A CFD model for gas uniform distribution in turbulent flow for the production of titanium pigment in chloride process☆

Yadong Li,Gang Xie ,Ting Lei,Chongjun Bao ,Lin Tian ,Yanqing Hou *

1 State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization,Kunming University of Science and Technology,Kunming 650093,China

2 Faculty of Metallurgical and Energy Engineering,Kunming University of Science and Technology,Kunming 650093,China

3 Kunming Metallurgical Research Institute,Kunming 650031,China

4 Key Laboratory of Pressure Hydrometallurgy Technology Associated Non-Ferrous Metal Resources,Kunming 650031,China

5 Kunming Metallurgy College,Kunming 650033,China

1.Introduction

Titanium pigment is an important inorganic pigment;it is non-toxic,has the best whiteness,a high refractive index,covering power and brightness[1].It is believed to be the best white pigment in the world[2].TiO2has various commercial applications,such as coatings,plastics,rubber,metallurgy,and cosmetics[3–5].With the development of coating industry,the demand for titanium pigment has increased in recent years[6,7],especially in China.The method to manufacture titanium pigment includes the sulfuric acid process and the chloride process at present[8–10].The chloride process has been the primary technology because of its advantages,such as short process flow,ease of expanding production capacity,high quality product,low pollution,and high automation[11].

The TiCl4gas phase oxidation reactor is the key equipment in the chloride process,in which the oxidation reaction(TiCl4+O2=TiO2+2Cl2)mainly occurred,and the side reaction can be ignored since the main reaction proceeds rapidly in milliseconds.If the mixing of two reactants is not good enough,the reaction rate will be unequal in the reacting zone.Thus,the size of generated particles may be distributed widely,which makes unqualified products.Furthermore,the concentrated reaction heat release may lead to sintering or scarring when the reacting mixture is not uniform.Therefore,the mixing of reactants is very important in order to obtain the required product[12].The mixing must be investigated in detail to reveal the hydrodynamic characteristics of feeding flow in the reactor.It has been noticed that the jet in cross flow(JIC)is usually used to increase the mixing rate[13].However,the non-uniform flow of fluid,which has a considerable effect on mixing of the feeding gases,may occur in the feeding torus of the oxidation reactor because of the pattern of jet hole distribution[13].Therefore,the flow characteristics and pressure distribution of gas in the annular channel should be investigated in detail to get a better design of the feeding torus.However,few relevantstudies have been published in the open literature.Zhang[14]investigated gas mixing and the reaction process through cold and hot experiments,and showed that tangential inlet ring should be used to improve the gas mixing.Lu et al.[13]and Cheng et al.[15,16]carried out an experiment regarding gas mixing with the aid of particle tracing and particle image velocimetry(PIV)techniques.They concluded that the main factors affecting the mixing conditions were the momentum ratio and the crossing angle of two flows.However,there is ambiguity regarding the appropriate momentum ratio,crossing angle and jet ring opening position.The pressure distribution of the variable mass flow of gas in the annular channel was theoretically derived from the momentum theorem by Cong[17,18]and Li et al.[19,20].Lu et al.[21]used the simplified onedimensional model to analyze the static pressure distribution of the variable mass flow in the annular channel.Nevertheless,the momentum exchange coefficient was introduced in these models,which needed to be determined through experimentation.Engineering momentum exchange coefficient data are scarce,and the flow structure of fluid in the annular channel is not clear.

Singh and Raman[22]studied the methane/air flame and interaction with synthesis TiO2by TiCl4oxidation using a two-dimensional numerical simulation based on a multi-step chemical reaction mechanism.Sung et al.[23]simulated the TiO2nucleation process through the application of a large eddy dissipation model.The simulated results are in accordance with the experimental results.Schild et al.[24]and Johannessen et al.[25]simulated the chemical reaction and particle formation kinetics of a TiO2formation using particle growth kinetics,establishing the particle formation kinetics model.Recently,Akroyd et al.[26]and Buddhiraju and Runkana[27]simulated the nano-TiO2formation process in turbulent reactions using a fully coupled model and a coupled flame dynamics–mono-disperse population balance model.

Researchers obtained many interesting results focused on gas flow in the feeding torus over the past years.However,there is ambiguity regarding the appropriate momentum ratio,crossing angle and jet ring opening position.In order to obtain the optimum gas mixing conditions and reveal the flow hydrodynamics in the feeding torus,a CFD simulation of turbulent feeding gas flow for the production of titanium pigment in the chloride process was performed.Based on the verified numerical model,the characteristics of velocity and pressure distribution in the feeding torus are analyzed.The flow distributions with five different rectifying rings have been achieved.

2.Mathematical Model

2.1.Distributor geometry

The present investigation considers the production of titanium pigment in a lab scale oxidation reactor.The geometry and input conditions are representative of industrial conditions for the chloride process.The structure of the distributor with different rectifying rings was developed.The inner diameters of the TiCl4tangential inlet pipe and the annular channel are 60 mm and 300 mm,respectively.The external diameter of the jet ring is 100 mm,where 24 jet holes with a diameter of 5 mmare uniformly distributed on the periphery.The feeding gas enters the reactor through these jet holes.The basic Case#0 without a rectifying ring has been studied as a reference structure as shown in Fig.1.Fig.1 also shows other 5 structures with different rectifying rings.The width of the feeding torus is 100 mm.In Cases#1,#2 and#3,the external diameter of the rectifying ring is 200 mm.The opening ratio of the rectifying ring is defined as φ=A0/Aa,where A0represents the total area of the hole on the rectifying ring and Aarepresents the total area of the rectifying ring.In Cases#1 and#2 small holes are uniformly distributed as shown in Fig.1,and the opening ratios of the rectifying ring are 8%(holes of?10)and 18%(holes of?5),respectively.In Case#3,the holes are 16 slits(60 mm×5 mm)on the rectifying ring with an opening ratio of 8%.The rectifying ring of a single baffle was adopted in Case#4 while double baffles were adopted in Case#5.The measurements are taken every 30°along the gas flow direction in the plane as shown in Case#0.R is the radial position of the measurement points,and θ is the circumferential angle of the measuring point along the gas flow direction with starting point θ of 0°being shown in Fig.1.

2.2.Mathematical model

The flow of TiCl4steam in the annular channel of the oxidation reactor is turbulent[21].It is necessary to solve the continuity equation,the momentum equation,the turbulent kinetic energy and the turbulent kinetic energy dissipation rate equation.The following three hypotheses have been made,since the characteristics of TiCl4flow in an annular channel were simulated under a steady state.(1)TiCl4flow reached a steady state.(2)TiCl4gas is pure without chemical reactions.(3)There is no temperature gradient in the whole annular channel.Under these assumptions,the governing equations[28,29]with the Realizable k-ε turbulence model are listed in Table 1.

TiCl4was considered as the main gas stream.The computational domain includes the inner area of the feeding torus,feeding pipe,and the jet holes in the reactor wall.The operating pressure was 400 kPa,and the pressure in the reactor is assumed to be 0 Pa.The boundary conditions are as follows:

(1)Inlet boundary:A simple velocity inlet was chosen,allowing specification of velocity,hydraulic diameter and turbulent intensity.Uniform velocity and temperature profiles were specified at the inlet.The inlet velocity varied from 10 to 30 m·s-1under different flow conditions,where the temperature was 700 K.

(2)Outlet boundary:The pressure-outlet boundary condition defined in Fluent was selected.

2.3.Numerical method

The commercial ANSYS Fluent 14.5.0 was used to solve the governing equations.Pressure–velocity coupling was achieved using the SIMPLEC algorithm for the steady state solution method.The default relaxation factors of Fluent were used.The standard discretization scheme was chosen to solve the pressure equation,whereas the second order upwind schemes were used to solve the momentum,turbulent kinetic energy and turbulent kinetic energy dissipation rate equations.The Realizable k-ε turbulence model with wall function of enhanced wall treatments[15,30,31]was selected for the flow field model.The computational domain is meshed by applying tetrahedrons with the following conditions:min size is 7.91×10-5m,max face size is 7.91×10-3m,max size is 1.58×10-2m,and growth rate is 1.57×10-2.The grid quality satisfies the research requirements with min orthogonality of 0.25 and max skewness of 0.74.The solution was convergent when the residual errors were less than 10-3.

3.Model Validation

In order to verify the model validity,the geometry and input conditions were identical with the geometric size of the experimental reactor in the open literature[32]as shown in Fig.2.The inlet gas was air at 298.15 K with a Reynolds number of 77000(Uin=15.7 m·s-1).

In order to verify the model,the calculated results and the experimental data of the velocity distribution at the radii of 175 mm and 215 mm were plotted as shown in Fig.3.The average relative error,

is usually employed to measure the closeness of the calculated results and experimental data.Here,vi-Caland vi-Expindicate the velocity of the calculated data and the experimental data at the measuring point,respectively.The average relative errors of velocity at the radii of 175 mm and 215 mm were 1.73%and 3.33%respectively,which indicates that the adopted model is valid.It is noted that the errors of velocity at 0°are large(up to about 20%)due to the following reasons:(1)the incompressible gas is employed in the model but the real gas is compressible;(2)the viscosity of the gas is assumed to be constant.The density of the gas increases because of merging flow at 0°which causes the velocity of gas calculated to be bigger than experimental data according to continuity equation of compressible gas[33]:

Fig.1.Structure of the distributor with different rectifying rings.

where ρ is the fluid density(kg·m-3),M is the Mach number,and v is the velocity magnitude of fluid(m·s-1).

However,the total pressure may decrease,which causes gas density decrease,with the action of reactor wall resistant loss because the reactor wall resistance affects energy loss more strongly than converging flow.According to Eq.(1),the velocity of gas calculated is smaller than experimental data at about 0°.

4.Results and Discussion

4.1.Velocity distribution in annular channel

Fig.4 shows velocity distribution in the annular channel of Case#1 when the flow flux of TiCl4gas was controlled to 101.736 m3·h-1.The density of the gas is 11.8 kg·m-3and the viscosity is 2.3×10-5m·s-1at 700 K.The gas flow direction in the outer channel was tangential to the inner wall of the rectifying ring.The rotational flow is formed in the outer channel,and the fluid is split into the inner channel via jetting.The converging flow forms between 0°and 60°,which reforms the circulating flow when the angle is greater than 60°.It is obvious that the flows in the inner and outer channels are different.The velocity magnitude is from 0.5 m·s-1to 0.8 m·s-1whenthe fluid flows into the inner channel.The flow is laminar flow since the Reynolds numberis calculated to be from 1536 to 2458 according to the equation Re=ρvd/μ,where d represents the diameter of rectifying ring.Therefore,the gas flow direction scatters when the fluid flows into the inner channel.The velocity in the inner channel is significantly smaller than that in the outer channel,and there is no rotating flow in the inner channel.The fluid is then divided into many small jetting streams by the rectifying ring.Since rectifying ring where small holes are uniformly distributed cannot distribute the fluid equally in the outer channel,the flow may be disturbed.However,the direction of the flow gradually tends to be radial when the fluid gets closer to the jet ring as shown in Fig.4.The velocity magnitude can be accelerated near the jet holes.The flow in the inner channel,thus,can be distributed to the jet ring radially.

Table 1 Governing equations

In order to investigate the change of velocity in the inner and outer channels in detail,the velocity distribution at different radial positions,with and with out the rectifying ring,was plotted,as shown in Fig.5.The flow flux of TiCl4gas was controlled to 101.736 m3·h-1.The density of the gas is 11.8 kg·m-3and the viscosity is 2.3×10-5m·s-1at 700 K.It can be seen from Fig.5(a)and(b)that the velocity increases sharply when the circumferential angle increases from 0°to 60°.The feeding gas merges with the circulating flow,which causes the gas velocity to sharply increase at the circumferential angle of 0°in the annular channel.The velocity reaches its maximum at the circumferential angle of 60°because the merge of the feed fluid and the circulating flow finish at θ=60°as mentioned above.After the merge at θ=60°,the gas stream mass flux begins to decrease.So does the resultant velocity,as shown in Fig.5(a)and(b).

Fig.3.The calculated and experimental velocity distribution in annular channel.

Comparison between Fig.5(c)and(a)shows that the velocity distributions with and without rectifying rings are similar.This indicates that the gas in outer channel will be distributed by the small holes in the rectifying ring,which is still a complex variable mass flow in the outer channel.However,in the inner channel,the flow rate and direction vary greatly at a radius of 80 mm,as shown in Fig.5(d),which indicates heavy disturbance of the flow near the inner wall of the rectifying ring.Ata radius of70 mm,the flow rate differs less while the flow direction is inconsistent.This indicates that the influence of the flow disturbance is not completely eliminated.As the gas flows to the jet ring,the velocity at a radius of 60 mm is close to about 5 m·s-1.The flow direction also tends to be radially inwards.At a radius of 52 mm,the velocity distribution curve is close to a horizontal line,as shown in Fig.5(d),where the velocity will accelerate to about 23 m·s-1(the velocity in the jet hole is about 55 m·s-1).The curve of the resultant velocity distribution gets smoother while setting up the rectifying ring by comparing Fig.5(d)with Fig.5(b),which indicates that the flow velocity tends to be nearly radial at all positions near the reactor wall.

4.2.Pressure distribution in annular channel

Fig.2.Structure of the distributor in the open literature[32].

The uniformity of the pressure distribution is important for jet holes to realize an equal distribution.In order for the simulation results to be applicable to the distributor with radial jet injection in cross flow,the dimensionless pressure of gas flow in the annular channel is defined as

where P0is the pressure(Pa)atθ=0°in Fig.1,Piis the pressure(Pa)at measurement point i,and q is the dynamic pressure defined as

where ρ is the fluid density(kg·m-3),v is the entrance velocity(m·s-1).Non-uniform pressure distribution corresponds to the increases in dimensionless pressure,which is unfavorable in realizing a uniform distribution.

Fig.4.Vector map of velocity in annular channel(The arrow and color in the figure represents the direction and value of the velocity,respectively.)

Fig.6 shows the pressure distribution in the annular channel of Cases#0 and#1(TiCl4gas flux 101.736 m3·h-1,density 11.8 kg·m-3,viscosity 2.3× 10-5m·s-1).It can be seen from Fig.6(a)that the pressure decreases first and then increases with the increasing circumferential angle under different Reynolds numbers.It was noted that the velocity increases when the circumferential angle increases from 0°to 60°,as mentioned above.The pressure,thus,should be decreased when circumferential angle increases from 0°to 60°according to the Bernoulli equation[33].The pressure reaches its minimum at the section near the 60–90°.Because of the decreasing velocity,the pressure increases when the circumferential angle is greater than 60°.It is obvious that the curves of the pressure distribution are almost coincident at the different TiCl4inlet Reynolds numbers.This indicates that the turbulence development in the annular channel is sufficient under the given flow rate,and the inlet Reynolds numbers have little effect on the pressure distribution[33].The trend of the curve for Cpversus the circumferential angle at a radius of 120 mm is the same as that at a radius of 52 mm,as shown in Fig.6(b).However,the curve of the pressure distribution at a radius of 120 mm is smoother than that of the curve at a radius of 52 mm.The radial pressure difference in the annular channel is the driving force of the gas radial flow.However,the non-uniformity of pressure distribution increases as the radial coordinate decreases,which is unfavorable in realizing an equal flow distribution.

Fig.6.Pressure distribution in annular channel.

By setting up the rectifying ring,the pressure distribution in the outer channel at a radius of 120 mm is consistent with that without the rectifying ring,as shown in Fig.6(d)and(b).The pressure also shows a non-monotonic variation trend.This indicates that the fluid in the outer channel is still a complex variable mass flow as mentioned above.However,in the inner channel,the velocity distribution curve at a radius of52 mm is close to a horizontal line,as shown in Fig.6(c).This indicates that the flow states in the inner channel have been significantly improved by setting up the rectifying ring.Therefore,the pressure distribution in the inner channel is more uniform,which is favorable when realizing an even distribution.

4.3.Influence of different rectifying rings on the pressure distribution

In Fig.7,the pressure distributions in the inner channel at a radius of 52 mm were plotted under the condition of different Reynolds numbers and different rectifying rings.The case with out a rectifying ring,Case#0,acts as a control,and represents the pressure distributions of the corresponding position.

The pressure distribution curves with different rectifying rings are all smoother than those of curves without rectifying rings,as shown in Fig.7(a).The curve amplitude change is at its minimum in Cases#1,#3 and#5,which indicates that the pressure distribution is more uniform.The uniformity of the pressure distribution has been significantly improved by setting up the rectifying ring as mentioned above.The pressure distribution curve of Case#1 is smoother than that of Case#2,which indicates that the smaller the opening ratio of the rectifying ring,the more favorable it is in achieving uniform distribution.The rectifying ring with slits or the double baffle can also significantly improve the uniformity of the pressure distribution.It is obvious that the curves of the pressure distribution are almost coincident at different TiCl4inlet Reynolds numbers,when comparing Fig.7(b)with Fig.7(a).This indicates that the inlet Reynolds numbers have little effect on the pressure distribution.

4.4.Non-uniformity of the pressure and outlet velocity

In order to more accurately describe the effects of the above five different rectifying rings on the flow uniform distribution,the nonuniformity of the pressure and outlet velocity is defined as

where φiis the pressure(Pa)of the measurement point or the velocity(m·s-1)at outlet of hole i,φ is the average value of 12 measurement points.The uniformity of the measurement points is higher when the value of M is smaller.

There are five kinds of rectifying rings with all the other identical structures,as shown in Fig.1.In the inner channel at a radius of 70 mm,the non-uniformity of the pressure(MP)and the outlet velocity(MV)with and without a rectifying ring are listed in Table 2 under the conditions of TiCl4gas flux 101.736 m3·h-1,density 11.8 kg·m-3,and viscosity 2.3×10-5m·s-1.

Table 2 Non-uniformity of the pressure and outlet velocity with different rectifying rings

It is noted that the better production can be obtained under the conditions of uniform gas mixing,which shows that the uniform pressure distribution is a prerequisite in the inner channel.It is evident in Table 2 that the non-uniformity of the pressure of Case#1 is the smallest.However,MVof Case#5 is the smallest in Table 2.In summary,the uniformity of the pressure and outlet velocity can be improved by decreasing the opening ratio of the rectifying ring.However,the rectifying ring of double baffle,which can be installed and adjusted easily,is more effective to realize the uniform distribution.

4.5.Influence of rectifying ring on the total pressure loss of the distributor

The gas velocity and pressure changes in the annular channel of the distributor are nonlinear.The rectifying ring can reduce the nonuniformity in the annular channel to achieve flow uniform distribution.However,this will increase the flow resistance and the energy consumption.The total pressure loss is the pressure drop from the TiCl4inlet to the outlet of the jet hole.From Table 3,it can be clearly observed that the total pressure loss of the distributor with the rectifying ring is lower than that of the original distributor under the same flow rate.For example,the total pressure loss is 3935 Pa in Case#0 while it is 2389 Pa in Case#1 under the flow flux of 101.736 m3·h-1.In fact,the resistance of the circulating flow in the annular channel consumes more energy since it flows with a high velocity.However,installing a rectifying ring divides the annular channel into two channels(the inner and outer channels),which significantly reduces the circulatingflow in the annular channel.The turbulent kinetic energy is 3.306 × 107m2·s-2in Case#0 while it is 3.025 × 107m2·s-2,1.008 × 107m2·s-2,1.867 × 107m2·s-2,2.009 × 107m2·s-2and 2.017 × 107m2·s-2in Cases#1,#2,#3,#4,and#5,respectively.The total pressure loss for cases with rectifying ring is smaller than Case#0 without rectifying ring,as shown in Table 3,so does the total energy loss.Therefore,in order to achieve flow uniform distribution,it is necessary to set up a rectifying ring in the annular channel of the distributor.

Table 3 Total pressure loss of the distributor with different rectifying rings

5.Conclusions

Three-dimensional CFD was used to study TiCl4gas uniform distribution in turbulent flow for the manufacture of pigment TiO2in the chloride process.The following conclusions were reached:

(1)The present CFD model for the gas flow of TiCl4in the annular channel of an oxidizing reactor was validated by good agreement between its prediction and the experimental data available from the literature.

(2)The distribution uniformity of the pressure and outlet velocity was significantly improved by setting up a rectifying ring.Compared to the case without the rectifying ring,the nonuniformity of the pressure and outlet velocity can be reduced by up to 91%and 69%respectively.The rectifying ring in Case#5,which can be installed and adjusted easily,is more conducive to realize equal distribution.

(3)The turbulent kinetic energy loss will decrease since the circulating flow in the annular channel can be effectively reduced by installing a rectifying ring.This will reduce the resistance loss in the annular channel and the total energy loss,which is enough to compensate for the energy consumption of the rectifying ring.

[1]G.Z.Deng,Titanium metallurgy,Metallurgical Industry Press,Beijing,2010(in Chinese).

[2]W.Zhang,Z.Zhu,C.Y.Cheng,A literature review of titanium metallurgical processes,Hydrometallurgy 108(3–4)(2011)177–185.

[3]S.Yuan,W.Chen,S.Hu,Fabrication of TiO2nanoparticles/surfactant polymer complex film on glassy carbon electrode,Mater.Sci.Eng.C 25(2005)479–483.

[4]J.Gambogi,Mineral commodity summaries 2014,USGS,Washington D.C.,2014 170.

[5]B.A.van Driel,P.J.Kooyman,K.J.van den Berg,A.Schmidt-Ott,J.Dik,A quick assessment of the photocatalytic activity of TiO2pigments—From lab to conservation studio!Microchem.J.126(2016)162–171.

[6]S.Middlemas,Z.Z.Fang,P.Fan,Life cycle assessment comparison of emerging and traditional titanium dioxide manufacturing processes,Hydrometallurgy 108-132(2013)107–113.

[7]S.Middlemas,Z.Z.Fang,P.Fan,A new method for production of titanium dioxide pigment,J.Clean.Prod.89(2015)137–147.

[8]N.Abdullah,S.K.Kamarudin,Titanium dioxide in fuel cell technology:An overview,J.Power Sources 278(2015)109–118.

[9]X.Cong,Z.F.Zhang,X.Q.Wang,Preparation of TiC14with the titanium slag containing magnesia and calcia in a combined fluidized bed,Chin.J.Chem.Eng.14(3)(2006)281–288.

[10]H.F.Liu,S.F.Ji,Y.Y.Zhang,M.Li,H.Yang,Porous TiO2-coated magnetic core–shell nanocomposites:Preparation and enhanced photocatalytic activity,Chin.J.Chem.Eng.21(5)(2013)569–576.

[11]J.Q.Meng,J.X.Wu,J.H.Zhu,F.Y.Wu,L.Q.Long,Worldwide patent distribution of titanium dioxide by chlorination process,Inorg.Chem.Ind.46(1)(2014)11–16(in Chinese).

[12]S.E.Pratsinis,P.T.Spicer,Competition between gas phase and surface oxidation of TiCl4during synthesis of TiO2particles,Chem.Eng.Sci.53(10)(1998)1861–1868.

[13]Z.M.Lu,C.Z.Li,D.Z.Cong,H.L.Liu,Y.Hu,Study on jet mixing characters in the oxidation reactor for titanium dioxide powder production by chloride process,Chem.Eng.China 29(3)(2001)25–28(in Chinese).

[14]S.M.Zhang,A cold model experimental study of titanium pigment,Paint Coat.Ind.30(5)(2000)20–22(in Chinese).

[15]Y.Cheng,A.W.Ye,F.Liu,F.Wei,Numerical simulation of swirling flows in oxidation reactors for TiO2manufacture,China Particuol.4(3–4)(2006)108–113.

[16]Y.Cheng,Z.Liu,P.C.Luo,F.Wei,Gas cross- flow mixing in TiO2oxidation reactor of chloride process,J.Chem.Ind.Eng.China 57(12)(2006)2840–2846(in Chinese).

[17]D.Z.Cong,The pressure distribution of variable mass flow in the annular channel(part one),Chem.Equip.Technol.16(6)(1995)1–5(in Chinese).

[18]D.Z.Cong,The pressure distribution of variable mass flow in the annular channel(part two),Chem.Equip.Technol.17(1)(1996)7–10(in Chinese).

[19]R.J.Li,Y.H.Chen,Z.B.Zhu,Analysis on uniform fluid distribution of circle distributor in multitube fixed-bed reactor,Chem.Eng.China 37(3)(2009)20–26(in Chinese).

[20]R.J.Li,C.Y.Chen,Y.Q.Wu,Z.B.Zhu,Parameters for uniform distribution of stream in large-scale radial flow reactor,Chem.Eng.China 37(10)(2009)28–31(in Chinese).

[21]Z.M.Lu,C.Z.Li,D.Z.Cong,Gas flow and distribution in circle distributor,Chem.Eng.China 29(2)(2001)26–30(in Chinese).

[22]R.Singh,V.Raman,Two-dimensional direct numerical simulation of nanoparticle precursor evolution in turbulent flames using detailed chemistry,Chem.Eng.J.207-208(2012)794–802.

[23]Y.Sung,V.Raman,R.O.Fox,Large-eddy-simulation-based multiscale modeling of TiO2nanoparticle synthesis in a turbulent flame reactor using detailed nucleation chemistry,Chem.Eng.Sci.66(19)(2011)4370–4381.

[24]A.Schild,A.Gutsch,H.Mühlenweg,S.E.Patsinis,Simulation of nanoparticle production in premixed aerosol flow reactors by interfacing fluid mechanics and particle dynamics,J.Nanopart.Res.1(2)(1999)305–315.

[25]T.Johannessen,S.E.Patsinis,H.Livbjerg,Computational analysis of coagulation and coalescence in the flame synthesis of titania particles,Powder Technol.118(3)(2001)242–250.

[26]J.Akroyd,A.J.Smith,R.Shirley,L.R.Mcglashan,M.Kraft,A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows,Chem.Eng.Sci.66(17)(2011)3792–3805.

[27]V.S.Buddhiraju,V.Runkana,Simulation of nanoparticle synthesis in an aerosol flame reactor using a coupled flame dynamics–monodisperse population balance model,J.Aerosol Sci.43(1)(2012)1–13.

[28]G.K.Batchelor,An introduction to fluid dynamics,Cambridge University Press,England,1967.

[29]T.H.Shih,W.W.Liou,A.Shabbir,Z.Yang,J.Zhu,A new k-ε eddy-viscosity model for high Reynolds number turbulent flows-model development and validation,Comput.Fluids 24(3)(1995)227–238.

[30]A.Guardoa,M.Coussiratb,M.A.Larrayoza,F.Recasensa,E.Egusquizab,Influence of the turbulence model in CFD modeling of wall-to- fluid heat transfer in packed beds,Chem.Eng.Sci.60(6)(2005)1733–1742.

[31]A.G.Dixona,M.E.Taskina,M.Nijemeislandb,E.H.Stittb,Wall-to-particle heat transfer in steam reformer tubes:CFD comparison of catalyst particles,Chem.Eng.Sci.63(8)(2008)2219–2224.

[32]Z.M.Lu,C.Z.Li,D.Z.Cong,H.C.Gu,H.L.Liu,Y.Hu,The mass variable flow and pressure distribution in a gas distributor,J.East China Univ.Sci.Technol.26(4)(2000)362–366(in Chinese).

[33]Z.Chen,P.Zhou,C.Mei,Principle of transport process,Central South University Press,Changsha,China,2011(in Chinese).