,may be partitioned into two layers:cmicthe micro-concentration of a species having been dispersed uniformly on the molecular scale and ready for chemical reaction,andcMacthe macro-concentration remaining in a segregated state and unavailable for chemical reaction:

Fig.5.Conception of the whole CFD simulation coupled with mechanistic micro-mixing model.

Both parts are transported separately across the reactor by the same bulk flow and turbulent dispersion,which can be resolved as the macro-mixing phenomena by macroscopic simulation of fluid flow on the reactor scale.Meanwhile,some ofcmicis transformed tocMacby micro-mixing mechanisms,which should be resolved by a suitable mechanism-based micro-mixing model in every computational control volume.

For convenience,the total concentration of a species is solved with

whereSis the source term due to non-uniformity of effective diffusivityDeff,andRAis the rate of production of species A due to chemical reactions including nucleation and crystal growth.The transition from macro-mixed to micro-mixed species occurs inside the same control volume,so it does not show up in the source term.

Similarly,the micro-concentrationcmicis controlled by

whereDeffis used since turbulence works also for micro-scale species transport,andSmicdescribes the rate of A transformed from the macro-scale to the micro-scale by micro-mixing.In fact,it is a function of position vector r,where many hydrodynamic parameters(u,k,ε)decide how fast the turbulence promotes the molecular diffusion,and the concentrations(C,cmic)decide the driving force for diffusion.By the way,Eq.(19)with Eq.(20)subtracted gives the equation forcMac.

The term ofSmicis what we must resolve by using micro-mixing mechanisms or models.The present popular micro-mixing models,like those listed in Fig.1 may be adopted or renovated for this purpose under the condition of correct partition ofcMacandcmic.Suggestion:Look at the following example(Fig.6).If the fluid mixing structure may be represented by a two-slab configuration,typically a slab diffusion model can be used to approximate the increased amount of micro-mixed A.At the starting point of a time-step,the micro-mixed concentrations of A and B are deemed uniform in a control volume,but the macro-mixed A and B exist in a segregated manner.During the time-step a suitable micro-mixing model governs the rate of transform ofreactants into the micro-mixed status,which contributes to the chemical reaction and the evolution ofcmicin the control volume.Along this line,more and perhaps accurate methods might be developed and tested,so that accurate mechanism-based micro-mixing models can be used for numerical simulation of micro-mixing dominated complex reactions.

Fig.6.Conceptual sketch of components Aand B in segregated and micro-mixed status in a control volume.

When employing such a strategy of simulation of micro-mixing dominated chemical reactions,the real intrinsic kinetic equations based on the micro-concentrations are required,but are often unavailable at present.One tentative option is to adopt the functional form of the literature kinetic equation,with the reaction orders unchanged,but the pre-exponential coefficients may be raised greatly to compensate to very low values of the real supersaturation in terms of the micro-concentrations.This realization needs certainly strict check-up against typical experimental results.It is believed that this proposition may lead to a step further to the final goal of full coupling of CFD with mechanistic micro-mixing models.

Comment:Almost all kinetic equations were developed in terms of the total or macroscopic concentrations,or the feed concentration was used when nothing can be resorted to.For seeking the true kinetics of biphasic hydroformylation of 1-dodecene,Zhanget al.[47]made an effort of combining the CFD simulation with kinetic experiment.To ease the CFD simulation part,the kinetic experiment was conducted in a simple cylindrical reactor and agitated properly so that the macromixing was fast enough but the flow remained laminar to assure the simulation being easy and accurate.With the present development level of computer capacity and numerical technique,some kinetic experiments may be worthy of being treated again using such a CFD-aided method of retrieving true reaction kinetics.

Although academic and industrial communities recognize that the micro-mixing does have important impact on complex reacting systems,there are scarcely reports on industrial application of micromixing study and simulation.One reason may be that the micromixing models are not developed sufficiently to become reliable prediction tools,in addition to the difficulty of accurate simulation of a large scale chemical reactor.Certainly,this task of commercial application is an urgent one,either for demonstrating the values of micro-mixing study or for pinpointing the new issues of micro-mixing in industrial reactors.

5.Topics in Two-phase Systems

Reports on the micro-mixing in two-phase reacting systems are scarce,probably because the accumulated understanding in singlephase reactors is not enough for further tackling such hard problems.The second phase introduced acts either as an inert phase for influencing the operating behavior of the original bulk phase,or as an active reagent in chemical reaction aiming to improve the product properties in industrial practice.

5.1.Effect of inert phase on micro-mixing

Experiments on the effects of micro-mixing in two-phase systems have been conducted in recent publications.These experimental studies suggest that the experimentally determined segregation in dices are possibly influenced by a few factors.Turbulence may be altered by the presence of a second phase.As mentioned above,small solid particles below a critical dimension suppress turbulence of the continuous phase,leading to greater index of segregationXS,while larger particles promote turbulence and in turn the micro-mixing,leading to lowerXS;gas introduction strengthens the turbulence in gas–liquid systems in most conditions;the enhancement of turbulence by the dispersed liquid phase shows a minimum at a certain intermediate fraction in immiscible liquid systems(refer to the review of Chenget al.[5]).Chenget al.[48]conducted precipitation experiments in single-feed semibatch stirred tanks and found that the BaSO4precipitation process was sensitive in probing the effect of micro-mixing in a precipitation process in a complex gas–liquid–liquid system.

Thus,it is interesting to acquire the quantitative description of modulation of micro-mixing efficiency by the features of turbulence in multi-phase systems.We are definitely sure on one thing:more special features need to be added in the micro-mixing models.For example,the shrinking slab model in a single phase system is depicted in Fig.2.In case of an inert fluid particle present,the micro-mixing by the mechanism of diffusion proceeds only on the continuous phase in subject to a stretchable interface(Fig.7(a)),and the two slabs would be stretched or retracted at the same speed.On the other hand,a solid particle makes the pair of liquid slab adhered to a rigid boundary(Fig.7(b)),which would make two slabs stretch at different paces.All these effects require a quantitative investigation in the future to fill the vacancy of micro-mixing model in two-phase systems.

5.2.Micro-mixing with reaction between two phases

In cases that several reactants originally residing in two immiscible phases are bought together to undergo chemical reactions,many scenarios could occur.A typical situation is the reaction proceeds at the phase boundary,if each reactant is soluble only in a specific phase.

An example is the two-phase reaction for preparation of caprolactam(CPL)by lactamination of CCA(cyclohexanecarboxylic acid,C7H12O2)with nitrosyl sulfuric acid(NOHSO4).The major chemical process may be expressed by a parallel-serial reaction system in a pseudo-homogeneous phase.

→k1 CPL+CO2+H2SO4 (R1)CCA+SO3→k2 CCAS(sulfonated byproduct) (R2)CPL+NOHSO4→k3 CPLS(degraded CPL) (R3)CCA+NOHSO4

The study of over all kinetics[49]indicated that the reaction order of key reactant NOHSO4is smaller in the main reaction(R1)than its counterpart in by-product reaction(R3).Therefore,the concentration of NOHSO4should be kept lower for better utilization of NOHSO4and higher selectivity to the desired CPL,if comprehended from the view point of micro-mixing in a homogeneous reacting system.

In a technical renovation,the distributor for feeding liquid containing NOHSO4and oleum(fuming sulfuric-acid)was modified so that this feed stream was dispersed into smaller drops and mixed faster with the bulky oil phase containing CCA.The renovation proved successful,and the yield of CPL was raised by above 4.5 percentage point.

Fig.7.Shrinking slab models in two-phase systems.

However,the success needs a thorough quantitative interpretation based on the real mechanism of liquid–liquid inter facial reaction(R1),because NOHSO4,oleum and product CPL are soluble in the drop of a heavy and polar phase,while CCA is the continuous phase of less polarity.In this liquid–liquid two-phase system,a simple physical model consists of an average spherical liquid drop ofNOHSO4immersed in its share of CCA solution as a shell around it(Fig.8(a)),as perceived in terms of a cell model[50].Reactions(R1)and(R2)occur at the drop interface and reaction(R3)proceeds inside the NOHSO4drop.The concentration pro files of reactants and products can be perceived as in Fig.8(b).

Numerical simulation of this reacting system by the cell model is itself a tough job,because the kinetics necessary for reaction rates on surface and liquid bulk is not available.Secondly,the effect of turbulence and flow structure should be included into the mathematical models,so that the macro-mixing in the two-phase system can be resolved.Last but not least,the effect of micro-mixing for such a fast complex reaction system must be incorporated in the models.For a small drop in a finely dispersed system where the internal laminar circulation is slow,the micro-mixing may not be significant.Micromixing effect can be very important for the continuous phase in turbulent flow on the other hand,and the micro-mixing could be retarded by the presence of the interface with a dispersed phase,but possibly enhanced by flow turbulence.It is unfortunate that so far little attention has been paid on the micro-mixing in multiphase reacting systems,particularly its modeling and simulation study.

Fig.8.A cell consisting of a central NOHSO4 drop and its corresponding CCA solution shell.

6.Conclusions and Perspectives

A few remarks follow the above survey:

(1)After more than 50 years of micro-mixing study,tremendous amount of understanding has been accumulated.Experimental evaluation of micro-mixing performance in chemical reactors are mainly aided by using complex competitive parallel or consecutive test reaction systems,because only the results from chemical methods include the information on the necessary step of molecular diffusion before chemical reactions occur.Recent progress in using chemical reaction-triggered laser-induced fluorescence provides a new technique with the potential to offering the local micro-mixing efficiency distribution and possible clue for quantitative resolution of the detailed interaction between micro-mixing and macro-mixing.This method needs further development,particularly the procedure for retrieving more illustrative and quantitative data needed for engineering application.

(2)Earlier empirical micro-mixing models presently lose their glamor,and attention is now shifted to mechanism-based modeling and improvement of existing micro-mixing models so as to be fully compatible with CFD simulation.Some micro-mixing models are flawed in the consensus of a Lagrangian point of view,and should be soundly renovated in a Lagrangian frame,so that they can couple well with the Eulerian simulation on the reactor scale.

(3)The closure of concentration fluctuation in the reacting turbulent flow remains a problem to tackle.It occurs when the time averaging is applied to calculate the reaction rate.In this case,sticking strictly to the Lagrangian approach seems necessary.

The study of micro-mixing in chemical reactors is being advanced steadily in academic research but slowly in applied aspects.Some topics listed below are worthwhile in the near future:

(1)As compared with numerous reports on academic matters of micro-mixing,the report on the application of micro-mixing models to industrial complex fast reaction systems is surprisingly rare.This implies the developed micro-mixing models and simulation methods have not surpassed the engineering experience in the accuracy and reliability.The industrial application of micro-mixing awaits a breakthrough.

(2)A full CFD micro-mixing simulation strategy seems in order.Physically,there exists a conceptual distinction between the macro-mixed and micro-mixed parts of chemical species,both of which are governed by the mass transfer and conservation laws.A renovated procedure of micro-mixing simulation based on more physics and less modeling simplification is believed to have better prospects in academic study and industrial application.

(3)Micro-mixing issues do exist in multiphase reacting systems,no matter if there are inert phases or not.At least,the effect of micro-mixing in the continuous phase awaits endeavors to the effects of existence of a dispersed phase.The present experimental reports have evidenced the issue of micromixing,but more delicate experimental techniques need to be developed so as to pinpoint the keys that lead to the observed micro-mixing phenomena and establish the micromixing models in multiphase systems.

[1]R.J.Santos,M.M.Dias,J.C.B.Lopes,Mixing through half a century of chemical engineering,in:R.Dias,R.Lima,A.A.Martins,T.M.Mata(Eds.),Chapter 4 in Single and Two-phase Flows on Chemical and Biomedical Engineering,Bentham Science Publishers 2012,pp.79–112.

[2]J.Baldyga,J.R.Bourne,Turbulent Mixing and Chemical Reactions,Wiley,New York,1999.

[3]J.Villermaux,R.David,Recent advances in the understanding of micromixing phenomena in stirred reactors,Chem.Eng.Commun.21(1–3)(1983)105–122.

[4]R.O.Fox,Computational Models for Turbulent Reacting Flows,Cambridge University Press,Cambridge,2003.

[5]J.C.Cheng,X.Feng,D.Cheng,C.Yang,Retrospect and perspective of micro-mixing studies in stirred tanks,Chin.J.Chem.Eng.20(1)(2012)178–190.

[6]P.Guichardon,L.Falk,M.Andrieu,Experimental comparison of the iodide–iodate and the diazo coupling micromixing test reactions in stirred reactors,Chem.Eng.Res.Des.79(8)(2001)906–914.

[7]J.R.Bourne,Comments on the iodide/iodate method for characterising micromixing,Chem.Eng.J.140(1)(2008)638–641.

[8]A.K?lbl,S.Schmidt-Lehr,The iodide iodate reaction method:The choice of the acid,Chem.Eng.Sci.65(5)(2010)1897–1901.

[9]H.Unadkat,Z.K.Nagy,C.D.Rielly,Investigation of turbulence modulation in solid–liquid suspensions using parallel competing reactions as probes for micro-mixing efficiency,Chem.Eng.Res.Des.91(11)(2013)2179–2189.

[10]J.R.Bourne,S.Y.Yu,Investigation of micromixing in stirred tank reactors using parallel reactions,Ind.Eng.Chem.Res.33(1)(1994)41–56.

[11]M.C.Fournier,L.Falk,J.Villermaux,A new parallel competing reaction system for assessing micromixing efficiency—Experimental approach,Chem.Eng.Sci.51(22)(1996)5053–5064.

[12]G.W.Chu,Y.H.Song,J.M.Chen,H.J.Yang,H.Chen,J.F.Chen,Modified expression of segregation index for iodide–iodate reaction system,J.Chem.Ind.Eng.China56(10)(2005)1856–1859(in Chinese).

[13]C.Habchi,T.Lemenand,D.D.Valle,M.Khaled,A.Elmarakbi,H.Peerhossaini,Mixing assessment by chemical probe,J.Ind.Eng.Chem.20(4)(2014)1411–1420.

[14]J.R.Bourne,F.Kozicki,P.Rys,Mixing and fast chemical reaction.I.Test reactions to determine segregation,Chem.Eng.Sci.36(10)(1981)1643–1648.

[15]K.Kling,D.Mewes,Two-colour laser induced fluorescence for the quantification of micro-and macromixing in stirred vessels,Chem.Eng.Sci.59(7)(2004)1523–1528.

[16]A.Lehwald,D.Thévenin,K.Z?hringer,Quantifying macro-mixing and micro-mixing in a static mixer using two-tracer laser-induced fluorescence,Exp.Fluids48(5)(2010)823–836.

[17]A.Lehwald,G.Janiga,D.Thévenin,Z?hringer,Simultaneous investigation of macroand micro-mixing in a static mixer,Chem.Eng.Sci.79(10)(2012)8–18.

[18]K.W.Mao,H.L.Toor,A diffusion model for reactions with turbulent mixing,AIChE J.16(1)(1970)49–52.

[19]J.M.Ottino,W.E.Ranz,C.W.Macosko,Alamellar model for analysis of liquid–liquid mixing,Chem.Eng.Sci.34(1979)877–890.

[20]J.Baldyga,J.R.Bourne,A fluid mechanical approach to turbulent mixing and chemical reaction.Part II.Micromixing in the light of turbulence theory,Chem.Eng.Commun.28(1984)243–258.

[21]X.Li,G.T.Chen,S.X.Rong,Studies on micromixing(III).Morphology and deformation of materials in the meso-scale,Chem.React.Eng.Technol.6(4)(1990)15–22(in Chinese).

[22]X.Li,G.T.Chen,Simplified framework for description of mixing with chemical reactions.II.Chemical reactions in the different mixing regions,Chin.J.Chem.Eng.4(4)(1996)322–332.

[23]R.A.Bakker,H.E.A.van den Akker,A cylindrical stretching vortex model of micromixing in chemical reactors,8th European Conference on Mixing,I Chem E Symp.Series,136,Cambridge,England 1994,pp.275–282.

[24]R.A.Bakker,H.E.A.van den Akker,A Lagrangian description of micromixing in a stirred tank reactor using 1D-micromixing models in a CFD flow field,Chem.Eng.Sci.51(11)(1996)2643–2648.

[25]J.Baldyga,J.R.Bourne,Simplification of micromixing calculations.I.Derivation and application of new model,Chem.Eng.J.42(1989)83–92.

[26]W.Angst,J.R.Bourne,R.N.Sharma,Mixing and fast chemical reaction-IV.The dimensions of the reaction zone,Chem.Eng.Sci.37(1982)585–590.

[27]S.B.Pope,Turbulent Flows,Cambridge University Press,Cambridge,2000.

[28]J.Baldyga,L.Makowski,CFD modelling of mixing effects on the course of parallel chemical reactions carried out in a stirred tank,Chem.Eng.Technol.27(3)(2004)225–231.

[29]Y.Han,J.J.Wang,X.P.Gu,L.F.Feng,Numerical simulation on micromixing of viscous fluids in a stirred-tank reactor,Chem.Eng.Sci.74(2012)9–17.

[30]X.X.Duan,X.Feng,C.Yang,Z.-S.Mao,Numerical simulation of micro-mixing in stirred reactors using the engulfment model coupled with CFD,Chem.Eng.Sci.140(2016)179–188.

[31]O.Akiti,P.M.Armenante,Experimentally-validated micromixing-based CFD model for fed-batch stirred-tank reactors,AIChE J.50(3)(2004)566–577.

[32]A.Brucato,M.Ciofalo,F.Grisa fi,R.Tocco,On the simulation of stirred tank reactors via computational fluid dynamics,Chem.Eng.Sci.55(2)(2000)291–303.

[33]J.Baldyga,W.Orciuch,Closure problem for precipitation,Chem.Eng.Res.Des.75(A2)(1997)160–170.

[34]J.Baldyga,W.Orciuch,Barium sulphate precipitation in a pipe—An experimental study and CFD modeling,Chem.Eng.Sci.56(7)(2001)2435–2444.

[35]D.Piton,R.O.Fox,B.Marcant,Simulation of fine particle formation by precipitation using computational fluid dynamics,Can.J.Chem.Eng.78(5)(2000)983–993.

[36]D.L.Marchisio,A.A.Barresi,R.O.Fox,Simulation of turbulent precipitation in a semibatch Taylor–Couette reactor using CFD,AIChE J.47(3)(2001)664–676.

[37]D.L.Marchisio,R.O.Fox,A.A.Barresi,On the simulation of turbulent precipitation in a tubular reactor via computational fluid dynamics(CFD),Chem.Eng.Res.Des.79(A8)(2001)998–1004.

[38]Q.H.Zhang,Z.S.Mao,C.Yang,C.J.Zhao,Numerical simulation of barium sulphate precipitation process in a continuous stirred tank with multiple-time-scale turbulent mixer model,Ind.Eng.Chem.Res.48(1)(2009)424–429.

[39]D.L.Marchisio,A.A.Barresi,CFD simulation of mixing and reaction:The relevance of the micro-mixing model,Chem.Eng.Sci.58(2003)3579–3587.

[42]A.A.?ncül,G.Janiga,D.Thévenin,Comparison of various micromixing approaches for computational fluid dynamics simulation of barium sulfate precipitation in tubular reactors,Ind.Eng.Chem.Res.48(2009)999–1007.

[40]Z.Wang,Q.H.Zhang,C.Yang,Z.-S.Mao,X.Q.Shen,Simulation of barium sulfate precipitation using CFD and FM-PDF modeling in a continuous stirred tank,Chem.Eng.Technol.30(12)(2007)1642–1649.

[41]Z.Wang,Z.S.Mao,C.Yang,X.Q.Shen,CFD approach to the effect of mixing and draft tube on the precipitation of barium sulfate in a continuous stirred tank,Chin.J.Chem.Eng.14(6)(2006)713–722.

[43]K.T.Yu,X.G.Yuan,Introduction to Computational Mass Transfer—With Applications to Chemical Engineering,Springer,Heidelberg,2014.

[44]J.R.Bourne,H.L.Toor,Simple criteria for mixing effects in complex reactions,AIChE J.23(4)(1977)602–604.

[45]J.Baldyga,A closure model for homogeneous chemical reactions,Chem.Eng.Sci.49(12)(1994)1985–2003.

[46]K.W.Mao,H.L.Toor,Second-order chemical reactions with turbulent mixing,Ind.Eng.Chem.Fundam.10(2)(1971)192–197.

[47]Y.Q.Zhang,Z.S.Mao,J.Y.Chen,Interfacial kinetics of biphasic hydroformylation of 1-dodecene catalyzed by water-soluble rhodium complex by a combined numerical and experimental approach,Ind.Eng.Chem.Res.40(21)(2001)4496–4505.

[48]D.Cheng,X.Feng,C.Yang,J.C.Cheng,Z.-S.Mao,Experimental study on micromixing in a single-feed semibatch precipitation process in a gas–liquid–liquid stirred reactor,Ind.Eng.Chem.Res.53(48)(2014)18420–18429.

[49]Z.H.Li,H.A.Luo,J.Wu,X.Yuan,Nitrosation kinetics of cyclohexane-carboxylic acid with nitrosyl sulfuric acid,J.Chem.Ind.Eng.China56(6)(2005)1015–1019(in Chinese).

[50]Z.-S.Mao,J.Y.Chen,2012.Numerical approach to the motion and external mass transfer of a drop swarm by the cell model,International Solvent Extraction Conference(ISEC'2002)2002,pp.227–232(Cape Town,South Africa).