Acta Mech Sin (2015) 31(3):292302DOI 10.1007/s10409-015-0398-5

REVIEW PAPER

A review of research on nanoparticulate flow undergoingcoagulation

Jianzhong Lin1,2 Linlin Huo1

Received: 12 September 2014 / Revised: 20 October 2014 / Accepted: 29 October 2014 / Published online: 15 May 2015 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag BerlinHeidelberg 2015

Abstract Nanoparticulate flows occur in a wide range ofnatural phenomena and engineering applications and, hence,have attracted much attention. The purpose of the presentpaper is to provide a review of the research conducted overthe last decade. The research covered relates to the Browniancoagulation of monodisperse and polydisperse particles, theTaylor-series expansion method of moment, and nanoparticledistributions due to coagulation in pipe and channel flow,jet flow, and the mixing layer and in the process of flamesynthesis and deposition.

Keywords Nanoparticulate flow Coagulation Review

1 Introduction

Nanoparticulate flows occur in a wide range of naturalphenomena and engineering applications and have beenextensively investigated during the past 20 years. Many phys-ical properties of nanoparticulate flows and their behaviorinvolving diffusion and condensation are strongly dependenton their size distribution. One of the most significant factorsaffecting particle size distribution is coagulation. Individualnanoparticles suspended in a fluid may come into contact,collide with one another, and stick together to form largerparticles because of their Brownian motion or as a result oftheir relative motion arising from gravity, electrical force,

B Jianzhong Linjzlin@sfp.zju.edu.cn

1 Institute of Fluid Measurement and Simulation,China JiliangUniversity, Hangzhou 310018, China

2 Institute of Fluid Engineering, Zhejiang University,Hangzhou 310027, China

hydrodynamic force, or other external forces. The net resultof particle coagulation is a decrease in particle number andan increase in particle size. Therefore, in actual applications,the evolution of particle number concentration and size dis-tribution in nanoparticulate flow undergoing coagulation isof fundamental importance and interest.

2 Brownian coagulation of monodisperse particles

2.1 Equation for particle number

Based on the assumption that particles will adhere when theycollide and that particle size changes slowly, Uchowski [1]first derived the coagulation equation for monodisperse par-ticles by solving the diffusion equation around a particle andby obtaining the flux of other particles toward it:

dndt

= k0n2, (1)

where n is the particle number, t the time, k0 the ideal ratewithout considering the interactions between particles, and is the collision efficiency and represents the probabilityof coagulation between two particles [2]. By integrating thecoagulation equation, the particle number concentration canbe explicitly determined as a function of time.

2.2 Particle collision efficiency

The collision efficiency in Eq. (1) is an important physicalquantity. Some studies in the literature address the collisionefficiency. By considering the van der Waals force, Russelet al. [2] computed the coagulation rate. The collision effi-

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A review of research on nanoparticulate flow undergoing coagulation 293

ciency was first derived by Han et al. [3] by considering theinterparticle forces and hydrodynamics and then combiningthe shift of the centroids of the curves for the particle sizedistribution with the total number of particles to quantify thecoagulation rate [4]. Shear-induced coagulation under theeffect of van der Waals interaction was studied by Vanni andBaldi [5]. An expression of the collision efficiency for variouscollision angles was derived by Chin et al. [6] by consideringthe electrostatic, van der Waals, magnetic dipole, and inter-particle forces and hydrodynamics. A geometric model wasproposed by Olsen et al. [7] to predict the collision efficiencyfor systems consisting of two oppositely charged species. Theeffects of van der Waals forces and noncontinuum lubricationforces on the coagulation process were explored by Chun andKoch [8]. An expression of the collision efficiency consider-ing elastic deformation and van der Waals forces was derivedby Feng and Lin [9]. New formulas for the central obliquecollision efficiency of dioctyl phthalate particles with diam-eters ranging from 100 to 800 nm were derived by Wanget al. [10] by considering the elastic deformation and van derWaals force. Those researchers showed that the elastic defor-mation between two particles should not be neglected. Thereexists an obvious difference in the collision efficiencies forcentral normal and central oblique collisions. New formulasof the collision efficiency of dioctyl phthalate particles withdiameters ranging from 50 to 500 nm under the effect of theStokes resistance, van der Waals, elastic deformation, andlubrication forces were built by Chen and You [11]. Theyfound that the collision efficiency increased with decreasingparticle size.

The collision efficiency in cases where a nanoparticlecollides with a wall was studied by Wang and Lin [12]by considering the Stokes resistance, van der Waals, andelastic deformation forces. New formulas for the collisionefficiency of dioctyl phthalate particles with diameters rang-ing from 100 to 800 nm were derived for different initialangles of attack. The results showed that, on the whole,collision efficiency increases with decreasing particle size,but there is a maxmum collision efficiency for particleswith diameters of around 550 nm. Whether or not the elas-tic deformation force is considered has a significant efffecton the collision efficiency, whereas the difference in thecollision efficiency in cases where particles collide with awall vertically or horizontally can be neglected. Molecu-lar dynamics simulation was used by Zhang et al. [13] toexplore the effect of interactions between two dipoles on thecoagulation of charge-neutral TiO2 nanoparticles in the freemolecule regime, and the results showed that particle coag-ulation increases with decreasing particle diameter becausethe dipoledipole force gradually becomes larger than thevan der Waals force. However, the effect of the interac-tion between two dipoles on coagulation drops drasticallywith as temperature increases because the average dipole

moment becomes correspondingly small. A model of cen-tral oblique collisions between two nanoparticles at differentinitial angles of attack was proposed by Wang and Lin [14]to derive the collision efficiency under the effect of elasticdeformation and van der Waals forces. New formulas forthe friction coefficient between two nanoparticles were firstproposed by Chen et al. [15] using a statistical method andthen modifying the relative diffusion coefficient, taking intoaccount the slip effect. The results showed that the modi-fied diffusion coefficient enhanced the collision efficiencyto some extent whether or not the van der Waals force wasconsidered.

3 Brownian coagulation of polydisperse particles

The coagulation equation for polydisperse particles becomesmuch more complicated and no explicit solution existsbecause the coagulation rate depends on the range of par-ticle sizes. The coagulation mechanisms of unequal sizedsilicon nanoparticles of volume ratios between 0.053 and 1were explored by Hawa and Zachariah [16] using molecu-lar dynamics simulation. It was found that the convectionprocess and the diffusion process dominate the coagulationprocess for liquidlike particles and for near solidlike parti-cles, respectively. For liquidlike particles, the deformationof smaller particles also has a significant effect on the coag-ulation process. Two particles with a much smaller ratio ofsizes coagulate much faster. The collision efficiency of twounequal sized dioctyl phthalate particles with diameters rang-ing from 100 to 750 nm was derived by Wang and Lin [17]by considering the elastic deformation and van der Waalsforces. They found that, on the whole, the collision effi-ciency increases as particle sizes decrease. A coagulationrate constant that is dependent on the time and particle sizewas derived by Kelkar et al. [18]. The results showed that thesize dependence of the coagulation rate constant affects inparticular predictions for initially polydisperse particle sys-tems.

To predict the coagulation of polydisperse particles moregenerally, the following population balance equation wasproposed by Mller [19] to represent changes in the poly-disperse particle size distribution:

n (v, t)

t= 1

2

v

0

(v1, v v1) n (v1, t) n (v v1, t) dv1

n (v, t)

0

(v1, v) n (v1, t) dv1, (2)

where n(v, t)dv is the number of particles whose volume isbetween v and v + dv, and (v1, v) is the collision kernel

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294 J. Lin, L. Huo

for two particles of volume v and v1, which depends on thecollision mechanism and the sizes of two colliding particles.The first and second terms on the right-hand side denote theincrease and decrease in particle number, respectively.

As far as the Brownian coagulation is concerned, for thoseparticles much smaller or much larger than the mean free pathlength of the gas molecules, the kinetic theory of gases or thecontinuum diffusion theory must be used to derive the col-lision kernel, respectively. Therefore, the Knudsen numberK n = l/r (l is the mean free path length of the gas moleculesand r is the particle radius) is usually used to define the par-ticle size regime. In the free molecule regime with K n > 50,the collision kernel is [20]

fm(v, v1) = B1(1/v + 1/v1)1/2(v1/3 + v1/31 )2, (3)

where B1 = (3/4 )1/6(6kbT/)1/2(kb is the Boltzmannconstant, T is the temperature, and is the particle density)is the Brownian coagulation coefficient.

In the continuum regime with K n < 1, the collision kernelis derived by the continuum diffusion theory as [20]

co(v, v1) = B2(

C(v)v1/3

+ C (v1)v

1/31

) (v1/3 + v1/31

), (4)

where B2 = 2kbT/ ( is the viscosity), and C = 1 +K n[1.142+0.558exp(0.999/Kn)] is the gas slip correctionfactor [21].

In the transition regime with 1 < K n < 50, the collisionkernel is described neither by the kinetic theory of gases norby the continuum diffusion theory. Based on the semiem-pirical solution of the collision kernel found by Fuchs [22],Otto et al. [23] proposed the collision kernel in the transitionregime as

entire(v1, v) = co f (K n), (5)

where f (Kn) is called the enhancement function.

4 Taylor-series expansion method of moment

The evolution of nanoparticle distribution can be traced bysolving the population balance equation (2). However, thenumerical calculations often become impractical because ofthe huge computational cost. Some methods have been usedto alleviate the computational cost; one of the common meth-ods is called the method of moment [2426], which has beenextensively used and has become a powerful tool owing tothe relative simplicity of its implementation and relativelylow computational cost. In the method of moment, Eq. (2) isfirst transformed into an ordinary differential equation withrespect to the moment mk by multiplying both sides of Eq. (2)

by vk and then integrating over the entire particle size range[27]:

mkt

= 12

0

0

[(v + v1)k vk

vk1] (v, v1) n (v, t) n (v1, t) dvdv1,

(k = 0, 1, 2, . . .), (6)

with

mk =

0vkn(v)dv. (7)

The moment equation (6) must be closed before being solved.A number of different methods for closing Eq. (6) haveemerged. The method of moment proposed by Pratsinis(PMM) [28] makes a prior assumption about the profile ofthe particle size distribution. The PMM can be used to obtainsuch important physical variables as total particle concentra-tion, average particle size, and polydispersity by solving thezero-, one-, and two-order moment equations. The quadraturemethod of moment proposed by McGraw [29] approximatesthe integral moment by an n-point Gaussian quadrature andcan be applied to cases governed by laws of particle growthand collision kernels. With this method, the moment equa-tion can be closed by the quadrature approximation withoutmaking a prior assumption for the profile of the particle sizedistribution when the population balance equation is derivedin terms of one internal coordinate.

The Taylor-series expansion method of moment (TEMOM)proposed by Yu et al. [30] uses the Taylor-series expansiontechnique to close the moment equation. The collision kernelin the free molecule regime is shown by Eq. (3), in which weuse the notation C = (1/v + 1/v1)1/2. Equation (6) is notclosed when we substitute Eq. (3) into Eq. (6) because thevariable C exists in quadratic form. For the PMM, Eq. (6)becomes integrable by using the following approximation[28,31]:

C = b(

1/v1/2 + 1/v1/21)

, (8)

in which b is strongly dependent on the range of the parti-cle size spectrum. For the TEMOM, Eq. (3) of the collisionkernel can be rewritten as

fm = B1 (v + v1)1/2(v1/6v1/21 + 2v1/6v1/61 + v1/2v1/61

), (9)

and then (v + v1)1/2 in Eq. (9) is expanded in a Taylor seriesabout point (v = u, v1 = u):

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A review of research on nanoparticulate flow undergoing coagulation 295

(v + v1)1/2 =

2u +

2 (v u)4

u+

2 (v1 u)

4

u

2 (v u)232u3/2

2 (v1 u) (v u)16u3/2

2 (v1 u)232u3/2

+ . (10)

Substituting Eqs. (9) and (10) into Eq. (6), we can close themoment equation (6).

The aforementioned method is also used to deal with col-lision kernels in the continuum regime and close the momentequation (6).

No prior assumption about the profile of the particle sizedistribution is necessary for TEMOM, and the number ofmoment equations required equals the order of the Taylor-series expansion. In TEMOM the precision of a solution isenhanced and the computational cost is increased as the orderof the Taylor-series expansion increases.

The TEMOM has been used to deal with various prob-lems in the past 5 years. The binary homogeneous nucleationof watersulfuric acid and the growth of nanoparticles wereexplored by Yu and Lin [32]. They analyzed the competi-tion between nucleation, coagulation, and condensation inboth cases with and without background particles and showedthat the production rate of sulfuric acid is an important fac-tor affecting nucleation kinetics and particle dynamics. Theeffect of particle diffusion on coagulation was studied byWang et al. [33], who showed the distributions of parti-cle number and mass concentration and the average particlevolume under coagulation and diffusion. The TEMOM wasmodified by Lin and Chen [34] to match the property of realself-preserved nanoparticles under coagulation based on thenumerical results given by the sectional method in both freemolecule and continuum regimes. The results showed thatwhen the particle size distribution attains a self-preservedstate or the coagulation time is long enough, the modifiedTEMOM can give more precise predictions on the results ofthe zeroth and second moments than the original TEMOM.They also proposed a special kind of coordinate diagramto describe qualitatively the errors produced by differentmethods of moment based on the zeroth, first and secondmoments. Meanwhile, a new set of moment equations in thefree molecule regime were built for the case in which theparticle size distribution attained a self-preserved state withlog-normal form. Some fundamental issues of the TEMOMwere investigated by Xie and He [35], who clarified theuniqueness of Taylor-series expansions, the availability ofclosuring moment equations with fractional moments, theconvergence of the analytical solutions, and so on. A directexpansion method of moment over the entire size regimewas proposed by Chen et al. [36] using the exact Dahnekesformula as the collision kernel. They showed that the direct

expansion method is can be used to describe the evolutionof the zeroth and second moments and predict more pre-cise results than those predicted by the quadrature methodof moment when the initial geometric standard deviation isrelatively small.

In Sects. 58 the nanoparticle distributions due to coag-ulation in the pipe and channel flow, jet flow, and mixinglayer and in the process of flame synthesis and depositionare demonstrated based on the Brownian coagulation ofmonodisperse and polydisperse particles partly through theTEMOM.

5 Pipe and channel flows

5.1 Straight pipe and channel flow

The formation, growth, and transport of particles with diam-eters ranging from 1 to 50 nm in a channel capacitivelycoupled radio-frequency silane discharge were investigatedby De Bleecker et al. [37], who found that the concertedactions of particle charging and transport greatly affect thelocation of particle growth because of coagulation. Particlecontamination can be controlled by thermophoretic force.The distributions of nanoparticle number, average size, andvolume at different pipe lengths and Reynolds numbers weremeasured by Yin and Lou [38]. The results show that forparticles with diameters ranging from 5.6 to 560 nm, theincrement of the pipe length and the turbulence intensity willaccelerate the deposition process. The smaller the particles,the lower the penetration efficiency. Nanoparticle migrationin a turbulent pipe flow was studied by Lin et al. [39]. Itwas found that particles distribute nonuniformly in a crosssection. Coagulation leads to a gathering of larger particlesat the pipe center. As the Schmidt number increases, theparticle mass and number concentrations, as well as the poly-dispersity, decrease in the pipe center. The increase in theDamkohler number leads to an increase in the particle sizeand geometric standard deviation. The transport and penetra-tion efficiency of particles with diameters ranging from 50to 450 nm in a turbulent flow were studied by Lin et al. [40]by considering the Brownian diffusion, turbulent diffusion,coagulation, and breakage. They showed that the particlenumber concentration is distributed nonuniformly in crosssections. with higher values in the pipe center. The particlediameter decreases gradually from wall to pipe center. Theparticle penetration efficiency changes from 65 % to 95 %when the Reynolds number changes from 4426 to 8500 andthe ratio of pipe length to diameter changes from 375 to 625.The particle penetration efficiency increases and decreaseswith increases in the particle size and the Reynolds num-ber, respectively. The researchers also derived the relation ofparticle penetration efficiency to related parameters.

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296 J. Lin, L. Huo

5.2 Curved pipe and channel flow

Nanoparticle transport and deposition in a curved pipe at dif-ferent Reynolds numbers and Dean numbers were simulatedby Lin and Lin [41]. It was found that the particle distrib-ution was symmetrical with respect to the pipe bottom andtop sides. There was a large and a small amount of deposi-tion on the outside and inside walls, respectively. Particlesat higher flow Reynolds numbers deposited quickly on thewall, whereas pipe curvature had an insignificant effect onparticle deposition. The researchers also studied the effectof Schmidt numbers on particle transport and deposition andfound that the Schmidt number, pipe curvature, and Reynoldsnumber had first-order, second-order, and forth-order effectson particle deposition, respectively [42]. Nanoparticle trans-port and coagulation in a curved pipe were simulated byLin et al. [43], who showed that the particle number andmass concentration, polydispersity, average diameter, andgeometric standard deviation increased with time. The par-ticle number concentration increased, whereas the particlepolydispersity, average diameter, and geometric standarddeviation decreased with increasing initial particle diametersand Reynolds numbers. The effect of particle coagulation onparticle distribution was more pronounced in the initial stagethan in subsequent stages.

5.3 Rotating curved pipe and channel flow

Nanoparticle transport and deposition in a rotating curvedpipe at different angular velocities, Schmidt numbers, andDean numbers were explored by Lin et al. [44]. It was foundthat the particle distribution was basically dependent on theaxial velocity when the Schmidt number was small, whereasthe secondary flow controlled the particle distribution whenthe Schmidt number was much larger than 1, as shown inFig. 1. The particle deposition became uniform along theentire edge of the wall with an increase in the Dean number.The rotation direction of the pipe had an important effect onthe particle deposition. There was a large amount of deposi-tion when secondary flow appeared in the pipe.

Fig. 1 Distribution of nanoparticle mass fraction at different Schmidtnumbers

The evolution of nanoparticle distribution in a rotatingcurved pipe at different rotation numbers, Schmidt numbers,and Reynolds numbers was explored by Lin et al. [45]. Theresults showed that the particles gathered in the region closeto the outside edge of the pipe when the Coriolis force andthe centrifugal force were in the same direction, while parti-cles gathered in the region close to the inside edge when theCoriolis force was in the opposite direction to the centrifugalforce and the Coriolis force was much larger than the centrifu-gal force. Particle mass and number concentrations increasedquickly in the early stage than that in subsequent stages andfinally attained a stable value. The particle distribution wasdominated by the competition between the pipe curvature andthe rotation number. The particle mass and number concen-trations increased, whereas particle polydispersity, diameter,and geometric standard deviation decreased with increasesin the Reynolds and Schmidt numbers.

6 Jet flows

6.1 Round jet

Nanoparticle coagulation and dispersion in a round jet weresimulated by Chan et al. [46], who showed that particle massconcentration in the jet core decreases with increases in thedistance from the jet exit. The particle number concentra-tion decreases quickly in the exit region of the jet and thendecreases slowly as the distance from the jet exit increases,until an asymptotic state is reached. The growth rate of parti-cle polydispersity is greatest in the exit region of the jet, whileparticles with the largest diameter are found in the core regionof the jet. The particle diameter changes slightly across thewidth of the jet, except in the interface region between thejet and the outside. The particle geometric standard devia-tion increases in the exit region of the jet and then slowlyapproaches an asymptotic value in the core region of the jet,as shown in Fig. 2. Across the width of the jet the particlediameter increases as the the Damkohler number increases,as shown in Fig. 3, and the particle number concentrationincreases as the Schmidt number increases.

Nanoparticle coagulation and dispersion was studied byLin et al. [47] on the basis of unimodal lognormal particle sizedistributions. It was found that coherent structures appearedin the interface region between the jet and the outside, whichpromoted particle coagulation and changed the particle con-centration distribution across the width of the jet, resultingin a nonuniform dispersion along the flow direction. Withincreases in the Schmidt number, the region occupied by par-ticles became narrower, the particle polydispersity decreased,and particle number concentration increased. Across thewidth of the jet, the particle size and geometric standarddeviation increased as the Damkohler number increased.

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A review of research on nanoparticulate flow undergoing coagulation 297

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

1.0

1.2

1.4

1.6

1.8

2.0x/D =0; x/D =2; x/D =4; x/D =6; x/D =8

g

r/D

Fig. 2 Distribution of geometric standard deviation across width of jet

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

1.0

1.2

1.4

1.6

1.8

2.0

Da =0.5 Da =1 Da =2

r/D

d p/(n

m)

Fig. 3 Distribution of particle diameter across width of jet

The formation of nanoparticles arising from the homoge-neous binary nucleation of H2SO4 and H2O vapors in twinround jets was simulated by Yin et al. [48], who showed thatthe velocity ratio of ambient wind to exhaust gas plays animportant role in affecting particle concentration and sizedistribution. The particle formation rate decreased and parti-cle size increased with increases in the ambient wind velocity.The formation of nanoparticles in twin impinging jets in dif-ferent spaces between the two jets at different distances fromnozzle exit to impingement plane was studied by Yin andLin [49]. It was found that the maximum particle size andthe maximum particle number concentration appeared in theregion of the free jet and in the region close to the imping-ing plane, respectively. The particle concentration and sizedistribution in the interface region of the two jets were depen-dent on the space between the two jets. The larger the spacebetween the two jets, the more the particles were produced.The greater the distance from the nozzle exit to the impinge-ment plane, the lower the particle number concentration,

and the smaller the number of particles distributed in theregion close to the impinging plane. The optimum dilutionconditions were explored by Fujitani et al. [50] by consid-ering particle growth via coagulation and other factors in adiesel exhaust. It was found that a short residence time couldprevent particles from growing via coagulation followingthe primary dilution or before the diluted exhaust reachedthe inhalation chamber following the secondary dilution. Theeffects of some related factors on the gas-to-nanoparticle con-version and distribution of gas released from the tailpipe of adiesel vehicle were explored by Chen et al. [51]. The resultsshowed that the coagulation process was slowest comparedwith nucleation and turbulent dispersion processes. A highervehicle velocity decreased the particle number concentration.Nanoparticle coagulation and dispersion in a turbulent roundjet were studied by Zhu et al. [52]. It was found that the par-ticle size was distributed uniformly and remained constantin the potential core but became large in the region of highturbulence intensities. The distribution of particle size alongthe flow direction changed slightly at first, and then quickly,finally attaining a steady state.

6.2 Planar jet

Nanoparticle coagulation and growth in a planar jet was stud-ied by Yu et al. [53], who showed that the distributionsof particle diameter, number intensity, and polydispersityare dominated by the formation and evolution of coherentstructures. Along the flow direction, the particle number con-centration decreases in the core region of the jet and increasesin the peripheral region. The particle mass concentrationremains unchanged on the whole flow, whereas particle sizeand geometric standard deviation increase and attain theirmaximum in the interface region between the jet and theoutside. The effect of Damkohler number and Schmidt num-ber on nanoparticle distribution was investigated by Yu et al.[54]. It was found that the larger the Schmidt number is, thenarrower the region across the width of the jet in which par-ticles are distributed. Particles with smaller diameters showa higher polydispersity because they coagulate and dispersemore easily and grow quickly. The characteristic time of par-ticle coagulation is so short that particle collision and coagu-lation happen frequently, which results in an increase in parti-cle size. Particle polydispersity is directly proportional to theDamkohler number. The distributions of nanoparticles aris-ing from the homogeneous binary nucleation of H2SO4 andH2O vapors in a submerged constraint jet was simulated byLiu [55]. It was found that the particle number concentrationclearly increased within the core region of the coherent vor-tex produced by the rolling up of the interface between jet andambient. The presence of a coherent vortex promoted particlecoagulation by increasing the possibility of particle collision.

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298 J. Lin, L. Huo

The effects of turbulence on nanoparticle growth wasexplored by Das and Garrick [56] by considering Browniancoagulation, nucleation, and condensation in a reacting flow.They showed that particles with large diameters appeared inregions away from the core region of the jet. The particleaverage diameter increased with increases in the precursorconcentration. The particle growth rate was higher withinthe core region of the eddy and increased continuously alongthe flow direction. The evolution of the TiO2 nanoparticledistribution in a turbulent reacting planar jet was simulatedby Loeffler et al. [57] by neglecting the effect of unresolvedfluctuations on coagulation. They suggested that neglect ofthe interactions between particles would result in an increasein the particle growth rate as the precursor concentrationincreased. The TiO2 nanoparticle synthesis in a planar, non-premixed diffusion flame was simulated by Garrick et al. [58]by considering coagulation, nucleation, coalescence, andcondensation. They explored the effect of mixing and finite-rate sintering on particle distributions and showed that highlyagglomerated particles were located on the periphery of thecoherent eddy, where particle collisions resulting in coagu-lation took place more quickly than particle coalescence.

7 Mixing layers

Based on the assumption that particle size has a lognor-mal distribution, nanoparticle coagulation in an isothermalmixing layer was studied by Settumba and Garrick [59],who discussed the spatiotemporal evolution of the particlenumber and mass concentration, mean size, and geometricstandard deviation for Damkohler numbers 0.2, 1, and 2.The researchers also performed a numerical simulation ofnanoparticle coagulation in a mixing layer [60] and found thatthe polydispersity calculated based on the diffusion coeffi-cient and the local average volume was slightly higher; hence,the method of diffusion coefficient and local average volumecan be used to reduce the spatial resolution, which allows formore affordable computations. Nanoparticle distribution inan isothermal shear layer under the influences of coagulation,convection, and diffusion at a Reynolds number of 200 andDamkohler numbers of 1 and 10 was investigated by Garricket al. [61], who showed that the nonuniformity of the particleconcentration distribution grew with time, which resulted inan increase in the particle geometric standard deviation. Withthe increase in the Damkohler number, the particle growthincreased and the particle size distributions became widerthan the self-preserving limit. The formation and growth ofTiO2 nanoparticles, up to and including particles 128 nm indiameter, in a mixing layer was studied by Wang and Gar-rick [62]. It was found that particle formation and growthwere limited by the mixing and particle size increased morequickly with increases in the initial reactant levels. The

Fig. 4 Distribution of particle mean diameter

Fig. 5 Distribution of volume concentration

particle geometric standard deviation showed a greater vari-ation throughout the mixing layer. A numerical simulation ofnanoparticles arising from a homogeneous binary nucleationof H2SO4 and H2O vapors in a mixing layer was performedby Lin and Liu [63]. It was found that the particle numberand volume concentration distributions, as well as the meandiameter, were dominated by the coherent vortex structure.The particle number concentration decreased, whereas thevolume concentration and mean diameter increased alongthe flow direction, as shown in Figs. 4 and 5. The particlenumber and volume concentration, as well as mean diam-eter, were distributed nonuniformly across the width of themixing layer. The coagulation process was longer than thenucleation process and was mainly affected by the numberconcentration.

Particle coagulation in a temporal mixing layer was stud-ied by Xie et al. [64]. The results showed that the particlenumber concentration decreased while the particle mean vol-ume increased with time. The particle mass and number con-centration and mean volume were distributed nonuniformlyowing to the existence of a coherent structure. The flowadvection had an insignificant effect on particle coagulationin the region far from the coherent structures. The particleshad an obvious wavelike distribution within the core regionof the coherent vortex. The formation and evolution of thecoherent structure had a greater effect on particle coagulation,which in turn affected the distributions of the particle massand number concentration, as well as the mean diameter.

8 Flame synthesis and deposition

The generalizations of a Gaussian quadrature-based nine-moment method was used by Rosner and Pyykonen [65]

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A review of research on nanoparticulate flow undergoing coagulation 299

to study the bivariate particle balance equation in a lami-nar flame reactor. The evolution of alumina nanoparticleswas predicted under the combined effects of coagulation,thermophoresis, and sintering. The formation and growthof polydisperse nonspherical silica nanoparticles in an oxy-hydrogen coflow diffusion flame was simulated Kim et al.[66] by considering particle generation, coagulation, diffu-sion, convection, and coalescence. The results showed thatthe distribution of nonspherical particle sizes was highlyspatially nonuniform. The attainment of size distributionsnarrower than those predicted by the self-preserving the-ory of nanoparticle coagulation was studied by Tsantilisand Pratsinis [67]. They simulated the TiO2 formation viatitanium-tetra-isopropoxide (TTIP) or TiCl4 oxidation byconsidering the gas/surface reactions and coagulation andillustrated the effects of temperature, pressure, and initialprecursor molar fraction on the distributions of TiO2 particlesize and geometric standard deviation. A new constitutive lawfor fractal aggregates resulting from coagulation was gener-alized by Kostoglou et al. [68], who added a restructuringmechanism to the population balance model. By solving thebivariate coagulation equation, they found that the presenceof restructure resulted in an evolution dynamics of the frac-tal aggregate distribution that is much richer than that givenby previous coagulation models. A mass-flow-type stochas-tic particle algorithm that is used to simulate nanoparticlegrowth in flames and reactors was derived and tested byMorgan et al. [69]. The newly derived algorithm combinesthe effect of coagulation with particle source and surfacegrowth. A particle model and stochastic methods to simulatenanoparticle size distributions in a premixed flame was usedby Morgan et al. [70]. The results provided further evidenceof the interplay among nucleation, coagulation, and surfacerates. The formation of monomodal polyelectrolyte complexnanoparticles under different ionic strengths and applied cen-trifugation regimes was studied by Starchenko et al. [71],who used a novel particle coagulation model to simulate theaggregation process under different typical colloidal parame-ters and analyzed the effects of these parameters on the meanfinal size and the size distribution function of the particles.

A new numerical algorithm was developed by Yu et al. [72]to explore the effects of precursor loading on TiO2 nanoparti-cle synthesis in a flame reactor. The precursor TiCl4 oxidationleading to particle formation is modeled using a one-stepchemical kinetics approach. It was found that the particlenumber concentration and diameter increased with increasesin precursor loading, as shown in Figs. 5 and 6. The particlesurface fractal dimension was weakly dependent on the inletprecursor loading. When the inlet precursor loading remainedunchanged, the larger the carrying gas rate was, the smallerthe agglomerated particles, the larger the total specific sur-face area, and the wider the particle size distribution. Theresearchers took the particle size and particle surface area

x

m

0 0.05 0.11016

1017

1018

1019

1020

1021

0

0.4 10-4

1.6 10-4

5.8 10-4

Inlet TiCl4 loading (mol/min)

Fig. 6 Distribution of particle number concentration

x

da

0 0.05 0.1

100

200

300

400

500Inlet TiCl4 loading (mol/min)

0.4 10-4

1.6 10-4

5.8 10-4

Fig. 7 Distribution of particle diameter

as two independent variables to investigate the distributionsof particle-volume-equivalent diameters, the particle numberevolution, the geometric standard deviation based on particlevolume and the fractal nature of the agglomerated particles,and the number of primary particles per agglomerate [73](Fig. 7) .

A low Reynolds number turbulent model was used byAristizabal et al. [74] to simulate the synthesis process ofaluminum nanoparticles by taking the particle coagulation asthe dominant mechanism in particle growth. They found thatthe flow changes from laminar to turbulent in scale-up with aconstant residence time, which has a significant/insignificanteffect on particle properties for the reactors of small/largelength-to-diameter ratios, respectively. The synthesis of TiO2nanoparticles in a low-pressure flat stagnation flames wasstudied by Zhao et al. [75] studied using TTIP as precur-sor. It was found that both larger aggregate particles andsmaller primary particles were produced at higher pressures.When the precursor-loading rates were higher, the aggregateparticle size was larger, whereas the primary particle sizeremained constant in the experiment and decreased slightlyin the numerical simulation. Particle growth processes in the

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postflame region of a premixed ethylene-air flame was inves-tigated by De Filippo et al. [76]. The particle size distributionswere initially unimodal and then became bimodal in the post-flame region. The results showed that the smaller mode couldbe satisfactorily predicted with a size-dependent coagulationmodel.

The aggregation and deposition kinetics of fullerene C60nanoparticles was studied by Chen and Elimelech [77]. Theresults showed that the aggregation kinetics of the parti-cles exhibited reaction-limited and diffusion-limited regimesat critical coagulation concentrations. The deposition rateincreased with increases in the electrolyte concentrations.The Brownian and coagulation rates and deposition coeffi-cients of nanoparticles in a closed chamber were measuredby Kim et al. [78]. The results showed that the depo-sition processes were overwhelmed by the coagulationprocesses in the case of high particle number concentra-tion. The larger turbulent coefficients made the turbulentcoagulation stronger. Particle coagulation rates were dif-ferent in the small particle size range. The removal ofdispersant-stabilized carbon nanotube suspensions by polya-luminum chloride alum was studied by Liu [79]. It wasfound that the polyaluminum chloride had a different effectwith alum in the removal of the suspensions. The reduc-tion of particle number concentration first increased thendecreased with increases in coagulant dosage. The floc-culation and coagulation of carbon nanotube suspensionsby polyaluminum chloride were regulated mainly by themechanism of adsorption charge neutralization, whereas thecoagulation by alum mainly involved electrical double-layercompression.

9 Conclusions

This review presented an overview of the research progressmade in the Brownian coagulation of monodisperse andpolydisperse nanoparticles, the TEMOM, nanoparticle dis-tributions due to coagulation in pipes and channel flows, jetflows, and mixing layers, and in the process of flame synthe-sis and deposition.

Future investigations must include the following aspects:

(1) Contributions to nanoparticle coagulation resulting fromfluctuating particle concentrations;

(2) Determination of a collision kernel for two nonsphericalnanoparticles;

(3) Distributions of nanoparticles under nucleation, convec-tion, diffusion, coagulation, and breakage in turbulentflows.

Acknowledgments The project was supported by the Major Programof the National Natural Science Foundation of China (Grant 11132008).

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A review of research on nanoparticulate flow undergoing coagulationAbstract1 Introduction2 Brownian coagulation of monodisperse particles2.1 Equation for particle number2.2 Particle collision efficiency

3 Brownian coagulation of polydisperse particles4 Taylor-series expansion method of moment5 Pipe and channel flows5.1 Straight pipe and channel flow5.2 Curved pipe and channel flow5.3 Rotating curved pipe and channel flow

6 Jet flows6.1 Round jet6.2 Planar jet

7 Mixing layers8 Flame synthesis and deposition9 ConclusionsAcknowledgmentsReferences